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Proceed to summarize the following text: uncertainty relations are a key item of the quantum theory . this is from fundamental reasons , but also regarding practical applications , since phase - number uncertainty relations are the heart of the quantum limits to the precision of signal detection schemes @xcite . typically , uncertainty relations are expressed in terms of variances and are derived directly from the heisenberg form of commutation relations . however , this approach is not always useful . on the one hand , variance may not be a suitable uncertainty measure . this is specially clear regarding periodic phase - angle variables @xcite . on the other hand , the phase may not admit a simple well - behaved operator description suitable to obey a heisenberg form of commutation relations with the number operator @xcite . this has lead to the introduction of alternative uncertainty relations @xcite , some of them involving characteristic functions @xcite . in this regard , a recent work has proposed an uncertainty relation for position and momentum based on characteristic functions , which is derived directly from the weyl form of commutation relations @xcite . in this work we translate this approach to phase - number variables . despite the problems that quantum phase encounters , a very fundamental approach admits without difficulties the weyl form of commutation relations and has well - defined characteristic functions . therefore , the approach in ref . @xcite is a quite interesting formulation particularly suited to phase - angle variables . we also show that this encounters fundamental ambiguities when contrasting different slightly different alternative implementations , as it also holds for other approaches @xcite . let us point out that the weyl form is equivalent to say that every system state experiences a global phase shift after a cyclic transformation in the corresponding phase space . this implies that the quantum structure including uncertainty relations might be traced back to a geometric phase @xcite . let us consider general systems describable in a finite - dimensional space as a spin @xmath0 . this admits very general scenarios , including especially the phase difference between two modes of the electromagnetic field . this is because the total number of photons @xmath1 is compatible with the phase difference and defines finite - dimensional subspaces of dimension @xmath2 , where @xmath1 plays the role of the spin modulus as @xmath3 @xcite . let us focus on a spin component @xmath4 and the canonically conjugate phase @xmath5 . to avoid periodicity problems we focus on the complex exponential of @xmath5 , we shall call @xmath6 , this is @xmath7 . the eigenvectors @xmath8 can be referred to as phase states @xcite , being latexmath:[\[| \tilde{m } \rangle = \frac{1}{\sqrt{2j+1 } } \sum_{m =- j}^j e^ { -i \frac{2 \pi}{2j+1 } m \tilde{m } } @xmath4 , as usual @xmath10 , and @xmath11 . likewise , we may define the exponential of @xmath4 as @xmath12 these exponentials @xmath6 and @xmath13 are quite suited to the weyl form of commutation relation @xcite @xmath14 for any @xmath15 . it is worth noting that the weyl form has a quite interesting meaning when expressed as @xmath16 this represents a cyclic transformation in the form of a closed excursion over a @xmath17 rectangle in the associated phase space for the problem . the result is that every system state acquires a global phase after returning to the starting point . following ref . @xcite we can construct the gram matrix @xmath18 for the following three vectors @xmath19 where @xmath20 is an arbitrary state assumed pure for simplicity and without loss of generality , so that @xmath21 involving the characteristic functions @xmath22 and @xmath23 which is the term invoking the weyl commutator ( [ ws ] ) . after the positive semi - definiteness of @xmath18 we get @xmath24 where @xmath25 . from this point we can follow exactly the same steps in ref . these involve to construct another gram matrix after replacing @xmath26 by @xmath27 and @xmath28 by @xmath29 , adding the two determinants , using eq . ( [ ws ] ) and then following some clever simple algebraic bounds . this leads to @xmath30 with @xmath31 therefore , most of the analysis and results found in ref . @xcite could be translated here . even , the limit of vanishing argument of the characteristic functions may be reproduced in the limit of very large @xmath0 . besides the sums , uncertainty relations can be also formulated as the products of uncertainty estimators . in our case from eq . ( [ urs ] ) we can readily derive a bound for the product of characteristic functions @xmath32 a rather interesting point is that this can lead to conclusions fully opposite to the sum relation ( [ urs ] ) . this is specially so regarding the minimum uncertainty states , as we shall clearly show by some examples below . the smallest value for the bound @xmath33 is obtained for @xmath34 . in such a case , the sum of the two gram matrices commented above leads directly to the uncertainty relation @xmath35 where the @xmath36 term is expressing phase - number correlations that in standard variance - based approaches is expressed by the the anti - commutator . if this correlation term is ignored we get the more plain relations : @xmath37 let us note that these relations might be called _ certainty _ instead of _ uncertainty _ relations since we get upper bounds for characteristic functions , that take their maximum value when there is full certainty about the corresponding variable . the most simple and illustrative example is provided by the case @xmath38 . the most general state is of the form @xmath39 where @xmath40 are the pauli matrices , @xmath41 is the identity , and @xmath42 is a three - dimensional real vector with @xmath43 . we can chose the basis so that @xmath44 and @xmath45 . the only nontrivial uncertainty relation holds for @xmath46 so that @xmath34 , @xmath47 @xmath48 and eqs . ( [ urs ] ) , ( [ urt ] ) , and ( [ urp ] ) become , respectively @xmath49 actually , @xmath50 is a well - known duality relation expressing complementarity @xcite . the minimum uncertainty both for eqs . ( [ urs ] ) and ( [ urt ] ) holds for every pure state @xmath51 with @xmath52 . turning our attention to the alternative product of characteristic functions in eq . ( [ urp ] ) we get that the minimum uncertainty states are those pure states with @xmath52 and @xmath53 . on the other hand , the states with @xmath52 and @xmath54 or @xmath55 are of maximum uncertainty , contrary to the predictions of the sum relations ( [ urs ] ) and ( [ urt ] ) . next we address the case of the number and phase for a single field mode . there is always the possibility of addressing this from the number and phase difference taking a suitable reference state in one of the modes @xcite , but the direct approach has also its advantages . one of them is that it faces the fact that , roughly speaking , there is no phase operator . the exponential of the phase is not unitary , but represented instead by the one - sided unitary susskind - glogower operator @xcite @xmath56 where @xmath57 are the eigenstates of the number operator @xmath58 , and @xmath59 are the phase states @xmath60 where @xmath61 is the orthogonal projector on the subspace with less than @xmath26 photons @xmath62 and @xmath63 is the identity . this does not prevent the existence of a proper probability distribution for the phase in any field state @xmath64 . this can be defined thanks to the phase states ( [ ps ] ) as @xmath65 , that lead to the characteristic function @xmath66 where the last equality holds because for all @xmath26 @xmath67 in spite of the fact that the phase states are not orthogonal . this is to say that the lack of unitarity is equivalent to a description of phase in terms of a positive - operator measure . despite the lack of unitarity of @xmath68 there is also a suitable weyl form of commutation relations @xmath69 since the operator @xmath6 is not unitary , the change of @xmath26 by @xmath27 is not trivial , so in order to follow the procedure in ref . @xcite we have to construct explicitly the two gram matrices . the first one for the vectors @xmath70 is @xmath71 with @xmath72 being in this case @xmath73 @xmath25 , and @xmath74 the second gram matrix corresponds to the change of @xmath5 by @xmath75 and @xmath26 by @xmath27 , so the three vectors are now @xmath76 leading to @xmath77 where eqs . ( [ wpn ] ) and ( [ lu ] ) have been used , being @xmath78 and the same @xmath79 , @xmath80 and @xmath36 in eqs . ( [ phitphi ] ) and ( [ om ] ) . this leads to the determinant @xmath81 for the same @xmath82 above . at this point several routes can be followed . for definiteness from now on we will focus always in the most stringent scenario of @xmath83 . in such a case we readily get from the sum of eqs . ( [ detg+ ] ) and ( [ detg- ] ) the following bound : @xmath84 the lack of unitarity of the exponential of the phase reflects in the presence of the @xmath85 term . thus , whenever this term is absent @xmath86 we recover the same expressions obtained for the spin - like systems . otherwise , this term might be also moved to the right - hand side of the relation meaning that the nonunitarity implies a lower upper bound in accordance with the noisy nature of positive - operator measures . for definiteness , on what follows we will consider the following forms @xmath87 and @xmath88 as well as the product @xmath89 readily simple examples are provided by the eigenstates of @xmath58 and @xmath6 . for the number states @xmath57 we get for all @xmath5 , @xmath26 and @xmath90 that @xmath91 , @xmath92 , and there is no effect of the @xmath85 term . thus these are minimum uncertainty states . note that we have the opposite conclusions regarding the uncertainty product . on the other hand , the phase states @xmath93 do not provide a suitable example since they are not normalizable . instead , we can use their normalized counterparts , that are also eigenstates of @xmath6 @xmath94 . these states can be suitably approached in practice via quadrature squeezed states @xcite . in this case it can be readily seen that @xmath95 and @xmath96 in fig . 1 we have represented the combinations @xmath97 , @xmath98 and @xmath99 in eqs . ( [ up ] ) and ( [ upp ] ) as functions of @xmath100 for @xmath101 and @xmath102 . the minima of these functions represent maximum uncertainty and hold for phase states with very small mean number of photons , i. e. , @xmath103 , and 1.3 , for @xmath97 , @xmath98 , and @xmath99 , respectively . on the other hand , when @xmath104 , this is when @xmath105 , we get @xmath106 , @xmath107 , and @xmath108 , as expected for ideal phase states , becoming minimum uncertainty states . ( solid ) , @xmath98 ( dashed ) , and @xmath99 ( dotted ) as functions of @xmath109 for @xmath101 and @xmath102 for the phase states ( [ pc]).,width=226 ] however when considering the product @xmath110 in eq . ( [ v ] ) again for @xmath101 and @xmath102 it can be easily seen after eq . ( [ pc1 ] ) that when @xmath104 and @xmath111 we get maximum uncertainty @xmath112 , while @xmath110 attains its maximum value ( i. e. , minimum uncertainty ) , @xmath113 , when @xmath114 , this is @xmath115 . thus we see another clear example where maximum and minimum uncertainty states exchange their roles depending on the assessment of joint uncertainty considered . states with gaussian statistics are usually minimum uncertainty states in typical variance - based uncertainty relations . then it is worth examining the case in which the number statistics can be approximated by a gaussian distribution . this will work provided that the distribution is concentrated in large photon numbers and that it is smooth enough so that the number @xmath90 can be treated as a continuous variable . thus let us consider a pure state @xmath116 with @xmath117 , \ ] ] where @xmath118 represents the mean number , @xmath119 is given by the inverse of the number variance @xmath120 , and @xmath121 provides phase - number correlations taking positive as well as negative values . consistently with the above approximations we shall consider @xmath122 as well as @xmath123 this situation includes the glauber coherent states @xmath124 for large enough mean photon numbers @xmath125 with @xmath126 . throughout @xmath127 will be assumed . in these conditions we readily get @xmath128 , \ ] ] and @xmath129 with @xmath130 . the first thing we can notice is that @xmath121 increases phase uncertainty . let us begin with the simplest case @xmath126 . we can focus first on the plain sum relation @xmath97 in eq . ( [ up ] ) , which is plotted in fig . 2 in solid line as a function of @xmath119 . we can see that @xmath131 is just a function of @xmath132 and that minimum uncertainty , this is maximum @xmath97 , holds for @xmath132 tending both to 0 and infinity : this is when the state tends to be phase or number state , respectively , in accordance with the above results . in between we get a maximum uncertainty state , i. e. , minimum @xmath97 , when @xmath133 that correspond to @xmath134 , this is uncertainty equally split between phase and number . similar results are obtained for @xmath98 in the same eq . ( [ up ] ) , as shown in fig . 2 in dashed line . ( solid ) and @xmath98 ( dashed ) as functions of @xmath135 for gaussian states with @xmath126.,width=226 ] on the other hand , the situation is quite the opposite for the certainty product @xmath110 in eq . ( [ v ] ) : we have maximum uncertainty @xmath112 for phase and number states @xmath136 , while we have minimum uncertainty , this is maximum @xmath110 , for @xmath137 . this is just the opposite of the conclusion of the sum of characteristics . for the case @xmath138 in fig . 3 we have plotted the certainty sums @xmath97 and @xmath98 as functions of @xmath121 for @xmath139 and @xmath101 , showing that from @xmath126 increasing @xmath121 increases uncertainty until reaching @xmath140 where a revival of @xmath98 is produced , reaching the same certainty values around @xmath126 . regarding the product @xmath110 we have the same behavior of the case @xmath126 but the minimum uncertainty state holds for @xmath141 . ( solid ) and @xmath98 ( dashed ) as functions of @xmath121 for @xmath101 and gaussian states with @xmath139.the two lines overlap around @xmath126.,width=226 ] looking for states with interesting phase - number relations we may consider the eigenstates of @xmath143 , where @xmath144 is a real parameter @xcite : @xmath145 that has the following solution , for @xmath146 for definiteness , @xmath147 are the corresponding modified bessel functions . it could be interesting to apply the previous approach to these states ( [ i1 ] ) looking for the @xmath144 that lead to minimum uncertainty . easily we obtain : @xmath148 and @xmath149 as before , we focus on the case @xmath101 and @xmath150 . performing the numerical computation , we obtain plots of @xmath97 and @xmath151 in eq . ( [ up ] ) similar to the ones obtained in previous cases , as we can see in fig . the maximum uncertainty states are given by the minimum value of @xmath97 and @xmath151 for a given @xmath26 . in the present case the values of @xmath144 which minimizes these functions are @xmath152 with @xmath153 , and @xmath154 with @xmath155 , respectively . here again we obtain opposite results for the certainty product @xmath110 in eq . ( [ v ] ) . ( solid ) and @xmath98 ( dashed ) as functions of @xmath144 for the states ( [ i1]).,width=226 ] joint uncertainty relations of two observables are often minimized by states with properties somewhat intermediate between the two observables . the simplest case is a readily coherent superposition of number and phase states of the form @xmath156 we focus on the case @xmath157 and @xmath158 with the idea that @xmath159 approaches the ideal phase states ( [ ps ] ) . in such a case it can be seen that the normalization condition is just @xmath160 and we shall consider @xmath101 and @xmath102 . thus , eq . ( [ com ] ) reads @xmath161 minimum uncertainty holds just in the limiting cases @xmath162 and @xmath163 , recovering the cases of phase and number states . on the other hand maximum uncertainty holds for the intermediate state @xmath164 . clearly , the situation is reversed if we consider the product @xmath110 so that maximum and minimum uncertainty are exchanged . we have successfully derived meaningful phase - number uncertainty relations from the weyl form of commutation relations . this can be applied to study phase - number statistical properties of meaningful field states , especially intermediate states that have already demonstrated interesting properties regarding uncertainty relations @xcite . moreover , this can be a suitable tool to explore quantum metrology limits . in typical interferometry @xmath58 is the generator of phase shifts . thus the characteristic function @xmath79 is actually expressing the distinguishability of the probe state before and after a phase shift , which should be naturally related to detection resolution . therefore this uncertainty relations may be connected to optimized signal detection schemes @xcite . we are grateful to ukasz rudnicki for stimulating discussions , ivn lvarez domenech for his selfless help , and demosthenes ellinas for valuable comments . g. d. gratefully thanks a collaboration grant from the spanish ministerio de educacin , cultura y deporte . l. thanks support from project fis2012 - 35583 of spanish ministerio de economa y competitividad and from the comunidad autnoma de madrid research consortium quitemad+ s2013/ice-2801 . e. breitenberger , uncertainty measures and uncertainty relations for angle observables , found . phys . * 15 * , 353 ( 1985 ) ; t. opatrn , mean value and uncertainty of optical phase - a simple mechanical analogy , j. phys . a * 27 * , 7201 ( 1994 ) . j. c. garrison and j. wong , canonically conjugate pairs , uncertainty relations and phase operators , j. math . phys . * 11 * , 2242 ( 1970 ) ; a. galindo , phase and number , lett . phys . * 8 * , 495 ( 1984 ) ; * 9 * , 263 ( 1984 ) . j.m . lvy - leblond , who is afraid of nonhermitian operators ? a quantum description of angle and phase , ann . 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we derive suitable uncertainty relations for characteristics functions of phase and number variables obtained from the weyl form of commutation relations . this is applied to finite - dimensional spin - like systems , which is the case when describing the phase difference between two field modes , as well as to the phase and number of a single - mode field . some contradictions between the product and sums of characteristic functions are noted .

You are an expert at summarizing long articles. Proceed to summarize the following text: after the discovery of a signal with a mass of about 125 gev in the higgs searches at the lhc @xcite , the prime goal is now to identify the underlying nature of the new state and to determine the mechanism of electroweak symmetry breaking . while the properties of the observed state are compatible with the ones predicted for the higgs boson of the standard model ( sm ) at the current level of precision , also a wide range of alternative interpretations could be possible , corresponding to very different underlying physics . in particular , in models with an extended higgs sector the observed state would be accompanied by several other higgs bosons , in contrast to the minimal formulation of the sm where a single @xmath0-doublet is responsible for electroweak - symmetry breaking . supersymmetry ( susy ) @xcite is commonly regarded as the most appealing extension of the sm , since it provides a solution for stabilising the huge hierarchy between the planck scale and the weak scale @xcite and offers further attractive features such as unification of the gauge couplings and a natural candidate for cold dark matter in the universe . a crucial prediction of supersymmetric extensions of the sm are their extended higgs sectors : the holomorphicity of the superpotential ( as well as the cancellation of gauge anomalies ) implies that at least two @xmath0 doublets with opposite hypercharge have to be present , so as to generate mass terms for both up- and down - quarks ( in a type ii 2-higgs - doublet - model fashion ) . the minimal supersymmetric extension of the sm ( mssm ) @xcite is based on the minimal higgs sector of this kind comprising two higgs doublets , whereas the higgs sector of the next - to - minimal supersymmetric extension of the sm ( nmssm ) @xcite contains an additional ( complex ) gauge - singlet . it has long been recognised that the nmssm provides an elegant solution @xcite to the `` @xmath1-problem '' @xcite of the mssm . in the context of the relatively high mass value of about 125 gev of the observed state , this model has received particular attention lately since the mass of the light doublet - like state receives an additional contribution at lowest order as compared to the mssm , which dominates at low values of @xmath2 ( the ratio of the vacuum expectation values , v.e.v.s , of the two higgs doublets ) . in this case significantly smaller higher - order corrections are required to obtain a higgs - boson mass in the appropriate range @xcite as compared to the mssm , where the lowest - order prediction for the mass of the light cp - even higgs boson is bounded from above by the mass of the z boson , @xmath3 . furthermore , also the singlet doublet mixing can give rise to an uplift of the mass of the doublet state , provided that the cp - even singlet state is lighter than the doublet state . it has been argued in this context that the relaxed requirement on the size of the higher - order corrections as compared to the mssm makes it possible to obtain a higgs - mass prediction of about 125 gev in a `` more natural '' way @xcite . in the following we will focus on the nmssm as a theoretically well - motivated alternative to the sm with a potentially rich phenomenology in the higgs sector . for simplicity , we will restrict to the cp - conserving case , for which the spectrum of physical higgs states of the nmssm consists of three cp - even , two cp - odd and a pair of charged higgs states ( while we do not explicitly consider cp - violating effects giving rise to a mixture between the five neutral states , it should be noted that cases where a cp - even and a cp - odd state are nearly mass - degenerate essentially mimick a scenario where a single state is an admixture of cp - even and cp - odd components ) . furthermore , while several versions of the nmssm can be formulated , depending on the form of the singlet and singlet - doublet interaction terms in the superpotential , we will focus on the @xmath4-conserving version only , where the solution to the `` @xmath1-problem '' is more immediate ( on the other hand , this simple model could lead to a domain wall problem @xcite but we will not address this question here ) . the higgs sector of this version of the nmssm is characterised by six parameters ( at tree - level ) , in contrast to the two parameters of the mssm . while we shall borrow most of our notations from @xcite , we recall the higgs terms entering the superpotential of the model : @xmath5 where @xmath6 denotes the singlet ( super)field , @xmath7 and @xmath8 the doublets , while @xmath9 stands for the @xmath10 product . when confronting the predictions of an extended higgs sector with the observed signal and the limits from the higgs searches at lep , the tevatron and the lhc , the most obvious interpretation of the signal at about 125 gev is to associate it with the lightest cp - even higgs boson of the considered model . the case where all other higgs ( as well as all new physics ) states of a supersymmetric extension of the sm ( and the same is true for various other extended higgs sectors ) are significantly heavier corresponds to the `` decoupling region '' of the model under consideration , where the couplings of the light higgs boson to gauge bosons and sm fermions are very close to the ones of the sm . revealing deviations of those couplings from their sm counterparts in such a case will require high - precision measurements , where in many cases the expected deviations do not exceed the level of a few per cent . an additional source of possible deviations from the sm could be decays of the sm - like state into new - physics particles . such a decay could in particular occur into a pair of dark matter particles , if the mass of the latter is less than half of the mass of the higgs state , i.e. below about 60 gev . this would give rise to an invisible decay mode of the observed state , providing a strong motivation for searches of decays of the observed signal into invisible final states . besides the interpretation of the observed state as the lightest cp - even higgs boson of an extended higgs sector , it is also possible , at least in principle , to identify the observed signal with the second - lightest state of an extended higgs sector . this interpretation would have the immediate consequence that there should be an ( or more generally at least one ) additional higgs state _ below _ the one observed at about 125 gev . the phenomenology of such a scenario is very different from the case of the decoupling limit discussed above , because of the presence of at least one more light state in the spectrum . within the mssm this interpretation is in principle possible @xcite , but gives rise to a rather exotic higgs sector where in fact all additional higgs bosons are light , i.e.in the vicinity of the state at about 125 gev or below . it is remarkable that a global fit within the mssm within this interpretation has resulted in an acceptable fit probability @xcite , but lately this interpretation , which implies in particular a light charged higgs boson below the mass of the top quark ( see in particular ref . @xcite ) , has come under increased pressure from the limits obtained in the charged higgs searches by atlas @xcite and , more recently , cms @xcite . the nmssm provides a well - suited and theoretically well motivated framework for investigating to what extent interpretations that go beyond the obvious case of a single light state in the decoupling limit are compatible with the latest experimental results both with respect to the properties of the observed state and to the limits obtained from higgs searches ( as well as other existing constraints ) . it is the purpose of the present paper to perform such an analysis . it is obvious that compatibility with the observed signal requires much more than just a higgs state ( or possibly more than one ) in the spectrum with a mass of about 125 gev . in order to properly take into account the latest experimental results from higgs search limits and from measurements of the properties of the observed state , we make use of the public tools higgsbounds @xcite and higgssignals @xcite , which incorporate a comprehensive set of results from atlas @xcite , cms @xcite and the tevatron @xcite . we do not explicitly impose limits from the direct searches for susy particles at the lhc . while some of the scenarios discussed in this paper could be affected by constraints from susy particle searches , we have checked that the qualitative features of the higgs phenomenology of those scenarios are maintained also for somewhat heavier susy particle spectra . note however that most of the susy spectra that we employ ( especially for coloured particles ) are beyond the mass - range tested in the run - i of the lhc . as mentioned above , already since the very early hints of a signal at about 125 gev the higgs sector of the nmssm has found a lot of attention in this context . besides the mass prediction in comparison with the mssm case @xcite , the possibility of modified rates has been discussed , particularly in the diphoton channel @xcite . furthermore , the case of universal ( or semi - universal ) susy - breaking conditions at the gut scale @xcite , gauge - mediation @xcite and other related scenarios @xcite have been considered in this context . the cp - violating version of the nmssm also received some attention @xcite . @xcite analysed the fine - tuning in a @xmath4-violating version of the nmssm and variants . other groups confronted the presence of a higgs state at this mass with direct searches for susy particles at the lhc or dark - matter constraints @xcite . scenarios with a light singlet - like state around @xmath11 gev have found considerable interest @xcite . another possibility involves a singlet and a doublet that are almost mass - degenerate at about 125 gev and may mix with each other , see ref . @xcite ( and the last reference of @xcite ) . several studies also suggested to exploit pair production processes at the lhc in order to distinguish the sm from the nmssm and/or to look for a light singlet in this fashion @xcite . scenarios with a very light cp - odd ( or cp - even ) higgs boson were addressed with several search proposals in direct production , unconventional light charged - higgs decays , or cascade decays from sm - like / light singlet states ; large higgs - to - higgs decays were also considered from the point of view of the sm - compatible nature of the observed state @xcite . recent studies of the properties of a light pseudoscalar in the nmssm @xcite have emphasized the relevance of indirect production modes for the investigation of this scenario at the lhc . in a different direction , the authors of @xcite focussed on nmssm higgs scenarios with a low - scale doublet sector . furthermore , @xcite , and more recently @xcite , studied the discovery prospects of nmssm higgs states in the lhc run at @xmath12 tev . in our analysis we go beyond the previous work in several respects : while many of the afore - mentioned analyses discussed scenarios which are compatible with existing limits , our inclusion of a fitting tool allows us to highlight the quality of the various scenarios in view of the available data . furthermore , we aim at a comprehensive discussion from the point of view of the nmssm higgs phenomenology , hence do not confine to a specific scenario ( within our assumptions on the model , perturbativity constraints and choices of simplicity with regards to the susy spectrum ) . we also focus on higgs physics and thus try , without spoiling the physical content , to avoid emphasis on questions of secondary importance with respect to this topic ( e.g. the details of the supersymmetric spectrum ) . finally , much experimental data has become available in the last few years , narrowing the possibilities in the higgs sector , and most of the recent developments are included within the tools on which our discussion is based . the paper is organised as follows : in sect . 2 we describe the framework that we use for the analyses in this paper , in particular the statistical approach used in our fits , the treatment of external constraints and the tools that we apply . as a first step of our analysis , in sect . 3 we briefly consider the sm case and the corresponding decoupling limits of the mssm and the nmssm . the sm result is used for comparison with @xmath13 analyses in different nmssm scenarios , which we perform in sects . [ lsing][ldoub ] . in sect . [ fp ] , we focus on specific points of the nmssm parameter space and discuss in more details the higgs phenomenology and the consequences for future searches , should the corresponding spectrum be realised in nature . in sect . [ gs ] a more global scan is carried out , and the features observed in the global scan are discussed in view of the results obtained for the specific nmssm scenarios that we have considered before . [ conc ] contains our conclusions . the nmssm parameter space is explored with the help of the spectrum generator ` nmssmtools_4.4.0 ` @xcite , computing the higgs masses up to leading two - loop double - log order ( we will be using the default mode only ) , in an effective potential approach . this code considers a certain number of phenomenological limits , several among which are kept within our analysis . the first class of such tests are consistency requirements and are ( necessarily ) included as hard cuts : * stability of the ewsb - vacuum : positivity of the scalar squared - masses , absence of deeper minimum ; * absence of landau poles below the gut scale : while this requirement is sometimes omitted in order to probe effects associated with large values of the parameter @xmath14 and under the assumption that new - physics or specific properties of the non - perturbative regime would smoothen the theoretical difficulty of the landau poles , we choose to keep this theoretical limit ; * requirement for higgs soft squared - masses at the tev scale : the potential - minimization procedure in nmssmtools trades these masses for the higgs v.e.v.s , so that the naturalness requirement that soft masses intervene at the tev scale must be checked explicitly ; * requirement for a neutralino lsp ( the impact of which , however , is of secondary importance in our discussion ) . * @xmath15 decay ( @xmath16 ) * mass lower limits on squarks ( @xmath17 gev , @xmath18 gev ) , gluino ( @xmath19 gev ) , sleptons ( @xmath20 gev ) , charginos ( @xmath21 gev ) ; * limits on @xmath22 , @xmath23 , @xmath24 . we remind the reader that lhc limits on susy searches are not considered in our analysis . however , the susy spectra that we employ are typically beyond the mass - range of the searches in the run - i . in this context , the inclusion of lep limits as mentioned above has only a minor impact . * limits from the bottomonium sector : non - observation of a signal in @xmath25 , excessive contribution to the @xmath26 mixing @xcite . only light cp - odd higgs below @xmath27 gev are concerned by these constraints and the limits are kept as a @xmath28 c.l . cut . * limits from @xmath29-factories ( under a strong minimal flavour violation hypothesis , i.e. neglecting all possible tree - level flavour - changing neutral currents ) : @xmath30 , @xmath31 , @xmath32 , @xmath33 , @xmath34 @xcite . instead of treating the limits as a hard cut , we combine them in a @xmath13 function relying on the central value and error bars computed in nmssmtools : @xmath35 the corresponding experimental central values @xmath36 and standard deviation @xmath37 are summarized in table [ expblim ] . the theoretical error estimate is the result of an involved calculation : errors relative to sm - like contributions are taken from the corresponding sm estimate ; the uncertainty on new - physics contributions is estimated to @xmath38 ( if only leading - order effects are included ) / @xmath39 ( if next - to - leading @xmath40 corrections are present ) of the total corresponding contributions and are added linearly to the sm error ; additional error sources are mostly ckm matrix elements ( taken from tree - level measurements exclusively ) and hadronic parameters ( decay constants , taken from lattice calculations ) ; to obtain the final theoretical uncertainty range , both higher - order and parametric uncertainties are varied within these previously - discussed limits . * @xmath41 @xcite : similarly to @xmath29-observables , we add a contribution to the @xmath13 with experimental - sm input shown in table [ expblim ] , where the errors have been added in quadrature . the theoretical uncertainty associated to new - physics contributions and higher orders is calculated as the sum of a fixed error @xmath42 , a @xmath43 error estimate on 1-loop contributions ( which do not involve coloured particles ) and a @xmath38 error estimate on 2-loop effects ( involving coloured particles ) . additionally , given that a candidate for the interpretation of the signal observed at the lhc seems necessary , we require that the spectrum produces one cp - even higgs state in the mass - range @xmath44 $ ] gev . * dark matter searches : relic - density ( via micromegas ) , xenon 100 @xcite . the reasons for not taking such limits into account come from the observation that they are strongly dependent on the susy spectrum , while we want to focus on the higgs sector : confining to collider constraints allows us to handle simple supersymmetric spectra , which play a secondary part in our analysis , while these would likely have to be finely adjusted if one were to include , e.g. , the relic - density bounds . we note also that the dark matter phenomenology may involve mechanisms ( e.g. light gravitino lsp ) which may alter the conclusions in the dark sector , while all such considerations are not the focus of our discussion . * lep higgs searches : @xmath45 , @xmath46 , @xmath47 , @xmath48 ( z width ) ; 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while the properties of the signal that was discovered in the higgs searches at the lhc are consistent so far with the higgs boson of the standard model ( sm ) , it is crucial to investigate to what extent other interpretations that may correspond to very different underlying physics are compatible with the current results . we use the next - to - minimal supersymmetric standard model ( nmssm ) as a well - motivated theoretical framework with a sufficiently rich higgs phenomenology to address this question , making use of the public tools ` higgsbounds ` and ` higgssignals ` in order to take into account comprehensive experimental information on both the observed signal and on the existing limits from higgs searches at lep , the tevatron and the lhc . we find that besides the decoupling limit resulting in a single light state with sm - like properties , several other configurations involving states lighter or quasi - degenerate with the one at about 125 gev turn out to give a competitive fit to the higgs data and other existing constraints . we discuss the phenomenology and possible future experimental tests of those scenarios , and compare the features of specific scenarios chosen as examples with those arising from a more global fit .

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Proceed to summarize the following text: bekenstein - hawking entropy @xmath0 has been derived from different points of view @xcite . but if it is considered thermal with a microscopic description , according to the statistical foundations of entropy , perhaps the most promising and appropriate formalism to explain @xmath0 is the entanglement entropy approach @xcite . in particular , entanglement entropy of black shells is required , because the usual thermal entropies for black holes are divergent and geometric in nature @xcite . in the first part of this paper we complete the concept of black shell presented in @xcite . thus , by simplicity we effectively model the significant features of the gravitational collapse , in terms of a massive thin spherical collapsing shell with respect to an external observer . in the second part of the text we consider euclidean approach for entropy of black shells , thinking in its complementary description of entanglement entropy . thermal entropy or entanglement entropy modeled by a black shell is a real physical model for @xmath0 , because it corresponds to the thermodynamics of hot quantum fields confined near the outside of the shell @xcite . unlike black holes that do nt have structure and require a picture of quantum gravity to describe its geometrical entropy as some unknown physics , black shells have structure and thermal physical entropy . we show below that entropy calculated from gibbons - hawking euclidean approach for spherical black shells , retrieving the horizon integral and discarding boundary at infinity , preserves the same expression for bekenstein - hawking entropy : @xmath1 so entanglement entropy for black shells according to an external observer do not need to consider quantum gravity criteria to explain @xmath0 . in the last part of the paper we complete the model by describing complementary quantum details . in sec . 2 we present the black shell model and review the darmois - israel formalism @xcite that is needed in order to obtain the motion equations of this model that we solve in sec . 3 . in sec . 4 we reproduce the well known bekenstein - hawking entropy from euclidean approach for a schwarzschild black hole @xcite . in sec . 5 we retrieve the horizon integral and discard boundary at infinity in order to calculate the entropy of a black shell . there we introduce a mathematical identity to support our two complementary descriptions of physics near an event horizon . in sec . 6 , for completeness , we return to entanglement entropy of black shells . so , we reproduce some results from @xcite and complete details of the corresponding hamiltonian formulation . 7 is devoted to discuss some conclusions about this paper . we present in this section a model where a black hole is formed by a thin contracting shell of dust of mass @xmath2 that contracts beginning from infinity and at last tends to the schwarzschild radius from the point of view of an external observer far from the shell in a gravitational collapse evolution of oppenheimersnyder kind @xcite , as we see in section 3 . + in this context we employ the darmois - israel formalism @xcite for a spherical thin shell @xmath3 which divides the spacetime in two regions : the interior region @xmath4 , described by flat minkowskian geometry and @xmath5 , the exterior geometry described by schwarzschild spacetime . both regions are described by spherical coordinates : @xmath6 , and we use geometric units in which : @xmath7 , and signature : @xmath8 . then the line element is : @xmath9 where : @xmath10 for @xmath5 and @xmath11 for @xmath4 . the shell hypersurface @xmath12 is represented in the chosen frame by the following parametric equation : @xmath13 where @xmath14 is the thin shell radius . the following intrinsic coordinates on the shell are used : + @xmath15 , and the 3-metric elements induced in the hypersurface are : @xmath16 we note that the relation between coordinates of @xmath17 and intrinsic coordinates on @xmath3 are : @xmath18 and the parametric equation for @xmath12 adopts the following general form : @xmath19 we assume that @xmath12 is non - null and the unit 4-normals to @xmath12 in @xmath20 are : @xmath21 the extrinsic curvature ( seccond fundamental form ) is defined by : @xmath22 using ( [ bs1 ] ) and ( [ f2 ] ) the normal ( [ f6 ] ) is : @xmath23 with the usual formula for the covariant derivative equations ( [ f4 ] ) and ( [ f7 ] ) yields the simple relation : @xmath24 raising indexes and considering that schwarzschild metric is diagonal , we arrive to the simple results : @xmath25=\frac{1}{2}f^{-\frac{1}{2}}\partial_{r}f,\ ] ] @xmath26=\left[k_{\varphi}^{\varphi}\right]=\frac{1}{r}f^{\frac{1}{2}}-\frac{1}{r},\ ] ] @xmath27=f^{-\frac{1}{2}}(\frac{1}{2}\partial_{r}f+\frac{2}{r}f - f^{\frac{1}{2}}\frac{2}{r}).\ ] ] where square brackets denotes a discontinuity across the layer , i.e. , @xmath28=f^{+}-f^{-}$ ] . our following task is to perform the calculations to obtain the motion equations of the shell and the bekenstein - hawking entropy of the shell . this is developed in sections 3 and 5 . in this section we review the thin shells junction formalism @xcite , in order to study the motion of a spherical shell of dust that contracts beginning at rest from infinity as seen by a distant observer whose proper time is the schwarzschild time @xmath29 . for this purpose we consider the surface energy tensor of a shell of dust : @xmath30 with the condition : @xmath31 and where @xmath32 is the rest mass surface density of the dust . if we consider a trajectory @xmath33 of an element of rest mass on the shell , then the 3-vector associated @xmath34 is tangent to @xmath3 . for a spherical shell , the region @xmath5exterior to the shell have the metric given by equation ( 2 ) . and the region @xmath35interior to the shell is minkowskian . the equation of the shell is : @xmath36 where @xmath37 is the proper time measured by the dust particles . the equation of motion of the shell could be obtained from the lanczos equation @xcite : @xmath38-g_{ij}\left[k_{ab}g^{ab}\right]\right).\ ] ] from equations ( [ bs1 ] ) , ( [ fc1 ] ) , ( [ fc2 ] ) and ( [ fc3 ] ) we obtain rearranging therms : @xmath39 we are interested in the velocity of the shell as measured by an external observer at rest at infinity , then we must replace : @xmath40 if the shell is initially at rest : @xmath41 because gravity is attractive it produces a flux entering to the thin gaussian region that encloses the shell and by this reason we must choose the negative sign . with these consederations we finally obtain : @xmath42 this equation is of difficult integration but fortunately we can do the following good approximation : + + @xmath43 @xmath44 where @xmath45 must be chosen in order to match these two expressions for @xmath46 , where @xmath47 from this equation we obtain : @xmath48 and @xmath49 . then the motion equation reduces to : @xmath50 this equation could be integrated analytically to obtain : @xmath51\right)}e^{\left(-\frac{t}{\tau}\right)}.\ ] ] where @xmath52 , @xmath53 is the initial radius and @xmath54 and @xmath53 are measured in units of schwarzschild radius . taking little variations of @xmath55 respect to @xmath53 we can obtain : @xmath56 this important result sais that from the point of view of an external observer at rest and far from the horizon , the shell approaches to schwarzschild radius asymptotically . this is that we call a black shell model of a black hole . for an observer comoving with the shell it is easy to see from ( [ fc5 ] ) , ( [ fc7 ] ) , ( [ fc8a ] ) and ( [ fc9 ] ) that : @xmath57 integrating this equation we obtain : @xmath58 then we see that the shell reaches the radius @xmath59 in the finite time : @xmath60 + this is that we call a black shell free gravitational collapse . in this section we review the statistical derivation of bekenstein - hawking entropy ( @xmath0 ) for stationary black holes , using analytic continuation to euclidean sector and imposing a period on euclidean time . according to gibbons - hawking derivation @xcite , and in order to obtain the bekenstein - hawking entropy , we calculate the action @xmath61 for the schwarzschild metric ( [ bs1 ] ) : @xmath62 where @xmath20 is the schwarzschild spacetime with ( in natural units ) : @xmath10 , @xmath59 and @xmath45 is the second fundamental form . using the equations ( [ f11 ] ) , ( [ f12 ] ) and ( [ f13 ] ) for @xmath63 and @xmath64 we obtain : + + @xmath65 @xmath66 where @xmath67 is the euclidean period . then for the black hole mass , @xmath68 , we obtain : @xmath69 this is the bekenstein - hawking entropy of a black hole as was derived by gibbons - hawking @xcite . for a black shell , according to section 3 above , the radius approaches to the schwarzschild one and by this reason we retrieve the horizon integral obtaining the euclidean action : + + @xmath65 @xmath70 where @xmath67 is the euclidean period . for a black shell the inner space is the empty space and by this reason : @xmath71 obtaining : @xmath72 both results : ( [ fo3 ] ) and ( [ euc13e ] ) , are mathematically equivalent derivations of @xmath0 . in order to relate both procedures , consider the following mathematical identity : @xmath73 @xmath74 this identity corresponds to a physical model that resolves one of the questions raised in mukohyama - israel @xcite , in the sense that @xmath0 is not a one - loop correction to the zero - loop gibbons - hawking contribution . indeed this entropy is a zero - loop black shell contribution . according to entanglement entropy model of black shells , thermal energy strongly concentrated near the exterior of a starlike object is clearly evident . this object has a reflecting surface , compressed to nearly ( but not quite ) its gravitational radius . following this model we may approximate the total stress - energy ( ground state and thermal excitations ) @xmath75 , near the wall , to the hartle - hawking stress - energy @xmath76 @xcite : @xmath77 where @xmath78 is boulware stress tensor and @xmath79 are thermal excitations . thermofield dynamics encodes a reflexive symmetry between twin subsystems in such a way that each of them becomes macroscopically indistinguishable from a hot body at a definite temperature . the reflection symmetry of these subsystems corresponds to the right and left regions of an eternal black hole , i.e. , kruskal sectors r and l see ( fig . [ mg10i ] ) . thus , the twin subsystems correspond to field modes propagating in these ( causally disjoint ) kruskal sectors , and the thermally entangled state of the total system corresponds to the ground state on the full kruskal manifold . in that sense , this section defines modes and ground states appropriate for the subsystems @xmath80 and for the global system ( full kruskal manifold ) , i.e. , for stationary observers in a static spacetime ( killing - boulware modes ) and for free falling observers at the horizon of a black hole spacetime ( kruskal - hartle - hawking modes ) , and establishes the relationship between them . + from the line element ( [ bs1 ] ) consider the generic situation of a real scalar field @xmath81 propagating on a geometrical background with static metric given by @xmath82 on the other hand , @xmath81 may be expanded in terms of kruskal - hartle - hawking ( kh@xmath83)-modes @xmath84 and killing - boulware ( kb)-modes @xmath85 , which are connected by the bogoliubov transformation @xcite @xmath86 with @xmath87 . the physical sense of ( [ ete30 ] ) is based in the invariant action @xmath88 $ ] and invariant hamiltonian @xmath89 under this transformation , according to the action @xmath90 = \int \mathcal{l}[\phi ] d^{4}x = \int l[\phi ] dt_{+ } = \int_{-\infty}^{\infty}dt \left(\sum_{\epsilon}\epsilon l^{(\epsilon)}(\phi)\right),$}\ ] ] where the integration regions for the first and second integrals corresponds in fig . [ mg10ii ] to the second and first graphs , respectively . there @xmath91 takes same value at two mirror points in @xmath92 and @xmath55 sectors . these two points are distinguished by different imaginary parts of @xmath93 : @xmath94 with @xmath95 and @xmath96 , the surface gravity . @xmath97 is analytic over the full kruskal manifold , with the property that @xmath98 is positive frequency in kruskal time . @xmath92 sector contribution to @xmath99 enters with negative sign , because @xmath29 and @xmath93 run backwards in this sector . modes in the @xmath92 sector and beneath the horizon have negative energies . we are talking about the continuous extension of the killing vector which is future timelike in the @xmath55 sector , and therefore becomes past - directed in the @xmath92 sector . physically , this definition of energy includes gravitational potential energy , which is negative and becomes large below the horizon . in the hartle - hawking state , positive energies in the @xmath55 sector are evenly balanced with negative energies in the @xmath92 sector . the state of the black hole is represented globally by the hartle - hawking state @xmath100 @xcite . the total energy of a quantum field in this state is zero , because of the balance between positive - energy modes in the r sector and negative - energy modes in the l sector . formally , computing the expectation value of the global energy operator @xmath101 in the state @xmath100 , @xmath102 where @xmath103 on the other hand , the ground state @xmath104 for a parochial observer in @xmath55 is energetically depressed below the hartle - hawking `` vacuum '' . if we form the expectation value of @xmath101 in the state @xmath104 , only the l - sector part ( @xmath105 in eq . ( [ ete54 ] ) ) of @xmath101 contributes , since there are no r - sector modes in this state . we thus find @xmath106 correspondingly , this same parochial observer , using his parochial energy operator @xmath107 , will perceive the hartle - hawking vacuum as energetically excited . if we form the expectation value of @xmath107 in the state @xmath100 , the operator will pick out only the positive - energy modes in @xmath55 , so @xmath108 where we have dropped zero - point energies . entanglement entropy actually allows us to think that entropy arises physically located near the horizon , given by @xcite @xmath109 where @xmath110 the expressions above are similars to brick wall model @xcite , where @xmath111 is defined by the external observer , which it is the main difference with respect to brick wall model . in addition @xmath112 is finite according to thermo field dynamics @xcite . according to this model , the integral ( [ ete191 ] ) is dominated by two contributions , for large @xmath113 and for small @xmath114 . the former corresponds to a volume term , proportional to @xmath115 , which represents the entropy and energy of a homogeneous quantum gas in a flat space at a uniform temperature @xmath116 . the latter is the contribution of gas near the inner wall @xmath117 . then , for this last contribution is required to introduce the ultrarelativistic approximations @xmath118 substituting eq . ( [ bw8 ] ) into eq . ( [ ete191 ] ) , the wall contribution to the total entropy is obtained @xmath119 where @xmath120 accounts for helicities and the number of particle species , @xmath121 is the wall area and @xmath122 is the proper radial distance from horizon to the shell . now , depending on @xmath122 , we can obtain the bekenstein - hawking entropy from eq . ( [ bw10 ] ) @xmath123 where @xmath122 can be adjusted by resorting to times determined by an external observer in the context of the black shell model . + unlike black holes , equation ( [ bw10 ] ) corresponds to a microscopic description in terms of quantum field modes . we have described in sections 2 and 3 the motion of a spherical shell of dust that contracts beginning at rest from infinity by using darmois - israel thin shells formalism @xcite , then we obtain from the point of view of an external observer far from the horizon , that shell approaches to schwarzschild radius . this result is one of the main characteristic of the black shell model , introduced above . 4 we reproduce geometrical bekenstein - hawking entropy of a schwarzschild black hole using the well known gibbons - hawking euclidean approach @xcite and derive in sec . 5 physical entropy of a black shell from gibbons - hawking euclidean approach retrieving horizon integral and discarding boundary at infinity . this important result was obtained considering that for a black shell the mass is outside the horizon . + for the sake of completeness , in sec . 6 we return to entanglement entropy of black shells , because it is a complementary description of gibbons - hawking euclidean approach for a black shell . so we have completed and extended the idea of black shell presented in @xcite . in particular , we reproduce some results about it and complete details of the corresponding hamiltonian formulation . + we agree with s. mukohyama and w. israel that entropy contributed by thermal excitations or entanglement is not a one loop correction to the zero - loop gibbons - hawking contribution . actually we may consider these two entropy sources as equivalent but mutually exclusive descriptions of what is externally the same physical situation @xcite . in equation ( [ euc13f ] ) we observe that in the euclidean sector of the black shell space - time there is an inner boundary , the black shell itself . thus inner boundary contribution to the euclidean action cancels that of the outer boundary at infinity . so the gibbons - hawking zero - loop contribution is zero in this sense . + with the right identification of the ground state , the back - reaction problem in t hooft brick wall model is resolved @xcite . in that sense for black shells , we show in ( [ ete88 ] ) that the divergent parts cancel each other . the boulware ground state contribution ( which is energetically depressed below the vacuum ) and thermal excitations cancel . + in summary we propose an effective model consisting on a massive thin spherical shell contracting toward its gravitational radius with respect to an external observer in order to describe significant features of a gravitational collapsing mass . the collapsing massive shell is compressed near its gravitational radius defining an natural cut - off between horizon and shell depending on external observer . from this model we can obtain a thermal and no divergent entanglement entropy that could explain @xmath0 . + + + * acknowledgements * + + we are indebted to werner israel , our collaborator of much of the work presented here , for many contributions , stimulating discussions and helpful comments on the manuscript . specific lagrangian @xmath92 is given by @xmath124 where @xmath125 @xmath135 = \sum_{{\omega } , { \epsilon}}{\epsilon}\ , { \frac{1}{2}}\ , |{\omega}|\left({b_{\omega}}^{({\epsilon})\dag}\ , { b_{\omega}}^{({\epsilon } ) } + { b_{\omega}}^{({\epsilon})}\ , { b_{\omega}}^{({\epsilon})\dag}\right)\ , , \ ] ] @xmath137 = \sum_{{\omega } , { \epsilon}}{\epsilon}\ , { \frac{1}{2}}\ , |{\omega}|\left({a_{\omega}}^{({\epsilon})\dag}\ , { a_{\omega}}^{({\epsilon } ) } + { a_{\omega}}^{({\epsilon})}\ , { a_{\omega}}^{({\epsilon})\dag}\right).\ ] ] w. israel , phys . 164 * , ( 1967 ) 1776 . b. carter , phys . * 26 * , ( 1971 ) 331 . d. c. robinson , phys . lett . * 34 * , ( 1975 ) 905 . s. w. hawking , commun . phys . * 25 * , ( 1972 ) 152 . s. w. hawking , commun . phys . * 43 * , ( 1975 ) 199 . g. t hooft , nucl . b * 256 * , ( 1985 ) 727 . w. israel , _ _ phys . lett . a__**57 * * , 107 ( 1976 ) l. bombelli , r. k. koul , j. lee , and r. d. sorkin , _ _ phys . rev . d__**34 * * , 373 ( 1986 ) s. mukohyama , w. israel , phys . d * 58 * , ( 1998 ) 104005 . g. w. gibbons and s. w. hawking , phys . d * 15 * , ( 1977 ) 2752 . v. p. frolov , d. v. fursaev , _ class . quantum grav . _ * 15 * , 2041 ( 1998 ) d. v. fursaev , phys nucl . , * 36 * , ( 2005 ) 81 g. t. horowitz , _ black holes and relativistic stars , _ part ii , 12 , 241 , edited by robert m. wald . the univesity of chicago press , ( 1998 ) s. w. hawking , _ black holes and relativistic stars , _ part ii , 11 , 221 , edited by robert m. wald . the univesity of chicago press , ( 1998 ) v. frolov and i. novikov , phys . d * 48 * , 4545 ( 1993 ) hep - th/9802085 . arenas , j.m . tejeiro , _ nuovo cimento _ * 125b n.10 * , 1223 ( 2010 ) j.r . oppenheimer and h. snyder , phys . rev . * 56 * , ( 1939 ) 455 . w. israel , _ nuovo cimento vol xliv b _ * n.1 * , ( 1966 ) p. musgrave and k. lake , junctions and thin shells in general relativity using computer algebra , gr - qc/9510052v3 ( 1995 ) e. winstanley : phys . d63 , 084013(2001 ) hep - th/0011176 f pretorius , d. vollick , w. israel , phys . rev . d**57 * * , 6311 ( 1998 ) . j. d. bekenstein , in _ proceedings of the seventh marcel grossmann meeting on general relativity , stanford , usa , 1994 _ , edited by r. t. jantzen and g. m. keiser ( world scientific , singapore , 1996 ) , p. 39 , gr - qc/9409015 . y. takahashi and h. umezawa , collective phenomena * 2 * , 55 ( 1975 ) . h. umezawa , _ advanced field theory _ ( aip press , new york , 1993 ) . r. m. wald , phys . d * 17 * , 1477 ( 1978 ) . s. w. hawking , in _ general relativity : an einstein centenary survey _ , edited by s. w. hawking and w. israel ( cambridge university press , cambridge , 1979 ) , p. 746 . w. israel , in _ current trends in relativistic astrophysics _ , edited by l. fernndez , l. m. gonzlez ( springer lecture notes in physics , 2003 ) , lnp 617 , 15 ( 2003 ) .

we introduce the concept of black shell , consisting on a massive thin spherical shell contracting toward its gravitational radius from the point of view of an external observer far from the shell , in order to effectively model the gravitational collapse . considering complementary description of entanglement entropy of a black shell and according to gibbons - hawking euclidean approach , we calculate the bekenstein - hawking entropy retrieving horizon integral and discarding boundary at infinity . + key words : black hole entropy , entanglement entropy , black hole thermodynamics . + pacs numbers : 04.70.dy

You are an expert at summarizing long articles. Proceed to summarize the following text: the confirmation of the temporal variation of the fundamental constants would be the first indication of the universe expansion influence on the micro physics @xcite . shlyakhter was the first who showed that the variation of the fundamental constants could lead to measurable consequences on the sm isotops concentrations in the ancient reactor waste @xcite . later damur and dyson @xcite for zones 2 and 5 and also fujii @xcite for zone 10 of reactor oklo made more realistic analysis of the possible shift of fundamental constants during the last @xmath6 years based on the isotope concentrations in the rock samples of oklo core . in this investigation the idealized maxwell spectrum of neutrons in the core was used . the efforts to take into account more realistic spectrum of neutrons in the core were made in works @xcite . new severe constraints on the variation of the fine structure constant have been obtained from reactor oklo analysis in work @xcite : @xmath7 we investigate here how these constraints confine the parameter of bsbm model @xcite of varying @xmath0 . this theory combines bekenstein extension of electrodynamics @xcite with varying alpha to include gravitational effects of new scalar field @xmath8 . it respects covariance , gauge invariance , causality and has only two free parameters : the fraction of electromagnetic energy @xmath4 in the total energy of matter including dark matter as well as the dimensional parameter @xmath3 which is having sense of characteristic length . as a result of our analysis we get the constraints on the combination of the parameters of bsbm model . bsbm theory @xcite is the extension of the bekenstein @xcite theory to include dynamics of the gravitational field . total action of this theory has a form : @xmath9 where @xmath10 and @xmath11 . a parameter @xmath12 here is definite as @xmath13 where dimensional parameter @xmath3 is having sense of characteristic length . fine structure constant expressed via @xmath8 with the equation : @xmath14 . varying @xmath8 we get the following equation : @xmath15 for pure radiation @xmath16 , so @xmath8 remains constant during radiation domination epoch . only in matter domination epoch changes in @xmath0 take place . the only contribution to variation of @xmath8 come mainly from pure electrostatic or magnetostatic energy . it is convenient to work in the following parameter : @xmath17 and according to @xcite @xmath18 and @xmath19 . varying the metric tensor and using friedmann metric we get the following friedmann equation : @xmath20,\ ] ] and the equation for @xmath8 takes form : @xmath21 where @xmath22 . we have also energy conservation equations : @xmath23 which have solutions : @xmath24 , and @xmath25 let use critical density : @xmath26 and use also the fractions of all densities relative to critical : @xmath27 . index @xmath28 will denote the present values of these fractions . we use the ordinary values for these fractions at present : @xmath29 , @xmath30 , and @xmath31 is determined from the condition that the universe is flat . then the friedmann equation takes form : @xmath32,\ ] ] and equation for @xmath8 : @xmath33 here constant @xmath34 is equal to @xmath35 . for negative @xmath4 this constant is positive and has the following dependence on the ratio of characteristic and plank lengthes : @xmath36 the result of the numerical integration of these equations is presented of fig.1 for the variation of different components of energy density with red shift @xmath37 , and on fig.2 for the variation of fine structure constant @xmath0 . here we use the notation : @xmath38 . we took the value of the characteristic length @xmath3 equal to @xmath39 during this analysis , and assigned the following value for the the parameter @xmath4 : @xmath40 where @xmath41 - is the fraction of energy density in the universe due to ordinary baryonic matter . the initial values of the scalar field @xmath8 for the second order differential equation ( 6 ) : the value of the scalar field @xmath8 and its derivative during the radiation epoch was taken in such a manner that the present value of the fine structure constant coincide with experiment , and it appeared that the initial value of the @xmath42 during the radiation domination epoch could be assigned a rather arbitrary value because the result of integration influenced rather weakly by this choice . - dash - dot line.,width=453 ] experimental results for keck telescope @xcite , closed circles - experimental results from vlt telescope ( data were taken from work @xcite ) , red circle at @xmath43 - oklo result.,width=453 ] as it is followed from figure [ dens ] , the scalar field @xmath8 influence rather weakly on the variation of the different components of the energy density with red shift . the total variation of alpha during the whole history of the universe is about @xmath44 ( as is followed from figure [ alpha ] ) which is not contradict big bang and radiation recombination constraints @xcite . on the other side the oklo analysis predict about zero result for @xmath45 with the experimental error which could be seen in figure [ alpha ] ) if we increase the scale of figure [ alpha ] one hundred times . we investigate the constraints on the parameters of bsbm model followed from oklo analysis in the next section . in analysis of oklo data @xcite we obtained the following constraints on the variation of the fine structure constant @xmath46 during the past @xmath47 years . the age of the reactor @xmath48 years corresponds to red shift parameter @xmath43 . we use here also previous constraints obtained in @xcite : @xmath49 and in @xcite : @xmath50 all these constraints are shown on figure [ oklo ] . to provide the solution of the equations ( 5 ) and ( 6 ) which does nt contradict the result of work @xcite ( see figure [ oklo ] ) , we have to set rather severe constraints on the combinations of the parameters of bsbm model . they have to satisfy the following inequality : @xmath51 for realistic value @xmath52 to fulfill this inequality we have to demand that : @xmath53 a theoretical framework under very general assumptions was worked out by bekenstein to admit the variation of the fine structure constant . a characteristic length @xmath3 enters into it . an experimental constraint rules out @xmath0 variability of any kind if it is in clear conflict with predictions of the framework for @xmath3 no shorter than the fundamental length @xmath39 ( @xcite ) . as a result of oklo analysis we get @xmath54 the oklo geophysical constraints strongly rule out all @xmath0 variability . in this analysis we have considered only the variation of electromagnetic fine structure constant @xmath0 . if other fundamental constants also varies the picture would be more complicated as well as the analysis of the oklo phenomenon and the analysis of the cosmological variation of @xmath0 . to do such analysis in our opinion would be too early because till now we havent had any convincing manifestations of the cosmological variations of the other fundamental constants @xcite . the author would like to express his gratitude to s. karshenboim and m.s . yudkevich for useful discussions and critical remarks . this work was partly supported by the rscf grant ( project 14 - 22 - 00281 ) .

new severe constraints on the variation of the fine structure constant have been obtained from reactor oklo analysis in our previous work . we investigate here how these constraints confine the parameter of bsbm model of varying @xmath0 . integrating the coupled system of equations from the big bang up to the present time and taking into account the oklo limits we have obtained the following margin on the combination of the parameters of bsbm model : @xmath1 where @xmath2 cm is a plank length and @xmath3 is the characteristic length of the bsbm model . the natural value of the parameter @xmath4 - the fraction of electromagnetic energy in matter - is about @xmath5 . as a result it is followed from our analysis that the characteristic length @xmath3 of bsbm theory should be considerably smaller than the plank length to fulfill the oklo constraints on @xmath0 variation .

You are an expert at summarizing long articles. Proceed to summarize the following text: quantum phenomenon has been observed by numerous experiments on microscopic scales . however , on macroscopic scales , it is difficult to find quantum effects , such as quantum superpositions . a lot of physicists have been looking up the root of quantum - to - classical transition for decades . the reason can be divided into two categories : coarsened measurement and decoherence@xcite . commonly viewpoint is that decoherence plays a prominent role in quantum - to - classical transition . there are two routes to explain decoherence : one route is that system interacts with external environments , the other is taken in wave function collapse@xcite , which need not external environments . the latter one is often inspired by general relativity and makes a fundamental modification on quantum theory . recently , igor at al.@xcite demonstrated the existence of decoherence induced by gravitational time dilation without any modification of quantum mechanics . this work motivates further study on decoherence due to time dilation . spontaneous emission between two atomic levels inevitably occurs . we research decoherence due to time dilation during spontaneous emission . without spontaneous emission , decoherence will not occur in our model only by time dilation . as we all know , spontaneous emission can induce decoherence . we find that gravitational time dilation can reduce or increase the decoherence due to spontaneous emission in different reference frames ( different zero potential energy point ) . it is attributed to the fact that in different reference frame , the distinguishability of emission photon from different positions is different . the direction of emission light also influences the coherence of quantum superpositions in fixed direction of gravitational field . in order to make the decoherence due to time dilation stronger than due to spontaneous emission , time - delayed feedback control@xcite is used . the rest of paper is arranged as follows . in section ii , we present the model about the decoherence of quantum superpositions due to time dilation during spontaneous emission . coherence of particle s position in different reference frame is explored in section iii . in section iv , we discuss the influence of different directions of emission light . in section v , a time - delayed feedback scheme is utilized to increase decoherence induced by gravitational time dilation . we deliver a conclusion and outlook in section vi . firstly , let us simply review the gravitational time dilation which causes clocks to run slower near a massive object . given a particle of rest mass @xmath0 with an arbitrary internal hamiltonian @xmath1 , which interacts with the gravitational potential @xmath2 . the total hamiltonian @xmath3 is described by@xcite @xmath4,\end{aligned}\ ] ] where @xmath5 is external hamiltonian . for a free particle , @xmath6 . in eq.(1 ) , the last term , @xmath7 , is simply the velocity - dependent special relativistic time dilation . the coupling with position , @xmath8 , represents the gravitational time dilation . when we consider slowly moving particles , @xmath9 , the gravitational time dilation will be the main source of time dilation . it will not be canceled by the velocity - dependent special relativistic time dilation . we consider that an atom with two levels is in superposition of two vertically distinct positions @xmath10 and @xmath11 . the atom is coupled to a single unidirectional light field , as depicted in fig . 1 . decoherence of an atom is induced by gravitational time dilation under the situation of spontaneous emission . the dotted line represents a homogeneous gravitational field @xmath12 , where @xmath13 is the gravitational acceleration on earth . initial atom is in the superposition state : @xmath14 . the direction of emitting photon is along the solid line , x - direction , which is contrary with the direction of gravitational field . ] the whole system interacts with a homogeneous gravitational field @xmath12 which generates the gravitational time dilation . the total system - field hamiltonian is described by ( @xmath15 ) @xmath16|x_1\rangle\langle x_1|\nonumber \\+\sqrt{\kappa_1/2\pi}(1+g x_1/c^2)|x_1\rangle\langle x_1|\int dw [ a b^\dagger(w)\exp(-i w x_1/c)+h.c]\nonumber\\ + [ mc^2+m g x_2 + w_1(1+g x_2/c^2)|1\rangle\langle1|+ w_2(1+g x_2/c^2)|2\rangle\langle2|]|x_2\rangle\langle x_2|\nonumber\\ + \sqrt{\kappa_2/2\pi}(1+g x_2/c^2)|x_2\rangle\langle x_2|\int dw [ a b^\dagger(w)\exp(-i w x_2/c)+h.c]+\int dw w b^\dagger(w)b(w),\end{aligned}\ ] ] where @xmath17 and @xmath18 ( @xmath19 ) are eigenvalues for the atomic level 1 and 2 , respectively , and operator @xmath20 . @xmath21 and @xmath22 denote the coupling constants in position @xmath10 and @xmath11 , respectively . without extra control , the two coupling constants should be same : @xmath23 . the last term in eq.(2 ) represents the free field hamiltonian , and the filed modes , @xmath24 , satisfy @xmath25=\delta(w - w')$ ] . using pauli operator , @xmath26 , to simplify the eq.(2 ) , we can obtain the new form of system - field hamiltonian @xmath27|x_1\rangle\langle x_1|+\sqrt{\kappa/2\pi}(1+g x_1/c^2)|x_1\rangle\langle x_1|\int dw [ a b^\dagger(w)\exp(-i w x_1/c)+h.c]\nonumber\\ + [ e_2+w_0/2(1+g x_2/c^2)\sigma_z]|x_2\rangle\langle x_2|+\sqrt{\kappa/2\pi}(1+g x_2/c^2)|x_2\rangle\langle x_2|\int dw [ a b^\dagger(w)\exp(-i w x_2/c)+h.c]\nonumber\\+\int dw w b^\dagger(w)b(w),\end{aligned}\ ] ] where @xmath28 for @xmath29 and @xmath30 . we consider the initial field in the vacuum state and the atom in the state @xmath31 . then , the atom will spontaneously emit photon . according to there being only a single excitation conservation between system and field@xcite , the system state in any time @xmath32 can be solved analytically , see appendix . the quantum coherence of particle s position state can be quantified by the interferometric visibility @xmath33 , as shown in eq.(27 ) in appendix . when the time @xmath32 satisfy @xmath34 and @xmath35 , the amplitude of excitation state @xmath36 and @xmath37 . then , we arrive at @xmath38 ^ 2+[w_0(\lambda_2-\lambda_1)]^2 } } \exp[-\lambda_1 ^ 2\kappa\tau],\end{aligned}\ ] ] where @xmath39 for @xmath29 . from the above equation , we can see that the decoherence comes from the spontaneous emission ( when @xmath40 ) and the gravitational time dilation . spontaneous emission can generate the decoherence due to the fact that photon is emitted from different positions , which leads to having a phase difference @xmath41 , where @xmath42 denotes the frequency of photon . and we achieve that coherence depends on the reference frame . different zero potential energy point ( different value of @xmath43 ) will give different coherence strength . the counterintuitive result occurs because in different frame the phase difference will become different so that the distinguishability of emitting photon from two positions is different . reducing the zero potential point ( increasing the value of @xmath43 ) , the phase difference will increase because of time dilation . for a fixed position difference @xmath44 , the quantum coherence can be rewritten @xmath45 ^ 2+(w_0\delta)^2 } } \exp[-\lambda_1 ^ 2\kappa\tau].\end{aligned}\ ] ] there is an optimal value of @xmath43 , which can give the maximal quantum coherence , as shown in fig . 2 . diagram of quantum coherence @xmath46 changing with @xmath43 . the quantum coherence depends on reference frame . the parameters are given by : @xmath47 , @xmath48 , @xmath49 . ] in an optimal reference frame , one can obseve the maximal coherence : @xmath46 is close to 1 . in order to observe the decoherence induced by gravitational time dilation , it need to satisfy that the decoherence effect from time dilation is stronger than only from spontaneous emission ( @xmath40 ) : @xmath50 ^ 2+(w_0\delta)^2}}\exp[-\lambda_1 ^ 2\kappa\tau ] \ll\exp[-\kappa\tau ] . _ { } \end{aligned}\ ] ] noting that the value of @xmath51 is generally small in experiment , the condition @xmath52\gg1 $ ] is necessary for observing decoherence mainly induced by gravitational time dilation . when one changes the direction of emitting photon , the quantum coherence will change accordingly , @xmath53 ^ 2+[w_0(\lambda_2-\lambda_1)]^2 } } \exp[-\lambda_2 ^ 2\kappa\tau].\end{aligned}\ ] ] it is due to that the phase difference changes with the direction of emitting photon , becoming @xmath54 . different directions of emitting photon in the fixed gravitational field will generate different quantum coherence @xmath46 . then , we consider general three - dimensional space : the emitting photon can be along any direction , as shown in fig . diagram shows that the photon can spontaneously emit in any direction . here @xmath55 denotes the angle between direction of photon and x - direction , which can change from 0 to @xmath56 . ] we obtain the quantum coherence of particle s position state as following : @xmath57 ^ 2+[w_0(\lambda_2-\lambda_1)]^2}}| \int_0^{\pi/2 } d\theta\sin\theta\cos^2\theta \exp[(iw_0\lambda_1-\lambda_1 ^ 2\kappa)\tau\cos\theta]+\exp[-(iw_0\lambda_2+\lambda_2 ^ 2\kappa)\tau\cos\theta]|,\\ & = \frac{3\kappa\lambda_1\lambda_2}{\sqrt{[\kappa(\lambda_1 ^ 2+\lambda_2 ^ 2)]^2+[w_0(\lambda_2-\lambda_1)]^2}}|[-2 + \exp(k_1 ) ( 2 - 2 k_1 + k_1 ^ 2)]/k_1 ^ 3+[-2 + \exp(k_2 ) ( 2 - 2 k_2 + k_2 ^ 2)]/k_2 ^ 3|,\\ & \textmd{in which},\nonumber\\ & k_j=[(-1)^jiw_0\lambda_j-\lambda_j^2\kappa]\tau , \ \textmd{for } \ j=1,2,\end{aligned}\ ] ] where the coupling strength between atom and light field changes with the direction of emitting photon , becoming @xmath58@xcite . for @xmath59 and @xmath60 , @xmath61 ^ 2+[w_0(\lambda_2-\lambda_1)]^2}}(1-\lambda_1 ^ 2\kappa\tau-\lambda_2 ^ 2\kappa\tau)<v'<v.$ ] it means that the quantum coherence in general three - dimensional space is smaller than in one - dimensional space of fixed direction . for @xmath62 and @xmath63 , @xmath61 ^ 2+[w_0(\lambda_2-\lambda_1)]^2}}|\cos3\varphi|(3/[(w_0 ^ 2\lambda_1 ^ 2+(\lambda_1 ^ 2\kappa\tau)^2]^{3/2}+3/[(w_0 ^ 2\lambda_1 ^ 2+(\lambda_1 ^ 2\kappa\tau)^2]^{3/2})\geq v,$ ] with @xmath64 . it means that in new condition the quantum coherence in general three - dimensional space is larger than in one - dimensional space of fixed direction . the root of generating @xmath65 is the phase difference changing from @xmath41 to @xmath66 . when one chooses the center of two positions as the zero potential point , the interferometric visibility reads @xmath67 ^ 2+(w_0\delta)^2 } } \exp[-(1-\delta/2)^2\kappa\tau].\end{aligned}\ ] ] in order to observe the decoherence from gravitational time dilation , not from the spontaneous emission , it is necessary to satisfy condition @xmath68 $ ] . however , the value of @xmath69 is very small in experiment . so , the condition is hard to meet . we can utilize the time delay feedback @xcite to increase the decoherence from gravitational time dilation . diagram of time delay feedback . the center of two positions is chosen as zero potential point . so , in the new reference frame , the position @xmath10 ( @xmath11 ) is transformed to be @xmath70 ( @xmath71 ) . here , we just consider that the photon emits along the fixed x - direction , which can be easily generalized to the case of three - dimensional space . at @xmath72 , a mirror is put to reflect the light field , leading to that a time - delay light field is fed back to system - field interaction . ] as shown in fig . 4 , the light field is reflected by a mirror . the whole system - field hamiltonian can be described by @xmath73\cos(2w r)+h.c\}\nonumber\\ & + \sqrt{\kappa/2\pi}(1+g x_2/c^2)|x_2\rangle\langle x_2|\int dw\{a b^\dagger(w)2\exp[-i w(r+\delta c^2/2g)]\cos [ w ( 2r+2\delta c^2/g)/c]+h.c\}\nonumber\\ & + \int dw w b^\dagger(w)b(w)+[e_1+w_0/2(1+g x_1/c^2)\sigma_z]|x_1\rangle\langle x_1|+[e_2+w_0/2(1+g x_2/c^2)\sigma_z]|x_2\rangle\langle x_2|,\end{aligned}\ ] ] using the way in appendix , we can obtain the quantum coherence at time @xmath74 . with the feedback , the spontaneous emission is suppressed due to superposition effect . the total system - field wave function can also be described by eq.(13 ) in appendix . when the conditions @xmath75 and @xmath76 hold , for @xmath74 the amplitudes @xmath77 $ ] and @xmath78 , where @xmath79 . when @xmath80 and @xmath81 , @xmath82 . so , we achieve that the quantum coherence @xmath83 . without gravitational time dilation , the quantum coherence is much larger than 0 . so , utilizing time - delayed feedback scheme can satisfy that the decoherence induced by the gravitational time dilation is far less than by spontaneous emission . we explore the decoherence of an atom s positions induced by the gravitational time dilation only in the situation of spontaneous emission . as the phase difference of photon emitted from two positions are different in different reference frames , the quantum coherence of superposition state of positions depends on the reference frame . so one can choose proper reference frame to observe the decoherence from the gravitational time dilation . it is worth mentioning that the direction of emitting photon will influence the quantum coherence . so comparing the case of fixed emitting direction with the case of any direction , there are some differences about quantum coherence . when one chooses the center of two positions as the zero potential point , the decoherence induced by the gravitational time dilation is difficult to be far larger than by spontaneous emission . the time delay feedback can be used to increase the decoherence from the time dilation with proper conditions . in this article , we only discuss the decoherence of an atom with two energy levels induced by gravitational time dilation . it is interesting to research the decoherence of many particles with many energy levels induced by time dilation with spontaneous emission . in this case we believe that it will increase the decoherence effect from the gravitational time dilation . and considering extra drive is the further research direction . in this situation , due to the fact that a single excitation conservation between system and field do not hold , the question will become complex and rich . this work was supported by the national natural science foundation of china under grant no . 11375168 . maximilian schlosshauer , rev . mod . phys . * 76 * , 1267 ( 2005 ) . w. h. zurk , rev . phys . * 75 * , 715 ( 2003 ) . s. raeisi , p. sekatski , and c. simon , phys . lett . * 107 * , 250401 ( 2011 ) . h. jeong , m. paternostro , and t. c. ralph , phys . lett . * 102 * , 060403 ( 2009 ) . h. jeong and t. c. ralph , phys . * 97 * , 100401 ( 2006 ) . p. caldara , a. la cognata , d. valenti , and b. spagnolo , int . j. quantum . inf . * 9 * , 119 ( 2011 ) . d. kast and j. ankerhold , phys . . lett . * 110 * , 010402 ( 2013 ) . hyunseok jeong , youngrong lim , and m. s. kim , phys . * 112 * , 010402 ( 2014 ) . r. penrose , gen . * 28 * , 581 - 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w)t]c_{1k},\\ i\partial_tc_{1w}=[e_1-w_0/2(1+g x_1/c^2)]c_{1k}+(1+g x_1/c^2)\sqrt{\kappa/2\pi}\int dw \exp[i(-wx_1/c+w)t]c_{1},\\ i\partial_tc_2=[e_2+w_0/2(1+g x_2/c^2)]c_1+(1+g x_2/c^2)\sqrt{\kappa/2\pi}\int dw \exp[i(wx_2/c - w ) t]c_{2k},\\ i\partial_tc_{2w}=[e_2-w_0/2(1+g x_2/c^2)]c_{2k}+(1+g x_2/c^2)\sqrt{\kappa/2\pi}\int dw \exp[i(-wx_2/c+w)t]c_{2}.\end{aligned}\ ] ] substituting @xmath93c_1 , c'_{1k}=\exp[-i(e_1-w_0/2(1+g x_1/c^2))t]c_{1k},$ ] @xmath94c_2 , c'_{2k}=\exp[-i(e_2-w_0/2(1+g x_2/c^2))t]c_{2k}$ ] into above equations , we arrive at the following simplified equations : @xmath95c'_{1w},\\ i\partial_tc'_{1w}=(1+g x_1/c^2)\sqrt{\kappa/2\pi}\int dw \exp[i(-wx_1/c+(w - w_0(1+g x_1/c^2))t]c'_{1},\\ i\partial_tc'_2=(1+g x_2/c^2)\sqrt{\kappa/2\pi}\int dw \exp[i(wx_2/c+(w_0(1+g x_2/c^2)-w)t]c'_{2w},\\ i\partial_tc'_{2w}=(1+g x_2/c^2)\sqrt{\kappa/2\pi}\int dw \exp[i(-wx_2/c+(w - w_0(1+g x_2/c^2))t]c'_{2}.\end{aligned}\ ] ] eq.(15 ) and eq.(17 ) are integrated formally and inserted into eq.(14 ) and eq.(16),respectively . utilizing the integral @xmath96=2\pi\delta(t),\end{aligned}\ ] ] we can analytically solve the set of partial differential equations . at time @xmath32 , using the initial values @xmath97 and @xmath98 , we obtain @xmath99,\\ c'_{1w}=\frac{1-\exp[-1/2\lambda_1 ^ 2\kappa t - i(\lambda_1w_0-w)t]}{\lambda_1 ^ 2\kappa+i(\lambda_1w_0-w)}\sqrt{\kappa/2\pi}\lambda_1\exp[iwx_1/c],\\ c'_2(t)=1/\sqrt{2}\exp[-1/2\lambda_2 ^ 2\kappa t],\\ c'_{2w}=\frac{1-\exp[-1/2\lambda_2 ^ 2\kappa t - i(\lambda_2w_0-w)t]}{\lambda_2 ^ 2\kappa+i(\lambda_2w_0-w)}\sqrt{\kappa/2\pi}\lambda_2\exp[iwx_2/c],\end{aligned}\ ] ] where @xmath39 for @xmath29 . the quantum coherence of position state can be quantified by the interferometric visibility @xmath100{c'_1}^*c'_2+\exp[i(x_2-x_1)w_2g / c^2t]\int dwc'^*_{1w}\int dw'c'_{2w'}|,\end{aligned}\ ] ] where the term @xmath101 can be integrated by residue theorem . we can arrive at @xmath102-\exp[-1/2\lambda_1 ^ 2\kappa t+i\lambda_1w_0t+i\xi(\tau - t)]-\nonumber\\ & \exp[-1/2\lambda_2 ^ 2\kappa t - i\lambda_2w_0t+i(w_0\lambda_1+i\lambda_1 ^ 2\kappa)(\tau+t)]+\exp[-1/2\lambda_1 ^ 2\kappa ( t+\tau)-1/2\lambda_1 ^ 2\kappa t+i(\lambda_2-\lambda_1)w_0t+i\lambda_1w_0\tau]\},\\ & \textmd{in which},\nonumber\\ & \tau=(x_2-x_1)/c\geq0,\\ & \textmd{for}\ t<\tau , \ \xi = w_0\lambda_1+i\lambda_1 ^ 2\kappa,\ \ \textmd{for}\ t\geq\tau,\ \xi = w_0\lambda_2-i\lambda_2 ^ 2\kappa.\end{aligned}\ ] ]

we investigate decoherence of quantum superpositions induced by gravitational time dilation and spontaneous emission between two atomic levels . it has been shown that gravitational time dilation can be an universal decoherence source . here , we consider decoherence induced by gravitational time dilation only in the situation of spontaneous emission . then , we obtain that the coherence of particle s position state depends on reference frame due to the time dilation changing the distinguishability of emission photon from two positions of particle . changing the direction of light field can also result in the difference about the coherence of quantum superpositions . for observing the decoherence effect mainly due to gravitational time dilation , time - delayed feedback can be utilized to increase the decoherence of particle s superpositions .

You are an expert at summarizing long articles. Proceed to summarize the following text: the large hadron collider ( lhc ) probes collisions of protons at very high energies , resulting in a multitude of final - state particles . with increasing energy , the probability that one hadron - hadron collision leads to more than one scattering process also increases . these additional scattering processes beside the primary hard scattering belong to the group of multi - parton interactions ( mpi ) . their estimation is important for the correct determination of background from standard model processes , for instance when the signal process consists of new physics particles . in particular , double parton scattering ( dps ) , where two distinct parton interactions arise from the same proton - proton collision , can become likely enough to compete with single parton scattering ( sps ) processes , see fig . [ fig : dpsfeyn ] . therefore , a thorough understanding of these additional contributions is needed for a precise theoretical description of the background at the lhc and will also help to explore the inner structure of protons and nucleons , not being accessible by perturbative calculations . double parton scattering has been searched for both in pre - lhc experiments like afs , ua2 , cdf , and d0 as well as by the lhcb and atlas collaborations , in 4-jet @xcite , @xmath6-jet @xcite , di-@xmath7-jets @xcite , @xmath8-jets @xcite , @xmath9 @xcite , @xmath10 @xcite , open charm @xcite , @xmath0+charm @xcite , @xmath11+charm @xcite , @xmath12 @xcite and @xmath13 @xcite final states . on the theoretical side the efforts have concentrated on improving the understanding of the underlying scattering mechanism as well as providing phenomenological predictions . in particular related issues such as correlations and interferences between the two hard scatterings , the role of the perturbative splitting contributions ( so - called `` 2v1 '' ) and the definition of double parton scattering cross section as well as double parton distributions have been addressed , see e.g. @xcite for a comprehensive review . a @xmath0 pair is a very good candidate to study double parton scattering at the lhc due to relatively high production rates and subsequent decays into muons giving a clear and easily distinguishable signal . results for the production of @xmath0 pairs have been published by lhcb in @xcite , by d0 in @xcite , and by cms in @xcite . correspondingly , since then there has been a considerable interest to improve theoretical predictions for double @xmath0 production both for the sps and dps production modes @xcite . the calculation of conventional single parton scattering contributions to @xmath0 pair - production is non - trivial and requires specific methods to account for the non - perturbative mechanisms involved in meson production as well as the short - distance effects . two widely applied approaches are the colour - singlet model ( csm ) @xcite and non - relativistic quantum chromodynamics ( nrqcd ) @xcite . in the framework of nrqcd , until not long ago , only the lo predictions for hadronic production in the colour singlet production mode @xcite , supplemented by the octet corrections @xcite , were known . recently , the effects of relativistic corrections @xcite , nlo corrections and selected nnlo qcd contributions @xcite as well as an application of the @xmath14 factorisation approach @xcite have been investigated . additionally , the importance of including contributions from all possible @xmath15 fock state configurations relevant for prompt double @xmath0 production has been pointed out in @xcite . this paper documents the predictions of sps and dps production of a pair of @xmath0 , delivered to the lhcb and atlas experiments for their ongoing studies of double parton scattering with run i data . the work presented here updates the study on @xmath0 pair - production reported in @xcite , which in turn was inspired by the first measurement of a double @xmath0 signal @xcite . furthermore , predictions for the current lhc run at a centre - of - mass energy of @xmath16 tev are provided . we also perform a comparison with cms data @xcite and more thoroughly with theoretical predictions for double @xmath0 production obtained by another group @xcite . the outline is as follows . in section [ sec : theo_setup ] , the theoretical setup of @xcite used for both the sps and dps cross section calculations is reviewed , followed by a listing of monte carlo parameters for event simulation in section [ sec : monte_sim ] . we present numerical results for total cross sections and kinematic distributions for a choice of experimentally accessible variables in section [ sec : kin_dis ] . at last , we conclude in section [ sec : conclusions ] . in this work , the sps contributions will be considered utilising a leading - order ( lo ) colour - singlet result presented in @xcite and including radiative corrections from parton showering . the details of the implementation are described in section [ sec : monte_sim ] and the sps results obtained in this way are compared to the nlo calculations of @xcite in section [ sec : complansberg ] . as it was pointed out in @xcite , the prompt production of @xmath0 mesons comprises feed - down from the decay of @xmath17 and @xmath18 at a non - negligible amount of roughly 85% . the sps calculation of @xcite is for direct production of @xmath0 pairs only , so in the following , all sps cross sections will be considered for prompt production , @xmath19 . the dps results implicitely include feed - down contributions due to the fit to experimental data . to include some higher - order effects in our sps predictions , in addition to using nlo pdfs , we enable initial - state radiation or parton showering within the ` herwig ` @xcite framework . furthermore , if denoted , we also add effects of intrinsic transverse momentum of the initial - state partons using a gaussian model in ` herwig ` with a root mean square @xmath20 of 2 gev . we have checked that the predictions do not depend strongly on the actual numerical value , and it will be seen in the following sections that the effect of the intrinsic transverse momentum is rather mild on the distributions . dps production of a @xmath0-pair is described using an approximation in which the dps cross section factorises into a product of two hard - scattering cross sections describing single-@xmath0 production which are independent from each other : @xmath21 this customary approximation assumes factorization of the transverse and longitudinal components in the generalized parton distribution function . we refer the reader to @xcite for a discussion of the validity of the approximation and the status of understanding factorization in dps . the sps cross section for single-@xmath0 production is given as @xmath22 with a sum over the initial - state flavours @xmath23 and the parton distribution functions @xmath24 and @xmath25 the partonic cross section for single @xmath0 production with the corresponding matrix elements @xmath26 and the phase space @xmath27 . @xmath28 denotes any additional final state which is not a @xmath0 , and therefore not of interest for @xmath0 production . the factor @xmath29 is assumed to only depend on the transverse structure of the proton , and should therefore be process and energy independent if the factorisation of eq . holds . it is the main quantity to be extracted by a dps experiment . theoretical description of single quarkonium production @xcite is challenging even within the nrqcd framework @xcite . given that lhcb can trigger over low @xmath30 muons it is important to describe the low @xmath30 production accurately . therefore in this work we choose to model the low @xmath30 region and use the same setup as in @xcite . it relies on the matrix element as in eq . given by : @xmath31^{-n } & \text{for } p_t > \langle p_t\rangle , \end{cases}\label{crystalball}\end{aligned}\ ] ] which describes a fit of data from the lhcb @xcite , atlas @xcite , cms @xcite , and cdf @xcite experiments to a crystal ball function . in eq . , @xmath32 , and the fit parameters are determined to be @xmath33 and @xmath34 for @xmath35 and @xmath36 gev . while the work of @xcite updated the fit parameters to include more recent measurements of single @xmath0 production , we have checked that the change in predictions for dps production is only moderate and well within the uncertainty on @xmath29 . we have also checked that for the available measurement of single @xmath0 production at 13 tev from the lhcb experiment @xcite with @xmath37 @xmath38b , the fit parameters still produce results at 13 tev consistent with the lhcb measurement , @xmath39 @xmath38b . the public monte carlo event generator ` herwig-7.0.3 ` @xcite has been used to simulate double @xmath0 production at the lhc via sps . the central values for the renormalisation and factorisation scales are chosen as the transverse mass of a single @xmath0 , @xmath40 with the physical @xmath0 mass @xmath41 gev @xcite . one parameter appearing in the calculation of the sps cross section of @xcite is the charm quark mass which we set to @xmath42 which corresponds to the lo choice of the hadron mass in a nrqcd calculation @xcite . another input parameter entering the sps calculation is the non - perturbative wave function of the @xmath0 meson at the origin . in the following computations , it is set to @xmath43 gev@xmath44 @xcite . it should be noted that a variation of this parameter can be achieved by multiplying the sps cross section by a factor of @xmath45 , where @xmath46 is the new value of the wave function . we use mstw2008 nlo parton distribution functions @xcite for the sps predictions including initial - state radiation and for dps . the parton distribution functions are accessed via the lhapdf 6 library @xcite . the @xmath0 mesons are assumed to decay isotropically into a pair of opposite - sign ( os ) muons with a branching ratio of @xmath47 . out of the two possible combinations of choosing os muon pairs , the one with an invariant mass closest to @xmath48 is chosen . from these pairs , properties of the @xmath0 are reconstructed . to optimise the data samples collected by the experiments for a dps analysis , a certain set of cuts on transverse momentum and ( pseudo-)rapidity of the @xmath0 as well as their decay products is applied . unless otherwise specified , we use the cdf value of @xmath49 mb @xcite . a more recent double-@xmath0 study by d0 reports a lower value of @xmath50 mb @xcite , but given that most of other experiments measure higher values , see e.g. @xcite , and the difficulty of theoretical modelling of @xmath29 , we choose the cdf value . with its relatively wide error bars it then accounts to a large extent for the observed span in values of @xmath51 . we also note that this value is in accordance with the phenomenological estimates @xcite taking into account in our framework ( eq . [ dpsfact ] ) the so - called `` 2v2 '' and `` 2v1 '' contributions , i.e. contributions from two separate parton ladders or from one ladder and another ladder created by a perturbative splitting of a single parton , respectively . as found out in @xcite , the shapes of the transverse momentum and rapidity distributions for the two types of production mechanisms remain very similar , justifying our effective approach of considering only the conventional 2v2 scattering . in a similar manner as for the non - perturbative wave function at the origin , the dps results for a different value of @xmath29 can be obtained by rescaling our dps cross section with a factor of @xmath52 . the dps predictions have been cross checked with two independent numerical in - house implementations . the lhcb experiment , being a forward spectrometer , mainly selects events in the forward - scattering region with low transverse momentum . the cuts are : * @xmath53 gev , * @xmath54 , with @xmath55 being the transverse momentum and @xmath56 the rapidity of a single @xmath0 . any further cuts on the muons as the decay products of the di-@xmath0 are not relevant for the theoretical predictions presented in this work , as they are already taken into account in the efficiency correction of the data . the atlas experiment probes the @xmath0 in the central region imposing a minimum transverse momentum and a more central rapidity region . additionally , several cuts are also applied to the muons : * @xmath57 gev , * @xmath58 , * @xmath59 gev , * @xmath60 , * at least 1 @xmath0 with both @xmath61 gev , with @xmath62 being the transverse momentum and @xmath63 the pseudorapidity of one muon . .total cross sections for sps and dps production of a @xmath0 pair for different centre - of - mass energies and cuts . ps+@xmath64(2 gev ) denotes the addition of initial - state radiation and intrinsic transverse momentum to the lo calculation in ` herwig ` . all numbers include the branching ratio factor of @xmath65 . the uncertainties on the ps+@xmath64(2 gev ) numbers correspond to a simultaneous variation of the renormalisation and factorisation scales up and down by a factor of two , while the uncertainty on the dps numbers corresponds to the uncertainty of our value of @xmath29 used , see section [ sec : monte_sim ] . [ cols="^,^,^",options="header " , ] also for the atlas predictions , shown in fig . [ fig : atlas_plots13 ] , the dominance of the dps contributions at 13 tev leads to an easier distinction between sps and dps . despite the transverse momentum distribution of the di-@xmath0 system in fig . [ fig : atlas_plots13 ] ( a ) now offering a clearer possibility to separate sps and dps contributions for low @xmath66 due to the dps contributions being larger than sps by almost a factor of 2 , we see that the increase in the centre - of - mass energy complicates the distinction of the shapes of sps and dps , and we again remark that higher - order corrections which are not included here can change the shape of the distribution significantly . cc tev for the invariant mass ( a ) , the rapidity separation ( b ) , and the transverse momentum of the di-@xmath0 system ( c ) . shown are bins which are normalised to the corresponding total cross sections of the different sps and dps distributions and the data.,title="fig : " ] & tev for the invariant mass ( a ) , the rapidity separation ( b ) , and the transverse momentum of the di-@xmath0 system ( c ) . shown are bins which are normalised to the corresponding total cross sections of the different sps and dps distributions and the data.,title="fig : " ] + ( a ) & ( b ) + + the cms experiment has recently measured @xmath0-pair production at @xmath1 tev @xcite . they applied the following cuts to their data : * @xmath67 gev for @xmath68 , * @xmath69 gev for @xmath70 , * @xmath71 gev for @xmath72 . the @xmath30 cut in point 2 scales linearly from 6.5 gev to 4.5 gev with the value of @xmath73 from 1.2 to 1.43 . no further cuts on the muons are applied . in fig . [ fig : cms_comp ] , we compare our predictions to the cms data . we show all bins normalised to the corresponding total cross section of a line to only compare the shape of the distributions and approximately remove the dependence on a specific pdf set . furthermore , for the theoretical sps and dps predictions , we only show the central values without the error bands as described in section [ sub : lhcb_predictions ] . shown are the invariant mass distribution , the transverse momentum of the di-@xmath0 system , and the rapidity separation of the two @xmath0 . we see that our predictions catch the bulk behaviour of the cms data , in particular also when further sources of uncertainty like the exact choice of the parameters which appear in the sps and dps calculations would be taken into account . especially for the invariant mass and the rapidity separation distributions , figs . [ fig : cms_comp ] ( a ) and ( b ) , we see that at the high end of the spectrum the dps contributions can not be neglected . at the same time , the existing discepancies between theory and data call for further improvements in the theoretical description of sps and dps distributions . at last , we compare our results to the recently published ones of lansberg and shao @xcite . the authors present predictions for similar scenarios of @xmath0-pair production at the lhcb and atlas experiments , with the difference of using full calculations of real gluon emission at nlo@xmath74 , the asterisk denoting the lack of virtual corrections . this method differs from ours by also taking into account hard gluon emission , while parton showering only considers soft gluons , however to all orders in @xmath75 . in this regard , it is interesting to see how the two approaches compare for the lhcb and atlas cuts . in order to minimise the sources of uncertainty , we choose parameters and pdf sets as close as possible to lansberg et al . these are the wave function of the @xmath0 meson at origin @xmath76 gev@xmath44 , the charm mass in the range @xmath77-@xmath78 gev , and the pdf sets cteq6l1 for lo @xcite , cteq6 m for sps ( ps+@xmath64 ) @xcite , and mstw2008 nlo for dps . the renormalisation and factorisation scales for sps production are set to @xmath79 , where @xmath80 . the error bands are obtained from a simultaneous variation of @xmath81 and @xmath82 as @xmath83;@xmath84;@xmath85 . for dps , we additionally use the effective dps cross section @xmath86 mb and the best fit parameters of @xcite with @xmath87 and @xmath88 . the factorisation scale in this case is set , as before , to the transverse mass of a single @xmath0 . the error bands are now obtained from the uncertainty of @xmath29 . [ fig : comp_lhcb_lansberg ] shows the comparison of distributions for three kinematic variables : the transverse momentum of the di-@xmath0 system @xmath66 , the rapidity separation between the two @xmath0 @xmath89 , and the invariant mass of the di-@xmath0 system @xmath90 , for the lhcb cuts at a collider energy of @xmath1 tev . we note that in @xcite , the sps predictions for the @xmath89 and @xmath90 distributions are only given at lo without taking into account additional gluon radiation . in fig . [ fig : comp_lhcb_lansberg ] ( a ) , it can be seen that the transverse momentum distributions of sps in our calculation and the one of @xcite agree within the error bands for intermediate values of @xmath66 , while they differ at low and high @xmath66 . we see at low @xmath66 the typical suppression from the all - order structure of parton showering , while the nlo@xmath74 prediction is growing towards small @xmath66 . we expect that the inclusion of parton showering describes the shape of the distribution at low @xmath66 better than a fixed - order calculation at nlo@xmath74 , since a major part of the contribution in this region comes from soft - collinear gluon emission , which is approximately taken into account at all orders in @xmath75 in the parton shower formalism . on the other hand , the high-@xmath66 region can not be described properly by parton showering due to the lack of hard gluon emission which dominates this region . we remark that the good agreement for intermediate @xmath66 is also related to the @xmath53 gev cut , effectively cutting off the high-@xmath66 region where hard gluon emission becomes important . the dps predictions for the transverse momentum distribution agree very well between our calculation and @xcite due to the same functional form of the cross section fit of eq . . cc system ( a ) , the rapidity separation ( b ) , and the invariant mass ( c ) . the `` sps nlo ( lansberg et al . ) '' and `` dps ( lansberg et al . ) '' uncertainty bands are read off from the corresponding plots in @xcite . the cross sections here are not multiplied by the squared branching ratio @xmath65.,title="fig : " ] & system ( a ) , the rapidity separation ( b ) , and the invariant mass ( c ) . the `` sps nlo ( lansberg et al . ) '' and `` dps ( lansberg et al . ) '' uncertainty bands are read off from the corresponding plots in @xcite . the cross sections here are not multiplied by the squared branching ratio @xmath65.,title="fig : " ] + ( a ) & ( b ) + + cc for the case of the atlas@xmath74 cuts . these differ from the atlas cuts defined in section [ subsec : atlascuts ] by imposing a lower @xmath55 cut of @xmath91 gev instead of @xmath57 gev . furthermore , the predictions are shown for @xmath92 tev instead of 8 tev.,title="fig : " ] & for the case of the atlas@xmath74 cuts . these differ from the atlas cuts defined in section [ subsec : atlascuts ] by imposing a lower @xmath55 cut of @xmath91 gev instead of @xmath57 gev . furthermore , the predictions are shown for @xmath92 tev instead of 8 tev.,title="fig : " ] + ( a ) & ( b ) + + the rapidity separation in fig . [ fig : comp_lhcb_lansberg ] ( b ) shows , as expected , a very good agreement between the sps and dps predictions from us and @xcite because of the lo calculations and the same parametrisation of eq . . for the invariant mass distributions of fig . [ fig : comp_lhcb_lansberg ] ( c ) , our sps and dps predictions agree well with @xcite for an invariant mass up to approx . 20 gev , while there are differences for sps in the last two bins and for dps in the last bin . these differences might be related to numerical precision , as the differential cross sections become very small for a high invariant mass . [ fig : comp_atlas_lansberg ] for the atlas@xmath74 predictions at a collider energy of @xmath1 tev shows the same set of distributions as for the lhcb predictions . it should be noted that the asterisk denotes a changed atlas cut with @xmath91 gev instead of @xmath57 gev . the transverse momentum distribution of fig . [ fig : comp_atlas_lansberg ] ( a ) displays a larger @xmath66 range than for the lhcb cuts , which shows that for large values of @xmath93 gev , the parton shower and nlo@xmath74 results of sps differ by a notable amount region ; instead the nrqcd framework should be used . ] while there is again an agreement for low @xmath94 - 15 gev within the error bands . we point out that the bulk of the cross section comes from the region for @xmath95 gev , as seen e.g. in fig . [ fig : atlas_plots ] ( b ) for a similar setup at 8 tev ( on a linear axis ) , so that the differences for @xmath93 gev between the parton shower and nlo@xmath74 results affect the description of only a small portion of events . interestingly , while there is a small difference for sps in the bin with small rapidity separation , fig . [ fig : comp_atlas_lansberg ] ( b ) , the two predictions agree well within the errors for @xmath96 . the invariant mass distribution of fig . [ fig : comp_atlas_lansberg ] ( c ) shows that , while there is again a difference for the smallest bin , the predictions for sps with parton shower and nlo@xmath74 corrections almost agree within the error bands ( and they in fact do for some bins ) , although it can be seen more clearly here that the lack of hard gluon emission leads to the parton shower result always being below the nlo@xmath74 result . the dps predictions agree very well for the transverse momentum distribution , while for the rapidity separation and the invariant mass , there are slight deviations in the high-@xmath89 and @xmath90 bins , possibly related to the difference in numerics and codes used to compute these predictions . from these comparisons , we see that , as one would expect , the rapidity separation distribution is most stable with respect to higher - order corrections from hard gluon emission that are not included in our approach , while the transverse momentum distribution of the di-@xmath0 system is most strongly affected by them . we remark that here the sps and dps predictions have been computed with different input pdfs ( cteq6l1 , cteq6 m , and mstw2008 nlo , respectively ) for the purpose of comparing to @xcite , while the comparison between the magnitudes of sps and dps presented in section [ sec : kin_dis ] avoids introducing pdf effects unrelated to the sps and dps calculations . precise predictions for multi - parton interations are a vital ingredient for the high - energy collisions at the lhc , in particular during the current run at a centre - of - mass energy of @xmath16 tev and future runs at higher energies , where the probability for such subleading scattering processes to happen is significantly increased . in this work we have documented sps and dps predictions for the production of @xmath0 pairs with the updated fiducial volume cuts for the lhcb analyses of run i data , and also for a new dps study of @xmath0-pair production at @xmath97 tev by the atlas experiment . the distributions show interesting indications that dps processes could contribute significantly to certain kinematic regions of the invariant mass and rapidity separation of the di-@xmath0 system , while the transverse momentum distribution is very susceptible to higher - order corrections . the predictions for a collider energy of @xmath16 tev show a considerable increase of the dps contributions with respect to sps . finally , the comparison to the results presented in @xcite indicate a good agreement for regions where it is reasonable to compare a parton shower to a nlo@xmath74 calculation , supporting the parton shower approach as a good approximation in these regions . + * note added : * in the final preparation stages of this report , we have become aware of a new atlas study of double @xmath0 production @xcite . in this study , our dps predictions presented in section [ sec : kin_dis ] are compared with the data - driven estimates of dps . a good agreement between dps theory and data is found for all differential distributions reported in @xcite . the full @xmath0 distributions measured by atlas are then compared with the sum of our dps predictions and nlo sps predictions of @xcite with the collision energy and fiducial volume adjusted according to the experimental analysis and normalised to the fraction of dps events found using data - driven model . we have checked that applying the same normalisation procedure to our predictions leads to a rather good agreement with the measured @xmath0 distributions , apart from the large end of the spectra ( and the first , low end bins in some cases ) , in accordance with observations in @xcite and section [ sec : complansberg ] . it needs to be checked if supplementing the theoretical predictions with full nlo corrections can eliminate the need for introducing the normalisation procedure , as results of @xcite would suggest . the authors thank c.h . kom for sharing his expertise during initial stages of the event simulation . part of this work has been performed on the high performance computing cluster palma maintained by the center for information technology ( ziv ) at wwu mnster , and on the high - 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section in pp collisions at @xmath114 = 8 tev with the atlas detector _ , '' atlas - conf-2016 - 047 .

double parton scattering ( dps ) is studied at the example of @xmath0 pair - production in the lhcb and atlas experiments of the large hadron collider ( lhc ) at centre - of - mass energies of @xmath1 , 8 , and 13 tev . we report theoretical predictions delivered to the lhcb and atlas collaborations adjusted for the fiducial volumes of the corresponding measurements during run i and provide new predictions at 13 tev collision energy . it is shown that dps can lead to noticeable contributions in the distributions of longitudinal variables of the di-@xmath0 system , especially at 13 tev . the increased dps rate in double @xmath0 production at high energies will open up more possibilities for the separation of single parton scattering ( sps ) and dps contributions in future studies . ms - tp-16 - 21 * double parton scattering in pair - production of @xmath0 mesons at the lhc revisited * + christoph borschensky@xmath2 and anna kulesza@xmath3 _ @xmath4 institute for theoretical physics , university of tbingen , auf der morgenstelle 14 , d-72076 tbingen , germany + @xmath5 institute for theoretical physics , wwu mnster , d-48149 mnster , germany _

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Proceed to summarize the following text: suprathermal charged particles scattering back and forth across the surface of a shock wave gain energy . the concept of stochastic energization due to randomly moving inhomogeneities was first proposed by fermi @xcite . in that original version , the acceleration process is easily shown to be efficient only at the second order in the parameter @xmath0 , the average speed of the irregularities in the structure of the magnetic field , in units of the speed of light . for non - relativistic motion , @xmath1 , the mechanism is not very attractive . the generalization of this idea to the case of a shock wave was first proposed in @xcite and is nicely summarized in several recent reviews @xcite , where the efficiency of the process was found to be now at the first order in @xmath0 . since these pioneering papers the process of particle acceleration at shock waves has been investigated in many aspects and is now believed to be at work in a variety of astrophysical environments . in fact we do observe shocks everywhere , from the solar system to the interplanetary medium , from the supernovae environments to the formation of the large scale structure of the universe . all these are therefore sites of both heating of the medium crossing the shock surface and generation of suprathermal particles . the two phenomena are most likely different aspects of the same process , also responsible for the formation of the collisionless shock itself . one of the major developments in the theory of particle acceleration at astrophysical shock waves has consisted of removing the assumption of _ test particle _ , namely the assumption that the accelerated particles could not affect the dynamics of the shocked fluid . two approaches have been proposed to treat this intrinsically non - linear problem : the _ two fluid models _ @xcite and the _ kinetic models _ @xcite , while numerous attempts to simulate numerically the process of particle acceleration have also been made @xcite . the two fluid models treat the accelerated particles as a separate fluid , contributing a pressure and energy density which enter the usual conservation laws at the shock surface . by construction , these models do not provide information about the spectrum of the accelerated particles , while correctly describing the detailed dynamics of the fluids involved . the kinetic models on the other hand have a potential predictive power in terms of both dynamics and spectral shape of the accelerated particles . all these considerations hold in principle for all shocks but in practice most of the work has been done for the case of newtonian shock waves ( however see @xcite for an extension to relativistic shocks ) . astrophysical studies have shown that there are plenty of examples in nature of fluids moving at relativistic speeds , and generating shock waves . the generalization of the process of particle acceleration to the relativistic case represents in our opinion the second major development of the theory ( baring , these proceedings ) . in this paper , we will not present a review of all the current efforts in the investigation of shock acceleration . we will rather concentrate our attention upon some recent work in the direction of accounting for the non - linear backreaction of the accelerated particles . the original theory of particle acceleration was based on the assumption that the accelerated particles represent a _ passive _ fluid , with no dynamical backreaction on the background plasmas involved . within the context of this approximation , several independent approaches @xcite give the spectrum of the accelerated particles in the form of a power law in momentum @xmath2 , where the slope @xmath3 is related in a unique way to the mach number @xmath4 of the upstream fluid as seen in the shock frame , through the expression @xmath5 ( here we asumed that the adiabatic index of the background gas is @xmath6 ) . this result is easily shown by using the diffusion - convection equation in one dimension for a stationary situation ( namely @xmath7 ) : @xmath8 - u \frac{\partial f ( x , p)}{\partial x } + \ ] ] @xmath9 where @xmath10 is the diffusion coefficient , @xmath11 is the distribution function of accelerated particles in phase space and @xmath12 is the injection function , which we will assume to be a dirac delta function at the shock surface @xmath13 in the downstream fluid ( @xmath14 ) . the function @xmath15 is normalized in such a way that the total number of accelerated particles is given by @xmath16 . as a first step , we integrate eq . [ eq : trans ] around @xmath13 , from @xmath17 to @xmath14 , which we denote as points `` 1 '' and `` 2 '' respectively , so that we get @xmath18_2 - \left [ d \frac{\partial f}{\partial x}\right]_1 + \frac{1}{3 } p \frac{d f_0}{d p } ( u_2 - u_1 ) + q_0(p)= 0,\ ] ] where @xmath19 ( @xmath20 ) is the fluid speed immediately upstream ( downstream ) of the shock and @xmath21 is the particle distribution function at the shock location . by requiring that the distribution function downstream is independent of the spatial coordinate ( homogeneity ) , we obtain @xmath22_2=0 $ ] , so that the boundary condition at the shock can be rewritten as @xmath18_1 = \frac{1}{3 } p \frac{d f_0}{d p } ( u_2 - u_1 ) + q_0(p ) . \label{eq : boundaryshock}\ ] ] we can now perform the integration of eq . ( [ eq : trans ] ) from @xmath23 to @xmath17 ( point `` 1 '' ) , in order to take into account the boundary condition at upstream infinity . ( [ eq : boundaryshock ] ) we obtain @xmath24 the solution of this equation for @xmath21 has the form of a power law @xmath25 with slope @xmath26 , where we introduced the compression factor @xmath27 at the shock . for a strong shock @xmath28 and we find the well known asymptotic spectrum @xmath29 , or @xmath30 in terms of energy ( here again we assumed that the adiabatic index of the background gas is @xmath31 . why should we expect this simple result to be affected by the assumption of test particles ? there are three physical arguments that may serve as plausibility arguments to investigate the effects of possible backreactions : 1 ) the spectrum @xmath32 is logarithmically divergent in its energy content , so that even choosing a maximum momentum , it is possible that the energy density in the form of accelerated particles becomes comparable with the kinetic pressure , making the assumption of test particles untenable ; 2 ) if the non thermal pressure becomes appreciable , the effective adiabatic index can get closer to @xmath33 rather than @xmath34 , making the shock more compressive and the spectrum of accelerated particles even more divergent ; 3 ) more divergent spectra imply larger fluxes of escaping particles at the maximum momentum , which make the shock radiative - like , again implying a larger compression and flatter spectra . all the three issues raised here point toward the direction of making the backreaction more severe rather than alleviating its effect , therefore a run - away reaction seems likely , which drives the shock toward a strongly non - linear cosmic ray modified configuration ( here the term _ cosmic rays _ is used in a general way to indicate the accelerated particles ) . we can describe the expected effects on the basis of the following simple argument : if , as is usually the case , the diffusion coefficient increases with the momentum of the particles , we can expect that particles with larger momenta will diffuse farther from the shock surface in the upstream section of the gas . at large distances from the shock , only the high energy particles will be present , while lower energy particles will populate the regions closer to the shock surface . there is some critical distance which corresponds to the typical diffusion length of the particles with the maximum momentum achievable , @xmath35 . at this distance , the pressure of the cosmic rays is basically zero and the fluid is unperturbed . on the other hand , moving inward , toward the shock , an increasing number of accelerated particles is present , and their pressure contributes to the local pressure budget by slowing down the fluid ( in the shock frame ) . this effect causes the fluid speed upstream to be space - dependent , and decreasing while approaching the shock surface . the region of slow decrease of the fluid velocity is usually called the _ precursor_. the shock , which may now be substantially weakened by the effect of the accelerated particles , is usually called _ subshock_. it is useful to introduce the two quantities @xmath36 and @xmath37 , which are respectively the compression factor at the gas subshock and the total compression factor between upstream infinity and downstream . here @xmath38 , @xmath19 and @xmath20 are the fluid speeds at upstream infinity , upstream of the subshock and downstream respectively . the two compression factors would be equal in the test particle approximation . for a modified shock , @xmath39 can attain values much larger than @xmath40 and more in general , much larger than @xmath41 , which is the maximum value achievable for an ordinary strong non - relativistic shock . the shape of the particle spectrum is still determined by some jump in the velocity field , but this quantity is now local : at low energies , the compression felt by the particles is @xmath42 , while at @xmath43 the effective compression is @xmath44 . it follows that , since @xmath45 , the spectrum at low energies is steeper than that at higher energies : the overall spectrum at cosmic ray modified shocks is therefore expected to have a concave shape . in the following we will describe the effects of the particle backreaction following the kinetic semi - analytical approach proposed in @xcite , and we will use the most general formalism , which includes the possible presence of seed pre - accelerated particles in the environment in which the shock propagates . we repeat here the steps illustrated above for the linear case . integrating again eq . [ eq : trans ] around @xmath13 , from @xmath17 to @xmath14 , we get eq . [ eq : boundaryshock ] , after invoking the homogeneity of the particle distribution downstream . performing now the integration of eq . [ eq : trans ] from @xmath23 to @xmath17 we obtain @xmath46 @xmath47 here @xmath48 represents the distribution of seed pre - accelerated particles possibly present at upstream infinity . we can now introduce the quantity @xmath49 defined as @xmath50 whose physical meaning is instrumental to understand the nonlinear backreaction of the accelerated particles . the function @xmath49 is the average fluid velocity experienced by particles with momentum @xmath51 while diffusing upstream away from the shock surface . in other words , the effect of the average is that , instead of a constant speed @xmath19 upstream , a particle with momentum @xmath51 experiences a spatially variable speed , due to the pressure of the accelerated particles that is slowing down the fluid . since the diffusion coefficient is in general @xmath51-dependent , particles with different energies _ feel _ a different compression coefficient , higher at higher energies if , as expected , the diffusion coefficient is an increasing function of momentum . with the introduction of @xmath49 , eq . ( [ eq : stepnl ] ) becomes @xmath52 + u_0 f_\infty + q_0(p ) = 0 , \label{eq : step1}\ ] ] the solution of this equation can be written in the following implicit form : @xmath53 @xmath54}{u_{\bar p } - u_2 } \times\ ] ] @xmath55 \right\}. \label{eq : solut}\ ] ] in the case of monochromatic injection with momentum @xmath56 at the shock surface , we can write @xmath57 where @xmath58 is the gas density immediately upstream ( @xmath17 ) and @xmath59 parametrizes the fraction of the particles crossing the shock which are going to take part in the acceleration process . in terms of @xmath40 and @xmath39 , introduced above , the density immediately upstream is @xmath60 . we can introduce the dimensionless quantity @xmath61 so that @xmath62 @xmath63 @xmath64 @xmath65 the structure of the fluid upstream of the shock and the corresponding spectrum of accelerated particles is determined if the velocity field @xmath61 is known . the nonlinearity of the problem reflects in the fact that @xmath66 is in turn a function of @xmath21 as it is clear from the definition of @xmath49 . in order to solve the problem we need to write the equations for the thermodynamics of the system including the gas , the reaccelerated cosmic rays , the cosmic rays accelerated from the thermal pool and the shock itself . the velocity , density and thermodynamic properties of the fluid can be determined by the mass and momentum conservation equations , with the inclusion of the pressure of the accelerated particles and of the preexisting cosmic rays . we write these equations between a point far upstream ( @xmath23 ) , where the fluid velocity is @xmath38 and the density is @xmath67 , and the point where the fluid upstream velocity is @xmath49 ( density @xmath68 ) . the index @xmath51 denotes quantities measured at the point where the fluid velocity is @xmath49 , namely at the point @xmath69 that can be reached only by particles with momentum @xmath70 . the mass conservation implies : @xmath71 conservation of momentum reads : @xmath72 where @xmath73 and @xmath74 are the gas pressures at the points @xmath23 and @xmath75 respectively , and @xmath76 is the pressure in accelerated particles at the point @xmath69 ( we used the symbol @xmath77 to mean _ cosmic rays _ , in the sense of _ accelerated particles _ ) . the mass flow in the form of accelerated particles has reasonably been neglected . our basic assumption , similar to that used in @xcite , is that the diffusion is @xmath51-dependent and more specifically that the diffusion coefficient @xmath78 is an increasing function of @xmath51 . therefore the typical distance that a particle with momentum @xmath51 moves away from the shock is approximately @xmath79 , larger for high energy particles than for lower energy particles increases with @xmath51 faster than @xmath49 does , therefore @xmath80 is a monotonically increasing function of @xmath51 . ] . as a consequence , at each given point @xmath69 only particles with momentum larger than @xmath51 are able to affect appreciably the fluid . strictly speaking the validity of the assumption depends on how strongly the diffusion coefficient depends on the momentum @xmath51 . the cosmic ray pressure at @xmath69 is the sum of two terms : one is the pressure contributed by the adiabatic compression of the cosmic rays at upstream infinity , and the second is the pressure of the particles accelerated or reaccelerated at the shock ( @xmath81 ) and able to reach the position @xmath69 . since only particles with momentum larger than @xmath51 can reach the point @xmath75 , we can write @xmath82 @xmath83 where @xmath84 is the velocity of particles with momentum @xmath51 , @xmath35 is the maximum momentum achievable in the specific situation under investigation , and @xmath85 is the adiabatic index for the accelerated particles . in eq . [ eq : cr ] the first term represents the adiabatic compression of the pressure of the seed particles advected from upstream infinity , while the second term represents the pressure in the freshly accelerated particles at the position @xmath69 . in the following we use @xmath86 ( see @xcite for a detailed discussion of the reasons for this choice ) . the pressure of cosmic rays at upstream infinity is simply @xmath87 where @xmath88 is some minimum momentum in the spectrum of seed particles . from eq . ( [ eq : pressure ] ) we can see that there is a maximum distance , corresponding to the propagation of particles with momentum @xmath35 such that at larger distances the fluid is unaffected by the accelerated particles and @xmath89 . we will show later that for strongly modified shocks the integral in eq . ( [ eq : cr ] ) is dominated by the region @xmath43 . this improves even more the validity of our approximation @xmath90 . this also suggests that different choices for the diffusion coefficient @xmath78 may affect the value of @xmath35 , but at fixed @xmath35 the spectra of the accelerated particles should not change in a significant way . assuming an adiabatic compression of the gas in the upstream region , we can write @xmath91 where we used mass conservation , eq . ( [ eq : mass ] ) . the gas pressure far upstream is @xmath92 , where @xmath93 is the ratio of specific heats for the gas ( @xmath6 for an ideal gas ) and @xmath94 is the mach number of the fluid far upstream . we introduce now a parameter @xmath95 that quantifies the relative weight of the cosmic ray pressure at upstream infinity compared with the pressure of the gas at the same location , @xmath96 . using this parameter and the definition of the function @xmath66 , the equation for momentum conservation becomes @xmath97 + \ ] ] @xmath98 using the definition of @xmath99 and multiplying by @xmath51 , this equation becomes @xmath100 = \ ] ] @xmath101 where @xmath21 depends on @xmath66 as written in eq . ( [ eq : laeffe ] ) . ( [ eq : eqtosolve ] ) is therefore an integral - differential nonlinear equation for @xmath66 . the solution of this equation also provides the spectrum of the accelerated particles . the last missing piece is the connection between @xmath40 and @xmath39 , the two compression factors appearing in eq . ( [ eq : solut ] ) . the compression factor at the gas shock around @xmath13 can be written in terms of the mach number @xmath102 of the gas immediately upstream through the well known expression @xmath103 on the other hand , if the upstream gas evolution is adiabatic , then the mach number at @xmath17 can be written in terms of the mach number of the fluid at upstream infinity @xmath94 as @xmath104 so that from the expression for @xmath40 we obtain @xmath105^{\frac{1}{\gamma_g+1}}. \label{eq : rsub_rtot}\ ] ] now that an expression between @xmath40 and @xmath39 has been found , eq . ( [ eq : eqtosolve ] ) basically is an equation for @xmath40 , with the boundary condition that @xmath106 . finding the value of @xmath40 ( and the corresponding value for @xmath39 ) such that @xmath106 also provides the whole function @xmath66 and , through eq . ( [ eq : solut ] ) , the distribution function @xmath107 for the particles resulting from acceleration and reacceleration in the nonlinear regime . when the backreaction of the accelerated particles is small , the _ test particle _ solution is recovered . in fig . 1 we show an example of the spectrum calculated for parameters which are typical of a supernova remnant ( solid line ) , as compared with the spectrum estimated according with the simple model of @xcite ( broken line ) and the result of a numerical simulation ( dashed line ) , also reported in@xcite . in this calculation no seed particles have been assumed to be present in the shock environment . the good agreement between the semi - analyical approach discussed here and the montecarlo simulations proves that the semi - analytical approach discussed here is quite effective in describing the behaviour of cosmic ray modified shock waves as particle accelerators . however the situation is in general more complex than this : previous approaches to the problem of cosmic ray modified shocks had already shown the appearance of multiple solutions . this was first discussed in @xcite in the context of two - fluid models and in @xcite by using a kinetic approach . multiple solutions are found with the method proposed here as well . in @xcite it was pointed out how the multiple solutions appear also in the case of reacceleration of seed particles . an example of the phenomenon is illustrated for the case of no seed particles in fig . 2 , where we plot @xmath108 , bound to be unity for the physical solutions , as a function of the total compression factor @xmath39 . here @xmath56 , @xmath35 and the shock mach number are all fixed . the solutions are identified by the points of intersections of the curves ( obtained for different values of @xmath59 , as indicated ) with the horizontal line at @xmath106 . one can see that for low values of @xmath59 ( approximately unmodified shock ) there is only one intersection at @xmath109 . however , when @xmath59 is increased the intersections may become three . all the three solutions are fully acceptable from the point of view of conservation laws . for large values of @xmath59 the shock is always strongly modified ( @xmath110 ) . for these cases , the asymptotic shape of the spectrum at large momenta is well described by the power law @xmath111 ( or @xmath112 ) . the comparison between the method described above and that of @xcite has been discussed in @xcite . in fig . 3 , extracted from @xcite , we illustrate the spectra and @xmath66 for a case in which three solutions appear ( in both approaches ) . the case corresponds to mach number @xmath113 , gas temperature at upstream infinity @xmath114 , injection momentum @xmath115 and maximum momentum @xmath116 . in the calculations of @xcite a specific form for the diffusion coefficient as a function of momentum is required . for reference we adopted a bohm diffusion coefficient @xmath117 . in fig . 3 , each panel corresponds to one solution . we plot in each panel the spectrum @xmath118 multiplied by @xmath119 ( the linear theory would predict @xmath120 ) . the solid lines show the spectra as calculated with the approach of @xcite , while the dashed lines are the corresponding spectra as obtained using the calculations of @xcite with bohm diffusion . the agreement between the two methods is excellent , despite the fact that the approach presented here does not require the detailed knowledge of the diffusion coefficient . the question arises of whether the appearance of multiple solutions is an artifact of our ignorance of the parameter @xmath59 , which defines the efficiency of the shock in injecting particles from the thermal pool . although this is probably not the all story , as confirmed by the fact that multiple solutions are present even in the case of reacceleration of pre - accelerated particles ( in that case @xmath59 is no longer a free parameter ) @xcite , it is likely that injection plays a crucial role . in order to show this , we adopt a simple physical recipe for the injection of particles at the shock . _ real _ shock fronts are not one - dimensional sheets but rather complex surfaces with a typical thickness that for collisionless shocks is expected to be of the order of the larmor radius of the thermal protons downstream of the shock . one should keep in mind that the temperature of the downstream gas is also affected by the non - linear modification induced by the accelerated particles , therefore the shock is expected to be thinner when the subshock is weaker . our recipe is the following : we assume that the particles which are injected at the shock are those with momentum @xmath121 , where we choose @xmath122 and @xmath123 is the momentum of the thermal particles in the downstream plasma ( we assume that the gas distributions are maxwellian ) , determined as an output of the non - linear calculations from the rankine- hugoniot relations at the subshock . this approximation is sometimes called _ thermal leakage _ @xcite . in physical terms , this makes @xmath59 an output of the calculations rather than a free parameter to be decided _ a priori_. in fig . 4 we plot @xmath108 calculated as described above , in the case in which @xmath59 is evaluated self - consistently from the prescription of thermal leakage . the different curves are obtained for @xmath124 ( from top to bottom ) for a fixed mach number @xmath125 . one can see that only single intersections with the horizontal line @xmath106 are present , namely the multiple solutions disappear if the shock is allowed to determine its own level of efficiency in particle acceleration . this calculation was repeated for different values of the parameters , but the conclusion was confirmed for all cases of physical interest @xcite . one can also see that large values of @xmath35 typically correspond to more modified shocks , and that the compression factor can reach large values , far from the test particle prediction . we discussed some aspects of particle acceleration in astrophysical collisionless shock waves , and showed that even when the fraction of particles that participate in the acceleration process is relatively small ( one in @xmath126 of the particles crossing the shock surface ) a large fraction of the incoming energy can be channelled into few non - thermal particles . this result , found previously by using several different approaches , is of the greatest importance for the physics of cosmic rays . not only the accelerated particles can keep a substantial fraction of the energy available at the shock , but the spectrum of the accelerated particles may substantially differ from a power law , showing a concavity which appears to be the clearest evidence for the appearance of cosmic ray modified shocks . despite the passive role that electrons are likely to play in the shock dynamics , the spectrum of accelerated electrons is expected to be determined by the ( cosmic ray modified ) velocity profile determined by the accelerated hadrons in the shock vicinity . a concavity in the spectrum of the radiation generated by relativistic electrons appears to be one of the possible evidences for shock acceleration in the non - linear regime . in the case of supernova remnants , there are hints that this concavity might have been observed @xcite . one of the aspects of particle acceleration that are more poorly understood is the injection of particles from the thermal pool of particles crossing the shock . this ignorance reflects in the difficulty of determining the fraction of particles that takes part in the acceleration process , and we argued that this might be the reason ( or one of the reasons ) why calculations of the non - linear shock structure may show the appearance of multiple solutions . on the other hand , assuming a simple recipe for the injection process ( thermal leakage ) is shown to result in the existence of only one solution . in other words , if the heating and acceleration processes are interpreted as two aspects of the same physical phenomenon , there seem to be no ambiguities in the way the shock is expected to behave . in this case , there is no doubt that strongly modified shocks are predicted . the efficient particle acceleration at strong shocks is also expected to result in the reduced heating of the downstream plasma , as compared with the heating achieved in the absence of accelerated particles . this effect should be visible in those cases in which it is possible to measure the temperatures of the upstream and downstream fluids separately , for instance through the x - ray emission of the thermal gases . when the shock is strongly modified by the accelerated particles , a large fraction of gas heating is due to adiabatic compression in the shock precursor , rather than to shock heating at the gasous subshock . in @xcite it was pointed out that if the shock propagates in a medium which is populated by seed pre - accelerated particles , the non - linear modification of the shock can be dominated by such seeds rather than by the acceleration of fresh particles from the thermal pool . this might be the case for shocks associated with supernova remnants , which move in the interstellar medium where the cosmic rays are known to be in rough pressure balance with the gas . the spectra of re - accelerated particles for modified shocks were calculated in @xcite and showed the usual concavity that is typical of cosmic ray modified shocks . there is an additional aspect of particle acceleration at shock waves that has not been discussed so far , namely the generation of a turbulent magnetic field in the upstream section , due to the streaming instability induced by the accelerated particles . the fact that the pressure in the form of accelerated particles may reach an appreciable fraction of the kinetic pressure at upstream infinity , @xmath127 , suggests that the magnetic field can also be amplified to a turbulent value which may widely exceed the background magnetic field , and approach the equipartition level . in @xcite the process of amplification has been studied numerically , and this naive expectation has been confirmed . one should however notice that the non - linear effects in particle acceleration , discussed in this paper , and in particular the spectral modification , are not included self - consistently in the calculations of the field amplification in the shock vicinity . all these issues are relevant for the investigations of the origin of ultra - high energy cosmic rays in many ways : 1 ) strongly modified shocks can be very efficient accelerators , so that the energy requirements for the sources we know might be substantially relaxed ; 2 ) the spectra of particles accelerated at strongly modified shocks are flatter than those expected in the linear theory . flat spectra generate a gzk feature which is milder than that due to steep spectra , therefore it may be a less severe problem to explain possible excesses of events at the highest energies ; 3 ) magnetic field amplification in the shock vicinity has been invoked in the case of snr s as a possible way to accelerate particles up to the ankle in these sources @xcite . for other classes of sources this may imply that it is easier to reach ultra - high energies in cases that are currently believed to have too low magnetic fields . this last point deserved deeper investigation .

the acceleration of charged particles at astrophysical collisionless shock waves is one of the best studied processes for the energization of particles to ultrarelativistic energies , required by multifrequency observations in a variety of astrophysical situations . in this paper we discuss some work aimed at describing one of the main progresses made in the theory of shock acceleration , namely the introduction of the non - linear backreaction of the accelerated particles onto the shocked fluid . the implications for the investigation of the origin of ultra high energy cosmic rays will be discussed .

You are an expert at summarizing long articles. Proceed to summarize the following text: the pauli exclusion principle ( pep ) is a consequence of the spin - statistics connection @xcite and plays a fundamental role in our understanding of many physical and chemical phenomena , from the periodic table of elements , to the electric conductivity in metals , to the degeneracy pressure , which makes white dwarfs and neutron stars stable , just to cite few ones . although the principle has been spectacularly confirmed by the number and accuracy of its predictions , its foundation lies deep in the structure of quantum field theory and has defied all attempts to produce a simple proof , as nicely stressed by r. feynman @xcite . given its basic standing in quantum theory , it seems appropriate to carry out precise tests of the pep validity and , indeed , in the last fifty years , several experiments have been performed to search for possible small violations @xcite . often , these experiments were born as by - products of experiments with a different objective ( e.g. dark matter searches , proton decay , etc .. ) , and most of the recent limits on the validity of pep have been obtained for nuclei or nucleons . concerning the violation of pep for electrons , greenberg and mohapatra @xcite examined all experimental data which could be related , directly or indirectly , to pep , up to 1987 . in their analysis they concluded that the probability that a new electron added to an antisymmetric collection of n electrons might form a mixed symmetry state rather than a totally antisymmetric state is @xmath2 . in 1988 , ramberg and snow @xcite drastically improved this limit with a dedicated experiment , searching for anomalous x - ray transitions , that would point to a small violation of pep in a copper conductor . the result of the experiment was a probability @xmath3 that a new electron circulating in the conductor would form a mixed symmetry state with the already present copper electrons . we have set up an improved version of the ramberg and snow experiment , with a higher sensitivity apparatus @xcite . our final aim is to lower the pep violation limit for electrons by at least 4 orders of magnitude , by using high resolution charge - coupled devices ( ccd ) as soft x - rays detectors @xcite , and decreasing the effect of background by a careful choice of the materials and sheltering the apparatus in an underground laboratory . in the next sections we describe the experimental setup , the outcome of a preliminary measurement performed in the frascati national laboratories ( lnf ) of infn in 2005 , along with a brief discussion on the results and the foreseen future improvements in the gran sasso national laboratory ( lngs ) of infn . the idea of the vip ( violation of the pauli exclusion principle ) experiment was originated by the availability of the dear ( da@xmath4ne exotic atom research ) setup , after it had successfully completed its program at the da@xmath4ne collider at lnf - infn @xcite . dear used charge - coupled devices ( ccd ) as detectors in order to measure exotic atoms ( kaonic nitrogen and kaonic hydrogen ) x - ray transitions . ccd s are almost ideal detectors for x - rays measurement , due to their excellent background rejection capability , based on pattern recognition , and to their good energy resolution ( 320 ev fwhm at 8 kev in the present measurement ) . the experimental method , originally described in @xcite , consists in the introduction of new electrons into a copper strip , by circulating a current , and in the search for x rays resulting from the @xmath5 anomalous radiative transition that occurs if one of the new electrons is captured by a copper atom and cascades down to the 1s state already filled by two electrons of opposite spin . the energy of this transition would differ from the normal k@xmath6 transition by about 300 ev ( 7.729 kev instead of 8.040 kev ) @xcite , providing an unambiguous signal of the pep violation . the measurement alternates periods without current in the copper strip , in order to evaluate the x - ray background in conditions where no pep violating transitions are expected to occur , with periods in which current flows in the conductor , thus providing `` fresh '' electrons , which might possibly violate pep . the fact that no pep violating transitions are expected to be present in the measurement without current is related to the consideration that any initial conduction electron in the copper that was in a mixed symmetry state with respect to the other copper electrons , would have already cascaded down to the @xmath7 state and would therefore be irrelevant for the present experiment . the rather straightforward analysis consists in the evaluation of the statistical significance of the normalized subtraction of the two spectra , with and without current , in the energy region where the pep violating transition is expected . the vip setup consists of a copper cylinder , 4.5 cm in radius , 50 @xmath8 m thick , 8.8 cm high , surrounded by 16 equally spaced ccd s @xcite . the ccd s are at a distance of 2.3 cm from the copper cylinder , grouped in units of two chips vertically positioned . the setup is shown in fig . the chamber is kept at high vacuum to minimize x - ray absorption and to avoid condensation on the cold surfaces . the copper target ( the copper strip where the current flows and new electrons are injected from the power supply ) is at the bottom of the setup . the ccd s surround the target and are supported by cooling fingers that start from the cooling heads in the upper part of the chamber . the ccd readout electronics is just behind the cooling fingers ; the signals are sent to amplifiers on the top of the chamber . the amplified signals are read out by adc boards in a data acquisition computer . more details on the ccd-55 performance , as well as on the analysis method used to reject background , can be found in @xcite the measurements reported in this paper have been performed in the period 21 november - 13 december 2005 , at the frascati national laboratories of infn , italy . two types of measurements were performed : * 14510 minutes ( about 10 days ) of measurements with a 40 a current circulating in the copper target ; * 14510 minutes of measurements without circulating current , where ccd s were read - out every 10 minutes . the two resulting x - ray spectra are shown in figure 2 a ) , with circulating current , and b ) , without current . the spectra refer to 14 ccd s ( out of 16 ) , due to noise problems in the remaining 2 . in order to obtain the number of x - rays due to the possible pep violating transitions , the spectrum without current was subtracted from the one with current . the resulting subtracted spectrum is shown in figure 3 a ) ( whole energy scale ) and b ) ( a zoom on the region of interest ) . the region of interest , from 7.564 to 7.894 kev , is defined by the ccd energy resolution ( 320 ev fwhm ) at the @xmath9 copper transition ( 8.04 kev ) , with an additional uncertainty of 10 ev , to account for the theoretical uncertainty in the calculation of the pep violating transition energy . the numbers of x rays in the region of interest were : * at i=40 a : n@xmath10 ; * for i=0 a : n@xmath11 ; * for the subtracted spectrum : @xmath12 . [ p18kev.eps ] for the parametrization of the results in a pauli principle violating theory , we use the notation of ignatiev and kuzmin @xcite , which has been incorporated in the paper of greenberg and mohapatra @xcite : even though the model of ignatiev and kuzmin has been later shown to be incompatible with quantum field theory @xcite , the parameter @xmath13 that measures the degree of pep violation has stuck and is still found in the literature , also because it is easy to show that it is related to the parameter @xmath14 of quon theory , by the relation : @xmath15 @xcite ( in quon theory , @xmath16 , where @xmath17 corresponds to fermions and @xmath18 corresponds to bosons , so that here @xmath14 must be close to -1 and @xmath19 must be very small , because we are dealing with electrons ) . moreover , we used this parametrization for an easy comparison of our results with the previous ramberg and snow ones @xcite , since the same has been used in that paper . in @xcite a pair of electrons in a mixed symmetry state has the probability @xmath20 for the symmetric component and @xmath21 for the usual antysymmetric one . the parameter @xmath22 is related , then , to the probability that an electron violates pep ( see also @xcite for further details ) . to determine the experimental limit on @xmath22 from our data , we used the same arguments of ramberg and snow , to compare the results . the number of electrons that pass through the conductor , which are new for this conductor , is : @xmath23 where @xmath24 is the electron electric charge , @xmath25 is the current intensity and @xmath26 represents the time duration of the measurement with current on . each new electron will undergo a large number of scattering processes on the atoms of the copper lattice . the minimum number of these internal scattering processes per electron , defined as @xmath27 , is of order @xmath28 , where @xmath29 is the length of the copper electrode ( 8.8 cm in our case ) and @xmath8 is the mean free path of electrons in copper . the latter parameter is obtained from the resistivity of the metal . we assume that the capture probability ( aside from the factor @xmath30/2 ) is greater than @xmath31 of the scattering probability . the acceptance of the 14 ccd detectors and the probability that an x ray of about 7.6 kev , the energy of the possible anomalous transition generated in the copper target , is not absorbed inside the copper itself , were evaluated by a monte carlo simulation of the vip setup , based on geant 3.21 . this probability turns out to be 2.1% . moreover , a ccd efficiency equal to 48% for a 7.6 kev x ray was considered . all these factors built up the so called @xmath32 ( @xmath33% ) . the number of x rays generated in the pep violating transition , @xmath34 , is then related to the @xmath20 parameter by : @xmath35 then , for @xmath36 c , @xmath37 cm , @xmath38 cm , @xmath39 c , we get @xmath40 the difference of events between the measurements with and without current , reported in the previous section , is @xmath41 . taking as a limit of observation three standard deviations , we get for the pep violating parameter : @xmath42 we can interpret this as a limit on the probability of pep violating interactions between external electrons and copper atoms : @xmath1 . we have thus improved the limit obtained by ramberg and snow by a factor about 40 . the paper reports a new measurement of the pep violation limit for electrons , performed by the vip collaboration at lnf - infn . the search of a tiny violation was based on a measurement of pep violating x - ray transitions in copper , under a circulating 40 a current . a new limit for the pep violation for electrons was found : @xmath1 , lowering by almost two orders of magnitude the previous one @xcite . we shall soon repeat the measurement in the gran sasso infn underground laboratory , at higher integrated currents . from preliminary tests , it appears that the x - ray background in the lngs environment is a factor 10 - 100 lower than in the frascati laboratories . a vip measurement of two years ( one with current , one without current ) at lngs , to start in spring 2006 , will then bring the limit on pep violation for electrons into the 10@xmath43 - 10@xmath44 region , which is of particular interest @xcite for all those theories related to possible pep violation coming from new physics . w. pauli , phys . * 58 * ( 1940 ) 716 . r. p. feynman , r. b. leighton , and m. sands : `` the feynman lectures on physics '' , vol . 3 , ( addison - wesley , reading , ma , 1963 ) . r. bernabei _ et al . _ , phys . lett . * b408 * ( 1997 ) 439 . h. o. back _ et al . _ ( borexino collaboration ) , eur . j. * c37 * ( 2004 ) 421 . r. c. hilborn and c. l. yuca , phys . * 76 * ( 1996 ) 2844 . nemo collaboration , nucl . b87 * ( proc . suppl . ) ( 2000 ) 510 . et al . _ , j. phys . g : nucl . part . * 17 * ( 1991 ) s355 . m. tsipenyuk , a. s. barabash , v. n. kornoukhov , and b. a. chapyzhnikov , radiat . phys . chem . * 51 * ( 1998 ) 507 . greenberg and r.n . mohapatra , phys . * 59 * ( 1987 ) 2507 . e. ramberg and g.a . snow , phys . b238 * ( 1990 ) 438 . the vip proposal , lnf - lngs proposal , september 2004 ( http://www.lnf.infn.it/esperimenti/vip ) . see e.g. j.l . culhane , nucl . instr . and meth . * a310 * ( 1991 ) 1 ; j .- p . egger , d. chatellard , e. jeannet , particle world * 3 * ( 1993 ) 139 ; g. fiorucci _ et al . instr . and meth . * a292 * ( 1990 ) 141 ; d. varidel _ et al . _ , instr . and meth . * a292 * ( 1990 ) 147 ; r.p . et al . _ , instr . and meth . * a372 * ( 1995 ) 372 . t. ishiwatari _ _ , phys . lett . * b593 * ( 2004 ) 48 ; g. beer _ et al . * 94 * ( 2005 ) 212302 . s. di matteo and l. sperandio , vip note , ir-04 , april 26 , 2006 ( the energy shift has been computed by p. indelicato - private communication ) . ccd-55 from eev ( english electric valve ) , waterhouse lane , chelmsford , essex , cm1 2qu , uk . t. ishiwatari _ et al . _ , instrum . and meth . research * a556 * ( 2006 ) 509 . ignatiev and v. a. kuzmin , yad . fiz . * 46 * ( 1987 ) 786 ; ictp preprint ic/87/13 ( 1987 ) ; a. yu ignatiev , arxiv : hep - ph/0509258 . a. b. govorkov , phys . lett * a137 * ( 1989 ) 7 . greenberg , phys . * d43 * ( 1991 ) 4111 . okun , comments nucl . ( 1989 ) 998 . i. duck and e.c.g . sudarshan , am . j. of physics * 66 * ( 1998 ) 284 .

the pauli exclusion principle ( pep ) is one of the basic principles of modern physics and , even if there are no compelling reasons to doubt its validity , it is still debated today because an intuitive , elementary explanation is still missing , and because of its unique stand among the basic symmetries of physics . the present paper reports a new limit on the probability that pep is violated by electrons , in a search for a shifted k@xmath0 line in copper : the presence of this line in the soft x - ray copper fluorescence would signal a transition to a ground state already occupied by 2 electrons . the obtained value , @xmath1 , improves the existing limit by almost two orders of magnitude . , , , , , , , , , , , , , , , , , , , , symmetrization principle , identical particles , tests of quantum field theories , anomalous atomic transitions , x rays , ccd 11.30.-j ; 03.65.-w ; 29.30.kv ; 32.30.rj

You are an expert at summarizing long articles. Proceed to summarize the following text: silicates are an important component of interstellar and circumstellar dust . many galactic sources show broad emission or absorption features at 10 and 18 @xmath0 m due to the si o stretching and o si o bending vibrations of silicate bonds . because of the width of these features , these silicates must have an amorphous structure . these broad amorphous silicate features have also been observed in a variety of extragalactic environments , including seyfert galaxies ( e.g. * ? ? ? * ; * ? ? ? * ) , quasars @xcite , luminous and ultra - luminous infrared galaxies @xcite and a sample of mid - infrared detected , optically invisible , high - luminosity galaxies with redshifts of 1.7 @xmath3 z @xmath3 2.8 @xcite . in recent years , observational evidence for the presence of _ crystalline _ silicates in various astrophysical environments has also emerged . in particular , infrared spectra have revealed that silicates in circumstellar environments often contain a significant crystalline fraction around both pre - main - sequence stars ( t - tauri stars and herbig aebe stars ) and post - main - sequence stars ( asymptotic giant branch ( agb ) stars , post - agb stars , planetary nebulae , luminous blue variables ( lbvs ) , red super giants ( rsgs ) and post - rsgs ) ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? ? * ; * ? ? ? these crystalline silicates are invariably magnesium rich ( e.g. , pyroxene ( mgsio@xmath4 ) and forsterite ( mg@xmath1sio@xmath2 ) . crystalline silicates are also known to be ubiquitous in the solar system , including primitive objects such as comets ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? because crystallization is inhibited by high - energy barriers , the origin and evolution of the crystalline silicate fraction in interstellar and circumstellar media has the potential to provide direct evidence of energetic processing of grains . silicates play an especially large role in shaping the mid - infrared spectral appearance of ultra - luminous infrared galaxies ( ulirgs ; l@xmath510@xmath6l@xmath7 ) . ulirgs are thought to represent the final stage in the merging process of gas - rich spirals , where the interaction has driven gas and dust towards the remnant nucleus , fueling a massive starburst and a nascent active galactic nucleus ( agn ) . here , we present evidence for the presence of _ crystalline _ silicates as part of the deep silicate absorption features observed towards a sample of twelve heavily obscured ulirgs . we are obtaining mid - infrared spectroscopy for a sample of 110 ulirgs , as part of the guaranteed time observation ( gto ) ulirg program of the infrared spectrograph ( irs)@xcite on the spitzer space telescope @xcite . seventy seven of these spectra have been analyzed so far . the twelve ulirg spectra presented in this paper were selected based upon the strength of spectral structure indicative of crystalline silicate absorption bands at 16 and 23@xmath0 m . table [ tab1 ] lists the basic properties of these targets , along with their observation dates and on - source integration times . the observations were made with the short - low ( sl ) and long - low ( ll ) modules of the _ irs_. the spectra were extracted from the flatfielded images provided by the spitzer science center ( pipeline version s11.0.2 ) . the images were background - subtracted by differencing the two sl apertures and for ll , by differencing the two nod positions . spectra were then extracted and calibrated using the _ irs _ standard star hr6348 for sl and the stars hr6348 , hd166780 , and hd173511 for ll @xcite . small wavelength corrections were made to compensate for known offsets in the s11 processed data . after extraction the orders were stitched to ll order 1 , requiring order - to - order scaling adjustments of typically 510% . the largest adjustment was made for arp220 to match sl order 1 and ll order 2 , which required sl1 to be scaled up by 21% . in the final step , the 537@xmath0 m spectra were scaled to match the observed _ irs _ blue or red peak - up flux . for those sources lacking a ( useful ) peak - up flux , the spectra were scaled down by 10% , the average scaling factor of the other spectra . in some spectra , most notably in iras205514250 , residual fringing , which is due to subpixel pointing errors , appears in ll order 1 ( between 20 and 30@xmath0 m ) . lccccc target & aor key & date observed & int . time & redshift & d@xmath8 + & & & min . & & mpc + 001837111 & 7556352 & 14 nov . 2003 & 15 & 0.327 & 1700 + 003971312 & 4963584 & 04 jan . 2004 & 14 & 0.262 & 1320 + 011992307 & 4964864 & 18 jul . 2004 & 14 & 0.156 & 737 + 063017934 & 4970240 & 11 aug . 2004 & 12 & 0.156 & 700 + 063616217 & 4970496 & 11 aug . 2004 & 14 & 0.160 & 760 + 08572 + 3915 & 4972032 & 15 apr . 2004 & 6 & 0.0584 & 258 + 15250 + 3609 & 4983040 & 4 mar . 2004 & 7 & 0.0554 & 244 + arp220 & 4983808 & 29 feb . 2004 & 5 & 0.0181 & 78 + 17068 + 4027 & 4986112 & 16 apr . 2004 & 14 & 0.179 & 858 + 18443 + 7433 & 4987904 & 5 mar . 2004 & 14 & 0.135 & 627 + 205514250 & 4990208 & 14 may 2004 & 4 & 0.0427 & 186 + 23129 + 2548 & 4991488 & 17 dec . 2003 & 23 & 0.179 & 858 + the top panel of figure[fig1 ] shows the average _ irs _ low - resolution spectrum for our sample , obtained after scaling the spectra to s@xmath9=1jy at 15@xmath0 m . the average spectrum is dominated by broad absorption features at 10 and 18@xmath0 m , which we attribute to absorption by amorphous silicates . the spectrum further shows weaker absorption features due to water ice ( 6.0@xmath0 m ) and hydrocarbons ( 6.90 and 7.25@xmath0 m ) . pah emission features can be seen at low contrast at 6.2 , 7.7 , and 11.2@xmath0 m . the features are especially weak in the spectrum of irasf001837111 @xcite and absent in the spectrum of iras08572 + 3915 ( figure[fig1 ] , lower panel ) . our spectra further show a few emission lines , most notably the h@xmath1s(3 ) line at 9.66@xmath0 m , the [ neii ] line at 12.81@xmath0 m and the h@xmath1s(1 ) line at 17.0@xmath0 m . these will be discussed elsewhere @xcite . upon close inspection , the spectra also reveal evidence for weak and narrow absorption features near 16 , 19 and 23 @xmath0 m , independent of the redshift of the source . emission and absorption features at these wavelengths are also know in galactic sources with strong silicate features and have been attributed to the presence of crystalline silicates . in order to investigate the presence of a crystalline component to the silicate absorption features in our sample , we infer the silicate optical depth spectrum by dividing out a local continuum from our _ irs _ spectra . we define the local continuum as a three point spline interpolation of continuum points at 5.6@xmath0 m ( 0.1@xmath0 m shortward of the wavelength range affected by the 6.0@xmath0 m water ice absorption feature ) , 7.1@xmath0 m ( in between the two hydrocarbon absorption bands at 6.90 and 7.25@xmath0 m ) , and the red cut - off of irs ll order 1 ( ranging from 29 to 36@xmath0 m , depending on the redshift of the source ) . for sources without discernable 7.7@xmath0 m pah emission ( e.g. iras08572 + 3915 and irasf001837111 ) , we replace the 7.1@xmath0 m pivot by the continuum at 7.98.0@xmath0 m . the resulting spline interpolated local continuum is illustrated for iras08572 + 3915 in the lower panel of figure[fig1 ] . the effect of small changes in the adopted local continuum on the resulting silicate profile is illustrated by the shaded areas around the spline - interpolated continuum in the lower panel of figure[fig1 ] and on the resulting silicate profile in the third panel of figure[fig2 ] . the resulting optical depth profiles for the twelve sources in our sample have apparent 10@xmath0 m optical depths ranging from 2.1 for iras17068 + 4027 to 4.2 for iras08572 + 3915 , with a mean value of 2.5 ( see figure[fig3 ] and table[tab2 ] ) . in order to derive the characteristics of these narrow features we have to account for the amorphous silicate features which dominate the absorption structure in this wavelength range . we have compared the general profile of the 10 and 18@xmath0 m amorphous silicate featues in these ulirgs with those in a sample of galactic background sources @xcite . however , because of the uncertainties in the placement of the continuum for these galactic sources , the derived interstellar 10 and @xmath10 m amorphous silicate features show differences in their relative strengths , and in the level of absorption between them . since fitting the ulirg data with a composite galactic spectrum would therefore introduce spurious artifacts , we have elected to represent the broad amorphous silicate features in the ulirg and galactic spectra with spline fits . deviations from these fits in the ulirg spectra , significantly larger than those seen among the galactic sources , would then argue strongly for a crystalline component . we have tested this procedure on the high signal - to - noise spectrum of the galactic center source gcs3 , and on laboratory absorption spectra of amorphous silicates . the residuals from the spline fits to these spectra are very small ( @xmath36% ) , validating our analysis to that level . the method is demonstrated in the top panel of figure[fig2 ] for laboratory spectra of amorphous olivines and pyroxenes @xcite . the spline fit traces the amorphous profile to better than 2% of the local optical depth , except for the 16@xmath0 m range , where the spline deviates by up to 6% . the resulting small artifacts are shown at the top of the panel . we also tested our method on the optical depth profile as seen towards the galactic background source gcs3 @xcite , depicted in the second panel of figure[fig2 ] . the silicates along this line of sight are thought to be @xmath1199% amorphous in composition @xcite . in contrast to the laboratory profiles , the fit residuals for the gcs3 spectrum are dominated by spectral noise rather than fitting artifacts . the third panel of figure[fig2 ] shows the fit to the silicate profile of the ulirg iras08572 + 3915 . the red curve smoothly fits the maximum depth of the broad 18@xmath0 m feature , while ignoring the strong , narrow substructure around 11 , 16 , 19 and 23@xmath0 m . after subtracting the spline - interpolated amorphous component from the optical depth spectrum , the features at 11 , 16 , 19 and 23@xmath0 m show up in the fit residual spectrum , clearly above the levels expected for artifacts introduced by the fitting procedure . in the lower panel of figure[fig2 ] we compare these residuals to known spectra of circumstellar and laboratory crystalline silicates . the adopted spline fits to the amorphous silicate profiles for the twelve ulirgs in our sample are overplotted in red in figure[fig3 ] , while the residual optical depth spectra are presented in figure[fig4 ] . the individual spectra in the latter figure are truncated at -0.1 optical depth in order not to be dominated by spurious optical depth structure introduced by emission features of 11.3@xmath0 m pah and 12.8@xmath0 m [ ne ii ] . these common features aside , the twelve spectra clearly show absorption structure at 16 , 19 and 23@xmath0 m that can not be attributed to observational artifacts ( i.e. fringing , flux calibration ) or imperfections in the spline fit to the amorphous component . a few sources ( e.g. iras063017934 , iras23129 + 2548 ) further show a weak feature at 2728@xmath0 m and only one source , iras08572 + 3915 , a strong feature at 11@xmath0 m ( see also figure[fig2 ] ) . the latter feature may also be present in the other spectra , but its detection is complicated by the presence of emission from 10.51@xmath0 m [ s iv ] and 11.3@xmath0 m pah in the same wavelength range ( see figure[fig4 ] ) . further note the presence of absorption bands of gas phase c@xmath1h@xmath1 and hcn at 13.7 and 14.05@xmath0 m in the spectra of several of our ulirgs , most notably iras15250 + 3609 and iras23129 + 2548 . these bands are otherwise seen in dense , warm molecular clouds surrounding massive galactic protostars @xcite . in figure[fig5 ] we show the average residual optical depth spectrum for our twelve sources and compare it to the observed crystalline silicate opacity profile of the young star is tau @xcite and the calculated opacity profile of forsterite @xcite . the relative strengths of the absorption bands in the ulirg spectra are different from those of the comparison profiles . in the ulirg spectra , the features get progressively weaker towards longer wavelength , whereas no such trend exists for the comparison profiles ( see also figures [ fig2 ] and [ fig5 ] ) . a possible explanation for this trend could be dilution of the crystalline absorption spectrum by cold dust emission from a less obscured cooler component , either the other nucleus or the circumnuclear environment . nevertheless , the similarities between the ulirg profile and the comparison profiles in peak position and width are striking . we therefore conclude that the features observed at 11 , 16 , 19 , 23 and 28@xmath0 m in the ulirg spectra are indeed caused by the presence of crystalline silicates in their nuclear medium . the peak position of the 16@xmath0 m band , in particular , is sensitive to the mg / fe ratio of olivines @xcite . the peak position of this band observed in the ulirg spectra ( 16.1@xmath0 m ) falls close to the laboratory measured position of forsterite , the mg - rich end member of the olivines ( mg@xmath1sio@xmath2 ; 16.3@xmath0 m ; * ? ? ? * ) and much to the blue of the peak position in fayalite , the iron - rich end member of the olivines ( fe@xmath1sio@xmath2 ; 17.7@xmath0 m ; * ? ? ? * ) . hence , in line with other observations of crystalline silicates in space , extragalactic olivines appear to be extremely mg - rich and fe - poor . lcccc target & @xmath12(peak ) & @xmath13(peak ) & @xmath13(peak)/@xmath12(peak ) & n@xmath14/n@xmath15 + 001837111 & 3.0 & 0.22 & 0.073 & 0.11 + 003971312 & 3.2 & 0.28 & 0.088 & 0.13 + 011992307 & 3.6 & 0.21 & 0.060 & 0.09 + 063017934 & 3.4 & 0.35 & 0.10 & 0.15 + 063616217 & 2.9 & 0.15 & 0.051 & 0.08 + 08572 + 3915 & 4.2 & 0.22 & 0.053 & 0.08 + 15250 + 3609 & 3.6 & 0.20 & 0.056 & 0.08 + arp220 & 3.3 & 0.16 & 0.048 & 0.07 + 17068 + 4027 & 2.1 & 0.17 & 0.082 & 0.12 + 18443 + 7433 & 3.2 & 0.25 & 0.078 & 0.12 + 205514250 & 2.9 & 0.18 & 0.062 & 0.09 + 23129 + 2548 & 3.2 & 0.25 & 0.078 & 0.12 + we infer the fraction of crystalline silicates in our sample from the peak optical depths of the 10@xmath0 m amorphous and the 16@xmath0 m crystalline silicate absorption bands using @xmath16 where @xmath17 and @xmath18 are the mass column densities of the amorphous and crystalline silicates . adopting @xmath19(peak ) = 2.4@xmath2010@xmath21 @xmath22/g as the peak mass absorption coefficient for amorphous silicates @xcite and @xmath23(peak ) = 1.6@xmath2010@xmath21 @xmath22/g as the peak mass absorption coefficient for forsterite @xcite , we find crystalline - to - amorphous silicate mass column density ratios ranging from 0.07 to 0.15 , with a median value of 0.11 ( see table[tab2 ] ) . it should be understood that these numbers are upper limits , since foreground emission and radiative transfer effects within the optically thick 10@xmath0 m silicate band may cause the apparent optical depth of the 10@xmath0 m silicate feature to be a lower limit to the true optical depth . our findings should be contrasted to the upper limit of crystalline silicates in the general ism of the milky way of @xmath31% @xcite and the crystalline - to - amorphous ratios of up to 0.75 observed in circumstellar environments @xcite . given the broad range in infered crystalline - to - amorphous silicate ratios in our sample , we have investigated the existence of correlations with the apparent 10@xmath0 m optical depth , the iras r(60,100 ) color , the 5.5@xmath0m to23@xmath0 m rest frame spectral slope and the 6.2@xmath0 m pah equivalent width , but found no significant trends . the fraction of crystallinity within this sample can therefore , at present , not be linked to the infrared spectral appearance . we have detected crystalline substructure in the silicate absorption features towards twelve strongly obscured ulirg nuclei . this is the first detection of crystalline silicates in any source outside the local group . space - based and ground - based observations have revealed the presence of infrared emission features of crystalline silicates around young stellar objects and evolved stars @xcite . comets also often show evidence for crystalline silicates in their spectra @xcite . in contrast , while stellar sources of silicate dust inject at least 5% of their silicates in crystalline form , the galactic _ interstellar _ silicate feature is exceedingly smooth , and there is no evidence for a crystalline absorption component in the galactic ism . this translates into an upper limit on the crystalline fraction in the galactic interstellar medium of 1% @xcite . this difference in crystallinity between silicates injected and silicates in the ism implies a rapid transformation of crystalline silicates into amorphous silicates in the interstellar medium ( @xmath24yr ; * ? ? ? * ; * ? ? ? this transformation has been attributed to energetic processing of the dust by heavy cosmic ray ions @xcite or to ion bombardment in high velocity ( v@xmath111000km / s ) shocks @xcite . in particular , based upon laboratory experiments and estimated cosmic ray fluxes in our galaxy , the timescale for amorphization is estimated to be only 70 million years @xcite , considerably shorter than the timescale at which ( crystalline ) silicates are injected into the galactic ism ( 4 billion years ; * ? ? ? * ) these same processes injection of crystalline silicates from stars and cosmic ray and shock amorphization will play a role in the interstellar media of these ulirgs as well . possible explanations for the much higher crystalline silicate fraction in these ulirgs as compared to the milky way include a higher fraction of crystalline silicates injected and/or a delay in the interstellar amorphization rate . in particular , ulirgs are characterized by a high rate of star formation driven by merging events . in contrast to the local ism , the enrichment of the dusty interstellar medium will be dominated by the rapidly evolving massive stars and the contribution by the more numerous low - mass stars will lag on the timescale associated with the ulirg - merger event ( @xmath25 yr ; * ? ? ? * ; * ? ? ? * ) . evidence for the presence of a large population of evolved massive stars in ulirg nuclei exists for the nearest ulirg , arp220 @xcite . spectroscopic studies of sources in our galaxy have revealed that massive stars are prominent sources of crystalline silicates during the supergiant phase . in particular , the more extreme examples of this class , such as afgl4106 , nmlcyg and irc+10420 , have crystalline silicate fractions of the order of 0.15 @xcite . likewise , crystalline silicates have been observed in the ejecta of some lbvs , such as r71 and agcar @xcite . this episodic mass - loss phase dominates the dust injection by the most massive stars ( m@xmath1150m@xmath7 ) in the galaxy . finally , it is currently unknown how much crystalline silicate dust is injected by supernovae . indeed it is unknown how much dust is injected by sne . however , it is conceivable that the sne resulting from this starburst also inject a large fraction of their freshly condensed dust in the form of crystalline silicates . hence , a starburst may rapidly increase the crystalline silicate fraction in the nuclear region . of course , the high supernova rate in a starburst will drive strong shock waves into the local environment , which will sputter the dust , returning the atoms from the solid to the gas phase @xcite . however , likely , the destruction rate of crystalline and amorphous silicates are similar and hence , to first order , this will not affect the crystalline - to - amorphous fraction . more importantly , supernovae are thought to be the dominant source of cosmic rays and this may increase the rate at which crystalline silicates are transformed into amorphous silicates . however , cosmic rays freshly accelerated in the supernova remnants may leak away to the rest of the galaxy on a timescale of @xmath2610@xmath27 yr @xcite , limiting the rate of this solid - state conversion in the starburst environment . thus , the crystalline - to - amorphous conversion time - scale may actually be quite similar to the value calculated for the local milky way ( 70 million years ; * ? ? ? we expect that over time , as the merging event ages and the starburst intensity decreases , the fraction of crystalline silicates will decrease on a similar time - scale . thus , we attribute the high fraction of crystalline silicates in these ulirgs as compared to others to the relative ` youth ' of these systems ; e.g. , the amorphitization process may lag the merger - triggered , star - formation - driven dust injection process . all 17 ulirgs with @xmath28(10@xmath0m)@xmath112.9 in our ulirg sample of 77 galaxies show crystalline silicate structure in their irs spectra . at lower silicate optical depth , the number of clear detections falls sharply . above @xmath28(10@xmath0m)=2 , 18 out of 23 ulirgs definitely show the 16@xmath0 m crystalline feature . above @xmath28(10@xmath0m)=1 , this is 20 out of 46 and for the entire ulirg sample only 21 out of 77 . we note that a complete analysis of this fraction is hampered by possible radiative transfer effects in the silicate features ( as evidenced by the strong variation in the ratio of 10to18@xmath0 m peak optical depths ) , the presence of 1618@xmath0 m uir emission in some sources with non - negligible pah emission @xcite and by the challenge to detect weak 16@xmath0 m absorption features at low contrast and low signal - to - noise . nevertheless , it is clear that crystalline silicates are a common component of the nuclear interstellar medium during the strongly obscured evolutionary phase of the ulirg phenomena . crystalline silicates can also be a sign of high - temperature grain processing . in a ulirg , both a central agn and hot , young stars are potential sources of high - energy photons capable of heating the dust in the ism . x - ray and far uv photons from a central agn will produce a hot , inner region . however , an agn as the source of grain processing in our ulirg sample is problematic for two reasons . first , interferometric 10@xmath0 m spectra of the seyfert-2 nucleus of ngc1068 do not show any indication for crystalline silicates in its inner warm 2pc region @xcite . second , the crystalline silicates are seen in absorption in our spectra , implying that they are located in the cooler , outer regions . this would require a large scale mechanism to transport the inner toroid material outwards and distribute it over the surrounding medium . the extended nature of a starburst does not require such a mechanism and therefore seems a more likely source of processed material than a central source . crystalline silicates are also a characteristic of the circumstellar planetary disks surrounding young stellar objects such as herbig aebe stars and t - tauri stars . again , in order to be seen in absorption , this material will have to be transported outwards into the surrounding cold medium , presumably through jets and winds @xcite . there is , however , no indication in galactic sources for such large scale transport of crystalline silicates . on the contrary , for herbig aebe stars the emission features of crystalline silicates are strongly concentrated towards the inner 2au of the disks ( e.g. , * ? ? ? hence , we deem high - temperature crystallization of existing amorphous silicate grains unlikely as the source of the high crystalline silicate fraction in ulirgs . in this work , we report the discovery of crystalline substructure at 11 , 16 , 19 , 23 and 28@xmath0 m in the amorphous silicate features towards twelve deeply obscured ulirg nuclei . these features indicate the presence of the mineral forsterite ( mg@xmath1sio@xmath2 ) . previously , crystalline silicates have only been observed in circumstellar environments . we infer the fraction of crystalline silicates in our sample from the peak optical depths of the 10@xmath0 m amorphous and the 16@xmath0 m crystalline bands and find a crystalline - to - amorphous ratio ranging from 0.07 to 0.15 , with a median value of 0.11 . these numbers are likely upper limits , since foreground emission and radiative transfer effects will cause the apparent 10@xmath0 m silicate optical depth to be a lower limit to the true optical depth . the crystalline - to - amorphous ratio in our twelve deeply obscured ulirgs is 715 times larger than the upper limit for this ratio in the interstellar medium of the milky way . this suggest that the timescale for injection of crystalline silicates into the ism is short in a merger - driven starburst environment ( eg . , as compared to the total time to dissipate the gas ) , pointing towards evolved massive stars ( red supergiants , lbvs and type ii supernovae ) as prominent sources of crystalline silicates . furthermore , the timescale for amorphitization of crystalline silicates , which is known to be fairly rapid in the ism of the milky way ( @xmath2610@xmath29 yr ) , is at most of similar order in starburst environments . we expect that over time , as the merging event ages and the starburst decreases in intensity , the fraction of crystalline silicates will decrease rapidly . thus , we attribute the high fraction of crystalline silicates in these ulirgs , as compared to others , to the relative ` youth ' of these systems . finally , other galaxy types may also exhibit crystalline silicate features far above the upper limits set for the crystalline silicate fraction in the ism of our galaxy ( @xmath31% ; * ? ? ? * ; * ? ? ? the possible detection of a 23@xmath0 m crystalline emission feature in the quasar pg1351 + 640 , reported by @xcite , may be a signpost of more detections to come . the authors wish to thank elise furlan , bill forrest , patrick morris , els peeters for discussions , and frank molster and sacha hony for sharing their iso sws data . support for this work was provided by nasa through contract number 1257184 issued by the jet propulsion laboratory , california institute of technology under nasa contract 1407 . hwws was supported under this contract through the spitzer space telescope fellowship program .

silicates are an important component of interstellar dust and the structure of these grains amorphous versus crystalline is sensitive to the local physical conditions . we have studied the infrared spectra of a sample of ultra - luminous infrared galaxies . here , we report the discovery of weak , narrow absorption features at 11 , 16 , 19 , 23 , and 28@xmath0 m , characteristic of crystalline silicates , superimposed on the broad absorption bands at 10 and 18@xmath0 m due to amorphous silicates in a subset of this sample . these features betray the presence of forsterite ( mg@xmath1sio@xmath2 ) , the magnesium - rich end member of the olivines . previously , crystalline silicates have only been observed in circumstellar environments . the derived fraction of forsterite to amorphous silicates is typically 0.1 in these ulirgs . this is much larger than the upper limit for this ratio in the interstellar medium of the milky way , 0.01 . these results suggest that the timescale for injection of crystalline silicates into the ism is short in a merger - driven starburst environment ( e.g. , as compared to the total time to dissipate the gas ) , pointing towards massive stars as a prominent source of crystalline silicates . furthermore , amorphization due to cosmic rays , which is thought to be of prime importance for the local ism , lags in vigorous starburst environments .

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Proceed to summarize the following text: -reinforcing brakes are a subject of intensive investigation during last years . working principle of such brakes is to use the wheelset s inertia momentum of a vehicle as the source of power for braking . one major advantage of self - reinforcing brakes is the energy consumption decrease which makes this research direction perspective . at present the development of a new brake concept of a self - energizing electro - hydraulic brake for a railway application is being carried out at the institute for fluid power drives and control ( ifas , rwth aachen university ) . working principle of the braking system can be found in [ 69 ] . the control task of the self - energizing electro - hydraulic brake is to track the reference signal of the pressure @xmath0 ( the output variable ) in the supply line of the brake system . the control system with a pure proportional controller allows reaching the goal , as shown in fig . 1 , where the reference signal represents a step function , whose values are the sequence \{27 , 59 , 91 , 59 , 27 bar } , which corresponds to brake forces of \{5 , 10 , 15 , 10 , 5 kn}. however , as seen from the figure , the oscillations of the supply line pressure appear in the system , which cause the oscillations of the brake force . such a behavior of braking , of course , is inadmissible for passenger trains . therefore , the main requirement for the closed - loop system behavior is to get an aperiodic process of supply line pressure changes independent of the desired brake force level and friction coefficient variations of brake pads . to reach the required behavior of the controlled variable a controller of nonlinear structure is proposed to synthesize on basis of the exact feedback linearization method [ 3 ] , [ 5 ] . mathematical description of the electro - hydraulic system can be represented by a nonlinear model of 10^th^ order [ 6 ] . high order of the model result in the analytically complicated controller structure , which is a problem for simulation and experiments . due to simplifications based on the results of the brake system analysis [ 6 ] as well as the approach used in [ 1 ] , the basic model can be reduced to a 4^th^ order nonlinear model . for positive ( * case a * ) and negative ( * case b * ) valve opening this model has the form * case a : * @xmath1 * case b : * @xmath2 where @xmath3 the load pressure of the brake actuator , the pressure in the supply line , the velocity and the control valve spool movement , respectively ; @xmath4 the ratio between piston areas of the brake actuator ; @xmath5 the pressure in the low pressure line ; @xmath6 the control valve parameters ; @xmath7 the known constant parameters ; @xmath8 the parameters dependent of the brake pads friction coefficient ; @xmath9 the control valve input signal . the output of the brake system is @xmath10 this model will be used for the controller synthesis in the following . the nonlinear model ( 1 ) ( 3 ) can be represented as & = f(x)+g(x)u + y & = h(x ) the idea of the exact feedback linearization method is to find a nonlinear transformation ( linearization algorithm ) of a control signal , for which the model ( 4 ) , ( 5 ) is linear or equivalent to a linear model in new coordinates @xmath11 , where @xmath12 is the coordinates transformation . an important notion of an input - output model is its relative degree . for linear systems , as known , the relative degree is the difference between the number of poles and zeros of the transfer function of a system . this is also the number of times an output needs to be differentiated in order that an input appears in the equation . for nonlinear systems it is defined in the similar manner . by differentiating the output @xmath13 and substituting ( 4 ) we get ( see [ 2 ] ) @xmath14 where @xmath15 and @xmath16 are lie derivatives of the function @xmath17 along the vector field @xmath18 and @xmath19 , respectively . for the electro - hydraulic brake , for positive and negative valve opening , @xmath20 for all @xmath21 in the range of operating points . therefore @xmath22 continuing in this way , we get @xmath23 where @xmath24 . consequently , @xmath25 the time derivative @xmath26 yields : @xmath27 where the lie derivative @xmath28 for all @xmath21 in the range of operating points and for positive and negative valve opening has the form * case a : * @xmath29 * case b : * @xmath30 thus , the equation ( 8) with the nonzero factor for @xmath9 describes the relation between the input @xmath9 and the output @xmath13 . here , according to the definition in [ 3 ] , the relative degree of the 4th order nonlinear model of the electro - hydraulic brake is 3 . for linear state - space systems , derivatives of an output are chosen as state variables of a plant . using the similar approach , we determine new coordinates as @xmath31 for positive and negative valve spool movement , the new coordinates @xmath32 can be expressed by the coordinates @xmath21 of the simplified brake model ( 1 ) ( 3 ) , i.e. * case a : * z_1 & = p_sup + z_2 & = t^a_sup x _ + z_3 & = t^a_sup ( t^a_sup - t^a_l ) x^2 _ + t^a_sup v _ * case b : * z_1 & = p_sup + z_2 & = t^b_sup x _ + z_3 & = t^b_sup ( t^b_sup - t^b_l ) x^2 _ + t^a_sup v _ in accordance with ( 11 ) , equations ( 5 ) ( 7 ) can be rewritten in the form @xmath33 where @xmath34 the function @xmath35 , for positive and negative valve spool movement , is determined by equations ( 9 ) , ( 10 ) , the function @xmath36 is expressed as * case a : * @xmath37 * case b : * @xmath38 if models ( 1 ) ( 3 ) , and ( 12 ) where equivalent , than the exact feedback linearization problem would be solvable . this , in its turn , would mean that a control signal of the form @xmath39 where @xmath40 is a new control signal , due to overall compensation of the nonlinear functions @xmath36 and @xmath35 , would lead the plant model ( 1 ) ( 3 ) , on the assumption of parametric certainty of the model and measurability of the state variables @xmath21 , to a system , whose behaviour is exactly identical with the behaviour of the linear model = 0 & 1 & 0 + 0 & 0 & 1 + 0 & 0 & 0 + z_1 + z_2 + z_3 + & + 0 + 0 + 1 + , + y= 1 & 0 & 0 + & z_1 + z_2 + z_3 + , whose output @xmath41 would coincide with the output @xmath42 of the simplified nonlinear model . however , since rank @xmath43 for all @xmath21 in the range of operating points , which is less then the dimension @xmath44 of the model ( 1 ) ( 3 ) , the mapping @xmath12 is not a diffeomorphism . let us note that a mapping @xmath45 , where @xmath46 and @xmath47 are smooth manifolds of dimension @xmath48 , is a diffeomorphism if @xmath49 is bijective and both @xmath49 and @xmath50 are smooth mapping ( see appendix a in [ 3 ] ) . considering aforesaid , the model ( 12 ) is not equivalent to the simplified nonlinear model ( 1 ) ( 3 ) . this is caused by the internal dynamics , i.e. behaviour of @xmath51-dimensional part of the nonlinear model , which , however , has no influence on the output ( 15 ) of the linear model ( 14 ) . in that case the problem of input - output linearization can be solved [ 4 ] . this means that the control ( 13 ) transforms the nonlinear model ( 1 ) ( 3 ) of the electro - hydraulic brake into a model whose input - output behaviour can be represented in the form ( 14 ) , ( 15 ) . additionally , one more variable @xmath52 has to be supplemented with the new coordinates @xmath32 . according to proposition 4.1.3 in [ 3 ] it is always possible to find such a function that the jacobian matrix of the mapping @xmath53 is nonsingular at some point @xmath54 . for the plant in question the load pressure of the brake actuator @xmath55 was chosen as the additional variable , i.e. @xmath56 . the jacobian matrix of the mapping @xmath57 is nonsingular at each operating point of the brake . the time derivative of @xmath58 for positive and negative valve opening has the form * case a : * @xmath59 * case b : * @xmath60 thus , the equivalent model in the new coordinates can be written as & = q(z ) , + = 0 & 1 & 0 + 0 & 0 & 1 + 0 & 0 & 0 + z_1 + z_2 + z_3 + & + 0 + 0 + 1 + ( ( , z)+(,z)u ) , + y&=z_1 , where the equation ( 18 ) for describes the zero - dynamics of the model ( 1 ) ( 3 ) . taking into consideration that @xmath61 , the functions @xmath62 and @xmath63 are expressed as @xmath64 the block diagram of the equivalent model is depicted in fig . 2 . thus we can conclude that for the self - energizing electro - hydraulic brake the controller synthesis based on the exact feedback linearization method is conventionally divided into the following steps . firstly , the control signal @xmath40 is determined for the linear model ( 14 ) , ( 15 ) , which describes the relationship between input and output variables of the nonlinear model ( 1 ) ( 3 ) . secondly , we take into consideration the internal dynamics of the plant , described by the equation ( 18 ) , which has no influence on the output of the linear model , see fig . 3 . finally , we generate the control signal @xmath9 of the 4^th^ order nonlinear model using the nonlinear transformation for the signal @xmath40 . the control @xmath9 is expressed as @xmath65 the obtained signal @xmath9 is the input signal for the real brake system , described by the nonlinear model of 10^th^ order . the linear model ( 14 ) gives the opportunity to use classical methods of linear systems theory to achieve the required control performance , i.e. to get an aperiodic behaviour of the output variable ( 15 ) of the model ( 14 ) . one such approach is the pole placement method which allows reaching any prescribed placement of closed loop system poles [ 10 ] . according to the method the control input signal @xmath40 for the linear model ( 14 ) is chosen in the form @xmath66 where is the error , i.e. the difference between the reference state vector and the real state vector . since the reference signal @xmath67 is assumed to be a piecewise constant function , the reference signal @xmath68 of the model ( 14 ) during each time period , where @xmath67 is constant , can be represented as @xmath69 the row - vector @xmath70 provides prescribed poles placement of the following closed loop system @xmath71 where @xmath72 since the electro - hydraulic brake system in question is a siso system , i.e. a single - input single - output system , only the output @xmath0 is measured . consequently , not all state vector @xmath21 is known . it means that both the state vector @xmath32 of the model ( 14 ) , ( 15 ) and the variable @xmath58 of the model ( 18 ) are unmeasured . instead of these variables their estimates have to be used . therefore , the control system is supplemented with a full - order state observer for ( 14 ) , ( 15 ) . by means of the estimate @xmath73 of the vector @xmath32 the estimate @xmath74 of the variable @xmath58 can be calculated ( see ( 16 ) and ( 17 ) ) . the full - order state observer model is determined on the basis of the model ( 14 ) , ( 15 ) and has the form : & = a + b+ k_obs(y- ) , + & = c^t , where @xmath75 is a column - vector of preset coefficients and . the vector @xmath75 is chosen such that the matrix of closed loop estimation system @xmath76 be hurwitz stable [ 10 ] . simulation results of the electro - hydraulic brake system of 10^th^ order with the controller of described nonlinear structure are shown in fig . one can see from the figure , that the nonlinear control algorithm synthesized on the basis of the exact feedback linearization method allows to get the desired closed - loop system behavior . the results of accomplished simulation demonstrated as well , that the control system is robust with respect to small variations of the friction coefficient . expansion of the robust stability ranges stimulates further development of the controller for the self - energizing electro - hydraulic brake . nonlinear control algorithm synthesis of the self - energizing electro - hydraulic brake on the basis of the exact feedback linearization method has been fulfilled . with the help of the obtained algorithm the problem of the desired closed - loop system behavior has been solved . the author would like to thank german academic exchange service ( daad ) and ministry of education and science of russian federation for supporting the research stay at the institute for fluid power drives and control in aachen ( project no . 7.375.c2007 ) . m. liermann and c. stammen . development of a self - energizing electro - hydraulic brake ( sehb ) for rail vehicles . in : _ the tenth scandinavian international conference on fluid power : sicfp07 _ , tampere : university of technology , institute of hydraulics and automation , 2007 . m. liermann , c. stammen and h. murrenhoff . development of a self - energizing electro - hydraulic brake ( sehb ) . _ paper no . 2007 - 01 - 4236 , sae commercial vehicle congress and exhibition 2007 _ , chicago , usa , 2007 . m. liermann , c. stammen and h. murrenhoff . pressure tracking control for a self - energizing hydraulic brake . in _ the 20th bath symposium on power transmission and motion control _ , bath : hadleys ltd . , p. 315330

nonlinear control algorithm for a self - energizing electro - hydraulic brake is analytically designed . the desired closed - loop system behavior is reached via a synthesized nonlinear controller . shell nonlinear system , nonlinear control , algorithm , electro - hydraulic brake .

You are an expert at summarizing long articles. Proceed to summarize the following text: one of the outstanding issues in @xmath35 meson physics is the semileptonic branching fraction puzzle . experimentally @xmath36 is measured to be ( @xmath37)% @xcite , whereas theoretical calculations have difficulties accommodating a branching fraction below @xmath38 @xcite . one way to reduce the theoretical expectations is through a two - fold enhancement in the assumed @xmath39 rate @xcite , which is estimated to be @xmath40 from the measured inclusive rates for @xmath41 and @xmath42 . recently , buchalla _ et al._@xcite and blok _ et al._@xcite have suggested that a significant fraction of the @xmath39 transition hadronizes into @xmath43 . this is supported by cleo s @xcite observation of `` wrong - sign '' @xmath44 mesons from @xmath35 decays , @xmath45 , where the @xmath44 comes from the virtual @xmath46 . the aleph @xcite and delphi @xcite collaborations have also observed sizeable @xmath47 decay rates . exclusive @xmath35 decays involving wrong - sign @xmath44 mesons can result from ( 1 ) resonant @xmath48 decays , where the @xmath46 hadronizes to an excited @xmath5 meson that decays into @xmath49 ; and ( 2 ) non - resonant @xmath50 decays . this paper explores one possibility in the first case , namely , the decays @xmath51 where @xmath52 is the narrow p - wave @xmath5 meson with @xmath53 . the `` upper - vertex '' production of @xmath52 from @xmath46 hadronization is shown in figure [ fig : feynman](a ) . in addition , @xmath52 mesons can be produced from `` lower - vertex '' decays @xmath54 with the creation of an @xmath55 quark pair , as shown in figure [ fig : feynman](b ) . this produces right - sign @xmath44 mesons ; however , the decay rate is expected to be small . throughout this paper charge conjugate states are implied . continuum @xmath52 production has been thoroughly studied @xcite . the @xmath52 is just above the @xmath56 mass threshold and decays dominantly into @xmath57 and @xmath58 . other possible decay channels are negligible : @xmath59 due to isospin conservation , @xmath60 due to ozi suppression @xcite , @xmath61 or @xmath62 due to angular momentum and parity conservation , and @xmath63 due to the small radiative decay rate . the data used in this analysis were selected from hadronic events collected by the cleo ii detector at the cornell electron storage ring ( cesr ) . the cleo ii detector @xcite is a large solenoidal detector with 67 tracking layers and a csi electromagnetic calorimeter that provides efficient @xmath64 reconstruction . the data consist of an integrated luminosity of 3.11 fb@xmath65 at the @xmath66 resonance , corresponding to @xmath67 @xmath68 events . to evaluate non-@xmath68 backgrounds we also collected 1.61 fb@xmath65 of `` continuum '' data 60 mev below the @xmath66 resonance . the inclusive @xmath69 decay is studied by reconstructing the decay channels @xmath70 and @xmath71 using the decay modes @xmath72 and @xmath73 . the @xmath74 is reconstructed using the decay modes @xmath75 and @xmath76 . hadronic events are required to satisfy the ratio of fox - wolfram moments @xcite @xmath77 to reduce the background from continuum events . charged tracks , except pions from @xmath78 decays , are required to be consistent with coming from the primary interaction point . charged kaon and pion candidates are identified using specific ionization ( @xmath79 ) and , when available , time - of - flight ( tof ) information . for kaon identification , we consider the relative probability for a charged track to be a kaon , @xmath80 , where @xmath81 is the @xmath82 probability for a given particle hypothesis . the requirement on @xmath83 depends on the decay mode of interest . pion candidates are identified by requiring the @xmath79 and , when available , tof information to be within 3 standard deviations ( @xmath84 ) of that expected for pions . we select @xmath78 candidates through the decay to @xmath85 by requiring a decay vertex displaced from the primary interaction point and a @xmath78 invariant mass within 10 mev / c@xmath86 of its nominal value . we reconstruct @xmath64 candidates through the decay to @xmath87 by requiring candidates to have an invariant mass within 2.5 standard deviations ( @xmath88 mev / c@xmath86 ) of the nominal @xmath64 mass . the @xmath89 and @xmath76 combinations are required to have a kaon identification of @xmath90 and @xmath91 , respectively , and an invariant mass within 15 and 25 mev / c@xmath86 ( @xmath92 ) of the nominal @xmath74 mass , respectively . in addition , we select regions of the @xmath93 dalitz plot to take advantage of the known resonant substructure @xcite . for the @xmath70 mode , the dalitz cut reduces the signal efficiency by 40% and the background by 80% . we relax the dalitz cut for the @xmath71 mode since the combinatoric background is substantially lower . the @xmath73 candidates are required to have a mass difference @xmath94 within 1.5 mev / c@xmath86 ( @xmath92 ) of the nominal value of 145.4 mev / c@xmath86 , where @xmath95 is the reconstructed invariant mass of @xmath96 . similarly , the @xmath72 candidates are required to have a mass difference @xmath97 within 1.5 mev / c@xmath86 ( @xmath92 ) of the nominal value of 142.1 mev / c@xmath86 . to form @xmath52 candidates charged kaons are combined with @xmath98 candidates and @xmath78 s are combined with @xmath99 candidates . since the primary kaons from @xmath70 decays have low momentum , we can impose a stringent @xmath100 requirement on the @xmath101 with negligible loss of efficiency . the @xmath52 candidates are required to have a scaled momentum @xmath102 , which is the kinematic limit for @xmath69 decays . ( we ignore the negligible contributions from @xmath103 decays . ) upper - vertex @xmath52 production results in a maximum @xmath104 of 0.35 , and this requirement is imposed when determining the @xmath52 decay constant . the @xmath52 decay channels with @xmath64 s in the final state often have multiple @xmath52 candidates per event . we select the candidate with the highest @xmath82 probability of being a @xmath52 , which is derived from the invariant masses of the reconstructed @xmath64 , @xmath74 and @xmath105 mesons . the @xmath52 signal is identified using the @xmath56 mass difference , @xmath106 and @xmath107 , where @xmath108 and @xmath109 are the known masses @xcite . the @xmath56 mass difference signal has a resolution that is two to four times smaller than the corresponding signal in the reconstructed @xmath56 invariant mass distribution . the @xmath110 and @xmath111 distributions are shown in figure [ fig : mass diff ] , where the @xmath75 and @xmath76 modes have been added together . the data is fit with a gaussian signal and a threshold background function . the gaussian width is fixed to that expected from a geant - based monte carlo simulation @xcite ( @xmath112 mev / c@xmath86 , depending on the mode ) and the mean is fixed to the measured @xmath52 mass difference from continuum data ( @xmath113 mev / c@xmath86 and @xmath114 mev / c@xmath86 . ) we observe @xmath115 signal events in the @xmath57 mode and @xmath116 events in the @xmath71 mode . however , when the @xmath57 candidates are further subdivided into the @xmath75 and @xmath76 decay channels there is a discrepancy in the @xmath52 yields . as shown in figure [ fig:2 d0 modes ] , we observe @xmath117 signal events in the @xmath110 distribution for the @xmath75 channel and @xmath118 @xmath52 signal events for the @xmath93 channel . after accounting for branching fractions and efficiencies , discussed below , this results in a @xmath119 discrepancy in the @xmath57 rates between the two @xmath74 modes . we can not rule out the fact that background sources may be contributing a false @xmath52 signal in the @xmath93 channel , but not in the @xmath75 channel . however , no such mechanism has been uncovered . to be conservative , we choose to quote only an upper limit for the decay @xmath69 . since the @xmath52 reconstruction efficiency increases rapidly with @xmath104 and the @xmath52 momentum distribution from @xmath35 decays is not known , we compute the inclusive @xmath69 branching fraction by dividing the data into four equal regions of @xmath104 from 0.05 to 0.45 and summing the efficiency corrected yields . the @xmath70 and @xmath58 branching fractions are equal according to isospin , and their ratio has been measured to be within 30% of unity @xcite . we measure the branching fraction @xmath69 to be @xmath120 from the @xmath57 mode and @xmath121 from the @xmath71 mode , where the error is statistical only . the two measurements are statistically consistent . the @xmath104 distribution for our @xmath52 candidates is shown in figure [ fig : xp dist ] . several cross - checks , shown in figure [ fig : cross checks ] , were performed to corroborate the validity of the @xmath52 signal . the scaled continuum background from data after satisfying all selection cuts is negligible , and there is no excess in the @xmath110 signal region ( @xmath122 events ) . the uncertainty in the continuum @xmath52 contribution is included in the systematic error . there is also no evidence of peaking in the @xmath110 signal region for wrong - sign @xmath123 combinations ( @xmath124 events ) , @xmath74 mass sidebands ( @xmath125 events ) , and @xmath98 mass sidebands ( @xmath126 events ) . we have also searched for the @xmath74 signal from @xmath127 candidates in the @xmath110 signal region , @xmath128 mev / c@xmath129 mev / c@xmath86 , by relaxing the @xmath74 mass cut and histogramming the invariant mass of all @xmath89 and @xmath76 combinations that satisfy the remaining selection criteria . in events with multiple candidates per @xmath74 decay mode we select the candidate with the highest @xmath82 probability , which is derived from the reconstructed @xmath64 and @xmath52 masses . we observe @xmath130 @xmath74 events . however , there are also real @xmath74 s in the random @xmath57 combinations under the @xmath52 peak ; after a @xmath110 sideband subtraction the @xmath74 invariant mass spectrum yields @xmath131 events ( see figure [ fig : d0 yield](a ) ) . this is consistent with our @xmath70 yield in figure [ fig : mass diff ] . similarly , we have studied the @xmath98 signal from @xmath127 candidates in the @xmath110 signal region . we observe @xmath132 @xmath74 events . as in the @xmath74 case there are also real @xmath98 s in the random @xmath57 combinations under the @xmath52 peak . after a @xmath110 sideband subtraction the @xmath98 mass difference spectrum yields @xmath133 events ( see figure [ fig : d0 yield](b ) ) , consistent with our @xmath70 yield . finally , we have studied the @xmath52 production from continuum @xmath134 events . the selection criteria is similar to that used to find @xmath52 from @xmath35 decays , but since continuum charm production has a hard fragmentation , we require @xmath135 . in addition , we remove the @xmath136 cut , relax the charged kaon identification to @xmath137 , and remove the dalitz cut for @xmath93 . the mass difference distribution for @xmath57 and @xmath71 combinations are shown in figure [ fig : continuum ] , where the @xmath75 and @xmath76 modes have been added together . we extract the @xmath52 signal by fitting the data with a gaussian signal and a threshold background function . the gaussian width is fixed to the value predicted by monte carlo ( 2.1 mev / c@xmath86 ) , and the mean is allowed to float . we observe @xmath138 events in the @xmath70 mode with a mass difference of @xmath139 mev / c@xmath86 ( statistical error only ) , and @xmath140 events in the @xmath141 mode with a mass difference of @xmath142 mev / c@xmath86 . the results are consistent with the previous cleo analysis @xcite . there are several sources of systematic error . we assign a systematic error of 16% to account for the @xmath119 discrepancy between the @xmath57 rates for the @xmath75 and @xmath76 modes . this accomodates different methods of computing the weighted average of the @xmath69 branching fraction from the four separate decay chains . uncertainties due to reconstruction efficiencies include 1.5% per charged track , 5% per @xmath64 , 5% for slow pions from @xmath105 , and 5% for @xmath78 . we also include systematic errors of 7% for monte carlo statistics , 5% for kaon identification and the dalitz decay cut efficiency , 4% for uncertainties in the yield for @xmath143 , and 8% for uncertainties in the continuum @xmath52 contribution that passes our selection criteria . the total systematic error is 24% . averaging the @xmath57 and @xmath71 modes together , we obtain @xmath144 . since the @xmath52 signal is observed largely in only one decay mode @xmath70 with @xmath93 , and since there is a discrepancy between this mode and the corresponding mode involving @xmath75 , we instead prefer to quote an upper limit on the branching fraction to be @xmath145 at the 90% c.l . @xcite this decay rate limit is small relative to the total rate expected for @xmath2 of about @xmath146 from the wrong - sign @xmath44 meson yield in @xmath35 decays @xcite . this is not surprising considering the @xmath147 system has appreciable phase space beyond the @xmath52 mass @xcite . also , cleo s @xcite recent observation of exclusive @xmath50 decays shows that the @xmath148 invariant mass distribution lies mostly above the @xmath52 mass . measurement of the @xmath69 decay rate also provides an estimate of the @xmath52 decay constant , @xmath149 , assuming that the @xmath52 comes dominantly from upper - vertex decays . the inclusive decay rate for @xmath35 mesons into ground state or excited @xmath5 mesons can be calculated assuming factorization @xcite , @xmath150 where @xmath151 is the bsw @xcite parameter for the effective charged current , and @xmath152 is a kinematic factor with @xmath153 and @xmath154 . for scalar or pseudoscalar @xmath155 mesons , @xmath156 , and for vector or axial - vector @xmath155 mesons , @xmath157 . we have tightened the @xmath104 requirement to @xmath158 since this is the kinematic limit for upper - vertex @xmath159 decays . the production of ground state and excited @xmath5 mesons from lower - vertex decays such as @xmath160 is expected to be suppressed . this is certainly true for @xmath161 decays where the fraction of @xmath5 produced at the lower - vertex is measured to be @xmath162 @xcite . moreover , there is no evidence of @xmath52 production in the region @xmath163 where lower - vertex production is likely to occur ( see figure [ fig : xp dist ] . ) with the assumption @xmath164 we can extract @xmath149 from the ratio of inclusive rates , @xmath165 many systematic errors cancel in the ratio . when computing the @xmath52 decay constant from the above equation , we use @xmath166 of the measured @xmath69 branching fraction to account for uncertainties in the upper and lower vertex contributions to @xmath52 . this accomodates the excess of @xmath69 candidates observed at low @xmath167 as seen in figure [ fig : xp dist ] . from our upper limit on @xmath69 and cleo s @xcite measurement of @xmath168 , we derive @xmath169 at the 90% c.l . the central value is @xmath170 , where the first error is due to the total error in the inclusive @xmath171 and @xmath69 branching fractions , and the second is the uncertainty in the non - factorizable and lower - vertex contributions to the @xmath69 decay rate . using the measured value of @xmath172 mev @xcite gives @xmath173 mev which corresponds to an upper limit of @xmath174 mev . this limit accomodates the prediction of @xmath175 mev by veseli and dunietz @xcite . in summary , we have searched for @xmath35 mesons decaying into the p - wave @xmath176 meson . the upper limit of @xmath177 at the 90% c.l . accounts for at most only a fraction of the total wrong - sign @xmath178 rate . assuming factorization , the decay constant @xmath149 is at least a factor of 2.5 times smaller than the decay constant for the pseudoscalar @xmath5 . we gratefully acknowledge the effort of the cesr staff in providing us with excellent luminosity and running conditions . , j.r.p . , and i.p.j.s . thank the nyi program of the nsf , m.s . thanks the pff program of the nsf , g.e . thanks the heisenberg foundation , k.k.g . , t.s . , and h.y . thank the oji program of doe , j.r.p . , m.s . and v.s . thank the a.p . sloan foundation , r.w . thanks the alexander von humboldt stiftung , m.s . thanks research corporation , and s.d . thanks the swiss national science foundation for support . this work was supported by the national science foundation , the u.s . department of energy , and the natural sciences and engineering research council of canada . kuhn , s. nussinov , and r. ruckl , z. phys . c * 5 * , 117 ( 1980 ) . m. bauer , b. stech , and m. wirbel , z. phys . c * 29 * , 637 ( 1985 ) . cleo collaboration , x. fu _ et al . _ , cleo conf-95 - 11 , eps0169 , contributed to the 1995 international europhysics conference on high energy physics , brussels , belgium .

we have searched for the decay @xmath0 and measured an upper limit for the inclusive branching fraction of @xmath1 at the 90% confidence level . this limit is small compared with the total expected @xmath2 rate . assuming factorization , the @xmath3 decay constant is constrained to be @xmath4 mev at the 90% confidence level , at least 2.5 times smaller than that of @xmath5 . 6.5 in 9.0 in -0.50 in 0.00 in 0.00 in m. bishai,@xmath6 j. fast,@xmath6 j. w. hinson,@xmath6 n. menon,@xmath6 d. h. miller,@xmath6 e. i. shibata,@xmath6 i. p. j. shipsey,@xmath6 m. yurko,@xmath6 s. glenn,@xmath7 s. d. johnson,@xmath7 y. kwon,@xmath8 s. roberts,@xmath7 e. h. thorndike,@xmath7 c. p. jessop,@xmath9 k. lingel,@xmath9 h. marsiske,@xmath9 m. l. perl,@xmath9 v. savinov,@xmath9 d. ugolini,@xmath9 r. wang,@xmath9 x. zhou,@xmath9 t. e. coan,@xmath10 v. fadeyev,@xmath10 i. korolkov,@xmath10 y. maravin,@xmath10 i. narsky,@xmath10 v. shelkov,@xmath10 j. staeck,@xmath10 r. stroynowski,@xmath10 i. volobouev,@xmath10 j. ye,@xmath10 m. artuso,@xmath11 f. azfar,@xmath11 a. efimov,@xmath11 m. goldberg,@xmath11 d. he,@xmath11 s. kopp,@xmath11 g. c. moneti,@xmath11 r. mountain,@xmath11 s. schuh,@xmath11 t. skwarnicki,@xmath11 s. stone,@xmath11 g. viehhauser,@xmath11 x. xing,@xmath11 j. bartelt,@xmath12 s. e. csorna,@xmath12 v. jain,@xmath13 k. w. mclean,@xmath12 s. marka,@xmath12 r. godang,@xmath14 k. kinoshita,@xmath14 i. c. lai,@xmath14 p. pomianowski,@xmath14 s. schrenk,@xmath14 g. bonvicini,@xmath15 d. cinabro,@xmath15 r. greene,@xmath15 l. p. perera,@xmath15 g. j. zhou,@xmath15 b. barish,@xmath16 m. chadha,@xmath16 s. chan,@xmath16 g. eigen,@xmath16 j. s. miller,@xmath16 c. ogrady,@xmath16 m. schmidtler,@xmath16 j. urheim,@xmath16 a. j. weinstein,@xmath16 f. wrthwein,@xmath16 d. w. bliss,@xmath17 g. masek,@xmath17 h. p. paar,@xmath17 s. prell,@xmath17 v. sharma,@xmath17 d. m. asner,@xmath18 j. gronberg,@xmath18 t. s. hill,@xmath18 d. j. lange,@xmath18 r. j. morrison,@xmath18 h. n. nelson,@xmath18 t. k. nelson,@xmath18 j. d. richman,@xmath18 d. roberts,@xmath18 a. ryd,@xmath18 m. s. witherell,@xmath18 r. balest,@xmath19 b. h. behrens,@xmath19 w. t. ford,@xmath19 h. park,@xmath19 j. roy,@xmath19 j. g. smith,@xmath19 j. p. alexander,@xmath20 c. bebek,@xmath20 b. e. berger,@xmath20 k. berkelman,@xmath20 k. bloom,@xmath20 v. boisvert,@xmath20 d. g. cassel,@xmath20 h. a. cho,@xmath20 d. s. crowcroft,@xmath20 m. dickson,@xmath20 s. von dombrowski,@xmath20 p. s. drell,@xmath20 k. m. ecklund,@xmath20 r. ehrlich,@xmath20 a. d. foland,@xmath20 p. gaidarev,@xmath20 l. gibbons,@xmath20 b. gittelman,@xmath20 s. w. gray,@xmath20 d. l. hartill,@xmath20 b. k. heltsley,@xmath20 p. i. hopman,@xmath20 j. kandaswamy,@xmath20 p. c. kim,@xmath20 d. l. kreinick,@xmath20 t. lee,@xmath20 y. liu,@xmath20 n. b. mistry,@xmath20 c. r. ng,@xmath20 e. nordberg,@xmath20 m. ogg,@xmath21 j. r. patterson,@xmath20 d. peterson,@xmath20 d. riley,@xmath20 a. soffer,@xmath20 b. valant - spaight,@xmath20 c. ward,@xmath20 m. athanas,@xmath22 p. avery,@xmath22 c. d. jones,@xmath22 m. lohner,@xmath22 c. prescott,@xmath22 j. yelton,@xmath22 j. zheng,@xmath22 g. brandenburg,@xmath23 r. a. briere,@xmath23 a. ershov,@xmath23 y. s. gao,@xmath23 d. y .- j . kim,@xmath23 r. wilson,@xmath23 h. yamamoto,@xmath23 t. e. browder,@xmath24 y. li,@xmath24 j. l. rodriguez,@xmath24 t. bergfeld,@xmath25 b. i. eisenstein,@xmath25 j. ernst,@xmath25 g. e. gladding,@xmath25 g. d. gollin,@xmath25 r. m. hans,@xmath25 e. johnson,@xmath25 i. karliner,@xmath25 m. a. marsh,@xmath25 m. palmer,@xmath25 m. selen,@xmath25 j. j. thaler,@xmath25 k. w. edwards,@xmath26 a. bellerive,@xmath27 r. janicek,@xmath27 d. b. macfarlane,@xmath27 p. m. patel,@xmath27 a. j. sadoff,@xmath28 r. ammar,@xmath29 p. baringer,@xmath29 a. bean,@xmath29 d. besson,@xmath29 d. coppage,@xmath29 c. darling,@xmath29 r. davis,@xmath29 s. kotov,@xmath29 i. kravchenko,@xmath29 n. kwak,@xmath29 l. zhou,@xmath29 s. anderson,@xmath30 y. kubota,@xmath30 s. j. lee,@xmath30 j. j. oneill,@xmath30 s. patton,@xmath30 r. poling,@xmath30 t. riehle,@xmath30 a. smith,@xmath30 m. s. alam,@xmath31 s. b. athar,@xmath31 z. ling,@xmath31 a. h. mahmood,@xmath31 h. severini,@xmath31 s. timm,@xmath31 f. wappler,@xmath31 a. anastassov,@xmath32 j. e. duboscq,@xmath32 d. fujino,@xmath33 k. k. gan,@xmath32 t. hart,@xmath32 k. honscheid,@xmath32 h. kagan,@xmath32 r. kass,@xmath32 j. lee,@xmath32 m. b. spencer,@xmath32 m. sung,@xmath32 a. undrus,@xmath33 r. wanke,@xmath32 a. wolf,@xmath32 m. m. zoeller,@xmath32 b. nemati,@xmath34 s. j. richichi,@xmath34 w. r. ross,@xmath34 and p. skubic@xmath34 @xmath6purdue university , west lafayette , indiana 47907 + @xmath7university of rochester , rochester , new york 14627 + @xmath9stanford linear accelerator center , stanford university , stanford , california 94309 + @xmath10southern methodist university , dallas , texas 75275 + @xmath11syracuse university , syracuse , new york 13244 + @xmath12vanderbilt university , nashville , tennessee 37235 + @xmath14virginia polytechnic institute and state university , blacksburg , virginia 24061 + @xmath15wayne state university , detroit , michigan 48202 + @xmath16california institute of technology , pasadena , california 91125 + @xmath17university of california , san diego , la jolla , california 92093 + @xmath18university of california , santa barbara , california 93106 + @xmath19university of colorado , boulder , colorado 80309 - 0390 + @xmath20cornell university , ithaca , new york 14853 + @xmath22university of florida , gainesville , florida 32611 + @xmath23harvard university , cambridge , massachusetts 02138 + @xmath24university of hawaii at manoa , honolulu , hawaii 96822 + @xmath25university of illinois , urbana - champaign , illinois 61801 + @xmath26carleton university , ottawa , ontario , canada k1s 5b6 + and the institute of particle physics , canada + @xmath27mcgill university , montral , qubec , canada h3a 2t8 + and the institute of particle physics , canada + @xmath28ithaca college , ithaca , new york 14850 + @xmath29university of kansas , lawrence , kansas 66045 + @xmath30university of minnesota , minneapolis , minnesota 55455 + @xmath31state university of new york at albany , albany , new york 12222 + @xmath32ohio state university , columbus , ohio 43210 + @xmath34university of oklahoma , norman , oklahoma 73019

You are an expert at summarizing long articles. Proceed to summarize the following text: figure [ fig : figure1 ] shows the point set of an optimal ( crossing minimal ) rectilinear drawing of @xmath8 , with an evident partition of the @xmath9 vertices into @xmath1 highly structured clusters of @xmath1 vertices each : [ fig : figure1 ] are clustered into @xmath1 sets.,width=113 ] a similar , natural , highly structured partition into @xmath1 clusters of equal size is observed in _ every _ known optimal drawing of @xmath0 , for every @xmath3 multiple of @xmath1 ( see @xcite ) . even for those values of @xmath3 ( namely , @xmath10 ) for which the exact rectilinear crossing number @xmath7 of @xmath0 is not known , the best available examples also share this property @xcite . in all these examples , a set @xmath11 of @xmath3 points in general position is partitioned into sets @xmath12 and @xmath13 , with @xmath14 with the following properties : \(i ) there is a directed line @xmath15 such that , as we traverse @xmath15 , we find the @xmath15orthogonal projections of the points in @xmath16 , then the @xmath15orthogonal projections of the points in @xmath17 , and then the @xmath15orthogonal projections of the points in @xmath13 ; \(ii ) there is a directed line @xmath18 such that , as we traverse @xmath18 , we find the @xmath18orthogonal projections of the points in @xmath17 , then the @xmath18orthogonal projections of the points in @xmath16 , and then the @xmath18orthogonal projections of the points in @xmath13 ; and \(iii ) there is a directed line @xmath19 such that , as we traverse @xmath19 , we find the @xmath19orthogonal projections of the points in @xmath17 , then the @xmath19orthogonal projections of the points in @xmath13 , and then the @xmath19orthogonal projections of the points in @xmath16 . * definition * a point set that satisfies conditions ( i)(iii ) above is @xmath1 _ decomposable_. we also say that the underlying rectilinear drawing of @xmath0 is @xmath1_decomposable_. a possible choice of @xmath20 , and @xmath19 for the example in figure [ fig : figure1 ] is illustrated in figure [ fig : figure2 ] . [ ht ] 1 cm 0.5 cm it is widely believed that all optimal rectilinear drawings of @xmath0 are @xmath1decomposable . one of our main results in this paper is the following lower bound for the number of crossings in all such drawings . [ thm : main ] let @xmath5 be a @xmath1decomposable rectilinear drawing of @xmath0 . then the number @xmath4 of crossings in @xmath5 satisfies @xmath21 the best known general lower and upper bounds for the rectilinear crossing number @xmath7 are @xmath22 ( see @xcite and @xcite ) . thus the bound given by theorem [ thm : main ] closes this gap by over 40% , under the ( quite feasible ) assumption of @xmath1decomposability . to prove theorem [ thm : main ] ( in section [ sec : proofmain ] ) , we exploit the close relationship between rectilinear crossing numbers and @xmath2sets , unveiled independently by brego and fernndez merchant @xcite and by lovsz et al . @xcite . recall that a @xmath2_set _ of a point set @xmath11 is a subset @xmath23 of @xmath11 with @xmath24 such that some straight line separates @xmath23 and @xmath25 . the number @xmath26 of @xmath2sets of @xmath11 is a parameter of independent interest in discrete geometry ( see @xcite ) , and , as we recall in section [ sec : proofmain ] , is closely related to the rectilinear crossing number of the geometric graph induced by @xmath11 . the main ingredient in the proof of theorem [ thm : main ] is the following bound ( theorem [ thm : mainksets ] ) for the number of @xmath2sets in @xmath1decomposable point sets . the bound is in terms of the following quantity ( by convention , @xmath27 if @xmath28 ) , @xmath29 where @xmath30 is the unique integer such that @xmath31 . [ thm : mainksets ] let @xmath11 be a @xmath1decomposable set of @xmath3 points in general position , where @xmath3 is a multiple of @xmath1 , and let @xmath32 . then @xmath33 the best general lower bound for @xmath26 is the sum of the first two terms in ( [ eq : ygriega ] ) ( see @xcite and @xcite ) . thus the third summand in ( [ eq : ygriega ] ) is the improvement we report , under the assumption of @xmath1decomposability . the proofs of theorems [ thm : main ] and [ thm : mainksets ] are in sections [ sec : proofmain ] and [ sec : proofmainksets ] , respectively . in section [ sec : concludingremarks ] we present some concluding remarks and open questions . let @xmath5 be a @xmath1decomposable rectilinear drawing of @xmath0 , and let @xmath11 denote the underlying @xmath3point set , that is , the vertex set of @xmath5 . besides theorem [ thm : mainksets ] , our main tool is the following relationship between @xmath2sets and the rectilinear crossing number ( see @xcite or @xcite ) : @xmath34 combining theorem [ thm : mainksets ] and eq . ( [ eq : aflov ] ) , and noting that both the @xmath35 in the factor @xmath36 and the summand @xmath37 in ( [ eq : ygriega ] ) only contribute to smaller order terms , we obtain : @xmath38 elementary calculations show that @xmath39 and @xmath40 . thus , @xmath41 since @xmath42 , then @xmath43 -0.6 cm the first step to prove theorem [ thm : mainksets ] is to obtain an equivalent ( actually , more general ) formulation in terms of circular sequences ( namely proposition [ prop : main ] below ) . all the geometrical information of a point set @xmath11 gets encoded in ( any halfperiod of ) the _ circular sequence _ associated to @xmath11 . we recall that a circular sequence on @xmath3 elements is a doubly infinite sequence @xmath44 of permutations of the points in @xmath11 , where consecutive permutations differ in a transposition of neighboring elements , and , for every @xmath45 , @xmath46 is the reverse permutation of @xmath47 . thus a circular sequence on @xmath3 elements has period @xmath48 , and all the information is encoded in an @xmath3_halfperiod _ , that is , a sequence of @xmath49 consecutive permutations . each @xmath3point set @xmath11 has an associated circular sequence @xmath50 , which contains all the geometrical information of @xmath11 @xcite . as we observed above , any @xmath3halfperiod @xmath51 of @xmath52 contains all the information of @xmath50 , and so @xmath3halfperiods are usually the object of choice to work with . in an @xmath3halfperiod @xmath53 , the _ initial _ permutation is @xmath54 and the _ final permutation _ is @xmath55 . not every @xmath3halfperiod @xmath51 arises from a point set @xmath11 . we refer the reader to the seminal work by goodman and pollack @xcite for further details . observe that if @xmath11 is @xmath1decomposable , then there is an @xmath3halfperiod @xmath51 of the circular sequence associated to @xmath11 , whose points can be labeled @xmath56 , so that : \(i ) the initial permutation @xmath54 reads @xmath57 ; \(ii ) there is an @xmath58 such that in the @xmath59st permutation first the @xmath60 s appear consecutively , then the @xmath61 s appear consecutively , and then the @xmath62 s appear consecutively ; and \(iii ) there is a @xmath63 , with @xmath64 , such that in the @xmath65st permutation first the @xmath60 s appear consecutively , then the @xmath62 s appear consecutively , and then the @xmath61 s appear consecutively . * definition * an @xmath3halfperiod @xmath51 that satisfies properties ( i)(iii ) above is @xmath1_decomposable_. a transposition that occurs between elements in sites @xmath45 and @xmath66 is an @xmath67_transposition_. an @xmath45_critical _ tranposition is either an @xmath67transposition or an @xmath68transposition , and a @xmath2_critical _ transposition is a transposition that is @xmath45critical for some @xmath69 . if @xmath51 is an @xmath3halfperiod , then @xmath70 denotes the number of @xmath2critical transpositions in @xmath51 . the key result is the following . [ prop : main ] let @xmath51 be a @xmath1decomposable @xmath3halfperiod , and let @xmath32 . then @xmath71 * proof of theorem [ thm : mainksets ] . * let @xmath11 be @xmath1decomposable , and let @xmath51 be an @xmath3halfperiod of the circular sequence associated to @xmath11 , that satisfies properties ( i)(iii ) above . then @xmath51 is @xmath1decomposable . now , for any point set @xmath23 and any halfperiod @xmath72 associated to @xmath23 , the @xmath2critical transpositions of @xmath72 are in one to one correspondence with @xmath2sets of @xmath23 . applying this to @xmath51 and @xmath11 , it follows that @xmath73 . applying proposition [ prop : main ] , theorem [ thm : mainksets ] follows . we devote the rest of this section to the proof of proposition [ prop : main ] . throughout this section , @xmath74 is a @xmath1decomposable @xmath3halfperiod , with initial permutation @xmath75 . in order to ( lower ) bound the number of @xmath2critical transpositions in @xmath1decomposable circular sequences , we distinguish between two types of transpositions . a transposition is _ homogeneous _ if it occurs between two @xmath61 s , between two @xmath60 s , or between two @xmath62 s ; otherwise it is _ we let @xmath76 ( respectively @xmath77 ) denote the number of homogeneous ( respectively heterogeneous ) @xmath78critical transpositions in @xmath51 , so that @xmath79 let us call a transposition an @xmath80_transposition _ if it involves one @xmath61 and one @xmath60 . we similarly define @xmath81 and @xmath82transpositions . thus , each heterogeneous transposition is either an @xmath80 or an @xmath81 or a @xmath82transposition . since in @xmath51 each @xmath80transposition moves the involved @xmath61 to the right and the involved @xmath60 to the left , then ( a ) for each @xmath83 , there are _ exactly _ @xmath45 @xmath45critical @xmath80 transpositions ; and ( b ) for each @xmath45 , @xmath84 , there are _ exactly _ @xmath85 @xmath45critical @xmath80transpositions . since the same holds for @xmath81 and @xmath82transpositions , it follows that for each @xmath83 , there are _ exactly _ @xmath86 @xmath45critical heterogeneous transpositions , and for each @xmath45 , @xmath87 , exactly @xmath88 @xmath45critical heterogeneous transpositions . # 1 # 1 therefore , for each @xmath89 , there are exactly @xmath90 @xmath2critical transpositions , and if @xmath91 , then there are exactly @xmath92 @xmath2critical transpositions . we now summarize these results . [ pro : heterogeneous ] let @xmath51 be a @xmath1decomposable @xmath3halfperiod , and let @xmath32 . then @xmath93 3\binom{n/3 + 1}{2 } + ( k - n/3)n & \hbox{\hglue 0.3 cm } \text{if $ n/3 < k < n/2 $ , } \end{cases}\ ] ] our goal here is to give a lower bound ( see proposition [ pro : homogeneous ] ) for the number @xmath76 of homogeneous @xmath2critical transpositions in a @xmath1decomposable @xmath3halfperiod @xmath51 . our approach is to find an _ upper _ bound for @xmath94 , which will denote the number of @xmath95transpositions that are _ not _ @xmath2critical ( @xmath96 and @xmath97 are defined analogously ) . since the total number of @xmath95transpositions is @xmath98 , this will yield a lower bound for the contribution of @xmath95transpositions ( and , by symmetry , for the contribution of @xmath99transpositions and of @xmath100transpositions ) to @xmath76 . [ rem : couldbe0 ] for every @xmath89 , it is a trivial task to construct @xmath3halfperiods @xmath51 for which @xmath101 . in view of this , we concentrate our efforts on the case @xmath102 . a transposition between elements in positions @xmath45 and @xmath66 , with @xmath103 , is _ valid_. thus our goal is to ( upper ) bound the number of valid @xmath95transpositions . let @xmath104 be the digraph with vertex set @xmath105 , and such that there is a directed edge from @xmath106 to @xmath107 if and only if @xmath108 and the transposition that swaps @xmath106 and @xmath107 is valid . for @xmath109 , we let @xmath110 ( respectively @xmath111 ) denote the outdegree ( respectively indegree ) of @xmath107 in @xmath104 . we define @xmath112 and @xmath113 analogously . the inclusion of the symbol @xmath51 in @xmath114 , etc . , is meant to emphasize the dependence on the specific @xmath3halfperiod @xmath51 . for brevity we will omit the reference to @xmath51 and simply write @xmath115 , and so on . no confusion will arise from this practice . the importance of @xmath116 , and @xmath117 is clear from the following observation . [ rem : digraph ] for each @xmath3halfperiod @xmath51 , the number of edges of @xmath118 _ equals _ @xmath94 . indeed , to each valid @xmath95transposition , that is , each transposition that contributes to @xmath94 , there corresponds a unique edge in @xmath118 . analogous observations hold for @xmath119 and @xmath117 . in view of remark [ rem : digraph ] , we direct our efforts to bounding the number of edges in @xmath118 . the essential observation to get this bound is the following : @xmath120 to see this , simply note that , @xmath121 , since @xmath122 is clearly the maximum possible number of valid moves in which @xmath107 moves right , and trivially @xmath123 , since there are only @xmath124 @xmath106 s with @xmath108 . [ pro : boundingnoedges ] if @xmath51 is a @xmath1decomposable @xmath3halfperiod , and @xmath91 , then @xmath118 has at most @xmath125edges . _ _ let @xmath126 denote the class of all digraphs with vertex set @xmath105 , with every directed edge @xmath127 satisfying @xmath108 and @xmath128 . we argue that any graph in @xmath126 has at most @xmath129 @xmath130 edges . this clearly finishes the proof , since @xmath131 . to achieve this , we note that it follows from the work in section 2 in @xcite that the maximum number of edges of such a digraph is attained in the digraph @xmath132 recursively constructed as follows . first define that all the directed edges arriving at @xmath133 are the edges @xmath134 for @xmath135 . now , for @xmath136 , once all the directed edges arriving at @xmath137 have been determined , fix that ( all ) the directed edges arriving at @xmath107 are @xmath127 , for all those @xmath138 that satisfy @xmath139 . since no digraph in @xmath126 has more edges than @xmath132 , to finish the proof it suffices to bound the number of edges of @xmath132 . this is the content of claim [ cla : theclaim ] below . [ cla : theclaim ] @xmath132 has at most @xmath140 @xmath141 edges . _ sketch of proof . _ since we know the exact indegree of each vertex in @xmath132 , we know the exact number of edges of @xmath132 , and so the proof of claim [ cla : theclaim ] is no more than a straightforward , but quite long and tedious , calculation . [ cor : alsobandc ] if @xmath51 is a @xmath1decomposable @xmath3halfperiod , and @xmath91 , then each of @xmath119 and @xmath117 has at most @xmath142edges . _ _ in the proof of proposition [ pro : boundingnoedges ] , the only relevant property about @xmath118 is that the a s form a set of @xmath143 points that in some permutation of @xmath51 ( namely @xmath54 ) appear all consecutively and at the beginning of the permutation . since @xmath51 is @xmath1decomposable , this condition is also satisfied by the set of b s and by the set of c s . we now summarize the results in the current subsection . [ pro : homogeneous ] if @xmath51 is a @xmath1decomposable @xmath3halfperiod , and @xmath91 , then @xmath144 _ proof . _ by remark [ rem : digraph ] , the number @xmath94 of @xmath95transpositions that are _ not _ @xmath2critical equals the number of edges in @xmath118 , which by proposition [ pro : boundingnoedges ] is at most @xmath145 . since the total number of @xmath95transpositions is @xmath98 , then the number of @xmath95transpositions that contribute to @xmath70 is at least @xmath146 @xmath147 . by corollary [ cor : alsobandc ] , @xmath99 and @xmath100transpositions contribute in at least the same amount to @xmath76 , and so the claimed inequality follows . proposition [ prop : main ] follows immediately from eq . ( [ eq : suma ] ) and propositions [ pro : heterogeneous ] and [ pro : homogeneous ] . all the lower bounds proved above remain true for point sets that satisfy conditions ( i ) and ( ii ) ( and not necessarily condition ( iii ) ) for @xmath1decomposability . 99 b.m . brego and s. fernndez merchant , a lower bound for the rectilinear crossing number , _ graphs and comb . _ , * 21 * ( 2005 ) , 293300 . brego , j. balogh , s. fernndez merchant , j. leaos , and g. salazar , an extended lower bound on the number of @xmath2-edges to generalized configurations of points and the pseudolinear crossing number of @xmath0 . submitted ( 2007 ) . o. aichholzer , j. garca , d. orden , and p. ramos , new lower bounds for the number of @xmath148-edges and the rectilinear crossing number of @xmath0 , _ discr . _ , to appear . o. aichholzer . on the rectilinear crossing number available online at http://www.ist.tugraz.at/ staff / aichholzer / crossings.html . j. balogh and g. salazar , @xmath149sets , convex quadrilaterals , and the rectilinear crossing number of @xmath0 , _ discr . * 35 * ( 2006 ) , 671690 . p. brass , w.o.j . moser , and j. pach , _ research problems in discrete geometry . _ springer , new york ( 2005 ) . j. e. goodman and r. pollack , on the combinatorial classification of nondegenerate configurations in the plane , _ j. combin . theory ser . a _ * 29 * ( 1980 ) , 220235 . l. lovsz , k. vesztergombi , u. wagner , and e. welzl , convex quadrilaterals and @xmath149sets . _ towards a theory of geometric graphs _ , ( j. pach , ed . ) , contemporary math . , ams , 139148 ( 2004 ) . * appendix : proof of claim [ cla : theclaim ] * since @xmath132 is a well defined digraph , and we know the exact indegree of each of its vertices , claim [ cla : theclaim ] is no more than long and tedious , yet elementary , calculation . the purpose of this appendix is to give the full details of this calculation . we prove claim [ cla : theclaim ] in two steps . first we obtain an expression for the _ exact _ value of the number of edges of @xmath132 , and then we show that this exact value is upper bounded by the expression in claim [ cla : theclaim ] . the exact number of edges in @xmath132 is a function of the following parameters . let @xmath150 be positive integers with @xmath151 . then : * @xmath152 is the ( unique ) nonnegative integer such that @xmath153 ; and * @xmath154 and @xmath155 are the ( unique ) integers satisfying @xmath156 and such that @xmath157 for brevity , in the rest of the section we let @xmath158 and @xmath159 . the key observation is that we know the indegree of each vertex in @xmath132 : [ ingrade ] for each integer @xmath160 , and each vertex @xmath161 of @xmath132 , @xmath162 . the number of edges of @xmath132 equals the sum of the indegrees over all vertices in @xmath132 . thus our main task is to find a closed expression for the sum @xmath163 . this is the content of our next statement . [ numberofedges ] the number @xmath164 of edges of @xmath132 is @xmath165 _ proof . _ we break the index set of the summation @xmath163 into three parts , in terms of @xmath166 and @xmath167 . we let @xmath168 , @xmath169 , and @xmath170 so that @xmath171 we calculate each of @xmath16 , @xmath17 , and @xmath13 separately . _ calculating @xmath16 _ if @xmath172 are integers such that @xmath173 and @xmath174 , we define @xmath175 and @xmath176 note that @xmath177is a partition of @xmath178 and that for each @xmath179 is a partition of @xmath180 note that @xmath16 can be rewritten as @xmath181 by proposition [ ingrade ] this equals @xmath182 . that is , @xmath183 since @xmath184 for all @xmath45 , and @xmath185 @xmath186 is a partition of @xmath187 , then @xmath188 thus , @xmath189 on other hand , for @xmath190 and @xmath191 , it is not difficult to verify that @xmath192 @xmath193 . this implies that @xmath194 by definition of @xmath187 we have @xmath195 by definition of @xmath196 we have @xmath197 substituting ( [ s_j ] ) and ( [ t_jl ] ) into ( [ a ] ) we obtain @xmath198 & = 2m^{2}\tbinom{\theb(m,\enetercios)+1}{3}+ \tbinom{\theb(m,\enetercios)+1}{2}\tbinom{m}{2}. \label{eq : fora}\end{aligned}\ ] ] _ calculating @xmath17 _ since @xmath199 for each @xmath200 , and @xmath201 @xmath202 , then @xmath203 therefore @xmath204 on other hand it is easy to check that @xmath205 for every @xmath149 such that @xmath206 since @xmath207 for every @xmath45 such that @xmath208 then @xmath209 @xmath210 is a partition of @xmath211 thus , @xmath212 we note that @xmath213 using this fact in ( [ b ] ) we obtain @xmath214 thus , @xmath215 _ calculating @xmath13 _ since @xmath199 ; @xmath216 for each @xmath45 such that @xmath217 ; and @xmath201 @xmath202 , it follows that @xmath218 from ( [ eq : a1 ] ) it follows that @xmath219 , and so @xmath220 now from ( [ eq : fora ] ) , ( [ eq : forb ] ) , and ( [ eq : forc ] ) , it follows that @xmath221 , and so proposition [ numberofedges ] follows from ( [ eq : decomp ] ) . first we bound the number of @xmath2edges in @xmath1decomposable @xmath3halfperiods in terms of the expression @xmath222 in proposition [ numberofedges ] . [ exact ] let @xmath51 be a @xmath1decomposable @xmath3halfperiod , and let @xmath223 . then @xmath224 3\binom{n/3 + 1}{2 } + ( k - n/3)n+3 \biggl ( \binom{n/3}{2 } - e(k , n)\biggr ) & \hbox{\hglue 0.3 cm } \text{if $ n/3 < k < n/2$. } \\[0.4 cm ] \end{cases}\ ] ] _ proof . _ obviously , @xmath225 and so the case @xmath89 follows from proposition [ pro : heterogeneous ] . now suppose that @xmath226 . recall that @xmath227 . now the total number of @xmath95 ( and @xmath99 , and @xmath100 ) transpositions is exactly @xmath98 , and so @xmath228 . thus it follows from remark [ rem : digraph ] and proposition [ numberofedges ] that @xmath229 . this fact , together with proposition [ pro : heterogeneous ] , implies that @xmath230 , as claimed .

the point sets of all known optimal rectilinear drawings of @xmath0 share an unmistakeable clustering property , the so called _ 3decomposability_. it is widely believed that the underlying point sets of all optimal rectilinear drawings of @xmath0 are @xmath1decomposable . we give a lower bound for the minimum number of @xmath2sets in a @xmath1decomposable @xmath3point set . as an immediate corollary , we obtain a lower bound for the crossing number @xmath4 of any rectilinear drawing @xmath5 of @xmath0 with underlying @xmath1decomposable point set , namely @xmath6 . this closes this gap between the best known lower and upper bounds for the rectilinear crossing number @xmath7 of @xmath0 by over 40% , under the assumption of @xmath1decomposability .

You are an expert at summarizing long articles. Proceed to summarize the following text: the spectral problem for possible sums of two random hermitian matrices with given spectra had been a long - standing problem , formulated as horn s conjecture , until it was completely solved only recently by klyachko , knutson and tao ( @xcite , @xcite ) . in @xcite , klyachko then observed a close connection between @xmath0-biinvariant random walks on @xmath1 and random walks on the additive group @xmath2 of all hermitian @xmath3-matrices with trace 0 whose transition probabilities are conjugation - invariant under @xmath0 . he used this connection to reduce the description of the possible singular spectra of products of random matrices from @xmath1 with given singular spectra to the spectral problem for sums of hermitian matrices . basically , klyachko s observation is a connection between the convolution algebras of the gelfand pairs @xmath5 and @xmath6 . it is closely related to a similar correspondence between the convolutions of conjugation - invariant measures on a compact lie group ( here @xmath0 ) on one hand and @xmath7-invariant measures on its lie algebra ( here @xmath8 ) on the other , in terms of the so - called wrapping map ( see @xcite and remark [ wrapping ] ) . klyachko noticed in @xcite , but did not not explain that his connection can be well expressed in terms of hypergroups ; his proof goes via random walks in the group @xmath0 and relies on various identities between the spherical functions of @xmath5 , the characters of @xmath0 and the euclidean group @xmath2 , as well as poisson s summation formula . the main purpose of the present paper is to clarify and simplify klyachko s approach @xcite by using so - called deformations of hypergroup convolutions by positive semicharacters as introduced in @xcite , @xcite . our description in particular implies that the banach algebra @xmath9 of @xmath0-biinvariant signed measures on @xmath1 with total variation norm is isometrically isomorphic to a certain banach subalgebra of @xmath10 , whereby probability measures are being prserved . we finally show how this leads to new proofs for ( known ) limit theorems for @xmath0-biinvariant random walks on @xmath1 . the explicit construction of this isomorphism runs as follows : 1 . @xmath0 acts on @xmath2 by conjugation as a group of orthogonal transformations . the space @xmath11 of all orbits can be identified with @xmath12 where @xmath13 represents the ordered eigenvalues of a matrix in @xmath2 . the banach algebra @xmath14 of all @xmath0-invariant bounded measures on @xmath2 can be identified with the banach algebra @xmath15 of the associated orbit hypergroup @xmath16 . the polar decomposition of @xmath17 shows that the double coset space @xmath18 may also be identified with @xmath19 where now for @xmath20 corresponds to the singular spectrum @xmath21 of some @xmath22 . this leads to a canonical banach algebra isomorphism between @xmath9 and the banach algebra @xmath23 of the double coset hypergroup @xmath24 . the characters of the hypergroups @xmath25 and @xmath26 are spherical functions of the corresponding symmetric spaces , and it is well - known that as functions on @xmath27 they only differ by a known factor @xmath28 ( see @xcite ) . we prove that such a connection between the characters of two hypergroup structures on @xmath19 implies that their convolutions are related by a so - called hypergroup deformation . in particular , the supports of the convolution products of two point measures are the same in both hypergroups . this immediately implies the equivalence of the two spectral problems for hermitian and unitary matrices , the main result of klyachko @xcite . the deformation isomorphism , however , is not isometric and not probability preserving . to achieve this , a final correction is needed : 4 . instead of looking at the banach algebra @xmath14 of all @xmath0-invariant measures on @xmath2 , we take a suitable exponential function @xmath290,\infty[$ ] and observe that the norm - closure of @xmath30 is a banach algebra which turns out to be isometrically isomorphic to @xmath23 . putting all steps together one obtains the claimed probability preserving isometric isomorphism . for @xmath31 this construction reflects the known close connection between the so - called naimark hypergroup @xmath32 and the bessel - kingman hypergroup @xmath33 ; see @xcite , @xcite . all results will be derived in the general setting of a complex connected semisimple lie group @xmath34 with finite center and maximal compact subgroup @xmath35 ; in the special situation above , @xmath36 . for further results concerning the associated hypergroup convolutions considered in this paper we refer to @xcite and @xcite . the paper is organized as follows : in section 2 we collect some relevant facts on commutative hypergroups and their deformations . in section 3 we then use this deformation to show how the banach algebras of all @xmath35-biinvariant bounded complex measures on @xmath34 appear as subalgebras of the banach algebra of bounded complex measures on some euclidean space . the final section is devoted to probabilistic applications . in this section we give a quick introduction to hypergroups . we in particular prove a general result on deformations of hypergroup convolutions in proposition [ deform ] which is crucial for step ( 3 ) of our construction . moreover , step ( 4 ) will be explained . we first fix some notations . for a locally compact hausdorff space @xmath37 , @xmath38 denotes the space of all positive radon measures on @xmath37 , and @xmath39 the banach space of all bounded regular complex borel measures with the total variation norm . moreover , @xmath40 is the set of all probability measures , @xmath41 the set of all measures with compact support , and @xmath42 the point measure in @xmath43 . the spaces @xmath44 of continuous functions are defined as usual . for details on the following we refer to @xcite and @xcite . a hypergroup @xmath45 consists of a locally compact hausdorff space @xmath35 and a convolution @xmath46 on @xmath39 such that @xmath47 becomes a banach algebra , where @xmath46 is weakly continuous and probability preserving and preserves compact supports of measures . moreover , there exists an identity @xmath48 with @xmath49 for @xmath43 , as well as a continuous involution @xmath50 on @xmath37 such that for @xmath51 , @xmath52 is equivalent to @xmath53 , and @xmath54 . here for @xmath55 , the measure @xmath56 is given by @xmath57 for borel sets @xmath58 . a hypergroup @xmath45 is called commutative if and only if so is the convolution @xmath46 . hence , for a commutative hypergroup @xmath45 the triple @xmath59 is a commutative banach-@xmath46-algebra with identity @xmath60 . [ ex_1 ] 1 . if @xmath34 is a locally compact group , then @xmath61 is a hypergroup with the usual group convolution @xmath46 . 2 . let @xmath35 be a compact subgroup of a locally compact group @xmath34 . then @xmath62 is a banach-@xmath46-subalgebra of @xmath63 with identity @xmath64 where @xmath65 is the normalized haar measure of @xmath35 embedded into @xmath34 . moreover , the double coset space @xmath66 is a locally compact hausdorff space , and the canonical projection @xmath67 induces a probability preserving , isometric isomorphism @xmath68 of banach spaces by taking images of measures . the transport of the convolution on @xmath69 to @xmath70 via @xmath71 leads to a hypergroup structure @xmath72 with identity @xmath73 and involution @xmath74 , and @xmath71 becomes a probability preserving , isometric isomorphism of banach-@xmath46-algebras . 3 . let @xmath75 be a finite - dimensional euclidean vector space and @xmath76 a compact subgroup of the orthogonal group of @xmath77 acting continuously on @xmath78 . for @xmath79 , denote the image measure of @xmath80 under @xmath81 by @xmath82 then the space of @xmath35-invariant measures @xmath83 is a banach-@xmath46-subalgebra of @xmath84 ( with the group convolution ) , with identity @xmath85 . moreover , the space @xmath86 of all @xmath35-orbits in @xmath78 is locally compact , and the canonical projection @xmath87 induces a probability preserving , isometric isomorphism @xmath88 of banach spaces and an associated so - called orbit hypergroup @xmath89 such that @xmath71 becomes a probability preserving , isometric isomorphism of banach-@xmath46-algebras . the involution on @xmath89 is given by @xmath90 . we next collect some data of a commutative hypergroup @xmath45 . by a result of r. spector , there exists a ( up to normalization ) unique haar measure @xmath91 which is characterized by @xmath92 for all @xmath93 and @xmath94 where we use the notation @xmath95 similar to the dual of a locally compact abelian group , one defines the spaces 1 . @xmath96 ; 2 . @xmath97 @xmath98 the elements of @xmath99 and @xmath100 are called semicharacters and characters , respectively . all spaces above are locally compact hausdorff spaces w.r.t . the topology of compact - uniform convergence . the fourier transform on @xmath101 is defined by @xmath102 similar , the fourier - stieltjes transform of @xmath55 is defined by @xmath103 @xmath104 . both transforms are injective , c.f . @xcite . in the following , we consider different hypergroup convolutions on @xmath37 , and we write @xmath105 etc . in order to specify the relevant convolution . [ ex_2 ] 1 . assume that in the situation of [ ex_1](ii ) , @xmath106 is commutative . then a @xmath35-biinvariant function @xmath107 with @xmath108 is by definition a spherical function of @xmath109 if @xmath110 multiplicative functions @xmath111 are in one - to - one correspondence with spherical functions on @xmath34 via @xmath112 for the projection @xmath67 . in this way , the fourier(-stieltjes ) transform on @xmath106 corresponds to the spherical fourier(-stieltjes ) transform . 2 . in the situation of example [ ex_1](iii ) , the functions @xmath113 are continuous multiplicative functions of the orbit hypergroup @xmath114 for @xmath115 , the complexification of @xmath78 , where @xmath116 holds if and only if @xmath117 . it is also well - known ( see @xcite ) that @xmath118 . in @xcite , positive semicharacters were used to construct deformed hypergroup convolutions . more precisely , the following was proven there : let @xmath119 be a positive semicharacter on the commutative hypergroup @xmath45 , i.e. , @xmath120 for @xmath43 . then the convolution @xmath121 extends uniquely to a bilinear , associative , probability preserving , and weakly continuous convolution @xmath122 on @xmath39 . moreover , @xmath123 becomes a commutative hypergroup with the identity and involution of @xmath45 . for @xmath124 , one has @xmath125 note that by eq.([deformconvo ] ) , the mapping @xmath126 establishes a canonical algebra isomorphism between @xmath127 and @xmath128 which usually when @xmath129 is unbounded can not be extended to @xmath39 . the hypergroup @xmath123 is called the deformation of @xmath45 w.r.t . @xmath129 . clearly , many data of @xmath123 can be expressed in terms of @xmath129 and corresponding data of @xmath45 . in the above setting , we have=-1pt 1 . @xmath130 is a haar measure of @xmath123 . the mapping @xmath131 is a homeomorphism ( w.r.t . the compact - uniform topology ) between @xmath132 and @xmath133 , and also between @xmath134 and @xmath135 . for ( i ) and the first part of ( ii ) see @xcite ; the second part of ( ii ) is analogous . we next turn to the following converse statement ; it will be crucial for this paper : [ deform ] let @xmath45 and @xmath123 be commutative hypergroups on @xmath37 . assume there is a positive semicharacter @xmath129 of @xmath45 such that the spaces of multiplicative continuous functions for @xmath45 and @xmath123 are related via @xmath136 then @xmath123 is the deformation of @xmath45 w.r.t . @xmath129 . let @xmath137 denote the deformation of @xmath45 via @xmath129 . . then by our assumption and the proposition above , @xmath139 is multiplicative w.r.t . @xmath140 as well , and the fourier - stieltjes transforms of @xmath141 and @xmath142 w.r.t . @xmath123 satisfy @xmath143 by the injectivity of the fourier - stieltjes - transform on @xmath39 , we obtain @xmath144 . thus the convolutions of @xmath123 and @xmath137 coincide , and so do the involutions , because they are uniquely determined by the convolutions . it is well - known ( see @xcite,@xcite,@xcite,@xcite ) that the double coset hypergroup @xmath32 may be realized as hypergroup @xmath145 with multiplicative functions @xmath146 via the correspondence of @xmath147 with @xmath148 , we have @xmath149 . on the other hand , the orbit hypergroup @xmath150 may be realized as the bessel - kingman hypergroup @xmath151 with multiplicative functions @xmath152 see @xcite , @xcite . clearly , @xmath153 is a positive semicharacter on @xmath45 abd @xmath154 for @xmath155 . proposition [ deform ] implies the known fact that @xmath123 is a deformation of @xmath45 , c.f . @xcite . we next consider further examples which explain step ( 4 ) in the introduction ; they are similar to a construction in @xcite . let @xmath156 be a euclidean vector space of finite dimension @xmath157 , @xmath76 a compact subgroup of the orthogonal group of @xmath78 , and @xmath89 the associated orbit hypergroup . fix @xmath158 with @xmath159 , and consider the exponential @xmath160 on @xmath78 . let further @xmath161 the multiplicativity of @xmath162 yields that @xmath163 , where @xmath46 denotes the group convolution on @xmath78 . hence @xmath164 is a subalgebra of the banach-@xmath46-algebra @xmath84 , and its norm - closure @xmath165 is a banach subalgebra of @xmath84 . notice that for @xmath166 this algebra is not closed under the involution on @xmath84 ; for instance , the @xmath157-dimensional normal distribution @xmath167 with density @xmath168 is contained in @xmath169 while this is not the case for @xmath170 . nevertheless , we prove that @xmath169 is isometrically isomorphic as a banach algebra to the banach algebra of measures of a suitable deformation of the orbit hypergroup @xmath89 . more precisely : [ euclidean - deform ] let @xmath158 with @xmath159 and define @xmath171 then the following hold : = -1pt 1 . @xmath129 is a positive semicharacter on @xmath172 2 . if @xmath173 is the deformation of @xmath89 w.r.t . @xmath129 , then the canonical projection @xmath174 induces ( by taking image measures ) a probability preserving isometric isomorphism of banach algebras from @xmath169 onto @xmath175 . 1 . @xmath129 is obviously continuous and positive . for @xmath176 , we further have @xmath177 moreover , by the condition on @xmath178 above and the properties of the haar measure of @xmath35 , @xmath179 therefore , @xmath129 is a positive semicharacter on @xmath89 . 2 . consider the diagram + m_c^0,k(v ) & ^ e _ & m_c^,k(v ) + & & + ( m_c(v^k ) , * ) & ^ _ 0 & ( m_c(v^k ) , ) + where @xmath180 always stands for a space of measures with compact support , and the vertical mappings are obtained by taking image measures w.r.t . the mapping @xmath71 restricted to @xmath181 and @xmath164 is probability preserving and isometric . moreover , for @xmath182 and @xmath183 we obtain by the @xmath35-invariance of @xmath80 and the definition of @xmath129 , @xmath184 + which proves that the diagram commutes . therefore , as both horizontal mappings and the left vertical mapping are algebra isomorphisms , the right vertical mapping is also a probability preserving isometric isomorphism of banach algebras from @xmath185 onto @xmath186 . a continuity and density argument shows that the banach algebras @xmath169 and @xmath175 are isomorphic via the probability preserving mapping @xmath71 . we here identify the banach algebra of all bounded borel measures on a connected semisimple noncompact lie group with finite center , which are biinvariant under a maximal compact subgroup , as a banach algebra of measures in some euclidean setting . for the general background we refer to @xcite . let @xmath34 be a complex , noncompact connected semisimple lie group with finite center and @xmath35 a maximal compact subgroup of @xmath34 . consider the corresponding cartan decomposition @xmath187 of the lie algebra of @xmath34 , and choose a maximal abelian subalgebra @xmath188 . @xmath35 acts on @xmath189 via the adjoint representation as a group of orthogonal transformations with respect to the killing form as scalar product . let further @xmath190 be the weyl group of @xmath35 , which acts on @xmath191 as a finite reflection group , with root system @xmath192 . here and lateron , @xmath191 is always identified with its dual @xmath193 via the killing form , which we denote by @xmath194 . we fix some weyl chamber @xmath195 in @xmath191 and denote the associated system of positive roots by @xmath196 . the closed chamber @xmath197 is a fundamental domain for the action of @xmath190 on @xmath191 . later on we shall need the half sum of roots , @xmath198 we now identify @xmath19 with the orbit hypergroup @xmath199 where each @xmath35-orbit in @xmath200 corresponds to its unique representative in @xmath201 . then in view of example [ ex_2](ii ) above and prop . iv.4.8 of @xcite , the multiplicative continuous functions of @xmath26 , considered as @xmath35-invariant functions on @xmath189 , are given by @xmath202 where @xmath148 runs through the complexification @xmath203 of @xmath191 . moreover , @xmath204 iff @xmath148 and @xmath80 are in the same @xmath190-orbit . this is a special case of harish - chandra s integral formula for the spherical functions of a cartan motion group . according to theorem ii.5.35 and cor . ii.5.36 of @xcite , they can also be written as @xmath205 with the fundamental alternating polynomial @xmath206 on the other hand , @xmath19 can be identified with @xmath106 where @xmath207 corresponds to the double coset @xmath208 . according to this identification and the explicit formula for the spherical functions in theorem iv.5.7 of @xcite , the multiplicative continuous functions on the commutative double coset hypergroup @xmath209 are ( as functions on @xmath191 ) given by @xmath210 with @xmath211 . thus in particular , @xmath212 notice that @xmath213 is a positive semicharacter of @xmath26 . by weyl s formula ( @xcite , prop . i.5.15 . ) , @xmath214 the square of @xmath213 ( called @xmath215 in @xcite ) determines the ratio of the volume elements in @xmath189 and @xmath216 . proposition [ deform ] and eq . ( [ zshg ] ) show that @xmath209 is the deformation of the orbit hypergroup @xmath217 via @xmath213 . moreover , we have @xmath218 with the half sum @xmath219 and @xmath220 . as the condition @xmath159 is satisfied , proposition [ euclidean - deform ] further implies that the banach algebra of measures of @xmath209 can be identified with @xmath221 , which is the closure of @xmath222 the claimed isomorphism between @xmath221 and @xmath223 is now given by taking image measures w.r.t . the canonical projection @xmath224 . [ wrapping ] the algebra isomorphism @xmath225 with the semicharacter @xmath226 is closely related to the so - called wrapping map for the compact lie group @xmath35 , see @xcite . in fact , as @xmath34 is complex , we have @xmath227 in the cartan decomposition @xmath228 . thus @xmath229 ( on which @xmath35 acts by conjugation ) is the dual symmetric space of @xmath216 . the wrapping map @xmath230 is defined by @xmath231 where @xmath232 and @xmath233 is the @xmath35-invariant extension of @xmath234 notice that @xmath235 . as shown in @xcite , @xmath236 is an algebra homomorphism from @xmath35-invariant measures in @xmath237 to conjugation - invariant measures in @xmath238 . the proof thereof is based on weyl s integration formula and kirillov s character formula for compact groups . in contrast to our situation , @xmath236 does not preserve positivity and can therefore not be associated with a hypergroup deformation . the group @xmath239 is a maximal compact subgroup of the connected semisimple lie group @xmath240 . in the cartan decomposition @xmath241 we obtain @xmath189 as the additive group @xmath2 of all hermitian @xmath3-matrices with trace @xmath4 , on which @xmath0 acts by conjugation . moreover , @xmath191 consists of all real diagonal matrices with trace 0 and will be identified with @xmath242 on which the weyl group acts as the symmetric group @xmath243 as usual . we thus may take @xmath244 this set parametrizes the possible spectra of matrices from @xmath2 . a system of positive roots corresponding to @xmath19 is @xmath245 where @xmath246 denotes the standard basis of @xmath247 . the root system is of type @xmath248 . in order to describe the probability preserving isometric isomorphism for this example explicitly , we realize the canonical projections @xmath249 and @xmath250 as follows : for @xmath251 , define @xmath252 as the tuple of eigenvalues of @xmath253 , ordered by size . for @xmath254 define @xmath255 as the element @xmath207 such that the singular spectrum @xmath256 of @xmath257 is @xmath258 . we have @xmath259 which implies that the banach algebra @xmath221 above may be described as the closure of @xmath260 with @xmath261\bigr).\ ] ] as discussed above , the mappings @xmath262 induce ( by taking images of measures ) probability preserving isometric isomorphisms from the banach algebras @xmath221 and @xmath263 onto @xmath264 for the double coset convolution @xmath122 on @xmath265 . let us finally come back to the two spectral problems studied by klyachko @xcite . the hypergroup deformation between the two convolutions @xmath46 and @xmath122 on @xmath19 gives the natural explanation for the close connection of these problems : first , for fixed @xmath266 the probability measure @xmath267 is the distribution of possible spectra of sums @xmath268 where the @xmath269 run through all matrices from @xmath270 with @xmath271 ( @xmath272 ) . on the other hand , @xmath273 describes the distribution of possible singular spectra @xmath274 of products @xmath275 where the @xmath276 run through all matrices from @xmath277 with given singular spectra @xmath278 . as @xmath279 we obtain : ( c.f . @xcite , theorem b ) for elements @xmath280 the following are equivalent:=-2pt 1 . there exist matrices @xmath281 with given spectra @xmath282 such that @xmath283 . 2 . there exist matrices @xmath284 with given singular spectra @xmath285 such that @xmath286 . we now briefly discuss the further classical series of complex , noncompact connected simple lie groups with finite center ( c.f . appendix c in @xcite . ) 1 . * the @xmath287-case . * for @xmath288 consider @xmath289 with the maximal compact subgroup @xmath290 . in this case @xmath191 may be identified with @xmath291 with standard basis @xmath292 , and we may choose @xmath293 and @xmath294 . the weyl group @xmath190 is isomorphic to the semidirect product @xmath295 , and @xmath296 2 . * the @xmath297-case . * for @xmath298 consider @xmath299 with the maximal compact subgroup @xmath300 . in this case , again @xmath301 with @xmath19 and @xmath190 as in the @xmath287-case . a positive root system is @xmath302 and @xmath303 . comparing this with the @xmath287-case , we see from eq . ( [ euclidean ] ) that the spherical functions @xmath304 of the orbit hypergroups @xmath305 are the same in both cases . the preceding results on hypergroup deformations therefore imply that the double coset hypergroups @xmath306 and @xmath307 are deformations of each other w.r.t . certain positive semicharacters . * the @xmath308-case . * for @xmath309 consider @xmath310 with the maximal compact subgroup @xmath311 . in this case @xmath301 with @xmath312 and @xmath313 . in this final section we use our previous results to translate limit theorems for random walks on @xmath189 into corresponding results for @xmath35-biinvariant random walks on @xmath34 . we first sketch the method . [ method ] let @xmath315 be a sequence on independent @xmath34-valued random variables with distributions @xmath316 ( @xmath317 ) . then @xmath318 with @xmath319 forms a @xmath35-biinvariant random walk on @xmath34 . using the canonical projection @xmath320 and the associated isomorphism @xmath321 of banach algebras , we see that @xmath322 is a markov process on @xmath19 with initial distribution @xmath85 and transition probabilities @xmath323 for @xmath324 , @xmath207 , and borel sets @xmath325 . this means that @xmath326 is a random walk on the hypergroup @xmath25 with transition probabilities @xmath327 . clearly , @xmath328 has distribution @xmath329 on the other hand , when considering the canonical projection @xmath330 , we find unique probability measures @xmath331 with @xmath332 for @xmath333 . if @xmath334 is a sequence on independent @xmath189-valued random variables with distributions @xmath335 , we get a random walk @xmath336 on @xmath189 whose projection @xmath337 is a random walk on @xmath25 with the same transition probabilities as @xmath338 , i.e. , @xmath337 and @xmath338 have the same finite dimensional distributions and therefore admit the same limit theorems . hence , limit theorems for the random walk @xmath339 on @xmath189 may be translated into limit theorems for @xmath338 . here is an example . assume that the @xmath340 are independent of @xmath157 , and that for the associated @xmath341 on @xmath189 the first moment vector @xmath342 exists . then by kolmogorov s strong law , @xmath343 almost surely as @xmath344 . in other words , @xmath345 a.s .. as @xmath330 is homogeneous of degree 1 and contractive ( see below ) , we obtain @xmath346 and thus @xmath347 a.s .. therefore , @xmath348 a.s . for @xmath344 . this strong law for @xmath338 has the computational drawback that @xmath349 is described in terms of @xmath350 . to overcome this , we introduce a suitable modified moment function @xmath351 such that for all probability measures @xmath350 having first moments , @xmath352 we shall show that @xmath353 is determined uniquely by ( [ momentenforderung ] ) and give an explicit formula . this allows to compute the limiting constant @xmath349 directly on @xmath19 . we next define @xmath353 . motivated by ( [ momentenforderung ] ) for @xmath359 with @xmath360 for @xmath207 , we put @xmath361 notice here that @xmath362 . the function @xmath353 is obviously @xmath35-invariant , continuous , and satisfies @xmath363 for @xmath364 with respect to the norm induced by the killing form . we first check that @xmath368 for @xmath366 . the definition of @xmath353 and the harish - chandra formula ( [ har ] ) show that for @xmath369 , @xmath370 where @xmath371 is the derivative in direction @xmath372 w.r.t . the open weyl chamber @xmath373 corresponding to @xmath19 is an orthogonal transversal manifold for the adjoint action of @xmath35 on @xmath189 ( @xcite , ch.ii , 3.4.(vi ) ) . therefore the orthogonal complement @xmath374 of @xmath191 in @xmath189 coincides with the tangent space of the orbit @xmath375 in @xmath178 . as @xmath376 is constant on @xmath35-orbits , @xmath377 for @xmath378 . hence @xmath379 . in order to check @xmath380 , we recall from ch . 3 of @xcite that @xmath381 on the other hand , an elementary rearrangement inequality ( theorem 368 of @xcite ) shows that for @xmath382 and @xmath383 , @xmath384 as @xmath385 , we obtain for @xmath383 that @xmath386 eq . ( [ charc ] ) now shows that @xmath380 . this completes the proof of part ( i ) . part ( ii ) follows from eqs . ( [ ableitungm1 ] ) and ( [ euclidean ] ) for the spherical functions @xmath304 on @xmath191 . [ moments ] let @xmath350 be a probability measure and @xmath387 . we say that @xmath341 admits @xmath388-th moments if @xmath389 , or equivalently , if @xmath390 for all @xmath391 . this condition can be translated into a corresponding condition for @xmath392 . in fact , as @xmath35 acts on @xmath189 as a group of orthogonal transformations , @xmath393.\ ] ] therefore , @xmath350 admits @xmath388-th moments if and only if @xmath394 admits @xmath388-th moments . proposition [ m1inc ] and the estimate @xmath395 for @xmath364 show that that this condition for @xmath396 implies that the modified moment vector @xmath397 exists . moreover , as @xmath398 we obtain from propos . [ m1inc ] that @xmath399 and hence , as claimed , @xmath400 [ strong - law ] let @xmath4010,2[$ ] and @xmath402 such that @xmath403 admits @xmath388-th moments . let @xmath315 be a sequence of i.i.d . @xmath34-valued @xmath80-distributed random variables . then @xmath404 for @xmath344 , with @xmath405 in case @xmath406 , while @xmath407 is arbitrary for @xmath4010,1[$ ] . let @xmath406 . by section [ moments ] , the @xmath388-th moment of the associated @xmath408 with @xmath409 exists . let @xmath334 be i.i.d . @xmath189-valued random variables with distribution @xmath341 , and @xmath410 the associated random walk on @xmath189 . the classical marcinkiewicz - zygmund law ( theorem 5.2.2 of @xcite ) yields that for all @xmath391 , @xmath411 as @xmath412 but this means that @xmath413 a.s . and hence , by [ m1inc](i ) and ( [ momentenforderung ] ) , @xmath414 as @xmath415 and @xmath416 have the same finite - dimensional distributions , the claim follows . the case @xmath4010,1[$ ] is similar . [ strong - law - spezial ] let @xmath422 and @xmath423 such that its projection @xmath403 admits @xmath388-th moments . then , for each sequence @xmath315 of i.i.d . @xmath1-valued and @xmath80-distributed random variables , @xmath424 the strong laws [ strong - law ] and [ strong - law - spezial ] are in principle well - known especially for @xmath425 ; see for instance @xcite and references cited therein . nevertheless our approach may be of some interest , because it uses the close connection between ( biinvariant ) random walks on @xmath34 and those on @xmath189 in a simple explicit way . t. koornwinder , _ jacobi functions and analysis of noncompact semisimple lie groups_. in : r. a. askey et al . ( eds . ) , special functions : group theoretical aspects and applications . dordrecht - boston - lancaster : d. reidel publishing company 1984 , pp . 1 - 85 .

we establish a deformation isomorphism between the algebras of @xmath0-biinvariant compactly supported measures on @xmath1 and @xmath0-conjugation invariant measures on the euclidean space @xmath2 of all hermitian @xmath3-matrices with trace @xmath4 . this isomorphism concisely explains a close connection between the spectral problem for sums of hermititan matrices on one hand and the singular spectral problem for products of matrices from @xmath1 on the other , which has recently been observed by klyachko @xcite . from this deformation we further obtain an explicit , probability preserving and isometric isomorphism between the banach algebra of bounded @xmath0-biinvariant measures on @xmath1 and a certain ( non - invariant ) subalgebra of the bounded signed measures on @xmath2 . we demonstrate how this probability preserving isomorphism leads to limit theorems for the singular spectrum of @xmath0-biinvariant random walks on @xmath1 in a simple way . our construction relies on deformations of hypergroup convolutions and will be carried out in the general setting of complex semisimple lie groups .

You are an expert at summarizing long articles. Proceed to summarize the following text: galaxy populations were first studied by hubble ( 1926 ) , who developed the familiar tuning fork diagram of ellipticals , spirals and barred spirals . most bright galaxies can be morphologically classified by their hubble type . however , many types of galaxy have been found that do nt fit the tuning fork . these occur both at low redshift and at high redshift where the galaxies can be intrinsically different due to evolution . some of these galaxies are shown in fig . 1 . the tuning fork can be extended to include these new types of galaxy as shown in fig . however , morphological classification only tells us what type of galaxies there are . it does not tell us what proportion are of each type or whether this varies over time . galaxy formation and evolution is a relatively new field and so far very little is known about what produced the galaxy population we see today . to understand this process it is first necessary to have some quantitative information about the local galaxy population . the luminosity is an easy to measure quantity which can be used to classify galaxies . the luminosity function ( lf ) ( peebles & hauser 1974 ) measures the space density of galaxies as a function of luminosity . this can be convolved with different evolutionary models and compared to number - magnitude counts ( driver et al . however , recent surveys have produced a large range in the measured lf , fig . 3 , with the variation between the surveys much greater than random errors . the systematic errors causing this wide variation must be understood before any progress is going to be made . disney ( 1976 ) pointed out that surface brightness selection effects are important to take into account when measuring the luminosity function . phillipps & disney ( 1986 ) and phillipps , davies & disney ( 1990 ) went on to calculate the effects of surface brightness . these calculations take into account light lost below the isophote , malmquist bias and surface brightness dimming due to cosmological expansion . here , we will discuss how we have used the `` two degree field galaxy redshift survey '' ( 2dfgrs ) to produce a bivariate brightness distribution ( bbd ) , which corrects for these effects . then we will describe some of the results and some of the selection effects that we have not corrected for . we will then briefly describe the millennium galaxy catalogue ( mgc ) , one of the projects in the isaac newton telescope wide field survey ( int - wfs ) . the mgc is a deep , imaging survey designed to remove some of the remaining selection effects . the selection effects are then modelled by convolving the bbd with different functions based on the visibility theory of phillipps , davies & disney 1990 ) adopting parameters appropriate for the eso slice project ( esp ) ( zucca et al . 1997 ) , the las campanas redshift survey ( lcrs ) ( lin et al . 1996 ) and the second southern sky redshift survey ( ssrs2 ) ( marzke et al . 1998 ) . the 2dfgrs is a @xmath0 redshift survey of the north galactic and south galactic polar ( ngp , sgp ) regions . when it is complete it will have redshifts for 250,000 galaxies with @xmath1 . at present more than 100,000 redshifts have been measured , making this the largest survey to date by a factor of 4 . the input catalogue is the apm catalogue ( maddox et al . 1990a , b ) . for more details on the 2dfgrs see the article in these proceedings by bridges et al . one method to remove surface brightness selection effects is to construct a bbd . this is a plot of the space density of galaxies versus the absolute magnitude , and the effective surface brightness . this is a simple extension of the lf , whereby galaxies of different surface brightnesses are treated separately . in cross et al . ( 2000 ) , we describe how to construct the bbd for the 2dfgrs . throughout this work , we make corrections based upon the data as much as possible . the data has been corrected for light lost below the isophote , malmquist bias , redshift incompleteness and clustering . in each case these corrections were functions of both absolute magnitude and effective surface brightness . the bbd was produced from 50,000 galaxies in the sgp . 4 shows the final malmquist corrected space density of galaxies as a function of @xmath2 and @xmath3 . there is a strong luminosity - surface brightness relation . l-@xmath4 relations are also seen in driver ( 1999 ) , binggeli ( 1993 ) and de jong and lacey ( 2000 ) . the shaded region on fig . 4b shows where the volume is less than 10,000mpc@xmath5 . this plot suggests that luminous lsbgs such as malin 1 are extremely rare . faint hsbgs such as m32 are likely to be rare as they are far from the l-@xmath4 axis . the luminosity density @xmath6mpc@xmath7 . the luminosity density is plotted as a function of absolute magnitude and surface brightness in fig . this shows clearly that the luminosity density is decreasing well before the selection boundaries imposed by the survey . this implies that we are indeed seeing the majority of the light emitted by local galaxies . the input catalogue completeness is a complex function of both surface brightness and magnitude . the curve on fig . 4b shows where the volume , over which a galaxy can be detected , is equal to 10,000mpc@xmath5 . outside this region , the detection of galaxies becomes increasingly difficult . the input catalogue is only complete in regions determined by the detection isophote ( @xmath8mag arcsec@xmath9 ) , the apparent magnitude limit ( @xmath10mag ) and the smallest isophotal radius ( @xmath11 ) . the smallest isophotal radius depends on the seeing ( @xmath12 for the apm ) . the apm suffers from less than ideal photometry which is only accurate to @xmath13mag . the data does not contain information on the light profiles of the galaxies such as disc - bulge separation parameters . to remove these problems we have undertaken a deep equatorial ccd survey over part of the 2dfgrs northern galactic polar region . this survey is described in more detail in these proceedings by lemon et al . it will push the boundaries of the @xmath14 plane that can be studied , giving us a quantitative measure of the population that we may have missed with the 2dfgrs . it will also improve the photometry of the galaxies we have already observed and allow us to do morphological classification and disc - bulge separation . fig . 6 shows the schechter function fits to three lfs shown in fig . these are the lcrs ( lin et al . 1996 ) , the ssrs2 ( marzke et al . 1998 ) and the esp ( zucca et al . 1997 ) . the lcrs and esp show the largest variation amongst the surveys . also plotted are three lines showing the lfs produced when the 2dfgrs bbd is convolved with three different visibility functions . the parameters used for the lcrs are @xmath15b@xmath16 , m@xmath17b , and d@xmath18 , calculated using a b - r colour of 1.5 ( driver et al . 1994 ) . the parameters used for the esp are @xmath19b@xmath16 , m@xmath20b , and d@xmath18 . the parameters used for the ssrs2 are @xmath19b@xmath16 , m@xmath21b , and d@xmath18 . this plot shows that the lfs can be reproduced well at the faint end . this demonstrates that surface brightness selection effects are a major systematic error in determining luminosity functions . to understand the effects of evolution , it is essential to have a good quantitative knowledge of the local galaxy population . the new technologies in this `` new era of wide field astronomy '' allow us to detect larger numbers of galaxies with higher precision . these new technologies are the multi - fibre spectrometers such as 2df and the large format ccds such as those in the int wide field camera . these new surveys allow us to properly tackle the selection effects that have dogged our surveys and therefore will allow us to finally pin down the space and luminosity densities of galaxies .

we discuss the quantification of the local galaxy population and the impact of the `` new era of wide - field astronomy '' on this field , and , in particular , systematic errors in the measurement of the luminosity function . new results from the 2dfgrs are shown in which some of these selection effects have been removed . we introduce an int - wfs project which will further reduce the selection biases . we show that there is a correlation between the surface brightness and the luminosity of galaxies and that new technologies are having a big impact on this field . finally selection criteria from different surveys are modelled and it is shown that some of the major selection effects are surface brightness selection effects .

You are an expert at summarizing long articles. Proceed to summarize the following text: stellar populations are fundamental tracers of the formation and evolution history of their parent galaxy . the accurate determination of their basic parameters ( as age , metallicity and distance ) are then crucial , but this is restricted to a number of relatively nearby stellar systems , for which single stars are spatially resolved and whose magnitudes can be accurately determined . at least the brightest portion of the red giant branch ( rgb ) is needed in order to have hits on metallicity and distance , while the detection of the main sequence turn - off ( ms - to ) is required to get the age . in this respect , detailed investigations of simple stellar populations ( ssp , _ i.e. _ coeval and chemically homogeneous stellar aggregates ) offer a unique opportunity to empirically calibrate suitable photometric indices and evolutionary features in terms of the overall metallicity of the system . stellar clusters represent the best approximation of ssp known in the universe , hence they are ideal tools for this purpose . an empirical method to get simultaneously metallicity and reddening from the morphology and location of the rgb in the @xmath0 color - magnitude diagram ( cmd ) has been presented a decade ago @xcite . more recently , this method has been extended to the @xmath1 plane by @xcite and further improved by @xcite and @xcite by adopting the @xcite and the global ( @xmath2 $ ] ) metallicity scale , respectively . the method presented here adopts the most recent calibrations as obtained from a large ir photometric database of gcs , collected by our group with different instruments at the eso telescopes ( la silla , chile ) and at the tng ( la palma , spain ) over the last decade in the framework of a long - term project devoted to study the photometric properties of the rgb @xcite . in particular , we have recently presented a complete set of photometric indices ( colors , magnitudes and slopes ) describing the location and the morphology of the rgb and their calibrations in terms of the global cluster metallicity @xcite , together with the empirical calibrations of the ir luminosity of the two major rgb evolutionary features , namely the rgb bump and tip @xcite . it is worth mentioning that , the calibration relations used in this study rely on a database , whose properties have been derived in a fully self consistent way . in fact , the adopted estimates of the cluster reddening , metallicity and distance are based _ i ) _ on a homogeneous photometric system ; _ ii ) _ on the @xcite distance scale which relies on the most recent and largest database of galactic gcs ; and _ iii ) _ on a uniform and high resolution metallicity scale @xcite . the calibration of suitable relations to derive metallicity , reddening and distance in the near ir plane is crucial in the study of extragalactic bulges , which can be characterized by high metallicities and can be affected by severe reddening . the current generation of ground - based ir instrumentation with high resolution and wide field coverage , and the future availability of the james webb space telescope will allow us to resolve the brightest giants in galaxies up to several mpc away , and to derive their overall metallicity , reddening and distance modulus with great accuracy . the paper is organized as follows . 2 discusses the adopted metallicity scale , while 3 and 4 present the equation sets for both the _ disk - like _ and _ bulge - like enrichment scenarios _ , respectively . 5 describes the computational routine which provides photometric estimates of metallicity , reddening and distance , while in 6 we test the described method on two template gcs in the galactic bulge and in the large magellanic cloud , namely ngc 6539 and ngc 1978 respectively , in order to demonstrate its reliability . as widely discussed in @xcite a correct parameterization of the rgb characteristics as function of the metal content of the population does require the knowledge of the so - called _ global _ metallicity , which takes into account the iron as well as the @xmath3element abundances . this is an important point since the location of the rgb strongly depends on the low ionization potential [ fe+mg+si / h ] abundance mixture @xcite rather than on [ fe / h ] abundance alone . in fact , low ionization potential elements are the main contributors to free electrons which generate the @xmath4 ion , the major component responsible for the continuum opacity in the rgb temperature range ( 30006000 k , * ? ? ? * ) . in halo / disk field stars the average [ @xmath3/fe ] abundance ratio shows a general enhancement of 0.30.5 dex with respect to the solar value up to [ fe / h]@xmath51 ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * and references therein ) and a linear decreasing trend towards solar [ @xmath3/fe ] with further increasing metallicity . a [ @xmath3/fe ] enhancement is also found in the metal poor halo gcs ( see e.g. * ? ? ? * ; * ? ? ? * and references therein ) . however , until a few years ago , only a few measurements were available in the high metallicity regime , to properly define the [ @xmath3/fe ] trend in gcs ( see e.g. * ? ? ? * ; * ? ? ? the actual position of the knee ( i.e. the metallicity at which [ @xmath3/fe ] begins to decrease ) depends on the type ia sn timescales and it is also a function of the star formation rate , while the amount of @xmath6 enhancement depends on the initial mass function of the progenitors of the type ii sne ( see * ? ? ? measurements of metal rich field and cluster giants towards the galactic bulge are a recent accomplishment of high resolution optical and ir spectroscopy . bulge stars are indeed ideal targets to study the behavior of the abundance patterns in the high metallicity domain , but foreground extinction is so great as to largely preclude optical studies of any kind , particularly at high spectral resolution . the most accurate abundance determinations obtained so far and based on high resolution optical spectroscopy refer to a sample of k giants in the baade s window @xcite and a few giants in two gcs , namely ngc 6553 and ngc 6528 @xcite . recently , high resolution ir spectroscopic measurements of m giants in the baade s window @xcite as well as of bulge gcs @xcite dramatically extended the sample of measured bulge stars . all these studies point toward a @xmath3-enhancement by a factor of 2 - 3 up to solar metallicity . in the computation of the _ global _ metallicity for a sample of 61 ggcs , @xcite used a constant @xmath7=0.28 $ ] for @xmath8<-1 $ ] and linearly decreasing to zero at @xmath9<0 $ ] . in the following we compute two independent sets of global metallicities and photometric relations , according to two different scenarios ( see fig . [ scenario ] ) : _ scenario ( 1 ) - _ _ disk - like _ enhancement ( accordingly to @xcite ) @xmath7=0.3 $ ] for @xmath8<-1 $ ] and @xmath7 $ ] linearly decreasing to zero for @xmath9\le0 $ ] . _ scenario ( 2 ) - _ _ bulge - like _ enhancement ( accordingly to carney s ( 1996 ) suggestions and the recent results on the bulge field and cluster populations ) @xmath7=0.30 $ ] constant over the entire range of metallicity ( @xmath10\le0 $ ] ) . the set of equations discussed in @xcite[hereafter paper i ] and @xcite[hereafter paper ii ] have been computed accordingly to _ scenario ( 1)_. here we present similar equations for _ scenario ( 2 ) _ , so the reader can choose the equation set which turns out to be the most suitable to the describe the stellar system . in both scenarios the contribution of the @xmath3element enhancement has been taken into account by simply re scaling standard models to the _ global _ metallicity [ m / h ] according to the following relation @xcite : @xmath11 = [ fe / h ] + log_{10 } ( 0.638 f_{\alpha}+0.362)\ ] ] where @xmath12 is the enhanced factor of the @xmath3elements . in this section we present and discuss the equation set adopted to construct the computational routine which provides metallicity , reddening and distance of a stellar population , by using the rgb observables in the @xmath13 ir plane . note that the validity of the relations has been extended up to [ m / h]@xmath14 + 0.4 dex , with respect to the results presented in paper i , including the recent results on the metallicity of ngc 6791 @xcite . hence the relations presented here have been obtained in the metallicity range : -2.16@xmath15[fe / h]@xmath15 + 0.4 dex . the overall metallicity of a stellar population can be linked to the slope of the rgb ( @xmath16 ) as defined in the near ir - cmd , by @xcite and @xcite , accordingly to the following relation : @xmath17 = -2.53 -20.83 ( slope^{jk}_{rgb})\ ] ] where @xmath18 is the rgb slope measured following the prescriptions discussed in paper i in the @xmath13-cmd . the rgb - tip luminosity is a quantity well predicted by theoretical models and it can be easily measured in populous stellar systems like galaxies . the luminosity of the rgb tip for stellar populations older than @xmath19 gyr is nearly independent from the population age . moreover , it turns out to be particularly bright in the near - ir ( @xmath20 ) hence , it is a very promising standard candle up to large distances . the main limitation of this method is the clear - cut determination of the rgb - tip luminosity that could be seriously affected by low - number statistics . in fact , the rapid star evolution near the end of the rgb , makes the brightest portion of the rgb intrinsically poorly populated . however , this problem is marginal in very populous stellar systems . the dependence of this feature from the overall metallicity of the stellar population has been empirically determined by using a sample of galactic gcs ( see paper ii ) and it turns out to be in good agreement with theoretical expectations . here we report the relation obtained in paper ii : @xmath21\ ] ] as well known ( see * ? ? ? * ; * ? ? ? * ) for clusters older than 2 gyr the level of the helium - burning red clump ( rc ) is mainly influenced by metallicity and it shows little dependence on the age . hence for relatively old metal rich populations ( @xmath22>-1 $ ] ) it is also possible to use the level of the rc as an additional distance indicator . we estimate the mean level of the rc by using a metal - rich ( -0.9@xmath23[fe / h]@xmath23 - 0.3 ) cluster subsample ( namely , 47 tuc , ngc 6342 , ngc 6380 , ngc 6441 , ngc 6440 , ngc 6553 , ngc 6528 , and ngc 6637 ) selected from the global database presented in paperi . the value turns out to be : @xmath24 note that most selected clusters can be considered coeval within 10 - 20% accordingly to @xcite who measured the relative ages of 47 tuc , ngc 6342 , ngc 6637 and @xcite who measured the relative ages of 47 tuc , ngc 6528 , ngc 6553 . the adopted uncertainty of @xmath25 in equation ( 3 ) , takes into account the metallicity dependence of the rc position in the metallicity range spanned by our metal rich gc subsample ( @xmath26\sim 0.6 dex$ ] ) . in fact , accordingly to table 1 by @xcite the @xmath27 level is expected to vary @xmath28 at @xmath29 over a such metallicty range . moreover in the age range 2 - 10gyr , at fixed metallicity ( @xmath2=-0.68 $ ] ) the @xmath27 varies 0.03 mag per gyr . hence the 0.2 mag uncertainty adopted as conservative estimate of the @xmath27 level is expected to be reasonable over the considered range of age and metallicity . the absolute magnitude of the rgb at fixed color is another observable that can be used to define useful relations with the overall metallicity . here we report the calibrations discussed in paper i : @xmath30\ ] ] finally , the entire set of equations describing the rgb location in @xmath31 color at different level of magnitudes ( @xmath32 ) are listed . @xmath33 = -4.37 + 3.84(j - k)^{-5.5}_0\ ] ] @xmath34 = -4.51 + 4.24(j - k)^{-5}_0\ ] ] @xmath35 = -4.87 + 5.20(j - k)^{-4}_0\ ] ] @xmath36 = -5.36 + 6.48(j - k)^{-3}_0\ ] ] an analogous set of 16 equations can be obtained in the _ bulge - like _ enrichment scenario . here we list the complete equation set based on the database presented in paper i and ii by adopting a constant @xmath37 $ ] over the entire range of metallicity . @xmath38 = -2.63 -22.50 ( slope^{jk}_{rgb})\ ] ] @xmath39\ ] ] @xmath40 @xmath41\ ] ] @xmath42 = -4.59 + 4.13(j - k)^{-5.5}_0\ ] ] @xmath43 = -4.74 + 4.55(j - k)^{-5}_0\ ] ] @xmath44 = -5.12 + 5.58(j - k)^{-4}_0\ ] ] @xmath45 = -5.64 + 6.94(j - k)^{-3}_0\ ] ] the computational routine requires as input parameters : * the observed rgb mean ridge line in the @xmath46 cmd ; * the observed rgb slope ( @xmath18 ) ; * the observed rgb - tip ( @xmath47 ) and/or the rc level ( @xmath48 ) . [ diagram ] shows the flow - diagram of the computational routine . in the following , references to eqs . 1 - 8 either equations in sect . [ diskeq ] or in sect . [ bulge ] , depending on the adopted enrichment scenario . since the rgb slope is independent of the cluster distance and reddening , we can use eq . ( 1 ) to get a first guess of the population metallicity . by using this first guess metallicity , it is possible to derive the expected level of the rgb tip from eq . hence , the comparison with the observed value ( @xmath47 ) will directly yield the first estimate of the distance modulus @xmath49 . for the ( closest ) metal - rich stellar systems for which the rc is observed , an independent estimate of the modulus can be also derived from the comparison with the mean rc levels reported in eq . ( 3 ) . by using the first guess metallicity and eq . ( 4 ) it is also possible to obtain the expected absolute magnitude ( @xmath50 ) at fixed color ( @xmath51 ) . now , the mean ridge line of the program population can be corrected by using the distance modulus obtained above : the observed color ( @xmath52 ) in correspondence of the derived magnitude ( @xmath50 ) will yield the first guess estimate of the reddening : @xmath53 where the extinction coefficients from @xcite have been adopted . the color of the input rgb mean ridge line can be now de - reddened and transformed into the absolute plane @xmath54 . once in the absolute plane , the color at fixed magnitude levels can be measured and inserted in eqs . ( 5 - 8 ) to get a new mean value of metallicity . the latter can be inserted in eq . ( 2 ) , and the overall procedure iterated until the values of the three output quantities , namely metallicity , reddening and distance , converge within suitable ranges , which can be specified as input tolerances . the formal error on each derived quantity can be estimate by using a simple monte carlo simulation . in doing this we randomly extracted a large number of values for each of the three main observables ( namely @xmath18 , @xmath55 and @xmath48 ) from a gaussian distribution peaked on the observed value and with @xmath56 equal to the uncertainty of each observable . these values are used ( following the scheme plotted in fig . [ diagram ] ) to derive metallicity , reddening and distance . the standard deviation of each set of values is the error associated to that specific quantity . however , beside the formal errors obtained from this procedure ( typically @xmath57 mag ; @xmath58 mag ; @xmath59=\pm0.10 - 0.15 $ ] dex ) we estimate that conservative uncertainties for the derived quantities are : @xmath60 mag ; @xmath61 mag ; @xmath59=\pm0.2 $ ] dex . to test the reliability of the proposed technique we have chosen two metal - rich clusters for which accurate determination of the metallicity have been recently obtained : the large magellanic cloud cluster ngc 1978 , and the galactic bulge cluster ngc 6539 . the cmds of ngc 1978 and ngc 6539 shown in fig . [ 1978 ] and [ 6539 ] are from @xcite and from @xcite , respectively . the rgb mean ridge lines computed following the prescriptions described in paper i are also overplotted as solid line onto the cmds of fig . [ 1978 ] and [ 6539 ] . in the case of ngc 1978 , the cmd shown in fig . [ 1978 ] has been used to measure the rgb slope ( @xmath18=0.104 ) and to estimate the observed rgb - tip ( k@xmath62=11.89 ) . it is worth noticing that this cluster is suspected to be a relatively young cluster ( @xmath63 ) , hence we do not use its rc level in deriving the distance modulus level has been used for this cluster . ] since for extreme young clusters the @xmath64 level is a sensible function of the age of the population ( see e.g. fig . 3 and table 1 by * ? ? ? * ) . by running the computational routine with the input parameters quoted above , a good convergence is reached after few iterations . in the _ bulge - like scenario _ , the computational routine leaded a reddening estimate of e(b - v)=0.09@xmath650.04 mag , an intrinsic distance modulus of ( m - m)@xmath66=18.53@xmath650.18 mag and a metallicity of [ m / h]=0.29@xmath650.14 dex . in the _ disk - like scenario _ the following solutions have been obtained : e(b - v)=0.09@xmath650.04 mag , ( m - m)@xmath66=18.55@xmath650.19 mag , and [ m / h]=0.36@xmath650.12 dex . these values are fully consistent with the results found by @xcite and @xcite for the extinction ( e(b - v)=0.100 , and e(b - v)=0.075 , respectively ) , by @xcite and @xcite for the distance ( ( m - m)@xmath66=18.48 and ( m - m)@xmath66=18.50 , respectively ) , and by @xcite and @xcite for the metallicity ( [ fe / h]=0.38 dex and [ fe / h]=0.42 dex , based on high - resolution optical spectra and on the calcium triplet , respectively ) . adopting the same strategy followed in the case of ngc 1978 , we have analyzed the cmd of ngc 6539 @xcite , finding a slope@xmath67=0.091 and k@xmath62=8.47 mag . from the derived k luminosity function we also estimated the rc level to be k@xmath68=13.60 mag . by running the computational routine within the _ bulge - like scenario _ , we found a reddening of e(b - v)=1.08@xmath650.06 mag , an intrinsic distance modulus of @xmath69 mag , and a global metallicity [ m / h]=0.57@xmath650.15 dex . the derived metallicity is fully consistent within 0.2 dex with the recent estimate , based on high - resolution ( r@xmath70 ) ir spectroscopy , by @xcite , who found [ fe / h]=0.76 dex . the obtained reddening value is also in excellent agreement with the estimate listed in the @xcite compilation ( e(b - 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we present an empirical method to derive photometric metallicity , reddening and distance to old stellar populations by using a few major features of the red giant branch ( rgb ) in near ir color magnitude diagrams . we combine the observed rgb features with a set of equations linking the global metallicity [ m / h ] to suitable rgb parameters ( colors , magnitudes and slope ) , as calibrated from a homogeneous sample of galactic globular clusters with different metallicities . this technique can be applied to efficiently derive the main population parameters of old stellar systems , in the view of using ground - based adaptive optics and space facilities to probe the stellar content of remote galaxies .

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Proceed to summarize the following text: discrete fractional transforms are the generalizations of the ordinary discrete transforms with one additional fractional parameter . in the past decades , various discrete fractional transforms including discrete fourier transform @xcite , @xcite , discrete fractional hartley transform @xcite , discrete fractional cosine transforms and discrete sine transform @xcite have been introduced and found wide applications in many scientific and technological areas including digital signal processing @xcite , image encryption @xcite , @xcite , @xcite and digital watermarking @xcite and others . different fast algorithms for their implementations have been separately developed to minimize computational complexity and implementation costs . in @xcite a discrete fractional hadamard transform for the vector of length @xmath3 was introduced , however a fast algorithm for the realization of this transform has not been proposed . in our previous paper @xcite we describe a rationalized algorithm for dfrht possessing a reduced number of multiplications and additions . analysis of the mentioned algorithm shows that not all of existing improvement possibilities have been realized . in this paper , we proposed a novel algorithm for the discrete fractional hadamard transform that require fewer total real additions and multiplications than our previously published solution . a hadamard matrix of order @xmath0 is a @xmath4 symmetric matrix whose entries are either 1 or @xmath5 and whose rows are mutually orthogonal . in this paper we will use the normalized form of this matrix and we will denote it by @xmath6 . for @xmath7 the hadamard matrices can be recursively obtained due to sylvester s construction @xcite : @xmath8\!,\ \mathbf{h}_{n}=\frac{1}{\sqrt{2 } } \left [ \begin{array}{cc } \mathbf{h}_{\frac{n}{2 } } & \mathbf{h}_{\frac{n}{2}}\\ \mathbf{h}_{\frac{n}{2 } } & -\mathbf{h}_{\frac{n}{2 } } \end{array}\right]\ ] ] for @xmath9 . definition of the discrete fractional hadamard ( dfrht ) transform is based on an eigenvalue decomposition of the dht matrix . any real symmetric matrix ( including the hadamard matrix ) can be diagonalized , e.g. written as a product @xcite @xmath10 where @xmath11 is a diagonal matrix of order @xmath12 , whose diagonal entries are the eigenvalues of @xmath6 @xmath13\ ] ] + @xmath14 $ ] - the matrix whose columns are normalized mutually orthogonal eigenvectors of the matrix @xmath6 . the eigenvector @xmath15 is related to the eigenvalue @xmath16 . a superscript @xmath17 denotes the matrix transposition . the dfrht matrix of order @xmath7 with real parameter @xmath18 was first defined in @xcite . this matrix can be regarded as a power of the dht matrix , where the exponent @xmath19 @xmath20 for @xmath21 the dfrht matrix is converted into the identity matrix , and for @xmath22 it is transformed into the ordinary dht matrix . generally the dfrht matrix is complex - valued . an essential operation , by obtaining the discrete fractional hadamard matrix , defined by ( [ eq : rozklad2 ] ) , is calculating the eigenvalues and the eigenvectors of the matrix @xmath6 . the only eigenvalues of the unnormalized hadamard matrix of order @xmath23 are known to be @xmath24 and @xmath25 @xcite , hence the normalized hadamard matrix @xmath26 has only the eigenvalues 1 and @xmath5 . a method for finding the eigenvectors of hadamard matrix was firstly presented in @xcite , but in @xcite a recursive method for calculation the eigenvectors of the hadamard matrix order @xmath27 based on the eigenvectors of the hadamard matrix of order @xmath12 has been proposed . we will use this method to obtain the dfrht matrix . here we will present it briefly . in @xcite it was proven that if @xmath28 ( @xmath29 ) is an eigenvector of hadamard matrix of order @xmath23 associated with an eigenvalue @xmath30 , then vector @xmath31 \label{eq : wektorhat}\ ] ] will be an eigenvector of the matrix @xmath32 associated with the eigenvalue @xmath30 . in @xcite it was proven that if @xmath28 is an eigenvector of @xmath6 associated with an eigenvalue @xmath30 , then the vector @xmath33 \label{eq : wektortilde}\ ] ] will be an eigenvector of the matrix @xmath32 associated with the eigenvalue @xmath34 . these two results allow as to generate the eigenvectors of hadamard matrix of order @xmath27 from the eigenvectors of hadamard matrix of order @xmath12 . knowing the straightforward calculated eigenvectors of the matrix @xmath35 @xmath36\;\ ; \mathbf{v}_{2}^{(1)}=\left [ \begin{array}{c } 1-\sqrt{2}\\ 1\\ \end{array } \right ] \label{eq : wektoryh2}\ ] ] associated with eigenvalues 1 and @xmath5 respectively , the eigenvectors for hadamard matrix of arbitrary order @xmath23 can be recursively computed . in @xcite it was also shown so this recursively computed eigenvectors of matrix @xmath26 will be orthogonal . it should be noted that for any @xmath23 there are only two distinct eigenvalues of hadamard matrix , so for @xmath37 the eigenvalues are degenerated . because of this fact the set of eigenvectors proposed in @xcite and @xcite is not unique . the igenvectors @xmath28 for @xmath38 , which are columns of the matrix @xmath39 ( after normalization ) , as well as their associated eigenvalues @xmath40 , can be however ordered in different ways . in @xcite it has been also established a method of ordering the eigenvectors . in many cases , including the case of discrete fractional transforms is used so - called sequency ordering of the eigenvectors . this means that the @xmath41-th eigenvector has @xmath41 sign - changes . the discrete hermite - gaussians , eigenvectors of discrete fourier transform matrix are ordered this way as well @xcite . we will show this method of ordering of the eigenvectors in example [ example1 ] . [ example1 ] the number of sign - changes in eigenvectors @xmath42 and @xmath43 of matrix @xmath35 , determined by ( [ eq : wektoryh2 ] ) , is equal to 0 and 1 respectively . using expressions ( [ eq : wektorhat ] ) and ( [ eq : wektortilde ] ) we obtain the eigenvectors of matrix @xmath44 : @xmath45\;\ ; \mathbf{\tilde v}_{4}^{(0)}= \left [ \begin{array}{c } -b\\ -b^2 \\ 1\\ b\\ \end{array } \right]\;\ ; \mathbf{\hat v}_{4}^{(1)}= \left [ \begin{array}{c } -b\\ 1 \\ -b^2\\ b\\ \end{array } \right]\;\ ; \mathbf{\tilde v}_{4}^{(1)}= \left [ \begin{array}{c } b^2\\ -b \\ -b\\ 1\\ \end{array } \right],\ ] ] where @xmath46 . the numbers of sign - changes in the above vectors are 0 , 1 , 3 , 2 respectively ( @xmath47 ) . therefore , a sequency ordered set of eigenvectors of matrix @xmath44 will be as follows : @xmath48 the corresponding eigenvalues will be equal to : @xmath49 the relations obtained in example [ example1 ] can be easily generalized as follows : @xmath50 \mathbf{v}_{2n}^{(4l+1)}=\mathbf{\tilde v}_{2n}^{(2l)}\\[6pt ] \mathbf{v}_{2n}^{(4l+2)}=\mathbf{\tilde v}_{2n}^{(2l+1)}\\[6pt ] \mathbf{v}_{2n}^{(4l+3)}=\mathbf{\hat v}_{2n}^{(2l+1 ) } \end{array } \right.\ ] ] for @xmath51 and @xmath52 for @xmath53 . both the eigenvectors of the matrix @xmath35 and the eigenvectors obtained for higher order hadamard matrices are not normalized . let the notation @xmath54 means the euclidean norm of vector @xmath28 . in @xcite it was shown that for any @xmath7 we have the relationship @xmath55 where @xmath38 and @xmath56 . if we take the designation @xmath57 , then normalized eigenvectors of hadamard matrix of order @xmath7 will take the form @xmath58 using the normalized and sequency ordered eigenvectors of the hadamard matrix , the eigenvalue decomposition ( [ eq : rozklad ] ) of the hadamard matrix can be written as follows : @xmath59 where @xmath11 is the diagonal matrix whose non - zero elements are @xmath60 for @xmath38 . hence the definition ( [ eq : rozklad2 ] ) of dfrht matrix will take the form : @xmath61 where @xmath62 for @xmath38 . our goal is to calculate the discrete fractional hadamard transform for an input signal @xmath63 in which the number of samples is equal to @xmath23 . by @xmath64 we denote an output signal which is calculated using the formula @xmath65 supposing that the matrix @xmath66 is given , to calculate the output signal it is necessary to perform @xmath1 complex multiplications and @xmath67 complex additions . if the input signal is real , then the number of real multiplications will be equal to @xmath68 , and the number of real additions will be equal to @xmath69 . if we use the decomposition ( [ eq : rozklad2n ] ) of the matrix @xmath66 by calculating ( [ eq : transformata ] ) and will perform the matrix - vector multiplication from the right side to the left , the most time - consuming operations are multiplications of matrices @xmath70 and @xmath71 by the vector , because those matrices are not diagonal . if we interchange the columns of the matrix @xmath71 in the prescribed manner , we obtain a matrix @xmath72 of special structure , which can be generated recursively . we will show it in example [ example2 ] . it will allow to reduce the number of arithmetical operations by calculating the products of matrices @xmath70 and @xmath71 by the vector . [ example2 ] the matrices @xmath71 for @xmath73 are as follows : @xmath74,\;\ ; \mathbf{v}_4= \left [ \begin{array}{cccc } 1&-b&b^2&-b\\ b&-b^2&-b&1\\ b&1&-b&-b^2\\ b^2&b&1&b\\ \end{array } \right],\;\;\ ] ] @xmath75.\ ] ] the matrix @xmath76 has some specific structure . now we consider the matrix @xmath77 . if in the matrix @xmath77 the second and fourth columns will be interchange and then the third and fourth columns will be interchange too , we obtain the following matrix : @xmath78 = \left [ \begin{array}{cc } \mathbf{v}_2&-b\mathbf{v}_2\\ b\mathbf{v}_2&\mathbf{v}_2\\ \end{array } \right].\ ] ] the matrix @xmath77 differs from the matrix @xmath79 only in the order of the columns . therefore , the matrix @xmath77 can be obtained by post - multiplying the matrix @xmath79 by the permutation matrix @xmath80 : @xmath81 where @xmath82.\ ] ] now we consider the matrix @xmath83 . if we perform the following permutation of columns of this matrix : @xmath84 as a result we obtain the following matrix : @xmath85= \left [ \begin{array}{cc } \mathbf{\overline{v}}_4&-b\mathbf{\overline{v}}_4\\ b\mathbf{\overline{v}}_4&\mathbf{\overline{v}}_4\\ \end{array } \right].\ ] ] as previously , we can write : @xmath86 where @xmath87.\ ] ] if we generalize the above considerations for @xmath7 we can write : @xmath88 for @xmath89 . for @xmath90 we can also write @xmath91 where @xmath92 is an identity matrix of order two @xmath93 the permutation matrix @xmath94 of order @xmath95 can be obtained recursively from the permutation matrix @xmath96 of order @xmath97 according to the following relation : @xmath98,\ \mathbf{p}_{n}=\mathbf{s}_{n } \left[\begin{array}{cc } \mathbf{p}_{\frac{n}{2}}\hspace{0.2 cm } & \mathbf{0}_{\frac{n}{2}}\\ \mathbf{0}_{\frac{n}{2}}\hspace{0.2 cm } & \mathbf{p}_{\frac{n}{2}}\mathbf{j}_{\frac{n}{2}}\\ \end{array}\right]\!\ ] ] for @xmath9 . @xmath99 is the perfect shuffle permutation matrix of order @xmath12 , @xmath100 is the counter - identity matrix of order @xmath101 and @xmath102 is zero matrix . the perfect shuffle permutation is the permutation that splits the set consisting of an even number of elements into two piles and interleaves them . it can be written as follows : @xmath103 for example @xmath104 , \ \mathbf{j}_4= \left [ \begin{array}{cccc } 0&0&0&1\\ 0&0&1&0\\ 0&1&0&0\\ 1&0&0&0\\ \end{array } \right].\ ] ] if we write the matrix @xmath71 as a product @xmath105 the expression ( [ eq : rozklad2n ] ) will take the form : @xmath106 the product @xmath107 is a diagonal matrix , which has the same diagonal entries as the matrix @xmath108 but in different order and for a chosen parameter @xmath109 it may be prepared in advance . if we denote this product multiplied by a factor @xmath110 by @xmath111 : @xmath112 the dfrht algorithm ( [ eq : transformata ] ) will take the following form : @xmath113 where the matrix @xmath114 can be generated recursively : @xmath115\;\ ; \mathbf{\overline{v}}_{2k}= \left [ \begin{array}{cc } \mathbf{\overline{v}}_k&-b\mathbf{\overline{v}}_k\\ b\mathbf{\overline{v}}_k&\mathbf{\overline{v}}_k\\ \end{array } \right ] \label{eq : defvn}\ ] ] for @xmath116 . the most time - consuming operations by calculating the dfrht transform according to ( [ eq : transformatan ] ) are multiplications of matrices @xmath117 and @xmath72 by the vector . since in the matrix @xmath72 occur only following powers of @xmath118 we can write this matrix as follows : @xmath119 in the figure [ fig : figure1 ] it was shown the way of calculating the matrix - vector product @xmath120 , using the expression ( [ eq : sum ] ) . in this paper , the graph - structural models and data flow diagrams are oriented from left to right . straight lines in the figures denote the operation of data transfer . we use the usual lines without arrows specifically so as not to clutter the picture . note that the circles in this figure shows the operations of multiplication by a number inscribed inside a circle . in turn , the rectangles indicate the matrix - vector multiplications by matrices , scaledwidth=70.0% ] although it may seem strange , we will see that such an operation allows to reduce the number of multiplication and additions by multiplying the matrix @xmath121 by a vector . it should be noted that because of the recursive relation ( [ eq : defvn ] ) between matrices @xmath122 and @xmath123 , the following recursive relation between the matrices @xmath124 , @xmath125 and @xmath126 occurs : @xmath127=\mathbf{i}_{n}\ ] ] @xmath128 \label{eq : defan}\ ] ] @xmath129\ ] ] for @xmath130 and @xmath131=log@xmath2 , where @xmath132=\mathbf{i}_{2},\;\;\ ; \mathbf{a}_{2}^{(1)}= \left [ \begin{array}{cc } 0&-1\\ 1&0\\ \end{array } \right].\ ] ] to clarify our idea we show the explicit form of expressions ( [ eq : sum ] ) and ( [ eq : defan ] ) for @xmath90 , @xmath133 and @xmath134 in an example [ example3 ] . [ example3 ] @xmath135 where the matrices @xmath136 and @xmath137 are presented above . @xmath138 where @xmath139= \left [ \begin{array}{cc } \mathbf{a}_{2}^{(0)}&\mathbf{0}_{2}\\ \mathbf{0}_{2}&\mathbf{a}_{2}^{(0)}\\ \end{array } \right]= \mathbf{i}_4,\ ] ] @xmath140= \left [ \begin{array}{cc } \mathbf{a}_{2}^{(1)}&-\mathbf{a}_{2}^{(0)}\\ \mathbf{a}_{2}^{(0)}&\mathbf{a}_{2}^{(1)}\\ \end{array } \right],\ ] ] @xmath141= \left [ \begin{array}{cc } \mathbf{0}_{2}&-\mathbf{a}_{2}^{(1)}\\ \mathbf{a}_{2}^{(1)}&\mathbf{0}_{2}\\ \end{array } \right ] .\ ] ] @xmath142 where @xmath143= \left [ \begin{array}{cc } \mathbf{a}_{4}^{(0)}&\mathbf{0}_{4}\\ \mathbf{0}_{4}&\mathbf{a}_{4}^{(0)}\\ \end{array } \right]= \mathbf{i}_8,\ ] ] @xmath144= \left [ \begin{array}{cc } \mathbf{a}_{4}^{(1)}&-\mathbf{a}_{4}^{(0)}\\ \mathbf{a}_{4}^{(0)}&\mathbf{a}_{4}^{(1)}\\ \end{array } \right],\ ] ] @xmath145= \left [ \begin{array}{cc } \mathbf{a}_{4}^{(2)}&-\mathbf{a}_{4}^{(1)}\\ \mathbf{a}_{4}^{(1)}&\mathbf{a}_{4}^{(2)}\\ \end{array } \right],\ ] ] @xmath146= \left [ \begin{array}{cc } \mathbf{0}_{4}&-\mathbf{a}_{4}^{(2)}\\ \mathbf{a}_{4}^{(2)}&\mathbf{0}_{4}\\ \end{array } \right].\ ] ] now we will evaluate the number of arithmetical operations , which are necessary to calculate the matrix - vector product @xmath147 . we note , that in a general case such an operation requires @xmath1 multiplications and @xmath67 additions . now we will calculate the numbers of multiplications and additions needed for this operation if we use the expression ( [ eq : sum ] ) for the matrix @xmath72 , i.e. @xmath148 since the non - zero entries of matrices @xmath149 , @xmath150 are only 1 and -1 , no multiplications are needed when calculating the matrix - vector products @xmath151 . the only multiplications we have to perform are multiplications of the vectors @xmath152 by the powers of @xmath153 : @xmath154 by @xmath153 , @xmath155 by @xmath156 by @xmath157 . because the number @xmath153 is constant and known , its powers @xmath158 , @xmath159 may be prepared in advance . thus , the number of multiplication by calculating the matrix - vector product @xmath160 is equal to @xmath161log@xmath0 . let us examine the number of additions , we need to perform , when calculating the matrix - vector product @xmath160 . the total number of additions consist of number of additions by calculating the matrix - vector products @xmath152 , and @xmath162 additions which are needed to calculate the sum of vectors : @xmath163 , @xmath164 , @xmath165 . since , according to ( [ eq : defan ] ) , the matrices @xmath166 have specific structures , the products @xmath152 can be obtained by subtracting the products @xmath167 , @xmath168 of twice smaller size and summing the products @xmath169 , @xmath170 ( excluding products @xmath163 and @xmath171 which can be obtained even in a simpler way ) . by @xmath172^t$ ] we denote the first half of the input vector @xmath173 and by @xmath174^t$ ] - the second half of this vector , as it was shown , for @xmath134 , in the figure [ fig : figure2 ] . using the products of twice smaller size : @xmath175 , @xmath176 , @xmath177 , @xmath178 for @xmath179,scaledwidth=100.0% ] it should be noted that the products @xmath167 and @xmath170 are used to calculate both products @xmath180 and @xmath181 . for example the products @xmath182 and @xmath183 are used to calculate @xmath184 and @xmath185 . it allows to reduce the number of additions , because the some products are used twice . of course , this procedure can be repeated and each of products @xmath167 , @xmath168 , @xmath169 , @xmath170 can be calculated by summing ( subtracting ) products of twice smaller size and so on . it can be continued until calculating products of matrices @xmath186 and @xmath187 by two - element sub - vectors of the vector @xmath63 . the whole process of going down by calculating the product @xmath147 is presented , for @xmath134 , in the figure [ fig : figure3 ] . , scaledwidth=70.0% ] the expression ( [ eq : rozklad4 ] ) can be also written as the matrix - vector product , as follows : @xmath188 where @xmath189,\ ] ] @xmath190\otimes\mathbf{i}_n,\ ] ] @xmath191 the symbol @xmath192 denotes the kronecker product of matrices , and @xmath193 is the matrix ( row vector ) whose all entries are equal to 1 . the matrix @xmath194 is responsible for multiplications of the matrices @xmath149 , @xmath195 , @xmath196 , @xmath197 by the input vector @xmath173 , the matrix @xmath198 - for multiplications of those products by the proper powers of @xmath153 , and the matrix @xmath199 - for aggregation of results . the process of going down by calculating the product @xmath160 , which has been presented in figures [ fig : figure2 ] and [ fig : figure3 ] , can be also described in the therm of matrices product . it means factorisation of the matrix @xmath194 into the product of @xmath131 matrices @xmath200 the matrices which occur on the right side of expression ( [ eq : rozklada ] ) have the following forms : @xmath201\ ] ] where @xmath202\otimes\mathbf{i}_{1}=\mathbf{a}_{2}^{(0)},\;\ ; \mathbf{\overline{a}}_{2\times 2}^{(1)}=\mathbf{a}_{2}^{(1)}\otimes\left [ 1\right]\otimes\mathbf{i}_{1}=\mathbf{a}_{2}^{(1)}.\ ] ] @xmath203,\ ] ] where @xmath204\otimes\mathbf{i}_{2},\;\ ; \mathbf{\overline{a}}_{4\times 8}^{(1)}=\mathbf{a}_{2}^{(1)}\otimes\left[0\;\ ; 1\right]\otimes\mathbf{i}_{2}\ ] ] and the matrix @xmath205 denotes the matrix @xmath206 which columns were circularly shifted by 2 positions to the right , and the matrix @xmath207 denotes the matrix @xmath206 which columns were circularly shifted by 2 positions to the left . the last matrix @xmath208 is defined as @xmath209\ ] ] where @xmath210\otimes\mathbf{i}_{n/2},\;\ ; \mathbf{\overline{a}}_{n\times nn}^{(1)}=\mathbf{a}_{2}^{(1)}\otimes\left[0\;\;0\;\;\ldots \;\;1\right]\otimes\mathbf{i}_{n/2}.\ ] ] using the expression ( [ eq : rozklada ] ) the algorithm ( [ eq : rozkladmac ] ) of calculating the product @xmath160 can be written as follows : @xmath211 the expression ( [ eq : productv ] ) allows for evaluating the total number of additions , which are needed to calculate the matrix - vector product @xmath160 . we assume that the input vector @xmath173 is real - valued . each of matrices @xmath212 , for @xmath213 , is the direct sum of @xmath214 identical blocks and the single block is the vertical concatenation of @xmath215 matrices . the firs , indicated by @xmath216 , and the last , indicated by @xmath217 , do not need any additions or subtractions by multiplying them by a vector . the @xmath218 others matrices , which are sums of @xmath216 and @xmath217 , after circularly shifting their columns , so multiplying each of them by a vector needs @xmath219 additions . to calculate the product @xmath220 we have to perform @xmath221 additions . the product of the matrix @xmath198 by a vector do not need any additions and the product of the matrix @xmath222 by a vector needs @xmath162 additions . thus the total number of additions by calculating the products @xmath160 , according to ( [ eq : productv ] ) , is equal to @xmath223 . example [ example4 ] shows the explicit form of the algorithm ( [ eq : productv ] ) with all occurring in it matrices for @xmath134 . [ example4 ] the algorithm ( [ eq : productv ] ) of calculating the product of matrix @xmath224 by the vector @xmath225 is as follows : @xmath226 where @xmath227= \left [ \begin{array}{cccccccc } 1&0&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0\\ 0&-1&0&0&0&0&0&0\\ 1&0&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&-1&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&-1&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&1\\ 0&0&0&0&0&0&0&-1\\ 0&0&0&0&0&0&1&0\\ \end{array } \right],\ ] ] @xmath228=\ ] ] @xmath229,\ ] ] @xmath230=\ ] ] @xmath231,\ ] ] @xmath232\otimes \mathbf{i}_{8},\ ] ] @xmath233 it is easy to check that in this case the total number of addition is equal to 48 and the number of multiplications is equal to 24 ( we can see it also in the figure [ fig : figure3 ] ) . now we return to the dfrht algorithm ( [ eq : transformatan ] ) . according to ( [ eq : sum ] ) the matrix @xmath121 can be written as the sum of the matrices @xmath149 , @xmath150 with coefficients @xmath234 . the transposed matrix @xmath235 can be written as the sum of the transposed matrices @xmath236 , @xmath237 with the same coefficients @xmath234 : @xmath238 since the matrices with the even indexes @xmath149 , @xmath239 are symmetric and the matrices with the odd indexes are asymmetric the expression ( [ eq : sumt ] ) can be written as follows : @xmath240 according to ( [ eq : rozkladmac ] ) the matrix @xmath121 from expression ( [ eq : sum ] ) can be transformed into the product @xmath241 so the matrix @xmath235 may by also transformed from ( [ eq : sumt1 ] ) into the following product : @xmath242 where the matrix @xmath243 is defined as follows : @xmath244 and @xmath245.\ ] ] the others matrices in the expression ( [ eq : rozkladmac1 t ] ) are the same as that in the expression ( [ eq : rozkladmac ] ) . taking into account the decompositions ( [ eq : rozkladmac1 ] ) and ( [ eq : rozkladmac1 t ] ) of matrices @xmath72 and @xmath117 respectively the dfrht algorithm ( [ eq : transformatan ] ) will take the form : @xmath246 where the matrix @xmath194 is decomposed according to ( [ eq : rozklada ] ) . for example , for @xmath134 this algorithm will take the following form : @xmath247 figure [ fig : figure4 ] shows a data flow diagram of the algorithm for 8 point dfrht . ) for @xmath134,scaledwidth=100.0% ] let us assess the computational complexity in term of numbers of multiplications and additions required for dfrht calculation . calculation of the discrete fractional hadamard transform for a real - valued vector @xmath63 of length @xmath7 , assuming that the matrix @xmath66 defined by ( [ eq : rozklad2 ] ) is given , requires @xmath248 multiplications of a complex number by a real number and @xmath249 complex additions . each multiplication of a complex number by a real number needs two real multiplications and each addition of two complex numbers requires two real additions . hence the numbers of real multiplications and real additions required for computing the dfrht using the naive method are equal to @xmath250 and @xmath251 respectively . let us now evaluate the computational complexity of the dfrht realization with the help of the procedure ( [ eq : transformata2 ] ) . as it was discussed in the section [ advantages ] , if we use the factorized representation of the matrices @xmath117 and @xmath72 , calculating the product of the real - valued matrix @xmath117 and the real - valued vector @xmath63 requires @xmath162 real multiplications and @xmath223 real additions . as a result , we again obtain the real - valued vector . then there is computed the product of the complex - valued diagonal matrix @xmath252 and the real - valued vector calculated previously ( we assume that for a predetermined parameter @xmath109 , the diagonal elements of this matrix were calculated in advance ) . the calculation of this product requires @xmath253 real multiplications . the resulting complex - valued vector is then multiplied by the factorized matrix @xmath72 . this operation requires @xmath254 real multiplications and @xmath255 real additions . the total numbers of arithmetic operations to compute dfrht of size @xmath12 using our new algorithm are @xmath256 real multiplications and @xmath257 real additions . it is easy to check that even for small @xmath131 the numbers of arithmetic operations required for realization of the proposed algorithm are several times less than in the naive method of computing . tables [ tab:1 ] and [ tab:2 ] display the numbers of multiplications and additions required for the dfrht transform implementation of the real - valued input signal of the length @xmath7 . these numbers were calculated for three methods of the transform implementation : the direct multiplication of the dfrht matrix by a vector of the input data , calculation according to authors algorithm described in the work @xcite , and according to the algorithm ( [ eq : transformata2 ] ) proposed in this article . it is easy to check that for @xmath258 the number of arithmetic operations , required for dfrht transform realization according to the proposed algorithm , is smaller than in the other two methods of dfrht computing . [ ! h ] rrrr @xmath7 & direct method & method @xcite & proposed algorithm + 2 & 8 & 6 & 10 + 4 & 32 & 18 & 32 + 8 & 128 & 54 & 88 + 16 & 512 & 162 & 224 + 32 & 2048 & 486 & 544 + 64 & 8192 & 1458 & 1280 + 128 & 32768 & 4374 & 2944 + 256 & 131072 & 13122 & 6656 + 512 & 524288 & 39366 & 14848 + 1024 & 2097152 & 118098 & 32768 + [ ! h ] rrrr @xmath7 & direct method & method @xcite & proposed algorithm + 2 & 4 & 5 & 6 + 4 & 24 & 25 & 36 + 8 & 112 & 95 & 144 + 16 & 480 & 325 & 480 + 32 & 1984 & 1055 & 480 + 64 & 8064 & 3325 & 1440 + 128 & 32512 & 10295 & 4032 + 256 & 130560 & 31525 & 10752 + 512 & 523264 & 95855 & 69120 + 1024 & 2095104 & 290125 & 168960 + the article presents the novel algorithm for the dfrht performing . the algorithm has a much lower computational complexity than the direct way of the dfrht implementation . the computational procedure for dfrht calculating is described in kronecker product notation . the kronecker product algebra is a very compact and simple mathematical formalism suitable for parallel realization . this notation enables us to represent adequately the space - time structures of an implemented computational process and directly maps these structures into the hardware realization space . for simplicity , we considered the synthesis of a fast algorithm for the dfrht calculation for @xmath259 . however it is clear that the proposed procedure was developed for the arbitrary case when the order of the matrix is a power of two . sylvester j.j . : thoughts on inverse orthogonal matrices , simultaneous sign successions , and tessellated pavements in two or more colours , with applications to newton s rule , ornamental tile - work , and the theory of numbers . philos . mag . 34 , 461 - 475 ( 1867 )

we present a novel algorithm for calculating the discrete fractional hadamard transform for data vectors whose size @xmath0 is a power of two . a direct method for calculation of the discrete fractional hadamard transform requires @xmath1 multiplications , while in proposed algorithm the number of real multiplications is reduced to @xmath0log@xmath2 . example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore

You are an expert at summarizing long articles. Proceed to summarize the following text: given a diagram @xmath2 of a link and a modulus @xmath3 , a ( fox ) coloring ( @xcite ) is an assignment of integers modulo @xmath0 to the arcs of @xmath2 such that at each crossing twice the color assigned to the over - arc equals the sum of the colors assigned to the under - arcs , modulo @xmath0 ( see figure [ fig : xtop ] ) . for each diagram and for each modulus @xmath3 there is always at least one solution to this problem namely by assigning the same color ( i.e. , integer modulo @xmath0 ) to each and every arc of the diagram ; thus there are exactly @xmath0 such solutions modulo @xmath0 . these are the trivial solutions modulo @xmath0 i.e. , the so - called trivial @xmath0-colorings of the diagram . the non - trivial @xmath0-colorings are the solutions , modulo @xmath0 , which involve at least two distinct colors . * remark*. we remark that it is well known that this system of equations is also a system of relations for the first homology group of the 2-fold branched covering along the link ( @xcite , theorem 3.3 ) . in fact , the fundamental group of the @xmath4-fold branched covering along a link is presented by labeling the arcs of the unoriented link diagram and having relations of the form @xmath5 read off at each crossing when @xmath6 is the label of the over - crossing line . it then follows that @xmath7 ( the first homology group of the @xmath4-fold branched covering along the link @xmath8 ) has presentation with @xmath9 , where @xmath10 are the corresponding elements in the abelianization of the fundamental group ( @xcite ) . should one set the color of one of the arcs equal to @xmath11 then there would be a bijective correspondence between this set of colorings and @xmath12 . it is interesting to remark that the fundamental group of the @xmath4-fold branched covering along the link is itself a non - abelian generalization of the fox coloring . while we do not use this aspect of the topology here , we are aware of it and it may be of use in later work . for more background on this material see @xcite . if a diagram endowed with an @xmath0-coloring undergoes a reidemeister move , there is a unique reassignment of colors to the arcs involved in the move such that the new assignment is an @xmath0-coloring of the resulting diagram . since these reassignments are reversible there is a bijection between the @xmath0-colorings before and after the performance of a finite number of reidemeister moves . furthermore , these reassignments preserve trivial @xmath0-colorings and thus they preserve also non - trivial @xmath0-colorings . therefore the number of @xmath0-colorings is a link invariant ; the fact that a diagram of a link admits or not non - trivial @xmath0-colorings is an invariant of that link . it is known that there are links which do not admit non - trivial colorings over a given modulus . for example , the trefoil only admits non - trivial colorings over moduli divisible by @xmath13 . in the course of our work on colorings , we have observed that for some choices of a modulus @xmath3 and a link admitting non - trivial @xmath0-colorings , the following occurs . there are distinct non - trivial @xmath0-colorings , @xmath14 ( realized on an otherwise arbitrary diagram @xmath2 of this link ) and there is a permutation @xmath15 of the @xmath0 colors such that , for each arc @xmath16 of @xmath2 , the colors assigned to @xmath16 in the coloring @xmath17 , say @xmath18 , and in the coloring @xmath19 , say @xmath20 , satisfy : @xmath21 two such colorings will be said `` related '' . an instance where this occurs is depicted in figure [ fig:9_40tri ] . on the other hand it is not true that any permutation transforms the colors of a coloring into the colors of another coloring ( see figure [ fig:9_40bis ] ) . moreover , given non - trivial @xmath0-colorings @xmath17 and @xmath19 , realized on the same diagram , it may happen that there is no permutation @xmath15 of the @xmath0 colors such that for each arc @xmath16 of @xmath2 @xmath21 we will then say `` @xmath19 is essentially distinct from @xmath17 '' , in the given modulus , and the colorings split into equivalence classes ( to be elaborated upon below ) . in figure [ fig:9_40 ] we list representatives of the distinct equivalence classes of the non - trivial @xmath22-colorings of @xmath23 . we will be primarily concerned with permutations that preserve the coloring equation at each crossing for these are the ones that actually give us a corresponding coloring of the link and we will show that the relation sketched above among @xmath0-colorings of a diagram is an equivalence relation ( see below ) . we remark that the articles @xcite and @xcite address the same topic as the current article . their definition of equivalent colorings assumes one has a list of all non - trivial @xmath0-colorings for a given diagram and states simply that any two of these colorings are equivalent provided there is a permutation of the @xmath0 colors that , for each arc in the diagram , sends the color in this arc in the source coloring to the color in the same arc in the target coloring . this is equivalent to our definition . unfortunately , for the purposes of counting equivalence classes of colorings in generic cases , the methodology in @xcite and @xcite seems to resort to generating classes of colorings by letting the symmetric group on the @xmath0 colors act on a given @xmath0-coloring . as we see in figure [ fig:9_40bis ] , there are assignments of colors to a diagram obtained in this way that do not constitute colorings . the formulas in the articles referred to above predict in general less equivalence classes than ours due to their over - counting of the elements on each orbit . the equivalence classes of colorings constitute a topological invariant and in this article we provide combinatorial information about them . we hope this will prove to be useful for topological purposes . in section [ sect : prelim ] we discuss preliminaries such as the nullity and the generating arcs of a coloring ( subsection [ subsect : nullity ] ) , and the definition of the equivalence classes ( subsection [ subsect : equivclass ] ) . in section [ sect : results ] we calculate the number of equivalence classes in an infinite number of instances . consider a link , @xmath8 , along with a diagram @xmath24 for that link . regarding the arcs of this diagram as algebraic variables we write the homogeneous system of linear equations consisting of the equations read off each crossing as illustrated in figure [ fig : xtop ] . we call the matrix of the coefficients of this homogeneous system of linear equations * the coloring matrix of @xmath24*. any coloring matrix is made up of integers . specifically , along each row one finds exactly two 1 s and one -2 , the rest being perhaps 0 s . thus , adding all the columns of a coloring matrix we obtain a column made up of 0 s . it follows that the determinant of any coloring matrix is 0 . upon performance of reidemeister moves on a diagram , the changes on the original coloring matrix are realized by operations that constitute a subset of the following operations on integer matrices . these operations are generated by 1 . multiplication of a row ( column ) by @xmath25 ; 2 . addition to one row ( column ) of integer linear combinations of other rows ( columns ) ; 3 . insertion ( deletion ) of a row and column made up of 0 s except for a 1 at the diagonal entry ; 4 . permutations of rows ( columns ) . these are the operations which relate equivalent matrices over the integers ( see @xcite , page 50 ) . so the equivalence class of a coloring matrix is a topological invariant of the link under study . for each of these equivalence classes of matrices over the integers there is an outstanding representative which is called the smith normal form ( see @xcite ) . although the smith normal form ( snf ) is a familiar object we elaborate here slightly about it in order to bring out some connections with colorings of knots which we do not find in the literature . an integer matrix in smith normal form is a matrix such that its entries are all zero except perhaps along the diagonal . along the diagonal the entries are non - negative ( without loss of generality ) and the @xmath26-th entry divides the @xmath27-th entry , up to a certain index @xmath28 , and after that , the entries are all @xmath11 s : @xmath29 the @xmath30 s are called the * invariant factors * of the equivalence class ; their name reflects the fact that the multi - set formed by them is an invariant of the equivalence class . this multi - set is then a topological invariant if it originates from a coloring matrix . moreover , the smith normal form of a coloring matrix is sure to have a @xmath11 at the last entry of the diagonal since we proved above that the determinant of a coloring matrix is @xmath11 . the product of the remaining entries of the diagonal of the smith normal form of a coloring matrix is * the determinant of the link * under study . this is also a topological invariant . ( in passing , it is known that for knots i.e. , @xmath31-component links , the determinant of the knot is an odd integer , see @xcite . ) we denote the smith normal form of a matrix @xmath32 by @xmath33 . being an element of the equivalence class of @xmath32 , @xmath33 is obtained by a finite number of the operations listed above . we may then collect all the information concerning the row operations into an invertible matrix called @xmath34 and likewise for the column operations into an invertible matrix called @xmath35 to state ( @xcite ) @xmath36 with the juxtaposition of pairs of consecutive symbols on the right - hand side of the equation denoting matrix multiplication . let us now fix an otherwise arbitrary link along with one of its diagrams . let us then relate the smith normal form ( and its invariant factors ) of the coloring matrix of this diagram to the corresponding system of linear homogeneous equations and its solutions . there are always solutions of this system of equations namely by assigning the same integer to each arc . this corresponds to the fact that the determinant of the coloring matrix is @xmath11 . one of the algebraic variables may take on any value and if there is no other zero entry along the diagonal of the smith normal form , then the remaining variables are uniquely determined once the former variable has been assigned a value . going back to the original system of equations we obtain the so - called trivial solutions i.e. , those solutions that assign the same value to each and every arc of the diagram . the invariant factors associated to our coloring matrix via its smith normal form allow us to do something else . suppose we choose a factor @xmath0 of one of these invariant factors and decide to work over the integers modulo @xmath0 . then our smith normal form in this new setting acquired at least one more @xmath11 along the diagonal . then , there is at least one more variable which can take on any value , modulo @xmath0 . going back to the original system of equations , there are at least two arcs which can take on any value modulo @xmath0 . hence , we now have polychromatic colorings i.e. , solutions where at least two distinct arcs take on two distinct colors that is , values modulo @xmath0 . had we chosen an @xmath0 which does not possess common factors with the invariant factors , then modulo @xmath0 there would have been only trivial colorings . [ prop : genarcsp]let @xmath37 be an odd prime . let @xmath2 be a link diagram . the number of @xmath11 s @xmath38modulo @xmath39 along the diagonal of the smith normal form of the coloring matrix , @xmath32 , of @xmath2 , equals the least number of arcs of @xmath2 that can independently receive colors modulo @xmath37 , and generate each @xmath37-coloring of @xmath2 . proof . if the smith normal form exhibits @xmath40 @xmath11 s modulo @xmath37 , this means that the space of solutions has dimension @xmath40 ; working modulo a prime implies we are doing linear algebra over a field so it makes sense to talk about dimensions of spaces and bases . then matrix @xmath35 in ( [ eqn : eqn ] ) above operates a change of basis taking us back to algebraic variables equivalent to the arcs of the original diagram . then @xmath40 of these arcs have to generate all the colorings ( i.e. , all the solutions of the indicated system of equations modulo @xmath37 ) in terms of a basis of coloring vectors . [ def : pnul ] the number @xmath40 in the proof of proposition [ prop : genarcsp ] is called the @xmath37-nullity @xmath38or the rank mod @xmath42 of the coloring matrix of @xmath2 . any set of @xmath40 arcs that can independently receive colors modulo @xmath37 and so generate each @xmath37-coloring of the diagram under study is said a set of generating arcs @xmath38of this diagram , with respect to this modulus@xmath43 . we keep the notation of proposition [ prop : genarcsp ] . if the @xmath37-nullity of a link is @xmath40 then there are @xmath44 @xmath37-colorings of the link , and @xmath45 non - trivial @xmath37-colorings of this link . there are @xmath37 integers mod @xmath37 so there are always @xmath37 trivial @xmath37-colorings and @xmath44 @xmath37-colorings . @xmath41 let @xmath0 be a composite positive integer . let @xmath2 be a link diagram . each zero @xmath38modulo @xmath46 along the diagonal of the smith normal form of the coloring matrix , @xmath32 , of @xmath2 contributes with a factor @xmath0 for the number of solutions . each zero divisor , @xmath47 , of @xmath0 along the diagonal contributes with a factor @xmath48 to the number of solutions . with @xmath49 for the number of @xmath11 s @xmath38modulo @xmath46 along the diagonal in @xmath33 , and @xmath50 for the set the of invariant factors of @xmath32 which are zero divisors of @xmath0 , the formula for the number of @xmath0-colorings of @xmath2 is : @xmath51 proof . the contribution of the @xmath49 zero s ( modulo @xmath0 ) along the diagonal of the smith normal form to the number of solutions is clear . for the contribution of the zero divisors along the diagonal to the number of solutions see @xcite , page 40 . this concludes the proof . @xmath41 in this section we introduce equivalence classes of colorings as orbits of actions of certain groups of permutations on the set of colorings of a diagram . in order for this notion to be topological we require a special kind of permutation which we call a * coloring automorphism*. these are permutations which comply with the coloring operation , @xmath52 in a pre - assigned modulus @xmath0 . this operation generalizes to the quandle operation , generalizing also the notion of coloring ( @xcite ) . in the particular instance @xmath53 we are dealing with the so - called dihedral quandles , one per integer modulus @xmath0 . [ def : auto ] given an integer @xmath54 , we define a coloring automorphism of @xmath55 to be a permutation , @xmath56 , of @xmath55 such that @xmath57 for all @xmath58 , with @xmath59 @xmath38mod @xmath60 , for every @xmath61 . in @xcite we find the following facts . for a given integer @xmath54 , each coloring automorphism of @xmath55 is given by : @xmath62 with @xmath63 and @xmath64 , the set of units of @xmath55 . the set of all these coloring automorphisms of @xmath55 equipped with composition of functions , constitutes a group isomorphic to the affine group over @xmath55 i.e. , isomorphic to the semi - direct product @xmath65 . we denote it @xmath66 . for any integer @xmath54 the inner coloring automorphism group of @xmath55 is generated by the automorphisms of the form @xmath67 . it is easy to see that this group consists of the elements of the form , @xmath68 if @xmath0 is even this subgroup is isomorphic to the dihedral group of order @xmath0 and @xmath69 can take on only `` even '' values from @xmath55 . if @xmath0 is odd , this subgroup is isomorphic to the dihedral group of order @xmath70 and @xmath69 can take on any value from @xmath55 . we denote it @xmath71 . this information about coloring automorphisms of @xmath55 is contained in @xcite . in the sequel , we will write `` automorphism '' ( respectively , `` inner automorphism '' ) instead of the longer `` coloring automorphism of @xmath55 '' ( respectively , `` inner coloring automorphism of @xmath55 '' ) since these are the only automorphisms of @xmath55 we consider in this article i.e. , the permutations of elements of @xmath55 that comply with the coloring operation . specifically , we will use the expression * automorphism * to designate a permutation of the form @xmath62 with @xmath63 and @xmath64 , and * inner automorphism * to designate a permutation of the form @xmath68 with @xmath69 taking on only `` even '' values from @xmath55 if @xmath0 is even ; with @xmath69 taking on any value from @xmath55 if @xmath0 is odd . we remark that it is well known that for a quandle @xmath72 and a diagram @xmath2 , the set of diagram colorings by elements of @xmath73 , @xmath74 is a q - quandle set , where the action of @xmath73 on @xmath74 is given by @xmath75 for a coloring @xmath76 and @xmath77 ( kamada was the first proponent of this language ) . our considerations for dihedral quandles are related to this . let @xmath78 be an integer . let @xmath8 be a link admitting non - trivial @xmath0-colorings . let @xmath2 be a diagram of @xmath8 . we let @xmath79 stand for the set of * non - trivial * @xmath0-colorings of @xmath2 . [ prop : action ] let @xmath78 be an integer . let @xmath8 be a link admitting non - trivial @xmath0-colorings and let @xmath2 be a diagram of @xmath8 . let @xmath80 be a subgroup of @xmath81 . then @xmath80 acts on @xmath82 by permutations . specifically , given @xmath83 and @xmath84 , an @xmath0-coloring of @xmath2 with colors @xmath85 , then @xmath86 is the @xmath0-coloring of @xmath2 obtained by replacing each color @xmath85 by @xmath87 . moreover , this action is faithful and if @xmath0 is prime this action is also free . * proof*. we keep the notation of the statement . we regard @xmath88 as the map which assigns colors to the arcs of @xmath2 in such a way that , @xmath89 , where @xmath90 designates the index of the over - arc of the crossing where under - arcs with indices @xmath26 and @xmath91 meet , see figure [ fig : xtop ] ( where now each @xmath92 should be read @xmath93 ) . so , given @xmath83 and @xmath88 , then @xmath86 is such that @xmath94 so @xmath86 is again an @xmath0-coloring of @xmath2 . clearly , the identity element @xmath95 is such that @xmath96 . furthermore , for any two @xmath97 , the composition of functions guarantees that @xmath98 . we now prove that this action is faithful i.e. , we prove that given a non - identity @xmath83 there exists a coloring @xmath88 such that @xmath99 . we recall that the elements of @xmath80 are , in particular , permutations of the elements of @xmath55 . so given a non - identity element of @xmath80 which moves @xmath100 , then the coloring obtained by assigning @xmath26 to one of the generating arcs of the diagram is transformed via @xmath101 into a coloring where now this generating arc is assigned @xmath102 . we now prove that this action is free i.e. , that if given @xmath103 there exists a coloring @xmath88 such that @xmath104 then @xmath105 . we recall that , for some @xmath106 and @xmath107 , @xmath108 , @xmath109 for any @xmath110 . since @xmath104 then there exists two distinct colors in @xmath55 , say @xmath111 such that @xmath112 and @xmath113 . more precisely , @xmath114 since @xmath115 and @xmath0 is prime . thus @xmath105 . this concludes the proof . @xmath41 [ def : gequivd ] let @xmath80 be a subgroup of @xmath66 . the @xmath80-equivalence classes of @xmath0-colorings of @xmath2 are , by definition , the @xmath80-orbits over @xmath82 . we will next prove that this notion provides topological invariants ( in particular , the number of equivalence classes of @xmath0-colorings of a link ) . it is well known that , given any two diagrams of the same link , there is a bijection between the two sets of @xmath0-colorings of these diagrams ( @xcite , @xcite , lemma @xmath116 ) . moreover , this bijection takes trivial @xmath0-colorings to trivial @xmath0-colorings and non - trivial @xmath0-colorings to non - trivial @xmath0-colorings . this bijection is realized by the `` colored reidemeister moves '' . the colored reidemeister moves apply to a diagram endowed with an @xmath0-coloring ; a reidemeister move is applied to the diagram and a local adjustment of the coloring is performed . these adjustements are unique and reversible thereby proving the bijection between the two sets of @xmath0-colorings of any two diagrams of the same link . [ prop : equivtop ] let @xmath0 be an integer greater than @xmath31 . let @xmath8 be a link admitting non - trivial @xmath0-colorings and let @xmath2 and @xmath117 be two diagrams of @xmath8 . let @xmath80 be a subgroup of @xmath66 . there is a bijection from @xmath82 to @xmath118 , which preserves the @xmath80-equivalence classes . * proof * : from @xcite we know that the colored reidemeister moves realize a bijection from the set of @xmath0-colorings of @xmath2 to the set of @xmath0 colorings of @xmath117 , taking non - trivial colorings to non - trivial colorings . we now prove that the colored reidemeister moves take distinct elements of @xmath82 along a @xmath80-equivalence class , to distinct elements of @xmath118 along a @xmath80-equivalence class . specifically , for @xmath83 and @xmath88 , we prove that the `` colored reidemeister moves '' take @xmath88 to @xmath119 and @xmath120 to @xmath121 . the proofs of these statements for the individual `` colored reidemeister moves '' of type i , ii , and iii are displayed in figures [ fig : r1 ] , [ fig : r2 ] , and [ fig : r3 ] . the @xmath122 associates horizontally colorings on distinct diagrams related by a colored reidemeister move . vertically we display colorings @xmath84 and @xmath86 ( @xmath123 and @xmath124 , respect . ) for diagram @xmath2 ( @xmath117 , respect . ) . in figures [ fig : r1 ] and [ fig : r2 ] circles with dotted lines were drawn to bring out the local nature of the transformation . this was not done in figure [ fig : r3 ] in order not to overburden the figure . @xmath41 * remark . * proposition 2.3 can also be seen by regarding mod - m colorings as elements of @xmath125 . [ cor : numbers ] let @xmath8 be a link and @xmath2 one of its diagrams . let @xmath3 be an integer . the number of @xmath80-equivalence classes of @xmath0-colorings of @xmath2 is a topological invariant . the multi - set whose elements are the number of @xmath0-colorings per @xmath80-equivalence class of @xmath0-colorings of @xmath2 is a topological invariant . * proof*. this is a straight - forward consequence of proposition [ prop : equivtop ] . @xmath41 we remark that in the sequel @xmath80 , the subgroup of @xmath66 , will be either @xmath66 itself or @xmath71 . figure [ fig : fig8 ] illustrates the fact that in general there are more inner equivalence classes than equivalence classes ( for the same link and for the same modulus ) . in this section we apply the theory developed above to specific situations . * we use @xmath37-nullity as in definition [ def : pnul ] . * [ prop : numberequivclasscol]let @xmath37 be an odd prime and @xmath40 an integer greater than @xmath31 . a link @xmath8 with @xmath37-nullity @xmath40 has @xmath126 equivalence classes of @xmath37-colorings . * proof*. as discussed right after definition [ def : auto ] , an automorphism of @xmath127 @xmath128 depends on two parameters @xmath129 and @xmath130 . since @xmath131 and @xmath132 , there are then exactly @xmath133 automorphisms for @xmath127 . now suppose @xmath17 is in @xmath134 , where @xmath2 is a diagram of @xmath8 . since the action of @xmath135 is free ( [ prop : action ] ) each orbit of the action has exactly @xmath133 elements . since there are @xmath45 elements in @xmath134 , there are then @xmath136 orbits of this action which is the number of equivalence classes of @xmath37-colorings for link @xmath8 . @xmath41 [ cor : mincol ] we keep the notation of proposition [ prop : numberequivclasscol ] . 1 . if a diagram @xmath2 of link @xmath8 admits a non - trivial @xmath37-coloring with the least number of colors ( over all diagrams , over all non - trivial @xmath37-colorings ) , then there are at least @xmath133 such @xmath37-colorings of @xmath2 . if the nullity of @xmath8 mod @xmath37 is @xmath4 and a diagram @xmath2 of @xmath8 admits a non - trivial @xmath37-coloring with @xmath137 colors , then any other non - trivial @xmath37-coloring of @xmath2 uses @xmath137 colors . in particular , if @xmath2 is a diagram of @xmath8 where a non - trivial @xmath37-coloring is realized with the least number of colors , then any other non - trivial @xmath37-coloring of this diagram uses also the least number of colors . proof . 1 . since the automorphisms are permutations of the @xmath37 colors they preserve the number of distinct colors so if a non - trivial @xmath37-coloring of a diagram uses @xmath137 colors , then along its equivalence class the non - trivial @xmath37-colorings use @xmath137 colors each and there are @xmath133 non - trivial @xmath37-colorings per equivalence class . if a diagram @xmath2 of link @xmath8 admits a non - trivial @xmath37-coloring with the least number of colors then along its equivalence class the non - trivial @xmath37-colorings use the same number of colors each . if the nullity of @xmath8 mod @xmath37 is @xmath4 then there is only @xmath31 equivalence class ( mod @xmath37 ) . then the arguing of @xmath138 is valid for the @xmath133 non - trivial @xmath37-colorings in this orbit . @xmath41 [ cor : d ] let @xmath8 be a link with the following property . the smith normal form of a@xmath38ny@xmath43 coloring matrix of @xmath8 has only one @xmath11 and only one @xmath139 along the diagonal . then for any prime @xmath37 such that @xmath140 , there is only one equivalence class of @xmath37-colorings . in particular , rational links satisfy this property . working mod @xmath37 the smith normal form of the coloring matrix will exhibit exactly two zeros . hence the @xmath37-nullity is @xmath4 and the result follows from proposition [ prop : numberequivclasscol ] . @xmath41 [ cor : otherd ] the following links have only one class of @xmath37-colorings for each prime @xmath37 for which they admit non - trivial @xmath37-colorings . 1 . links whose determinant is prime . links of non - zero determinant whose smith normal form of the coloring matrix displays different primes on different diagonal entries @xmath38besides the @xmath11 entry and possible @xmath31s@xmath43 . knots whose knot group can be presented using one relator @xmath38 in particular , torus knots@xmath43 . proof . @xmath138 and @xmath141 are particular cases of corollary [ cor : d ] . as for @xmath142 , since the deficiency of knot groups is one then knot groups which can be presented with one relator only need two generators . then the smith normal form of the coloring matrix is @xmath143 where @xmath144 is the determinant of the knot . @xmath41 let @xmath37 be an odd prime and @xmath40 an integer greater than @xmath31 . a link @xmath8 with @xmath37-nullity @xmath40 has @xmath145 inner - equivalence classes of @xmath37-colorings . * proof*. as discussed right after definition [ def : auto ] , an inner - automorphism of @xmath127 is of the form @xmath146 with @xmath130 . there are then exactly @xmath147 inner - automorphisms for @xmath127 . the rest of the proof goes through as in the proof of proposition [ prop : numberequivclasscol ] leading to the following number of inner - orbits @xmath148 @xmath41 [ cor : innerd ] let @xmath8 be a link with the following property . the smith normal form of a@xmath38ny@xmath43 coloring matrix of @xmath8 has only one @xmath11 and only one @xmath139 along the diagonal . then for any prime @xmath37 such that @xmath140 , there are @xmath149 equivalence class of @xmath37-colorings . in particular , rational links satisfy this property . proof . adapt the proof for corollary [ cor : d ] . @xmath41 [ cor : innerotherd ] the following links have @xmath149 classes of @xmath37-colorings for each prime @xmath37 for which they admit non - trivial @xmath37-colorings . 1 . links whose determinant is prime . links of non - zero determinant whose smith normal form of the coloring matrix displays different primes on different diagonal entries @xmath38besides the @xmath11 entry and possible @xmath31s@xmath43 . knots whose knot group can be presented using one relator @xmath38 in particular , torus knots@xmath43 . adapt the proof for corollary [ cor : otherd ] . in the context of quandles this work has to do with homomorphisms from the fundamental quandle of the knot to the dihedral quandles ( @xcite ) . we organize these homomorphisms into equivalence classes . in future work we plan to generalize this work to other classes of target quandles , other than the dihedral quandles . j.g . thanks for support from nsfc ( grant no . 11171279 and no . 11271307 ) . p.l . acknowledges support from fct ( fundao para a cincia e a tecnologia ) , portugal , through project number ptdc / mat/101503/2008 , `` new geometry and topology '' . p.l . also thanks the school of mathematical sciences at the university of nottingham for hospitality . l. h. kauffman , _ knots and physics _ , first edition , second edition , third edition , fourth edition , series on knots and everything * 1 , 53 * , world scientific publishing co. pte . ltd . , singapore and river edge and hackensack , new jersey 1991,1994,2001,2013 .

for any link and for any modulus @xmath0 we introduce an equivalence relation on the set of non - trivial @xmath0-colorings of the link ( an @xmath0-coloring has values in @xmath1 ) . given a diagram of the link , the equivalence class of a non - trivial @xmath0-coloring is formed by each assignment of colors to the arcs of the diagram that is obtained from the former coloring by a permutation of the colors in the arcs which preserves the coloring condition at each crossing . this requirement implies topological invariance of the equivalence classes . we show that for a prime modulus the number of equivalence classes depends on the modulus and on the rank of the coloring matrix ( with respect to this modulus ) . keywords : links , colorings , equivalence classes of colorings msc 2010 : 57m27

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Proceed to summarize the following text: a significant theory regarding inequalities and exponential convexity for real valued functions , has been developed @xcite . the intention to generalize such concepts for the @xmath0-semigroup of operators , is motivated from @xcite . + in the present article , we shall derive a jessen s type inequality and the corresponding adjoint - inequality , for some @xmath0-semigroup and the adjoint - semigroup , respectively . + + the notion of banach lattice was introduced to get a common abstract setting , within which one could talk about the ordering of elements . therefore , the phenomena related to positivity can be generalized . it had mostly been studied in various types to spaces of real - valued functions , e.g. the space @xmath1 of continuous functions over a compact topological space @xmath2 , the lebesque space @xmath3 or even more generally the space @xmath4 constructed over measure space @xmath5 for @xmath6 . we shall use without further explanation the terms , order relation ( ordering ) , ordered set , supremum , infimum . + firstly , we shall go through the definition of vector lattice . any ( real ) vector space @xmath7 with an ordering satisfying ; @xmath8 : : : @xmath9 implies @xmath10 for all @xmath11 @xmath12 : : : @xmath13 implies @xmath14 for al @xmath15 and @xmath16 is called an _ ordered vector space_. the axiom @xmath8 , expresses the translation invariance and therefore implies that the ordering of an ordered vector space @xmath7 is completely determined by the positive part @xmath17 of @xmath7 . in other words , @xmath9 if and only if @xmath18 . moreover , the other property @xmath12 , reveals that the positive part of v is a convex set and a cone with vertex @xmath19 ( mostly called the _ positive cone _ of v ) . * an ordered vector space @xmath7 is called a _ vector lattice _ , if any two elements @xmath20 have a supremum , which is denoted by @xmath21 and an infimum denoted by @xmath22 . + it is trivially understood that the existence of supremum of any two elements in an ordered vector space implies the existence of supremum of finite number of elements in @xmath7 . furthermore , @xmath23 implies @xmath24 , so the existence of finite infima therefore implied . * few important quantities are defined as follows @xmath25 * some compatibility axiom is required , between norm and order . this is given in the following short way : @xmath26 the norm defined on a vector lattice is called a lattice norm . now , we are in a position , to define a banach lattice in a formal way . a _ banach lattice _ is a banach space @xmath7 endowed with an ordering @xmath27 , such that @xmath28 is a vector lattice with a lattice norm defined on it . + a banach lattice transforms to _ banach lattice algebra _ , provided @xmath29 implies @xmath30 . @xmath31 + a linear mapping @xmath32 from an ordered banach space @xmath7 into itself is _ positive _ ( denoted by : @xmath33 ) if @xmath34 , for all @xmath35 . the set of all positive linear mappings forms a convex cone in the space @xmath36 of all linear mappings from @xmath7 into itself , defining the natural ordering of @xmath36 . the absolute value of @xmath32 , if it exists , is given by @xmath37 thus @xmath38 is positive if and only if @xmath39 holds for any @xmath15 . * [ @xcite , p.249 ] * a bounded linear operator @xmath32 on a banach lattice v is a positive contraction if and only if @xmath40 for all @xmath15 . @xmath31 + an operator @xmath41 on @xmath7 satisfies the positive minimum principle if for all @xmath42 , @xmath43 @xmath44 a ( one parameter ) @xmath0-semigroup ( or strongly continuous semigroup ) of operators on a banach space @xmath7 is a family @xmath45 such that ( i ) : : @xmath46 for all @xmath47 . ( ii ) : : z(0)=i , the identity operator on v. ( iii ) : : for each fixed @xmath15 , @xmath48(with respect to the norm on v ) as @xmath49 . where @xmath50 denotes the space of all bounded linear operators defined on a banach space v. the ( infinitesimal ) generator of @xmath51 is the densely defined closed linear operator @xmath52 such that @xmath53 @xmath54 where , for @xmath55 , @xmath56f}{t}\,\,\,\,(f\in v).\ ] ] @xmath31 + let @xmath51 be the strongly continuous positive semigroup , defined on a banach lattice v. the positivity of the semigroup is equivalent to @xmath57 where for positive contraction semigroups @xmath51 , defined on a banach lattice v we have ; @xmath58 the literature presented in @xcite , guarantees the existence of the strongly continuous positive semigroups and positive contraction semigroups on banach lattice v , with some conditions imposed on the generator . the very important amongst them is , that it must always satisfy ( [ pmp ] ) . + a banach algebra @xmath59 , with the multiplicative identity element @xmath60 , is called the _ unital banach algebra_. we shall call the strongly continuous semigroup @xmath51 defined on @xmath59 , a _ normalized semigroup _ , whenever it satisfies @xmath61 the notion of normalized semigroup is inspired from normalized functionals @xcite . the theory presented in next section , is defined on such semigroups of positive linear operators defined on a banach lattice @xmath7 . in 1931 , jessen @xcite gave the generalization of the jensen s inequality for a convex function and positive linear functionals . see ( @xcite , pp-47 ) . we shall prove this inequality for a normalized positive @xmath0-semigroup and convex operator , defined on a banach lattice . + throughout the present section , @xmath7 will always denote a unital banach lattice algebra , endowed with an ordering @xmath27 . let @xmath62 be a nonempty open convex subset of @xmath7 . an operator @xmath63 is convex if it satisfies @xmath64 whenever @xmath65 and @xmath66 . @xmath31 + let @xmath67 denotes the set of all differentiable convex functions @xmath68 . * ( jessen s type inequality ) * let @xmath51 be the positive @xmath0-semigroup on @xmath7 such that it satisfies ( [ nsg ] ) . for an operator @xmath69 and @xmath70 ; @xmath71 * proof : * since @xmath68 is convex and differentiable , by considering an operator - analogue of [ theorem a , pp-98,@xcite ] , we have for any @xmath72 , there is a fixed vector @xmath73 such that @xmath74 using the property ( [ nsg ] ) along with the linearity and positivity of operators in a semigroup , we obtain @xmath75 in this inequality , set @xmath76 and the assertion ( [ jti ] ) follows . @xmath31 + the existence of an identity element and condition ( [ nsg ] ) , imposed in hypothesis of the above theorem is necessary . we shall elaborate the said , by following examples . let @xmath77 , @xmath51 be the left shift semigroup defined on x and @xmath78 taking the mirroring along @xmath79-axis . the identity function does not contain a compact support and therefore is not in @xmath59 . if we now take a bell - shaped curve like @xmath80 , @xmath81 . then f is positive , @xmath82 , and @xmath83 has maximum at @xmath84 , and it is between 0 and 1 elsewhere . on the other hand , @xmath85 has a maximum at @xmath86 and it is immediate that we can not compare the two functions in the usual ordering . see figure 1(a ) . let @xmath87 , and @xmath88 . the rotation semigroup @xmath51 is defined as , @xmath89 , @xmath90 . the identity element @xmath91 , s.t . for all @xmath92 , @xmath93 . then @xmath94 . or we can say that any complex number @xmath95 is mapped to @xmath96 . @xmath97 satisfies ( [ nsg ] ) , only when @xmath98 is a multiple of @xmath99 . let @xmath100 , then @xmath101 , hence @xmath102 . on the other hand , @xmath103 , and @xmath104 . hence , the equality holds in ( [ jti ] ) when @xmath98 is a multiple of @xmath99 , but the two sides are not comparable in general . it can easily be verified that @xmath105 is a subgroup of @xmath106 , as @xmath107 . therefore @xmath108 is a normalized semigroup . see figure 1(b ) . + @xmath31 + in previous section , a jessen s type inequality has been derived , for a normalized positive @xmath0-semigroup @xmath51 . this gives us the motivation towards knowing the behaviour of its corresponding adjoint semigroup @xmath109 on @xmath110 . as the theory for dual spaces gets more complicated , we do not expect to have the analogous results . it may ask for a detail introduction towards a part of the dual space @xmath110 , for which an adjoint of jessen s type inequality makes sense . given two banach spaces @xmath59 and @xmath111 and a bounded linear operator @xmath112 , recall that the adjoint @xmath113 is defined by @xmath114 for a strongly continuous positive semigroup @xmath51 on a banach space @xmath59 , by defining @xmath115 for every @xmath98 , we get a corresponding adjoint semigroup @xmath109 on the dual space @xmath116 . in @xcite , it is obtained that , the adjoint semigroup @xmath109 fails in general to be strongly continuous . the investigation @xcite , shows that @xmath109 acts in a strongly continuous way on ; @xmath117 this is the maximal such subspace on @xmath116 . the space @xmath118 was introduced by philips in 1955 , and latter has been studied extensively by various authors . at the present moment , we do not necessarily require the strong continuity of the adjoint semigroup @xmath109 on @xmath116 . + if @xmath59 is an ordered vector space , we say that a functional @xmath119 on @xmath59 is positive if @xmath120 , for each @xmath121 . by the linearity of @xmath119 , this is equivalent to @xmath119 being order preserving . i.e. @xmath122 implies @xmath123 . the set @xmath124 of all positive linear functionals on @xmath59 , is a cone in @xmath116 . + we are mainly interested in the study of the space @xmath110 , where in our case @xmath7 is a banach lattice algebra . let us consider the regular ordering among the elements of @xmath110 . i.e. @xmath125 , whenever @xmath126 , for each @xmath127 . + consider the convex operator ( [ co ] ) . in case of the equality , @xmath128 is simply a linear operator and the adjoint @xmath128 can be defined as above . but how can it be defined in other case ? this question has already been answered . + in @xcite , some kind of adjoint has been associated to a nonlinear operator @xmath128 . in fact , this is possible for lipschitz continuous operators only . consider the banach space @xmath129 of all lipschitz continuous operators @xmath130 satisfying @xmath131 , equipped with the norm @xmath132_{lip}= sup_{x_1\neq x_2}\frac { \|f(x_1)-f(x_2)\|}{\|x_1-x_2\|},\quad x_1,x_2\in x.\ ] ] where @xmath133 is the identity . it is easy to see that the space @xmath134 of all bounded linear operators from x to y is a closed subspace of @xmath129 . in particular , we set @xmath135 and call @xmath136 the pseudo - dual space of @xmath59 ; this space contains the usual dual space @xmath116 as closed subspace . + for @xmath137 , the pseudo - adjoint @xmath138 of @xmath128 is defined by ; @xmath139 this is of course a straightforward generalization of ( [ co ] ) ; in fact , for linear operators @xmath140 we have @xmath141 . i.e. the restriction of the pseudo - adjoint to the dual space is the classical adjoint . + for the sake of convenience , we shall denote the adjoint of the operator @xmath128 by @xmath142 , throughout the present section . either it s a classical adjoint or the pseudo - adjoint ( depending upon the operator @xmath128 ) . + similarly , the considered dual space of the vector lattice algebra @xmath7 will be denoted by @xmath110 , which can be the intersection of the pseudo - dual and classical dual spaces in case of a nonlinear convex operator . let @xmath128 be the convex operator on a banach space @xmath59 , then the adjoint operator @xmath142 on the dual space @xmath116 is also convex . * * proof:**for @xmath121 and @xmath143@xmath144 where @xmath145 . by putting @xmath146 , for @xmath147 and using the convexity of the operator @xmath128 we finally get @xmath148 hence , @xmath142 is convex on @xmath116 . * ( adjoint - jessen s inequality ) * let @xmath109 be the adjoint semigroup on @xmath110 such that the original semigroup @xmath51 , the operator @xmath78 and the space @xmath7 are same as in theorem ( 1 ) . for a convex operator @xmath149 and @xmath150 @xmath151 * proof : * for @xmath15 and @xmath150 , consider @xmath152,f ) & = & ( z^\ast(t)f^\ast,\phi(f ) ) \\ & = & ( f^\ast , z(t)(\phi f ) ) \\ & \geq & ( f^\ast,\phi(z(t)f ) ) \\ & = & ( \phi^\ast ( f^\ast),z(t)f ) \\ & = & ( z^\ast(t)[\phi^\ast f^\ast],f ) \end{aligned}\ ] ] therefore , the assertion ( [ ajti ] ) is satisfied . in this section we shall define the exponential convexity of an operator . moreover , few complex structures , involving the operators from a semigroup , will be proved to be exponentially convex . let @xmath7 be a banach lattice endowed with ordering @xmath27 . an operator @xmath153 is exponentially convex if it is continuous and for all @xmath154 @xmath155 where @xmath156 such that @xmath157 , @xmath158 . * * proof:**@xmath163 + take any @xmath156 and @xmath164 , @xmath162 . since the interval @xmath165 is convex , the midpoints , @xmath166 . now set @xmath167 , for @xmath162 . then we have , @xmath168 , for all @xmath158 . therefore , for all @xmath154 , we can apply @xmath169 to get , @xmath170 @xmath171 + let @xmath172 , such that @xmath173 , for @xmath158 . define @xmath174 , so that @xmath175 . therefore , for all @xmath154 , we can apply @xmath176 to get , @xmath177 @xmath31 + let @xmath159 be an exponentially convex operator . writing down the fact for @xmath178 , in ( 10 ) , we get that @xmath179 , for @xmath180 and @xmath15 . for @xmath181 , we have @xmath182 hence , for @xmath183 and @xmath184 , we have @xmath185 i.e. @xmath153 , does indeed satisfy the condition of convexity . @xmath31 + for @xmath186 , let us assume that @xmath63 is continuously differentiable on @xmath62 . i.e. the mapping @xmath187 , is continuous . moreover @xmath188 , will be a continuous linear transformation from @xmath7 to @xmath189 . a bilinear transformation @xmath190 defined on @xmath191 is symmetric if @xmath192 for all @xmath20 . such a transformation is * positive definite [ nonnegative definite ] * , if for every nonzero @xmath15 , @xmath193 [ @xmath194 . then , @xmath188 is symmetric wherever it exists . see [ @xcite , pp-69 ] . [ condif ] let @xmath128 be continuously differentiable and suppose that second derivative exists throughout an open convex set @xmath186 . then @xmath128 is convex on @xmath62 if and only if @xmath188 is nonnegative definite for each @xmath195 . and if @xmath188 is positive definite on @xmath62 , then @xmath128 is strictly convex . let @xmath7 be a unital banach algebra . for @xmath15 , a family of operators @xmath199 is defined as @xmath200 then @xmath201 . whenever , @xmath35 , @xmath202 , therefore by theorem 3 , the mapping @xmath203 is convex . [ t4 ] let @xmath51 be the positive @xmath0-semigroup , defined on a unital banach lattice algebra @xmath7 , such that it satisfies ( [ nsg ] ) . let @xmath15 , such that @xmath204 , for @xmath205 , @xmath206 , if @xmath207 and @xmath208 , if @xmath209 . let us define @xmath210 then ( i ) : : for every @xmath154 and for every @xmath211 , @xmath212 , @xmath213_{i , j=1}^n\geq 0.\ ] ] ( ii ) : : if the mapping @xmath214 is continuous on @xmath215 , then it is exponentially convex on @xmath215 . * * proof:**consider the operator @xmath216 for @xmath217 , @xmath218 and @xmath219 where @xmath220 . then @xmath221 so , @xmath222 is a convex operator . therefore by applying ( [ jti ] ) we get @xmath223 and the assertion ( [ a ] ) follows . assuming the continuity and using the * proposition 1 * we have also exponential convexity of the operator @xmath214 . @xmath31 + * proof : * similar to the proof of theorem ( [ t4 ] ) . @xmath31 + + * competing interests * + the authors declare that they have no competing interests . + * author s contribution * + all authors contributed equally and significantly in writing this paper . all authors read and approved the final manuscript . + * acknowledgement * + authors of this paper are grateful to prof . andrs btkai for his generous help in construction of examples . 20 , _ convexity , subadditivity and generalized jensen s inequality _ , vol 4 , no . 2 , 183194 . 2013 . , _ exponential convexity , positive semi - definite matrices and fundamental inequalities _ , j. math . , vol 4 , no 2 , 171 - 189 . 2010 . , _ nonlinear spectral theory _ , walter de gruyter . new york , 2004 . , _ inequalities _ , springer - verlag , berlin , 1961 . , _ interpolation and approximation _ , dover , new york , 1975 . , _ ber die entwicklung realen funktionen in reihen mittelst der methode der kleinsten quadrate _ , j. reine angewendte math . , no 94 , 4173 . 1883 . , _ cauchy type means on one - parameter @xmath0-group of operators _ , j. math . vol 9 , no . 2 , 631639 . 2015 . , _ bemaerkinger om konvekse funktioner og uligheder imellem middelvaerdier i _ , muf . tidsskrifr , 17 - 28 . 1931 . , _ konano dimenzionnalni vektorski prostori i primjene _ , tehnike knjige , zagreb , 1990 . , _ classical and new inequalities in analysis _ , kluwer academic publishers , the netherlands , 1993 . , _ one - parameter semigroups of positive operators _ , lect . notes in math . 1184 , springer - verlag , 1986 . , _ convex functions , partial orderings , and statistical applications _ , academic press , inc . new york , 1992 . , _ the adjoint semi - group _ , pac . j. math . no 5 , 269 - 283 , 1955 . , _ convex functions _ , academic press , new york and london , 1973 . , _ the exponential and logarithmic functions on commutative banach algebras _ , int . journal of math . analysis , vol . 42 , 2079 - 2088 . 2010 . , _ the adjoint of a positive semigroup _ , compositio mathematica , no 90 , 99 - 118 , 1994

in this paper the jessen s type inequality for normalized positive @xmath0-semigroups is obtained . an adjoint of jessen s type inequality has also been derived for the corresponding adjoint - semigroup , which does not give the analogous results but the behavior is still interesting . moreover , it is followed by some results regarding positive definiteness and exponential convexity of complex structures involving operators from a semigroup . example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore

You are an expert at summarizing long articles. Proceed to summarize the following text: smart and responsive complex materials can be achieved by self - organization of simple building blocks . by now , a broad range of functionalized colloidal and polymeric building blocks have been proposed and designed . @xcite this comprises synthetic colloidal structures , e.g. , patchy or janus colloids @xcite or biological molecules such as dna duplexes . @xcite these building blocks are able to self - organized into gel - like structures , e.g. , hydrogels , which are able to undergo reversible changes in response to external stimuli.@xcite thereby , rodlike molecules , such as viruses @xcite or telechelic associative polymers , @xcite exhibit novel scaffold - like structures , and theoretical and experimental studies have been undertaken to unravel their structural and dynamical properties in suspensions . here , polymer flexibility and end - interactions are the essential parameters to control the properties of the self - assembled network structures . @xcite the appearing structures can be directed and controlled by external parameters , specifically by the application of external fields such as a shear flow.@xcite here , a fundamental understanding of the nonequilibrium response of a network structure is necessary for the rational design of new functional materials and that of already existing synthetic and biological scaffold - like patterns . @xcite computer simulations are an extremely valuable tool to elucidate the self - organized structures of functionalized polymers . monte carlo @xcite and molecular dynamics simulation @xcite studies of coarse - grained models of end - functionalized flexible , semiflexible , and rodlike polymers in solution have shown that in thermal equilibrium self - organized scaffold - like network structures form above a critical attraction strength and within a range of concentrations . this network formation is strongly affected by the polymer flexibility , because flexible polymers can span a larger range of distances between connections points , even form loops , and deform easily thereby generating softer networks . the molecular dynamics simulation studies of telechelic polymers of ref . predict flower - like micellar aggregates for flexible polymers . for stiffer polymers , significant morphological changes appear , with liquid - crystalline - like order of adjacent polymers and inter - connected structures.@xcite recent nonequilibrium simulations of end - functionalized rodlike polymers exhibit further structural changes under shear flow . @xcite at low shear rates , the scaffold structure compactifies , while at intermediate shear rates novel bundle - like structures appear with nematically ordered rods . in the limit of very strong flows , all structures are dissolved and the rodlike polymers align in a nematic fashion . in this article , we extend the previous studies and investigate the influence of shear flow on the scaffold - like network structure of end - functionalized _ semiflexible _ polymers . both , the structure properties under shear flow as well as the rheological properties are analyzed for various shear rates . we find that an initial scaffold structure breaks up and densified aggregates are formed at low shear rates , while the structural integrity is completely lost at high shear rates . thereby , flexibility gives rise to particular compact aggregates at intermediate shear rates . in addition , the relaxation behavior of shear - induced structures after cessation of flow is analyzed in part in order to elucidate the reversibility of the shear - induced structures . we apply a hybrid simulation approach , which combines the multiparticle collision dynamics ( mpc ) method for the fluid , @xcite which accounts for hydrodynamic interactions , @xcite with molecular dynamics simulations for the semiflexible polymers . @xcite the mpc method has successfully been applied to study the equilibrium and nonequilibrium dynamical properties of complex systems such as polymers , @xcite colloids , @xcite vesicles and blood cells , @xcite as well as various active systems . @xcite the combination of coarse - grained modeling of end - functionalized polymers and a particle - based mesoscale hydrodynamic simulation technique is ideally suited for such a study . on the one hand , we want to elucidate the general principles of structure formation under nonequilibrium conditions . the achieved insight will be useful to understand the behavior of a broad spectrum of experimental systems , ranging from highly flexible synthetic polymers , e.g. , telechelics , to stiff biological macromolecules , such as dna segments . on the other hand , mesoscale hydrodynamic simulation approaches are essential , because only they allow to reach the large length and time scales , which are required to capture the long structural relaxation times in shear flow with typical shear rates of @xmath0 hz . @xcite in addition and most importantly , particle - based hydrodynamic simulation approaches naturally include thermal fluctuations , which are indispensable for a proper description of polymer entropy and entropic elasticity . of course , coarse - grained modeling has its limitations in predicting the behavior of particular experimental systems quantitatively . here , additional simulations of atomistic models are required to predict binding energies and bending rigidities . this paper is organized as follows . the simulation approaches are introduced in section [ sec2 ] . the deformation of the polymer network under shear and rheological properties are discussed in section [ sec3 ] , and the dependence on the polymer flexibility is addressed . relaxation of shear - induced structures is discussed as well . section [ sec4 ] summarizes our findings . our hybrid simulation approach combines the multiparticle collision dynamics method for the fluid with molecular dynamics simulations for the semiflexible polymers . @xcite in the mpc method , the fluid is represented by @xmath1 point particles of mass @xmath2 , which interact with each other by a stochastic process . @xcite the dynamics proceeds in two steps streaming and collision . in the streaming step , the particles move ballistically and their positions are updated according to @xmath3 here , @xmath4 and @xmath5 are the position and velocity vector of the _ _ i__th particle , and @xmath6 is the time between collisions . in the collision step , the particles are sorted into cells of a cubic lattice with lattice constant @xmath7 , and their velocities are rotated relatively to the center - of - mass velocity @xmath8 of the cell @xmath9,\ ] ] where @xmath10 is the rotation matrix for the rotation around a randomly oriented axis by the fixed angle @xmath11 . the orientation of the axis is chosen independently for every collision cell and collision step . a semiflexible polymer is modeled as a linear sequence of @xmath12 mass points of mass @xmath13 . these monomers are connected by harmonic springs with bond potential @xmath14 where @xmath15 is the position of monomer @xmath16 , @xmath17 is the equilibrium bond length , and @xmath18 is the spring constant . semiflexibility is implemented by the bending potential @xmath19 here , @xmath20 is the bending rigidity , where @xmath21 is the boltzmann constant , @xmath22 is the temperature , and @xmath23 is the persistence length . excluded - volume interactions between monomers are taken into account by the shifted and truncated lennard - jones potential ( lj ) @xmath24 & r < { r_c } \\ 0 & r \ge { r_c } \end{array } \right . , \ ] ] where @xmath25 is the diameter of a monomer and @xmath26 is the interaction strength . aside from the polymer ends , all monomer interactions are purely repulsive , with the cutoff distance @xmath27 and the shift @xmath28 . for the attractive ends , the cutoff is set to @xmath29 and @xmath30 is varied according to the desired attraction strength . the polymer - solvent coupling is implemented by including the monomers in the collision step . hence , the particle center - of - mass velocity of a cell containing monomers is @xmath31 where @xmath32 and @xmath33 are the number of solvent and monomer particles in the cell , respectively . @xcite we consider a cubic simulation box of side length @xmath34 . the parameters for the mpc fluid are @xmath35 , @xmath36 , and the mean number of fluid particles in a collision cell @xmath37 . we choose the bond length @xmath38 as length unit and set for the collision - cell size @xmath39 . moreover , we set @xmath40 , @xmath41 , and @xmath42 . the latter ensures that the bond lengths remain close to the equilibrium value even under shear flow for all considered shear rates . the equations of motion for the monomers are solved by the velocity - verlet algorithm with time step @xmath43 . @xcite in total , 2000 polymers of length @xmath44 are considered . approximating a polymer by a cylinder of length @xmath45 , the polymer - volume fraction is @xmath46 . initially , the polymers are distributed randomly in the simulation box and are equilibrated without end - attraction . then , the end - attraction is turned on and the system is again equilibrated until expectation values reach a steady state . shear flow is imposed on equilibrium structures by lees - edwards boundary conditions , @xcite with the flow direction along the @xmath47 axis and the gradient along the @xmath48 axis of the cartesian reference system . shear is characterized by the weissenberg number @xmath49 , where @xmath50 is the shear rate and @xmath51 is the end - to - end vector relaxation time of a polymer in dilute solution . @xcite explicitly , the values of the relaxation time are @xmath52 and @xmath53 for the persistence lengths @xmath54 and @xmath55 , respectively.@xcite in the following , we will refer to polymers with @xmath56 and @xmath55 as _ semiflexible _ and _ rodlike _ , respectively . for an efficient simulation of the polymer and mpc fluid dynamics , we exploit a graphics - processing - unit ( gpu ) based version of the simulation code . @xcite and the end - attraction strengths @xmath57 . open symbols correspond to the numbers at equilibrium without flow . error bars display the magnitude of the fluctuation in the steady state . the ines are guides for the eye.,width=321 ] , the end - attraction strength @xmath58 and various shear rates . the dashed lines are fits to guide the eye , with a gaussian function for @xmath59 and an exponential function for @xmath60.,width=321 ] for an end - end attraction strength @xmath61 , scaffold structures appear under equilibrium conditions . @xcite this equilibrium scaffold - like network structure undergoes severe structural rearrangement under shear flow . this is illustrated in fig . [ fgr : shear ] , where polymer configurations are shown for the persistence length @xmath56 , the end - attraction strength @xmath62 , and various shear rates . as shear flow is applied , the network breaks up and for low shear rates ( @xmath59 ) densified aggregates are formed . the scaffold structure persists , but the network phase separates into polymer - rich and polymer - poor domains . at intermediate shear rates @xmath63 , smaller , partially connected domains are formed , which are reminiscent to micellar structures . @xcite finally , for high shear rates @xmath60 , the structural integrity is completely lost and polymers are aligned in a nematic - like manner along the flow direction . the initial separation ( for @xmath64 ) into polymer - rich and polymer - poor domains appears in a similar fashion for rodlike polymers.@xcite hence , it seems to be a generic feature of such network structures . however , the shear - induced micellar structures are only observed for more flexible polymers . here , the flow is sufficiently strong to bend the polymers and induce an attraction between the ends of the same polymer . the nematic alignment at high shear rates is again similar to rodlike polymers . it is caused by the shear forces and appears also for dilute solutions of flexible polymers.@xcite to characterize these structures , we determine the average coordination number @xmath65 , which is defined as the number of end - beads in proximity of each other , i.e. , within distances @xmath66 . figure [ fgr : bond]a shows @xmath65 as a function of the shear rate for the end - attraction strengths @xmath67 and @xmath68 . for the lowest value @xmath69 , no scaffold is formed at equilibrium . @xcite in addition , the @xmath70 is independent of shear rate , which indicates that there is no shear - induced network structure either . naturally , the polymers are aligned by the flow , in a similar fashion as non - attractive polymers . @xcite in systems with scaffold structures , the equilibrium coordination number at zero shear exceeds that of disordered systems considerably , as discussed in more detail in ref . . this equilibrium scaffold structure is gradually broken by the shear flow for @xmath71 , and the average coordination number decreases . the number of free ends @xmath72 , which are not adjacent to any other end - bead , increases simultaneously . in contrast , for @xmath62 , the average coordination number first increases with increasing shear rate and passes through a maximum at @xmath73 . this is associated with the compactification of the scaffold structure visible in fig . [ fgr : shear ] . the attraction is evidently so strong that the shear - induced structural changes lead to an enhanced binding of polymer ends . the values of @xmath70 decrease rapidly with increasing shear rate for @xmath74 , and the value of an equilibrium non - attractive assembly of polymers is assumed . simultaneously , the number of free end - beads @xmath72 increases as the shear rate increases , as shown in fig . [ fgr : bond]b . we present the distribution of the coordination number for various shear rates in fig . [ fgr : bnd ] . the dashed lines are fits to guide the eye , with a gaussian function for @xmath59 and an exponential function for @xmath75 . evidently , nodes with a larger number of end - beads are induced at low shear rates ( @xmath59 ) compare to the distribution without flow ( @xmath76 ) . for high shear rates ( @xmath74 ) , the coordination number is significantly small . corresponding mean values are shown in fig . [ fgr : bond]a . a qualitative similar behavior of @xmath77 is found for systems at equilibrium and various attraction strengths.@xcite for attraction strengths @xmath78 , @xmath77 decreases exponentially with increasing @xmath79 . for larger values of @xmath30 , a maximum of the distribution function appears , as also shown in fig . [ fgr : bnd ] . the exponential decay indicates the lack of a network structure either due to too weak attraction or too strong external forces . ) for the end - attraction strength @xmath62 , and different persistence lengths , ( a ) @xmath56 , @xmath80 and ( b ) @xmath81 , @xmath82 . only beads with the slice @xmath83 are shown . the color code corresponds to the number of adjacent ends . ( multimedia view),width=321 ] = 4.0 and the persistence lengths @xmath84.,width=321 ] between bundles for the end - attraction strength @xmath85 = 4.0 and the persistence lengths @xmath84.,width=321 ] polymer flexibility strongly affects the appearing shear - induced structures . this is reflected in fig . [ fgr : shear_f ] ( multimedia view ) , where structures are displayed for the persistence lengths @xmath56 and @xmath55 . for semiflexible polymers ( @xmath86 ) , the original scaffold network breaks up and micellar structures are formed . in contrast , rodlike polymers ( @xmath87 ) are strongly aligned along the flow direction and form thick bundles , an effect already observed for various end - attraction strengths in ref . . in both cases , the end - beads assemble in nodes . for the semiflexible polymers , this can be achieved by significant shear - induced conformational changes of an individual polymer , which gives rise to micellar - like aggregates . the two ends of a polymer can even meet at the same node . @xcite this is not possible for rodlike polymers . their two ends can only participate in two different nodes.@xcite in consequence , more dense structures are formed with well aligned rods . the respective coordination number distributions are shown in fig . [ fgr : bnd_f ] . rodlike polymers form nodes with a large number of end - beads , in agreement with the thick bundles ( cf . [ fgr : shear_f]b ) . to further characterize the shear - induced structure , fig . [ fgr : theta_f ] presents the distribution @xmath88 of angles @xmath89 between bundles for the two different persistence lengths . here , a bundle is defined as a connection of two neighboring nodes by two or more polymers . for semiflexible polymers ( @xmath86 ) , the distribution exhibits a broad peak at @xmath90 . note that a peak at @xmath90 is a characteristics of a scaffold network , @xcite which is more pronounced in equilibrium structure without flow , @xcite while a peak at @xmath91 indicates parallel alignment of bundles along the flow direction . for rodlike polymers ( @xmath87 ) , the peak at @xmath91 is much more pronounced , as expected for bundles . peaks at @xmath90 and @xmath92 are also present for rodlike polymers , which implies that the initial scaffold - like connectivity is not completely lost . , the end - attraction strength @xmath62 , and various shear rates . , width=340 ] ) as a function of shear rate for different persistence lengths and end - attraction strengths . , width=321 ] in the small - strain region for the persistence length @xmath56 , the end - attraction strength @xmath62 , and various shear rates . , width=321 ] in the small - strain region for the persistence length @xmath56 , the end - attraction strength @xmath62 , and various shear rates . the inset shows the number including large - strain region . , width=321 ] the structural rearrangement under shear flow affects the rheological properties of the system . @xcite figure [ fgr : vel ] shows average monomer velocity profiles along the flow - gradient direction for @xmath56 and @xmath62 . the flow profiles are non - monotonic for shear rates @xmath59 , which has also been observed in previous studies of rodlike polymers . @xcite the bands in the velocity profile can be understood as a consequence of the structural inhomogeneity under shear flow . the low - shear - rate regions correspond to polymer - rich domains , where a densified network resists the applied shear . in contrast , polymer - poor domains can flow easily , which yields high - shear - rate regions . for higher shear rates , the velocity profile becomes smoother and we observe a linear monotonic profile for @xmath93 . here , the structural integrity is lost and polymers are aligned along the flow direction ( cf . [ fgr : shear ] ) . for both , rodlike@xcite and semiflexible polymers , a monotonic velocity profile is observed for weak end - attraction strengths ( @xmath94 ) , where the network is either not formed or not strong enough to resist flow . we present the polymer contribution to the shear viscosity @xmath95 as a function of shear rate in fig . [ fgr : strall ] . the polymer contribution to the shear stress @xmath96 is determined by the virial expression @xmath97 where the forces @xmath98 follow from the potentials of eqs . ( [ eq : eq3 ] ) , ( [ eq : eq4 ] ) , and ( [ eq : eq5 ] ) . @xcite the viscosity is then calculated as @xmath99 . for semiflexible polymers ( @xmath56 ) , the viscosity increases with increasing attraction strength for all shear rates ( cf . [ fgr : strall ] ) . in particular , for @xmath62 , the viscosity of systems of rodlike networks ( @xmath81 ) is somewhat larger than those comprised of semiflexible polymers ( @xmath56 ) . evidently , the rodlike nature enhances polymer end contacts , and thus , leads to more stable structures . the systems exhibit shear - thinning behavior for the range of applied shear rates , and a newtonian plateau is observed for weak end - attraction strength ( @xmath100 ) at low shear rates . the shear stress @xmath96 in the small - strain region @xmath101 is plotted in fig . [ fgr : str ] for @xmath62 . the stress increases initially in a linear manner . the end of this elastic regime is reached at the strain @xmath102 . for larger strains , the network deforms plastically and reaches its maximum strength for @xmath103 . for even larger strains , the stress decreases again . the initial elastic response and yield suggests that there is no newtonian viscosity plateau for large attraction strengths . to shed light on the structural change in the vicinity of the maximum strength , we present the average coordination number as a function of strain in fig . [ fgr : bondt ] . initially ( @xmath104 ) , @xmath70 is constant for low shear rates ( @xmath59 ) . in this regime , the network structure is stable and the deformation energy is stored , i.e. , the structure behaves elastically . as strain increases , @xmath70 starts to decrease and reaches a minimum at @xmath103 , where the network structure breaks up . for @xmath105 , @xmath70 increases again slowly ( cf . inset of fig . [ fgr : bondt ] ) , which implies that shear - induced aggregates form . for high shear rates ( @xmath74 ) , @xmath70 decreases monotonically and an asymptotic low steady - state value is assumed . here , the network breaks up continuously as shear flow is applied . and the end - attraction strength @xmath62 . polymers are relaxed without flow after sheared with ( a ) @xmath80 and ( b ) @xmath106 . only beads with the slice @xmath83 are shown . the color code corresponds to the number of adjacent ends . , width=321 ] and the end - attraction strength @xmath62 . polymers are relaxed without flow after sheared with different shear rates . the dashed lines are fits to guide the eye , with a gaussian function , width=321 ] in order to elucidate the uniqueness of the observed structures , we allow the shear - induced structures to relax after cessation of flow . figure [ fgr : relx ] shows snapshots of structures after relaxation from initially sheared states for @xmath56 and @xmath62 . shear - induced aggregates , which are formed at low shear rates ( @xmath107 ) remain after relaxation , and the initial scaffold - like network structure is not fully recovered . when the structural connectivity is fully destroyed for larger shear rates ( @xmath108 ) , the system relaxes back to a scaffold - like network . the coordination number distributions for the two structures are shown in fig . [ fgr : bnd_relx ] . in addition , the distribution of @xmath109 of the initial , non - sheared structure is displayed . the recovered structure after high shear rates ( @xmath110 ) shows a similar distribution of the coordination number as the initial scaffold - like network . however , the coordination number is clearly larger for the relaxed structure after application of a low shear rate ( @xmath111 ) . here , a new ( equilibrium ) structure is formed , which is at least metastable . our studies of networks with weak end - attraction strengths ( @xmath94 ) reveal that the scaffold - like network structure is recovered after relaxation regardless of pre - applied shear rate . the dependence of the network structure on the initial configuration for strong end - attraction ( @xmath112 ) has also been observed at equilibrium . @xcite hence , care has to be taken on equilibrated state of the system . the nonequilibrium structural and dynamical properties of end - functionalized semiflexible polymer suspensions have been investigated by mesoscale hydrodynamic simulations . under flow , the scaffold - like network structure of polymers breaks up and densified aggregates are formed at low shear rates , while the structural integrity is completely lost at high shear rates . we find that network deformation is strongly affected by the polymer flexibility . shear - induced aggregates , which are formed at low shear rates and strong end - attraction , show different structures depending on the polymer flexibility . for semiflexible polymers , the scaffold network breaks up under shear and micellar structures are formed . in contrast , rodlike polymers are more strongly aligned along the flow direction and form thick bundles of smectic - like stacks . for high attraction strengths @xmath113 , we find that shear - induced dense aggregates remain after relaxation , while the system relaxes back to a scaffold - like network when the structural connectivity is fully destroyed under high shear . for lower attraction strengths , the equilibrium structure is fully recovered . our studies shed new light on the nonequilibrium properties of self - organized scaffold structures , specifically their formation and deformation under flow . we expect this knowledge to be useful and provide the basis for further theoretical and experimental studies of such systems . 73ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevlett.109.238301 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1039/c5sm01678a [ ( ) , 10.1039/c5sm01678a ] @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , , ) @noop * * , ( )

the nonequilibrium dynamical behavior and structure formation of end - functionalized semiflexible polymer suspensions under flow are investigated by mesoscale hydrodynamic simulations . the hybrid simulation approach combines the multiparticle collision dynamics method for the fluid , which accounts for hydrodynamic interactions , with molecular dynamics simulations for the semiflexible polymers . in equilibrium , various kinds of scaffold - like network structures are observed , depending on polymer flexibility and end - attraction strength . we investigate the flow behavior of the polymer networks under shear and analyze their nonequilibrium structural and rheological properties . the scaffold structure breaks up and densified aggregates are formed at low shear rates , while the structural integrity is completely lost at high shear rates . we provide a detailed analysis of the shear - rate - dependent flow - induced structures . the studies provide a deeper understanding of the formation and deformation of network structures in complex materials .

You are an expert at summarizing long articles. Proceed to summarize the following text: the interaction of driven particles , flexible lines and membranes etc . with disorder is an important topic in condensed matter physics @xcite . usually , this disorder is taken to be _ quenched _ , or frozen , such that its properties do not change within the relevant time scales . however , under certain conditions , this changes as in the case of the diffusion of solute atoms in metallic alloys @xcite or oxygen vacancies in superconductors @xcite . the mobile impurities play an important role in the dynamics of such systems , as evidenced for example by the portevin - le chatelier ( plc ) effect in solid solutions @xcite . there , within a certain range of temperatures and applied strain rates , the dynamic interaction of lattice dislocations and diffusing solute atoms result in phenomena such as negative strain rate sensitivity of the flow stress , giving rise to macroscopic serrations in the stress strain curve and strain localization in the form of bands of activity of various types @xcite . here , we consider the simple test problem of a single particle interacting with a cloud of diffusing impurities , with the dynamics constrained in one dimension ( a line ) . we restrict ourselves to the region of the parameter space in which the impurities have a vanishingly small probability to escape from the vicinity of the particle . despite its apparent simplicity , such a system exhibits rich dynamics , but at the same time has features that make the problem analytically tractable . in the absence of external forces , we consider the statistics of _ monotonic excursions _ @xmath0 of the particle , i.e. the distances the particle moves to a particular direction ( here `` left '' or `` right '' along the one - dimensional line ) without changing direction , see also figure [ fig : trajectories ] . we find that these obey power law distributions @xmath4 with the exponent @xmath5 . the same is true for the monotonic changes of the external force @xmath2 when the particle is driven with a slow constant velocity . this paper is organized as follows : in the next section , we consider the interaction of a particle with a single mobile impurity , in the absence of external forces . then we generalize this to the case with more impurity particles . in section [ driven ] , the effect of external drive is studied . finally , section [ concl ] finishes the paper with conclusions . as a starting point of our analysis , we consider the dynamics of a single particle interacting with one diffusing impurity particle . the equations of motion for the system are @xmath6 where @xmath7 and @xmath8 are the positions of the particle and the impurity particle , respectively . @xmath9 is the interaction force between the particle and the impurity particle , @xmath10 defines the relative mobilities of the impurity and the particle and @xmath11 is gaussian white noise with standard deviation @xmath12 mimicking the effect of temperature . the dynamics of the particle can be analyzed by considering the stochastic process for the velocity @xmath13 . by differentiating the equation of motion of the particle with respect to time and using the equation of motion for @xmath8 , one obtains @xmath14.\ ] ] close to @xmath15 , the force @xmath9 can be taken to be linear in @xmath16 and thus the derivative of the force can be approximated by a constant , @xmath17 , with @xmath18 . with @xmath19 and @xmath20 , equation ( [ eq : x1 ] ) can then be rewritten in the form of an ornstein - uhlenbeck process for @xmath13 , @xmath21 for the process @xmath13 , equation ( [ eq : x2 ] ) describes brownian motion pushed toward the origin by a linear damping term . this problem has been considered e.g. in @xcite , and the scaling exponents are known . in particular , the first return times @xmath22 to origin of @xmath13 have a probability distribution scaling as @xmath23 with @xmath24 and the cut - off scale @xmath25 . similarly , the average shape of an excursion , @xmath26 , scales for @xmath27 as @xmath28 , with @xmath29 . then , by using the scaling relation @xmath30 @xcite , one obtains the distribution of the lengths @xmath31 of monotonic excursions of the particle , @xmath32 with @xmath1 . the cut - off scaling can be found as follows : the cut - off of the first return time distribution of equation ( [ eq : x2 ] ) is given by @xmath25 @xcite . here , both @xmath33 and @xmath34 in equation ( [ eq : x2 ] ) depend on @xmath35 and @xmath10 . due to the relation @xmath36 with @xmath29 , @xmath37 is expected to scale like @xmath38 we check this result in numerical simulations , where we take for simplicity @xmath9 to be of the form @xmath39 , corresponding to @xmath40 , i.e. @xmath41 . we integrate the equations of motion ( [ eq : motion ] ) with the euler algorithm . the strength of the thermal noise was chosen to be sufficiently weak such that the impurity can not escape from the neighborhood of the particle . figure [ fig : trajectories ] shows an example of the trajectories of the particle and the impurity . figure [ fig : onesolute ] displays the distribution of @xmath0 for various values of @xmath42 . for a weak enough interaction strength a , the distributions display the expected scaling with @xmath1 . the cut - off is found to scale as @xmath43 , in agreement with our results above . this implies that the motion of the particle is reminiscent of _ truncated _ levy flight , with a step length distribution given by equation ( [ eq : dist ] ) . however , the steps are not instantaneous - their durations @xmath22 exhibit power law scaling as well , equation ( [ eq : pt ] ) . by interpreting these step durations as waiting times between instantaneous steps , one would obtain for early times the scaling @xmath44 . however , here @xmath0 and @xmath22 are not independent ( due to the relation @xmath45 ) . as the early time behavior is dominated by a single large step , one must consider instead the effect of a single step given its duration , @xmath46 ^ 2 p(t ) dt = \int_0^t t^{2\gamma-\tau_t } dt \sim t^{2\gamma-\tau_t+1},\ ] ] corresponding to @xmath47 . due to the truncated nature of the levy flight , one expects a cross - over to diffusive behavior with @xmath48 for long times . this is verified in figure [ fig : diffusion ] . the case of a fixed number @xmath49 of impurity particles is a straightforward generalization of that presented in the previous subsection . the equations of motion become @xmath50 from these equations , with the same procedure as above , one obtains an equation of the form of equation ( [ eq : x2 ] ) by setting @xmath51 and @xmath52 . thus , the same scaling , i.e. @xmath53 is expected . one should notice , however , that the cut - off scale @xmath37 is getting smaller with increasing @xmath54 . with similar arguments as above , one finds that @xmath55 this is verified by the numerical results presented in figure [ fig : manysolutes ] . next we proceed to study the effect of a weak external force @xmath56 on the dynamics of the particle . in this context a _ constant velocity _ drive is perhaps the more interesting form of driving as a small _ constant force _ with @xmath57 does not change the dynamics from the non - driven case : equation ( [ eq : x1 ] ) remains the same even if a constant force term is introduced in equation ( [ eq : motion ] ) . in particular , we consider a particle driven by a force given by @xmath58 , where @xmath59 is the driving velocity and @xmath60 is a spring constant characterizing the response of the driving mechanism . the equations of motion read @xmath61 in systems like this driven with a constant velocity , the interesting quantity is the statistics of the external force fluctuations . to this end , we consider the stochastic process @xmath62 . with a similar approach as above , one can write @xmath63\partial_t f + \frac{kc}{\mu}\sum_i \eta_i \nonumber \\ & & + \frac{kc}{\mu}\left[v(n+\mu)-f\right],\end{aligned}\ ] ] where the relation @xmath64 has been used . in the steady state , the last term in equation ( [ eq : f1 ] ) has a zero mean , as one can write for the average steady state force @xmath65 , where @xmath66 is the magnitude of the average retarding force acting on the particle due to a single impurity . in the steady state the condition @xmath67 holds , and thus @xmath68 . assuming that the fluctuations @xmath69 = \delta f$ ] are small compared to those of the white noise term in equation ( [ eq : f1 ] ) , i.e. @xmath70 , equation ( [ eq : f1 ] ) can be approximately written as @xmath71\partial_t f + \frac{kc}{\mu}\sum_i \eta_i,\ ] ] which is again of the same form as equation ( [ eq : x2 ] ) . thus , the monotonic changes of the external force @xmath72 are expected to be distributed according to a power law @xmath73 , with the exponent @xmath74 and the cut - off scale @xmath75 scaling as @xmath76^{3/2}}.\ ] ] notice that the condition @xmath70 implies that @xmath77^{3/2 } } \ll 1.\ ] ] for most of the relevant parameter values condition ( [ eq : cond ] ) is fulfilled , only for @xmath78 this is not the case . + next we check these predictions numerically for different values of the parameters satisfying the condition ( [ eq : cond ] ) . figure [ fig : force ] shows an example of the behaviour of the force as a function of time . after an initial transient , the system reaches a steady state in which the force fluctuates around a constant average value . in figure [ fig : forcedists ] , we show the distributions of the monotonic changes of the external force @xmath2 in the steady state , for different values of the various parameters . power law scaling of the distributions consistent with the exponent value @xmath74 is observed , with a cut - off of the distributions in agreement with equation ( [ eq : cutoff ] ) . the different @xmath2-values appear to be uncorrelated in time . in this paper we have studied the dynamics of a single particle interacting with a cloud of diffusing impurities . in the absence of external forces , the problem can be mapped to brownian motion in a potential within the harmonic approximation of the attractive particle - impurity interaction . thus , the monotonic excursions of the particle are distributed as a power law , @xmath79 , with @xmath1 . for a particle driven with a small constant velocity ( such that the particle is dragging the impurity cloud without escaping from it ) , the external force fluctuations follow the same dynamics , which makes it possible to derive the probability distribution for the monotonic changes of the external force scaling as @xmath80 , again with @xmath74 . while a typical experimentally relevant scenario would correspond to either a higher dimensional object such as a flexible line or a large number of interacting particles interacting with mobile impurities , the simple setup of the present study serves as a convenient starting point for such considerations illustrating the relevant phenomena in a transparent manner . one interesting observation is that already at the level of a single particle interacting with one or more mobile impurities the dynamics has scale free features , arising from the properties of simple random walks . physical situations in which these kind of considerations could be relevant include dislocations interacting with solute atoms , a subject that has recently attracted considerable attention @xcite . in many of these studies , more realistic interaction forces between dislocations and solute atoms have been used , but the focus has been on different quantities such as the _ average _ velocity of the dislocation . the most intriguing phenomena associated with the presence of diffusing impurities are the collective effects arising from the simultaneous interaction of large number of entities with each other and with mobile impurities . an example of such a system is provided by interacting dislocation ensembles interacting with diffusing solute atoms in solid solutions , giving rise to phenomena such as the portevin - le chatelier effect . in the plc effect , large numbers of dislocations synchronize their motion to form macroscopic deformation bands of various kinds . as this is widely believed to be due to the dynamic interaction of the dislocations with the diffusing solute atoms , a natural future line of research would be to study such effects by considering numerically the dynamics of a large number of interacting dislocations interacting with diffusing impurities . one further motivation for such studies could be the recent observation that purely stochastic effects can induce switching between collective motion states @xcite . ll and mja gratefully thank the financial support of the european commissions nest pathfinder programme trigs under contract nest-2005-path - com-043386 . they also acknowledges the financial support from the center of excellence program of the academy of finland . 10 nattermann t 1983 _ phys . status solidi _ b * 119 * 209 huse d a and henley c l 1985 _ phys . lett . _ * 54 * 2708 kardar m and zhang y c 1987 _ phys . lett . _ * 58 * 2087 cahn j w 1962 _ acta metall . _ * 789 blavette d , cadel e , fraczkiewicz a , and menand a 1999 _ science _ * 286 * 2317 chudnovsky e m 1998 _ europhys . lett . _ * 43 * 445 portevin a and le chatelier f 1923 _ c. r. acad . paris _ * 176 * 507 ananthakrishna g , noronha s j , fressengeas c , and kubin l p 1999 _ phys . rev . _ e * 60 * 5455 lebyodkin m a , brechet y , estrin y , and kubin l p 1995 _ phys . lett . _ * 74 * 4758 hhner p , ziegenbein a , rizzi e , and neuhuser h 2002 _ phys . _ b * 65 * , 134109 colaiori f , baldassaarri a , and castellano c 2004 _ phys . _ e * 69 * 041105 lubeck s 2004 _ int j. mod . _ b * 18 * 3977 wang y , srolovitz d j , rickman j m , and lesar r 2000 _ acta mater . _ * 48 * 2163 rickman j m , lesar r , and srolovitz d j 2003 _ acta mater . _ * 51 * 1199 deo c s , srolovitz d j , cai w , and bulatov v 2005 _ phys . _ b * 71 * 014106 kolpas a , moehlis j , and kevrekidis i g 2007 _ pnas _ * 104 * 5931

the dynamics of a test particle interacting with diffusing impurities in one dimension is investigated analytically and numerically . in the absence of an applied external force , the dynamics of the particle can be characterized by a distribution of monotonic excursions @xmath0 , which scales as a power law with an exponent @xmath1 . when the particle is driven at a slow constant velocity , there is again a power law distribution for the monotonic changes of the force @xmath2 , which is characterized by a similar exponent @xmath3 . these results can be understood from the theory of random walks .

You are an expert at summarizing long articles. Proceed to summarize the following text: in experiments on ultracold trapped fermi gases , there are many situations where the system is out of thermal equilibrium . the first one is of course the trapping and cooling stage , i.e. , before the system has reached its equilibrium state which is usually the starting point for the actual experiment . then , in some experiments the system is excited in order to observe its dynamical behavior . for instance , many experiments studied collective oscillations of the system @xcite , another example being a recent experiment at mit where the collision of two atom clouds ( both in equilibrium ) was studied @xcite . finally , often the system is not imaged directly during the experiment , but only after the trap was switched off and the system has expanded for a certain time , in order to increase its size . the modeling of such time - dependent processes from the theoretical point of view can be quite complicated . for practical reasons , only semiclassical approaches are suitable for the description of time - dependent phenomena involving typically several @xmath0 atoms in a three - dimensional , non - uniform geometry . in some cases , it is possible to use hydrodynamic approaches : superfluid hydrodynamics describes the expansion @xcite and the collective modes @xcite of superfluid systems at zero temperature . hydrodynamics is also applicable in the normal - fluid phase if the mean time between collisions is much shorter than all other time scales of the process under consideration , so that the system can always be considered to be in a local equilibrium @xcite . superfluid and normal hydrodynamics can be combined to two - fluid hydrodynamics in order to describe superfluid systems at finite temperature @xcite . however , in many cases hydrodynamic approaches are not sufficient . in all cases where it is important that even locally the distribution @xmath1 of the atoms is not an equilibrium one , the boltzmann equation allows a very general description , provided the system is in the normal phase . if the system is superfluid , a more elaborate theory is necessary which couples the dynamics of the quasiparticle distribution function to the dynamics of the superfluid order parameter @xcite . in the past , several authors used the boltzmann equation for the investigation of collective oscillations in normal - fluid trapped fermi gases @xcite . in most cases , the boltzmann equation was not solved directly in order to find the distribution function @xmath2 , but semi - analytical approximate solutions were found by using the scaling ansatz @xcite or the method of moments @xcite . these methods rely explicitly or implicitly on the assumption that the collision term can be treated in the relaxation - time approximation , with a single relaxation time @xmath3 which is independent of the position in the trap . an exception is the work by toschi et al . @xcite , where the boltzmann equation was solved numerically , using a test - particle method very similar to the one we are using here . the test - particle method for the solution of the boltzmann equation has been used for many years in nuclear physics for the simulation of heavy - ion collisions @xcite . in the context of trapped atoms , it was also used for the simulation of the dynamics of the thermal cloud in a bose - einstein condensate @xcite and ( without collision term ) of the normal - fluid component in a superfluid fermi gas @xcite . for the collision term of the boltzmann equation , it is important to know the cross section which in principle can be modfied by in - medium effects . in the work by riedl et al . @xcite , it was shown that by using in the boltzmann equation instead of the free cross - section the in - medium one , the agreement between theoretical and experimental frequencies and damping rates of different collective modes is deteriorated . the reason is that the in - medium cross section is larger than the free one , so that the relaxation time is dramatically reduced . in our previous work @xcite , our aim was to include in addition to the in - medium cross section medium effects into the mean - field potential . this mean field resulted in better density profiles and allowed us to understand the shift of the quadrupole frequency in the collisionless regime at very low temperature observed in @xcite . however , it did not help to improve the agreement between theory and experiment in the region of higher temperatures , where the properties of the collective modes are completely dominated by collisions . hence , one of our motivations for the present work was to check the validity of the relaxation - time approximation which is implicitly made in the method of moments . here we will restrict ourselves to the case without mean field and with the free cross section . as we will show in sec . [ sec : modes ] , the numerical solution of the boltzmann equation gives indeed a significantly longer relaxation time than the method of moments . as we will show , this discrepancy is due to the restriction of the method of moments to second - order moments in the existing literature @xcite . once fourth - order moments are included , the results of the method of moments and of the numerical solution are in good agreement . however , already in the simplest case of a spherical harmonic trap without mean field , the inclusion of fourth - order moments is a very tedious task , while the numerical method can be generalized to more realistic cases . in addition , there are some other reasons why we felt the necessity for a numerical method . for example , there are damping effects due to the anharmonicity of the trap potential which can not be described by the method of moments . another advantage of the numerical method is that it offers the possibility to simulate not only the oscillation of the cloud , but also the subsequent expansion after the trap has been switched off . in sec . [ sec : numericaldescription ] of the present paper , we give a detailed description of the method . in particular , we explain in detail how the collisions are simulated , since our method is somewhat different from that of ref . moreover we discuss some tests we made in order to estimate up to which precision we can trust our simulation . then , in sec . [ sec : modes ] , we come to the main point of our article and calculate the properties of some collective modes for a system in a spherical harmonic trap . while the sloshing and breathing modes are rather trivial , the frequency and damping rate of the quadrupole mode are very sensitive to the collisions . we compare the numerical results with those of the method of moments . finally , in sec . [ sec : conclusions ] we summarize and give an outlook to future studies . throughout the paper , we will use units with @xmath4 ( @xmath5 reduced planck constant , @xmath6 = boltzmann constant ) . the strength of the interaction is characterized by the dimensionless quantity @xmath7 , where @xmath8 is the scattering length . concerning the fermi momentum @xmath9 and the fermi energy @xmath10 we follow the usual convention that these quantities are defined by the corresponding ones of an ideal fermi gas at zero temperature , i.e. , @xmath11 , @xmath12 being the atomic mass , and @xmath13 , where @xmath14 is the number of atoms and @xmath15 the trap frequency . temperatures will be measured in units of the fermi temperature @xmath16 ( since @xmath17 ) . we study a two - component ( @xmath18 ) gas of fermionic atoms of mass @xmath12 in a potential @xmath19 with attractive interaction @xmath20 . we assume that the system is in the normal phase and that it can be described semiclassically by phase - space distribution functions @xmath21 . in this paper , we will restrict ourselves to the case that the distribution functions of both spin states are equal ( @xmath22 ) , but the generalization of the method to the cases of different distribution functions , more than two components , or components with different masses is straight - forward . the time evolution of the distribution function @xmath2 is governed by the boltzmann equation @xcite @xmath23\,,\ ] ] where the left - hand side ( lhs ) describes the particle propagation , with @xmath24 and @xmath25 $ ] on the right - hand side ( rhs ) denotes the collision term which will be discussed later . the potential felt by the particles is the trap potential that contains a static part and a time dependent one ( which will be used to simulate the excitation of the collective modes ) @xmath26 . the density per spin state is related to the distribution function by @xmath27 and the number of atoms is given by @xmath28 the basic idea of the test - particle method ( also called pseudoparticle method ) for solving the boltzmann equation consists in replacing the continuous distribution function by a sum of delta functions , @xmath29 where @xmath30 is the number of `` test particles '' . this allows one to express the average of an arbitrary single - particle observable @xmath31 in the simple form @xmath32 in order to sample the six - dimensional phase space , it is necessary to choose a sufficiently large number of test particles @xmath30 ( usually @xmath33 ) . neglecting the collision term @xmath25 $ ] for the moment , it is easy to see that eq . ( [ eq : defdistri ] ) satisfies the boltzmann equation ( eq . ( [ eq : eqboltzmann ] ) ) if the positions @xmath34 and momenta @xmath35 of each test particle @xmath36 follow the classical equations of motion , eq . ( [ eq : eqmotion ] ) . in practice , the delta functions in eq . ( [ eq : defdistri ] ) can pose some problems . for instance , they do not result in a continuous density @xmath37 . therefore it is often useful to replace them by gaussians of width @xmath38 and @xmath39 in position and momentum space , respectively : @xmath40 with @xmath41 the widths @xmath38 and @xmath39 must be adapted such that they smooth out the fluctuations due to the finite number of test particles , but not the structure of the distribution function @xmath2 . the statistical fluctuations are of the order of @xmath42 , i.e. , the first condition is equivalent to @xmath43 the second condition implies of course that @xmath44 and @xmath45 , where @xmath46 and @xmath47 are the thomas - fermi radius and the fermi momentum , respectively , but this is not always sufficient . at low - temperature , it is crucial to resolve the rapid change of the distribution function around the fermi surface , i.e. , @xmath48 in practice , as the computation time increases as @xmath49 , it turns out that the conditions ( [ conditionn ] ) and ( [ conditiont ] ) can not simultaneously be satisfied at too low temperatures . in the absence of collisions , the numerical task consists only in solving simultaneously the classical equations of motion ( [ eq : eqmotion ] ) for the @xmath30 test particles . we do this by using the velocity verlet algorithm @xcite , which contrary to the original verlet algorithm @xcite uses the positions @xmath50 and velocities @xmath51 as starting point for the time step from @xmath52 to @xmath53 . the propagation from @xmath52 to @xmath54 is done according to @xmath55 where @xmath56 is the acceleration of the @xmath36-th test particle . if it is written in this way , it is obvious that the velocity verlet algorithm is identical to the leap - frog algorithm @xcite . note that the accelerations @xmath57 can be reused in the next time step , so that the algorithm needs only one evaluation of the acceleration per time step , but nevertheless its global error is of the order @xmath58 . this allows us to obtain a good accuracy for reasonable time steps @xmath59 . a good test of the particle propagation is to check the energy conservation : typically we find @xmath60 for each test particle and for all times considered . the rhs of the boltzmann equation ( [ eq : eqboltzmann ] ) describes the collisions between particles of opposite spin . it thus depends on the scattering cross section @xmath61 and reads @xcite @xmath62=\int \frac{d^3 p_1}{(2 \pi)^3 } \int d\omega \frac{d\sigma}{d\omega } -f^\prime f_1^\prime ( 1-f ) ( 1-f_1 ) ] \ , . \label{eq : collisionterm}\end{gathered}\ ] ] in the first term , @xmath63 and @xmath64 are the incoming momenta , @xmath65 and @xmath66 are the outgoing ones , @xmath67 is the solid angle formed by the incoming relative momentum @xmath68 and the outgoing relative momentum @xmath69 , and @xmath70 , @xmath71 , etc . in the second term , the role of incoming and outgoing momenta is exchanged . momentum and energy conservation implies @xmath72 and @xmath73-wave scattering , in which the cross section is isotropic , i.e. , @xmath74 . in principle the cross section is modified by medium effects @xcite , but in the present paper we will only use the free cross - section ( i.e. , the cross - section for the scattering of two atoms of opposite spin in free space ) which is given by @xcite @xmath75 where @xmath76 . in our numerical simulation , the collision term is included by allowing the test particles to collide with each other . the cross section of the test particles , @xmath77 , is related to the cross section of the atoms by @xmath78 ( since @xmath30 test particles represent @xmath79 atoms of a given spin ) . whether a pair @xmath80 of test particles collides in a time step @xmath52 or not is determined as follows : first , we determine if the two particles are at their closest approach in the present time step . explicitly , if we write @xmath81 and @xmath82 , the closest approach is reached at @xmath83 and we check if @xmath84 . if yes , we calculate the corresponding minimal distance by @xmath85 and check if @xmath86 . in this case , the collision is classically allowed . we then propagate both test particles to @xmath87 , change the direction of their relative velocity @xmath88 in a random way ( thus conserving the total momentum and the total energy ) , and propagate them back to the original time @xmath52 . finally , in order to take into account the pauli - blocking factors in eq . ( [ eq : collisionterm ] ) , we calculate the occupation numbers @xmath89 and @xmath90 at the new positions and momenta ( @xmath91 etc . ) using eq . ( [ eq : defdistri ] ) with gaussians instead of delta functions , see eq . ( [ eq : gaussians ] ) . with probability @xmath92 the collision is allowed and we keep the new positions and momenta , otherwise the collision is blocked and we keep the old ones . we checked that the total energy is still well conserved when collisions are switched on : typically we find better than @xmath93 for all times considered . before the simulation can start , the test - particle positions and momenta have to be initialized . here we assume that the system is initially in equilibrium . a suitable equilibrium distribution is given by the distribution function within the thomas - fermi or local - density approximation ( lda ) , @xmath94 since it is a stationary solution of the boltzmann equation @xcite . this distribution has two parameters , namely the inverse temperature @xmath95 and the chemical potential @xmath96 . the temperature @xmath97 is an input parameter , whereas the chemical potential @xmath96 is determined by demanding that the integral of eq . ( [ eq : equilibrium ] ) over @xmath98 and @xmath63 gives the right number of atoms . having determined the chemical potential @xmath96 , we randomly generate the test - particle positions and momenta in such a way that the probability to be at position @xmath98 and to have momentum @xmath63 is proportional to @xmath99 . in practice , we do this by first generating the positions according to the density profile obtained from eq . ( [ eq : density ] ) with @xmath100 . then we generate the momenta according to @xmath101 . in this subsection we describe two main tests we made to be sure that our code is reliable . here , we assume the potential to be static and , as in the rest of the paper , we use a spherical harmonic potential @xmath102 this potential defines naturally a time scale @xmath103 , a length scale @xmath104 , an energy scale @xmath15 , and so on . let us consider the energy distribution of the atoms , @xmath105 in equilibrium , the distribution should be given by @xmath106 , where @xmath107 is the density of states ( including the degeneracy factor 2 ) . in the present case of a spherical harmonic oscillator , we have @xmath108 . in the absence of collisions , energy conservation automatically implies that the distribution stays constant , but in the presence of collisions this test is a non - trivial check of the pauli blocking in the simulation . within the test particle method , @xmath109 is obtained by counting the test particles in energy bins . in fig . [ fig : distri ] energy distribution of the atoms ( divided by the density of states ) for @xmath110 ( see text for details ) . the system consists of @xmath111 atoms with a scattering length @xmath112 ( @xmath113 ) . the parameters of the simulation are : @xmath114 , @xmath115 , @xmath116 , and @xmath117.,width=302 ] we show , for @xmath110 , the initial fermi distribution ( solid line ) and the stationary distribution obtained in the numerical simulation after @xmath118 ( filled circles ) . the agreement between the distribution generated by the simulation and the initial fermi one is not perfect , but satisfactory . in order to show that this is not a trivial result , let us see what happens if we switch off the pauli blocking in the simulation of the collision term . in this case , already after a relatively short time @xmath119 , the distribution in the numerical simulation ( empty circles ) has converged to a boltzmann distribution with the same number of atoms and total energy ( dashed line ) . so , the stability of the fermi distribution in our full simulation shows clearly that pauli blocking is correctly implemented . the small deviations from the ideal fermi distribution are a consequence of the fact that with the chosen widths of the gaussians ( @xmath115 and @xmath116 ) , the condition ( [ conditiont ] ) is not well satisfied at @xmath110 . when we did the same kind of comparison at higher temperatures , we found that the agreement between the simulation and the fermi distribution improves : at @xmath120 , it is already perfect . the test described above is independent of the actual number of collisions . in order to check the latter , let us look at the collision rate @xmath121 , where @xmath122 denotes the number of collisions of test particles per unit time . although in equilibrium the net effect of collisions is zero , the collision rate in equilibrium is a good test for the simulation because it can be compared with the exact result [ eq . ( [ eq : mcblock ] ) , see appendix [ app : collrate ] ] . for testing purposes , it is useful to compare also the total rate of allowed and blocked collisions with the exact result [ eq . ( [ eq : mcnoblock ] ) ] . in fig . [ fig : collrate ] , the collision rates ( with and without blocking ) ) and eq . ( [ eq : mcnoblock ] ) ( solid lines ) , for a gas of 10000 atoms with interaction strength @xmath123 ( top ) and @xmath124 ( bottom).,title="fig:",width=302 ] + ) and eq . ( [ eq : mcnoblock ] ) ( solid lines ) , for a gas of 10000 atoms with interaction strength @xmath123 ( top ) and @xmath124 ( bottom).,title="fig:",width=302 ] of the simulation are shown together with the exact results as functions of the temperature for two different values of the scattering length . in the case of relatively weak interaction , @xmath125 ( upper panel ) , we see that the agreement between the simulation and the exact result is excellent for temperatures above @xmath126 . below that temperature , the collision rate in the simulation with pauli blocking gradually becomes too high since the finite widths of the gaussians ( @xmath127 , @xmath128 ) do not satisfy any more the condition ( [ conditiont ] ) and act in the pauli - blocking factors like an enhanced temperature . in the rest of this paper , we will therefore restrict ourselves to temperatures above @xmath129 . near unitarity ( @xmath130 ) , we consider only temperatures above @xmath131 because this is close to the superfluid transition temperature at unitarity @xcite . as it can be seen in the lower panel of fig . [ fig : collrate ] , the agreement between the collision rate obtained in the simulation and the exact one is satisfactory in the temperature range considered . the agreement is not as good as for @xmath132 at high temperature because of the larger cross section which leads to collisions between test particles which are further apart . the sloshing mode is an oscillation of the center of mass of the system . it plays a special role because in a harmonic trap it is undamped and its frequency is equal to that of the trap , independently of the number of atoms , of the temperature , and of the interaction between the atoms ( kohn mode @xcite ) . this is why it is often used for the experimental determination of the trap frequency @xcite . within the test - particle method , this general theorem is satisfied and it is easy to see why : let us first neglect collisions . from the equations of motion of the individual test particles in the harmonic potential ( [ eq : harmonic ] ) , @xmath133 it is evident that the averages @xmath134 and @xmath135 obey analogous equations of motion , @xmath136 let us now consider the effect of a collision of two test particles . of course , the trajectories of the colliding test particles will not obey any more the original equations of motion ( [ eq : eqmotionharmonic ] ) , but the collision has absolutely no effect on the averages : since the positions do not change during the collision , @xmath134 remains unchanged , and since the total momentum of the two colliding test particles is conserved , the average @xmath135 is not changed either . so , the equations of motion ( [ eq : eqmotionsloshing ] ) for the averages @xmath134 and @xmath135 remain valid in the presence of collisions . their solution is of course an undamped oscillation of the center of mass @xmath134 with frequency @xmath15 . this is confirmed by the numerical result shown in the upper panel of fig . [ fig : sloshingbreathing ] . by displacing all test particles by @xmath137 in the @xmath138 direction . bottom : simulation of the breathing mode . the mode was excited by changing at @xmath139 all test - particle momenta according to @xmath140 ( @xmath141 ) . both simulations were done for a system of @xmath142 particles at @xmath143 and @xmath144.,title="fig:",width=302 ] + by displacing all test particles by @xmath137 in the @xmath138 direction . bottom : simulation of the breathing mode . the mode was excited by changing at @xmath139 all test - particle momenta according to @xmath140 ( @xmath141 ) . both simulations were done for a system of @xmath142 particles at @xmath143 and @xmath144.,title="fig:",width=302 ] a couple of experiments studied the damping of the longitudinal and radial breathing modes @xcite in elongated traps . in a spherical trap , there is only one breathing mode ( monopole mode ) , corresponding to an oscillation of the mean - square radius @xmath145 around its equilibrium value @xmath146 . in a spherical harmonic trap , this mode is undamped and its frequency @xmath147 is independent of the number of collisions , like in the case of the sloshing mode . again , this is easy to see . consider the average kinetic and potential energies , @xmath148 and @xmath149 . in equilibrium , both are equal ( virial theorem ) . now let us assume that the system is compressed or expanded , such that @xmath150 . using again the equations of motion ( [ eq : eqmotionharmonic ] ) , one obtains @xmath151 obviously , these two equations describe an undamped oscillation with frequency @xmath147 . let us now look if they stay valid in the presence of collisions . since the collisions do not change the positions of the particles and conserve the total kinetic energy , it is clear that @xmath152 and @xmath153 are not affected . now let us write the difference of @xmath154 before and after a collision of two test particles @xmath36 and @xmath155 : @xmath156 where @xmath157 and @xmath158 are the relative momenta ( e.g. , @xmath159 ) before and after the collision . in the original collision term as written in eq . ( [ eq : collisionterm ] ) , particles have to be at the same position to collide , i.e. , @xmath160 , such that @xmath154 is not changed . in our simulation this is somewhat different , since the test particles can collide at a distance of up to @xmath161 . this adds a small noise to @xmath154 . in all practical cases , however , this noise is completely negligible . as an example we show in the lower panel of fig . [ fig : sloshingbreathing ] the oscillation of the mean - square radius of the cloud as a function of time . as one can see , it is a perfectly undamped harmonic oscillation with frequency @xmath147 . for the theoretical investigation of collective modes , it is convenient to consider a system which is in equilibrium until it is excited by a short pulse at @xmath139 . formally , this is achieved by adding to the time - independent trap potential a perturbation term of the form @xmath162 the reason for this choice , which is of course different from the experimental way of exciting a collective mode , is the following : provided the perturbation @xmath163 is small enough ( such that the system reacts linearly to it ) , the response to a perturbation with arbitrary time dependence , @xmath164 , can easily be obtained by folding the result for the perturbation ( [ eq : deltapulse ] ) with the function @xmath165 . by integrating the boltzmann equation over the ( infinitesimal ) duration of the pulse , one can show that the effect of the perturbation ( [ eq : deltapulse ] ) is to change the distribution function as @xmath166 where @xmath167 and @xmath168 denote the limits @xmath169 from above and below , respectively . in the numerical simulation , this means that all test particles get a kick at @xmath139 , @xmath170 whereas their positions are not changed by the perturbation . from now on we will study the quadrupole mode as an example for a collective mode with non - trivial properties . we write the perturbation as @xmath171 corresponding to a kick at @xmath139 of @xmath172 and @xmath173 . the parameter @xmath174 determines the amplitude of the perturbation . if @xmath174 is chosen too small , it is difficult to separate the oscillation of the mode from fluctuations ; if it is chosen too large , one is not in the linear - response regime . all the following results were obtained with @xmath175 , corresponding to moderate amplitudes . by varying @xmath174 within reasonable limits , we checked that the amplitude of the resulting oscillation scales linearly with @xmath174 . ) with @xmath175 for different temperatures . the system has @xmath176 atoms and @xmath177 . lower panel : imaginary part of the corresponding fourier transforms.,title="fig:",width=302 ] + ) with @xmath175 for different temperatures . the system has @xmath176 atoms and @xmath177 . lower panel : imaginary part of the corresponding fourier transforms.,title="fig:",width=302 ] after the excitation of the radial quadrupole mode , we can look at the time evolution of the quadrupole moment @xmath178 as a function of time . results for different temperatures are displayed in fig . [ fig : respmode ] . contrary to the sloshing and breathing modes , the quadrupole mode is damped and the system approaches equilibrium ( @xmath179 ) after a certain time . at high temperatures ( @xmath180 ) , the system gets so dilute that it is in the collisionless regime ( @xmath181 , @xmath182 being the mean time between collisions of one atom ) . in this case , it takes many oscillations before the system returns to equilibrium . for lower temperatures , the mode is damped because of the high collision rate ( @xmath183 ) , but the system is not yet in the hydrodynamic regime ( @xmath184 ) where the mode would become undamped again . for the analysis of the results , it is useful to apply a fourier transform @xmath185 the so - called response function is the imaginary part of @xmath186 and can easily be obtained from the numerical results for @xmath187 by using a fast fourier transform ( fft ) algorithm @xcite . as an example , the fourier transforms of the results discussed above are shown in the lower panel of fig . [ fig : respmode ] . from the fourier transform one can clearly see that the spectrum of the mode in the collisionless regime , i.e. , at high temperature , has a sharp maximum at @xmath188 , as it should be in an ideal fermi gas , whereas at lower temperature the spectrum is broadened and the centroid of the spectrum is shifted to lower frequencies . this can be understood since at lower temperature the system is closer to the hydrodynamic regime , where the frequency should be @xmath189 . of course , one would like to give numbers @xmath190 and @xmath191 corresponding to the frequency and damping rate of the quadrupole mode in order to quantify these effects . the simplest way to obtain such numbers would be to fit the response function @xmath187 with a damped oscillation of the form @xmath192 . however , in the case of strong damping , this ansatz fits very badly the numerical results for @xmath187 . this can be understood by looking at the fourier transforms : the fourier transform of this ansatz function is a lorentzian , which has a line shape quite different from that obtained in our numerical simulation for @xmath193 or @xmath194 , cf . lower panel of fig . [ fig : respmode ] . hence , in order to analyze our numerical results , we need some physically motivated ansatz for the fit . in most of the theoretical work on collective modes in normal - fluid fermi gases , the boltzmann equation was not solved numerically , but approximate analytical solutions were found with the help of the method of moments @xcite . for a detailed description of the method , see e.g. ref . @xcite . applying the method of moments to the case of a perturbation of the form ( [ eq : deltapulse ] ) with @xmath195 according to eq . ( [ eq : potexcquad ] ) , one obtains a theoretical prediction for the response function @xmath196 . a brief description of the derivation is given in appendix [ app : mmresp ] , the final result reads @xmath197 where @xmath198 is the mean energy per atom in equilibrium , and @xmath3 is the relaxation time as defined in refs . @xcite , and depends on the cross section ( i.e. , the interaction strength ) , and the equilibrium distributions , cf . ( [ eq : deftau ] ) . one can see from eq . ( [ eq : imqmoment ] ) that in the collisionless and hydrodynamic limits the quadrupole mode has the frequencies @xmath199 and @xmath200 , respectively . the shape of the response function is completely determined by a single parameter , @xmath3 . by looking for the poles of eq . ( [ eq : imqmoment ] ) , one can calculate the inverse fourier transform which gives @xmath187 . the result has the form @xmath201 i.e. , it is a superposition of a damped oscillation with frequency @xmath190 and damping @xmath191 , and a non - oscillating , exponentially decaying term . the explicit expressions for @xmath202 , @xmath191 , and @xmath190 as functions of @xmath3 as well as for the amplitudes @xmath203 and @xmath204 are given in appendix [ app : qmom ] . we will refer to @xmath190 and @xmath191 as the frequency and damping rate of the quadrupole mode . note that in experiments determining these quantities , the data are usually fitted with a function that is similar to eq . ( [ eq : qmoment_t ] ) @xcite . in fig . [ fig : fittf ] we compare the response function obtained from the ) where the parameter @xmath3 is obtained by the method of moments ( dots ) , or by a fit to the simulation ( long dashes ) . the short dashes represent the response obtained with the extended method of moments including fourth - order moments . the system is a gas of @xmath176 particles at @xmath205 and @xmath206.,width=302 ] numerical simulation ( solid line ) with the result obtained from the method of moments , eq . ( [ eq : imqmoment ] ) ( dotted line ) . as one can see , the height of the peak and its general shape are in good agreement , but the position of the maximum is at different frequencies . however , if we try to fit the numerical result with a function of the form of eq . ( [ eq : imqmoment ] ) , using @xmath3 as fitting parameter , we can very well reproduce the numerical response function ( long - dashed line ) . it is remarkable that by adjusting only one parameter , @xmath3 , one can simultaneously reproduce the position , the height and the width of the peak , and also the shape far away from the maximum . however , surprisingly , the fitted value of @xmath3 is larger by approximately @xmath207 than the one obtained by the method of moments . as a consequence , the frequency @xmath190 and damping rate @xmath191 obtained from the fit of the response function deviate significantly from those predicted by the method of moments . these results are summarized in table [ tab : omegamma ] . .[tab : omegamma ] relaxation time , frequency , and damping of the quadrupole mode as obtained from the method of moments and fitting the results of the numerical simulation with a function of the form ( [ eq : imqmoment ] ) , corresponding to the dotted and dashed curves in fig . [ fig : fittf ] . [ cols="^,^,^,^",options="header " , ] in the existing literature @xcite , the method of moments was limited to second - order moments , as described in appendix [ app : mmresp ] . however , as we have seen above , this implies that the system is characterized by a single relaxation time @xmath3 , whereas in the spirit of a local - density approximation one would expect that in a trapped system the relaxation time should be position - dependent , @xmath208 . for instance , one could imagine that the gas in the center of the trap is more or less hydrodynamic ( short relaxation time ) , whereas far away from the trap center it gets very dilute and hence collisionless ( long relaxation time ) . in the case of the quadrupole mode , this means that the fermi - surface deformation is stronger at larger radii than in the trap center . it seems therefore natural to include into the ansatz for the perturbed distribution function in addition to the standard term @xmath209 describing the fermi - surface deformation , a term @xmath210 . more generally speaking , we should go beyond the standard approximation to include only second - order moments , and include also fourth - order ( or perhaps even higher ) moments . the task of extending the method of moments to the next higher order is in principle straight - forward but in practice very tedious : in the case of the quadrupole mode , the number of moments is increased from three to twelve . some details are given in appendix [ app : fourthorder ] . the resulting response function is shown in fig . [ fig : fittf ] as the short - dashed line . surprisingly , its shape is still similar , but now the position of the maximum agrees rather well with the result of the numerical simulation ( solid line ) . the agreement is even better at higher temperature ( see fig . [ fig : compfourth0.7 ] ) . for a temperature @xmath211 . for clarity , the fit of the simulation is not shown . the system is a gas of @xmath176 particles at @xmath206.,width=302 ] this nicely confirms the correctness of our numerical simulation and shows explicitly that the method of moments , if truncated at the lowest order , is insufficient . by doing calculations for various interaction strengths and temperatures , we found that the relaxation time from the simulation is systematically longer than that from the method of moments ( without fourth - order moments ) , eq . ( [ eq : deftau ] ) . results for weaker and stronger interactions ( @xmath212 and @xmath124 ) are displayed in fig . [ fig : tauinv ] . , as obtained from the simulation ( crosses ) and from the method of moments eq . ( [ eq : deftau ] ) ( solid lines ) , as a function of temperature . the system consists of @xmath213 atoms with @xmath132 ( upper panel ) and @xmath124 ( lower panel).,title="fig:",width=264 ] , as obtained from the simulation ( crosses ) and from the method of moments eq . ( [ eq : deftau ] ) ( solid lines ) , as a function of temperature . the system consists of @xmath213 atoms with @xmath132 ( upper panel ) and @xmath124 ( lower panel).,title="fig:",width=264 ] we see that the general behavior of @xmath3 as a function of temperature is the same within the simulation and the method of moments , but quantitatively there is a discrepancy of the order of @xmath207 in the whole range of temperatures where our numerical simulation is very accurate ( @xmath214 , cf . [ fig : collrate ] showing the temperature dependence of the collision rate ) . note that at lower temperatures , the determination of the pauli - blocking factors in the simulation of the collisions is not completely accurate , as discussed below fig . [ fig : collrate ] , such that the collision rate below @xmath215 is slightly too high . nevertheless the inverse relaxation time is too small . from this one can conclude that if we could improve the pauli blocking in the simulation , the discrepancy between the simulation and the method of moments ( without fourth - order moments ) would be even worse . the fourth order is thus important for the determination of the relaxation of the system and particularly for the frequency and the damping of collective modes . as we have just seen , the numerical simulation gives systematically a longer relaxation time @xmath3 than the method of moments . as the frequency @xmath190 and damping rate @xmath191 of the quadrupole mode are parametrized in terms of @xmath3 ( see appendix sec . [ app : qmom ] ) , one can ask the question how strongly this difference in @xmath3 will affect the results for @xmath190 and @xmath191 . since we are mainly interested in the intermediate regime @xmath216 between the hydrodynamic and collisionless limits , a difference of @xmath207 in @xmath3 can completely change the temperature dependence of @xmath190 and @xmath191 . this is shown in fig . [ fig : quauni ] , where the crosses are the results of the simulation may be wrong by 15% . the system consists of @xmath176 particles close to unitarity ( @xmath217).,title="fig:",width=264 ] of the simulation may be wrong by 15% . the system consists of @xmath176 particles close to unitarity ( @xmath217).,title="fig:",width=264 ] obtained from the simulation , whereas the solid lines are the results from the method of moments . one can clearly see that the numerical results stay close to the collisionless limit to much lower temperatures than the results obtained by the method of moments . to estimate the resulting precision of our numerical result on @xmath190 and @xmath191 , we show in fig . [ fig : quauni ] the error bands ( short - dashed lines ) which we obtain if we assume that our simulation may give a @xmath3 which is wrong by at most @xmath218 . this error includes numerical uncertainties which can be estimated from the scattering of the points in fig . [ fig : tauinv ] and the systematic deviation of the collision rate shown in fig . [ fig : collrate ] . if we include the fourth order moments , a global relaxation time @xmath3 does not exist anymore but we could define an effective one by fitting the response function as we did for the simulation . this effective relaxation time agrees very well with the one of the simulation so that both results give very similar frequency and damping . however , from a theoretical point of view , the definition of these quantities should come from the zeroes of the determinant of the matrix @xmath219 defined in appendix [ app : fourthorder ] . such a discussion is postponed to a forthcoming publication @xcite . in this paper , we presented a test - particle method for solving numerically the boltzmann equation for trapped fermi gases . while such methods have been popular in other fields of physics for many years , there have been only a few applications to ultracold atomic gases @xcite . our method is similar to that of refs . @xcite with some differences in the treatment of the collision term . in order to compute the occupation numbers in the pauli - blocking factors in the collision term , we represent each test particle by a gaussian in @xmath98 and @xmath63 space . the minimum value of the width of the gaussian is dictated by the statistical fluctuations due to the finite number of test particles , limiting the applicability of the method to temperatures above @xmath220 . as a first application of the method we discussed some collective modes . for simplicity , we considered only toy systems consisting of @xmath221 atoms in a spherical harmonic trap and neglected the mean - field potential and medium modifications of the cross section . as expected , the sloshing and monopole modes are undamped and independent of the collisions . in contrast , the quadrupole mode is very sensitive to collisions . in the hydrodynamic limit , its frequency should approach @xmath222 , while it is @xmath147 in the collisionless limit . in our simulations , we never reach the hydrodynamic regime , but the collisionless regime can be realized at high temperature due to the diluteness of the gas . surprisingly , the frequency and damping rate of the quadrupole mode obtained within the numerical simulation are quite different from those obtained within the widely used method of moments including moments up to second order in @xmath98 and @xmath63 . the method of moments predicts a relaxation time @xmath3 which is significantly shorter than the one obtained within the simulation . the reason is that the @xmath98 dependence of the relaxation time is neglected if only the @xmath223 moment is taken into account for the description of the fermi - surface deformation . we have shown that if the method of moments is extended to moments up to fourth order in @xmath98 and @xmath63 , e.g. , the @xmath224 moment , the agreement with the simulation becomes very good . the focus of the present paper was mainly to explain the test - particle method and to show its usefulness . for instance , the deficiency of the method of moments up to second order would not have been detected without the comparison with the numerical result . in future studies , we plan to apply the method to more realistic cases . in particular , in order to reach the typical numbers of atoms in the experiments , we will have to increase @xmath14 by a factor of @xmath225 . however , this should not pose a big problem : according to eq . ( [ conditiont ] ) , if we increase @xmath14 but keep the ratio @xmath226 fixed , the widths @xmath38 and @xmath39 may be chosen larger ( @xmath227 ) . this means that eq . ( [ conditionn ] ) stays satisfied with the same number of test particles , @xmath30 , i.e. , with a reduced ratio @xmath228 . the computation time will only grow because of the increased collision rate ( due to the larger test - particle cross section @xmath229 ) . another point is the trap geometry . the traps in the experiments are usually not spherical , but elongated . concerning the propagation of the test particles , this does not cause any difficulty , but in the calculation of the occupation numbers , it will probably be necessary to replace the width @xmath38 of the gaussian in @xmath98 space by different widths @xmath230 , @xmath231 , and @xmath232 in the three space directions . another important advantage of the numerical method is that an anharmonicity of the trap potential , which is always present in real experiments , can easily be included . finally , the mean field @xcite and medium modifications of the cross section @xcite should be included . the mean field , which originally depends on the chemical potential @xmath96 and the temperature @xmath97 , can be expressed as a function of the local density and energy density , which are both obtainable in the simulation . however , as shown previously @xcite , the mean field is not just proportional to the density : this leads to a huge numerical effort which is beyond the scope of this paper . the in - medium cross section is also difficult to be included because it depends on too many variables to be tabulated : @xmath233 . one possible solution of this problem is to replace the full @xmath234 and @xmath235 dependence of the in - medium cross section by a simple parametrization which results in the same local relaxation time @xmath236 . work in this direction is already in progress . in refs . @xcite , within the method of moments up to second order , the use of the in - medium cross section spoiled the agreement with experimental data because the resulting relaxation times were too short . since the present work shows that the numerical simulation gives a longer relaxation time than the method of moments , we hope that this problem can be solved . further important extensions of the present work are the generalization to polarized fermi gases and to superfluid systems . these questions , however , require more fundamental theoretical studies before they can be tackled numerically . replacing the distribution functions in eq . ( [ eq : collrate ] ) by equilibrium distribution functions @xmath237 , one obtains after some algebra the equilibrium collision rate @xmath238 where @xmath239 , @xmath240 , @xmath241 and @xmath242 . the total rate of allowed and blocked collisions is , in turn , given by eq . ( [ eq : collrate ] ) but without the factor @xmath243 in the integrand , leading to @xmath244 if the trap potential is spherically symmetric or harmonic , the spatial integrals can be reduced to one - dimensional ones . the remaining three - dimensional integrals are evaluated numerically with a monte - carlo algorithm . in the case of a weak perturbation , we can write the deviation of the distribution function from the equilibrium one in the form @xmath245 inserting this expression into the boltzmann equation and keeping only terms linear in the perturbation , one obtains ( see eq . ( 36 ) of ref . @xcite , for the case without mean field but with an external perturbation ) : @xmath246 . \label{eq : boltzlin}\end{gathered}\ ] ] here , @xmath247 $ ] is the linearized collision term as defined in eq . ( 37 ) of ref . @xcite ( up to a factor @xmath248 since here we are using a different normalization of @xmath2 ) : @xmath249 = \int \frac{d^3p_1}{(2 \pi ) ^3}\int d\omega \frac{d\sigma}{d\omega } |{\bm{\mathrm{v}}}-{\bm{\mathrm{v_1}}}| f_{\mathit{eq}}f_{{\mathit{eq}}\,1}\\ \times ( 1-f_{\mathit{eq}}^\prime)(1-f_{{\mathit{eq}}\,1}^\prime ) ( \phi+\phi_1-\phi^\prime-\phi_1^\prime)\ , . \label{collisionterm}\end{gathered}\ ] ] the perturbation @xmath250 is given by eqs . ( [ eq : deltapulse ] ) and ( [ eq : potexcquad ] ) . the usual approximation consists in making the ansatz @xmath251 with time - dependent coefficients @xmath252 and @xmath253 , @xmath254 , and @xmath255 , i.e. , only quadratic moments are considered . evaluating the moments @xmath256 , one obtains a system of equations for the fourier transformed coefficients : @xmath257 with @xmath258 \bigg\rbrack\end{gathered}\ ] ] and @xmath259 where @xmath260 are the poisson brackets . using the virial theorem , we obtain explicitly : @xmath261 where the relaxation time @xmath3 is defined by @xcite @xmath262\ , . \label{eq : deftau}\ ] ] solving this system of equations , we find @xmath263 and similar expressions for @xmath264 and @xmath265 . however , only the coefficient @xmath266 contributes to @xmath267 : with eq . ( [ eq : f1phi ] ) , and using again the virial theorem , we obtain @xmath268 , or explicitly : @xmath269 taking the imaginary part , we obtain eq . ( [ eq : imqmoment ] ) . in order to compute the fourier transform of eq . ( [ eq : qomega ] ) , let us start by factorizing the denominator : @xmath270 the expressions for the roots @xmath271 can be given in closed form . defining @xmath272 and @xmath273 we can write the roots @xmath271 as @xmath274 with @xmath275 now it is straight - forward to evaluate the inverse fourier transform of the response ( [ eq : imqmoment ] ) using the residue theorem . the result is given by eq . ( [ eq : qmoment_t ] ) with @xmath276 taking fourth order moments into account , we extend the previous ansatz eq . ( [ eq : ansatzmom2 ] ) as follows : @xmath277 which can be written as @xmath278 with , for example , @xmath279 . following the same steps as explained in appendix [ app : mmresp ] , we obtain now a system of twelve equations . the matrix @xmath219 can be computed explicitly . contrary to the second order calculations , the virial theorem can no longer be used to reduce the number of unknown quantities so that the system now depends on @xmath280 , @xmath281 and @xmath282 . in the matrix elements of the collision term , more parameters appear , generalizing the single parameter @xmath3 of the second order method . * j. kinast , s.l . hemmer , m.e . gehm , a. turlapov , and j.e . thomas , phys . lett . * 92 * , 150402 ( 2004 ) . j. kinast , a. turlapov , and j.e . thomas , phys . a * 70 * , 051401(r ) ( 2004 ) . m. bartenstein , a. altmeyer , s. riedl , s. jochim , c. chin , j.h . denschlag , and r. grimm , phys . lett . * 92 * , 203201 ( 2004 ) . a. altmeyer , s. riedl , m.j . wright , c. kohstall , j.h . denschlag , and r. grimm , phys . a * 76 * , 033610 ( 2007 ) . a. altmeyer , s. riedl , c. kohstall , m.j . wright , r. geursen , m. bartenstein , c. chin , j.h . denschlag , and r. grimm , phys . lett . * 98 * , 040401 ( 2007 ) . s. riedl , e r. sanchez guajardo , c. kohstall , a. altmeyer , m.j . wright , j.h . denschlag , r. grimm , g m. bruun , and h. smith , phys . a * 78 * , 053609 ( 2008 ) . s. nascimbne , n. navon , k.j . jiang , l. tarruell , m. teichmann , j. mckeever , f. chevy , and c. salomon , phys . lett . * 103 * , 170402 ( 2009 ) . m. zwierlein , talk given at the conference bec 2009 , sant feliu de guixols ( spain ) , sept . 5 - 11 , 2009 . c. menotti , p. pedri , and s. stringari , phys . rev . lett . * 89 * , 250402 ( 2002 ) . m. cozzini and s. stringari , phys . lett . * 91 * , 070401 ( 2003 ) . p. pedri , d. gury - odelin , and s. stringari , phys . a * 68 * 043608 ( 2003 ) . e. taylor and a. griffin , phys . a * 72 * , 053630 ( 2005 ) . m. urban and p. schuck , phys . a * 73 * , 013621 ( 2006 ) . m. urban , phys . a * 75 * , 053607 ( 2007 ) , phys . a * 78 * , 053619 ( 2008 ) . f. toschi , p. vignolo , s. succi , and m.p . tosi , phys . a * 67 * , 041605(r ) ( 2003 ) . f. toschi , p. capuzzi , s. succi , p. vignolo , and m.p . tosi , j. phys . b * 37 * , s91 ( 2004 ) . p. massignan , g.m . bruun , and h. smith , phys . a * 71 * , 033607 ( 2005 ) bruun and h. smith , phys . rev . a * 76 * , 045602 ( 2007 ) s. chiacchiera , t. lepers , d. davesne and m. urban , phys . a * 79 * , 033613 ( 2009 ) . bertsch and s. das gupta , phys . rep . * 160 * , 189 ( 1988 ) . b. jackson and e. zaremba , phys . a * 66 * , 033606 ( 2002 ) . e.m . lifshitz and l.p . pitaevskii , _ physical kinetics _ , l.d . landau and e.m . lifshitz course of theoretical physics vol . 10 ( pergamon , oxford , 1980 ) . landau and e.m . lifshitz , _ quantum mechanics _ , course of theoretical physics vol . 3 ( pergamon , london , 1958 ) . w.c . swope , h.c . andersen , p.h . berens , and k.r . wilson , j. chem . phys . * 76 * , 637 ( 1982 ) . l. verlet , phys . rev . * 159 * , 98 ( 1967 ) . w.h . press , teukolsky , w.t . vetterling , and b.p . flannery , _ numerical recipes in fortran : the art of scientific computing _ , 2nd edition ( cambridge university press , 1992 ) . w. kohn , phys . rev . * 123 * , 1242 ( 1961 ) . l. brey , n.f . johnson , and b.i . halperin , phys . b * 40 * , 10647 ( 1989 ) . thomas lepers , institut de physique nuclaire , universit de lyon , phd thesis , to be published . t. lepers , s. chiacchiera , m. urban , and d. davesne in preparation .

we numerically solve the boltzmann equation for trapped fermions in the normal phase using the test - particle method . after discussing a couple of tests in order to estimate the reliability of the method , we apply it to the description of collective modes in a spherical harmonic trap . the numerical results are compared with those obtained previously by taking moments of the boltzmann equation . we find that the general shape of the response function is very similar in both methods , but the relaxation time obtained from the simulation is significantly longer than that predicted by the method of moments . it is shown that the result of the method of moments can be corrected by including fourth - order moments in addition to the usual second - order ones and that this method agrees very well with our numerical simulations .

You are an expert at summarizing long articles. Proceed to summarize the following text: it has been recognized that important particle physics ingredients including gauge symmetry and chiral matter can be realized in type ii superstring models using the mechanism of intersecting d - branes @xcite . more recently , this method has been extensively progressed and it is now possible to investigate some stringy effects which could give rise to deeper details seen in particle physics . in particular , d - brane instantons wrapping non trivial cycles in the internal manifold have been particularly explored in this regard @xcite . they give non - perturbative superpotential corrections which could explain the large mass hierarchies including the smallness of neutrino masses in the standard model ( sm ) @xcite . many works have been done along these lines using configurations realizing mssm - like orientifolded models based on quiver approach . in this way , rather than considering full string theory models , working at the level of quivers allow for dealing with many important physical effects . indeed , one could examine the possible presence or absence of couplings and other physical effects by considering quantum numbers associated with the quiver configuration data . more precisely , many investigations have been performed for mssm - like orientifolded models satisfying necessary constraints allowing for hierarchical mass terms for all three families of quarks and leptons . this can be approached using either extra singlet fields that string theory compactifications contain or via euclidean d - brane wrapping non trivial cycles producing e - instantons which we are interested in here @xcite . they exhibit the so called uncharged modes consisting of fermion modes @xmath5 preserving half of the bulk supersymmetry breaking @xmath6 to @xmath7 and the four bosonic modes @xmath8 , which are associated with breakdown of four - dimensional poincar invariance . these kinds of uncharged zero modes appear if the instantons wrap non - rigid three - cycles in the internal manifold . however , the charged fermionic zero modes @xmath9 appearing at the intersection between e - instantons and d - branes are crucial in the determination of superpotential corrections to four dimensional effective field theories . the non - perturbative contributions are given by performing the path integral over all fermionic zero modes . to get superpotential terms , one must ensure that all uncharged fermionic zero modes , apart from @xmath8 and @xmath10 , are projected out or lifted . in type iia superstring for instance , this can be done by considering e2-instantons obtained from d2-branes wrapping rigid orientifold - invariant 3-cycles embedded in the internal space @xcite . these are referred to as rigid @xmath11 instantons , which will be discussed here . in this picture , when instanton effects do not give rise to phenomenological inconveniences , they therefore represent a way in which one can give rise to viable mass hierarchies for the quarks and leptons as well as the smallness of neutrino masses . in this paper we contribute to these activities by performing a discussion of mass hierarchies based on a type iia four - stack quiver orientifolded model in the presence of e2-instanton effects . more precisely , we focus on @xmath12 quiver gauge theory based on intersecting d6-brane configurations with instanton corrections to the corresponding effective superpotential . for simplicity , we consider @xmath11-instantons wrapping rigid oriontifold - invariant cycles and carrying the right global @xmath13 charges through the @xmath14 . then we give the corresponding mssm - like quiver with the instanton corrections . compared to perturbatively allowed superpotential terms of the heavy fields , the scales of the induced terms of the light fields depend on the suppression factors of e2-instantons which may induce them with some combinations with the string scale @xmath15 through higher order terms . this mechanism offers a stringy framework to explain the fermion mass hierarchies in the quiver method . including the left - handed neutrino masses in the analysis , we get the order of magnitude of the relevant instanton effects and recover the string scale upper bound @xmath16 . the organization of this paper is as follows . in section 2 , we build a quiver gauge theory realizing mssm - like orientifolded models based on four stacks of intersecting d6-branes . in section 3 , we study the fermion mass hierarchies by introducing the possible stringy corrections to the corresponding superpotential using o(1 ) e2-instantons then we give the corresponding quiver . their suppression magnitudes have been estimated . section 4 will be devoted to our discussions . in this section , we will give a quiver model based on four - stack of d6-branes embedded in iia superstring moving on orientifolded geometry . the gauge theory lives on d6-branes wrapping four - dimensional minkowski space and non trivial 3-cycles in the internal manifold @xcite . it is recalled that d6-branes carry ramond - ramond charges that should be cancelled by the introduction of orientifold geometries related to fixed point loci of an antiholomorphic involution acting on the internal space . in such quiver models , the bifundamental chiral matter arises at the non - trivial intersection of two generic d6-branes . it turns out that , one can have symmetric or anti - symmetric tensor representations where a d6-brane intersects its image brane under the orientifold action . furthermore , two given d6-branes might intersect in multiple points on the compact internal space , giving rise to multiple families , where the number of families is the topological intersection number of two 3-cycles belonging to middle dimensional cohomology . indeed , a stack of @xmath17 d6-branes wrapping 3-cycles gives rise to @xmath18 gauge symmetry . however , @xmath17 d6-branes wrapped on cycles which are homologically - fixed or pointwise - fixed by the orientifold action give rise to @xmath19 and @xmath20 gauge symmetry , respectively . since @xmath21 is isomorphic to @xmath22 , the @xmath23 of the mssm - like models can be realized as a @xmath21 arising from a d6-brane stack wrapping an orientifold invariant 3-cycle . in this regard , the sm gauge symmetry and the matter content can be accommodated in the gauge group @xmath24 ) obtained from three stacks of d6-branes . enlarging the gauge group can make the model richer . this can be done by assuming that one has an appropriate middle dimensional cohomology generated by non trivial 3-cycles . the model we consider here relies on a four stacks of d6-branes giving rise to the following gauge symmetry @xmath25with a gauged flavor group @xmath26 distinguishing various quarks from each others . roughly , the tadpole conditions imply the vanishing of non - abelian anomalies , while abelian and mixed anomalies are canceled via the green - schwarz mechanism . generically , the anomalous @xmath27 acquire a mass and survive only as global symmetries , which forbid various couplings on the perturbative level . since the sm gauge symmetries contain the abelian symmetry @xmath28 , and the @xmath21 does not exhibit a @xmath29 which could contribute to the hypercharge , one requires that a linear combination of @xmath30 remains massless . thus , the resulting gauge group in four - dimensional spacetime can be written as @xmath31given the above gauge symmetry based on d6-brane configurations , we can construct a quiver describing mssm - like orientifolded model . indeed , vanishing of anomalies which we require to be satisfied are used to fit the matter content involving two up quarks , one down quark charged under the @xmath32 gauge symmetry and the opposite arrangement charged under @xmath33 the model we present here relies on a particular intersection numbers of 3-cycles in middle dimensional cohomology . for that , we choose the following intersections ] @xmath34the other intersection numbers are set to zero . the chiral spectrum and the gauge symmetry can be represented in the table 1 together with their identification with sm matter fields . .the spectrum and their @xmath35 charges for @xmath36 . the index @xmath37 denotes the family index [ cols="^,^,^,^,^,^,^,^,^",options="header " , ] in this description , all leptons and the two higgs doublets are charged under @xmath38 symmetries while for quarks only their right handed partners are . the model with four stacks can be encoded in a quiver where each node represents a d6-brane and the links between them indicate their chiral intersections . the quiver summarizing the above spectrum with the two higgses is shown in figure 1 ) gauge group . , width=340 ] since any realistic string vacua has to exhibit the phenomenologically desired terms in four dimensional effective superpotential , we require the presence of all the mssm yukawa couplings and the absence of the phenomenologically undesired couplings terms on perturbative level . in what follows , the @xmath38 symmetries will be used to select the candidate terms for the superpotential . taking into account the charges presented in the table 1 , we can write down the all allowed interaction terms generating the possible yukawa couplings . it turns out that , the only fields involved in the allowed perturbative interaction terms are those charged under the @xmath39 gauge symmetry . we interpret them as the the heavy quarks @xmath40 , leptons @xmath41 and the @xmath42 term for @xmath43 @xmath44 . this could be seen at the level of their associated quantum numbers @xmath45here the index @xmath46 denotes the family index and the subscript indicates the charge under the @xmath47 symmetries . our quiver naturally allows for the following couplings @xmath48where @xmath49 are coupling constants accounting for hierarchies between these terms . the absent couplings , which are phenomenologically desired , corresponding to fermions charged under @xmath26 violate the @xmath38 symmetries . these will be refered to the light fermions @xmath50 that will be yielded massless , unless some kind of new effects in the defined effective field theory break the remnants abelian symmetries @xmath38 . in string theory , the natural candidate non - perturbative effects to violate these @xmath38 symmetries are instantons arising from euclidean d - branes coupling to these fields @xcite . they can potentially destabilize the vacuum or lead to new effects in the four - dimensional effective action . in what follows , we will discuss the implication of such non perturbative effects in our quiver model . in particular , we will consider orientifolded invariant 3-cycles on which d2-branes can wrap to make rigid @xmath11 instantons . then , we discuss fermion mass hierarchies in such instanton configurations . in this section we introduce the effects of spacetime instantons on the above described model . in type iia superstring model , such non - perturbative effects are generated in principle from d0 , d2 and d4-branes wrapping one , three and five - cycles respectively . however , since a calabi - yau manifold does not have any continuous one and five - cycles , the only relevant instantons are d2-branes wrapped on three - cycles in the internal manifold @xcite . roughly , the resulting d2-instanton action takes the follwing form @xmath51 in this equation , @xmath52 describes all terms involving fermionic instanton zero modes and four dimensional charged matter fields @xmath53 while @xmath54 is the dirac born - infeld action on d2-brane wrapping a 3-cycle @xmath55 in the presence of the wz term . this action reads as @xmath56where @xmath57 is the r - r 3-from coupled to the d2-brane . in four dimensions , the instanton contributions to the low energy effective theory can be obtained by performing the grassmann path integral over all fermionic zero modes @xmath58 : @xmath59 e^{-s_{e}}={\underset{n}{\pi } } \phi _ { n}e^{-s_{e}^{cl}}. \label{eq8}\]]the classical instanton action @xmath60 can absorb the @xmath61 s charge excess of the matter fields operator @xmath62 @xcite . under such abelian symmetries the transformation property of the exponential instanton action is @xmath63where @xmath64 represents the amount of the u(1)-charge violation by the @xmath65-instantons . their microscopic origin is in the extra fermionic zero modes @xmath58 living in the intersection of the e2-branes with the @xmath66 branes . instead of being general , let us consider a concrete configuration with @xmath65-intantons wrap rigid orientifold - invariant 3-cycles . in this case , the @xmath67 and @xmath68 are identified . so , an @xmath65-instanton intersecting @xmath69-branes can induce the desired couplings of the matter fields operator @xmath70 @xmath71 @xmath72 . the @xmath38 charges which are carried by the instanton action have their origin in the intersection @xmath73 pattern . for an instanton wrapping a 3-cycle with an appropriate number of times , this can be made exactly opposite to the total charge of the operator @xmath74 , so the coupling ( 8) is also @xmath38-invariant . examining the @xmath75 charge - excess for each term , we determine the intersection @xmath73 pattern giving rise to the right charged fermionic zero modes @xmath76 @xmath77 compensating the @xmath75 charge - excess of the desired couplings . indeed , the up - quark coupling term @xmath78violate the @xmath38 symmetries by one unit as @xmath79 then is generated by an instanton @xmath80 intersecting @xmath69 branes with the following intersection numbers @xmath81which gives rise to two charged modes @xmath82 @xmath83 @xmath84these charges compensate the charge - excess of the up - quark term . in this case , the corresponding e2-instanton action takes the following form @xmath85performing the integration over the all fermionic zero mode , one get the desired term @xmath86analogous analysis can be done for the down and strange quark coupling terms @xmath87this can be generated by an @xmath88-instanton intersecting the @xmath69-branes with the following intersection numbers @xmath89and producing the two charged modes @xmath90 @xmath91 . for the sector of neutrinos , the following higher order term @xmath92could be generated by a @xmath93-instanton intersecting @xmath69-branes . the corresponding intersection numbers are given by @xmath94this leads to the charged modes @xmath95 @xmath96 . integrating over the all above fermionic zero modes as in ( eq14 ) , we get the missing superpotential terms at the four - dimensional theory @xmath97where the coupling constants @xmath98 can account for hierarchies between the @xmath99 and @xmath100 quarks having the same instanton suppression . the neutrinos superpotential term is highly suppressed by @xmath101 factor and once higgs fields get a vev @xmath102 this operator gives rise directly to majorana masses for the left - handed neutrinos depending on the scale @xmath15 taken as the low string scale at which neutrino masses have origin . the quiver illustrating these instanton intersection patterns with their appropriate charged fermionic zero modes is depicted in the figure2 . -instantons and dotted lines indicate their intersections with the @xmath69-branes.,width=340 ] recall that the exponential suppression effects can be derived from the classical instanton actions . their magnitudes depend on the internal geometry on which the quiver model is based . in particular , they depend on the complex structure moduli space described by the volume of the 3-cycles wrapped by the relevant instantons . to have an idea about the induced mass hierarchies , we need to estimate the different instanton factors . although such approach forces to consider deeper details of the fully defined string model , we can get an order of magnitude of the different suppression factors by refereing to the known data . effectively , after the higgs fields break the electroweak symmetry at the usual scale @xmath103 @xmath104 , we take the quark couplings terms of eq([eq19 ] ) together with the quark coupling terms of eq ( [ eq5 ] ) with some assumption on their scalar - fermion couplings to derive the expected magnitudes . for that , using some combination of the quark masses where their net scalar - fermion couplings effect could be neglegted , one allows for getting approximate values of the corresponding suppression factors . for the @xmath105-quark suppression factor , assuming @xmath106 and @xmath107 , we find @xmath108while for the @xmath99 and @xmath100-quark suppression factors with @xmath109 , we get @xmath110similarly , using the known data with neutrino masses upper bound @xmath111 @xmath112 we find the neutrino high suppression factor@xmath113considering the usual value of the string scale , namely @xmath114 , the e2-instanton induced operator only leads to subleading corrections to the neutrino masses . their observed order can be obtained by lowering the string scale due to large internal dimensions @xcite . lowering the string scale down to the @xmath115 scale , this leads to the interesting features at the lhc . from ( [ eq21 ] ) , we can deduce the string scale upper bound @xmath116investigating the above instanton suppression factors ( [ eq20 ] ) and ( eq21 ) and specializing the case that the d6-branes and the e2-branes wrap factorizable three - cycles of toroidal orientifolds , the volume of these instantons could give an insight of the detailed description of geometric background of the internal manifold . we believe that this connection deserves to be studied further . in this paper , we have discussed the fermion mass hierarchies in a local four - stack intersecting mssm - like d - brane quiver models using non perturbative stringy effects . more precisely , we have focused on a four - stack of d6-brane configurations in type iia orientifolded geometries giving rise to the mssm - like spectrum without right handed - neutrinos including rigid o(1 ) e2-instantons . in particular , we have given a quiver model from which we have shown that some perturbatively absent coupling terms can be generated from d2-branes wrapping 3-cycles belonging to middle dimensional cohomology of the internal space . analyzing the quiver allows for the determination of perturbatively and non - perturbatively contributions to the superpotential using the abelian symmetries obeyed at the perturbative regime . in this approach , attributing the perturbatively realized terms to the heavy fermions and the remaining missing desired terms to the light fermions reflects interesting hierarchies . the latters are induced and exponentially suppressed by e2-instantons carrying the right charged fermionic zero modes appearing at the e2-d6 intersections . these stringy corrections do not induce the perturbatively forbidden dangerous proton decay terms through the dimension 5 operators @xmath117 and @xmath118 . such kinds of local intersecting d - brane models have been discussed extensively in @xcite . these models , which include phenomenological constraints of fast proton decay are listed in @xcite . however , the presented model does not appear in that classification since it does not involve right - handed neutrinos in addition to the lepton number violation that arises through @xmath119 and @xmath120 terms induced by the @xmath80 instanton required to generate the missing up - quark mass term . despite , it should be interesting to make contact with the models given in @xcite . we believe that this connection deserves more study . this will be reported elsewhere . using the known data with neutrino masses @xmath1 @xmath2 @xmath3 @xmath121 , we have given the magnitudes of the relevant instanton effects and the string scale upper bound @xmath122 @xmath123 imposed to match with the observed order of the known data . in the end of this work , a possible discussion could be done in terms of middle dimensional cohomology of the internal space describing its complex structure . the analysis depends on the volume of the 3-cycles on which d2-branes wrap to make rigid o(1 ) instantons . it should be very nice to find a geometrical interpretation in terms of quiver data encoded in the middle dimensional cohomology . we believe that it will be useful to explore extended dynking geometries involving more than bosonic vertices . this seems to be promising in type iib d5-branes set - up in the presence of d - strings wrapping 2-cycles making instantons .

using instanton effects , we discuss the problem of fermion mass hierarchies in an mssm - like type iia orientifolded model with @xmath0 gauge symmetry obtained from intersecting d6-branes . in the corresponding four - stack quiver , the different scales of the generated superpotential couplings offer a partial solution to fermion mass hierarchies . using the known data with neutrino masses @xmath1 @xmath2 @xmath3 @xmath4 , we give the magnitudes of the relevant scales . * keywords * : type iia superstring , instanton effects and yukawa couplings .

You are an expert at summarizing long articles. Proceed to summarize the following text: it has long been thought that t hooft s @xmath3 limit @xcite of @xmath4 gauge theory might be usefully described by some sort of string theory . however , there is an apparently devastating argument , that this `` qcd string '' ( a.k.a . a tower of glueballs ) is _ not _ fundamental string ( of `` string theory '' ) : the graviton appears in the spectrum of the latter , contradicting the well - known folk theorem @xcite forbidding massless spin 2 bound states in a poincar invariant quantum field theory . a way to evade this argument has been shown by the conjectured equivalence between classical iib superstring theory on an ads@xmath5 background and @xmath6 supersymmetric @xmath7 yang - mills on flat 4 dimensional minkowski space - time @xcite . the point is that in this example the graviton lives in 5 space - time dimensions and the flat space - time global symmetry ( poincar(3,1 ) ) is only a subgroup of poincar(4,1 ) , which is realized locally , not globally . thus the `` massless '' 5 dimensional graviton is a composite of the quanta of a flat - space quantum field theory in 4 dimensional space - time . there is no massless spin 2 particle in this 4d quantum field theory and no folk theorems are violated . in the @xmath6 conformally invariant example , the projection of the 5d graviton onto the 4d minkowski boundary of ads@xmath5 is a multi - gluon continuum state . but if the mechanism can be extended to the non - conformally invariant gauge theory describing the gluon sector of qcd , the graviton 4d remnant would presumably be a massive spin 2 glueball . to move these statements beyond conjecture , one clearly needs to establish the `` dual '' description starting from either of the supposedly equivalent theories . i think it is clear that the best starting point for such a project is the flat space quantum field theory . unlike the previous conjectured dualities in string theory , which asserted the equivalence of pairs of poorly defined theories , this duality asserts the equivalence of a poorly defined theory ( string or quantum gravity ) to a perfectly well defined theory ( asymptotically free or conformally invariant quantum gauge theory on flat 4d space - time ) . indeed , i am inclined to regard this `` duality '' as more analogous to the alternate descriptions of superconductivity given by bcs theory on the one - hand and landau - ginzburg theory on the other . if this metaphor holds , the flat space quantum field theory should be embraced as the long sought microscopic formulation of string / quantum gravity . as t hooft showed in his pioneering paper @xcite , the @xmath3 limit of @xmath4 yang - mills theory reduces to a certain sum of planar feynman diagrams . elegant techniques , involving the explicit elimination of the off - diagonal matrix elements of the matrix field , have been used to obtain this limit in matrix theories of extremely low space - time dimension ( namely d=0,1 ) @xcite , but these methods have failed to deal with theories with space - time dimension @xmath8 . at the moment , i see no better approach to the @xmath8 case than setting up a framework to carry out the direct summation of planar graphs . in the mid-1970 s , motivated by the success of light - cone quantization of string theory @xcite , i proposed @xcite that planar diagram sums be carried out by using light - cone parameterization @xmath9 and that a convenient way to digitize the sum was to discretize the momentum conjugate to @xmath10 , @xmath11 with @xmath12 , and imaginary light - cone time @xmath13 , with @xmath14 . in those first papers i restricted attention to scalar field theories , but brower , giles , and i soon made a first attempt to extend the approach to qcd @xcite . in our setup , the strong t hooft coupling limit @xmath15 favors the fishnet @xcite diagrams that lead to a light - cone string interpretation . of course , by its very nature a strong coupling limit probes the microscopic details of discretization , and can at best show only rough qualitative resemblance to the continuum theory . even so , there were a number of loose ends and unsatisfactory features of this first discretization of qcd which needed to be addressed . motivated by the goal of discovering a more definitive string description of large @xmath0 qcd klaus bering , joel rozowsky , and i set out to remedy these shortcomings , and in this talk i would like to tell you about the results of our efforts @xcite . as you will see , we have obtained a much improved discretization setup , but have just begun to explore its usefulness in capturing a string picture of the sum of planar diagrams . an evolving string sweeps out a world sheet @xmath16 in space - time . one can choose the parameters so that @xmath17 , and so that @xmath18 , the density of @xmath1 , is a constant @xmath19 . then evolution in @xmath2 is generated by the hamiltonian @xmath20,\end{aligned}\ ] ] where for brevity we have called @xmath21 and the prime denotes differentiation with respect to @xmath22 . of course @xmath23 is the momentum conjugate to @xmath24 . a key novelty here is that @xmath25 measures the quantity of string present , and its interpretation as a component of momentum is secondary and derivative . in this way the string is seen to enjoy a galilei invariant dynamics as it moves only in the @xmath26 dimensional transverse space . as shown by mandelstam @xcite , interactions are easily introduced by using the path history form of quantum mechanics and including histories in which strings break and join . technically this is accomplished by first obtaining the imaginary time @xmath27 path integral representation of @xmath28 for free string . the action in this case is just @xmath29.\end{aligned}\ ] ] here the action is seen to be an integral over a simply connected rectangular domain for an open free string and a cylinder for a closed string . a diagram describing an arbitrary number of splits and joins is obtained by allowing some number of cuts , each at constant @xmath22 but of varying length , within the domain . the ends of these cuts mark the splitting or joining points , and @xmath30 is discontinuous across them . calling the generic such domain @xmath31 , the complete amplitude is then given by @xmath32 in order to give a nonperturbative definition of the path integrals appearing in the previous section , roscoe giles and i introduced a lattice version @xcite of the domains @xmath31 . it was only necessary to discretize @xmath33 and @xmath22 . so we set @xmath34 and @xmath35 , with @xmath36 and @xmath37 fixed positive integers . then @xmath38 is replaced by @xmath39 , and the functional integration by ordinary integrals . finally , the action is simply replaced by @xmath40 where @xmath41 labels a link on the lattice . links between nearest neighbor sites in the @xmath33 direction are all present , but those in the @xmath22 direction are occasionally absent reflecting the possibility of splits and joins . it is a highly nontrivial fact that this apparently noncovariant setup turns out , after the continuum limit , to be fully compatible with poincar invariance in the critical dimension . to understand how feynman diagrams look in light - cone parameterization , consider the mixed representation of the scalar field propagator : @xmath42 for simplicity we may establish the convention that each line propagates forward in @xmath2 , and correspondingly @xmath43 . also we may pass to imaginary time , @xmath44 and then write @xmath45 where the second form is in transverse coordinate representation and @xmath46 is the dimensionality of transverse space . discretizing @xmath47 and @xmath48 , @xmath49 , the coordinate propagator becomes provides a universal ultraviolet cutoff , and every diagram will therefore be finite . in contrast , the dlcq industry @xcite keeps time continuous , and must regulate ultraviolet divergences in some other fashion . ] @xmath50 comparing to the previous section , we see that we can crudely think of a planar feynman diagram as a ( coarsely ) discretized world sheet , with a dynamical link dependent string tension @xmath51 . each link has its own independent @xmath52 which are each summed over all positive integers . of course only the large fishnet diagrams will bear any actual resemblance to a continuous world sheet ! the sum over all planar diagrams would then define the qcd string dynamics as including an average over all such discretizations ranging from coarse to fine . also note that a good world sheet path integral should have ( effectively ) only positive weights . for instance , for @xmath53 theory each vertex contributes a minus sign if @xmath54 . a good world sheet interpretation requires @xmath55 , the attractive unstable sign . gauge theories have vertices of both signs , complicating a straightforward world sheet interpretation . to illustrate the effect of our discretization on a standard diagrammatic calculation , consider the sum of those 2 to 2 scattering diagrams in @xmath53 theory shown in fig . [ twototwo ] . we work in the transverse center of mass frame . assume that the ( discrete ) @xmath56 of the initial ( final ) particles is @xmath57 ( @xmath58 ) . let @xmath59 be the initial energy ( @xmath60 ) . fix the discrete time of the first vertex at @xmath61 , and sum over all diagrams in which the final particles both propagate to time @xmath62 , multiply by @xmath63 and sum over all @xmath62 from 1 to @xmath64 . the result for the off - energy shell s - matrix is then @xmath65 compared to standard formal scattering theory , we see that instead of a factor @xmath66 which acts in wavepackets like the standard energy conserving delta function , the use of discrete time has rendered this as @xmath67 . this replacement is easy to understand because with discrete imaginary time , the amplitudes should be periodic in @xmath68 with period @xmath69 . it is thus apparent that the scattering amplitude should be identified with the coefficient of this factor . for @xmath55 , the scattering amplitude shows a bound state pole at a real negative value of @xmath68 : @xmath70.\ ] ] in the continuum limit , @xmath71 , one can make the pole location stay finite by tuning @xmath72 to vanish logarithmically as @xmath73 , showing dimensional transmutation in an asymptotically free theory . the discretization of qcd initially attempted by brower , giles , and me @xcite , was based on a literal transcription of the feynman rules in light - cone gauge . the transverse gauge field can be described in the complex basis @xmath74 when it takes on the guise of a complex scalar field , described diagrammatically by attaching an arrow to each line of a feynman diagram . the primitive quartic vertex conserves arrows , but the cubic vertices can act as sources or sinks of arrows . the longitudinal gauge field @xmath75 does not propagate and can be integrated out to yield an induced quartic vertex , which depends upon the @xmath1 values of the incoming legs in a singular way : @xmath76 upon discretization , we adopted the drastic prescription of simply dropping the infinite contribution at @xmath77 . furthermore all tadpole diagrams had to be dropped , because of our insistence that no line propagate 0 time steps . then the strong coupling limit singled out large planar diagrams involving only the primitive quartic couplings , thus leading to an evaluation similar to the @xmath78 example of the previous section . unfortunately , these quartic couplings have mixed signs : an attractive interaction between gluons of parallel spin and repulsive between gluons of antiparallel spin . this ferromagnetic interaction pattern meant that our discretization led to a formal strong coupling limit in which the only long string that could form in the limit would have huge total spin . the essential problem is that attractive interactions between gluons of opposite spin arise in qcd from gluon exchange @xcite , and the discretization chosen in @xcite prevents anti - ferromagnetic gluon exchange from competing at strong coupling with the ferromagnetic quartic interaction . this , together with our drastic prescription for all of the @xmath79 ills of light - cone quantization , points to the need for a more refined discretized model to adequately describe the strong coupling behavior of large @xmath0 qcd . clearly what is needed is a prescription that either enhances strong coupling gluon exchange or suppresses the strong coupling quartic interactions . we found it most natural to arrange the latter by abolishing all quartic interactions , primitive and induced , and replacing them with the exchange of short lived fictitious particles . this is shown for the primitive quartic interaction in fig.[4vertexto3vertex ] . the dashed line is associated with the fictitious two - form propagator @xmath80 where the @xmath81 are tunable parameters which are required to vanish rapidly with @xmath82 , the number of time steps propagated . we treat the induced quartic interaction in a similar fashion , giving the non - dynamical field @xmath75 a short - time propagator @xmath83 the presence of the tunable parameters @xmath84 and @xmath81 is very welcome , because they can be adjusted to arrange the cancelation of cut - off artifacts that can typically spoil poincar invariance in the continuum limit . as a bonus , we find that our prescription provides the appealing interpretation of tadpole diagrams indicated in fig . [ tadpole2 ] . the first two diagrams cancel exactly ( which is fortunate since they ca nt be drawn in our discrete model ! ) leaving the third diagram which _ can _ be drawn . our complete set of feynman rules is neatly summarized in fig . [ newrules ] taken from our paper @xcite . [ cols="^,^,^,^ " , ] note that to avoid clutter we have suppressed the double line notation so these rules are completely sufficient for all graphs of planar or cylindrical topology ( @xmath3 ) . for general diagrams , the double line notation must be restored in order to properly account for the @xmath85 corrections . an easy way to understand why a set of rules with only cubic vertices is possible , is to apply light - cone gauge to the yang - mills lagrangian in first order form . the upshot is the lagrangian @xmath86\nonumber\\ & & -ig\tr{1\over\partial_-}{\hat a}[a_k,\partial_-a_k ] -ig\tr\phi_{kn}[a_k , a_n],\end{aligned}\ ] ] where @xmath87 . this makes it clear why we called the fictitious scalar represented by the dashed lines a 2-form : it is a ( pseudo ) scalar only in 3 + 1 dimensions . quite apart from its use as a facilitator for a strong coupling expansion , our discretization can also serve as a novel way to regulate the diagrams of weak coupling perturbation theory . to illustrate this aspect , we quote the result for the one loop gluon self - energy , with discretization in place : @xmath88-{(m-1)(11m-1)\over3 m } \right)\nonumber\\ & & -\sum_{k=1}^\infty u^k\sum_{l=1}^{m-1}{lh_k(l)\over mk } -\sum_{k=1}^\infty f_k{u^k\over k } \left(4m[\psi(m)+\gamma]-{7(m-1)\over2}\right ) \bigg ] , \label{pi2new}\end{aligned}\ ] ] where @xmath89 . in order to cancel lattice artifacts in the continuum limit @xmath90 , we find the constraints @xmath91 then we obtain @xmath92\ln{q^2\over2mt_0}+{4\over3 } \right\}.\end{aligned}\ ] ] here @xmath93 functions as a uv cut - off . with this understanding our result for @xmath94 agrees exactly with the known light - cone gauge result @xcite . notice that the parameters @xmath95 which specify our discretization enter weak coupling physics only through their _ moments _ , for example @xmath96 . in contrast , the strong coupling limit is sensitive to the values of these parameters at low @xmath62 . thus the two limits give complementary constraints on these parameters . our main aim in developing this discretization formalism is to establish a framework for handling the sum of all the planar diagrams of @xmath3 gauge theories . as yet we do nt have any dramatic results to report . however we have begun studying how the machinery works in simpler situations than full - blown gauge theories . in our paper @xcite we worked out the sum of the densest ( strong coupling `` fishnet '' ) planar diagrams of @xmath97 scalar field theory . as expected the result leads to the light - cone quantized bosonic string ( with all of its usual pathologies , including the tachyon ) . the presence of tachyons is not particularly surprising , since the energy density of the theory is unbounded below . we have not made analogous progress on the corresponding diagrams of gauge theories . but we have studied the latter theory in the sectors with @xmath98 . this is not particularly difficult , since the limitation on @xmath36 reduces the sum of all possible diagrams to a geometric series . nevertheless , it is interesting , for example , that the @xmath98 gluon propagator summed to all orders in perturbation theory displays no additional poles beyond that of the massless gluon itself . in contrast the @xmath98 propagator for the fictitious 2-form field shows a bound state pole at sufficiently strong coupling @xmath99 . this indicates that the 2-form field may be particularly important for the understanding of the strong coupling limit . another relatively simple testing ground for our formalism is quantum field theory in low space - time dimensions , the simplest being the t hooft model ( qcd in one space and one time dimensions ) . rozowsky and i are just finishing up a study of this model . since this model is well understood even in the continuum limit , we used it mainly as a test of how our model approaches the continuum theory . one interesting feature of our simultaneous discretization of @xmath1 and @xmath2 is that one can approach the continuum in different directions . for example the approach to continuum at fixed @xmath100 is different from the approach studied in conventional dlcq . the latter keep @xmath2 continuous throughout ( in our language this means taking @xmath73 _ first _ followed by @xmath101 ) . we have confirmed that the same continuum physics emerges in both cases . i would like to conclude this talk with some remarks on longer term prospects and goals . our renewed efforts to sum planar graphs have been directly stimulated by the ads / cft duality @xcite proposed in the last couple of years . this duality in turn was recognized @xcite to be a higher dimensional realization of t hooft s concept of holography @xcite : the vision that a consistent quantum theory of gravity requires our apparently 3 dimensional spatial world to be 2 dimensional . i have advocated string bits as a way to realize holography in t hooft s original 2 dimensional sense : the two dimensions being the transverse dimensions of light - cone string . however , the qcd gluons of this talk really live in 3 dimensions in spite of their description on the light - cone : the longitudinal dimension hasnt disappeared . rather , it has been disguised as a variable newtonian mass . ( in the ads / cft duality this third dimension gets interpreted as a fifth dimension , whence holography is the statement that a 4 + 1 dimensional effective theory arises from a 3 + 1 dimensional quantum field theory . ) the defining character of string bits is that they have a _ fixed _ newtonian mass , in sharp contrast to the gluons we have been describing . to understand 3 + 1 gauge theories as part of a string bit theory , a gluon with @xmath11 must in reality be a composite of @xmath102 string bits : the gluon vertices would then be effective fission / fusion amplitudes as in nuclear physics , rather than fundamental interactions . s. mandelstam , _ nucl . phys . _ * b64 * ( 1973 ) 205 ; _ phys _ * 46b * ( 1973 ) 447 ; _ nucl . phys . _ * b69 * ( 1974 ) 77 ; see also his lectures in _ unified string theories _ , ed . m. green and d. gross ( world scientific ) 1986 .

i describe renewed efforts to establish a string description of large @xmath0 qcd by summing large `` fishnet '' diagrams . earlier work on fishnets indicated that the usual relativistic ( zero thickness ) string theory can arise at strong t hooft coupling , at best yielding a highly idealized description , which fails to incorporate such salient features of continuum qcd as asymptotic freedom and point - like constituents . the recently conjectured ads / cft correspondence is compatible with such limitations because it also gives a simple picture of large @xmath0 gauge theory only at strong coupling . in order to better understand how string theory could emerge from large @xmath0 qcd at strong coupling , klaus bering , joel rozowsky , and i have developed an improved implementation of my effort of the late seventies to digitize the planar diagrams of large @xmath0 light - cone quantized qcd by discretizing both @xmath1 and @xmath2 . this discretization allows a strong coupling limit of the sum of planar diagrams to be defined and studied . it also provides a natural framework to explore the possible dual relationship between qcd in light - cone gauge and string theory quantized on the light - cone . ufift - hep-00 - 9 + hep - th/0004129 1.5 cm * string from large n gauge fields + via graph summation on a @xmath1 - @xmath2 lattice * 2.cm charles b. thorn 0.5 cm _ institute for fundamental theory + department of physics , university of florida , gainesville , fl 32611 _ ( ) .5 cm

You are an expert at summarizing long articles. Proceed to summarize the following text: , response to the emerging carbon emissions constrained world , the usage of renewable energy sources is increasing . the overall increase in penetration of renewable energy resources in the u.s . is depicted in fig . [ fig : renewablepercentage ] . such growth of renewable energy resource is not limited to the u.s . globally , installed global renewable electricity capacity has continued to increase and represented 28.5% of total electricity capacity in 2014 @xcite @xcite . renewable energy sources are generally characterized as _ variable energy resources _ ( ver ) due to their variability and uncertainty @xcite . while there has been efforts for better control of resources such as wind farms as doubly fed induction generators @xcite , their limited controllability and lack of predictability pose new challenges for the operation of the power system . price - responsive demand , or _ demand response _ ( dr ) is a key mechanism to achieve system balancing . one avenue is by modifying consumption patterns through economically exposing customers to time - varying pricing that reflect supply - demand balancing status . a number of programs on dr have been implemented or proposed @xcite @xcite . while such price - responsive demand can potentially provide the key to system operability under high penetration of vers , the presumed benefits of dr programs substantially depend on how responsive demand actually is to price : the price elasticity of demand @xcite . though price elasticity of demand is critical for the effectiveness of dr programs , previous empirical works based on data - driven _ static _ analysis of demand suggests that even load labeled as price - responsive is fairly inelastic @xcite . on the other hand , viewing dr as a _ dynamical system _ , our previous work @xcite analyzing industrial and commercial loads of which is directly exposed to _ electric reliability council of texas _ ( ercot ) real - time wholesale market makes two observations : 1 . the consumer s response to large prices ( over the 95%-quantile : $ 144.42 ) can be modeled as a _ hammerstein system _ , i.e. , a static nonlinearity followed by a linear transfer function @xcite . after accounting for this nonlinear transformation , which is typically concave since the response is sublinear , the response exhibits a reduction after a delay of about 0.75 - 2.5 hours , before subsequently reverting back to normal levels . the response to moderate prices ( up to $ 144.42 ) can be modeled as a linear stochastic system , specifically as an _ autoregressive exogenous _ ( arx ) system , i.e. , an autoregressive ( ar ) system with exogenous input and white noise . this paper extends our previous work @xcite , by analyzing the economic effect of characteristics of consumer behavior that prevent real - time retail electricity pricing from optimal signaling and respon . the rest of this paper is organized as follows . in section [ sec : litreview ] , previous works analyzing the models and benefits of price responsive demand , mostly conducted in the economics literature , are reviewed . we introduce our observations on consumer behavior from empirical load data from ercot in section [ sec : consmobs ] . in section [ sec : effrtp ] , on the basis of our empirical observation we discuss the implication of our observations , presenting an alternative analysis of the potential benefits of dr in comparison to previous literature . concluding remarks followed in section [ sec : conc ] . while the idea of dr is currently attracting wide interest as a solution for system operability under high penetration of vers , the necessity of dr has been advocated for decades by economists from a market efficiency perspective . the volatility of load that has been challenge for system operators to cope with also entails abrupt and drastic changes in electricity price in the wholesale market . though extreme price fluctuation is widely observed in today s restructured electricity wholesale competitive markets , retail customers in most regions do not face frequent price change . while wholesale electricity prices vary from hour to hour , retail prices do not change for months in most electricity markets . such discordance between rapid fluctuation in wholesale prices and near flat retail prices not only incurs economic inefficiency in terms of social welfare , but also creates price - inelastic wholesale demand that severely exacerbates the volatility of wholesale electricity prices . the combination of inelastic demand with the inherent real - time nature of the market makes electricity markets vulnerable to the exercise of market power @xcite . as a method to achieve price responsive demand , there has been a consensus on the potential benefits of _ real - time retail pricing _ ( rtrp ) among economists @xcite @xcite @xcite @xcite @xcite @xcite . the first potential benefit most discussed in the literature is the allocative efficiency improvement resulting from resolving the market inefficiency caused by ( near ) constant retail electricity prices @xcite @xcite @xcite @xcite @xcite @xcite . the second benefit studied is the increased robustness of the market with rtrp forestalling the exercise of market power @xcite @xcite @xcite @xcite . the last benefit considered is that the mitigation of demand volatility induced by real - time price signals will also relieve the need for excessive reserve requirement which incurs a large portion of the societal costs @xcite @xcite @xcite . however , all the potential economic benefits of rtrp substantially depend on how responsive demand is to price , i.e. , the price elasticity of demand @xcite @xcite . the efficiency improvement of rtrp is well analyzed in the literature @xcite @xcite @xcite @xcite , as depicted in figure [ fig : econinefffixrate ] . since the demand curve has a time variant property , it is not likely to happen that the fixed rate meets @xmath0 or @xmath1 , which are the optimal market clearing prices in terms of social welfare maximization . thus , the shaded triangles @xmath2 and @xmath3 are the deadweight loss , the economic inefficiency caused by the fixed rate @xmath4 . due to the instantaneousness of electricity , it is reasonable to assume that the electricity at each time slot is a distinct commodity . thus , rtrp advocates argue that the ultimate real - time retail price is the optimal pricing policy@xcite in terms of economic efficiency . although the analysis shown in figure [ fig : econinefffixrate ] seems reasonable , it requires a crucial assumption to be justified : _ demand converges to @xmath0 or @xmath1 almost immediately , in at most one time slot as determined by the market rules . _ however , this assumption is controversial when the market is fast - paced . additionally , a fundamental limitation in the demand - supply curve model is that it is difficult to obtain any insight concerning dynamic behavior from the demand curve , which makes it difficult to estimate and predict demand from this static model . the goal of our work is to develop a model for demand response as a stochastic dynamical system where both past prices and past consumption influence future consumption probabilistically . in this section , we introduce our prior work @xcite that poses the problem of modeling price responsive demand at wholesale level . this work is based on an analysis of the data from an anonymous commercial / industrial ( c / i ) load in houston , purchasing its power directly from ercot real - time wholesale market , gleaned over nine months ( jan.1 - sep . 30 , 2008 ) . based on the empirical data , a dynamical model of consumer behavior is presented . the c / i load and prices from houston measured at intervals of 15 minutes from jan . 1 , 2008 to sep . 30 , 2008 is presented with respect to time in fig . [ fig : pq - hourlyplot ] . the first notable point observed here is that the plot on price ( fig . [ fig : price - boxplothourly ] ) shows many outliers while the plot on load rarely has any . this is called the spiky " nature of electricity prices , an irregular sudden extreme price change for a very short duration of 15 - 30 minutes ( fig . [ fig : price - sample ] ) . this gives the price a highly non - normal heavy - tail distribution . the fundamental reason for the spiky nature of prices is explained in section [ sec : litreview ] . the other property we see in fig . [ fig : pq - hourlyplot ] is that the time - series of load shows a depressed demand in `` peak hours '' ( afternoons ) , over time intervals that overlap with the time intervals exhibiting frequent large outliers in the price time series . here , we surmise that the depressed demand is a manifestation of demand response , and that this demand response is highly connected to the outliers of price , because the depression is not likely to be explained by the plot of the median prices ( fig . [ fig : price - medianhourly ] ) . .statistics of price ( p ) and load ( @xmath5 ) [ cols="^,^,^,^,^ " , ] on the basis of the above observations , we establish a simple dynamic model between the magnitude of the price surge and the load , in the case of high price surges . taking into account the long - tailed characteristic of prices , we consider a linear model for a concave transformation @xmath6 , instead of @xmath7 . in this paper , we present a tf for a specific time period , from 2:00pm to 2:30pm , due to the innate time - dependency on dr . the estimation results for the arx model of dr at high price are presented in tables [ tab : dr - peak1 ] , [ tab : dr - peak2 ] , and [ tab : dr - peak3 ] . the estimated tf of the arx model is : @xmath8 which accounts for 51.2% of the variance of @xmath9 . the notable feature we find here is that the accuracy of the ar model for @xmath9 is severely degraded ( @xmath10 ) in table [ tab : dr - peak1 ] , compared to the ar model for the moderate price regime ( table [ tab : dr - moderate1 ] ) . however , we observe that a relatively high portion ( 27% ) of the variance of @xmath11 is explained by the estimated model of @xmath11 shown in table [ tab : dr - peak2 ] , from which we conclude that the innovation from the price information is significant to improve @xmath12 of the arx model up to 51.2% , as presented in table [ tab : dr - peak3 ] . ccccc + * coeff . & * estimate & * se & * tstat & * pvalue + @xmath13 & 748.26 & 233.72 & 3.2015 & 0.0025097 + @xmath14 & 0.40153 & 0.11763 & 3.4133 & 0.0013678 + @xmath15 & -0.23826 & 0.1461 & -1.6308 & 0.10992 + @xmath16 & 0.25124 & 0.11516 & 2.1816 & 0.0344 + & + & + * * * * * ccccc + * coeff . & * estimate & * se & * tstat & * pvalue + @xmath17 & 1213.4 & 293.68 & 4.1316 & 0.00014688 + @xmath18 & -220.1 & 52.774 & -4.1707 & 0.00012965 + & + & + * * * * * cccc + * coeff . & * estimate & * coeff . & * estimate + @xmath14 & 0.40153 & @xmath18 & -220.1 + @xmath15 & -0.23826 & @xmath19 & 1961.66 + @xmath16 & 0.25124 + & + * * * * in fig . [ fig : dr - prediction ] , the validity of our model is shown by sample load forecast . [ fig : dr - error - probplot ] and [ fig : dr - qqhat ] delineate the errors in the load forecast at 3:15pm after a price surge at 2:15pm . we see that the forecasted @xmath20 and the actual @xmath9 at @xmath21 are reasonably well correlated ( correlation ( @xmath22 ) in fig . [ fig : dr - qqhat ] , and that the errors exhibit normality ( kurtosis = 3.1809 ) in fig . [ fig : dr - error - probplot ] . our empirical study suggests that ( 1 ) _ the demand only responds to high price surges at peak hour _ and ( 2 ) _ there exists a demand response delay consequent on a high price surge . _ the second finding shows that there exists a certain `` inertia '' in consumption , resulting in a certain time delay to reduce power consumption after a peak price observation . in this section , we shall further analyze the results from the data . we will first examine the rationality of consumer behavior . then we will show that the observed delay in demand response changes classical arguments about the role of prices and the equilibrium process , as well as classical efficiency results of markets . one of our observations in section [ sec : consmobs ] is that price responsive demand exhibits delayed response to price shock at peak hours . it may seem to be irrational to decrease one s demand after a price shock has already occurred . however , if we consider that consumer behavior is based on _ prediction _ of price , rather than the current price itself , then we can explain the delayed response based on the inertia of demand . in this sense , decreasing one s demand after a price spike , specifically , during or after the price plummets after price surge , can be well explained as a rational behavior if there is a high chance of a price increase after price spike . the chance of such a price increase relapse after a price spike is presented in figure [ fig : spikeprob ] . figure [ fig : spikeprob ] shows a comparison of the estimated conditional probability of a high price in different situations , based on the obtained data from houston . we can easily discern from figure [ fig : spikeprob ] that the conditional probability of a second price spike following the occurrence of a price spike quickly reduces in off - peak hours . however , we also observe that the conditional probability of a price spike after the occurrence of a price spike during peak - hours remains at a significantly higher level than the probability of price spike without any conditioning . this rationalizes our observation that _ if we assume a consumer has limited ability for immediate load reduction _ , then a rational consumer adjusts its load in response to price spike in spite of its inertia , because the relative chance of repeated price surge after a price spike is significantly high , and the demand is still not able to respond quickly to that following price surge . this also explains why demand is responsive to price only when it is during peak hours . the rigorous analysis of the consumer rationality behind such behavior will be presented in subsequent work . based on the assumed rationality of a consumer , the premise that demand is an optimal choice in consumption space given a price bundle under individual s budget constraint , is the major basis for classical arguments regarding the market benefits of price responsive demand . from the viewpoint of classical arguments , the demand curve as a function of price always represents an optimal solution that maximizes ( aggregated ) consumer utility given a budget constraint . however , the existence of inertia of demand suggests that such inertia may result in a restriction of feasible choices in consumption space , which then prevents consumer from engaging in the optimal consumption choice predicted by classical consumer theory . this implies that an instantly observed static demand curve may not fully reflect consumers utility structure regardless of consumers rationality , which may impact negatively on overall market efficiency under previously suggested pricing solutions for electricity markets . while there may exist a _ nominal demand curve _ well reflecting a consumer s utility , the inability to respond instantly creates a temporal distortion in demand curve in two respects : ( 1 ) since the instantaneous price elasticity of demand is near zero , an instantaneous demand curve during the time slot with a sudden price surge may be represented by a vertical line in a classical quantity - price plot , so that it may not coincide with the nominal demand curve ; and ( 2 ) the demand hedged against the risk of a repeated price surge would not respond to subsequent price reduction , so that an instantaneous demand curve after an unruffled price may not coincide with the nominal demand curve for a certain subsequent period of time . it is generally expected that dr will be beneficial both in terms of system operability as well as economical perspective , under the high penetration of vers . however , the distortion in the demand curve due to demand inertia can degrade the overall expected benefits from dr programs designed incautiously without consideration of demand inertia . an important example of the latter is high - frequency rtrp . in this subsection , we discuss the limitation of potential benefits from high - frequency rtrp under supply fluctuation caused by vers . while the analysis on the allocative efficiency of rtrp compared to fixed price under volatile demand analyzed in previous literature , is depicted in figure [ fig : econinefffixrate ] , this analysis can be extended to the allocative efficiency of rtrp compared to fixed price under supply fluctuation by vers as depicted in figure [ fig : econinefffixratever ] . analogously , the shaded triangles @xmath2 and @xmath23 are the deadweight loss , representing the economic inefficiency caused by the fixed rate @xmath4 , or , conversely , the expected allocative efficiency benefit from rtrp compared to a fixed tariff . according to the classical arguments on rtrp , more frequent price change would be more beneficial , because they would more accurately the balance supply and demand in real time , so that it is more informative for consumers to make an optimal decision . however , our observation suggests that such allocative efficiency is not likely to be achievable because of the distortion in demand curve caused by demand inertia . the inability of customers to respond instantly may distort the demand curve to a vertical line in quantity - price plot . noting this distortion , the demand behavior in practice may be realized as if it is exposed to fixed price . the impact on allocative efficiency from the demand curve distortion is presented in figure [ fig : ineffver ] . figure [ fig : ineffverdropoff ] depicts the situation where ver drops due to events such as sudden diminution of wind or cloud cover blocking the sun . suppose that the market equilibrium point before the ver drop is @xmath24 . then , the ver drop shifts the supply curve to the left . after the supply curve shift , the optimal market clearing point maximizing social welfare is @xmath0 on the nominal demand curve . however , the actual market clearing point is realized at @xmath25 due to demand curve distortion . hence , the shaded triangle @xmath2 is the deadweight loss , exhibiting the economic inefficiency following from demand inertia . in figure [ fig : ineffverrestoration ] , the analysis of the situation where ver is restored by incidents e.g. the increase in wind generation caused by a gust of wind , or that of solar generation following a cloud gap is provided . suppose the market equilibrium point before ver restoration is @xmath24 . the increase in generation followed by ver restoration event results in the newly formed supply curve to lie on the right hand side of the previous one . while @xmath0 on the nominal demand curve is the optimal market clearing point after the supply curve shift , the demand curve distortion caused by the hedging of demand against the risk of a repeated ver drop - off may result in the actual market clearing point being realized at @xmath25 . again , the shaded triangles @xmath2 indicates the deadweight loss , the economic inefficiency from demand inertia . the allocative ( in)efficiency analysis of rtrp under demand inertia suggests that there is a fundamental limitation to achieving market efficiency that can be expected from traditional market efficiency analysis without consideration of demand inertia . this necessitates a redefinition of market efficiency from an optimal control theoretical perspective . in addition , the demand behavior under rtrp is as if it is exposed to a fixed price , leading to another crucial implication that rtrp may not significantly resolve the vulnerability of markets from the exercise of market power . the differences in the ability of various market participants to control their behavior endows different market powers to each market participant ; the more instantaneously responsive market participant has an advantage over market participants with larger delay . such a combination of differentially endowed market powers makes market more vulnerable to the exercise of market power or market manipulation . a similar argument is found in financial markets with high - frequency trading ( hft ) practices , in terms of the robustness with respect to market manipulation and market fairness @xcite . moreover , the inability of demand response to instantaneously respond also suggests that rtrp may contribute negatively to demand volatility mitigation , so that the savings in the cost of maintaining reserve capacity may be less than expected under previous literature . a market is a dynamical system that is designed to proceed toward an optimal state as its equilibrium . however , such a process necessarily requires a certain amount of time to reach its equilibrium . while dynamic modeling and control on the generation side in power systems has been well understood , the understanding of dynamic behavior on the demand side in response to price has been unclear . in this paper , we consider a consumer s dynamic behavior in response to real time price change , by studying empirical data on a price - responsive load in the ercot area . out empirical study suggests the following : ( 1 ) _ the price responsiveness of demand may exhibit qualitatively different behavior in response to `` normal price '' and `` high price '' ; _ and ( 2 ) _ there exists a demand response delay consequent on a high price surge at peak hours . _ this idea provides important guidance for designing two fundamental factors in time - varying retail electricity prices , _ frequency _ and _ timeliness_. here frequency of price " is the frequency at which retail prices change , and timeliness of price " is the time lag between when a price is set and when it is effective @xcite . it is generally assumed among economists that rtrp with high frequency and just - in - time timeliness would be ideal in terms of economic efficiency in the electricity market , as rtrp is an attempt to provide more accurate signals closely reflecting the actual supply / demand status in the market . however , the inference based on our work is that neither argument is necessarily right . the inherent delay in the responsiveness of loads to high price volatility exacerbates the predictability of price , thereby making demand less responsive to rtrp , which in fact worsens economic efficiency . consumers which are more exposed to market volatility stiffen their demand to be more inelastic and tend to be more conservative due to the inertial nature of demand . this suggests that there exists a trade - off between controllability of demand and observability of markets , so that there may exist an optimal frequency and timeliness which should be carefully considered for optimal pricing design . this also supports the importance of relatively long - term contract markets such as day - ahead electricity markets . market efficiency should be re - analyzed taking into consideration the trade - off between the controllability of demand and the observability of the market . in subsequent work , we aim to provide a rigorous analysis of consumer rationality and develop a quantitative prediction model for demand response . frank a. wolak , `` managing demand - side economic and political constraints on electricity industry restructuring processes , '' _ economic reform in india , cambridge university press _ , cambridge , uk , ch . 455 - 498 , 2013 . c. aubin , d. fougere , e. husson , and m. ivaldi , `` real - time pricing of electricity for residential customers : econometric analysis of an experiment , '' _ journal of applied econometrics _ 171191 . dec .

in this paper , we study the price responsiveness of electricity consumption from empirical commercial and industrial load data obtained from texas . employing a dynamical system perspective , we show that price responsive demand can be modeled as a hybrid of a hammerstein model with delay following a price surge , and a linear arx model under moderate price changes . it is observed that electricity consumption therefore has unique characteristics including ( 1 ) qualitatively distinct response between moderate and extremely high prices ; and ( 2 ) a time delay associated with the response to high prices . it is shown that these observed features may render traditional approaches to demand response and retail pricing based on classical economic theories ineffective . in particular , ultimate real - time retail pricing may be limitedly beneficial than as considered in classical economic theories . demand response , electricity market , dynamic system modeling .

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Proceed to summarize the following text: squeezing of the radiation field is one of the distinct features of the quantum theory of light @xcite . squeezed states of light , being nonclassical in origin @xcite , have a reduced variance in one of the quadrature components of the electric field . it is well known that the resonance fluorescence from a driven atomic system can serve as a source of squeezed radiation . theoretical investigations on two - level and three - level atoms demonstrated squeezing either in the total variance of phase quadratures or in the frequency ( spectral ) components of the fluorescence radiation @xcite . for a driven two - level atom , walls and zller first predicted that squeezing can occur in the in - phase / out - of - phase quadrature component of the fluorescent light [ 4a ] . the noise spectrum was shown to exhibit single- and two - mode squeezing in the weak- and strong - excitation regimes @xcite . some experimental evidences of squeezing have also been reported in the phase - dependent spectra of two - level atoms @xcite . unlike in two - level systems , two - photon coherences play a significant role in the dynamics of three - level atoms driven by coherent fields @xcite . _ examined the role of atomic coherences and studied the maximum squeezing that can be obtained in the fluorescence from three - level systems @xcite . a detailed study by gao _ @xcite has shown that ultranarrow squeezing peaks may appear in the spectrum of driven three - level atoms in @xmath0- and v - type configurations . however , the role of two - photon coherence is seen to destroy the spectral - component squeezing in the fluorescent field @xcite . one of the interesting developments in the study of resonance fluorescence is the possibility of modifying spectral properties of the atoms via quantum interferences in spontaneous decay channels . the interference in spontaneous emission occurs when the atomic transitions are coupled by same vacuum modes . the early work of agarwal on this subject demonstrated population trapping and generation of quantum coherence between the excited states in a v - type atom @xcite . since the fluorescence properties of a driven atomic system result from its spontaneous emission , studying the influence of interference in such processes has become an important topic of research @xcite . much attention has been paid to study the fluorescence spectrum of driven atoms @xcite . all these theoretical studies assume non - orthogonal dipole moments of the atomic transitions for the interference to exist in decay processes @xcite . however , in real atomic systems , it is difficult to meet this condition . different schemes were later proposed to bypass the condition of non - orthogonal dipole moments @xcite . experimentally , coherence between the ground states arising from spontaneous emissions has been reported using electron spin polarization states in quantum dots @xcite and zeeman sub - levels in atomic systems @xcite . the squeezing characteristics of the emitted fluorescent light was also discussed extensively @xcite . in a three - level v - type atom interacting with a coherent field , the interference is shown to enhance the spectral component squeezing in the presence of standard / squeezed vacuum @xcite . _ studied the squeezing spectrum of a four - level atom in the @xmath1-type scheme and showed that unusual squeezing properties appear due to interference if the fluorescent field is detected on the slow decaying transition @xcite . further , gonzalo _ et al . _ have examined a driven three - level atom of the @xmath1 configuration with particular attention to the squeezing in spectral components @xcite . recently , the effects of spontaneously generated interferences have been investigated in the context of enhancing self - kerr nonlinearity @xcite , soliton formation @xcite , and preserving bi - partite entanglement @xcite . in this paper , we consider a four - level atom in the y - type configuration interacting with two coherent fields ( as shown in fig . the excited atomic states are assumed to be near degenerate and decay spontaneously via the same vacuum modes to the intermediate state . the atom in the intermediate state can make spontaneous transitions to the ground state . since the cascade decays from the excited atomic states lead to an emission of the same pair of photons , quantum interference exists in decay processes . the role of the interference was investigated in the fluorescence spectrum of this system in ref . @xcite . in the present work , we study the squeezing spectrum and examine the interference effects on squeezing properties of the fluorescence fields . the paper is arranged as follows . in sec . ii , we present the atomic density matrix equations , describing the interaction of a y - type atom with two coherent fields , when the presence of quantum interference in decay channels is included . the formula for the squeezing spectrum is then derived using atomic correlation operators in sec iii . in sec . iv , we analyze the numerical results of the squeezing spectrum and identify the origin of interference effects using the dressed - state picture . finally , the main results are summarized in sec . we consider a four - level atom in the y - type configuration as shown in fig . 1 . in this scheme , the atom has two closely lying excited states @xmath2 and @xmath3 with energy separation @xmath4 . it is assumed that the excited atomic states are coupled by common vacuum modes to decay spontaneously to the intermediate state @xmath5 with rates @xmath6 and @xmath7 . the atom in the intermediate state @xmath5 is further allowed to undergo spontaneous emissions to the ground state @xmath8 with decay rate @xmath9 . the direct transitions between the excited states @xmath10 and that between the excited and ground states @xmath11 of the atom are forbidden in the dipole approximation . we assume that the transition frequencies @xmath12 of the upper transitions differ widely from that of the lower transition @xmath13 . this leads to a situation in which the vacuum modes coupling the upper and lower atomic transitions are totally different . in addition to spontaneous decays , two coherent fields are applied on the atom as shown schematically in fig . 1 . the upper transitions @xmath14 are driven by a coherent field of frequency @xmath15 ( amplitude @xmath16 ) and another field of frequency @xmath17 ( amplitude @xmath18 ) couples the lower transition @xmath19 . the rabi frequencies of the atom - field interaction are denoted as @xmath20 , @xmath21 , and @xmath22 with @xmath23 being the dipole moment of the atomic transition from @xmath24 to @xmath25 . the system is studied in the interaction picture using time independent hamiltonian @xmath26 here , the operators @xmath27 represent the atomic population operators for @xmath28 and transition operators for @xmath29 . to include decay processes in the dynamics , we use the master equation framework . with the inclusion of the decay terms , the time evolution of the density matrix elements in the interaction picture obeys @xcite @xmath30 @xmath31 @xmath32 @xmath33 @xmath34 @xmath35 \rho_{23 } + i \omega_2 ( \rho_{33 } - \rho_{22 } ) \nonumber \\ & & - i \omega_1 \rho_{21 } - i \omega_3 \rho_{24 } - p \sqrt{\gamma_1 \gamma_2}~ \rho_{13},\end{aligned}\ ] ] @xmath36 @xmath37 \rho_{14 } + i \omega_1 \rho_{34 } - i \omega_3 \rho_{13 } \nonumber \\ & & - p \sqrt{\gamma_1 \gamma_2}~ \rho_{24},\end{aligned}\ ] ] @xmath38 \rho_{24 } + i \omega_2 \rho_{34 } - i \omega_3 \rho_{23 } \nonumber \\ & & - p \sqrt{\gamma_1 \gamma_2}~ \rho_{14}. \label{rho9}\end{aligned}\ ] ] here , @xmath39 corresponds to the detuning between the atomic frequency @xmath40 of the @xmath41 transition and the frequency of the applied field @xmath16 . similarly , @xmath42 denotes the detuning of the field acting on the lower transition . in writing eqs . ( [ rho1])-([rho9 ] ) , we have assumed that the trace condition @xmath43 is satisfied . the cross - coupling term @xmath44)-([rho9 ] ) . it reflects the fact that population can be transferred between the excited states by the vacuum field . when @xmath45 , the decays from the excited states @xmath2 and @xmath3 are coupled and the interference effects are maximal . if the atomic dipole moments are orthogonal @xmath46 , there is no interference effect in spontaneous emission . for convenience in the calculation of the squeezing spectrum , we rewrite the density matrix equations ( [ rho1])-([rho9 ] ) in a more compact matrix form by the definition @xmath47 substituting eq . ( [ psidef ] ) into eqs . ( [ rho1])-([rho9 ] ) , we get the matrix equation for the variables @xmath48 @xmath49 where @xmath50 is the @xmath51-th component of the column vector @xmath52 and the inhomogeneous term @xmath53 is also a column vector with non - zero components @xmath54 in eq . ( [ matrix ] ) , @xmath55 is a 15@xmath5615 matrix whose elements are time independent and can be found explicitly from eqs . ( [ rho1])-([rho9 ] ) . the steady - state solutions of the density matrix elements can be obtained by setting the time derivative equal to zero in eq . ( [ matrix ] ) : @xmath57 we now proceed to the study of the squeezing spectra of the driven atom . since the atom is driven by two coherent fields , each field induces its own atomic dipole moment which then generates a scattered field . however , the fields scattered by the upper and lower transitions in the atom will have no correlations because the applied fields @xmath58 are of quite different carrier frequencies @xmath59 . assuming that the point of observation lies perpendicular to the atomic dipole moments , the positive - frequency part of the fluorescent fields in the radiation zone can be written as @xmath60\exp(-i\omega_a\hat{t } ) , \nonumber \\ \vec{e}^{(+)}_b(t ) & = & g(r ) \vec{\mu}_{34 } a_{43}(\hat{t } ) \exp(-i\omega_b\hat{t } ) , \label{electric}\end{aligned}\ ] ] where @xmath61 , @xmath62 , @xmath63 , and @xmath64 is the distance of the detector from the atom . the index @xmath65 @xmath66 in eq . ( [ electric ] ) refers to the fluorescent light of central frequency @xmath15 @xmath67 . in squeezing measurements , the two - time expectation value of a particular quadrature component of the electric field is the quantity of interest . we consider squeezing in the fluorescent light exclusively emitted by the upper and lower transitions in the atom . the slowly varying quadrature components with phase @xmath68 are defined as @xmath69 the spectrum of squeezing is defined by the fourier transformation of the normal and time - ordered correlation of the quadrature component @xmath70 : @xmath71 where @xmath72 , @xmath73 represents the time ordering operator , and the steady - state limit @xmath74 is considered . the calculations of the correlation functions in eq . ( [ spect ] ) can be carried out easily with the help of the quantum regression theorem and the density matrix equations ( [ matrix ] ) . for this purpose , we introduce column vectors of two - time averages @xmath75^{t } , m , n = 1,2,3,4 . \label{corre}\end{aligned}\ ] ] here , @xmath76 are the deviations of the atomic operators from its steady state values eq . ( [ steady ] ) . according to the quantum regression theorem @xcite , the column vectors ( [ corre ] ) satisfy @xmath77 now , following the procedure explained in refs . @xcite for the time ordering of operators in eq . ( [ spect ] ) , the squeezing spectrum can be obtained as @xmath78 e^{i2(\theta + \omega_a r / c ) } \label{specta } \\ & & + \sum_{k = 1}^{15 } \lim_{t \rightarrow \infty } \left [ \hat{m}_{11,k } \left ( { |\vec{\mu}_{13}|}^2 \hat{u}^{31}_k(t,0 ) + \vec{\mu}_{13}.\vec{\mu}_{23 } \right . . \nonumber \\ & & \left . \times \hat{u}^{32}_k(t,0 ) \right ) + \hat{m}_{12,k}\left ( { |\vec{\mu}_{23}|}^2 \hat{u}^{32}_k(t,0 ) \right . \nonumber \\ & & + \left . \left . \vec{\mu}_{13}.\vec{\mu}_{23}\hat{u}^{31}_k(t,0)\right ) \right ] \bigg\ } , \nonumber\end{aligned}\ ] ] where @xmath79 denotes the @xmath80 element of the matrix @xmath81 $ ] . similarly , as a function of @xmath82 for the parameters @xmath83 , @xmath84 , @xmath85 , @xmath86 , @xmath87 , and @xmath88 ( a ) and @xmath89 ( b ) . the solid ( dotted ) curves are for @xmath90 @xmath91 . for clarity , the dotted curve in ( a ) has been displaced by 2 units along the @xmath82-axis.,title="fig:",height=226 ] as a function of @xmath82 for the parameters @xmath83 , @xmath84 , @xmath85 , @xmath86 , @xmath87 , and @xmath88 ( a ) and @xmath89 ( b ) . the solid ( dotted ) curves are for @xmath90 @xmath91 . for clarity , the dotted curve in ( a ) has been displaced by 2 units along the @xmath82-axis.,title="fig:",height=226 ] as a function of @xmath82 for the parameters @xmath83 , @xmath84 , @xmath92 , @xmath86 , @xmath87 , and @xmath88 . the solid ( dotted ) curves are for @xmath90 @xmath91 . for clarity , the dotted curve has been displaced by 2 units along the @xmath82-axis.,height=226 ] @xmath93\bigg\ } , \label{spectb}\end{aligned}\ ] ] with the elements of the matrix @xmath94 as defined in eq . ( [ specta ] ) . the squeezing spectra of the fluorescence fields can be obtained numerically using eqs . ( [ specta ] ) and ( [ spectb ] ) . in the following , we assume equal dipole moments @xmath95 and decay rates @xmath96 for the upper atomic transitions . all the frequency parameters such as decay rates , detuning , and rabi frequencies are scaled in units of @xmath97 . the numerical results are presented by considering both the presence @xmath98 and absence @xmath91 of quantum interference . * a. squeezing spectrum : @xmath99 * we first consider spectral squeezing in the fluorescence field generated by the upper atomic transitions . a selected quadrature component @xmath68 is said to exhibit spectral squeezing if the squeezing spectrum is negative , @xmath100 , at a certain frequency @xmath82 . we analyze the squeezing spectrum @xmath99 calculated using eq . ( [ specta ] ) for the case of strong driving fields @xmath101 @xcite . in the calculations , we assume @xmath102 and scale the spectrum in units of @xmath103 . the numerical results of the spectrum @xmath99 are displayed in fig . 2 for the in - phase @xmath104 quadrature component @xcite . it is seen that the presence of quantum interference @xmath98 enhances ( three times ) the spectral squeezing @xmath105 in outer sidebands [ see fig . 2(a ) ] . for a suitable choice of parameters as in fig . 3 , the squeezing can be enhanced in inner sidebands as well . note that the amount of maximal squeezing due to quantum interference is appreciable only when the excited atomic levels decay much slower than the middle level @xmath106 a physical understanding of the numerical results can be obtained if the squeezing spectrum ( [ specta ] ) is rewritten as a sum of two contributions , i.e. , @xmath107 . here , the terms @xmath108 and @xmath109 referred to as noninterfering and interfering terms , are expressed as @xmath110,\end{aligned}\ ] ] as a function of @xmath82 from different contributions : @xmath111 [ dotted curve ( a ) ] , @xmath112 [ solid curve ( a ) ] , @xmath113 [ dotted curve ( b ) ] , and @xmath114 [ solid curve ( b ) ] . the parameters are those used to produce fig . 2(a ) with @xmath90 . for clarity , the dotted curve has been displaced by 2 units along the @xmath82-axis.,title="fig:",height=226 ] as a function of @xmath82 from different contributions : @xmath111 [ dotted curve ( a ) ] , @xmath112 [ solid curve ( a ) ] , @xmath113 [ dotted curve ( b ) ] , and @xmath114 [ solid curve ( b ) ] . the parameters are those used to produce fig . 2(a ) with @xmath90 . for clarity , the dotted curve has been displaced by 2 units along the @xmath82-axis.,title="fig:",height=226 ] to explore further the origin of interference induced effects , we go to the dressed state description of the atom - field interaction . the dressed states are defined as eigenstates of the time independent hamiltonian @xmath120 in eq . ( [ ham ] ) . in the general parametric conditions , it is difficult to find analytical solutions to the eigenvalue problem @xmath121 . however , the eigenvalues @xmath122can be obtained numerically by solving a quartic equation . the eigenstates @xmath123 can be expanded in terms of the bare atomic states as @xmath124 where the expansion coefficients are explicitly given by @xmath125 with the normalization constant @xmath126^{\frac{1}{2 } } . \nonumber\end{aligned}\ ] ] in order to interpret the numerical results , we consider transitions between the dressed states with the inclusion of decay processes . the allowed transitions between the dressed states @xmath127 give the peaks in the fluorescence spectrum at the frequencies @xmath128 . specifically , for the parameters of fig . 2 , the eigenvalues ( in units of @xmath97 ) obtained numerically are @xmath129 , @xmath130 , @xmath131 , and @xmath132 . the squeezing found at the outer sideband in fig . 2 can be seen as arising from the transitions @xmath133 . in the high field limit @xmath134 , the squeezing spectrum @xmath119 can be worked out to be @xmath135 } { \gamma_{\alpha\beta}^2 + { ( \omega \mp \omega_{\alpha \beta})}^2 } , \label{dressa}\ ] ] where the subindex + ( - ) stands for the positive @xmath136 [ negative @xmath137 part of the spectrum and @xmath138 represents the population of the dressed state @xmath139 . the term @xmath140 denotes the decay rate of coherence between the dressed states and is given explicitly in appendix . by using the numerical values of expansion coefficients ( [ ecoeff ] ) , the formula eq . ( [ dressa ] ) accounts very well for the spectral squeezing displayed in fig . 2 . as a function of @xmath82 for the parameters @xmath83 , @xmath84 , @xmath85 , @xmath141 , @xmath142 , and @xmath143 the solid ( dotted ) curves are for @xmath90 @xmath91.,height=226 ] * b. squeezing spectrum : @xmath144 * we next consider the spectrum of fluorescence field generated by the lower atomic transitions . the squeezing spectrum calculated using eq . ( [ spectb ] ) is displayed in fig . 5 for the in - phase @xmath104 quadrature @xcite . the spectrum is scaled in units of @xmath145 . we also assume @xmath146 and consider the case of fast decaying upper levels @xmath147 of the atom . the effect of quantum interference @xmath98 is now seen to reduce the squeezing in spectral components as evident from fig . 5 . for slow decay of the upper levels @xmath148 , the reduction in squeezing is not appreciable ( graph not shown ) . as a function of the interference parameter @xmath149 . the other parameters for the calculation are the same as in fig . 5.,height=226 ] in order to understand this result , the spectrum is further analyzed in the dressed state picture . the contribution to the squeezing spectrum ( [ spectb ] ) originating from the dressed - state transitions @xmath150 can be given as @xmath151 , \label{dressb}\end{aligned}\ ] ] where the different terms have their meanings as in eq . ( [ dressa ] ) . for the parameters of fig . 5 , the numerical values of eigenvalues ( in units of @xmath97 ) obtained are @xmath152 , @xmath153 , @xmath154 , and @xmath155 . the spectrum eq . ( [ dressb])is a pair of lorentzians centered at the frequencies @xmath156 . since the numerators of the lorentzians are negative , the graph ( fig . 5 ) has negative peaks of height proportional to the inverse of decay rate @xmath140.the reduction in squeezing observed in fig . 5 may be traced to the increase in the decay rate @xmath140 of the dressed atomic transition . to this end we study the variation of the decay rate @xmath140 as a function of the interference parameter @xmath149 . as seen in fig . 6 , the decay rate attains a maximum in the presence of full quantum interference @xmath157 , thereby reducing the spectral squeezing observed in fig . finally , we note that the spectrum @xmath144 exhibits pure two - level squeezing for weak applied fields @xmath158 independent of the quantum interferences . the spectrum in this case is centered around the laser frequency @xmath159 and the maximum squeezing is obtained for the out - of - phase quadrature @xmath160 similar to two - level atoms @xcite . in this paper , we investigated the squeezing spectrum of the resonance fluorescence from a driven y - type atom when the presence of interference in spontaneous decay channels is important . in particular , we considered the atom to be driven by two coherent fields and examined the squeezing spectrum of fluorescence radiation from both the upper- and lower transitions in the atoms . it was shown that the decay - induced interference enhances squeezing in the spectrum of upper transitions for off - resonance and strong driving fields . a detailed analysis using dressed - states was presented to bring out the role of interferences . further , the interference was also shown to degrade the spectral squeezing if the fluorescence is detected on the lower atomic transitions . this has been explained as due to a fast decay of the dressed state in the atom . in the secular approximation , the time evolution of the coherence @xmath161 between the dressed states @xmath162 and @xmath163 obeys @xmath164 with @xmath165 obtained by diagonalizing the hamiltonian ( [ ham ] ) . the decay rate @xmath166 is given by @xmath167 where @xmath168 @xmath169 @xmath170 @xmath171 99 j. mod . opt . 34 ( 1987 ) ( 6/7 ) , special issue on squeezed light , edited by r. loudon and p.l . knight ; j. opt . b * 4 * ( 1987 ) ( 10 ) , special issue on squeezed light , edited by h.j . kimble and d.f . walls . hong , and l. mandel , j. opt . b 4 ( 1987 ) 1574 . squeezing , photon antibunching , and sub - poissonian photon statistics are essential phenomena demonstrating the non - classical states of radiation . for a discussion of the photon statistics of non - classical fields , lai , v. buek , and p.l . knight , phys . a 43 ( 1991 ) 6323 ; fam le kien , anil k. patnaik , and k. hakuta , phys . rev . a 68 ( 2003 ) 063803 . 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the squeezing spectrum of the resonance fluorescence is studied for a coherently driven four - level atom in the y - type configuration . it is found that the squeezing properties of the fluorescence radiation are modified significantly when quantum interference of the spontaneous decays channels is included . we show a considerable enhancement of steady - state squeezing in spectral components for strong and off - resonant driving fields . the squeezing may be increased in both the inner and outer sidebands of the spectrum depending upon the choice of parameters . we also show that the interference can degrade the spectral squeezing by increasing the decay rates of atomic transitions . an analytical description using dressed states is provided to explain the numerical results . + + keywords : resonance fluorescence ; squeezing ; quantum interference

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Proceed to summarize the following text: in the treatment of the motion of _ extended _ bodies in classical field theory , the derivation of radiation reaction forces is based upon certain expansions of the retarded field potentials in powers of the retardation @xcite . the resulting local equations of motion involve derivatives of the acceleration and generally suffer from the existence of unphysical runaway solutions . under certain model circumstances , we trace the origin of these problems to the expansion of the functions of the retarded arguments resulting in the replacement of the original nonlocal delay equations of motion by local higher - derivative equations that exhibit runaway solutions . in this general context , the properties of the delay equations that appear in classical field theory were first studied by l. bel @xcite . although our approach is rather general , for the sake of concreteness we discuss physical situations involving only the gravitational interaction . consider , for instance , inspiraling compact binaries that are expected to be promising sources of gravitational radiation . for a binary that is comprised of two compact objects neutron stars or black holes with , say , approximately equal masses @xmath0 and @xmath1 in nearly circular orbits about each other , the relative orbital radius decays because orbital energy is emitted in the form of gravitational radiation . the dynamics of a usual binary system can be adequately described using the post - newtonian approximation scheme that is valid in case the gravitational field is everywhere ` weak ' and the motion is slow , that is @xmath2 , where @xmath3 is the characteristic orbital speed and @xmath4 is the speed of light . although einstein s equations have a hyperbolic character associated with the retarded gravitational interaction , the standard post - newtonian approximation scheme of general relativity deals with functions of instantaneous coordinate time @xmath5 rather than the retarded time @xmath6 , where ( for the binary system ) @xmath7 is the effective distance between the bodies ( approximately the relative orbital radius ) . the gravitational potentials , which are originally functions of the retarded time , are expanded in taylor series about @xmath5 using the effective small parameter in this expansion that can be written as @xmath8 , where @xmath9 is the relative orbital frequency and @xmath10 is the binary period . because the gravitational waves emitted by the binary have an effective frequency of @xmath11 and wavelength @xmath12 , the small parameter in the expansion can be reduced to the ratio @xmath13 . due to the observational fact that in typical astronomical systems @xmath14 , the first few terms of such an expansion can be used to derive the post - newtonian equations of motion that describe , for instance , the orbital evolution of the binary pulsars discovered by hulse and taylor @xcite . the post - newtonian equations of motion of binary stars are similar to the abraham - lorentz form in electrodynamics @xcite but , because of the tensorial character of the gravitational field , these equations involve not only the third , but the fourth and fifth derivatives of the stars positions with respect to time as well . schematically , the equation of the relative orbital motion reads @xmath15 where @xmath16 is the radius vector connecting the stars , the overdot denotes differentiation with respect to time , @xmath17 and per reduced mass @xmath18 is the newtonian force , @xmath19 is the post - newtonian force , @xmath20 is the post - post - newtonian force and @xmath21 is the gravitational radiation reaction force responsible for the decay of the orbital period ( @xmath22 ) associated with the emission of gravitational waves by the binary system . in the quadrupole approximation under consideration here , the gravitational waves carry away energy and angular momentum , but not linear momentum @xcite ; therefore , the total momentum of the binary system is conserved and this fact is responsible for the absence of a force term proportional to @xmath23 in . moreover , all tidal , spin - orbit , and spin - spin interactions are neglected in ; the only parameters in equation are the masses and the separation between the centers of mass of the members of the binary system . we mention that relativistic hydrodynamical ( euler ) equations similar to system have been derived to describe the motion of the fluid elements of the stars @xcite . the higher time - derivative equations of the form can not be used directly to predict the dynamical evolution of a physical system because of the existence of so - called runaway modes that have been much discussed in the literature on electrodynamics but not in connection with astrophysical problems involving gravitational radiation reaction and the calculation of templates of the gravitational waves emitted by coalescing binaries @xcite . in analogy with electrodynamics , the existence of these runaway modes suggests that the truncated equations of the form may not correctly predict the qualitative behavior of the solutions of the original true dynamical delay - type equations @xcite that involve the retarded time @xmath6 . moreover , the existence of runaway modes can cause serious difficulties for numerical integration in addition to the problems associated with the inaccuracies inherent in the approximation of higher - order derivatives by finite differences @xcite . in case the post - newtonian expansion parameter @xmath24 is sufficiently small , we will provide in the following section a theoretical basis for eliminating the runaway solutions by replacing system with a new model that is a system of second - order ordinary differential equations . within this theory , high - order vector differential equations like equation are not the desired approximate equations of motion , and they should not be used for numerical integration . rather , system must be viewed merely as an intermediate step in the derivation of the physically correct , second - order model equation with no runaway solutions that faithfully approximates the dynamics of the underlying delay - type equation . for illustration purposes , we apply this approach in section [ sec:3 ] to a discussion of one - dimensional gravitational dynamics of a two - body system . as expected , the reduced model predicts the correct motion of the binary except possibly when @xmath24 is not small and residual terms that have been neglected in equation start to play a significant role . indeed , for an inspiraling binary system , the effective delay @xmath25 increases to some noticeable finite value as the system approaches coalescence . motivated by this physical scenario , we introduce a simple model involving variable delay in section [ sec:4 ] that can be expressed as a duffing - type differential - delay equation . this model is then analyzed to show some specific behavior of this delay equation that is not predicted by expansion in powers of the delay . finally , section [ sec:5 ] contains a discussion of our results . the delay - type equations of motion with retarded arguments are usually too complicated for mathematical analysis ; therefore , we limit our discussion in this section to equations with constant delays . although this is an unrealistic restriction in general , we note that for an astrophysical binary system consisting of compact point - like neutron stars or black holes moving around each other along circular orbits , the delay is almost constant . in fact , a close approximation to this delay is the ratio @xmath26 , where @xmath7 , the radius of the relative orbit , is changing very slowly due to the emission of gravitational energy in the form of gravitational waves . taking into account the last remark , let us consider a family of delay differential equations of the form @xmath27 where @xmath28 is viewed as a real dimensionless parameter and @xmath29 is a variable in @xmath30 ; intuitively , the constant delay @xmath28 corresponds in effect to @xmath31 . the members of this family are examples of a more general and widely studied class called retarded functional differential equations ( see @xcite ) . using the delay equation as an abstraction for the retarded - time model that is supposed to be approximated by a system of the form , we will discuss an approach for extracting the ` correct ' dynamical equations of motion from system that eliminates the runaway solutions . our approach assumes the existence of an attractor for the underlying delay - type equation . we will rely on the work of bel @xcite for ( numerical ) evidence in favor of the existence of attractors in the retarded equations of motion with space - dependent delays that appear in electrodynamics ; but , we know of no mathematical proof for the existence of attractors for these equations or for the similar delay - type equations of astrophysics . indeed , the proof of the existence of attractors for delay - type equations with space - dependent delays remains a challenging mathematical problem of physical significance . for the delay equation , however , if @xmath32 is sufficiently small , then the corresponding member of the family has a global @xmath33-dimensional attractor such that the restriction of the delay equation to this attractor is equivalent to a first - order system @xmath34 of ordinary differential equations . we will eventually outline a proof of this result . but , let us first discuss our approach to eliminating the runaway solutions . the solutions of the delay equation approach the attractor exponentially fast ; therefore , the system @xmath34 on this attractor determines ( asymptotically ) the true dynamical behavior of the system , hence we consider it to be the ` correct ' physical model . on the other hand , it is easy to see that if equation is expanded in the small parameter @xmath28 and truncated at some order @xmath35 , then an @xmath35th - order ordinary differential system @xmath36 akin to system is obtained such that the coefficient of the @xmath35th - order time derivative of @xmath29 contains the factor @xmath37 and is therefore singular in the limit as @xmath38 . for @xmath39 and sufficiently small , the high - order differential equation @xmath36 has an equivalent first - order system @xmath40 that has an @xmath33-dimensional ( invariant ) slow manifold . moreover , this slow manifold has corresponding stable and unstable manifolds ; in effect , the first - order system has ( physical ) solutions that are asymptotically attracted to the slow manifold and ( unphysical or runaway ) solutions that are asymptotically repelled from the slow manifold . the restriction of the first - order singularly perturbed system of differential equations to its slow manifold is of course an @xmath33-dimensional first - order system of ordinary differential equations @xmath41 on this @xmath33-dimensional manifold . our main result states that _ in appropriate local coordinates , the system @xmath41 on the slow manifold agrees to order @xmath35 in @xmath28 with the first - order system @xmath34 on the global attractor of the underlying delay differential equation ; therefore , the system @xmath41 , which can be obtained directly from the high - order differential equation @xmath36 , is a faithful approximation of the ` correct ' physical model . _ generalizing to the abraham - lorentz type equation ( analogous to @xmath36 ) , the correct physical model is obtained as the system of ordinary differential equations ( analogous to @xmath41 ) that determines the motion on the slow manifold of a corresponding first - order system that is viewed as being singularly perturbed relative to the small parameter @xmath24 . while there is evidence that the mathematical assertions in the scenario just proposed are valid , some of these assertions have not yet been rigorously justified in full generality , even for the case of fixed delays . in the remainder of this section we will provide some evidence , in the case of fixed delays , for the existence of a global attractor and for the claim that the dynamical system on this attractor is well approximated by the dynamical systems on the slow manifolds of singularly perturbed first - order systems obtained by truncations of the expansion of the delay equation in powers of the delay . our approach for the elimination of runaway solutions is equivalent to the procedure of iterative reduction ( also called order reduction ) that is often used to eliminate runaway solutions by means of the evaluation of the higher time - derivative terms in equations like ( [ 5der ] ) by the repeated substitution of the equations of motion and the subsequent reduction of the resulting equation to one of the second order ( cf . thus , our approach provides a theoretical framework for the rigorous justification of iterative reduction ( cf . @xcite ) , a procedure that has been justified so far by physical intuition . for _ conservative _ higher time - derivative systems , the order reduction procedure has been investigated within the frameworks of lagrangian and hamiltonian dynamics by a number of authors ( see @xcite and the references cited therein ) . in particular , it can be shown that under suitable conditions relevant , for instance , to euler - lagrange equations analogous to equation truncated at the fourth order higher - derivative lagrangians can be iteratively reduced by redefinitions of position variables @xcite . returning to the delay equation , we note that it has an infinite - dimensional state space of initial conditions . for example , if the delay @xmath28 is a fixed positive number , then the natural state space of initial conditions is the infinite - dimensional vector space of continuous @xmath30-valued functions on the interval @xmath42 $ ] . this space endowed with the supremum norm is a banach space that we will denote by @xmath43 . note that for an arbitrary continuous @xmath30-valued function @xmath44 defined on the interval @xmath45 $ ] , the function @xmath46 given by @xmath47 is in @xmath43 . under the assumption that @xmath48 is a smooth function and @xmath49 , there is a unique continuous solution @xmath50 of the corresponding delay equation in the family such that @xmath50 is uniquely defined for @xmath51 and @xmath52 ( see , for example , @xcite ) . the state of the system at time @xmath53 is defined to be the function @xmath54 in @xmath43 . to see that there is an attractor for the family in case @xmath28 is sufficiently small , it is convenient to introduce the fast time @xmath55 , valid for @xmath56 , so that with @xmath57 the family takes the form @xmath58 and each member of this family , parametrized by @xmath28 , has the same state space the continuous functions on the interval @xmath59 $ ] . for each @xmath56 the delay equation is equivalent to the corresponding member of the family . for @xmath60 the corresponding differential equations are not equivalent , but this is of no consequence because we are only interested in the solutions of the family for @xmath56 . by viewing the unperturbed system , namely @xmath61 , as a delay equation with unit delay , it is clear that the solution with initial state @xmath62 is given by @xmath63 on the interval @xmath59 $ ] and by the constant @xmath64 for @xmath65 . the initial state in @xmath43 thus evolves at time @xmath66 to its final constant state , the function defined on the interval @xmath59 $ ] with the constant value @xmath67 . thus , we conclude that the @xmath33-dimensional space of constant functions on @xmath59 $ ] is a global attractor for the delay equation @xmath61 . moreover , the convergence to this attractor is faster than any exponential ( the solution reaches the attractor in finite time ) , and the dynamical system on this attractor is given by the ordinary differential equation @xmath61 . if @xmath48 is appropriately bounded and @xmath28 is sufficiently small , then , because the contraction rate to the attractor is exponentially fast , the attractor persists in the family in analogy with the persistence of attractors in finite - dimensional dynamical systems ; in fact , each corresponding member of the family has an @xmath33-dimensional attractor in the state space @xmath43 and the restriction of the dynamical system to this attractor is an ordinary differential equation . in particular , the family has a corresponding family of invariant manifolds ( that is , manifolds consisting of a union of solutions ) that depend smoothly on the parameter @xmath28 . to identify the dynamical system on an attractor of a delay equation , let us suppose that the delay equation has a family of @xmath33-dimensional invariant manifolds parametrized by @xmath28 . moreover , let @xmath68 denote the local coordinate on these invariant manifolds , and let @xmath69 denote the solution with the initial condition @xmath70 on the invariant manifold corresponding to the parameter value @xmath28 . because these solutions satisfy the delay equation , we have that @xmath71 therefore , the generator of the dynamical system on the attractor is the vector field @xmath72 under our assumption that this vector field is analytic in @xmath28 , it can be expanded as a taylor series about @xmath60 by differentiating the function @xmath73 with respect to @xmath28 . to this end , we note that the partial derivatives of @xmath74 with respect to its first argument can be evaluated using equation , and partial derivatives with respect to its third argument vanish at @xmath60 since @xmath70 ; moreover , its mixed partial derivatives can be evaluated by differentiation of equation with respect to @xmath28 . as a concrete and instructive example of the construction of the dynamical system on an attractor , let us consider a simple case of equation by replacing it with @xmath75 where the function @xmath76 is the scalar linear function given by @xmath77 so that the associated family of delay equations is @xmath78 in this case , there is a corresponding family of solutions given by @xmath79 where @xmath80 is the unique _ real _ root of the equation @xmath81 , a fact that is easily checked by direct substitution of equation into the delay equation . the dynamical system on the invariant manifold is generated by the family of vector fields @xmath82 . by an application of the lagrange inversion formula @xcite , the taylor series expansion of @xmath83 about @xmath60 is @xmath84 and its radius of convergence is @xmath85 . the qualitative behavior of solutions of the system for small @xmath28 is clear : all solutions are attracted to a one - dimensional invariant manifold on which the dynamical system is the linear ordinary differential equation @xmath86 . for example , if @xmath87 and @xmath28 is sufficiently small , then all solutions are attracted to the trivial solution @xmath88 . let us now turn to the standard approach in physics where an underlying delay equation is expanded in powers of the delay to obtain an ordinary differential equation of motion . to illustrate this , let us consider a special scalar case of the delay equation given by @xmath89 and let us suppose , in analogy with the true dynamical delay - type equations that might arise in theories of electromagnetism and gravitation , that the true equation of motion for some process is the delay equation . the result of expanding equation to order @xmath90 is the second - order differential equation ( an analogue of equation ) @xmath91,\ ] ] where a prime denotes differentiation with respect to @xmath29 . although we only write the second - order expansion , we note that the coefficient of the @xmath35th - order time derivative of @xmath29 in the @xmath35th - order expansion is @xmath92 . hence , the corresponding @xmath35th - order ordinary differential equation is singular in the limit as @xmath38 . also , if we assume that system has a ( smooth ) family of attractors parametrized by @xmath28 , then the corresponding family of vector fields generating the dynamical systems on these attractors is given to second order in @xmath28 by @xmath93 ^ 2\\ & & \mbox{}+f'(x)[3f'(x)+g'(x)][f(x)+g(x)]\big\}+o(\tau^3).\end{aligned}\ ] ] the ` correct ' model ( that is , the dynamical system on the attractor in the original delay equation ) can be obtained by treating the expanded and truncated system akin to system as a singular perturbation problem , which can be analyzed using fenichel s geometric theory of singular perturbations @xcite . a basic result of this theory states that if an @xmath35th - order singular perturbation problem with small parameter @xmath28 is recast as a first - order ( ` fast ' ) system and the corresponding unperturbed system has an invariant manifold that satisfies certain conditions ( normal hyperbolicity ) , then for sufficiently small @xmath28 each member of the family of perturbed first - order systems has an invariant slow manifold . the dynamical system on this slow manifold for the perturbed first - order family obtained from the @xmath35th - order truncation of the delay equation is the desired faithful approximation to the correct model . for example , let us recast the second - order ordinary differential equation as the first - order singular perturbation problem @xmath94 . \end{aligned}\ ] ] using fenichel s theory , it is easy to show that each member of this family , corresponding to a sufficiently small value of @xmath32 , has a slow manifold . also , it is possible to prove that the family of vector fields on these manifolds agrees to order @xmath90 with the family of vector fields on the attractor in the state space of the underlying family of delay equations . we note that these results are valid for the vector case of delay equation as well . for the delay equation , and also for more general families of delay equations where the delay is viewed as a small parameter , we conjecture that the slow vector field , for an appropriately defined first - order system that is equivalent to the @xmath35th - order truncation of the expansion of the family in powers of the small delay , agrees to order @xmath35 with the vector field on the attractor in the state space of the original delay equation . we have just mentioned that this conjecture is true for the delay equation in case @xmath95 . it can be shown that the conjecture is true in general for the linear delay equation @xmath96 , where @xmath29 is a variable in @xmath30 and @xmath97 is a nonsingular @xmath98 matrix @xcite . as we have already discussed , singular equations of motion like system generally have unphysical runaway solutions . to eliminate these solutions and leave only the physical solutions , the singular system must be replaced by the dynamical system on the corresponding slow manifold . in effect , the truncated equations obtained from the underlying delay equation after expansion in the small delay must be replaced by the system obtained using iterative reduction ; this system is equivalent to the dynamical system on the slow manifold . without this replacement , the appearance of spurious runaway modes in inevitable , and their existence will cause overflows in numerical simulations . to illustrate the singular perturbation procedure described in section [ sec:2 ] as a method for the elimination of runaway solutions , we examine an application of this approach to a one - dimensional abraham - lorentz equation of the form . let us consider an ideal linear quadrupole oscillator ( that is , two masses @xmath0 and @xmath1 connected by a spring of negligible mass ) , where the only source of damping is the gravitational radiation reaction force associated with the emission of gravitational radiation due to the variable quadrupole moment of the system . a model for the ( dimensionless ) relative position @xmath99 of these particles , with gravitational radiation damping included , is the fifth - order ordinary differential equation @xmath100 where the small parameter is given by @xmath101 , @xmath102 is the reduced mass ( @xmath103 ) , @xmath104 is the spatial scale parameter and @xmath105 is the frequency of the ideal linear oscillator such that @xmath106 is the temporal scale parameter . in equation , we have neglected the newtonian gravitational interaction between @xmath0 and @xmath1 as well as all relativistic effects except for radiation damping . let us note that this oscillator has an equilibrium solution @xmath107 . according to our general approach described in section [ sec:2 ] , this system can be treated as a singular perturbation problem , where the physically correct dynamical system would be the system defined on the slow manifold of a corresponding , appropriately chosen , first - order system . we emphasize that the fifth - order differential equation is not the correct physical model ; for example , to specify a solution , the initial position and its first four time derivatives must be given . even with the obvious choice for these initial conditions that is , the initial conditions for a sinusoidal oscillation a numerical integration shows that such solutions do not oscillate ; rather , they are divergent . to obtain a system with the expected dynamical behavior of an under - damped oscillator , the differential equation must be replaced by its restriction to an appropriate slow manifold . we will not carry out the complete reduction procedure here @xcite . we note , however , that the system matrix for the linearization of system at the steady state solution @xmath108 has five distinct eigenvalues that are given to lowest order in the small parameter by @xmath109 for small @xmath110 , the first three eigenvalues are ` fast ' and the last two are ` slow ' . this suggests that the nonlinear system has a two - dimensional slow manifold . in fact , in accordance with our general scheme , the restriction of the dynamical system to this invariant manifold is a second - order system that gives the correct post - newtonian dynamics . in this case , the dynamical system on the slow manifold to first order in the small parameter is given by the second - order differential equation @xmath111 a unique solution of this equation is obtained by specifying only the initial relative position and velocity of the oscillating masses . for @xmath99 near the equilibrium state @xmath108 , the expected dynamics for the radiating system is revealed : the relative motion is an under - damped oscillator . numerical integration of this equation using standard algorithms is stable and produces the expected result . the iterative reduction procedure can also be used to obtain equation from equation . even our simple example illustrates the necessity of reducing the higher - order equations of motion involving radiation reaction before numerical integration . for the more realistic hydrodynamic equations that include conservative post - newtonian terms as well as radiation reaction , the corresponding euler equation must involve these forces in the _ reduced _ form , that is , they should contain at most the position and the velocity of the fluid element @xcite . it is important to point out that the reduction procedure described in sections [ sec:2 ] and [ sec:3 ] can not in general be expected to produce a good approximation to the true dynamics for ` large ' delays . as a simple but revealing example , let us reconsider the scalar linear delay equation @xmath112 with @xmath113 . for small @xmath32 , we have already shown that all orbits in the state space are attracted to a one - dimensional attractor on which the dynamical system is given by the vector field with @xmath114 . for @xmath115 less than the radius of convergence of this series @xmath116 , the correct dynamical behavior of the delay equation is predicted by this vector field . because , in this case , the zeroth - order approximation @xmath117 already has a hyperbolic structure ( that is , all solutions are attracted to the rest point at the origin exponentially fast ) , even the zeroth order approximation determines the qualitative dynamics for these values of @xmath28 . by inspection of this delay equation , it might seem natural to conclude that the fixed delay @xmath28 does not influence the behavior for sufficiently large @xmath5 and the approximation @xmath117 remains valid for all fixed delays . this is not true . for instance , if @xmath118 , then the delay equation has the two - parameter family of exact solutions @xmath119 therefore , the qualitative behavior of the delay equation @xmath120 is certainly not predicted by the ordinary differential equation @xmath117 , or by the corrections to this equation within the radius of convergence of the slow vector field . the transition of the dynamical behavior of this delay equation from a stable rest point to a periodic regime as @xmath28 increases is easily seen to be the result of a degenerate hopf bifurcation @xcite . indeed , we recall that @xmath121 is a solution of the delay equation under consideration if @xmath122 is a solution of the characteristic equation @xmath123 . for @xmath124 , the solutions of this equation have negative real parts and all such solutions are therefore attracted to the zero solution . if @xmath118 , then the characteristic equation has a pair of pure imaginary roots that give rise to the two - parameter family of periodic solutions . for @xmath125 , the characteristic equation has roots with positive real parts ; therefore , there are solutions that grow without bound . nevertheless , for these values of @xmath28 , the delay equation has an attractor . in fact , for @xmath126 , there is a two - dimensional attractor and the dynamical system on the attractor has the form @xmath127 corresponding to the roots @xmath128 of the characteristic equation @xmath129 with positive real parts . as @xmath28 increases further , the dimension of the attractor increases discontinuously by two at each @xmath130 , where @xmath131 . the hopf bifurcation for delay equations with constant delays has been studied in detail . for instance , a more sophisticated analysis ( see , for example , @xcite ) shows that @xmath132 is a supercritical hopf bifurcation value for the nonlinear scalar delay equation @xmath133x(t-\tau).\ ] ] moreover , this system has a nontrivial periodic orbit for each @xmath134 . the delay - type equations of astrophysics generally do not have constant delays . but as we have mentioned , for two coalescing neutron stars with nearly equal masses and on nearly circular orbits , the delays involved are almost constant ; in fact , this ` fast ' periodic motion evolves as a result of radiation damping on a timescale that is much longer than @xmath10 . during this ` slow ' evolution , the delay increases as the radius of the binary decreases due to the emission of gravitational waves . motivated by this astrophysical scenario , we have studied an oscillator model with a time - dependent delay . this example is not intended to be a realistic model , rather it is meant to illustrate some of the bifurcation phenomena that occur in delay equations with time - dependent delays . our example is the second - order differential - delay equation @xmath135 where @xmath136 , @xmath137 and @xmath138 are constant system parameters and @xmath29 is viewed as the state variable of a ( duffing ) oscillator with variable delay @xmath139 such that @xmath140 . here @xmath141 and @xmath142 are constants ; hence , @xmath143 is an exponentially decreasing or increasing function of time depending on whether @xmath141 is positive or negative , respectively . in any case , we have a dynamic delay that is asymptotic to the constant value @xmath142 . note that if @xmath144 , the delay is constant ; in this case , the corresponding second - order differential equation on the slow manifold ( to first order in @xmath145 ) is given by @xmath146 a form of van der pol s equation . in case @xmath147 , this differential equation typically has a stable limit cycle for @xmath148 . but for @xmath149 ( that is when all forces are retarded ) , it is easy to prove that no periodic orbits exist and most solutions are unbounded . a typical plot of @xmath29 versus @xmath5 for system for @xmath150 is given in figure [ fig:1 ] , where the delay increases from its initial value @xmath151 to @xmath152 . the initial response of the system ( where the delay is small ) is characterized by an oscillation as expected from equation , which follows from the expansion of equation to first order in @xmath153 . but as @xmath145 increases , the qualitative behavior of the system is affected by three additional bifurcations not accounted for by equation . at the third bifurcation , the stable oscillation disappears . additional bifurcations of the same type occur if @xmath142 is set to a larger value . numerical experiments suggest that these bifurcations are not hopf bifurcations ; instead , they are ` center bifurcations ' , where at some parameter value there is a rest point of center type and one of the periodic orbits surrounding this rest point continues to exist as the parameter is changed . the family with the parameter values as in figure [ fig:1 ] has a bifurcation of this type as @xmath145 increases through @xmath154 . the behavior depicted in figure [ fig:1 ] is suggested by an analysis of the roots of the characteristic equation , @xmath155 for the linearization of the delay equation . the bifurcation points ( corresponding to the existence of centers ) are given by @xmath156 where @xmath157 is a non - negative integer . these are the values of @xmath145 such that the characteristic equation has pure imaginary roots . a computation shows that if @xmath157 is even , then as @xmath145 increases a pair of pure imaginary roots crosses the imaginary axis into the right half - plane , and if @xmath157 is odd , then the roots cross into the left half - plane . under the assumption that the bifurcations are supercritical , a stable limit cycle appears after the bifurcation in the first case ; in the second case , a stable limit cycle disappears . for the parameter values used to obtain figure [ fig:1 ] , the bifurcation values computed from equation are ( approximately ) @xmath158 , @xmath159 , @xmath160 , @xmath161 for @xmath162 such that @xmath163 . at @xmath164 a limit cycle appears , at @xmath165 the limit cycle disappears , and so on . thus , these bifurcations account for the appearance and disappearance of oscillations in figure [ fig:1 ] . we note that a similar sequence of bifurcations occurs whenever @xmath150 . on the other hand , if @xmath166 ( for example if @xmath149 ) , then all bifurcation points correspond to roots of the characteristic equation crossing into the right half - plane . in this case , the bifurcations can be subcritical . indeed , for @xmath149 , numerical simulations indicate that no limit cycle appears . as a result , solutions starting near the unstable rest point become unbounded . the slow dynamical system , obtained by reduction from a truncation of an expansion of a delay equation in powers of the delay , approximates the dynamics on the global attractor of the delay equation as long as the delay is sufficiently small ; but , as our examples show , the ordinary differential equations obtained by expansion , truncation , and reduction _ can not _ be used in general to predict the correct dynamical behavior for sufficiently large delays . we have mentioned , for example , that the dimension of the attractor of a family of delay equations , parametrized by the delay , can increase in dimension so that that the corresponding slow vector field is no longer defined on the attractor . but this is not the only possible scenario for the appearance of new attractor dynamics ; for example , the attractor could cease to exist or be a manifold for some values of the delay . the abraham - lorentz type equation can be used , after reduction to a slow manifold , to predict the relative orbital motion of a relativistic binary system in the regime where the delay is sufficiently small . the size of the maximum allowed delay would have to be computed on a case - by - case basis using the explicit form of the delay equation that models the dynamics of a coalescing pair of neutron stars . the results of this section show that for sufficiently large delay the attractor does not in general correspond to the slow manifold . the question remains whether such a divergence of behaviors could ever occur in the case of retarded equations of classical field theory . this is an interesting open problem . it is expected that interferometric gravitational wave detectors that are presently under construction will be able to detect signals from massive coalescing binary systems . for the analysis of such forthcoming data , it is important to have theoretically predicted wave forms ( ` templates ' ) for the relevant astrophysical processes . to this end , extensive computations are necessary that need to take gravitational radiation reaction into account @xcite . the standard approach leads to higher time - derivative equations that involve runaway modes and inevitably produce incorrect results . we have determined the source of the difficulty by investigating delay equations , which are essentially nonlocal , and the higher time - derivative equations that are obtained by truncations of the ( post - newtonian ) expansions in powers of the delay . for sufficiently small delays , a proper justification is provided for the usual method of replacing terms with higher - derivatives by terms with at most first derivatives using repeated substitution of the equations of motion ( ` iterative reduction ' ) . we have shown that in the investigation of the solutions of higher - derivative equations that represent phenomena involving radiation reaction , it is essential to reduce such equations to the corresponding slow manifolds before numerical analysis . our work suggests , however , that unexpected nonlocal phenomena could occur for sufficiently large delays that can not be predicted using the local equations of motion even after iterative reduction .

starting from delay equations that model field retardation effects , we study the origin of runaway modes that appear in the solutions of the classical equations of motion involving the radiation reaction force . when retardation effects are small , we argue that the physically significant solutions belong to the so - called _ slow manifold _ of the system and we identify this invariant manifold with the attractor in the state space of the delay equation . we demonstrate via an example that when retardation effects are no longer small , the motion could exhibit _ bifurcation _ phenomena that are not contained in the local equations of motion .

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Proceed to summarize the following text: low - energy hadron - hadron scattering plays an important role in the understanding of non - perturbative physics of strong interaction . due to its genuine non - perturbative nature , such problems can only be studied from first principles using non - perturbative methods like lattice qcd . lscher has outlined a finite - size formalism which enables us to calculate the elastic scattering phase shifts using lattice simulations @xcite . over the years , extensive numerical simulations have been carried out to the study on hadron - hadron scattering using lscher s formalism , both within the quenched approximations and using gauge field configurations with dynamical quarks @xcite . in lattice study on hadron spectroscopy and hadron - hadron scattering , the most important physical quantity is the energy of the system which is obtained via the measurements of various correlation functions . however , since a quantum field theory does not conserve particle numbers in general , the distinction between single- and multi - particle states becomes an important and delicate issue in lattice calculations . in the infinite volume , the difference is obvious since they have different kinematic behaviors : single - particle states have discrete energy eigenvalues when viewed in their rest frame while multi - particle states usually have continuous spectrum starting from the corresponding threshold . however , when performing a lattice simulation in a finite volume , all energy eigenvalues in the finite box become discrete . therefore , other means have to to be applied in order to identify the particle nature of a corresponding state . in principle , differences between single- and multi - particle states still persist in a finite volume . for example , although both have discrete spectra , the level spacing between neighboring multi - particle scattering states becomes infinitesimally small while the level spacing between the neighboring single - particle states remains finite as the volume goes to infinity . however , it is difficult to utilize this difference as a practical criteria since this requires the computation of excited energy eigenvalues in monte carlo simulations which is usually quite challenging . another method suggested by various authors is to use the so - called spectral weight as the identifier . this is the quantity which can be measured directly ( and relatively easily ) from monte carlo simulations , together with the corresponding energy eigenvalue . in a finite volume , the volume dependence of the spectral weight for a eigenstate is expected to show different behavior for single- and multi - particle states . for example , one expects the following empirical rule : the spectral weight to show little volume dependence for a single particle state ( if properly normalized ) , while for a two - particle state , it is expected to show a @xmath0 dependence where @xmath1 being the size of the cubic box . this expected difference in volume dependence can be measured in lattice simulations by performing the same calculation in two distinct volumes . as an example , this strategy has been used in ref . @xcite to study the possible penta - quark state . using this technique , the authors concluded that the expected penta - quark ( single - particle ) states measured in their lattice calculations are in fact kaon - nucleon two - particle scattering states . however , this conclusion is not so settled even in the first - principle lattice qcd calculations @xcite . therefore , the volume dependence of the spectral weight indeed can provide us useful information about the particle nature of the corresponding state . in a previous model study , we have shown that the above mentioned empirical rule to distinguish single- and multi - particle states are in fact only valid for stable particles and narrow resonances . using a solvable model , the lee model , we showed that this rule is violated for broad resonances @xcite . a general formula for the spectral weight was obtained which can show either single- or two - particle volume behavior depending whether the width of the resonance is narrow or broad . in this paper , we attempt to generalize this conclusion that we obtained in the lee model , to the case of general massive quantum field theory . for this purpose , the general lscher s formalism is adopted . in previous studies , people have been focusing mainly on the energy eigenvalue ( which directly enters the famous lscher s formula ) of the system within lscher s formalism . however , since the spectral weight @xmath2 of a given state is intimately related to the overlap of the exact energy eigenfunction with the free scattering states , we have to study the wavefunction of a energy eigenstate in a finite volume . in this paper , our study focuses on the wavefunction in the @xmath3 sector and a formula for the spectral weight is thus obtained within the non - relativistic quantum mechanics model . by studying the volume dependence of the spectral weight in the large volume limit , we arrive at the same conclusion as we drew from the previous lee model study . then , following lscher s arguments , this result is generalized to massive quantum field theory . our results also show a possibility of extracting the resonance parameters from the spectral weight function on various volumes . this paper is organized as follows . in sec . [ sec : model ] , we briefly review the quantum - mechanical model in the infinite volume . in sec . [ sec : model_torus ] , the quantum - mechanical model is studied on a three - dimensional torus of size @xmath1 . in this section , we derive the relevant formulae for the spectral weight function and study its volume dependence . it is found that similar conclusion is reached as in our previous study using the lee model . we then argue that , under the same restrictions as in lscher s formula , our results found in the quantum - mechanical model can be generalized to massive quantum field theory . the possibility of extracting resonance parameters from spectral weight is also discussed . in sec . [ sec : conclude ] , we will conclude with some general remarks . details on the evaluation of a function @xmath4 are listed in the appendix . consider a quantum mechanical model whose hamiltonian is given by : [ eq : hamiltonian ] h= -12m^2+v(r ) , where the potential @xmath5 is zero for @xmath6 with some @xmath7 . we now discuss the energy eigenstates satisfying : @xmath8 . one can expand the eigenfunction in terms spherical harmonics : ( r)=_lm(r)y_lm(n ) . with : @xmath9 and @xmath10 is the radial wave - function satisfying the radial schrdinger equation : [ eq : radial ] ( d^2dr^2 + 2rddr - l(l+1)r^2 + k^2 - 2mv(r))_lm(r)=0 . where @xmath11 being the energy eigenvalue of the state . it is well - known that , there exist only one solution to the radial schrdinger equation that is bounded near the origin . this solution will be denoted as : @xmath12 . to fix the normalization , we impose the condition : [ eq : normalize_u ] _ r0 r^-lu_l(r;k)=1 , and the solution to the radial schrdinger equation then has the form : _ lm(r)=b_lmu_l(r;k ) , with some constant @xmath13 to be fixed by other conditions ( normalization , boundary conditions , etc . ) . in the region @xmath6 where the interaction vanishes , the solution @xmath12 are expanded in terms of spherical bessel functions : [ eq : ul_expand ] u_l(r;k ) = _ l(k)j_l(kr ) + _ l(k)n_l(kr ) . the coefficients @xmath14 and @xmath15 have simple relation with the scattering phase shift : e^2i_l(k)=_l(k)+i_l(k ) _ l(k)-i_l(k ) , _ l(k)=_l(k)_l(k ) . in the low - energy limit : @xmath16 , one normally defines : ^0_l=_k0 k^l_l(k ) , ^0_l=_k0 k^-l-1_l(k ) , and the threshold parameters : which is usually the case . ] in particular , @xmath17 for @xmath18 is referred to as the @xmath19-wave scattering length . other @xmath20 s for @xmath21 are sometimes also called scattering lengths in the corresponding channel , although they do not have the dimension of a length . ) , it is easy to verify that the spectral parameters @xmath20 has the length dimension of @xmath22 . ] the threshold parameters @xmath20 are important because they characterize the behaviors in low - energy scattering processes . for example , we have : _ l(k ) a_l k^2l+1+o(k^2l+3 ) , ( ) . we now enclose the system we discussed in the previous section in a large cubic box and impose the periodic boundary condition in all three spatial directions . the potential itself is also modified to @xmath23 by periodically extending over the whole space : @xmath24 . for later convenience , we define the the so - called outer region " as : = \{r : |r+nl|>a , n^3}. this is the region where the potential vanishes identically . we assume the size of the box is @xmath1 which is much larger than any of the physical scale in the system . in particular , we need to have @xmath25 so that the outer region admits free spherical wave solutions ( asymptotic states ) . we now would like to study the change in the energy eigenvalues , the corresponding wave - functions and their possible connections with the scattering phase shifts in the infinite volume . our discussion here will focus on the case of a cubic box whose relevant symmetry group being the cubic group @xmath26 . generalization to an arbitrary rectangular box can be performed easily by changing the symmetry group to the corresponding ones ( @xmath27 or @xmath28 , etc . ) . since the boundary condition breaks rotational symmetry explicitly , we anticipate that energy eigenstates of the system will not have a definite angular momentum in general . to be specific , the original eigenstate in the @xmath19-wave will acquire mixtures from higher angular momentum modes ( mainly @xmath29 for a cubic box ) . however , since the original radial wave - function @xmath12 and the spherical harmonics forms a complete set in the functional space , we may still expand the true eigenfunction in the box in terms of them : [ eq : eigenfunction_exact ] ( r;k)=_lm b_lmu_l(r;k ) y_lm(n ) . where the coefficients are to be determined by boundary conditions and normalization . in the outer region @xmath30 , the solution are those singular , periodic solutions for the helmholtz equation . thus we may write : ( r;k)|_r=_lmv_lmg_lm(r;k^2 ) . in the meantime , the outer solution can also be expanded in terms of spherical harmonics and the spherical bessel functions @xmath31 and @xmath32 : [ eq : expand_ylm ] g_lm(r;k^2)=(-)^lk^l+14 , the explicit expression for @xmath33 is given in ref . @xcite which we quote here : [ eq : calm - zeta ] m_lm;js(k^2 ) & = & _ lm(-)^si^j - lz_lm(1,q^2 ) ^3/2q^l+1 + & & ( ccc l & l & j + 0 & 0 & 0 ) ( ccc l & l & j + m & m & -s ) . here we have used the wigner s @xmath34-symbols and @xmath35 . the zeta function @xmath36 is defined as : [ eq : zeta_def ] z_lm(s , q^2)= _ n y_lm(n ) ( n^2-q^2)^s . from the analytically continued formula , it is obvious from the symmetry of @xmath26 that , for @xmath37 , the only non - vanishing zeta functions at @xmath38 are : @xmath39 , and @xmath40 . this is in accordance with the fact that @xmath19-wave and @xmath41-wave mixes with each other in a cubic box . in what follows , we will focus on the @xmath19-wave eigenfunction . in the remaining part of this paper , we will be only concerned with the energy eigen - functions in the @xmath3 sector , which is the analogue of @xmath19-wave in a cubic box . a good approximation for the @xmath19-wave dominated eigenfunction can be written as a superposition of @xmath18 and @xmath29 spherical harmonics with the @xmath19-wave component much larger than that of @xmath41-wave . to explicitly construct this type of wave - functions , we notice that the eigen - function in @xmath3 sector has to be invariant under cubic symmetries . it is easy to verify that , there are only two homogeneous harmonic polynomials which are invariant under cubic symmetry up to @xmath42 . they can be conveniently expressed as : y_00=1 , y_40 + 14(y_4,4+y_4,-4 ) = 154(x^4+y^4+z^4 - 35 r^4 ) . so , we may write the eigen - function in @xmath3 sector as : [ eq : psi_a1 ] ^(a^+_1)(r;k ) = b_00u_0(r;k)y_00 + b_40u_4(r;k)(y_40 + 14(y_4,4+y_4,-4))+ , with @xmath43 in the large volume limit . in other words , to ensure cubic symmetry , the general coefficients @xmath13 at @xmath29 with different @xmath44 values must have definite ratios . in the outer region , using relation ( [ eq : ul_expand ] ) , we have : [ eq : psi_in_out ] ^(a^+_1)(r;k)|_r & = & b_00[_0j_0(kr)+_0n_0(kr)]y_00(_r ) + & + & b_40[_4j_4(kr)+_4n_4(kr ) ] ( y_40 + 14(y_4,4+y_4,-4 ) ) + . on the other hand , we know that , in the outer region @xmath30 , the eigen - function can also be expanded into singular periodic solutions of helmholtz equation . since @xmath45 with @xmath46 being rotationally invariant , we see that in order to keep the eigen - function invariant under cubic symmetry , we must have the combination : @xmath47 in the expansion . thus we may write : and an extra factor of @xmath48 for the coefficient of @xmath49 . ] [ eq : psi_into_glm ] ^(a^+_1)(r;k)|_r= ( 4k ) v_00 . the fact that such a combination respects cubic symmetry can also be checked explicitly . using the expressions ( [ eq : expand_ylm ] ) and ( [ eq : calm - zeta ] ) , we may write the expansion for @xmath50 as : [ eq : g00 ] g_00 = k4 , where we have introduced : @xmath51 and @xmath52 for later convenience ( see ref . @xcite for the notation ) . similarly , for the higher angular momentum functions , we have : [ eq : g_expand ] g_40 & = & k^54 , + g_4,4+g_4,-4 & = & k^54 , in the above expansions , we have also utilized the following properties of the matrix elements @xmath53 : m_lm;lm=m_lm;lm = m_l ,- m;l,-m. note that in the expansion of @xmath49 and @xmath54 in eq . ( [ eq : g_expand ] ) , there are terms with @xmath55 , @xmath56 spherical harmonics . however , when we construct the combination @xmath57 , the terms with @xmath55 cancel out explicitly since : @xmath58 which can be checked by looking into table e.1 in ref . therefore we finally have : [ eq : g40 ] g_40 + 14(g_44+g_4,-4)&= & k^54 where @xmath59 . at this stage , it is worthwhile to point out that , @xmath60 , @xmath61 and @xmath62 that we introduced here are exactly those reduced matrix elements of @xmath63 in the @xmath3 sector . please refer to ref . @xcite for further detailed explanations ( especially table e.1 and table e.2 in the reference ) . collecting relevant information from the expansions obtained thus far , i.e. eq . ( [ eq : psi_into_glm ] ) , eq . ( [ eq : g00 ] ) and eq . ( [ eq : g40 ] ) , we have : [ eq : psi_out_out ] & & ^(a^+_1)(r;k)|_r = v_00 + , we should now match the two solutions given by eq . ( [ eq : psi_in_out ] ) and eq . ( [ eq : psi_out_out ] ) in the outer region @xmath30 . this yields the following set of linear equations : v_00 & = & b_00_0 , v_00(m_00+v_40m_04)=b_00_0 , + v_00v_40&=&b_40_4 , v_00(m_04+v_40m_44)=b_40_4 . these four equations can be viewed as a set of homogeneous linear equations for the four coefficients : @xmath64 , @xmath65 , @xmath66 and @xmath67 . demanding a non - trivial solution to exist requires the corresponding determinant of the @xmath68 matrix to vanish . another simple way to proceed is to divide the second equation by the first and similarly divide the fourth one by the third . this will eliminate all coefficients except for @xmath69 . we then arrive at : ^(0)=m_00+v_40m_04 , ^(4)=m_44+m_04/v_40 . eliminating @xmath69 from the above two equations then yields : [ eq : match2 ] ( ^(0)-m_00 ) ( ^(4)-m_44)= m_04m_04 . this is exactly the equation obtained by general lscher s method when we only consider the mixing between @xmath18 and @xmath29 waves @xcite . therefore , using more explicit construction , not only have we recovered lscher s formula , we also obtained an explicit approximate expression for the energy eigen - function in the @xmath3 channel which is given by eq . ( [ eq : psi_a1 ] ) in general and given by eq . ( [ eq : psi_into_glm ] ) in the outer region . now we would like to derive a formula for the spectral weight function which can be measured in a monte carlo simulation . instead of working with general states , we will focus on the single- and two - particle states . these states naturally arise in the lattice study of hadron - hadron scattering and hadron spectrum . in such simulations , one constructs an operator ( also known as the interpolating field operator ) , or operators if more than one is needed , within a specific symmetry sector of the theory . the correlation matrix among these operators are then computed by ensemble averaging over different gauge field configurations that is generated in a monte carlo simulation . for this purpose , we pass over to the second - quantized version of our quantum mechanical scattering model . in this model , two distinguishable particles scatter via a potential @xmath5 where @xmath70 being the distance between them . the center - of - mass coordinate of the two - particle system is separated out and the mass parameter @xmath44 in the hamiltonian ( [ eq : hamiltonian ] ) refers to the reduced mass of the two - particle system . for each type of particle , a local scalar field operator @xmath71 , with @xmath72 designating different types of particles , is introduced together with its momentum space counterpart : [ eq : fourier_def ] _ i(x , t)=1_p_i(p , t)e^ ipx , _ i(p , t)=1d^3x_i(x , t)e^-ipx they satisfy the usual equal - time commutation relations : @xmath73=\delta_{{{\mathbf p}}{{\mathbf k}}}\delta_{ij}$ ] . using free states made up of two particles , one from each type , one can form a state : |=o^(0)|0=1l^3/2_p(p ) ^_1(p,0)^_2(-p,0 ) |0 , with the interpolating operator @xmath74 defined by : o(t)=1_p ^*(p)_1(p , t)_2(-p , t ) . requiring such a state to be normalized as : @xmath75 yields the condition : _ p|(p)|^2=1 . if such a state were a bound state of two particles , @xmath76 would be the corresponding momentum - space wavefunction normalized according to the above equation . we can now define the corresponding correlation function : c(t)=0|o(t)o^(0)|0 = _ e |e|o^(0)|0|^2 e^-et , where @xmath77 and @xmath78 represents the eigenvalue and eigenstate of the full hamiltonian , respectively . by fitting the time - dependence of the correlation function obtained from monte carlo simulations , the exact eigenvalue @xmath77 , and the corresponding spectral weight function @xmath79 , which is the coefficient in front of the exponential , is obtained . if we denote the overlap of two wavefunctions : [ eq : overlap ] o(e)=e|o^(0)|0 = d^3r_1d^3r_2e|r_1,r_2 r_1,r_2|o^(0 ) |0 , the spectral weight function is simply given by : [ eq : w_def ] w(e)=|e|o^(0)|0|^2=|o(e)|^2 . at this point , it is worthwhile to point out that the spectral weight function @xmath79 defined above depends explicitly on the normalization of @xmath80 . due to translational symmetry , the exact wave - function @xmath81 will only depend on the relative coordinate @xmath82 . it is independent of the center - of - mass coordinate @xmath83 . this means that , if the eigenstate @xmath78 is normalized according to @xmath84 as it should , the wave - function @xmath85 should be normalized according to : [ eq : a1_normalize ] _ t_3 d^3r|r|e|^2= _ t_3 d^3r|^(a^+_1)(r;k)|^2=1l^3 . therefore , in order to compute the volume dependence of the spectral weight function , we first have to fix the normalization of @xmath86 according to this convention . as discussed in the previous subsection , the wavefunction in the @xmath3 sector in eq . ( [ eq : psi_a1 ] ) must be normalized properly on the torus @xmath87 according to eq . ( [ eq : a1_normalize ] ) . the integral of the eigen - function on the torus runs over two regions : the inner region where the explicit form of the wavefunction is not known and the outer region @xmath30 where an approximate form of the function is given by eq . ( [ eq : psi_into_glm ] ) . although we do not know the exact form of the eigen - function in the inner region , we do know that the eigenfunction is bounded in this region . since it is assumed that the interaction region is of size @xmath88 with @xmath89 , therefore the integral in the normalization condition ( [ eq : a1_normalize ] ) is dominated by the integral of the function in the outer region @xmath30 . therefore , we may modify the normalization condition to : _ d^3r|^(a^+_1)(r;k)|^2 . since in the large volume limit , the eigen - function is dominated by the @xmath19-wave contribution , we may use the first term in eq . ( [ eq : psi_into_glm ] ) and write : [ eq : norm_approx ] ( 4k)^2|v_00|^2(_t_3 d^3r|g_00(r;k)|^2 -_b d^3r|g_00(r;k)|^2 ) , where the second integral is over the interaction ball region : @xmath90 . we now use the definition for @xmath50 : [ eq : g00_explicit ] g_00(r;k)=1l^3_pe^ip^2-k^2 , where the summation of @xmath91 is for all three - dimensional integers : @xmath92 . substituting this expression into the first term and eq . ( [ eq : psi_out_out ] ) into the second integral in eq . ( [ eq : norm_approx ] ) we get : _ p1(p^2-k^2)^2 - k^216 ^ 2 ^a_0 r^2dr ( n_0(kr)+m_00j_0(kr))^2 . the integral in the second term maybe evaluated directly within @xmath93 . we thus obtain : [ eq : v00_two_terms ] 1|v_00|^2l^3_p1(p^2-k^2)^2 - a2k^2 ^ 2 , where we have utilized the definition : @xmath94 . in the large volume limit , the first term on the r.h.s . of the above equation is much larger than the second ( see appendix [ app : f ] for the explanation of this assertion ) . if we drop the second term , we then arrive at : [ eq : v00_f ] ( 4k)^2|v_00|^2l^3 4(1l^3_p1(p^2-k^2)^2)^-1 , where we have defined the function : f(k^2)=1l^3_pf(p^2)^2-k^2 , where we have introduced a cutoff function @xmath95 to regulate possible ultra - violet divergences . the property of this function in the large volume limit is addressed in appendix [ app : f ] . the relevant formula for us is given by eq . ( [ eq : fprime ] ) . we now evaluate the spectral weight using eq . ( [ eq : w_def ] ) with the exact energy eigen - function given approximately by : @xmath96 . the overlap of the two wave - function is approximately given by : o=(4k)v^*_001 _ p(p)^2-k^2 . using the expression ( [ eq : v00_f ] ) and the expression in eq . ( [ eq : fprime ] ) , we finally obtain @xmath97 as : [ eq : w_formula ] w(e)=8k |_l(k^2)|^2 _ 0(k ) + 2k^2p^2 ^ 2_0(k ) = 8k |_l(k^2)|^2 _ 0(k ) + 2ee^2_0(k ) , where the function @xmath98 is defined as : _ l(k^2)=1l^3_p(p ) ^2-k^2 . in the large volume limit , following similar derivation as in our discussion of function @xmath4 , this function goes over to : _ ( k^2)=p+ k(k^2)4_0(k ) . thus the function @xmath98 has little volume dependence in the large volume limit . therefore , the explicit volume dependence of the spectral weight function @xmath79 comes mainly from the denominator in eq . ( [ eq : w_formula ] ) . normally , if @xmath99 is not changing rapidly , the second term in the denominator of eq . ( [ eq : w_formula ] ) , which is proportional to @xmath100 , dominates the result and one finds that the spectral weight is proportional to @xmath0 . this is the typical two - particle spectral weight function . however , if there exists a rather narrow resonance at energy @xmath101 , then close to this resonance energy , one has approximately : ( e ) , where @xmath102 is the physical width of the resonance . in this case , we obtain : [ eq : resonance ] w(e ) . if @xmath103 , then the quantity in the denominator is dominated by the first term and the spectral weight shows a typical single - particle behavior . this means that an extremely narrow resonance behaves like a stable particle . if on the other hand @xmath104 , which is always true for an extremely large volume ( assuming the width of the resonance remains finite ) , the denominator is dominated by the second term and the spectral weight itself is roughly proportional to @xmath0 which is typical for a two - particle scattering state . we therefore arrive at the conclusion that the volume dependence of the spectral weight near a resonance is controlled by the ratio @xmath105 . our results on the volume dependence of the spectral weight is obtained within a quantum mechanical model . in this subsection , we would like to generalize these results to massive quantum field theory , following the line of arguments in lscher s formalism @xcite . using an effective schrdinger equation ( derived from the bethe - salpeper equation ) @xcite , lscher has argued that , if the size of the box is large enough such that all quantum field theory effects are suppressed exponentially , the results obtained within the quantum - mechanical model can be carried over to the case of massive quantum field theory literally @xcite . here , we will assume that the same conditions are satisfied and thus our results obtained within the quantum - mechanical model are expected to be valid for massive quantum field theory . the relation established in eq . ( [ eq : resonance ] ) opens up a possibility for extracting the width of a resonance if the spectral weight can be measured in the simulation . assuming that there exists a single resonance in the energy region that we are interested in , and the contribution from this single resonance dominates the scattering , we simply rewrite eq . ( [ eq : resonance ] ) as : ( e_-e + e_e ) . therefore , by fitting the function @xmath106 for different @xmath77 and @xmath1 ( hence different @xmath107 as well ) , it is possible to extract the width parameter @xmath102 together with the resonance position @xmath108 of the resonance . note that in previous lattice calculations , focus has been mainly put on the energy levels , i.e. the values of @xmath77 , only . no attention is paid to the associated spectral weight function @xmath2 which in fact can be obtained from the fitting procedure of the corresponding correlation functions _ with almost no extra costs_. the study in this paper indicates that , the spectral weight function at various volumes also contains valuable information about the scattering and might also be utilized in some way . in fact , it can be used as an cross - check for the scattering phase obtained from the energy levels . of course , this is only a possibility at this stage . the feasibility of this method has to be check in realistic simulations . in this paper , we have studied the volume dependence of the spectral weight function which is accessible in monte carlo lattice simulations . motivated by our previous study in the lee model , it is expected that the spectral weight function shows little volume dependence for a stable or narrow resonance while for a broad resonance , it exhibits a typical @xmath0 dependence , the same as a two - particle scattering state . to verify this scenario , lscher s formalism is adopted . it is first shown in a quantum mechanical model and then generalized to any massive quantum field theory , assuming that the polarization effects are exponentially suppressed following lscher s arguments . in particular , we expect this scenario to be true also for qcd which governs the scattering of hadrons and therefore our result is relevant for lattice qcd simulations . our final result for the spectral weight is summarized in eq . ( [ eq : w_formula ] ) which exhibits either single- or two - particle volume dependence depending on the value of @xmath109 where @xmath102 is the physical width of the resonance and @xmath107 is the typical level spacing near the resonance in the finite volume . possibilities of using the formula to extract the width of a resonance is also discussed . the author c. liu would like to thank prof . liu from university of kentucky , dr . j.p.ma from itp , academia sinica , dr . y. chen from ihep , academia sinica , prof . b . liu from nankai university , prof . j .- b . zhang from zhejiang university , prof . h. q. zheng , prof . s. h. zhu and prof . s. l. zhu from peking university for valuable discussions . to study the normalization of the wavefunction @xmath86 in the large volume limit , we define the function : f(k^2)=1l^3_pf(p^2)^2-k^2 , where we have introduced a cutoff function @xmath95 . the relevant function appearing in the normalization condition ( [ eq : v00_f ] ) is given by the derivative of @xmath4 with respect to @xmath110 : f(k^2)=1l^3_pf(p^2)(p^2-k^2)^2 . we now follow the argument in ref . @xcite to estimate the value of @xmath4 for arbitrary value of @xmath110 in the large @xmath1 limit . we separate the summation into two parts with : @xmath111 and @xmath112 . the first part goes smoothly to the principle - valued integral @xmath113 while the second summation may be written as : _ p,|p^2-k^2|<1p^2-k^2 & = & 1l^3^_n=-1 p^2_+np^2-k^2 + & = & -l^3p^2 , where @xmath114 is the value of @xmath115 that is closest to @xmath110 ; @xmath116 is the typical level spacing between neighboring @xmath115 values which can be estimated by : 2p^2 = 1 l^3p^2=(2)^2 therefore we obtain : f(k^2)=(k^2)-k4 . since it is easy to verify that : f(k^2)=z_00(1;q^2)2 ^ 3/2 l _ 0(k ) , where we have utilized the approximate relation ( lscher s formula ) : _ 0(k)=z_00(1;q^2)^3/2 q. we therefore seem to have : @xmath117 in which case we recover the dewitt s formula : _ 0(k)=-(k^2-p^2_p^2 ) . if one evaluate @xmath113 explicitly , one gets : ( k^2)=p = 4 + 2k|-k+k| , with a sharp momentum cutoff @xmath118 . this expression indeed goes to zero if we drop the constant term and taking @xmath119 . consequently we have for the function @xmath120 : [ eq : fprime ] f(k^2 ) & = & -18k+ k4p^2 ^2 + & = & 18k_0(k ) + k4p^2 ^ 2_0(k ) , where in the second line we have used dewitt s formula . since @xmath121 , we find that @xmath122 in the large volume limit . this justifies the assertion made after eq . ( [ eq : v00_two_terms ] ) in the main text .

it has been suggested that the volume dependence of the spectral weight could be utilized to distinguish single and multi - particle states in monte carlo simulations . in a recent study using a solvable model , the lee model , we found that this criteria is applicable only for stable particles and narrow resonances , not for the broad resonances . in this paper , the same question is addressed within the finite size formalism outlined by lscher . using a quantum mechanical scattering model , the conclusion that was found in previous lee model study is recovered . then , following similar arguments as in lscher s , it is argued that the result is valid for a general massive quantum field theory under the same conditions as the lscher s formulae . using the spectral weight function , a possibility of extracting resonance parameters is also pointed out . , , , spectral weight , finite - size technique , lattice qcd .

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Proceed to summarize the following text: confinement of quarks is a very old problem that until now has not been solved . at low energies perturbative qcd can not be applied because the coupling constant is large . effective theories and lattice qcd have given important contributions to the understanding of this problem . a very useful approach to the confinement problem is the study of large n gauge theories because of the possibility of obtaining analytical and numerical results for large ( t hooft ) coupling in a simple way via the ads / cft correspondence @xcite . this correspondence relates large @xmath4 gauge theories on a @xmath5 dimensional manifold @xmath6 to 10-d string theories on @xmath7 being @xmath6 the boundary of @xmath3 and @xmath8 some compact space . in particular , in the context of ads / cft correspondence witten showed that @xmath9 super yang - mills ( sym ) theory in @xmath10 at strong coupling experiments a confinement / deconfinement transition that corresponds holographically to a gravitational transition between @xmath11 and black hole @xmath11 defined in global coordinates @xcite . this gravitational transition was studied before by hawking and page @xcite . the confinement criteria stated in @xcite is the behavior of the free energy of large n @xmath12 sym theory in @xmath13 . the theory is confining at low temperatures because the free energy is of order 1 ( color singlet contribution ) while is deconfining at high temperatures because the free energy is or order @xmath14 ( gluons contribution ) . this criteria can be extended to the cases @xmath0 . finite temperature qcd lives in @xmath15 which is the boundary of @xmath11 in poincar coordinates . sym theory in @xmath15 is not confining as can be seen using the wilson loop prescription @xcite . this prescription relates a wilson loop of the gauge theory to the world - sheet area of a static string living in @xmath16 in poincar coordinates . in particular , the free energy of a quark anti - quark pair in @xmath15 can be calculated from the correlator of two polyakov loops and exhibits a coulombian behavior . this behavior can be interpreted as deconfining because the free energy vanishes when the quark anti - quark distance goes to infinity . the wilson loop prescription was generalized to other curved spaces and a confinement criteria was established @xcite . the quark anti - quark free energy at high temperatures for the space @xmath15 was calculated in @xcite by considering an @xmath11 black hole in poincar coordinates . other wilson loop calculations involving gauge / string duality can be found at @xcite . there are some phenomenological models that introduce confinement in @xmath15 . a well succeeded one is the hard wall model which consists on introducing a hard wall on poincar @xmath11 space interpreted holographically as an infrared cut - off for the gauge theory . this model was motivated on the calculation of scattering amplitudes @xcite and leads to confinement at low temperatures @xcite . at high temperatures there is a confinement / deconfinement transition corresponding to a hawking - page transition between ( poincar ) @xmath3 and ( poincar ) black hole @xmath3 . other calculations of quark anti - quark potential using phenomenological models can be found at @xcite . in spite of being a little far from real world , large @xmath4 gauge theories on @xmath17 are physically interesting . for instance , it is possible to use perturbation theory in the small radius regime so that thermal transitions can be studied and connect them with the hawking - page and hagedorn transitions which are important in string theory and quantum gravity @xcite . the mean of this work is to calculate the free energy of a quark anti - quark pair living in @xmath17 space where @xmath18 represents the imaginary time coordinate with a period @xmath19 identified with the inverse of the temperature . for this purpose we consider a wilson loop containing two temporal lines and two spatial lines . the quark anti - quark free energy can be extracted from the correlator of two polyakov loops . in order to relate our wilson loop to the polyakov loop correlator , the size of the temporal lines must be @xmath19 . the dual description consists in a static string with the quark and anti - quark at the end points . this string configuration gives the dominant contribution to the connected part of the polyakov loop correlator ( see for example @xcite ) . then what we mean by potential in this article is the quark anti - quark free energy obtained from the connected part of the polyakov loop correlator . we follow the procedure of @xcite to calculate this potential . the space @xmath17 is the boundary of two spaces : global @xmath3 and black hole @xmath3 in @xmath20 dimensions . we first calculate explicitly the @xmath21 case where asymptotic limits for the potential are studied and then we generalize our results to the cases @xmath22 . we find a coulomb - like behavior of the free energy in the case of global @xmath1 and show that this behavior can be interpreted as confining . at high temperatures we calculate the quark anti - quark free energy for the two possible black hole solutions : the big black hole and small black hole in @xmath1 . in the case @xmath23 our results for the big black hole agree with those obtained before by landsteiner and lopez @xcite where it was defined a screening length for the quark anti - quark free energy . here , we calculate this screening length as a function of the temperature and the horizon position . for the big black hole the screening length never reaches the maximal distance while for the small black hole this could happen . this will imply ( for @xmath24 ) that the big black hole @xmath1 is indeed deconfining at any ( high ) temperature while the small black hole could give confinement in the limit @xmath25 . the hawking - page criteria states that the small black hole solution is instable , so deconfinement is guaranteed at high temperatures . our results indicate a confinement / deconfinement transition for the quark anti - quark free energy in @xmath17 corresponding to a hawking - page transition from @xmath1 to black hole @xmath1 . in this section let us calculate the quark anti - quark free energy on the compact space @xmath26 . this space is the boundary of the @xmath27 space in global coordinates . the metric of global @xmath27 with euclidean signature is @xmath28 where @xmath29 , @xmath30 and @xmath31 . the time coordinate is compact : @xmath32 with @xmath33 . for simplicity we consider @xmath34 and define a new coordinate @xmath35 with dimension of length . the metric then reads @xmath36 where @xmath37 . we consider the rectangular wilson loop of fig [ fig1 ] living in the @xmath3 boundary ( @xmath38 ) . the quark and anti - quark are localized at @xmath39 and @xmath40 . the distance between the quarks @xmath41 has a maximum value @xmath42 because of the compactness of the @xmath43 coordinate . the dual configuration consists on a static string living on @xmath3 with the quarks at the end points . the nambu - goto action for this string is @xmath44 where @xmath45 and we parameterized the string configuration by @xmath46 and @xmath47 valid for @xmath48 ( the other region should be parameterized in other way but it is not necessary to work there because of the symmetry of the problem ) . the solutions for the static string configuration ( corresponding to a minimum action ) represent geodesics of the ads space where the string reaches a minimum for the radial coordinate @xmath49 ( see fig 2 ) . using the prescription of @xcite we find @xmath50^{1/2 } \nonumber\\ & = & \ , \int_{1}^{\infty } \frac{2 r \sqrt{1+\lambda^2}dy}{y^4\sqrt{\lambda^2+\frac{1}{y^2}}\sqrt{1-\frac{1}{y^2 } } \sqrt{\frac{1+\lambda^2}{y^2}+\lambda^2}}\end{aligned}\ ] ] where @xmath51 and @xmath52 . the regulated free energy is given by @xmath53\end{aligned}\ ] ] where the second integral of the first line represents a regulator interpreted as the sum of the quark anti - quark masses . the integrals for @xmath41 and @xmath54 are elliptical so we calculate them numerically . we find a free energy with coulombian behavior at small distances and that goes to zero only when @xmath41 goes to @xmath55 . we will explain in the last section why this behavior for the free energy can be interpreted as confining . our results are shown in figs . [ fig4 ] and [ fig5 ] . it is interesting to analyze the behavior of the free energy in the asymptotic limits @xmath56 and @xmath57 . for this purpose it is convenient to define the variable @xmath58 . then @xmath59 @xmath60\!\ ! -\ ! { \lambda r \over \pi \alpha'}\ ] ] now we can evaluate the asymptotic limits . in this case @xmath61 is very large , then we have @xmath62 where @xmath63 . the integral for the free energy f is @xmath64- { \lambda r \over \pi \alpha'}= -{a \over l } \label{coulomb}\ ] ] where @xmath65 with @xmath66 . this result is similar as the one obtained working in the poincar @xmath3 metric @xcite . for small distances @xmath43 this metric can be thought as the asymptotic limit @xmath67 of our global @xmath3 metric , which is @xmath68 but note that in this case the coordinate @xmath43 has to be small . the behavior of this free energy is analogous to that of the phenomenological cornell potential for a heavy quark anti - quark pair at small distance . in this case @xmath69 is small then the integral for l is @xmath70 \label{l}\end{aligned}\ ] ] the integral for the free energy is given by @xmath71 \ ! \!- \ ! \!{\lambda r \over \pi \alpha ' } \nonumber \\ & = & { \lambda r \over 4 \pi \alpha ' } [ { l \over r } - \pi ] \end{aligned}\ ] ] where we approximated the term @xmath72 from equation ( [ l ] ) we have that @xmath73 so we finally obtain @xmath74 this result could be obtained considering the asymptotic behavior @xmath75 of the metric : @xmath76 with the quarks in a brane localized at @xmath77 . note that when @xmath41 reaches @xmath78 the free energy takes its maximal value @xmath79 . the black hole @xmath27 metric in global coordinates is defined by @xmath80 where @xmath29 , @xmath30 and @xmath32 . the factor @xmath81 is included so that @xmath82 is the mass of the black hole . also , the spacetime is restricted to the region @xmath83 , with @xmath84 the largest solution of the equation @xmath85 the region @xmath86 is the horizon of the black hole metric . the temperature of the black hole is related to @xmath87 by @xmath88 note that there is a minimum temperature @xmath89 for the existence of this black hole that corresponds to a critical horizon @xmath90 . this critical horizon divides two possible black hole solutions : the big black hole ( bbh ) with an horizon that grows with temperature and the small black hole ( sbh ) with an opposite behavior . the hawking - page criteria states that for temperatures @xmath91 the big black hole is stable only for @xmath92 while the small black hole is always instable @xcite . we again consider @xmath34 and define @xmath93 so the black hole metric reads @xmath94 then @xmath95 curve @xmath96 , correspond a u - type geodesic and @xmath97 a box - type geodesic . , width=207 ] now , we put a quark at @xmath39 and an anti - quark at @xmath98 . in this space we have two possible solutions for the static string : the first one is a u - type solution similar to the one obtained before for the @xmath3 where the string reaches a minimum for the radial coordinate @xmath49 , the second solution is a string reaching the horizon ( see fig [ graficocaixote ] ) . depending on the value of @xmath41 one of those solutions will have the minimum free energy . the distance and free energy for the quark anti - quark pair are in this case @xmath99^{1/2 } } \\ & & \nonumber\\ & & \nonumber \\ f & = & \frac{r\lambda}{\pi \alpha ' } \int_0^{1 } dt \ , t^{-2}\big [ \frac{\sqrt{\lambda^2+t^2-\frac{\gamma}{\lambda } t^3}}{\sqrt{1-t}[(1+t)(\lambda^2 + ( 1+\lambda^2)t^2)-\frac{\gamma}{\lambda } t^3]^{1/2}}-1 \big ] - { r\over \pi \alpha ' } \big(\lambda -\lambda_+\big ) \ , \label{eqenergybh}. \end{aligned}\ ] ] where @xmath100 , @xmath51 and we defined vs @xmath101 for black hole @xmath27 at different temperature values . [ energiabh],width=340 ] @xmath102 with @xmath103 . note that @xmath104 because @xmath105 . we also used the mass regulator @xmath106 from eq ( [ eqenergybh ] ) it is not difficult to show that the box - type solution has always zero free energy . the free energy of the u - type solution is negative for small @xmath41 and exhibits the same coulombian behavior found in eq . ( [ coulomb ] ) . when we increase the distance the free energy increases and reaches the zero value for some length @xmath107 that depends on @xmath108 . when @xmath109 the free energy of the u - type solution becomes positive and is inestable compared with the box - type solution ( which has zero free energy for every @xmath41 ) . so we find a transition for the free energy corresponding to a transition between the u - type and the box - type solutions being @xmath107 a screening length for which the free energy reaches the zero value . we show in fig . [ energiabh ] the free energy as a function of the distance for the two black hole solutions at different values of the temperature . note that the big black hole solution has a screening length that is always lower than the maximal distance @xmath110 while the small black hole has a confining limit corresponding to @xmath111 . vs @xmath112 for black hole @xmath27 . [ tlcrit],width=264 ] this limit is achieved when @xmath25 that corresponds to @xmath113 when the small black hole @xmath27 becomes @xmath27 ( which is confining ) . in fig . [ tlcrit ] we show the temperature as a function of the screening length . it is interesting to obtain the screening length dependence on @xmath108 for the big black hole . from fig . [ lcritlambdamais ] we see that for large @xmath108 the product @xmath114 is constant and approximately equal to @xmath115 . this behavior was also obtained for the poincar black hole @xcite . vs @xmath108 for black hole @xmath27 . [ lcritlambdamais],width=321 ] the quark anti - quark potential in @xmath0 can be calculated in a similar way as the @xmath117 case . at low temperatures we have to deal with the global @xmath1 space . the metric of this space for @xmath118 is given by @xmath119 where @xmath120 @xmath121 , @xmath30 , @xmath31 and @xmath32 . if we choose @xmath122 and define @xmath123 we arrive at the same metric of eq.([metricaadsx ] ) so we find the same free energy found before working in @xmath27 ( fig [ fig5 ] ) . at high temperatures , we use the black hole @xmath1 ( @xmath124 ) with metric @xmath125 where @xmath126 . the temperature is now related to @xmath87 by @xmath127 now the critical horizon is @xmath128 , the hawking - page horizon is always @xmath129 which is always above @xmath130 so the big black hole ( @xmath131 ) is stable while the small black hole is always instable @xcite . choosing again @xmath122 and defining @xmath123 we obtain @xmath132 so we find in this case @xmath133 leading to @xmath134^{1/2 } } \nonumber\\ & & \nonumber \\ & & \nonumber \\ \ ! \ ! \ ! \ ! \ ! f \ ! \ ! \ ! & = & \ ! \ ! \!\frac{r \!\lambda}{\pi \alpha ' } \ ! \ ! \ ! \ ! \int_0^{1}\ ! \ ! \ ! \ ! \ ! dt \ , t^{^{\ ! -2 } } \!\left[\ ! \!\frac{\sqrt{\ ! \lambda^2\ ! + \ ! t^2 \ ! - \ ! \frac{\gamma}{\lambda^{n-2 } } t^n } } { [ ( 1-t^2 ) ( ( 1\ ! + \lambda^2 ) t^2+\lambda^2)+\frac{\gamma}{\lambda^{n-2}}(t^4-t^n)]^{1/2}}\ ! -\ ! 1\ ! \right ] \ ! \ ! -\ ! \frac{r } { \pi \alpha ' } \big(\lambda - \lambda_+ \big ) \!\nonumber \\ & & \label{eqenergybh2}\end{aligned}\ ] ] where @xmath135 @xmath136 vs @xmath137 for black hole in @xmath27,@xmath11 and @xmath138 + ( @xmath139 ) . [ energiabh2],width=8 ] these integrals are slightly different from the integrals found before but the result for the free energy is very similar . we find ( again ) a transition between a u - type and a box - type solution and a screening length @xmath107 so that the free energy is negative below @xmath107 and zero above . we compare in fig [ energiabh2 ] the potential energies as a function of the distance for @xmath140 and @xmath141 at a fixed horizon position @xmath142 . we see that the energies coincide at small distances and there is a little diminution of the screening length when the dimension grows up . then the screening length will never reach the maximum distance @xmath143 in the big black hole . this means that the corresponding gauge theory in @xmath0 is deconfining at high temperatures for any @xmath24 . for the special case @xmath23 which corresponds to sym theory in @xmath144 our results agree with those obtained before by landsteiner and lopez @xcite where this screening length was defined we calculated the free energy of a quark anti - quark pair in a compact space @xmath0 with the period of @xmath145 being the inverse of the temperature . for the global @xmath3 we find a quark anti - quark free energy that has a coulomb - like behavior and that goes to zero only at the maximum distance @xmath55 in the circle of fig [ fig2 ] . so if the quark is in a fixed position the only way to obtain a zero free energy is to put the anti - quark at a distance @xmath146 . this means that the anti - quark does not have the freedom to move in the @xmath43 coordinate without feeling the potential so this corresponds to a confinement scenario . this scenario is in accordance with the one obtained in @xcite . at high temperatures the static string in the black hole metric leads to a coulomb - like free energy that goes to zero at a screening length @xmath107 . we calculate the dependence of @xmath107 on the horizon position and temperature and show that it only reaches @xmath55 for the small black hole solution which is instable by the hawking - page criteria . for the ( stable ) big black hole we found that @xmath107 is always lower than @xmath55 so considering a quark in a fixed position the anti - quark has some freedom to move in the @xmath43 coordinate with corresponds to a deconfinement scenario . we conclude that the transition of the quark anti - quark free energy in @xmath0 when one goes from @xmath3 to black hole @xmath3 is indeed a confinement / deconfinement transition when included the hawking - page criteria of stability . this confirms the interpretation of the confinement / deconfinement transition in @xmath0 as the dual of the hawking - page transition supported by the entropy jump of the spaces . * acknowledgments * : we would like to thank nelson braga and henrique boschi for important suggestions and comments . we also thank the cbpf for the hospitality . the authors are partially supported by cnpq and claf . h. boschi - filho , n. r. f. braga and c. n. ferreira , phys . d * 73 * , 106006 ( 2006 ) [ erratum - ibid . d * 74 * , 089903 ( 2006 ) ] [ arxiv : hep - th/0512295 ] . s. j. rey , s. theisen and j. t. yee , nucl . b * 527 * , 171 ( 1998 ) [ arxiv : hep - th/9803135 ] . a. brandhuber , n. itzhaki , j. sonnenschein and s. yankielowicz , phys . b * 434 * , 36 ( 1998 ) [ arxiv : hep - th/9803137 ] . h. boschi - filho , n. r. f. braga and c. n. ferreira , phys . d * 74 * , 086001 ( 2006 ) [ arxiv : hep - th/0607038 ] . o. andreev and v. i. zakharov , jhep * 0704 * , 100 ( 2007 ) [ arxiv : hep - ph/0611304 ] . h. dorn and h. j. otto , jhep * 9809 * , 021 ( 1998 ) [ arxiv : hep - th/9807093 ] . s. a. hartnoll and s. prem kumar , phys . d * 74 * , 026001 ( 2006 ) [ arxiv : hep - th/0603190 ] . o. aharony , j. marsano , s. minwalla , k. papadodimas and m. van raamsdonk , phys . rev . d * 71 * , 125018 ( 2005 ) [ arxiv : hep - th/0502149 ] . o. aharony , j. marsano , s. minwalla , k. papadodimas and m. van raamsdonk , adv . * 8 * , 603 ( 2004 ) [ arxiv : hep - th/0310285 ] . d. bak , a. karch and l. g. yaffe , jhep * 0708 * , 049 ( 2007 ) [ arxiv:0705.0994 [ hep - th ] ] .

we investigate the finite temperature quark anti - quark problem in a compact space @xmath0 by considering static strings in global @xmath1 space with @xmath2 . for high temperatures we work in the black hole metric where two possible solutions show up : the big black hole and the small black hole . using the ads / cft correspondence , we calculate the quark anti - quark potential ( free energy ) as a function of the distance . we show that this potential can be intrepeted as confining for the @xmath3 space and deconfining for the big black hole . we find for the small black hole a confining limit for the potential but this solution is instable following the hawking - page criteria . our results for the free energy reinforce the witten interpretation of the confinement / deconfinement transition as the dual of the well - known hawking - page transition .

You are an expert at summarizing long articles. Proceed to summarize the following text: neutron stars are objects of extremely rich internal structure . although their interior structure is still very uncertain , it seems that observations and theoretical studies of neutron stars are quite in agreement concerning the structure of their exterior parts . more specifically , there is agreement that neutron stars , 1 - 2 minutes after their formation , are cold enough to solidify their exteriors and form a crystal crust thanks to coulomb forces between the various atomic nuclei . the crust is covered by a very thin fluid ocean , while the interior is formed by a super - fluid mantle ( up to 5 km in size ) . the composition of the core is highly uncertain . the crystal crust extends from the neutron star s atmosphere @xmath3 km down where densities reach nuclear densities around @xmath4 gr/@xmath5 . the coulomb forces of the crystal ions forming the crust can be described via the shear modulus @xmath6 which is inversely proportional to the 4th power of the ion spacing . since the restoring force is the coulomb force , the periods of the torsional modes will strongly depend on the shear modulus , and its values will characterize the spectrum . up to now , there are only two detailed calculations of the crust equation of state @xcite . both were based on an approximate formulation @xcite leading to a typical value for the shear modulus of @xmath7 @xcite . in the newtonian limit and in the absence of strong magnetic fields , @xcite found that the period of the fundamental torsional modes @xmath8 depends mainly on the radius of the star @xmath9 , the speed of the shear waves @xmath10 and the angular index @xmath11 via the following relation @xmath12^{1/2 } v_s / r \label{eq : sigma}\ ] ] where @xmath13 , and @xmath6 , @xmath14 are the shear modulus and the density , respectively . torsional modes , which are axial - type oscillations , are thought to be more easily excited during a fracturing of the crust since they only involve oscillations of the velocity . the velocity field of torsional oscillations is actually divergence - free without a radial component . torsional modes ( _ t_-modes ) are labeled as @xmath15 , where @xmath11 is the angular index , while the index @xmath16 corresponds to the number of radial nodes in the eigenfunctions of the overtones for a specific @xmath11 . the shear and torsional modes are well studied in newtonian theory , see e.g. @xcite , while there are only a few studies in general relativity @xcite . relativistic effects have been found to increase significantly the fundamental @xmath17 torsional mode period by roughly 30% for a typical @xmath18 , @xmath19 model . the aim of this work is the study of the effect of rotation on torsional modes . up to now this is studied only in newtonian theory in a single study by @xcite , see also @xcite . that study suggested that the frequency of a torsional mode in a rotating star is given by @xmath20 where @xmath21 is the fundamental frequency of the torsional mode of a non - rotating star and @xmath22 is its rotational frequency . since the typical frequency of the torsional modes of non - rotating stars @xmath21 is of the order of 25 - 40 hz , it is obvious that they will be subject to the so called chandrasekhar - friedman - schutz ( cfs ) instability , even for small rotational frequencies @xcite . this is a quite interesting result for rotating neutron stars since the instability ( together with that of the r - modes ) might lead to further fracturing and/or melting of the crust . recently , there is increased interest in the study of torsional modes because of the belief that soft gamma repeaters ( sgrs ) could be magnetars experiencing star - quakes that are connected ( through the intense magnetic field ) to gamma ray flare activity . magnetar star - quakes may be driven by the evolving intense magnetic field which accumulates stress and eventually leads to crust fracturing . there are three sgr events detected up to now . the first event was detected already in 1979 from the source sgr 0526 - 66 @xcite , the second in 1998 from sgr 1900 + 14 @xcite , while the third and most energetic one was observed in december 2004 from the source sgr 1806 - 20 @xcite . analysis of the tail oscillations , for a full discussion see @xcite , revealed the presence of oscillations from a few tenths of hz up to about 2 khz . in an attempt to fit the observed frequencies to the torsional modes of various eos for the core and the crust , @xcite suggested that neutron star models should have stiff equations of state and a mass between 1.6 - 2 @xmath23 . still these results depend critically on the interpretation of the order of each mode . for this , see the discussion in @xcite . in the present work , we derive the perturbation equations for torsional oscillations of rotating stars in general relativity following the approach by @xcite in the cowling approximation which neglects perturbations of the space - time . the article is organized as follows . in the next section , we describe the general - relativistic equations that have been used to describe the background stellar configuration , and we derive the perturbation equations for torsional oscillations of rotating stars in the cowling approximation . in the third section , we describe the numerical techniques used to calculate the modes together with a toy problem . the last is meant to explain the numerical results . the article closes with a summary and discussion . we consider a slowly rotating relativistic star described by the metric @xmath24 where @xmath25 , @xmath26 and @xmath27 ( the frequency of the local frame ) are functions of the radial coordinate @xmath28 . up to first order in @xmath22 , the background 4-velocity of the star is given by @xmath29 \label{bvelocity}\ ] ] where @xmath22 is the angular velocity of the star . the background stellar models are solutions of the tov equations and an equation describing the dragging of inertial frames @xmath30 where @xmath31 . we assume that the star consists of a perfect fluid described by the energy - momentum tensor @xmath32 we also assume that the star is isotropic , therefore , the background shear tensor vanishes . due to the spherical symmetry of the background , the perturbations of the background configuration can be decomposed into spherical harmonics . this leads to a large system of partial differential equations @xcite . here the space - time perturbations are omitted ( cowling approximation ) and also the coupling between spheroidal ( polar ) and toroidal ( axial ) perturbations since they only marginally affect the eigenfrequencies of the torsional modes . under these approximations , the radial component of the perturbed velocity field and the variations of pressure and density will remain unaffected since they are polar perturbations . the perturbation of the 4-velocity @xmath33 is related to the displacement vector @xmath34 through the relation @xmath35 . for an observer co - rotating with the star , this is translated to : @xmath36 in other words , the velocity perturbations are described by the time derivative a function @xmath37 related to the displacement vector @xmath34 , here the dot stands for the temporal derivative . the perturbed energy - momentum tensor , which includes the contribution from shear , is given by @xmath38 where @xmath39 is the shear tensor , @xmath39 is defined by @xmath40 . here @xmath41 is the rate of shear given by the lie derivative of the shear along the world lines @xcite @xmath42 and @xmath43 is the projection tensor @xmath44 the speed of shear waves on the crust depends on the shear modulus @xmath6 , the density @xmath45 and the pressure @xmath46 of the star according to @xmath47 . a typical value of the speed of shear waves is @xmath48 . the perturbation equation for the energy - momentum tensor @xmath49 provides a single equation for the axial perturbations @xmath50 + 2 { \mbox{\rm i}}m \varpi \left[{1 \over \lambda } + v_s^2 \left ( 1 - { 2 \over \lambda } \right ) \right ] \dot{z}\end{aligned}\ ] ] where @xmath51 . in the absence of shear @xmath52 , an inertial observer obtains @xmath53 this gives the well known relation for the r - mode frequency in the newtonian limit ( @xmath54 ) , while in the relativistic case it leads to a continuous spectrum @xcite . in the last case couplings between higher @xmath55 and polar perturbations need to be included for a proper study of the spectrum . in this section , we present an approximate solution of the boundary value problem and describe the numerical techniques used in the calculation of the frequencies of torsional modes of rotating stars . for the numerical estimation of the frequencies , we use two different techniques . the first approach assumes a harmonic time - dependence of the perturbations . this leads on a boundary value problem . the second approach is based on a direct time evolution of equation ( [ eq : t_master ] ) followed by a fourier transform in time of the obtained values at a fixed radial position . we will describe only the first approach . the second approach has only been used for the verification of the results . the fourier transform of equation ( [ eq : t_master ] ) , i.e. , the assumption that @xmath56 , leads to the following differential equation @xmath57 - e^{2\nu } { \lambda - 2\over r^2 } \right\ } z = 0 \ , \ , .\end{aligned}\ ] ] the boundary conditions in the center ( or at the lower end of the crust ) and on the stellar surface are @xmath58 where @xmath59 is the distance of the lower end of the crust from the center . the above system of equations defines an eigenvalue problem for the frequencies of the torsional modes @xmath60 . for its solution , we use two approaches . the first is an approximate analytic solution , and the second is a numerical solution . here we provide an approximate analytic solution to the eigenvalue problem based on bessel functions . in order to be able to use bessel functions and to treat the problem semi - analytically , we need to simplify equation ( [ eque0 ] ) . for this we make the following simplifying assumptions , @xmath61 , @xmath62 , @xmath63 and @xmath64 . this reduces equation ( [ eque0 ] ) to a bessel equation @xmath65 z = 0 \label{eq : toy4}\ ] ] where @xmath66 together with the boundary conditions ( [ eq : bc1 ] ) , this equation leads on a simpler boundary value problem . actually , for high frequencies , the eigenvalues could easily be estimated using the wkb method . on the other hand , since the frequency of the fundamental torsional mode is relatively small , it is unclear whether the wkb approximation is applicable , in particular in the discussion of cfs instability . the last involves the investigation of the limit @xmath67 . the general solution of equation ( [ eq : toy4 ] ) can be given in the form of bessel functions @xmath68 where @xmath69 and @xmath70 are arbitrary constants . since the condition for regularity in the center demands that @xmath71 , the contribution of the bessel functions @xmath72 is excluded since it is divergent for @xmath73 . the second boundary condition demands that @xmath74 which leads to the following transcendental equation @xmath75 where @xmath76 . the roots of this equation can be found by elementary numerical methods . this leads to the following values for @xmath77 @xmath78 the eigenfrequencies can be obtained from the roots of the equation @xmath79 by setting @xmath80 ( the frequency in the absence of rotation ) . in this way , we get an approximate form of the torsional mode frequency @xmath81 in the rotating frame @xmath82 ^ 2 } + \frac{m \omega}{\ell \left(\ell+1\right ) } \ , \label{eq : toy3c}\ ] ] which for an inertial observer ( @xmath83 ) has the form @xmath82 ^ 2 } - \frac{m \omega \left(\ell^2+\ell-1\right)}{\ell \left(\ell+1\right ) } \ , \ , . \label{eq : toy3d}\ ] ] both relations agree with the newtonian results of @xcite , equation ( [ eq : toy3c ] ) and of @xcite equation ( [ eq : toy3d ] ) . note that @xmath84 leads to a well known form of the frequency of the fundamental torsional mode for non - rotating newtonian stars , see equation ( [ eq : sigma ] ) . as function of the rotation ( both in hz ) . the dashed line corresponds to the frequency of the toy model and the continuous line to the frequency of the relativistic model achieved by a direct numerical solution of the eigenvalue problem . the data are for a stellar model with a realistic eos a @xcite for the fluid core and an eos for the crust given by @xcite ( model @xmath85 ) . this stellar model has radius @xmath86 km @xmath87 and crust thickness @xmath88% . , width=377 ] the numerical solution of the eigenvalue problem defined by equations ( [ eque0 ] ) - ( [ eq : bc1 ] ) is solved by a shooting method . its results have been tested by direct numerical evolution of the time dependent equation ( [ eq : t_master ] ) . the numerical results verify the suggestion by @xcite that the torsional modes _ are cfs unstable_. in figure 1 , we plot the frequency of the fundamental torsional mode ( @xmath89 for @xmath90 ) as a function of the stellar rotation frequency @xmath22 . we also show the results derived by the approximate analytic method described earlier . it is clear that there is a difference in the results of the order of 30% . mainly , this is due to the fact that in one case we used the approximate newtonian form of the equation , while in the other case we used its exact relativistic form . it is obvious that the torsional modes become secularly ( cfs ) unstable even for very slowly rotating relativistic stars . in a newtonian study @xcite torsional mode frequencies , measured in a co - rotating reference frame , can be described approximately by the relation ( [ eq : sigma ] ) , where @xmath91 . moreover , an inertial observer will measure @xmath92 the cfs instability sets in when the frequency of the torsional mode for the inertial observer @xmath93 or , equivalently , when the phase velocity of the mode is equal to the rotational frequency @xmath94 , i.e. , when the critical rotation frequency of the star is given by @xmath95 the approximate results derived earlier ( [ eq : toy3c])-([eq : toy3d ] ) verify the newtonian results . also the numerical results agree extremely well on the effect of rotation on the frequencies of the torsional modes , i.e. , @xmath96 with a typical error of the order of 2 - 5% , depending on the compactness of the star . the perturbation equations have been solved for a number of different equations of state for the fluid core and the crust which are listed in table 1 of @xcite . the overall picture is the same , i.e. , shows a 20 - 30% difference of the frequencies of the relativistic and the newtonian equations , and a 2 - 5% difference in the rate of the frequency as function of rotation ( @xmath97 ) . the frequencies plotted in figure 1 are for a stellar model with a realistic eos a @xcite for the fluid core together with an eos for the crust given by @xcite ( model @xmath85 ) . this stellar model has radius @xmath86 km @xmath87 and crust thickness @xmath88% . in this article , we showed by means of numerical and semi - analytic methods that the torsional modes of rotating relativistic stars are subject to the cfs instability as suggested by @xcite for newtonian stars . this instability might be only of academic interest since viscosity works against it , and therefore it will probably never prevail . we would like to emphasize that the cfs instability does not operate in the up to now observed sgrs because they are very slowly rotating stars with periods of the order of seconds . still it is possible that the torsional modes of the newly born neutron stars ( soon after they form a crust ) will be cfs unstable because rotation periods of the order of tenths or hundreds of hz are expected . the instability of torsional modes in the crust and the unstable r - modes in the fluid core might work together in the direction of breaking or melting the crust . if strong magnetic fields are present , then the accumulated stress might enhance the above scenario suggesting that the young neutron stars with strong magnetic fields will probably have a more frequent flare activity . cfs type rotational instabilities of magnetic field modes will also be present in rotating neutron stars , and their effect in the stellar flare activity is part of an extension of this work . finally , rotation ( as well as the presence of strong magnetic fields ) will produce frequency shifts towards both lower and higher frequencies which poses extra difficulties for the identification of the various observed frequencies from sgrs . we are grateful to h. sotani , n. stergioulas , s. yoshida and j.l . friedman for helpful discussions . this work is supported by the greek gsrt programs heracleitus and pythagoras ii and by the german science foundation ( dfg ) , via a sfb / tr7 .

we study the effects of rotation on the torsional modes of oscillating relativistic stars with a solid crust . earlier works in newtonian theory provided estimates of the rotational corrections for the torsional modes and suggested that they should become cfs unstable , even for quite low rotation rates . in this work , we study the effect of rotation in the context of general relativity using elasticity theory and in the slow - rotation approximation . we find that the newtonian picture does not change considerably . the inclusion of relativistic effects leads only to quantitative corrections . the degeneracy of modes for different values of @xmath0 is removed , and modes with @xmath1 are shifted towards zero frequencies and become secularly unstable at stellar rotational frequencies @xmath2 20 - 30 hz . [ firstpage ] relativity methods : numerical stars : neutron stars : oscillations stars : rotation

You are an expert at summarizing long articles. Proceed to summarize the following text: the inner few degrees of the galactic center show a large concentration of molecular , atomic hydrogen and dust clouds ( pierce - price et al . 2000 ; lang et al . 2010 ; molinari et al . 2011 ) . the molecular gas toward the galactic center is considered to reside in the so - called central molecular zone ( cmz ) and consists of a mixture of diffuse and dense components ( morris and serabyn 1996 ; martin et al . 2004 ; sawada et al . 2004 ; oka et al . 2005 ; yusef - zadeh et al . radio continuum emission from this region is also extended and is produced by a mixture of thermal and nonthermal processes ( nord et al . 2004 ; yusef - zadeh et al . 2004 ; law et al . 2008 ) . given the confusing region of the inner galaxy due to large number of foreground and background sources along the line of sight as well as the complex motion of the gas in the galactic center , it is difficult to use the kinematic distance method to identify molecular clouds associated with hii regions and supernova remnants . we describe a new technique to identify neutral clouds that show a deficiency in the distribution of radio continuum emission . these clouds are embedded in a bath of radiation or cosmic ray particles produced by thermal or nonthermal sources , respectively . the strong radiation field in the environment of cloud complexes with high column densities , such as infrared dark clouds ( irdcs ) , allow us to identify their dark counterparts in radio continuum images at cm and mm wavelengths . the origin of radio dark clouds ( rdcs ) is unlike the x - ray shadowing and irdcs which are caused by strong absorption of background light by dense clouds ( egan et al . 1998 ; andersen et al . rdcs are also unlike optically thick hii regions seen in absorption against the strong background nonthermal emission at low frequencies . this effect is due to a free - free absorption coefficient which increases at low frequencies as @xmath2 ( nord et al . the origin of the deficiency in radio continuum emission at high frequencies is due to the high column of embedded molecular gas that does not allow an external radiation field or cosmic ray particles to penetrate through the cloud . this implies that this subset of molecular clouds is interacting with its surrounding hot medium . we first demonstrate the physical situation in which radio dark clouds are produced followed by five examples demonstrating the reality of radio dark clouds . since background radio continuum radiation is transparent when passing through neutral clouds , one would expect uniform background emission across the face of neutral clouds . however , if a cloud is surrounded by hot synchrotron or thermal emitting plasma , the continuum emission is depressed due to the shorter path length of the continuum emission integrated along the line of sight toward the center of the molecular cloud . one possibility involves external heating and ionization of neutral gas by ultraviolet continuum radiation that falls off rapidly from the edge to the center of the cloud with where the visual extinction ( a@xmath3 ) is much larger than one magnitude . thus , free - free radio emission is substantially reduced in clouds with high column densities and their imprint can be identified as dark features in radio continuum images . in other words , atomic or molecular gas clouds or dust clouds suppress the continuum emission and create the appearance of a `` hole '' in their distribution . it is expected that neutral clouds embedded in a hot plasma are edge brightened outlining the boundary of the cloud , thus can be distinguished from a cavity devoid of gas . the presence of spectral line and/or continuum dust emission can also distinguish rdcs . we consider a cloud with a diameter @xmath4 located at a distance d from us and is embedded within an ionized medium characterized to have electron density @xmath5 ( @xmath6 ) and with the emission measure @xmath7 @xmath8 along a path length @xmath9 . the surface brightness toward the center and away from the cloud are defined as s@xmath10 and s@xmath11 , respectively , at a given frequency @xmath12 . the flux deficiency @xmath13 is the difference between the flux density of the ambient gas toward and away from the cloud . the ratio of the diameter of the cloud to the path length @xmath9 is @xmath14 the differential emission measure @xmath15em between the cloud center and the ionized medium can be estimated by @xmath16 where t@xmath17 is the electron temperature in k , @xmath18 is the beam size in arcsecond and @xmath19 is the flux deficiency in mjy . if the electron density of the ionized medium is measured independently , then the depth of the molecular cloud @xmath4 along the line of sight can be estimated . another possibility that could produce rdcs is the deflection of non - thermal particles as they diffuse inside a molecular cloud . in this case , the path length over which nonthermal particles travel are limited by the magnetic field geometry of the cloud which could shield the electrons penetrating into the cloud . the ionization losses of nonthermal particles could also suppress the emission from high energy particles as they interact with the gas . thus , the flux of nonthermal emission at high frequencies is expected to be reduced with respect to the background nonthermal emission . if we assume that the magnetic field is in equipartition with the particles , the ratio of the magnetic field in the diffuse medium to the molecular cloud is @xmath20 where @xmath21 and @xmath22 are the magnetic fields in the ambient medium and dense cloud , respectively . the spectral index of the emission @xmath23 , where s@xmath24 , is assumed to be constant . multi - wavelength images presented here are based on observations that have already been described elsewhere . the data that we have used are taken by mopra telescope ( jones et al . 2011 ) , green bank telescope ( gbt ) of the national radio astronomy observatory ( nrao ) , ( law et al . 2008 ) , very large array ( vla ) ( yusef - zadeh et al . 2004 ) , antarctic submillimeter telescope and remote observatory ( ast / ro ) ( martin et al . 2004 ) , irac on spitzer space telescope ( arendt et al . 2008 ) , scuba on james clark maxwell telescope ( pierce - price et al . 2000 ) , nobeyama radio observatory ( nro ) ( tsuboi et al . 2011 ) , and nicmos of hubble space telscope ( hst ) ( yusef - zadeh et al . 2001 ) . we present below multi - wavelength observations of five sources toward the inner galaxy . * g359.750.13 : * figure 1a shows the distribution of velocity integrated hcn ( 1 - 0 ) line emission from g359.75 - 0.13 which runs for @xmath25 parallel to the galactic plane ( jones et al . this cloud is part of a layer of molecular gas associated with the central molecular zone . this elongated cloud is detected at mid - ir and submm images as an irdc ( molinari et al . 2011 ; arendt et al . 2008 ; yusef - zadeh et al . figure 1b shows the radio continuum counterpart to the molecular cloud at 3.5 cm . a dearth of emission coincides with the cloud tracing hcn line emission . figure 1c shows a composite color image of hcn ( 1 - 0 ) emission and 3.5 cm continuum image . radio continuum emission is present to the north and south of the elongated molecular gas layer . to determine the anti - correlation between radio continuum and molecular line emission , cross cuts along a line drawn on fig . 1a are made across the hcn and 3.5 cm radio continuum images and are presented in figure 1d . the deficiency in the flux density of radio continuum emission is @xmath260.1 jy implying that the depth of the ionized gas and molecular gas is similar to each other . these images demonstrate a clear anti - correlation in the distribution of molecular gas that is detected as an irdc and radio dark cloud . the cross cuts combined with images of this cloud at radio and millimeter wavelengths suggest that the ionized and molecular gas in g359.750.13 are in the same environment , most likely in the galactic center region . this is because diffuse radio continuum emission from the galactic center is much stronger than in the galactic disk . * g0.13 - 0.13 and the radio arc : * adjacent to the nonthermal filaments of the arc near galactic longitude @xmath27 lies the molecular cloud g0.13 - 0.13 we compare the distribution of radio continuum and molecular line emission from g0.13 - 0.13 . figure 2a - b show the distributions of a 20 cm continuum emission and integrated intensity of hcn ( 1 - 0 ) line , respectively . the vertical filaments associated with the radio arc are known to be magnetized structures running perpendicular to the galactic plane . a 3.5 cm image of the same region mapped by the gbt shows similar morphology to that of the 20 cm continuum image . using hcn line intensity and 3.5 cm continuum images , figure 2c shows cross cuts , along a line drawn on fig . 2b , demonstrating the depression in the continuum flux across the length of the cloud . the dip at the location of g0.13 - 0.13 corresponds to @xmath28 mjy at 3.5 cm . the molecular cloud g0.13 - 0.13 is surrounded by vertical nonthermal filaments of the arc to the east ( yusef - zadeh et al . 1987 ) and to the west of g0.13 - 0.13 ( reich 2003 ) . the images and cross cuts made from hcn line and radio continuum data show anti - correlation between molecular line and radio continuum distributions . the largest deficiency of radio continuum emission is located where molecular line emission peaks . these intensity profiles imply the interaction of nonthermal radio filaments and g0.13 - 0.13 and are consistent with earlier studies measurements ( tsuboi et al . 1997 ) . the flux of the continuum emission at 3.5 cm is reduced by a factor 2 where the molecular cloud g0.13 - 0.13 is located . this implies that the region of nonthermal emission surrounding rdc is twice the depth of the molecular cloud . * g359.160.04 and the snake filament : * one of the most prominent nonthermal filamentary structure in the galactic center region is the snake filament which extends for more than 20@xmath29 and runs almost perpendicular to the galactic plane ( e.g. , gray et al . figure 3a shows the northern half of this striking filament terminating at the radio continuum source g359.16 - 0.04 ( uchida et al . this 6 cm continuum image shows two dark features rdc-1 and rdc-2 to the e. and w. edges of the extended continuum source giving the appearance of a butterfly " . rdc-2 lies to the north of the well - known radio jet 1e1740 - 2942 ( mirabel et al . 1992 ) . given that the 6 cm data is produced by the vla and that incomplete _ uv _ coverage may be responsible for producing dark features , figure 3b shows a grayscale 3.5 cm continuum emission mapped by the gbt . we note that the dark features are also identified in single dish observations , thus establishing the reality of rdcs in interferometric images . contours of integrated emission of co ( 4 - 3 ) line are superimposed on the 6 cm continuum image , as shown in figure 3c . given the mismatch between the 2@xmath29 resolution of molecular line data based on ast / ro observations and radio continuum data , the distribution of molecular gas is similar to that of dark radio clouds . the continuum image reveals detailed morphological structures of a giant molecular cloud on arcsecond spatial resolution especially at the boundary of the molecular cloud near the continuum source . we also made cross cuts at b=@xmath30 across the 6 cm and molecular line images , as shown on figure 3d . the cross cuts show two peaks of molecular line emission coinciding with two dips in the 6 cm image with deficient radio flux of roughly 400 and 300 mjy . these dips correspond to rdc-1 and rdc-2 , and imply that the diameters of rdc-1 and rdc-2 are roughly 4/5 and 3/5 of the depth of the ionized medium surrounding the cloud . * g0.6 - 0.0 sgr b2 : * the sgr b2 cloud is a well - studied giant molecular cloud which lies near the galactic center ( reid et al . 2009 ) and is part of a continuous dust ridge that is viewed in absorption at mid - ir wavelengths ( lis & carlstrom 1994 ; molinari et al . figure 4a shows the irdc associated with sgr b2 to the n. and ne of the cluster of stars and the nebula . figure 4b , c show the distribution of 20 and 6 cm continuum emission from the same region , respectively . the brightness of the 20 cm continuum emission is saturated to bring out an elongated dark feature near b@xmath31 . radio continuum images show rdc-1 which appears to coincide with the southern edge of irdc that surrounds the cluster of uc hii regions in sgr b2 ( de pree et al . we also note a decrease in the radio flux at 6 and 20 cm between sgr b2 and the isolated radio continuum feature g0.73 - 0.10 . however , the image quality in this region is poor , thus may show artifacts . cross cut plots along a line drawn on fig . 4c across the rdc-1 are made using the 6 cm continuum and 450@xmath32 m images . the profiles of emission , as shown in figure 4d , show flux deficiencies of 5 and 2 mjy which imply that the depths of the ionized gas for the two dips are similar to and twice larger than those of molecular gas , respectively . because of the large number of bright compact hii regions in sgr b2 , the cross cuts also show fluctuations due to emission from individual compact sources . * sgr a east snr & sgr a west : * given its gaseous environment with a strong radiation field , the inner ten parsecs of the galactic center is an excellent site to search for rdcs tracing molecular clouds that interact with the strong radiation field or with expanding supernova remnants . the strong radiation field in the galactic center produces a thick layer of ionized gas at the edge of dark clouds . one of the most prominent giant molecular clouds in the galactic center is the 50 g0.02 - 0.07 . this cloud is physically interacting with the expanding shell of the sgr a east snr g0.0 + 0.0 which is thought to lie near the galactic center ( e.g. , tsuboi et al . this complex region is also the site of young massive star formation as a chain of compact hii regions that lie to the east of the sgr a east remnant . figures 5a - b show the inner 4@xmath29 of the galactic center at 6 cm hosting the shell - type snr g0.0 - 0.0 and four compact hii regions a - d at its eastern boundary ( zylka et al . the 1.3 mm continuum distribution traces dust emission from the 50 molecular cloud . we note several dark features rdc-1 to 4 surrounding the snr shell of figure 5a , all of which coincide with dust and molecular features ( tsuboi et al . the largest scale dark cloud rdc-1 is noted to the ne of the chain of hii regions where we note an oval - shaped structure with an extent of 2@xmath33.1 . figure 5c shows contours of integrated emission of sio ( 2 - 1 ) superimposed on the 6 cm image , supporting the suggestion that rdc-1 coincides with an oval - shaped molecular gas . another chain of hii regions to the ne of rdc-1 near @xmath34 might be associated with a second site of star formation in this cloud . to estimate the level at which the continuum emission is suppressed , figure 5d shows cross cuts of rdc-3 made at 1.3 mm and 6 cm , supporting a clear anti - correlation between dust and radio continuum emission . the largest deficient flux at 6 cm is 25 mjy implying that the size of the hot plasma is twice the size of the molecular cloud . on a smaller scale , the inner pc of the galactic center hosts diffuse ionized gas sgr a west orbiting the massive black hole ( e.g. , ferriere et al . sgr a west consists of three arms of ionized gas and is surrounded by the circumnuclear molecular ring ( cmr ) . figure 5e , f show a 3.5 cm and 1.87@xmath32 m continuum images of the eastern half of sgr a west whereas figures 5 g show molecular h2 1 - 0 s(1 ) counterparts ( yusef - zadeh et al . several dark features are noted in the 3.5 cm continuum image , especially in the region between the n. and e. arms . it turns out rdc-1 and rdc-2 lying between the two arms coincide with an extinction feature , as revealed in figure 5h where a drop in stellar density is noted . cross cuts across the 3.5 cm and h2 images show that the h2 emission coincides with a dip in the continuum peak at 3.5 cm . the deficient flux at 3.5 cm suggests that the depth of the ionizing gas is similar to that of the molecular gas . from equation ( 2 ) , the depth of the cloud is estimated to be 0.11 pc if @xmath35 @xmath6 and t@xmath36k . the association of a `` tongue '' of neutral gas ( rdc-3 ) , an extinction cloud with the n. arm of sgr a west ( jackson et al . 1993 ; yusef - zadeh et al . 2001 ) , as well as the association of the e. arm with neutral gas suggest that there is considerable molecular gas in the ring and that the arms of sgr a west trace the ionized surface of neutral clouds . if these clouds are massive , the presence of molecular gas inside the cmr can have important consequences in the formation of stars and the dynamics of stars near sgr a*. * conclusions : * we have presented five examples of dark features in radio continuum images of molecular complexes toward the inner galaxy . we illustrated that these dark features are anti - correlated with molecular line and dust emission , thus implying that cold gas associated with radio continuum features can be detected in radio images . given the new generation of radio telescopes with their broad band capability , future continuum measurements could potentially be effective in identifying cold gas clouds in star forming sites in the local and distant universe . some of the dark radio clouds are surrounded by nonthermal radio emission , thus , radio continuum imaging can identify the interaction of nonthermal particles from radio jets or jet - driven outflows or supernova remnants with the cold ism or the igm . lastly , rdcs can potentially be used as off positions in total power technique of observations with radio telescopes . acknowledgments : i am grateful to m. wardle , d. roberts , r. arendt , w. cotton , m. royster , m. tsuboi and other colleagues for discussions and for providing me with their data over the years . this work is partially supported by the grant ast-0807400 from the nsf . . _ ( b ) middle - _ similar to ( a ) except showing the distribution of radio continuum emission at 3.5 cm based on gbt observation with a spatial resolution of 88@xmath37 . _ ( c ) bottom - _ a color composite image showing ( a ) and ( b ) in red and green , respectively . _ ( d ) right - _ a cross cut , as drawn on ( a ) , made at constant l=16@xmath38 showing the intensity profiles of hcn line emission in red and and of 3.5 cm continuum data in blue . , title="fig : " ] _ ( b ) middle - _ similar to ( a ) except showing the distribution of radio continuum emission at 3.5 cm based on gbt observation with a spatial resolution of 88@xmath37 . _ ( c ) bottom - _ a color composite image showing ( a ) and ( b ) in red and green , respectively . _ ( d ) right - _ a cross cut , as drawn on ( a ) , made at constant l=16@xmath38 showing the intensity profiles of hcn line emission in red and and of 3.5 cm continuum data in blue . , title="fig : " ] . _ ( b ) middle - _ similar to ( a ) except showing the distribution of radio continuum emission at 3.5 cm based on gbt observation with a spatial resolution of 88@xmath37 . _ ( c ) bottom - _ a color composite image showing ( a ) and ( b ) in red and green , respectively . _ ( d ) right - _ a cross cut , as drawn on ( a ) , made at constant l=16@xmath38 showing the intensity profiles of hcn line emission in red and and of 3.5 cm continuum data in blue . , title="fig : " ] . _ ( b ) middle - _ similar to ( a ) except showing the distribution of radio continuum emission at 3.5 cm based on gbt observation with a spatial resolution of 88@xmath37 . _ ( c ) bottom - _ a color composite image showing ( a ) and ( b ) in red and green , respectively . _ ( d ) right - _ a cross cut , as drawn on ( a ) , made at constant l=16@xmath38 showing the intensity profiles of hcn line emission in red and and of 3.5 cm continuum data in blue . , title="fig : " ] . _ ( b ) top right - _ the integrated emission of the hcn line over velocities between 0 and 50 with a spatial resolution of 39@xmath37 ( jones et al . _ ( c ) bottom left - _ cross cuts aslong a line drawn on ( b ) are made at constant l@xmath39 showing the profiles of hcn line emission in red and of 3.5 cm continuum emission based on gbt observation in blue . , title="fig : " ] . _ ( b ) top right - _ the integrated emission of the hcn line over velocities between 0 and 50 with a spatial resolution of 39@xmath37 ( jones et al . _ ( c ) bottom left - _ cross cuts aslong a line drawn on ( b ) are made at constant l@xmath39 showing the profiles of hcn line emission in red and of 3.5 cm continuum emission based on gbt observation in blue . , title="fig : " ] and rms noise 10@xmath32jy . _ ( b ) bottom left - _ similar to ( a ) except that a 3.5 cm continuum image taken with the gbt with a spatial resolution of 89@xmath37 . _ ( c ) top right - _ a weighted average contours of co ( 4 - 3 ) line emission integrated between -150.5 and -140.5 with a resolution of @xmath40 superimposed on ( a ) . _ ( d ) bottom - _ cross cuts along a line drawn on ( a ) at b=@xmath30 show two co peaks and two 6 cm continuum dips , corresponding to rdc-1 and rdc-2 . the vla image at 6 cm is primary beam corrected . , title="fig : " ] and rms noise 10@xmath32jy . _ ( b ) bottom left - _ similar to ( a ) except that a 3.5 cm continuum image taken with the gbt with a spatial resolution of 89@xmath37 . _ ( c ) top right - _ a weighted average contours of co ( 4 - 3 ) line emission integrated between -150.5 and -140.5 with a resolution of @xmath40 superimposed on ( a ) . _ ( d ) bottom - _ cross cuts along a line drawn on ( a ) at b=@xmath30 show two co peaks and two 6 cm continuum dips , corresponding to rdc-1 and rdc-2 . the vla image at 6 cm is primary beam corrected . , title="fig : " ] and rms noise 10@xmath32jy . _ ( b ) bottom left - _ similar to ( a ) except that a 3.5 cm continuum image taken with the gbt with a spatial resolution of 89@xmath37 . _ ( c ) top right - _ a weighted average contours of co ( 4 - 3 ) line emission integrated between -150.5 and -140.5 with a resolution of @xmath40 superimposed on ( a ) . _ ( d ) bottom - _ cross cuts along a line drawn on ( a ) at b=@xmath30 show two co peaks and two 6 cm continuum dips , corresponding to rdc-1 and rdc-2 . the vla image at 6 cm is primary beam corrected . , title="fig : " ] m with a resolution of @xmath41 _ ( b ) top right - _ a continuum image of sgr b2 at 20 cm based on combining data taken from the c and d configurations of the vla and the gbt before the image was convolved to a resolution of 30@xmath37 . _ ( c ) bottom left - _ similar to ( b ) ( both vla and gbt data combined ) but at a resolution of 10.8@xmath42 _ ( d ) bottom right - _ cross cuts centered at @xmath43 along a line drawn on ( c ) show profiles of emission at 450@xmath32 m and 6 cm in blue and red , respectively . ] , stellar subtracted h2 1 - 0 s(1 ) emission at 2.12@xmath32 m , an image of 1.87@xmath32 m emission from stars and ionized gas , the cross cuts along the line drawn on ( f ) show the emission profiles of 3.5 cm and molecular h2 line . , title="fig : " ] , stellar subtracted h2 1 - 0 s(1 ) emission at 2.12@xmath32 m , an image of 1.87@xmath32 m emission from stars and ionized gas , the cross cuts along the line drawn on ( f ) show the emission profiles of 3.5 cm and molecular h2 line . , title="fig : " ]

we show radio continuum images of several molecular complexes in the inner galaxy and report the presence of dark features that coincide with dense molecular clouds . unlike infrared dark clouds , these features which we call `` radio dark clouds '' are produced by a deficiency in radio continuum emission from molecular clouds that are embedded in a bath of uv radiation field or synchrotron emitting cosmic ray particles . the contribution of the continuum emission along different pathlengths results in dark features that trace embedded molecular clouds . the new technique of identifying cold clouds can place constraints on the depth and the magnetic field of molecular clouds when compared to those of the surrounding hot plasma radiating at radio wavelengths . the study of five molecular complexes in the inner galaxy , sgr a , sgr b2 , radio arc , the snake filament and g359.75 - 0.13 demonstrate an anti correlation between the distributions of radio continuum and molecular line and dust emission . radio dark clouds are identified in gbt maps and vla images taken with uniform sampling of _ uv _ coverage . the level at which the continuum flux is suppressed in these sources suggests that the depth of the molecular cloud is similar to the size of the continuum emission within a factor of two . these examples suggest that high resolution , high dynamic range continuum images can be powerful probes of interacting molecular clouds with massive stars and supernova remnants in regions where the kinematic distance estimates are ambiguous as well as in the nuclei of active galaxies . & pdflatex # 1 ( g10 ^ 3 g_0 ) ^#1 # 1 10^#1 km s@xmath0 # 1 10^#1 # 1 # 2 ^#2 u#1 @xmath1 i v

You are an expert at summarizing long articles. Proceed to summarize the following text: observations of spatial patterns at various length scales frequently are the only point where the physical world meets theoretical models . in many cases these patterns consist of a number of comparable objects distributed in space such as pores in a sandstone , or craters on the surface of a planet . another example is given in figure [ fig : kerscher_galaxies - circles ] , where we display the galaxy distribution as traced by a recent galaxy catalogue . the galaxies are represented as circles centered at their positions , whereas the size of the circles mirrors the luminosity of a galaxy . in order to test to which extent theoretical predictions fit the empirically found structures of that type , one has to rely on quantitative measures describing the physical information . since theoretical models mostly do not try to explain the structures individually , but rather predict some of their generic properties , one has to adopt a _ statistical point of view _ and to interpret the data as a realization of a random process . in a first step one often confines oneself to the spatial distribution of the objects constituting the patterns and investigates their clustering thereby thinking of it as a realization of a _ point process_. assuming that perspective , however , one neglects a possible linkage between the spatial clustering and the intrinsic properties of the objects . for instance , there are strong indications that the clustering of galaxies depends on their luminosity as well as on their morphological type . considering figure [ fig : kerscher_galaxies - circles ] , one might infer that luminous galaxies are more strongly correlated than dim ones . effects like that are referred to as _ mark segregation _ and provide insight into the generation and interactions of , e.g. , galaxies or other objects under consideration . the appropriate statistical framework to describe the relation between the spatial distribution of physical objects and their inner properties are _ marked point processes _ , where discrete , scalar- , or vector - valued marks are attached to the random points . + in this contribution we outline how to describe marked point processes ; along that line we discuss two notions of independence ( section [ sec : kerscher_basic ] ) and define corresponding statistics that allow us to quantify possible dependencies . after having shown that some empirical data sets show significant signals of mark segregation ( section[sec : kerscher_data ] ) , we turn to analytical models , both motivated by mathematical and physical considerations ( section [ sec : kerscher_models ] ) . + contact distribution functions as presented in the contribution by d. hug et al . in this volume are an alternative technique to measure and statistically quantify distances which finally can be used to relate physical properties to spatial structures . mark correlation functions are useful to quantify molecular orientations in liquid crystals ( see the contribution by f. schmid and n. h. phuong in this volume ) or in self - assembling amphiphilic systems ( see the contribution by u. s. schwarz and g. gompper in this volume ) . but also to study anisotropies in composite or porous materials , which are essential for elastic and transport properties ( see the contributions by d. jeulin , c. arns et al . and vogel in this volume ) , mark correlations may be relevant . the empirical data the positions @xmath1 of some objects together with their intrinsic properties @xmath2 are interpreted as a realization of a marked point process @xmath3 ( stoyan , kendall and mecke , 1995 ) . for simplicity we restrict ourselves to homogeneous and isotropic processes . + the hierarchy of joint probability densities provides a suitable tool to describe the stochastic properties of a marked point process . thus , let @xmath4 denote the probability density of finding a point at @xmath5 with a mark @xmath6 . for a homogeneous process this splits into @xmath7 where @xmath8 denotes the mean number density of points in space and @xmath9 is the probability density of finding the mark @xmath6 on an arbitrary point . later on we need moments of this mark distribution ; for real - valued marks the @xmath10th - moment of the mark - distribution is defined as @xmath11 the mark variance is @xmath12 . + accordingly , @xmath13 quantifies the probability density to find two points at @xmath14 and @xmath15 with marks @xmath16 and @xmath17 , respectively ( for second - order theory of marked point processes see @xcite ) . it effectively depends only on @xmath16 , @xmath17 , and the pair separation @xmath18 for a homogeneous and isotropic process . two - point properties certainly are the simplest non - trivial quantities for homogeneous random processes , but it may be necessary to move on to higher correlations in order to discriminate between certain models . in the following we will discuss two notions of independence , which may arise for marked point patterns . for this , consider two renaissance families , call them the sforza and the gonzaga . they used to build castles spread out more or less homogeneously over italy . in order to describe this example in terms of a marked point process , we consider the locations of the castles as points on a map of italy , and treat a castle s owner as a discrete mark , @xmath19 and @xmath20 , respectively . there are many ways how the castles can be built and related to each other . [ [ independent - sub - point - processes ] ] independent sub - point processes : + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + for example , the sforza may build their castles regardless of the gonzaga castles . in that case the probability of finding a sforza castle at @xmath14 and a gonzaga castle at @xmath15 factorizes into two one - point probabilities and we can think of the sforza and the gonzaga castles as uncorrelated sub - point processes . in the language of marked point processes this means , e.g. , that @xmath21 for any @xmath22 . if all the joint @xmath23-point densities factorize into a product of @xmath24-point densities of one type each , then we speak of _ independent sub - point processes_. dependent sub - point processes indicate _ interactions _ between points of different marks ; for instance , the gonzaga may build their castles close to the sforza ones in order to avoid that a region becomes dominated by the other family s castles . [ [ mark - independent - clustering ] ] mark - independent clustering : + + + + + + + + + + + + + + + + + + + + + + + + + + + + a second type of independence refers to the question whether the different families have different styles to plan their castles . for instance , the gonzaga may distribute their castles in a grid - like manner over italy , whereas the sforza may incline to build a second castle close to each castle they own . rather than asking whether two sub - point processes ( namely the gonzaga and the sforza castles , respectively ) are independent ( `` independent sub - point processes '' ) , we are now discussing whether they are _ different _ as regards their statistical clustering properties . any such difference means that the clustering _ depends _ on the intrinsic mark of a point . + whenever the two - point probability density of finding two objects at @xmath14 and @xmath15 depends on the objects intrinsic properties we speak of _ mark - dependent clustering_. it is useful to rephrase this statement by using bayes theorem and the conditional mark probability density @xmath25 in case the spatial product density @xmath26 does not vanish . @xmath27 is the probability density of finding the marks @xmath16 and @xmath17 on objects located at @xmath14 and @xmath15 , given that there are objects at these points . clearly , @xmath27 depends only on the pair separation @xmath28 for homogeneous and isotropic point processes . we speak of _ mark - independent _ clustering , if @xmath29 factorizes @xmath30 and thus does not depend on the pair separation . that means that regarding their marks , pairs with a separation @xmath31 are not different from any other pairs . on the contrary , mark - dependent clustering or _ mark segregation _ implies that the marks on certain pairs show deviations from the global mark distribution . + in order to distinguish between both sorts of independencies , let us consider the case where we are given a map of italy only showing the gonzaga castles . if the distribution of castles in italy can be understood as consisting of independent sub - point processes , we can not infer anything about the sforza castles from the gonzaga ones . however , if @xmath32 , sforza castles are likely to be found close to gonzaga ones . here , @xmath33 and @xmath34 are the probabilities that a castle belongs to the sforza or gonzaga family . if , on the other hand , mark - independent clustering applies , typical clustering properties such as the spatial clustering strength are equal for both castle distributions , and the gonzaga castles are in the statistical sense already representative of the whole castle distribution in italy . that means in particular that , if the gonzaga castles are clustered , so are the sforza ones . before we turn to applications , we have to develop practical test quantities in order to test for segregation effects in real data and to describe them in more detail . to investigate correlations between sub - point processes , suitably extended nearest neighbor distribution functions or @xmath35-functions have been employed @xcite . also the ( conditional ) cross - correlation functions can be used ( see eq . [ eq : kerscher_cond - crosscorr ] ) , for a further test see @xcite , p. 302 . here we consider a multivariate extension of the @xmath36-function @xcite , as suggested by @xcite . + for this , consider the nearest neighbor s distance distribution from an object with mark @xmath2 to other objects with mark @xmath37 , @xmath38 ( `` @xmath39 to @xmath40 '' , for details see @xcite ) . let @xmath41 denote the distribution of the nearest neighbor s distance from an object of type @xmath39 to any other object ( denoted by @xmath42 ) . finally , @xmath43 is the nearest neighbor distribution of all points . similar extensions of the empty space function are possible , too . let @xmath44 denote the distribution of the nearest @xmath39-object s distance from an arbitrary position , whereas @xmath45 is the nearest object s distance distribution from a random point in space to any object in the sample . we consider the following quantities : @xmath46 they are defined whenever @xmath47 . if two sub - point processes , defined by marks @xmath48 , are independent then one gets @xcite @xmath49 note , that the @xmath50 depend on higher - order correlations functions , similar to the @xmath36-function @xcite . suitable estimators for these @xmath36-functions are derived from estimators of the @xmath51 and @xmath20-functions @xcite . in order to quantify the mark - dependent clustering or to look for the mark segregation , it proves useful to integrate the conditional probability density @xmath29 over the marks weighting with a test function @xmath52 @xcite . this procedure reduces the number of variables and leaves us with the weighted pair average : @xmath53 the choice of an appropriate weight - function depends on whether the marks are non - quantitative labels or continuous physical quantities . 1 . for labels only combinations of indicator - functions are possible , the integral degenerates into a sum over the labels . supposed the marks of our objects belong to classes labelled with @xmath54 , the conditional cross - correlation functions are given by @xmath55 with the kronecker @xmath56 for @xmath57 and zero otherwise . mark segregation is indicated by @xmath58 for @xmath59 and @xmath60 , where @xmath61 denotes the number density of points with label @xmath39 . the @xmath62 are cross - correlation functions under the _ condition _ that two points are separated by a distance of @xmath31 ( compare @xcite , p. 264 , for applications see the martian crater distribution studied in sect . [ sec : kerscher_martian ] and figure [ fig : kerscher_mars - markcorr ] in particular ) . 2 . for positive real - valued marks @xmath6 , the following pair averages prove to be powerful and distinctive @xcite : + 1 . one of the most simplest weights to be used is the mean mark : @xmath63 quantifies the deviation of the mean mark on pairs with separation @xmath31 from the overall mean mark @xmath64 . a @xmath65 indicates mark segregation for point pairs with a separation @xmath31 , specifically their mean mark is then larger than the overall mark average . + closely related is stoyan s @xmath66 function using the squared geometric mean of the marks as a weight @xcite @xmath67 2 . accordingly , higher moments of the marks may be used to quantify mark segregation , like the mark fluctuations @xmath68 or the mark - variogram @xcite : @xmath69 3 . the mark covariance @xcite is @xmath70 mark segregation can be detected by looking whether @xmath71 differs from zero . a @xmath71 larger than zero , e.g. , indicates that points with separation @xmath31 tend to have similar marks . sometimes the mark covariance is normalized by the fluctuations @xcite : @xmath72 . + these conditional mark correlation functions can be calculated from only three independent pair averages @xcite : @xmath73 , @xmath74 , and @xmath75 . thus the above mentioned characteristics are not independent , e.g.@xmath76 . + we apply these mark correlation functions to the galaxy distribution in section [ sec : kerscher_gal ] ( figure [ fig : kerscher_ssrs - lum ] ) , to martian craters in section [ sec : kerscher_martian ] ( figure [ fig : kerscher_mars - markcorr ] ) and to pores in sandstones considered in section [ sec : kerscher_sandstone ] . 3 . also vector - valued information @xmath77 , describing , e.g. , the orientation of an anisotropic object at position @xmath1 may be available . it is therefore interesting to consider vector marks such as done by @xcite who use a mark correlation function to quantify the alignment of vector marks . here we suggest three mark correlation functions quantifying geometrically different possibilities of an alignment . in order to ensure coordinate - independence of our descriptors , we focus on scalar combinations of the vector marks in using the scalar product @xmath78 and the cross product @xmath79 . different from the case of scalar marks , it is a non - trivial task to find a set of vector - mark correlation functions which contain all possible information ( at least up to a fixed order in mark space ) . we provide a systematic account of how to construct suitable vector - mark correlation functions in a complete and unique way for general dimensions in the appendix . + here we only cite the most important results . for that we need the distance vector between two points , @xmath80 , the normalized distance vector , @xmath81 , and the normalized vector mark : @xmath82 with @xmath83 . the following conditional mark correlation functions will be used to quantify alignment effects : 1 . @xmath84 quantifies the @xmath85lignment of the two vector marks @xmath86 and @xmath87 : @xmath88 it is proportional to the cosine of the angle between @xmath86 and @xmath87 . we normalize with the mean @xmath89 . for purely independent vector marks @xmath84 is zero , whereas @xmath90 means that the marks of pairs separated by @xmath31 tend to align parallel to each other . in some applications , e.g. for the orientations of ellipsoidal objects , the vector mark is only defined up to a sign , i.e. @xmath91 and @xmath92 mean actually the same . in this case the absolute value of the scalar product is useful : @xmath93 for uncorrelated random vectors we get @xmath94 . @xmath85 and @xmath95 can readily be generalized to any dimension @xmath96 , where we expect @xmath97 @xmath98 for uncorrelated random orientations . in two dimensions @xmath95 is proportional to @xmath99 as defined by @xcite . @xmath100 quantifies the @xmath101ilamentary alignment of the vectors @xmath86 and @xmath87 with respect to the line connecting both halo positions : @xmath102 @xmath100 is proportional to the cosine of the angle between @xmath86 and the distance vector @xmath103 connecting the points . for uncorrelated random vector marks , we expect again @xmath104 ; @xmath100 becomes larger than that , whenever the vector marks of the objects tend to point to objects separated by @xmath31 an example is provided by rod - like metallic grains in an electric field : they concentrate along the field lines and orient themselves parallel to the field lines . 3 . @xmath105 quantifies the @xmath106lanar alignment of the vectors and the distance vector . @xmath105 is proportional to the volume of the rhomb defined by @xmath86 , @xmath87 and @xmath103 : @xmath107 quite obviously , this quantity can not be generalized to arbitrary dimensions ; the deeper reason for that will become clear in the appendix . we get @xmath108 for randomly oriented vectors , whereas it is becoming larger for the case that @xmath87 is perpendicular to @xmath86 as well as to @xmath103 . + applications of vector marks can be found in section [ sec : kerscher_halos ] ( figure [ fig : kerscher_halos - orientation ] ) where we consider the orientation of dark matter halos in cosmological simulations . but one can think of other applications : mark correlation functions may serve as orientational order parameters in liquid crystals in order to discriminate between nemetic and smectic phases ( see the contribution by f. schmid and n. h. phuong in this volume ) . they can also quantify the local orientation and order in liquids such as the recently measured five - fold local symmetry found in liquid lead @xcite . as a further application one could try to measure the signature of hexatic phases in two - dimensional colloidal dispersions and in 2d melting scenarios occurring in experiments and simulations of hard - disk systems ( for a review on hard sphere models see @xcite . finally , the orientations of anisotropic channels in sandstone ( see the contribution by c. arns et al . in this volume ) are relevant for macroscopic transport properties , therefore their quantitative characterization in terms of mark correlation functions might be interesting . before we move on to applications a few general remarks are in order : first , the definition of these mark characteristics based on the conditional density @xmath109 leads to ambiguities at @xmath31 equal zero as discussed by @xcite , but there is no problem for @xmath110 . furthermore , suitable estimators for our test quantities are based on estimators for the usual two - point correlation function @xcite . + mark - dependent clustering can also be defined at any @xmath23-point level . mark - independent clustering at every order is called the random labelling property @xcite . mark correlation functions based on the @xmath23-point densities may be used . for discrete marks the multivariate @xmath36-functions ( see eq . ) are an interesting alternative , sensitive to higher - order correlations . the random labelling property then leads to the relation @xmath111 which may be used as a test @xcite . in many cases already the question whether one or the other type of dependence as outlined above applies to certain data sets is a controversial issue . in the following we will apply our test quantities to a couple of data sets in order to probe whether there is an interplay between some objects marks and their positions in space . other applications to biological , ecological , mineralogical , geological data can be found in @xcite . the distribution of galaxies in space shows a couple of interesting features and challenges theoretical models trying to understand cosmological structure formation ( see e.g.@xcite ) . there has been a long debate , whether and how strongly the clustering of galaxies depends on their luminosity and their morphological type ( see , e.g.@xcite ) . the methods which have been used so far to establish such claims were based on the spatial two - point correlation function ; it was estimated from different subsamples that were drawn from a catalogue and defined by morphology or luminosity . however , some authors claimed that the signal of luminosity segregation observed by others was a spurious effect , caused by inhomogeneities in the sample and an inadequate choice of the statistics @xcite . @xcite could show that methods based on the mark - correlation functions , as discussed in sect . [ sec : kerscher_mark - segregation ] , are not impaired by inhomogeneities , and found a clear signal of luminosity and morphology segregation . + in order to quantify segregation effects in the galaxy distribution we consider the southern sky redshift survey 2 ( ssrs 2 , @xcite ) , which maps a significant fraction of the sky and provides us with the angular sky positions , the distances ( determined via the redshifts ) , and some intrinsic properties of the galaxies such as their flux and their morphological type . as marks we consider either a galaxy s luminosity estimated from its distance and flux , or its morphological type . in the latter case we effectively divide our sample into early - type galaxies ( mainly elliptical galaxies ) and late - type galaxies ( mainly spirals ) . in order to analyze homogeneous samples , we focus on a volume - limited sample of @xmath112 depth equals roughly @xmath113 million light years . the number @xmath114 accounts for the uncertainty in the measured hubble constant and is about @xmath115 . volume - limited samples are defined by a limiting depth and a limiting luminosity . one considers only those galaxies which could have been observed if they were located at the limiting depth of the sample . ] @xcite . + in a first step we ask whether the early- and the late - type galaxies form independent sub - processes . in figure [ fig : kerscher_morph_ind ] we show @xmath116 as function of the distance @xmath31 being far away from the value of one . recalling eq . , we conclude that the morphological types of galaxies are not distributed independently on the sky . not surprisingly , the inequality @xmath117 indicates positive interactions between the galaxies of both morphological types ; indeed galaxies attract each other through gravity irrespective of their morphological types . + after having confirmed the presence of interactions between the different types of galaxies , we tackle the issue whether the clustering of galaxies is different for different galaxies . we consider the luminosities as marks ( see fig . [ fig : kerscher_galaxies - circles ] ) . in figure [ fig : kerscher_ssrs - lum ] we show some of the mark - weighted conditional correlation functions . already at first glance , they show evidence for luminosity segregation , relevant on scales up to @xmath118 . to strengthen our claims , we redistribute the luminosities of the galaxies within our sample randomly , holding the galaxy positions fixed . in that way we mimic a marked point process with the same spatial clustering and the same one - point distribution of the luminosities , but without luminosity segregation . comparing with the fluctuations around this null hypothesis , we see that the signal within the ssrs 2 is significant . + the details of the mark correlation functions provide some further insight into the segregation effects . the mean mark @xmath119 indicates that the luminous galaxies are more strongly clustered than the dim ones . our signal is scale - dependent and decreasing for higher pair separations . the stronger clustering of luminous galaxies is in agreement with earlier claims comparing the correlation amplitude of several volume - limited samples @xcite . + the @xmath120 being larger than the mark variance of the whole sample , @xmath121 , shows that on galaxy pairs with separations smaller than @xmath118 the luminosity fluctuations are enhanced . the fact that the mark segregation effect extends to scales of up to @xmath118 is interesting on its own . in particular , it indicates that galaxy clusters are not the only source of luminosity segregation , since typically galaxy clusters are of the size of @xmath122 . + the signal for the covariance @xmath71 , however , could be due to galaxy pairs inside clusters . it is relevant mainly on scales up to @xmath123 indicating that the luminosities on galaxy pairs with small separations tend to assume similar values . our results in part confirm claims by @xcite , who compared the correlation functions @xmath124 for different volume - limited subsamples and different luminosity classes of the ssrs 2 catalog ( see also @xcite ) . many structures found in the universe such as galaxies and galaxy clusters show anisotropic features . therefore one can assign orientations to them and ask whether these orientations are correlated and form coherent patterns . here we discuss a similar question on the base of numerical simulations of large scale structure ( e.g. , @xcite ) . + in such simulations the trajectories of massive particles are numerically integrated . these particles represent the dominant mass component in the universe , the dark matter . through gravitational instability high density peaks ( `` halos '' ) form in the distribution of the particles ; these halos are likely to be the places where galaxies originate . in the following we will report on alignment correlations between such halos @xcite , for a further application of mark correlation functions in this field see @xcite . + the halos used by @xcite stem from a @xmath0-body simulation in a periodic box with a side length of 500 . the initial and boundary conditions were fixed according to a @xmath125cdm cosmology ( for a discussion of cosmological models see @xcite ) . halos were identified using a friend - of - friends algorithm in the dark matter distribution . not all of the halos found were taken into account ; rather the mass range and the spatial number density of the selected halos were chosen to resemble the properties of observed galaxy clusters in the reflex catalogue @xcite . typically our halos show a prolate distribution of their dark matter particles . + for each halo the direction of the elongation is determined from the major axis of the mass - ellipsoid . this leads to a marked point set where the orientation @xmath77 is attached to each halo position @xmath1 as a vector mark with @xmath126 . details can be founds in @xcite . + in fig . [ fig : kerscher_halos - orientation ] the vector - mark correlation functions as defined in eqs . , , and are shown . since only the orientation of the mass ellipsoids can be determined , we use @xmath127 ( eq . [ eq : kerscher_def - vector - corr - absolute ] ) instead of @xmath84 . the signal in @xmath127 indicates that pairs of halos with a distance smaller than 30 show a tendency of parallel alignment of their orientations @xmath128 . the deviation from a pure random alignment is in the percent range but clearly outside the random fluctuations . the alignment of the halos orientations @xmath128 with the connecting vector @xmath103 quantified by @xmath100 is significantly stronger ; it is particularly interesting that this alignment effect extends to scales of about 100 . + in a qualitative picture this may be explained by halos aligned along the filaments of the large scale structure . indeed such filaments are prominent features found in the galaxy distribution @xcite and in @xmath0-body simulations @xcite , often with a length of up to 100 . the lowered @xmath105 indicates that the volume of the rhomboid given by @xmath128 and @xmath103 is reduced for halo pairs with a separation below 80 . already a preferred alignment of @xmath128 along @xmath103 leads to such a reduction , similar to a plane - like arrangement of @xmath129 . for the halo distribution the signal in @xmath105 seems to be dominated by the filamentary alignment . + the question whether there are non - trivial orientation patterns for galaxies or galaxy clusters has been discussed for a long time . @xcite reported a significant alignment of the observed galaxy clusters out to 100 . @xcite , however claimed that this effect is small and likely to be caused by systematics ; @xcite find no indication for alignment effects at all . subsequently several authors purported to have found signs of alignments in the galaxy and galaxy cluster distribution ( see e.g.@xcite ) . our fig . [ fig : kerscher_halos - orientation ] shows that from simulations significant large - scale correlations are to be expected in the orientations of galaxy clusters , in agreement with the results by @xcite . these results are also supported by a simulation study carried out by @xcite . let us now turn to another , still astrophysical , but significantly closer object : the mars ( see figure [ fig : kerscher_mars_eg ] ) . many planets surfaces display impact craters with diameters up to @xmath130 km and a broad range of inner morphologies . these craters are surrounded by ejecta forming different types of patterns . the craters and their ejecta are likely to be caused by asteroids and periodic comets crossing the planets orbits , falling down onto the planet s surface , and spreading some of the underlying surfaces material around the original impact crater . a variety of different crater morphologies and a wide range of ejecta patterns can be found . in principle , either the different impact objects ( especially their energies ) or the various surface types of the planet may explain the repertory of patterns observed . whereas the energy variations of impact objects do not cause any peculiarities in the spatial distribution of the craters ( apart from a possible latitude dependence ) , geographic inhomogeneities are expected to originate inhomogeneities in the craters morphological properties . + we try to answer the question for the ejecta patterns origin using data collected by @xcite who already found correlations between crater characteristics and the local surface type employing geologic maps of the mars . complementary to their approach , we investigate two - point properties without any reference to geologic mars maps . we restrict ourselves only to craters which have a diameter larger than @xmath131 km and whose ejecta pattern could be classified , ending up with @xmath132 craters spread out all over the martian surface . we use spherical distances for our analysis of pairs . + in a first step we divide the ejecta patterns into two broad classes consisting of either the simple patterns ( single and double lobe morphology , i.e. sl and dl in terms of the classification by @xcite ; we speak of `` simple craters '' ) or the remaining , more complex configurations ( `` complex craters '' ) . using our conditional cross correlation functions @xmath62 as defined in equation , we see a highly significant signal for mark correlations ( figure [ fig : kerscher_mars - crosscorr ] ) . at small separations , crater pairs are disproportionally built up of simple craters at the expense of cross correlations . this can be explained assuming that crater formation depends on the local surface type : if the simple craters are more frequent in certain geological environments than in others , then there are also more pairs of them to be found as far as one focuses on distances smaller than the typical scale of one geological surface type . cross pairs are suppressed , since typical pairs with small separations belong to one geological setting where the simple craters either dominate or do not . only a small , positive segregation signal occurs for the complex craters . hence our analysis indicates that the broad class of complex craters is distributed quite homogeneously over all of the geologies . on top of this there are probably simple craters , their frequency significantly depending on the surface type . + if the ejecta patterns were independent of the surface , no mark segregation could be observed ( other sources of mark segregation are unlikely , since the martian craters are a result of a long bombardment history diluting any eventual peculiar crater correlations ) . in this sense , the signal observed indicates a surface - dependence of crater formation . this result is remarkable , given that we did not use any geological information on the mars at all . the picture emerging could be described using the random field model , where a field ( here the surface type ) determines the mark of the points ( see below ) . + in a second step , we analyze the interplay between the craters diameters and their spatial clustering . now the diameter serves as a continuous mark . the results in figure [ fig : kerscher_mars - markcorr ] show a clear signal for mark segregation in @xmath133 and @xmath134 at small scales . the latter signals that pairs with separations in a broad range up to @xmath135 km tend to have similar diameters ; this is in agreement with the earlier picture : as @xcite showed , the simple craters are mostly small - sized . pairs with relatively small separations thus often stem from the same geological setting and therefore have similar diameters and similar morphological type . + also the signal of @xmath133 seems to support this picture : since the simple craters are more strongly clustered than the other ones and since they have smaller diameters , one could expect @xmath136 . as we shall see in sect . [ sec : kerscher_models ] , however , a @xmath137 contradicts the random field model ; therefore , the mark - dependence on the underlying surface type ( thought of as a random field ) can not account for the signal observed . thus , we have to look for an alternative explanation : it seems reasonable , that , whenever a crater is found somewhere , no other crater can be observed close nearby ( because an impact close to an existing crater will either destroy the old one or cover it with ejecta such that it is not likely to be observed as a crater ) . this results in a sort of effective hard - core repulsion . this repulsion should be larger for larger craters . thus , pairs with very small separations can only be formed by small craters , therefore @xmath136 for tiny @xmath31 . the scale beyond which @xmath138 should somehow be hidden within the crater diameter distribution . indeed , at about @xmath139 km the segregation vanishes , which is about twice the largest diameter in our sample . taking into account that the ejecta patterns extend beyond the crater , this seems to be a reasonable agreement . as shown in sect . [ sec : kerscher_boolean - depletion ] a model based on these consideration is able to produce such a depletion in the @xmath140 . this effect could also in turn explain part of the cross correlations observed earlier in figure [ fig : kerscher_mars - crosscorr ] . a similar effect is to be expected for the mark variance . close pairs are only accessible to craters with a smaller range of diameters ; therefore , their variance is diminished in comparison to the whole sample . however , an effect like this is barely visible in the data . + altogether , the crater distribution is dominated by two effects : the type of the ejecta pattern and the crater diameter depend on the surface , in addition , there is a sort of repulsion effect on small scales . now we turn to systems on smaller scales . sandstone is an example of a porous medium and has extensively been investigated , mainly because oil was found in the pore network of similar stones . in order to extract the oil from the stone one can try to wash it out using a second liquid , e.g. water . therefore , one tries to understand from a theoretical point of view , how the microscopic geometry of the pore network determines the macroscopic properties of such a multi - phase flow . especially the topology and connectivity of the microcaves and tunnels prove to be crucial for the flow properties at macroscopic scales . details are given , for instance , in the contributions by c. arns et al . , h .- j . vogel et al . and j. ohser in this volume . a sensible physical model , therefore , in the first place has to rely on a thorough description of the pore pattern . + one way to understand the pore network is to think of it as a union of simple geometrical bodies . following @xcite , one can identify distinct pores together with their position and their pore radius or extension . this allows us to understand the pore structure in terms of a marked point process , where the marks are the pore radii . + in the following , we consider three - dimensional data taken from one of the fontainbleau sandstone samples through synchrotron x - ray tomography . these data trace a @xmath141 mm diameter cylindrical core extracted from a block with bulk porosity @xmath142 , where the bulk porosity is the volume fraction occupied by the pores . a piece with @xmath143 mm length ( resulting in a @xmath144 mm@xmath145 volume ) of the core was imaged and tomographically reconstructed @xcite . further details of this sample are presented in the contribution by c. arns et al . in this volume . based on the reconstructed images the positions of pores and their radii were identified as described in @xcite . + in our results for the mark correlation functions a strong depletion of @xmath140 and @xmath146 is visible for @xmath147 m in fig . [ fig : kerscher_sand ] . this small - scale effect may be explained similarly to the martian craters : large pores are never found close to each others , since they have to be separated by at least the sum of their radii . the histogram of the pore radii in fig . [ fig : kerscher_sand - hist ] shows that most of the pores have radii smaller than 100@xmath148 m , and consequently this effect is confined to @xmath147 m . in sect . [ sec : kerscher_boolean - depletion ] we discuss the boolean depletion model which is based on this geometric constraints and is able to produce such a reduction in the @xmath140 . this purely geometric constraint also explains the reduced @xmath146 and increased covariance @xmath149 . for separations larger than @xmath150 m there is no signal from the covariance , but both @xmath140 and @xmath146 show a small increase out to @xmath151 m . this indicate that pairs of pores out to these separations tend to be larger in size and show slightly increased fluctuations . however , this effect is small ( of the order of @xmath152 ) and may be explained by the definition of the holes , which may lead to `` artificial small pores '' as `` bridges '' between larger ones . this hypothesis has to be tested using different hole definitions . in any case the main conclusion seems to be that apart from the depletion effect at small scales there are no other mark correlations . given the significant mark correlations found in various applications , one may ask how these signals can be understood in terms of stochastic models . a thorough understanding of course requires a physical modeling of the individual situation . there are , however , some generic models , which we will focus on in the following : in sect . [ sec : kerscher_boolean - depletion ] we introduce the boolean depletion model , which is able to explain some of the features observed in the distribution of craters and pores in sandstone . another generic model is the _ random field model _ where the marks of the points stem from an independent random field ( section [ sec : kerscher_rfm ] ) . in sect . [ sec : kerscher_cox - rf - model ] we generalize the idea behind the random field model further in order to get the _ cox random field model _ , which allows for correlations between the point set and the random field . other model classes and their applications are discussed by e.g.@xcite . in our analysis of the martian craters and the holes in sandstone , we found that for small separations only small craters , or small holes in the sandstone , could be found . we interpreted this as a pure geometric selection effect . the boolean depletion model is able to quantify this effect , but also shows further interesting features . + the starting point is the boolean model of overlapping spheres @xmath153 ( see also the contributions by c. arns et al . and d. hug in this volume as well as @xcite ) . for that , the spheres centers @xmath1 are generated randomly and independently , i.e. according to a poisson process of number density @xmath154 . the radii @xmath155 of the spheres are then chosen independently according to a distribution function @xmath156 , i.e. with probability density @xmath157 . the main idea behind the depletion is to delete spheres which are covered by other spheres . to make this procedure unique we remove only those spheres which are completely covered by a ( notably larger ) sphere . the positions and radii of the remaining spheres define a marked point process . note , that this depletion mechanism is minimal in the sense that a lot of overlapping spheres may remain . this boolean depletion model may be considered as the low - density limit of the well - known widom - rowlinson model , or ( more generally ) of non - additive hard sphere mixtures ( see @xcite ) . + the probability that a sphere of radius @xmath155 is not removed is then given by @xmath158 with the step function @xmath159 for @xmath160 and @xmath161 otherwise , and the volume of the @xmath96-dimensional unit ball @xmath162 ( @xmath163 , @xmath164 , @xmath165 ) . the limit in eq . is performed by keeping @xmath166 constant , with @xmath0 the initial number of spheres and @xmath167 the volume of the domain . + the number density of the remaining spheres reads @xmath168 where the one - point probability density @xmath169 that a sphere has radius @xmath155 is given by @xmath170 the probability that one or both of the spheres @xmath171 and @xmath172 are not removed is given by @xmath173 with @xmath174 and @xmath175 at this point we have to consider the set union of two spheres with radii @xmath176 and @xmath177 , respectively ; the volume of this geometrical configuration can be calculated ; in three dimensions , e.g. , we have : @xmath178 for @xmath179 . otherwise this volume reduces either to the volume of the larger sphere ( @xmath180 ) or to the sum of both spherical volumes ( @xmath181 ) . + similarly as in eq . the spatial two - point density turns out to be @xmath182 such that the conditional two - point mark density simply reads @xmath183 from this we can derive all of the mark correlation functions from sect . [ sec : kerscher_mark - segregation ] . in order to get an analytically tractable model we adopt a bimodal radius distribution in the original boolean model and start therefore with @xmath184 where we assume that @xmath185 . due to the depletion the number density @xmath8 of the spheres as well as the probability @xmath186 to find the smaller radius @xmath187 at a given point are then lowered ; we get @xmath188 with @xmath189 . altogether , the bimodal model can be parameterized in terms of the radii @xmath190 @xmath187 , the ratio @xmath191 $ ] and the density @xmath192 . the latter two quantities , however are not observable from the final point process , therefore we convert them into the parameters @xmath193 $ ] and @xmath194 , so that all other quantities can be expressed in terms of these , for instance , @xmath195 , and @xmath196 , + from eq . we determine the mean mark , i.e. the mean radius of the spheres @xmath197 and from eq . the spatial product density @xmath198 1+\alpha^2 \left[\exp\left(ni(x)\right)-1\right ] & 1 \leq x < 2 , \\[1ex ] 1 & 2 \leq x , \end{cases}\ ] ] with the normalized inter - sectional volume @xmath199 of two spheres and @xmath200 . finally , using eq . one can calculate the mark correlation functions , e.g. @xmath201 1-\alpha^2(1-\alpha ) \frac{r_2-r_1}{\overline{r } } \frac{\exp\left(n i(x ) \right)-1 } { 1+\alpha^2\left[\exp\left(n i(x ) \right)-1\right ] } & 1 \leq x < 2 , \\[1ex ] 1 & 2\leq x . \end{cases}\label{eq : kerscher_km_depl}\ ] ] in fig . [ fig : kerscher_depletion ] the @xmath140 function from the boolean depletion model is shown . the model with the solid line illustrates that a reduced @xmath140 for small radii can be obtained by simply removing smaller spheres . at least qualitatively this model is able to explain the depletion effects we have seen both in the distribution of martian craters ( fig . [ fig : kerscher_mars - markcorr ] ) and in the distribution of pores in sandstone ( fig . [ fig : kerscher_sand ] ) . the jump at @xmath202 is a relict of the strictly bimodal distribution with only two radii . [ fig : kerscher_depletion ] also shows that the boolean depletion model is quite flexible , allowing for a @xmath203 , but also @xmath119 is possible . + without ignoring the considerable difference of this boolean depletion model to the pore size distribution in real sandstones ( see figures [ fig : kerscher_sandstone_eg]-[fig : kerscher_sand ] ) one may still recognize some interesting similarities : this simple model explains naturally a decrease of @xmath140 if the distribution of the radii is symmetric ( @xmath204 ) . as visible in figure [ fig : kerscher_sand - hist ] this is approximately the case for the pore radii . moreover , note that even quantitative features are captured correctly indicating that the decrease of @xmath140 visible in figure [ fig : kerscher_sand ] is indeed due to a depletion effect . for instance , the decrease starts at @xmath205 where @xmath206 is the largest occurring radius ( see the histogram in figure [ fig : kerscher_sand - hist ] ) and the value of @xmath207 at @xmath208 is in accordance with equation assuming that @xmath209 and the normalized density of pores @xmath210 necessary for a connected network . of course a more detailed analysis is necessary based on eqs . and and the histogram shown in figure [ fig : kerscher_sand - hist ] . the `` random - field model '' covers a class of models motivated from fields such as geology ( see , e.g. , @xcite ) . the level of the ground water , for instance , is thought of as a realization of a random field which may be directly sampled at points ( hopefully ) independent from the value of the field or which may influence the size of a tree in a forest . + in general , a realization of the random field model is constructed from a realization of a point process and a realization of a random field @xmath211 . the mark of each object located at @xmath1 traces the accompanying random field via @xmath212 . the crucial assumption is that the point process is stochastically independent from the random field . + we denote the mean value of the homogeneous random field by @xmath213=\overline{u^1}$ ] and the moments by @xmath214 , with the one - point probability density @xmath215 of the random field and @xmath216 the expectation over realizations of the random field . the product density of the random field is @xmath217 $ ] with @xmath28 . for a general discussion of random field models , see @xcite . + in this model the one - point density of the marks is @xmath218 , and @xmath219 etc . the conditional mark density is given by @xmath220,\ ] ] where @xmath221 is the dirac delta distribution . clearly , this expression is only well - defined under a suitable integral over the marks . with eq . one obtains @xmath222 and the mark - correlation functions defined in sect . [ sec : kerscher_mark - segregation ] read @xmath223 therefore , there are some explicit predictions for the random field model : an empirically determined @xmath133 significantly differing from one not only indicates mark segregation , but also that the data is incompatible with the random field model . looking at figure [ fig : kerscher_ssrs - lum ] we see immediately that the galaxy data are not consistent with the random field model . similar tests based on the relation between @xmath66 and the mark - variogram @xmath224 were investigated by @xcite and @xcite . the failure of the random field model to describe the luminosity segregation in the galaxy distribution allows the following plausible physical interpretation : the galaxies do not merely trace an independent luminosity field ; rather the luminosities of galaxies depend on the clustering of the galaxies . we shall try to account for this with a better model in the following section . in the random field model , the field was only used to generate the points marks . in the cox random field model , on the contrary , the random field determines the spatial distribution of the points as well . as before , consider a homogeneous and isotropic random field @xmath225 . the point process is constructed as a cox - process ( see e.g. @xcite ) . the mean number of points in a set @xmath226 is given by the intensity measure @xmath227 where @xmath228 is a proportionality factor fixing the mean number density @xmath229 . the ( spatial ) product density of the point distribution is @xmath230 where again @xmath231 denotes the product density of the random field . @xmath232 is the normalized two - point cumulant of the random field ( see below ) . we will also need the @xmath23-point densities of the random field : @xmath233.\ ] ] like in the random field model , the marks trace the field , but this time rather in a probabilistic way than in a deterministic one : the mark @xmath2 on a galaxy located at @xmath1 is a random variable with the probability density @xmath234 depending on the value of the field @xmath235 at @xmath1 . this can be used as a stochastic model for the genesis of galaxies depending on the local matter density . + in order to calculate the conditional mark correlation functions we define the conditional moments of the mark distribution given the value @xmath236 of the random field : @xmath237 the spatial mark product - density is @xmath238.\ ] ] and with eq . @xmath239 , \ ] ] for @xmath240 and zero otherwise . the mark correlation functions can therefore be expressed in terms of weighted correlations of the random field : @xmath241,\nonumber\\ { \left\langle m^2 \right\rangle_{\rm p}}(r ) = \frac{1}{\rho_2^u(r ) } { \mathbb{e}}\big [ \overline{m^2}(u({\mathbf{x}}_1))\ u({\mathbf{x}}_1 ) u({\mathbf{x}}_2)\big],\\ { \left\langle m_1 m_2 \right\rangle_{\rm p}}(r ) = \frac{1}{\rho_2^u(r ) } { \mathbb{e}}\big [ \overline{m}(u({\mathbf{x}}_1))\overline{m}(u({\mathbf{x}}_2))\ u({\mathbf{x}}_1 ) u({\mathbf{x}}_2)\big].\nonumber\end{gathered}\ ] ] to proceed further , we have to specify @xmath242 . as a simple example we choose @xmath2 equal to the value of the field @xmath235 at the point @xmath1 , such as in the random field model . thinking of the random field as a mass density field and the mark of a galaxy luminosity , that means that the galaxies trace the density field and that their luminosities are directly proportional to the value of the field . with @xmath243 the conditional mark moments become @xmath244 . the moments of the unconstrained mark distribution read @xmath245 , and the three basic pair averages are @xmath246 hence , the mark correlation functions defined in sect . [ sec : kerscher_mark - segregation ] are determined by the higher - order correlations of the random field . with the cox random field model we go beyond the random field model , e.g. @xmath247 is not equal to one any more . at this point , we have to specify the correlations of the random field @xmath211 . the simplest choice , a gaussian random field , is not feasible here , since a number density ( cp . [ eq : kerscher_cox - definition ] ) has to be strictly positive , whereas the gaussian model allows for negative values . instead , we will use the hierarchical ansatz : we first express the two- and three - point correlations in terms of normalized cumulants @xmath124 and @xmath248 ( see , e.g.,@xcite ) , @xmath249 in order to eliminate @xmath250 we use the hierarchical ansatz ( see e.g. @xcite ) : @xmath251 this ansatz is in reasonable agreement with data from the galaxy distribution , provided @xmath252 is of the order of unity ( @xcite ) . several choices for @xmath253 and @xmath252 lead to well - defined cox point process models based on the random field @xmath211 @xcite . now we can express @xmath140 from eq . entirely in terms of the two - point correlation function @xmath254 of the random field : @xmath255 where we made use of the fact that @xmath256 . inserting typical parameters found from the spatial clustering of the galaxy distribution we see from fig . [ fig : kerscher_km - coxrf ] that the cox random field model allows us to qualitatively describe the observed luminosity segregation in fig.[fig : kerscher_ssrs - lum ] . but the amplitude of @xmath133 predicted by this model is too high . the cox random field model , however , is quite flexible in allowing for different choices for @xmath242 ; also different models for the higher - order correlations of the random field may be used , e.g. a log - normal random field @xcite . clearly more work is needed to turn this into viable model for the galaxy distribution . whenever objects are sampled together with their spatial positions and some of their intrinsic properties , marked point processes are the stochastic models for those data sets . combining the spatial information and the objects inner properties one can constrain their generation mechanism and their interactions . + developing the framework of marked point processes further and outlining some of their general notions is thus of interest for physical applications . let us therefore look at mark correlations again from both a statistical and a physical perspective . we focused on two kinds of dependencies . + on the one hand , one can always ask , whether objects of different types `` know '' from each other . from a statistical point of view , this is the question whether the marked point process consists of two completely independent sub - point processes . physically , this concerns the question whether the objects have been generated together and whether they interact with each other . + on the other hand , it is often interesting to know whether the spatial distribution of the objects changes with their inner properties . for the statistician , this translates into the question whether mark segregation or mark - independent clustering is present . for the physicist such a dependency is interesting since one can learn from them whether and how the interactions distinguish between different object classes or whether the formation of the objects mark depends on the environment . + we discussed statistics capable of probing to which extent mark correlations are present in a given data set , and showed how to assess the statistical significance . applying our statistics to real data , we could demonstrate , that the clustering of galaxies depends on their luminosities . large scale correlations of the orientations of dark matter halos were found . using the mars data we could validate a picture of crater generation on the martian surface : mainly , the local geological setting determines the crater type . we also could show that the sizes of pores in sandstone are correlated . + in order to understand empirical data sets in detail , we need models to compare to . as generic models the boolean depletion model , the random field model and its extension , the cox random field models are of interest . + further application of the mark correlations properties may inspire the development of further models . it seems therefore that marked point processes could spark interesting interactions between physicists and mathematicians . certainly , the distributions of physicists and mathematicians in coffee breaks at the wuppertal conference were clustered , each . but could one observe positive cross - correlations ? using mark correlations we argue , that , even more , there is lots of space for positive interactions. . we would like to thank andreas faltenbacher , stefan gottlber and volker mller for allowing us to present some results from the orientation analysis of the dark matter halos ( sect . [ sec : kerscher_halos ] ) . for providing the sandstone data ( sect . [ sec : kerscher_sandstone ] ) and discussion we thank mark knackstedt . herbert wagner provided constant support and encouragement , especially we would like to thank him for introducing us to the concepts of geometric algebra as used in the appendix . in order to form versatile test functions for describing mark segregation effects , we integrated the conditional mark probability density @xmath29 twice in mark space thereby weighting with a function of the marks @xmath52 ( see eq . [ eq : kerscher_def - paverage ] ) . such a pair - averaging reduces the full information present in @xmath29 . so one may ask , whether or in which sense the mark correlation functions give a complete picture of the present two - point mark correlations . + for scalar marks @xmath2 this task is trivial . with a polynomial weighting function @xmath257 ( @xmath258 ) we consider moments of @xmath29 , hence , we can be complete only up to a given polynomial order in the marks @xmath16 and @xmath17 . at first order there is only the mean @xmath73 . at second order we have @xmath75 and @xmath259 . all the mark correlation functions discussed in sect . [ sec : kerscher_mark - segregation ] can be constructed from these three pair averages and @xmath259 at the two - point level , however , does not imply that one should not consider linear combinations of them . for instance , it may well be the case , that only certain linear combinations yield significant results . ] . higher - order moments of the marks involve more and more cross - terms . + for vector - valued marks , however , it is not obvious that the test quantities proposed in sect . [ sec : kerscher_mark - segregation ] trace all possible correlations between the vectors up to third order . to settle this case we have to consider the framework of geometric algebra , also called clifford algebra . a detailed introduction to geometric algebra is given in @xcite , shorter introductions are @xcite . in geometric algebra one assigns a unique meaning to the geometric product ( or clifford product ) of quantities like vectors , directed areas , directed volumes , etc . the geometric product @xmath260 of two vectors @xmath261 and @xmath262 splits into its symmetric and antisymmetric part @xmath263 here @xmath264 denotes the usual scalar product ; in three dimensions , the wedge product @xmath265 is closely related to the cross product between these two vectors . however , @xmath265 is not a vector like @xmath266 , but a bivector a directed area . higher products of vectors can be simplified according to the rules of geometric algebra ( for details see @xcite ) . + let us consider the situation where objects situated at @xmath14 and @xmath15 bear vector marks @xmath86 and @xmath87 , respectively , and let the normalized distance vector be @xmath267 . note , that @xmath103 is not a mark at all , rather it can be thought of as another vector which may be useful for constructing mark correlation functions . + for many applications it is reasonable to assume isotropy in mark space , i.e. all of the mark correlation functions are invariant under common rotations of the marks . for galaxies , e.g. , there does not seem to be an a priori preferred direction for their orientation . in more detail we have then @xmath268 and so on , where @xmath155 is an arbitrary rotation in mark space . this means that the mark correlation functions depend only on rotationally invariant combinations of the vector marks . therefore , only rotationally invariant combinations of vectors are sensible building blocks for weighting functions . we thus can restrict ourselves to scalar weighting functions , which result in coordinate - independent vector - mark correlation functions . + again we proceed by considering mixed moments as basic combinations . we restrict ourselves to scalar quantities being polynomial in the vector components . one may also discuss moments in a broader sense allowing for vector moduli . in this wider sense , for example , @xmath269 or @xmath270 would be allowed . we do not consider such quantities here , because they are not polynomial in the vector components . their squares anyway appear at higher orders . furthermore , it turns out that the characterization we will provide depends on the embedding dimension . the first- and second - order moments are identical in two and three dimensions , but at the third order they start to differ . 1 . in the strict sense of scalar quantities being linear in the vector components there are no first - order moments for vectors . at second order we encounter the following products : @xmath271 , @xmath272 , @xmath273 , @xmath274 . note , that , e.g. , @xmath275 and @xmath276 do not make any difference as regards the mark correlation functions , since the pair averages implicitly render the indices symmetric ; moreover , although the geometrical product is non - commutative , @xmath277 and @xmath278 do not lead to different mark correlation functions . furthermore , @xmath279 . @xmath280 provides us with higher moments of the modulus of the vectors . to investigate these kinds of correlations already scalar marks would be sufficient . new information is encoded in the other products . + consider @xmath281 . the symmetric part @xmath282 is clearly a scalar and defines the alignment @xmath283 ( eq . [ eq : kerscher_align ] ) . the antisymmetric part @xmath284 is a bivector . its unique modulus ( see again @xcite ) , @xmath285 , may be useful , but is no longer a polynomial in the vector components . @xmath286 appears at the fourth order . in a completely analogous way we can treat @xmath287 . the symmetric part @xmath288 defines @xmath100 . hence at second order , the only possible vector - mark correlation functions are @xmath84 and @xmath100 . 3 . at third order we have to consider products of three vectors . in general the product of three vectors @xmath289 splits into @xmath290 i.e. , a vector ( consisting of the three first terms ) , and a pseudo - scalar , a directed volume . in two dimensions the pseudo - scalar @xmath291 vanishes . + now we have to form all possible products of the three vectors @xmath129 and to derive scalars . in three dimensions the only new combination is the pseudo - scalar @xmath292 giving the oriented volume @xmath293 . unfortunately , this oriented volume averages out to zero . thus , in a strict sense , there are no interesting third - order quantities . closely related , however , is the modulus of the pseudoscalar @xmath294 proportional to our @xmath105 . this expression is invariant under permutations of the vectors . 4 . at third order and in two dimensions all of the relevant combinations are products of first- and second - order combinations ; no specifically new combination appears . this is different from the case of three dimensions , where at third order an entirely new geometric object , the pseudo - scalar @xmath292 can be constructed . there is a general scheme behind this argument : since in @xmath96 dimensions any geometrical product of more than @xmath96 vectors vanishes , all relevant combinations of vectors at orders higher than @xmath96 are essentially products of combinations of lower - order factors . baddeley , a. j. 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mark correlations provide a systematic approach to look at objects both distributed in space and bearing intrinsic information , for instance on physical properties . the interplay of the objects properties ( marks ) with the spatial clustering is of vivid interest for many applications ; are , e.g. , galaxies with high luminosities more strongly clustered than dim ones ? do neighbored pores in a sandstone have similar sizes ? how does the shape of impact craters on a planet depend on the geological surface properties ? in this article , we give an introduction into the appropriate mathematical framework to deal with such questions , i.e. the theory of marked point processes . after having clarified the notion of segregation effects , we define universal test quantities applicable to realizations of a marked point processes . we show their power using concrete data sets in analyzing the luminosity - dependence of the galaxy clustering , the alignment of dark matter halos in gravitational @xmath0-body simulations , the morphology- and diameter - dependence of the martian crater distribution and the size correlations of pores in sandstone . in order to understand our data in more detail , we discuss the boolean depletion model , the random field model and the cox random field model . the first model describes depletion effects in the distribution of martian craters and pores in sandstone , whereas the last one accounts at least qualitatively for the observed luminosity - dependence of the galaxy clustering . draft

You are an expert at summarizing long articles. Proceed to summarize the following text: the low enrichment of @xmath3mo alloy dispersed in an @xmath4 matrix is a prototype for new experimental nuclear fuels @xcite . when these metals are brought into contact , diffusion in the @xmath5 interface gives rise to interaction phases . also , when subjected to temperature and neutron radiation , phase transformation from @xmath6 to @xmath7 occurs and intermetallic phases develop in the u@xmath8mo@xmath9al interaction zone . fission gas pores nucleate in these new phases during service producing swelling and deteriorating the alloy properties @xcite . an important technological goal is to delay or directly avoid undesirable phase formation by inhibiting interdiffusion of @xmath4 and @xmath10 components . some of these compounds are believed to be responsible for degradation of properties @xcite . housseau _ et al . _ @xcite , based on the effective diffusion coefficients values calculated from their experimental permeation tests , have demonstrated that these undesirable phases have not influence on the mobility of @xmath10 in @xmath4 . on the other hand , bierlin and green @xcite have reported the activation energy values of @xmath10 mobility in @xmath4 , based on the maximum rate of penetration of @xmath10 into @xmath4 . on the other hand , brossa _ et al . _ @xcite , have produced couples and triplets structures using deposition methods to study the efficient diffusion barriers that should have simultaneously , a good bonding effect and a good thermal conductivity . the practical interest of a @xmath11 barrier is shown by several publications concerning with the diffusion in the systems @xmath12 , @xmath13 and @xmath14 . the study of the @xmath0 binary system was , limited to solid samples of the sandwich - type , clamped together by a titanium screw and diffusion treatments have been carried out . results from this work @xcite , have inspired present calculations . therefore it is important to study carefully and with special attention the initial microscopic processes that originate these intermetallic phases . in order to deal with this problem we started studying numerically the static and dynamic properties of vacancies and interstitials defects in the @xmath4(@xmath10 ) bulk and in the neighborhood of a @xmath15 interface using molecular dynamics calculations @xcite . here , we review our previous works @xcite , performing calculation of the tracer diffusion coefficients in binary @xmath0 and @xmath1 alloys , using analytical expressions of the diffusion parameters in terms of microscopical magnitudes . we have summarized the theoretical tools needed to express the diffusion coefficients in terms of microscopic magnitudes such as , the jump frequencies , the free vacancy formation energy and the vacancy - solute binding energy . then we start with non - equilibrium thermodynamics in order to relate the diffusion coefficients with the phenomenological onsager @xmath16-coefficients . the microscopic kinetic theory , allows us to write the onsager coefficients in term of the jump frequency rates @xcite , which are evaluated from the migration barriers and the phonon frequencies under the harmonic approximation . the lattice vibrations are treated within the conventional framework of vineyard @xcite that corresponds to the classical limit . the jump frequencies are identified by the model developed further by le claire in ref . @xcite , known as the five - frequency model for f.c.c lattices . the method includes the jump frequency associated with the migration of the host atom in the presence of an impurity at a first nearest neighbor position . all this concepts need to be put together in order to correctly describe the diffusion mechanism . hence , in the context of the shell approximation , we follow the technique developed by allnatt in refs . @xcite to obtain the corresponding transport coefficients , which are related to the diffusion coefficients through the flux equations . a similar procedure for f.c.c . structures was performed by mantina et al . @xcite for @xmath17 , @xmath18 and @xmath19 diluted in @xmath4 but using density functional theory ( dft ) . also , using dft calculations for b.c.c . structures , choudhury _ et al . _ @xcite have calculated the tracer self - diffusion and solute diffusion coefficients in diluted @xmath20 and @xmath21 alloys including an extensive analysis of the onsager @xmath16-coefficients . in the present work , we do not employ dft , instead we use a classical molecular statics technique coupled to the monomer method @xcite . this much less computationally expensive method allows us to compute at low cost a bunch of jump frequencies from which we can perform averages in order to obtain more accurate effective frequencies . although we use classical methods , we have also reproduced the migration barriers for @xmath0 with dft calculations coupled to the monomer method @xcite . we proceed as follows , first of all we validate the five - frequency model using the @xmath0 system as a reference case for which there is a large amount of experimental data and numerical calculations @xcite . since , the @xmath1 and @xmath0 systems share the same crystallographic f.c.c . structure , the presented description is analogous for both alloys . the full set of frequencies are evaluated employing the economic monomer method @xcite . the monomer is used to compute the saddle points configurations from which we obtain the jumps frequencies defined in the five - frequency model . for the @xmath0 system case , our results of the tracer solute and self - diffusion coefficients are in good agreement with the experimental data . in this case we found that @xmath4 in @xmath11 , at diluted concentrations , migrates as a free specie in the full range of temperatures here considered . in the case of @xmath1 , present calculations show that both , the tracer and self - diffusion coefficients agree very well with the available experimental data in ref . @xcite , although a vacancy drag mechanism could occur at temperatures below 500k , while , for at high temperatures the solute @xmath10 migrates by direct interchange with the vacancy . the paper is organized as follows : in section [ s1 ] we briefly introduce a summary of the macroscopic equations of atomic transport that are provided by non - equilibrium thermodynamics @xcite . in this way analytical expressions of the intrinsic diffusion coefficients in binary alloys in terms of onsager coefficients are presented . section [ s2 ] , is devoted to give the way to evaluate the onsager phenomenological coefficients following the procedure of allnat @xcite in terms of the jumps frequencies in the context of the five - frequency model . in section [ s3 ] we show the methodology used to evaluate the tracer diffusion coefficients for the solvent and solute atoms , as well as , the so called solvent enhancement factor . finally , in section [ s5 ] we present our numerical results using the theoretical procedure here summarized , which show a perfect accuracy with available experimental data , also we give an expression for the vacancy wind parameter which gives essential information about the flux of solute atoms induced by vacancy flow . the last section briefly presents some conclusions . isothermal atomic diffusion in binary @xmath22 alloys can be described through a linear expression between the fluxes @xmath23 and the driving forces related by the onsager coefficients @xmath24 as , @xmath25 where @xmath26 is the number of components in the system , @xmath23 describes the flux vector density of component @xmath27 , while @xmath28 is the driving force acting on component @xmath27 . the second range tensor @xmath24 is symmetric ( @xmath29 ) and depends on pressure and temperature , but is independent of the driving forces @xmath28 . from ( [ eq.2 ] ) the @xmath30 fick s law , which describes the atomic jump process on a macroscopic scale , can be recovered . on the other hand , for each @xmath27 component , the driving forces may be expressed , in absence of external force , in terms of the chemical potential @xmath31 , so that @xcite , @xmath32 in ( [ eq.3 ] ) @xmath33 is the absolute temperature , and the chemical potential @xmath31 is the partial derivative of the gibbs free energy with respect to the number of atoms of specie @xmath27 that is , @xmath34 where @xmath35 , is the activity coefficients , which is defined in terms of the activity @xmath36 and @xmath37 , is the molar concentration of specie @xmath27 . for the particular case of a binary diluted alloy @xmath38 with @xmath26 available lattice sites per unit volume , containing molar concentrations @xmath39 for host atoms , @xmath40 of solute atoms ( impurities ) and @xmath41 vacancies , the fluxes in terms of the onsager coefficients are expressed as , @xmath42 @xmath43 and @xmath44 from ( [ jal ] ) and ( [ jbl ] ) , we define @xmath45 @xmath46 in the case of @xmath47 , the diffusion coefficient for the vacancy is given by , @xmath48 in ( [ dateq ] ) and ( [ dbteq ] ) , @xmath49 and @xmath50 are the intrinsic diffusion coefficients for solvent @xmath51 and solute @xmath52 respectively , while @xmath53 is the vacancy diffusion coefficient @xcite . in ( [ dateq ] ) and ( [ dbteq ] ) the quantities @xmath54 are the thermodynamic factors , @xmath55 murch and qin @xcite have shown that the standard intrinsic diffusion coefficients in ( [ dateq ] ) and ( [ dbteq ] ) can be expressed in terms of the tracer diffusion coefficients @xmath56 , @xmath57 which are measurable quantities , and the collective correlation factor @xmath58 ( @xmath59 ) as : @xmath60\phi_a = d^{\star}_{a}\left [ \frac{f_{aa}}{f_a } - \left(\frac{c_a}{c_{s}}\right)\frac{f^{(a)}_{as}}{f_a}\right]\phi_a , \label{dateq1}\ ] ] @xmath61\phi_b = d^{\star}_{s}\left [ \frac{f_{ss}}{f_s } - \left(\frac{c_s}{c_{a}}\right)\frac{f^{(s)}_{as}}{f_s}\right]\phi_b . \label{dbteq2}\ ] ] the intrinsic diffusion coefficients in ( [ dateq1 ] ) and ( [ dbteq2 ] ) are known as the modified darken equations , where @xmath62 ( @xmath63 ) are the diffusion coefficients of atoms of specie @xmath64 in a complete random walk performing @xmath65 jumps of length @xmath66 per unit time . the collective correlation factors @xmath58 are related to the @xmath24 coefficients through , @xmath67 and for the mixed terms , @xmath68 the tracer correlation factors @xmath69 , @xmath70 are defined as the ratios @xmath71 and @xmath72 respectively . the term in square brackets in the second term of equations ( [ dateq1 ] ) and ( [ dbteq2 ] ) , is the vacancy wind factor @xmath73 @xcite . in the next sections , we present the onsager coefficients in terms of the atomic jump frequencies taken from ref . in order to understand the effect of different vacancy exchange mechanisms on solute diffusion , we adopt an effective five frequency model la le claire @xcite for f.c.c . lattices , assuming that the perturbation of the solute movement by a vacancy @xmath74 , is limited to its immediate vicinity . figure [ fig1 ] defines the jump rates @xmath75 ( @xmath76 ) considering only jumps between first neighbors . for them , @xmath77 implies in the exchange between the vacancy and the solute , @xmath78 when the exchange between the vacancy and the solvent atom lets the vacancy as a first neighbor to the solute ( positions denoted with circled 1 in figure [ fig1 ] ) . the frequency of jumps such that the vacancy goes to sites that are second neighbor of the solute is denoted by @xmath79 ( sites with circled 2 ) . the model includes the jump rate @xmath80 for the inverse of @xmath79 . jumps toward sites that are third and forth neighbor of the solute are all denoted with @xmath81 and @xmath82 respectively while @xmath83 and @xmath84 are used for their respective inverse frequency jumps . the jump rate @xmath85 is used for vacancy jumps among sites more distant than forth neighbors of the solute atom . in this context , that enables association ( @xmath80 ) and dissociation reactions ( @xmath79 ) , i.e the formation and break - up of pairs , the model include free solute and vacancies to the population of bounded pairs . it is assumed that a vacancy which jumps from the second to the third shell , with @xmath86 , will never return ( or returns from a random direction ) . as in ref . @xcite we express @xmath87 and @xmath88 the six symmetry types of vacancy sites that are in the first coordination shell ( first neighbor with the solute ) or in the second coordination shell ( sites accessible from the first shell by one single vacancy jump ) are shown in figure [ fig2 ] . sites that are equally distant from the solute atom @xmath52 at the origin , and that have the same abscissa ( x - coordinate in fig.[fig2 ] ) share the same vacancy occupation probability @xmath89 , equivalently for @xmath90 . table [ t2 ] resumes the sites probability with @xmath91 where for @xmath92 there is only one index @xmath64 that is given in crescent order with the distance to the solute atom @xmath52 in a positive abscissa , while @xmath93 denote sites with negative @xmath94 coordinate . for the sites in the @xmath95 plane ( @xmath96 ) , the sites are denoted with two indexes as @xmath97 , where the second index @xmath98 is given in crescent order of the distance to the solute atom @xmath52 . table [ t2 ] denotes the number of different types of sites and the distance of them to the @xmath94 axis . + .probability of occurrence of the vacancy at a site of the subset @xmath99 . [ cols="^,^,^,^,^,^,^,^,^",options="header " , ] in figure [ fig26 ] , we establish a comparison of our calculations for @xmath100 with the experimental data in table [ dexp ] , for a molar uranium concentrations @xmath101 . we see that , experimental values ( filled stars ) in the temperature range of @xmath102^{\circ}c$ ] are in perfect agreement with @xmath100 obtained with the here described procedure . in the temperature range where there are available experimental data , the @xmath10 mobility is mainly due to direct interchange between the @xmath10 atom and the vacancy . on the other hand , the diffusion of @xmath10 in @xmath4 was also calculated in a study of the maximum rate of penetration of @xmath10 into @xmath4 , in the temperature range @xmath103^{\circ}c$ ] @xcite . the maximum penetration coefficient values in ref . @xcite were , @xmath104 , @xmath105 and @xmath106 @xmath107 for @xmath108 , @xmath109 and @xmath110 , respectively . from the expression @xmath111 , the activation energy @xmath112 was @xmath113 in cal per mole in the temperature range of @xmath103^{\circ}c$ ] , where @xmath114 is expressed in calories per @xmath115 per mole , and @xmath116 is a proportionality constant . the plot @xmath117 vs @xmath118 provides a convenient basis for expressing and comparing penetration coefficients . as a final comment , a recent work by leenaers _ @xcite , presents a great quantity of experimental findings for a real system , where the present model can also be applied . also performed but not shown here , for the @xmath0 , we have reproduced all the microscopical parameters with @xmath119 atoms using the classical molecular static technique and the siesta code coupled to the monomer method @xcite . in the literature several researchers have studied the solvent atom - vacancy exchange in terms of the jump frequencies @xmath120 and @xmath121 , in the framework of the random alloy model , as for example in ref . the authors have performed an extensive monte carlo study of the tracer correlation factors in simple cubic , b.c.c . . binary random alloys . on the other hand , the kinetic formalism of moleko _ et al . _ @xcite , also describes the behavior of the tracer correlation factors for slow and faster diffusers . in summary , in this work we present the general mechanism based on non - equilibrium thermodynamics and the kinetic theory , to describe the diffusion behavior in f.c.c diluted alloys . non equilibrium thermodynamic , through the flux equations , relates the diffusion coefficients with the onsager tensor , while the kinetic theory relates the onsager coefficients in terms of microscopical magnitudes . in this way we are able to write expressions for the diffusion coefficients only in terms of microscopic magnitudes , i.e. the jump frequencies . the five frequency model has also been of great utility in order to discriminate the relevant jump frequencies , evaluated from the migration barriers under the harmonic approximation in the context of the conventional treatment by vineyard corresponding to the classical limit . hence , we have calculated the full set of phenomenological coefficients from which the full set of diffusion coefficients are obtained through the flux equation . in this respect , the jump frequencies have been calculated from the migration barriers which are obtained with an economic static molecular techniques ( cmst ) namely the monomer method , that searches saddle configurations efficiently . although in this work we have performed the treatment for the case of f.c.c . latices where the diffusion is mediated by vacancy mechanism , a similar procedure can be adopted for other crystalline structures or different diffusion mechanism ( for example , interstitials ) . we have exemplified our calculations for the particular cases of diluted @xmath0 and @xmath1 f.c.c . binary alloys . we have found that the tracer diffusion coefficient are in very good agreement with the available experimental data , for both alloys . present calculations show that qualitatively a vacancy drag mechanism is unlikely to occur for the @xmath0 system . in the case of @xmath1 , a vacancy drag mechanism could occur at temperatures below @xmath2k , while above this temperature the solute migrates by a direct interchange mechanism with the vacancy , such as was corroborated in the comparison with the available experimental data . we have demonstrated that , the cmst is appropriate in order to describe the impurity diffusion behavior mediated by a vacancy mechanism in f.c.c . this opens the door for future works in the same direction where a similar procedure will be used that includes interstitial defects . i am particularly grateful to dr . roberto c. pasianot for help on calculations of the attempt jump frequencies , to dr . rivas for comments on the manuscript , and to martn urtubey for figure [ fig2 ] . this work was partially financed by conicet pip-00965/2010 . references http://www.rertr.anl.gov/ a.m. savchenko , a.v . vatulin , i.v . dobrikova , g.v . kulakov , s.a . ershov , y.v . konovalov , proc . of the international meeting of the rertr , * s12 * -5 ( 2005 ) . mirandu , s.f.aric , s.n.balart , l.m . gribaudo , mat . charact . , * 60 * , 888 ( 2009 ) . n. housseau , a. van craeynest , d. calais , journal of nuclear materials , * 39 * -2 , 189 - 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impurity diffusion coefficients are entirely obtained from a low cost classical molecular statics technique ( cmst ) . in particular , we show how cmst is appropriate in order to describe the impurity diffusion behavior mediated by a vacancy mechanism . in the context of the five - frequency model , cmst allows to calculate all the microscopic parameters , namely : the free energy of vacancy formation , the vacancy - solute binding energy and the involved jump frequencies , from them , we obtain the macroscopic transport magnitudes such as : correlation factor , solvent - enhancement factor , onsager and diffusion coefficients . specifically , we perform our calculations in f.c.c . diluted @xmath0 and @xmath1 alloys . results for the tracer diffusion coefficients of solvent and solute species are in agreement with available experimental data for both systems . we conclude that in @xmath0 and @xmath1 systems solute atoms migrate by direct interchange with vacancies in all the temperature range where there are available experimental data . in the @xmath1 case , a vacancy drag mechanism could occur at temperatures below @xmath2k . diffusion , moddeling , numerical calculations , vacancy mechanism , diluted alloys , @xmath0 and @xmath1 systems .

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Proceed to summarize the following text: in this paper , we study binary linear @xmath0 $ ] codes ( codes of length @xmath1 and dimension @xmath2 ) used for error detection on the binary symmetric channel . a comprehensive introduction to the field is given in @xcite . the basic definitions are given in section [ sub : errordetection ] . a main quantity is the _ probability of undetected error _ of a code . if the probability of undetected error is an increasing function on the interval @xmath3 $ ] , the code is known as _ proper _ for error detection . it is believed that proper codes exist for all lengths @xmath1 and dimensions @xmath2 . however , this has been shown only for some cases . in particular , proper @xmath0 $ ] codes are known to exist for any given @xmath2 when @xmath1 is sufficiently large . the best known result in this direction was given by klve and yari @xcite who showed that proper codes exist for @xmath4 in this paper , we study a particular class of @xmath0 $ ] codes where @xmath5 . one of our results is that these codes are proper for many values of @xmath1 and @xmath2 where the existence of proper codes was previously unknown . in particular , we improve the bound ( [ kybound ] ) . we first consider @xmath1 in the range @xmath6 . the hamming bound proves that the dual of an @xmath0 $ ] code in this case has minimum distance at most 3 . moreover , an @xmath7 $ ] code with minimum distance 3 can be obtained by shortening the @xmath8 $ ] hamming code . two @xmath0 $ ] codes @xmath9 and @xmath10 are equivalent if there exists a permutation @xmath11 of @xmath12 such that @xmath13 if two codes are equivalent , then it may happen that the corresponding ( repeatedly ) punctured codes are not equivalent . let @xmath14 be some @xmath15 matrix having as columns all possible nonzero vectors of length @xmath2 . the code generated by @xmath14 is the simplex code @xmath16 , and the code having @xmath14 as parity check matrix is the well - known hamming code . note that the order of the columns is not specified ; all the equivalent codes are named hamming codes . however , when we want to puncture the code , the order is very important . we remind the reader that puncturing a code is equivalent to shortening the dual code . davydov et al . @xcite determined an ordering of the columns in @xmath14 such that any of the corresponding ( repeatedly ) shortened codes contains a minimal number of codewords of weight three ; the shortened @xmath7 $ ] codes are obtained by removing @xmath17 columns from @xmath14 to get a @xmath18 matrix @xmath19 and use this matrix as the parity check matrix for the @xmath7 $ ] code . they showed that a possible choice of @xmath19 is to have as columns the vectors that are the binary representation of the numbers from @xmath20 down to @xmath21 . for example @xmath22 we let @xmath23 denote the code generated by @xmath19 . for our investigation , we will consider codes that are equivalent ( but not equal ) to these codes ; we will denote them by @xmath24 . a main reason for considering @xmath24 rather than @xmath23 is that the determination of the weight distribution is easier for @xmath24 . in this paper , we investigate the performance of the codes @xmath24 when they are used for error detection . we compute their weight distribution that , in turn , permits us to calculate the undetected error probability @xmath25 . however , when the code length @xmath1 is large ( @xmath26 ) , the polynomial expressing @xmath25 may be difficult to evaluate , even when the weight distribution is known . for this reason , in the general case , we also find bounds on the length and dimension such that a necessary condition for codes to be satisfactory does not hold , using a method similar to the one proposed in @xcite . the paper is organized as follows : in section [ sub : errordetection ] we give some preliminaries on error detection ; in section [ sub : dkstconstruction ] we describe the construction of @xmath24 and show that @xmath24 and @xmath23 are equivalent ; in section [ sec : weights - distribution ] we determine the weight distribution of @xmath24 ; in section [ sec : prob ] we study the undetected error probability of @xmath24 and its dual code ; in section [ sec : approximate - analysis ] we give an asymptotic analysis ; in section [ sec : generalization ] we give a generalization of the construction to lenghts @xmath27 , finally , in section [ sec : results ] we summarize our results . we start by defining @xmath28 , the undetected error probability for an @xmath0 $ ] code @xmath9 when used on the binary symmetric channel with error probability @xmath29 : @xmath30 where @xmath31 is the number of codewords having hamming weight @xmath32 , see e.g. ( * ? ? ? * section 2.1.2 ) . one can also express this polynomial in terms of the weight distribution of the dual code , see e.g. ( * ? ? ? * theorem 2.4 ) . if @xmath33 is the number of codewords having hamming weight @xmath32 in the dual code @xmath34 , we have : @xmath35 as mentioned in the introduction , if @xmath28 is an increasing function on @xmath36 $ ] , the code @xmath9 is called _ proper _ for error detection . if @xmath37 for every @xmath38 $ ] , @xmath9 is called _ good _ for error detection . if @xmath39 for every @xmath38 $ ] , @xmath9 is called _ satisfactory _ for error detection , see @xcite . a code that is not satisfactory is called _ ugly_. when a code is proper then it is satisfactory ; so , if it is ugly it is clearly not proper ( nor good ) . we first describe a particular parity check matrix @xmath14 for the hamming code . for @xmath40 , let @xmath41 be the @xmath42 matrix constructed as follows : * the first @xmath43 rows are all - zero vectors . * row @xmath44 is the all - one vector . * in the @xmath45 matrix consisting of the last @xmath46 rows , the columns are ordered lexicographically . then @xmath47.\ ] ] we illustrate this with an example . for @xmath48 , we get @xmath49 @xmath50 and so @xmath51 \\ = & \begin{bmatrix } 111111110000000\\ 000011111111000\\ 001100110011110\\ 010101010101011 \end{bmatrix}.\end{aligned}\ ] ] we let @xmath52 denote the @xmath18 matrix containing the first @xmath1 columns of @xmath14 . for example @xmath53 we let @xmath24 denote the code generated by @xmath52 . we see that @xmath54 is the first order reed - muller code and @xmath55 , the simplex code . both of these codes are known to be proper ( and this is easy to show ) . the hamming code is @xmath56 . the code having @xmath52 as parity check matrix is a shortened hamming code which we denote by @xmath57 . we note that @xmath58 . in the rest of the paper ( except section [ sec : generalization ] ) we will assume that @xmath59 . [ dkst - bs ] the codes @xmath24 and @xmath23 are equivalent . we first illustrate by the example @xmath48 and @xmath60 , that is , the matrices ( [ h4 ] ) and ( [ d4 ] ) . adding the second row in ( [ h4 ] ) to the third and forth rows , we get @xmath61 this is an alternative generator matrix for @xmath62 . the last three columns are the same in ( [ hh4 ] ) and ( [ d4 ] ) , and the first eight columns of ( [ hh4 ] ) are a permutation of the first eight columns in ( [ d4 ] ) . hence , @xmath62 and @xmath63 are equivalent . in the general case , if @xmath64 $ ] for some @xmath46 , @xmath65 , we add row @xmath66 in @xmath52 to all the rows below . this gives an alternative generator matrix @xmath67 for @xmath24 . the first @xmath68 columns of @xmath67 are a permutation of the binary representations of @xmath69 $ ] , the next @xmath70 columns of @xmath67 are a permutation of the binary representations of @xmath71 $ ] , etc . the final @xmath72 columns in @xmath67 and @xmath19 are the same . hence , @xmath24 and @xmath23 are equivalent . if @xmath73 for some @xmath46 , @xmath74 , the same argument shows that the columns of @xmath52 are a permutation of the columns of @xmath19 , and so again @xmath24 and @xmath23 are equivalent . the main question we consider is : for which @xmath1 and @xmath2 is @xmath24 proper for error detection ? we will also in some cases consider the simpler question : for which @xmath1 and @xmath2 is @xmath24 satisfactory for error detection ? we note that this is equivalent to the question : for which @xmath1 and @xmath2 is @xmath57 satisfactory for error detection ? the reason is the following known lemma . * theorem 2.8 ) . [ sdual ] a code is satisfactory if and only if the dual code is satisfactory . to determine the probability of undetected error for @xmath24 , we have to determine its weight distribution . this is done in this section . we break the argument down into a number of lemmas . we first give some further notations . we observe that the matrix @xmath75\ ] ] has length @xmath76 for a given @xmath1 , let @xmath46 be determined by @xmath77 since @xmath78 , we have @xmath65 . let @xmath79 denote the last column of @xmath52 . [ alpha - lemma ] let @xmath80 . then @xmath81 and @xmath82 are determined by @xmath83 the last column in @xmath52 is a column in @xmath84 . hence ( [ alm ] ) follows . moreover , its number in @xmath84 is @xmath85 when we count the first column as number zero . the columns in @xmath84 are ordered lexicographically and so ( [ alr ] ) follows . let @xmath86 denote the weight of the @xmath87-th row in @xmath52 . as usual , @xmath88 denotes the largest integer less than or equal to @xmath89 . [ w - lemma1 ] let @xmath80 . then @xmath90 and @xmath91 if @xmath92 and @xmath93 , then @xmath94 if @xmath92 and @xmath95 , then @xmath96 all the rows of @xmath14 have weight @xmath68 . the first @xmath46 rows of @xmath52 are obtained from rows in @xmath14 removing some zeros . hence @xmath97 for @xmath98 . before we go on with the proof , let us take a closer look at @xmath41 . row @xmath99 consists of consecutive blocks of zeros and ones , each block of length @xmath100 . we use the term _ double block _ for a zero - block combined with the following one - block ; it has length @xmath101 . now , let @xmath102 then row @xmath87 in @xmath52 consists of @xmath103 double blocks of length @xmath101 , each of weight @xmath100 , followed by an incomplete double block of length @xmath104 that has to be considered further ( when @xmath105 , the incomplete double block is , of course , a full double block ) . if @xmath93 , the incomplete double block is all zero , and so ( [ wm2 ] ) follows . if @xmath95 , the incomplete double block consists of a full block ( of length @xmath100 ) of zeros followed by an incomplete block of ones of length @xmath106 . hence @xmath107 this proves ( [ wm3 ] ) . the proof of ( [ wm1 ] ) is similar ( and even simpler ) . [ lem1 ] consider sums of rows from @xmath52 . \a ) any of the @xmath108 non - zero sums of some of the first @xmath46 rows have weight @xmath68 . \b ) any of the @xmath109 sums containing row @xmath66 and zero or more previous rows have weight @xmath110 . \c ) for @xmath111 , @xmath112 sums containing row @xmath87 and some previous rows have weight @xmath113 and the other @xmath112 sums have weight @xmath114 . for each sum of rows from the first @xmath46 , the corresponding sum of rows in @xmath14 are codewords in the simplex code @xmath16 . these always have weight @xmath68 . since only zeros have been removed to get the corresponding rows in @xmath52 , their sum also has weight @xmath68 . this proves a ) . let @xmath115 . we note that in the set of positions of a double block in row @xmath115 in @xmath116 for @xmath117 , the elements of any previous row are all zero or all one . therefore , the weight of these positions in any sum of row @xmath87 and a combination of previous rows is @xmath100 . it remains to consider the contribution to the weight from the last @xmath104 positions ( where @xmath104 is defined by ( [ nu ] ) ) . case i ) @xmath93 . in this case , all the last @xmath104 elements of row @xmath87 are zeros . any previous row has all zeros or all ones in these positions , and so the weight of the elements in these positions in any sum is either 0 or @xmath104 . hence , the weight of the sum is either @xmath86 or @xmath118 , where @xmath86 is given by ( [ wm2 ] ) . moreover , row @xmath66 has all ones in the last @xmath104 positions . hence , half of the @xmath119 sums has weight @xmath86 and the other half has weight @xmath118 . case ii ) @xmath95 . in this case , all the last @xmath104 elements of row @xmath87 are @xmath100 zeros followed by @xmath106 ones . the weight of the last @xmath104 elements in a sum is therefore @xmath106 or @xmath100 . hence , the weight of a sum is @xmath86 or @xmath118 , where now @xmath86 is given by ( [ wm3 ] ) . as done above , considering sums containing row @xmath66 , we can see that the multiplicities of these two weights are the same . this proves c ) . finally , consider row @xmath66 . any previous row has all zeros in the last @xmath104 positions . hence , the weight of any sum involving row @xmath66 and previous rows is @xmath110 . this proves b ) . we next give an alternative expression for @xmath86 . [ w - alt ] let @xmath80 . + a ) if @xmath92 and @xmath93 , then @xmath120 b ) if @xmath92 and @xmath95 , then @xmath121 c ) further , @xmath122 from ( [ alr ] ) we get @xmath123 hence @xmath124 where @xmath125 hence @xmath126 and so , by ( [ wm2 ] ) , @xmath127 this proves a ) . similarly , if @xmath128 , ( [ wm3 ] ) gives @xmath129 since @xmath95 . this proves b ) . finally , c ) follows directly by substituting the expression for @xmath1 in the expression for @xmath110 in ( [ wm1 ] ) . [ ex0 ] consider @xmath130 , where @xmath131 . we have @xmath132 using lemma [ w - alt ] , we see that @xmath24 has @xmath108 codewords of weight @xmath133 and @xmath134 codewords of weight @xmath135 in particular , the minimum distance is @xmath136 . hence , the code @xmath24 is proper ( see ( * ? ? ? * theorem 2.2 ) ) . [ w - diff ] let @xmath137 and @xmath92 . \a ) if @xmath138 , then @xmath139 . \b ) if @xmath93 and @xmath140 , then @xmath141 . c1 ) if @xmath95 , @xmath142 , and @xmath143 for all @xmath144 such that @xmath145 , then @xmath139 . c2 ) if @xmath95 , @xmath142 , and @xmath146 for at least one @xmath147 , then @xmath141 . \d ) in all cases , @xmath148 in particular , the minimum distance @xmath149 of @xmath24 is @xmath150 . \e ) @xmath151 . \a ) if @xmath152 , then lemma [ w - alt ] gives @xmath153 if @xmath154 , then lemma [ w - alt ] gives @xmath155 \b ) if @xmath93 and @xmath140 , then lemma [ w - alt ] gives @xmath156 \c ) if @xmath95 and @xmath142 , then lemma [ w - alt ] gives @xmath157 we have @xmath158 with equality if and only if @xmath143 for @xmath145 . \d ) we have @xmath159 and so @xmath160 further , both for @xmath161 and @xmath162 , lemma [ w - alt ] gives @xmath163 for @xmath92 , a ) , b ) , c1 ) , and c2 ) show that @xmath164 . \e ) equation ( [ wm1 ] ) implies that @xmath165 let @xmath166 the midpoint of the interval @xmath167 $ ] . [ dmin ] let @xmath149 be the minimum distance of @xmath24 . \a ) if @xmath168 , then @xmath169 \b ) if @xmath170 , then @xmath171 \a ) we have @xmath172 hence @xmath173 . by lemma [ w - alt]a ) , @xmath174 \b ) we have @xmath175 . from ( [ alr ] ) and lemma [ w - alt]b ) , @xmath176 [ admin ] a ) if @xmath177 , then @xmath178 \b ) if @xmath179 and @xmath115 is given by @xmath180 then @xmath181 in particular , @xmath182 in all cases . the conditions imply that @xmath183 has value @xmath149 exactly for @xmath184 ( where @xmath185 for case a ) ) . hence @xmath186 from ( [ eq : puecode ] ) , ( [ eq : puedual ] ) , and lemma [ lem1 ] , we get the following theorems . [ th1 ] let @xmath80 . then @xmath187 [ th1dual ] let @xmath80 . then @xmath188 [ ex1 ] consider @xmath189 where @xmath190 . then @xmath191 using lemmas [ w - lemma1 ] and [ w - alt ] we get @xmath192 hence , the weight distribution of @xmath24 is given by table [ tabnr ] . @xmath193 for @xmath194 and @xmath195 in example [ ex1 ] we get @xmath196 and @xmath197 in fig . [ s320 ] we give the graphs of @xmath198 and the terms @xmath199 and @xmath200 . the contributions from the last two terms , @xmath201 and @xmath202 are so small that they are not visible on the graph . the graph illustrates that @xmath203 is ugly . for small @xmath29 ( @xmath29 up to approximately 0.42 ) , @xmath199 is the dominating term ; in this region the difference @xmath204 is so small that it is not visible on the graph . for @xmath29 close to 0.5 , the term @xmath200 dominates . ( solid line ) , @xmath199 ( dashed line ) , @xmath200 ( dotted line ) , and @xmath205 ( long dashed ) . ] theorem [ th1 ] can be used to determine if the code @xmath24 is proper and theorem [ th1dual ] if the code @xmath57 is proper . we just compute @xmath206 and check the presence or absence of real roots in @xmath207 . for moderate values of @xmath1 and @xmath2 ( e.g. @xmath208 ) , this is feasible in a reasonable time . before we give a main general result , we quote two lemmas from @xcite . [ incr ] a ) @xcite theorem 2.2 : if @xmath209 , then @xmath210 is increasing on @xmath3 $ ] . \b ) @xcite lemma 3.5 : if @xmath211 , then @xmath212 is increasing on @xmath3 $ ] . let @xmath213 @xmath214 [ s - proper ] for @xmath215 , if @xmath216 or @xmath217 then @xmath24 is proper . for @xmath73 and @xmath130 , @xmath24 is proper by example [ ex0 ] . for @xmath218 , ( [ wm ] ) , lemma [ w - diff]d ) , and lemma [ w - diff]e ) imply that @xmath219 and so @xmath220 is increasing on @xmath3 $ ] for @xmath221 by lemma [ incr]a ) . now , consider @xmath222 , where @xmath223 . by lemma [ dmin]a ) @xmath224 by lemmas [ w - diff]d ) and [ incr]b ) , @xmath225 is increasing for @xmath92 if @xmath226 , that is , if @xmath227 this is equivalent to @xmath228 and @xmath229 solving this for @xmath230 , we get @xmath231 we see that if ( [ kcon1 ] ) is satisfied , then all the terms in @xmath232 are increasing on @xmath3 $ ] . consequently , @xmath24 is proper . next , let @xmath233 where @xmath234 . then , by lemma [ dmin]b ) , @xmath235 we want @xmath236 solving for @xmath230 , we get @xmath237 and so @xmath238 as above , if ( [ kcon2 ] ) is satisfied , then @xmath24 is proper . when @xmath24 is proper , then it is satisfactory , and so , by lemma [ sdual ] , @xmath57 is satisfactory . hence we get the following corollary . [ c - kange ] if @xmath1 is in the range defined by ( [ kcon1 ] ) or ( [ kcon2 ] ) for some @xmath239 , then @xmath57 is satisfactory . [ p - kange ] a ) if @xmath240 then @xmath24 is proper for all @xmath241 $ ] . \b ) @xmath24 is proper for all @xmath242 $ ] . we have @xmath243 if and only if @xmath244 we observe that @xmath245 decreases with increasing @xmath46 and @xmath246 increases with increasing @xmath46 . . then ( [ full ] ) is equivalent to the following sequence of inequalities @xmath248 @xmath249 @xmath250 for @xmath251 we get @xmath252 and so @xmath253 for all @xmath239 . however , for @xmath254 we get @xmath255 and so @xmath256 hence , ( [ full ] ) is satisfied if and only if @xmath257 , that is when ( [ mb ] ) is satisfied . therefore , if ( [ mb ] ) is satisfied , then @xmath24 is proper for all @xmath258 $ ] . next , since @xmath259 , we see that if ( [ full ] ) is satisfied , then @xmath260 hence , @xmath261 , and so @xmath24 is proper also for all @xmath262.\ ] ] this , combined with the result above , proves a ) . since @xmath24 is proper for @xmath263\ ] ] for all @xmath264 , b ) follows . based on the previous theorems , we have found a set of values of @xmath1 for which @xmath24 is proper and , hence , satisfactory . for other values of @xmath1 , the existence of real roots of @xmath265 in @xmath266 must be checked . however , for large values of @xmath2 and @xmath1 in general it may be difficult to numerically compute the polynomial s real roots , or even just to determine the existence of real roots ( e.g. using sturm s chain ) . however , in many cases we can decide that the code @xmath24 ( and hence @xmath57 ) is not satisfactory ( i.e. ugly ) by showing that @xmath267 for some value of @xmath29 . how should the value of @xmath29 be chosen ? there is no theory that can give an exact answer to this question . however , it is known that if the minimum distance of the code is @xmath149 , then @xmath268 is often the dominating term of @xmath28 , except for large @xmath29 . this is well illustrated by the example of @xmath203 given in fig . [ s320 ] . since @xmath269 has its maximum for @xmath270 , a good choice for @xmath29 may be @xmath270 . this gives the following sufficient condition for @xmath24 to be ugly : @xmath271 where @xmath272 is the binary entropy function . we can reformulate this to the following well - known sufficient condition for a code to be ugly ( see e.g. * theorem 2.11 ) or @xcite ) : @xmath273 we showed in example [ ex0 ] that @xmath274 is proper for all @xmath275 . in general , @xmath24 and @xmath57 may be ugly for some values of @xmath1 when @xmath276 . we have checked that @xmath24 is proper for all @xmath277 when @xmath278 . when @xmath279 , @xmath24 is ugly for some values of @xmath1 . an example is @xmath203 in fig . [ s320 ] . [ var ] for a given @xmath2 , let @xmath280 then @xmath281 is increasing with @xmath1 on @xmath282 $ ] and decreasing with increasing @xmath1 on @xmath283 $ ] , where @xmath284 was given in ( [ nrm ] ) . from the definitions of @xmath285 and @xmath281 , we get @xmath286 by lemma [ dmin]a ) , @xmath149 is constant for @xmath287 $ ] . considering @xmath1 as a real variable for the moment , direct calculations gives @xmath288 since @xmath289 , we get @xmath290 . similarly , for @xmath291 $ ] , @xmath292 is constant by lemma [ dmin]b ) , and so @xmath293 note : the weight distribution of @xmath294 was given in table [ tabnr ] . in particular , @xmath295 for @xmath296 . moreover , @xmath182 for all @xmath297 $ ] . hence , ( [ per ] ) is satisfied for some such @xmath1 if and only if it is satisfied for @xmath296 . for @xmath298 we have , from table [ tabnr ] , that @xmath299 and so @xmath300 for a fixed @xmath46 , let @xmath301 where we consider @xmath2 a real variable , and where @xmath302 then ( [ per ] ) , for @xmath296 , can be rewritten as @xmath303 we get @xmath304 to analyze @xmath305 further , we first give some relations for @xmath285 . [ h - lemma ] for @xmath306 , we have @xmath307 and @xmath308 using taylor s theorem , we get @xmath309 for @xmath310 . the upper bound ( [ ent ] ) follows immediately since @xmath311 next , from ( [ ht ] ) , we get @xmath312 since @xmath313 for @xmath314 and @xmath315 , we get @xmath316 [ u - lemma ] for @xmath239 , we have @xmath317 where @xmath318 we have @xmath319 hence , from ( [ u - def ] ) and ( [ ent ] ) we get @xmath320 this proves the lower bound on @xmath321 . similarly , ( [ u - def ] ) and ( [ entl ] ) imply the upper bound on @xmath321 . in particular , ( [ kb ] ) and lemma [ u - lemma ] imply that @xmath322 for all @xmath239 . since @xmath323 for all @xmath2 and @xmath324 when @xmath325 , we see that @xmath326 has a unique root in @xmath327 , we denote it by @xmath328 . further , @xmath329 for @xmath330 . also , @xmath331 and @xmath305 is decreasing for @xmath332 . [ not - int ] for @xmath239 , @xmath328 is not an integer . for @xmath298 , we have @xmath299 @xmath333 and @xmath295 . by definition , @xmath334 if @xmath335 or equivalently , @xmath336 hence , if @xmath334 were an integer , then the exact powers of 2 dividing the two sides of ( [ eq2 ] ) would be the same . we will show that this is not the case . the exact power of 2 dividing @xmath337 is @xmath338 the exact power of 2 dividing @xmath339 is @xmath340 that is , we have a contradiction . hence , @xmath328 is not an integer . let @xmath341 be the smallest integer @xmath2 such that @xmath331 . then @xmath342 and so we have the following : @xmath343 in table [ k0tab ] we give the values of @xmath341 for @xmath344 . @xmath345 & [ 5,14 ] & [ 15,36 ] \\ \hline k(m ) & 9 & 2m+8 & 2m+9 & 2m+10 \\ \hline \hline m \in & & [ 37,81 ] & [ 82,172 ] & [ 173,356 ] \\ \hline k(m ) & & 2m+11 & 2m+12 & 2m+13 \\ \hline \end{array}\ ] ] @xmath346&&&\\ \hline 10 & [ 599,676]&&&\\ \hline 11 & [ 1140,1396]&&&\\ \hline 12 & [ 2219,2878 ] & [ 3286,3367]&&\\ \hline 13 & [ 4331,5853 ] & [ 6458,6844]&&\\ \hline 14 & [ 8540,11878 ] & [ 12717,13888 ] & [ 14812,14883]&\\ \hline 15 & [ 16870,23966 ] & [ 25208,28006 ] & [ 29371,30013]&\\ \hline 16 & [ 33486,48290 ] & [ 50034,56408 ] & [ 58305,58368]\,[58370,60396]\ , [ 60416,60461 ] & [ 62460,62468]\\ \hline 17 & [ 66546,66560]\,[66593,97028 ] & [ 99602,113304 ] & [ 116102,121434 ] & [ 124378,125472 ] \\ \hline 18 & [ 132560,194804 ] & [ 198432,227423 ] & [ 231354,231424]\,[231451,243631]\,[243712,243730 ] & [ 247954,251741 ] \\ \hline \end{array}\ ] ] we will next determine good bounds on @xmath328 . these can in turn be used to determine @xmath341 . we use the notations @xmath347 [ k1eks ] let @xmath348 . \a ) if @xmath349 then @xmath350 \b ) if @xmath351 then @xmath352 let @xmath353 . since @xmath354 , we have @xmath355 by lemma [ u - lemma ] , @xmath356 if ( [ bb ] ) is satisfied , then @xmath331 and so @xmath357 . this proves a ) and the proof of b ) is similar . let @xmath358 [ k1eksc ] if @xmath359 , then @xmath360 first we consider the upper bound . let @xmath361 . then @xmath362 for @xmath363 , that is , @xmath364 , i.e. @xmath365 . for @xmath366 we get @xmath367 also . in particular , ( [ bb ] ) is satisfied for all @xmath359 . the upper bound therefore follows from lemma [ k1eks]a ) . the proof of the lower bound is similar . for @xmath368 we can show it by direct computation . some calculus shows that @xmath369 for all @xmath359 and @xmath370 for @xmath371 . we skip the details . the lower bound therefore follows from lemma [ k1eks]b ) . from corollary [ k1eksc ] we immediately get the following result . [ k0main1 ] we have @xmath372 when @xmath373 . [ k0main2 ] for all @xmath359 , we have @xmath374 or @xmath375 in particular , if there is no integer between @xmath376 and @xmath377 , then @xmath378 since @xmath379 , ( [ k01 ] ) and corollary [ k1eksc ] imply that @xmath380 further , if there is no integer between @xmath376 and @xmath377 , then @xmath381 . the difference @xmath382 is small , except for small @xmath46 . hence , to have an integer between @xmath376 and @xmath377 , @xmath377 must be close to and above an integer . if we denote this integer by @xmath383 , then @xmath46 must be close to @xmath384 . we will make this statement more precise in the following lemma . [ theta - n ] let @xmath385 be a positive integer . \a ) if @xmath386 then @xmath387 . \b ) if @xmath388 then @xmath389 . in this proof , we let @xmath46 be a positive real variable . we note that @xmath377 and @xmath376 are still well defined . moreover , simple calculus shows that @xmath390 and @xmath391 for all @xmath392 . therefore , a ) is equivalent to @xmath393 and similarly for b ) . proof of a ) . let @xmath394 . then @xmath395 , where @xmath396 and so @xmath397 hence @xmath398 if @xmath399 the inequality ( [ om - con ] ) is equivalent to @xmath400 which in turn is equivalent to @xmath401 this is satisfied for all @xmath402 . for @xmath403 , we can show a ) directly by numerical computation . this completes the proof of a ) . the proof of b ) is similar . we give a sketch , leaving out some details . let @xmath404 . then @xmath395 , where @xmath405 and so @xmath406 we have @xmath407 we observe that the function @xmath408 is increasing with @xmath89 . hence , @xmath409 if @xmath410 that is , @xmath411 this is satisfied for @xmath412 . direct computation shows that b ) is true also for @xmath413 . combining lemmas [ theta - n]a ) and b ) , we see that if there is an integer between @xmath376 and @xmath377 , then this integer is @xmath383 for some integer @xmath385 , and @xmath414 where @xmath415 we have checked ( [ int ] ) for @xmath416 . for @xmath417 , there is no integer satisfying ( [ int ] ) . for @xmath418 , there actually is an integer @xmath419 satisfying ( [ int ] ) . however , in these cases , direct computations show that @xmath420 . we can therefore conclude that @xmath421 for @xmath422 . this is , of course , far beyond what is needed for any practical application . whether there are any @xmath423 such that there is an integer satisfying ( [ int ] ) remains an open question . however , the length of the interval in ( [ int ] ) is @xmath424 and @xmath425 therefore , it is highly unlikely that there is an integer satisfying ( [ int ] ) for some @xmath423 . based on this , we conjecture that @xmath421 for all @xmath359 . if @xmath426 , we define @xmath427 to be the smallest integer and @xmath428 the largest integer such that : * @xmath429 , * ( [ per ] ) is satisfied for @xmath430 . in the next section , we give estimates for @xmath427 and @xmath428 . for @xmath431 , we have computed the values of @xmath1 in the range @xmath432 $ ] for which ( [ per ] ) is satisfied . for @xmath278 , this never happens ; and we have checked that @xmath57 is always proper for @xmath278 . for @xmath433 , the values of @xmath1 for which ( [ per ] ) is satisfied are given in table [ pek - n ] . since @xmath434 , we only have to consider @xmath435 when @xmath431 . we see that , in general , for any given @xmath2 and @xmath46 the set of @xmath1 where ( [ per ] ) is satisfied consists of zero or more intervals . typically , we have several intervals , except for small values of @xmath2 . we describe @xmath436 , @xmath195 as an example to illustrate why this is the case . we first give a small list of values in table [ tab17 ] . @xmath437 we see that ( [ per ] ) is satisfied for @xmath438 , but not for @xmath439 . since ( [ per ] ) turns out to be satisfied for @xmath440 , we get @xmath441 . for @xmath1 in the range @xmath442 $ ] we have @xmath443 . since @xmath444 is decreasing with increasing @xmath1 , ( [ per ] ) is not satisfied for @xmath1 in the range @xmath445 $ ] . however , we see that for @xmath446 , we have a jump in the value of @xmath447 compared to @xmath448 , and ( [ per ] ) is again satisfied for all @xmath1 in the range @xmath449 . for @xmath450 , ( [ per ] ) is again not satisfied . [ ex16 ] an interesting example occurs when @xmath451 and @xmath452 ; @xmath453 is an isolated value of @xmath1 for which ( [ per ] ) is not satisfied . we have @xmath454 we have @xmath455 and @xmath456 and so ( [ per ] ) is not satisfied . however , @xmath457 and @xmath458 so the contribution from this term alone is sufficient to conclude that @xmath459 is not satisfactory after all . this shows that if ( [ per ] ) is not satisfied , but the second lowest weight of @xmath24 is close the minimum weight @xmath149 , it may be a good idea to consider the contribution from this weight also . for @xmath460 we have checked if the codes @xmath24 and @xmath57 are proper when ( [ per ] ) is not satisfied . it turns out that this is always the case for @xmath57 , but not for @xmath24 . as an illustration , in table [ tab : tab3 ] we give the range of values @xmath1 for @xmath195 and @xmath461 , such that @xmath24 is proper . @xmath462 & [ 331,384 ] \\ \hline 10 & [ 513,587 ] & [ 688,768 ] \\ \hline 11 & [ 1025,1124 ] & [ 1424,1536 ] \\ \hline 12 & [ 2049,2195 ] & [ 2904,3072 ] \\ \hline 13 & [ 4097,4298 ] & [ 5908,6144 ] \\ \hline 14 & [ 8193,8489 ] & [ 11937,12288 ] \\ \hline 15 & [ 16385,16798 ] & [ 24081,24576 ] \\ \hline 16 & [ 32769,33376 ] & [ 48422,49152 ] \\ \hline 17 & [ 65537,66388 ] & [ 97245,98304 ] \\ \hline 18 & [ 131073,132321 ] & [ 195096,196608 ] \\ \hline \end{array}\ ] ] in table [ pek - n ] we see that for a given @xmath46 , an increasing fraction of the codes are ugly when @xmath2 increases . define @xmath463 and @xmath464 by @xmath465 and @xmath466 clearly , @xmath467 . let @xmath468 [ ny - con ] we have @xmath469 let @xmath470 , where @xmath471 . by lemma [ dmin ] , @xmath472 , and by lemma [ admin ] , @xmath182 . by ( [ per ] ) we see that if @xmath473 then @xmath474 , and so @xmath24 is ugly . we have @xmath475 by ( [ ent ] ) , @xmath476 hence @xmath477 by ( [ perapx ] ) , if @xmath478 then @xmath24 is ugly . since ( [ par2 ] ) is equivalent to @xmath479 , and @xmath480 is an integer , that is @xmath481 , we can conclude that @xmath482 . this proves the theorem . _ remark_. we see that @xmath483 is an upper bound on the number of @xmath1 in @xmath484 $ ] such that @xmath24 is satisfactory . therefore , a main corollary of theorem [ ny - con ] is that , for any fixed @xmath46 , @xmath485 converges to 0 exponentially fast when @xmath2 increases . for @xmath486 , we have a similar result . let @xmath487 [ ny - con2 ] we have @xmath488 the proof is similar to the proof of theorem [ ny - con ] . let @xmath489 where @xmath490 . from lemma [ dmin ] we get @xmath491 and so @xmath492 therefore , analogously to ( [ par2 ] ) we get @xmath493 is a sufficient condition for @xmath24 to be ugly . since this is equivalent to @xmath494 , theorem [ ny - con2 ] follows . [ cor ] let @xmath495 be the number of @xmath496 $ ] such that @xmath24 is satisfactory . then @xmath497 the number @xmath498 of satisfactory codes for @xmath1 in the interval @xmath499 $ ] is at most those for @xmath500 $ ] plus those for @xmath501 $ ] . the number of @xmath1 in the first interval is @xmath502 the number of @xmath1 in the second interval is @xmath503 we see that , for @xmath65 , we have @xmath504 hence @xmath505 and so @xmath506 @xmath507 when @xmath325 . theorem [ cor ] shows that @xmath57 is ugly for most values of @xmath1 . on the other hand , @xmath57 is satisfactory for many values of @xmath1 as shown by corollary [ c - kange ] . the matrix @xmath14 was defined by concatenating @xmath508 for @xmath509 . we can generalize this by concatenating @xmath510 copies of @xmath511 , followed by @xmath512 copies of @xmath513 , @xmath514 copies of @xmath515 , etc . for any sequence @xmath516 of positive integers . most of the previous results carries over , with obvious modifications . for now , we only consider the construction with @xmath517 for @xmath314 , and we write @xmath518 . as before , we use the notation @xmath24 for the codes generated by the first @xmath1 columns of the matrix . for large @xmath519 , these codes have low rate . the dual codes will have very high rate and minimum distance 2 . consider @xmath520 generated by @xmath521 , and let @xmath24 , where @xmath522,\ ] ] be the code generated by the matrix @xmath523 we see that we get a code @xmath24 for each @xmath277 . also , given @xmath1 , the values of @xmath519 and @xmath524 are uniquely determined by ( [ tn ] ) . from its definition , we immediately get the following lemma . [ lenging ] a ) the weight of the first row of @xmath52 is @xmath525 larger than the weight of the first row of @xmath521 . \b ) for any other non - zero codeword in @xmath24 , the weight is @xmath526 larger than the weight of the corresponding codeword in @xmath521 . in particular , we see that * the minimum distance of @xmath24 is @xmath526 larger than the minimum distance of @xmath521 . * for a non - zero codeword of @xmath24 of weight @xmath32 , either @xmath209 or there is a unique other codeword in the code of weight @xmath527 . this last property was used to prove theorem [ s - proper ] . therefore , this theorem can be directly generalized by a similar proof . let @xmath528 @xmath529 note that @xmath530 and @xmath531 . [ s - propek - t ] for @xmath532 and @xmath215 , if @xmath533 or @xmath534 then @xmath24 is proper . the proof is similar to the proof of theorem [ s - proper ] and is omitted . [ tp - kange ] a ) if @xmath239 and @xmath535 then @xmath24 is proper for all @xmath536.\ ] ] \b ) @xmath24 is proper for all @xmath537.\ ] ] similarly to theorem [ p - kange ] , we see that if @xmath538 where @xmath247 , then @xmath24 is proper for all @xmath539.\ ] ] we see that if @xmath540 , then @xmath541 and so ( [ tx ] ) is not satisfied . however , if @xmath542 then @xmath543 and so @xmath544 for @xmath545 , that is , all @xmath239 . since ( [ txx ] ) is equivalent to @xmath546 , we get the theorem . [ skr ] for @xmath547 there exists an integer @xmath548 such that @xmath24 is proper for all @xmath549 . for @xmath547 and @xmath550 , we get @xmath551 by theorem [ tp - kange]b ) , @xmath24 is proper for all @xmath552.\ ] ] this implies that @xmath24 is proper for all @xmath553 [ skr1 ] if @xmath547 , then @xmath24 is proper for all @xmath554 by theorem [ skr ] , @xmath24 is proper for all @xmath555 . next , theorem [ tp - kange]b ) for @xmath556 shows that @xmath24 is proper for @xmath557.\ ] ] finally , theorem [ s - propek - t ] for @xmath556 , @xmath547 , and @xmath195 implies that @xmath24 is proper for @xmath558.\ ] ] it remains to show that @xmath559 we have @xmath560 and so , for @xmath547 , @xmath561 this proves ( [ tau - verdi ] ) . until recently , the best general result of this kind was ( * ? ? ? * theorem 2.64 ) : if @xmath562 and @xmath563 then there exists a proper @xmath0 $ ] code . this bound was recently improved in @xcite to the following : if @xmath562 and @xmath564 then there exists a proper ( and self complementary ) @xmath0 $ ] code . clearly , theorem [ skr ] above gives a further improvement and is now the best known such bound . we can also find lower bounds on @xmath565 . we consider @xmath24 for @xmath1 in the middle of the interval with @xmath195 , that is @xmath566 where @xmath567 was defined in ( [ nrm ] ) . when we consider only the term in @xmath24 of lowest degree , we know that the case @xmath568 is the worst case ( cfr . lemma [ var ] ) . moreover , this term of lowest degree is the dominating one in @xmath24 . therefore , it is reasonable to consider these values of @xmath1 when we look for non - proper @xmath24 . [ t - low ] for @xmath547 we have @xmath569 proof a ) . we see that if @xmath570 is not proper , then by the definition of @xmath565 , @xmath571 . therefore , ( [ bound1 ] ) follows . proof b ) . again , consider @xmath24 for @xmath572 . then , by table [ tabnr ] and lemma [ lenging ] , @xmath573 since @xmath574 , ( [ per ] ) implies that the code is ugly if @xmath575 we have @xmath576 and so , by ( [ ent ] ) , @xmath577 hence , if @xmath578 that is @xmath579 then @xmath24 is ugly . therefore , @xmath571 , and the theorem follows . we now give a lemma that is useful for studying when @xmath24 codes in general are proper for a given @xmath2 . [ small - n2 ] let @xmath580 and let @xmath32 be the weight of the first row of @xmath52 . if @xmath581 is increasing on @xmath3 $ ] , then @xmath582 is proper for all integers @xmath583 . the weight of the first row of @xmath584 is @xmath585 . the weight of any other non - zero codeword in @xmath582 is @xmath586 larger than the corresponding codeword in @xmath24 . hence @xmath587 by assumption , @xmath581 is increasing on @xmath3 $ ] . since @xmath588 and @xmath589 are increasing on @xmath3 $ ] , we can conclude that @xmath590 is increasing on @xmath3 $ ] , that is , @xmath582 is proper . for the use of this lemma , it is useful to observe that the conclusion of theorem [ s - propek - t ] can be improved : if ( [ tcon1 ] ) or ( [ tcon2 ] ) hold , then @xmath581 is increasing on @xmath3 $ ] . the proof carries over immediately . using lemma [ small - n2 ] and computations , we have determined @xmath565 for @xmath591 . these values are given in table [ t0tab ] together with the lower and upper bounds on @xmath592 in theorems [ t - low ] and [ skr ] . we have also included the bounds for @xmath593 . we see that the upper bound is very loose , but @xmath565 equals the implicit lower bound @xmath594 for all @xmath595 . we conjecture that this may be the case for all @xmath2 . table [ t0tab ] shows that the explicit lower bound @xmath594 is also loose ( but substantially better than the upper bound ) and the ratio @xmath596 is increasing slowly with @xmath2 . for @xmath597 the ratio is 1.375 , for @xmath451 it is 1.485 , and for @xmath598 it is 1.555 . theorem [ tilde ] below shows that the ratio is always less than 2 . the lower bound @xmath594 has the advantage that it is explicit and that it shows that @xmath565 grows exponentially with @xmath2 . in appendix 1 we prove the following theorem . [ tilde ] let @xmath547 and @xmath599 . let @xmath600 be the positive real number defined by @xmath601 where @xmath602 then @xmath603 combining theorem [ tilde]a ) and theorem [ tilde]b ) , we get the following corollary . we have @xmath604 moreover , the first alternative is the most likely . we have included @xmath605 in table [ t0tab ] . for the range of values we have computed , i.e. @xmath606 , we have @xmath607 . if the conjecture that @xmath608 is true , then @xmath605 is a sharp upper bound on @xmath565 . further , if the conjecture is true , then @xmath24 is proper for all @xmath609 , a substantially stronger result than what we have been able to show in corollary [ skr1 ] . @xmath610 @xmath611 for @xmath547 , let @xmath612 be the number of @xmath613 $ ] such that @xmath24 is proper . we have shown by direct computation and the use of lemma [ small - n2 ] that for @xmath614 , @xmath24 is proper for all @xmath615 $ ] . hence , @xmath616 for @xmath614 . for @xmath617 we know that there are values of @xmath1 where @xmath24 is not proper . for given @xmath519 , @xmath2 , @xmath46 , let @xmath618 be the set of @xmath619 $ ] for which @xmath24 is not proper . the set @xmath618 may be empty . in particular , theorem [ tp - kange]a ) shows that @xmath620 if ( 38 ) is satisfied . on the other hand , table [ tab : tab3 ] shows that @xmath621 for @xmath433 . by direct computation and the use of lemma [ small - n2 ] , we have shown that the values given in table [ noprop ] are the only values @xmath580 for which @xmath24 is not proper . the computations have been extended up to @xmath436 . in general , the values in @xmath618 are not necessarily consecutive . for example @xmath622\cup [ 93181,93184].\ ] ] this is similar to what we have in table [ pek - n ] , and the underlying reason is the same . @xmath623 \\ \hline 10 & 1 & [ 588,687],\,[1127,1175],\,[1661,1667 ] \\ \hline 11 & 1 & [ 1125,1423 ] , \,[2200,2402],\,[3255,3397],\\ & & [ 4306,4396],\,[5353,5398],\,[6399,6401 ] \\ & 2 & [ 1661,1667 ] \\ \hline 12 & 1 & [ 2196,2903],\,[4301,4902],\,[6393,6893 ] , \\ & & [ 8500,8901],\ , [ 10582,10917],\ , [ 12661 , 12935 ] , \\ & & [ 14738 , 14955],\,[16813 , 16977],\,[18887 , 19000 ] , \\ & & [ 20959 , 21024],\,[23030 , 23050 ] \\ & 2 & [ 3255,3397],\,[5353,5398 ] \\ \hline \end{array}\ ] ] the conjecture that @xmath624 can be reformulated as follows : if @xmath625 then @xmath626 . in general , we conjecture that if @xmath627 , then @xmath628 . from table [ noprop ] we get the explicit values of @xmath612 given in table [ theta - tab ] . we have included in the table the lower bound given in theorem [ del ] below . @xmath629 [ del ] when @xmath547 , @xmath24 is proper for @xmath630 of the values of @xmath613 $ ] . the proof is given in appendix 2 . _ comment . _ we clearly have @xmath631 . the discussion on @xmath565 above indicates that we may have @xmath632 . if this is the case , then we have @xmath633 . in this paper we have analyzed the codes @xmath24 and their duals @xmath57 ( which are shortened hamming codes ) to investigate if they are proper or satisfactory codes for error detection . we have determined the weight distribution of the codes @xmath24 and computed the undetected error probability . for @xmath634 , the codes @xmath24 are proper for all @xmath65 . however , for @xmath2 greater than 8 , there are values of @xmath1 such that @xmath24 and @xmath57 are ugly ( not satisfactory ) for error detection . for increasing @xmath2 , the percentage of such codes is increasing . on the other hand , we have shown that the number values @xmath1 such that @xmath24 is proper grows exponentially with @xmath2 . we have given a generalization of the construction which defines codes @xmath24 for all lengths greater than @xmath68 , and we have shown that @xmath24 is proper for all @xmath635 . moreover , @xmath24 is proper for at least @xmath636 of the shorter lengths . a plausible conjecture ( @xmath608 ) implies that @xmath24 probably is proper for all @xmath637 . an open question for future work is the following : is it the case that if there is an @xmath638 $ ] for which @xmath24 is not proper , then @xmath639 is such an @xmath1 ? if the answer is yes ( which we believe it is ) , then this would in particular imply the conjecture referred to above . further work may concern searching for modifications of the construction that will extend the range of lengths where the codes are proper or satisfactory . in particular , one line of investigation could be to consider lengths less than @xmath68 by looking at the the duals of the best known codes of minimum distance 4 . let @xmath640 the expression for @xmath641 that was used to determine the bound @xmath642 is easily obtained by combining table [ tabnr ] ( with @xmath195 ) and lemma [ lenging ] . since @xmath599 , we have : @xmath643^{4rt+r } \nonumber \\ & + 4\,p^{4rt+2r}(1-p)^{4rt } + p^{8rt}(1-p)^{2r}. \label{exp}\end{aligned}\ ] ] we note that @xmath641 is well defined for any positive real number @xmath519 . for large values of @xmath519 , @xmath641 is increasing on @xmath3 $ ] . for small values of @xmath519 , @xmath641 is first increasing , then decreasing , then again increasing . there is a limiting @xmath519 such that , for this @xmath519 , there is a @xmath644 such that @xmath645 and @xmath646 for all other @xmath29 in @xmath647 . in particular , this implies that @xmath648 . in principle , the two equations @xmath645 and @xmath648 can be used to determine @xmath644 and @xmath649 . however , the equations are complicated , and we will consider an approximation which is easier to handle . we remark at this point that for @xmath519 close to @xmath650 , @xmath648 is not possible . in this appendix we often drop @xmath2 from @xmath565 and write just @xmath651 when the value of @xmath2 should be clear from the context . similarly we write @xmath652 for @xmath653 , @xmath654 for @xmath655 , etc . in @xmath641 , the last three terms are increasing on @xmath3 $ ] whereas the first term is increasing on @xmath656 $ ] and decreasing on @xmath657 $ ] . for all @xmath658 $ ] , the first two terms are dominating . therefore , we first consider the sum of these two terms : @xmath659^{4rt+r}\ ] ] and determine the @xmath660 and @xmath661 such that @xmath662 . we expect @xmath660 to be a good approximation to @xmath654 ( and @xmath661 to be a good approximation to @xmath644 ) . the remaining two terms in @xmath663 are much smaller . moreover , they are increasing on @xmath3 $ ] . therefore , @xmath664 . in particular , @xmath665 . this proves theorem [ tilde]a ) . we have @xmath666^{4rt+r-1}(1 - 2p ) \nonumber \\ & = 4r\,p^{4rt-1}(1-p)^{4rt+r-1 } \nonumber \\ & \quad \cdot \bigl\{[2t-(4t+1)p](1-p)^r \nonumber \\ & \qquad \quad + ( 4r-2)(4t+1)(1 - 2p ) p^r \bigr\}.\label{fdir}\end{aligned}\ ] ] _ remark . _ we see that @xmath667 for all @xmath519 . hence , the equation @xmath668 will , in addition to @xmath661 , have at least one more solution in @xmath647 . this second solution will be closer to @xmath669 . however , it does not reflect a property of @xmath663 , but only the approximation @xmath670 . since @xmath671 and @xmath672 , we have @xmath673 where @xmath674 similarly , @xmath675 where @xmath676,\end{aligned}\ ] ] and so @xmath677 combining ( [ equation1 ] ) and ( [ equation2 ] ) , we get @xmath678 since @xmath679 , we have @xmath680 , and so @xmath681 solving this for @xmath29 , we get two solutions @xmath682 @xmath683 where @xmath684 we see that if @xmath685 , then the roots are not real . this reflects the fact that @xmath686 is not possible in this case . the smaller of the two roots is the @xmath661 we are looking for , the larger @xmath687 occurs because we have neglected the two smallest terms in ( [ exp ] ) as explained above . therefore , it is not relevant for our analysis of @xmath663 . since @xmath688 , ( [ fdir ] ) implies @xmath689(1-p_1)^r+(4r-2)(4{\vartheta}+1)(1 - 2p_1)p_1^r=0.\ ] ] substituting the value of @xmath661 into ( [ llikk ] ) and simplifying , we get ( [ tildef ] ) . we can not find a closed expression for @xmath660 , but , for a given @xmath2 , we can determine the value numerically . we note , however , that @xmath690 ( which is the quantity we want ) actually is the least integer @xmath519 such that @xmath691 where @xmath692 and this observation simplifies the numeric determination of @xmath690 since we do not have to solve the equation ( [ tildef ] ) , but only search for @xmath690 . to prove theorem [ tilde]b ) , we first give a couple of lemmas . [ kat1 ] if @xmath693 , then @xmath694 the function @xmath695 is decreasing when @xmath89 is increasing . we have @xmath696 and so @xmath697 . hence , @xmath698 } > 1-\frac{1}{2t}.\end{aligned}\ ] ] [ kat2 ] if @xmath693 , then @xmath699 we have @xmath700 from lemmas [ kat1 ] and [ kat2 ] , we see that if @xmath701 then ( [ lik2 ] ) is satisfied and so @xmath702 . taking logarithms of ( [ ll2 ] ) , we get the equivalent expression @xmath703 since @xmath704 for @xmath705 , we see that if @xmath706 then ( [ ll2 ] ) is satisfied , and so @xmath702 . since @xmath599 , we have @xmath707 . solving ( [ ll4 ] ) for @xmath519 , we get the following relation : @xmath708 in the proof of lemma [ kat2 ] , we used that @xmath709 . however , using ( [ tilweak2 ] ) we get a better bound on @xmath710 and hence a stronger version of lemma [ kat2 ] and a better bound on @xmath652 . [ kat3 ] if @xmath711 then @xmath712 by ( [ nyb ] ) @xmath713 for @xmath714 . hence , @xmath715 therefore , if @xmath716 then @xmath702 . taking logarithms and solving as above , we get theorem [ tilde]b ) exactly as we obtained ( [ tilweak2 ] ) from ( [ ll2 ] ) . for @xmath717 , we can show that theorem [ tilde]b ) is true by direct computation . to prove theorem [ tilde]c ) , we first give another lemma . [ kat1a ] @xmath718 using ( [ deltaskr ] ) , we get @xmath719 } { 8r\vartheta + 3r+ r[1 - 4/(k-1 ) ] } \\ & = 1-\frac{k-3}{2(k-1)\vartheta + ( k-2)}.\end{aligned}\ ] ] this is lemma [ kat1a]a ) . moreover , this implies that @xmath720 by theorem [ tilde]b ) @xmath721 for @xmath722 , and so @xmath723 . direct computations show that @xmath724 also for @xmath725 . hence lemma [ kat1a]b ) is proved . to prove theorem [ tilde]c ) , let @xmath726 and @xmath727 . we get @xmath728p^{4r(t-\varepsilon ) } ( 1-p)^{4r(t-\varepsilon ) + 2r-1 } \\ & \quad + ( 16r-8)[4r(t-\varepsilon ) + r ] \\ & \qquad \cdot [ p(1-p)]^{4r(t-\varepsilon ) + r-1}(1 - 2p ) \\ & = f'_{t}(p)-8r\varepsilon p^{4r(t-\varepsilon ) -1}(1-p)^{4r(t-\varepsilon ) + 2r } \\ & \quad + 8r\varepsilon p^{4r(t-\varepsilon ) } ( 1-p)^{4r(t-\varepsilon ) + 2r-1 } \\ & \quad -4r\varepsilon ( 16r-8)[p(1-p)]^{4r(t-\varepsilon ) + r-1}(1 - 2p).\end{aligned}\ ] ] since @xmath729 and @xmath730 , we get @xmath731.\end{aligned}\ ] ] let @xmath732 then @xmath733 . we get @xmath734\pi^{2r } -16r({\vartheta}-\varepsilon)\pi^{2r+1 } \\ & \quad \quad + 8r({\vartheta}-\varepsilon)\pi^{4r({\vartheta}-\varepsilon ) } -2r\pi^{4r({\vartheta}-\varepsilon)+1}\bigr\ } \\ & < 8r(1-p_1)^{8r({\vartheta}-\varepsilon ) + 2r-1 } \pi^{4r({\vartheta}-\varepsilon)-1 } \\ & \quad\cdot \bigl\ { \bigl [ 2{\vartheta}(1-\pi)+1\bigr]\pi^{2r } + \vartheta \pi^{4r({\vartheta}-\varepsilon)}\bigr\}.\end{aligned}\ ] ] clearly , @xmath735 if @xmath736\pi^{2r } \le \varepsilon(1-\pi),\ ] ] and @xmath737 equation ( [ pico1 ] ) is equivalent to @xmath738 we choose the @xmath739 which gives equality in ( [ pico11 ] ) , that is @xmath740 equation ( [ pico2 ] ) is equivalent to @xmath741 by ( [ vardef ] ) , @xmath742 further , by theorem [ tilde]b ) , @xmath743 and , finally , @xmath744 and so @xmath745 hence , ( [ pico2 ] ) is also satisfied . therefore @xmath746 where @xmath739 is given by ( [ vardef ] ) . by lemma [ kat1a ] , @xmath747 combining ( [ eqvar ] ) and ( [ epp ] ) we get theorem [ tilde]c ) for @xmath714 . direct computations show that it is true also for @xmath717 . this completes the proof of theorem [ tilde ] . let @xmath748 be the number of @xmath1 in @xmath749\ ] ] such that @xmath24 is proper . clearly , @xmath750 [ tau - bounds ] let @xmath547 and @xmath65 . \a ) if @xmath751 and @xmath752 , then @xmath753 \b ) if @xmath751 and @xmath754 , then @xmath755 \c ) if @xmath756 and @xmath532 , then @xmath755 by theorem [ s - propek - t ] , @xmath757 in the proof of theorem [ tp - kange ] we showed that if @xmath758 then @xmath759 this proves b ) and c ) . now , consider @xmath751 and @xmath752 . then @xmath760 and so @xmath761 similarly , @xmath762 combining these two inequalities with ( [ tet ] ) , a ) follows . [ sum3 ] for @xmath547 we have @xmath763 first we see that @xmath764 hence @xmath765 similarly , @xmath766 the lemma follows from these results and lemma [ tau - bounds]a ) . [ sum1 ] for @xmath547 we have @xmath767 and @xmath768 the result follows directly from lemmas [ tau - bounds]c ) and [ tau - bounds]b ) respectively . we can now combine these results into a proof of theorem [ del ] . from ( [ tetsum ] ) and lemmas [ tau - bounds][sum1 ] , we get @xmath769 theorem [ del ] follows from this expression . the authors are grateful to mario blaum for pointing out reference @xcite . the construction of @xmath52 is also essentially due to him . a. a. davydov , l. n. kaplan , yu . b. smerkis , and g. l. tauglikh , `` optimization of shortened hamming codes '' , _ problems of inform . 4 , pp . 261267 , oct .- dec . 1981 ( transl . from russian ) .

binary linear @xmath0 $ ] codes that are proper for error detection are known for many combinations of @xmath1 and @xmath2 . for the remaining combinations , existence of proper codes is conjectured . in this paper , a particular class of @xmath0 $ ] codes is studied in detail . in particular , it is shown that these codes are proper for many combinations of @xmath1 and @xmath2 which were previously unsettled . error detection , proper codes , satisfactory codes , simplex codes , punctured codes , ugly codes .

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Proceed to summarize the following text: the compact radio continuum sources comprising w51 ( @xcite ) have long been recognized to constitute some of the most luminous star formation regions in the disk of the galaxy ( @xcite ; @xcite ) . the high luminosity , the large number of inferred o type stars ( @xcite ) , and the location of these sources within a molecular cloud ( @xcite ) all suggest that the w51 region represents the early formation stages of an ob association . besides the intrinsic interest in the properties of w51 , this region represents one of the closest analogs in the disk of the milky way to the more vigorous star forming sites found in other galaxies ( e.g. 30 doradus ) . since these latter regions are quite distant , w51 affords many advantages in investigating the detailed properties of luminous star forming sites and inferring how these regions may originate . one key to understanding the formation and evolution of any star forming region is establishing the properties of the molecular cloud out of which the stars form . while the molecular gas in the w51 region has been the subject of numerous studies , the interpretation of the results remain controversial . scoville & solomon ( 1973 ) , primarily on the basis of small strip maps in @xmath1co(10 ) , identified several molecular line components toward w51 and derived a minimum mass of @xmath12 m and a diameter @xmath92030pc for the molecular cloud that they associated with the most intense radio component at @xmath136 cm ( g49.5 - 0.4 ; @xcite ) . they further suggested that this cloud might be physically related to the several thermal radio continuum sources that make up the w51 hii - region complex ( @xcite ; @xcite ; @xcite ) . subsequent studies of the molecular gas toward w51 have confirmed the existence of a large molecular cloud ( @xcite ; @xcite ; @xcite ; @xcite ; @xcite ; @xcite ) , although various models continue to be proposed for the relationship of the multiple spectral features seen in the molecular gas lines and their association with the different hii regions . the primary difficulty in establishing a definitive model of this region is the unique location of w51 in the galaxy with respect to the sun . the w51 region has classically been associated with the tangent point of the sagittarius spiral arm , which is oriented such that the line of sight toward w51 intersects the spiral arm over several kiloparsecs of path length ( @xcite ; @xcite ) . much of the uncertainty surrounding the w51 region stems from establishing whether the numerous radio continuum sources and molecular clouds represent a single , large massive star forming region , or the chance projection of unrelated star forming sites viewed down a spiral arm . to better place the w51 region in context with respect to its location in the galactic plane , figure [ fig1 ] displays the integrated @xmath1co(10 ) emission in 10km0.2em s@xmath7 velocity bins covering longitudes 4055 from the massachusetts - stony brook @xmath1co survey ( @xcite ) . the w51 region is distinguished by bright @xmath1co emission extending over a 1@xmath31 area centered on ( @xmath14 ) @xmath9(49.5 , -0.2 ) at velocities @xmath1555km0.2em s@xmath7 . a `` 3d '' view of the ( @xmath16 ) @xmath1co data cube covering the region surrounding w51 is shown in figure [ fig2 ] . the @xmath1co isointensity contour surface in this figure clearly illustrates both the relatively large number of smaller molecular clouds with typical internal velocity dispersions of @xmath1735km0.2em s@xmath7 , and the large concentration of @xmath1co emission extending over a @xmath920km0.2em s@xmath7 interval in the w51 region . much of the @xmath1co emission in this area has centroid velocities that exceed the maximum velocity permitted by pure circular rotation ( i.e. @xmath18 @xmath95457km0.2em s@xmath7 ; @xcite ) . such velocities have long been noted in 21 cm hi surveys at longitudes near @xmath19 , and have been attributed to large - scale streaming motions of gas in a spiral density wave ( e.g. @xcite ; @xcite ) . in principle the extent and properties of the molecular clouds located in the w51 region can be established by using the kinematic information in the molecular line data to isolate individual clouds . in practice , previous surveys have had either poor resolution or sparse sampling to make such an attempt feasible . therefore , we have obtained full beam sampled maps of the w51 region in both @xmath1co(10 ) and @xmath2co(10 ) at subarcminute resolution in order to determine the relationship between the various molecular components . these maps permit us to disentangle the blends of unrelated clouds along the line of sight and to obtain more accurate mass estimates of the molecular gas . these data can also be compared with similar maps of more nearby clouds that have recently been obtained by us and others . the outline of this paper is as follows . in section [ obs ] , the observing procedures are described and channels maps of the @xmath1co and @xmath2co emission are presented . analysis of the different spectral features observed in these maps and a more thorough discussion of the main features associated with the compact radio continuum sources in w51 is given in section [ analysis ] . in section [ discussion ] , we discuss the current massive star formation events in the region with respect to the various molecular components and comment on the evolution of the w51 giant molecular cloud ( gmc ) . our conclusions are summarized in section [ summary ] . a 1.39 x 1.33 region ( 100 x 96 pixels ) toward the w51 region was mapped in @xmath1co(10 ) ( 115.271203 ghz ) and @xmath2co(10 ) ( 110.201370 ghz ) in april 1995 using the quarry receiver array ( @xcite ) on the fcrao 14 m telescope . the full width at half maximum beam size of the 14 meter antenna at these frequencies is 45 and 47 at 110 and 115 ghz respectively . the data were taken in position switching mode and calibrated with the standard chopper wheel method of observing an ambient temperature load and sky emission . the backends for each pixel of the array consisted of an autocorrelator spectrometer set to span the velocity range from @xmath9 0100km0.2em s@xmath7 at 78 khz sampling ( 0.20km0.2em s@xmath7 @ 115 ghz ) and 94 khz resolution ( 0.24km0.2em s@xmath7 ) . during data reduction the spectra were smoothed to a velocity resolution of 0.5km0.2em s@xmath7 . previous fcrao measurements indicate that the spillover and scatter efficiency ( @xmath20 ) of the telescope and radome is @xmath90.7 at the observed frequencies . the observed antenna temperatures corrected by @xmath20 are presented as @xmath21 ( @xcite ) . a further correction , the source coupling efficiency ( @xmath22 ) , accounts for the coupling of the beam to the source . for a uniform source that fills only the main beam of the 14 m telescope , @xmath22 is @xmath90.7 ( i.e. 30% of the power is scattered on angular sizes much greater than the fwhm beam size ) , while for sources with uniform intensity over a diameter of 30 , @xmath22 is 1.0 . in practice , the observed structures in the @xmath1co and @xmath2co maps span a range of sizes and shapes , and applying a single coupling efficiency for the entire map is incorrect . for simplicity , we present and analyze the data in the @xmath21 temperature scale . the typical rms noise in the @xmath1co and @xmath2co maps in 0.5km0.2em s@xmath7 channels is @xmath23@xmath21 @xmath90.7 k and 0.6 k respectively . images of the integrated @xmath1co and @xmath2co intensity ( @xmath24 in 2km0.2em s@xmath7 wide intervals are presented in figures [ fig3 ] and [ fig4 ] respectively in the velocity range from 40 to 70km0.2em s@xmath7 . the values printed in the upper left corner of each figure panel denote the centroid velocity of the particular interval . extended @xmath1co and @xmath2co emission was detected between 0km0.2em s@xmath7 and 25km0.2em s@xmath7 , but these data are not presented here . this low velocity emission most likely originates from local molecular clouds and is not related to the w51 region of interest here . little emission was observed between 25 and 35km0.2em s@xmath7 and at velocities in excess of 75km0.2em s@xmath7 ( see [ analysis ] ) . similar velocity structure is also observed in the 21 cm hi emission lines ( @xcite ) . the following section analyzes the velocity structure in the molecular line maps and identifies individual molecular clouds . the @xmath1co and @xmath2co emission toward the w51 region contains a number of discrete velocity components that overlap in projection both spatially and kinematically . to identify and isolate the emission from these velocity components , multiple gaussians convolved with the spectrometer channel widths were fitted to each spectrum in an automated manner . the free parameters for each gaussian were the amplitude of the spectral line , the mean velocity , and the line width . the number of gaussians fitted to each spectrum was determined by searching for contiguous channels that contain an integrated intensity with a signal to noise ratio @xmath25 3 . channels containing a local antenna temperature maxima ( denoted here as channels @xmath26 ) in each such section were then identified . a local maximum at channel @xmath27 was deemed a `` significant '' peak if the antenna temperature in any channel between c@xmath28 and the neighboring local maximum at channels @xmath29 and @xmath30 decreased by more than 2@xmath31 from the antenna temperature at channel @xmath27 . the number of significant peaks corresponded to the number of gaussians fitted to that section . each spectrum was smoothed to a velocity resolution of 1.5km0.2em s@xmath7 prior to identifying the peaks , although the fits were performed on the 0.5km0.2em s@xmath7 resolution data . spectra with large residuals with respect to the gaussian fits were visually inspected and additional gaussians were added as appropriate . the @xmath2co data were easily decomposed into gaussians in this manner , but it often became difficult to reliably identify the velocity features in the heavily blended @xmath1co lines . also , toward the compact hii regions , some of the structure in the @xmath1co and @xmath2co line profiles can be attributed to the absorption of radiation from hot molecular gas by colder foreground material ( see @xcite ) . away from these compact regions , absorption effects are not as significant , and over most of the cloud , the peaks in the spectral lines should accurately represent the velocity structure along the line of sight . the results from the gaussian decomposition of the line profiles are synthesized in figure [ fig5 ] . the upper panel shows histograms of the mean velocities for the @xmath1co ( thick lines ) and @xmath2co ( thin lines ) gaussians , and the lower panel shows the total integrated intensity in the gaussians as a function of the mean velocity . most of the emission is confined to velocity intervals of 025km0.2em s@xmath7 and 3575km0.2em s@xmath7 . we identify the 025km0.2em s@xmath7 emission as originating from nearby molecular material and the 3575km0.2em s@xmath7 emission with molecular gas in the sagittarius spiral arm . several velocity components occur repeatedly in both the @xmath1co and @xmath2co gaussian fits as signified by the histogram peaks shown in the top panel in figure [ fig5 ] . in particular , velocity components at 7 , 1525 , 44 , 49 , 53 , 60 , 63 , and 68km0.2em s@xmath7 are readily apparent . in terms of the @xmath1co and @xmath2co integrated intensity , the two major features are the 60km0.2em s@xmath7 and 63km0.2em s@xmath7 components . a 58km0.2em s@xmath7 component is indicated as well since that is the centroid velocity of the molecular line emission toward the brightest radio continuum source in the w51 region ( @xcite ) . note , however , that the 58km0.2em s@xmath7 component is not a prominent feature as judged from figure [ fig5 ] . the following discussion briefly highlights the morphology of the individual velocity components . the 7km0.2em s@xmath7 velocity component contains weak , narrow lines over nearly the entire mapped region and is undoubtedly a nearby molecular cloud . the 1525km0.2em s@xmath7 interval appears to contain a few distinct velocity features ( see fig . [ fig5 ] ) , but it is unclear whether or not these components are physically related . the 44km0.2em s@xmath7 cloud is elongated parallel to the galactic plane at @xmath32 , although this cloud may form part of a larger structure that extends to lower galactic latitudes . the molecular gas at these lower latitudes occurs at velocities of @xmath940km0.2em s@xmath7 , which is outside the velocity range assigned to this feature . the @xmath1co emission from the 49 and 53km0.2em s@xmath7 components is more fragmented than the other features mentioned so far . these fragments may represent either individual clouds or the remnants of a once larger cloud in the galactic plane . the 53km0.2em s@xmath7 cloud is distinguished by bright @xmath1co emission near @xmath33 @xmath9@xmath34 that is associated with the compact hii region g49.4 - 0.3 ( see [ dis : hii ] ) . the 63km0.2em s@xmath7 component extends for nearly a degree in length and is found mainly in the central and eastern part of the mapped region . this is best observed in the 66km0.2em s@xmath7 panel shown in figure [ fig3 ] , which represents the line wing emission of this velocity component ( as well as emission from the 68km0.2em s@xmath7 cloud discussed below ) . the 60km0.2em s@xmath7 component consists predominantly of a diffuse patch of emission that extends into a filament to the east , and a second filament to the south . the 60km0.2em s@xmath7 and 63km0.2em s@xmath7 velocity components , along with the 68km0.2em s@xmath7 cloud discussed below , likely correspond to the `` high velocity stream '' of 21 cm hi emission ( @xcite ) that has been attributed to the streaming motions of gas down the sagittarius spiral arm . careful inspection of the channel maps indicates that the spatial distribution of the 60km0.2em s@xmath7 and 63km0.2em s@xmath7 components generally do not overlap . for example , the western edge of the 63km0.2em s@xmath7 component closely matches the eastern edge of the 60km0.2em s@xmath7 emission . this is best seen in figure [ fig4 ] and the 60km0.2em s@xmath7 and 66km0.2em s@xmath7 velocity panels in figure [ fig3 ] . further , the extended emission from the eastern portion of the 63km0.2em s@xmath7 component is just above the filamentary extension of the 60km0.2em s@xmath7 component . such interfaces are unlikely to occur by chance from two unrelated clouds along the line of sight , and suggest that the 60 and 63km0.2em s@xmath7 components represent kinematic structure within a single molecular cloud . velocity differences of this magnitude are commonly observed in nearby molecular clouds ( e.g. @xcite ) . the 58km0.2em s@xmath7 component is dominated by bright , compact molecular emission and does not contain the diffuse extended emission that characterizes the 60 and 63km0.2em s@xmath7 features . inspection of the channel maps suggests that this velocity component also reflects the interval kinematic structure within a single cloud encompassing the 60km0.2em s@xmath7 and 63km0.2em s@xmath7 clouds . for example , the emission from the filament protruding to the southern portion of the mapped region contains primarily a centroid velocity of 58km0.2em s@xmath7 closest to bright compact @xmath1co and @xmath2co emission . further away from this bright , compact emission region , the velocity of the filament changes to @xmath960km0.2em s@xmath7 . similar velocity patterns are observed in emission features along the eastern and western edges of the map . these results suggest that the emission constituting the 58 , 60 , and 63km0.2em s@xmath7 components represent the internal velocity structure within a single molecular cloud . koo ( 1997 ) reached similar conclusions concerning the atomic hydrogen clouds at these velocities based upon hi absorption observations toward the radio continuum sources . since the bright @xmath1co emission associated with the 58km0.2em s@xmath7 and 60km0.2em s@xmath7 components are coincident with the brightest radio continuum source in the w51 region ( g49.5 - 0.5 ; see [ dis : hii ] ) , we henceforth refer to the 586063km0.2em s@xmath7 components as the w51 molecular cloud . the 68km0.2em s@xmath7 component extends for @xmath91 east west across the image and also likely constitutes part of the `` high velocity stream '' identified in hi surveys ( @xcite ) . interestingly , the 68km0.2em s@xmath7 filament is located at the southern edge of the diffuse emission associated with the w51 molecular cloud at velocities @xmath1563km0.2em s@xmath7 ( see figs . [ fig3 ] and [ fig4 ] ) . again , such a clear truncation of the w51 molecular cloud at the location of the 68km0.2em s@xmath7 cloud is unlikely to occur from two random clouds along the line of sight , and strongly suggests that these two clouds are physically related objects at a common distance . nonetheless , the elongated appearance of the 68km0.2em s@xmath7 cloud is in stark contrast to the roughly circular shape of the w51 cloud , indicating that these two objects are best treated as individual structures rather than a single molecular cloud . the physical properties of the molecular clouds in the w51 region can be determined from the gaussian decomposition of the line profiles . we single out these two clouds since they contain four of the five bright hii regions found in radio continuum surveys ( see [ dis : hii ] ) . in deriving the properties , the distance to the w51 cloud is assumed to be 7.0 @xmath35 1.5 kpc as determined from proper motion studies of the w51main @xmath36 maser in the g49.5 - 0.4 dense core ( @xcite ) . the 68km0.2em s@xmath7 cloud was assumed to have the same distance based on its apparent association with the w51 cloud as discussed above . the other clouds are not included in this analysis since their distances are unknown . the properties of the w51 and 68km0.2em s@xmath7 clouds are summarized in table [ tbl-1 ] . the cloud size represents a visual estimate of the extent of the detectable @xmath1co emission along the major ( @xmath37 ) and minor ( @xmath38 ) axis of the cloud . the cloud line width , @xmath39 , is the full width at half maximum of the sum of the gaussian fits comprising the respective clouds . two estimates of the cloud mass are provided in table [ tbl-1 ] . the virial mass , @xmath40 , was calculated using the expression @xmath41 , where @xmath42 is the full width at half maximum line width in kilometers per second and @xmath43 is the cloud radius ( @xmath44 ) in parsecs at the zero intensity level . while the cloud size is actually measured at a finite @xmath1co intensity level , we find it unlikely that the clouds are appreciably larger at lower intensities , and no correction to the observed cloud size was applied . note that the above expression for the virial mass is appropriate for a uniform density , spherical cloud . the equivalent expression for a @xmath45 and @xmath46 density cloud would decrease the constant factor in the virial mass expression to 188 and 125 respectively . a second mass estimate can in principal be obtained from the @xmath1co and @xmath2co data using the lte analysis ( @xcite ) . however , in blended regions , it is often difficult to associate @xmath1co gaussian fits with analogous @xmath2co features . therefore , the h@xmath11 column densities were estimated by applying a constant conversion factor to the @xmath1co integrated intensities ( @xcite ; @xcite ; @xcite ) . to ensure that the galactic conversion factor is indeed valid for the w51 region , the conversion factor was estimated from lines of sight with unblended @xmath1co and @xmath2co lines in the 5671km0.2em s@xmath7 velocity interval that defines the w51 and 68km0.2em s@xmath7 clouds . the h@xmath11 column densities for these lines of sight were estimated using the procedure outlined by dickman ( 1978 ) and assuming a @xmath2co / h@xmath11 abundance of 1.5 x 10@xmath47 . a histogram of the ratio of the h@xmath11 column densities to @xmath1co integrated intensities is strongly peaked with a mean value of 1.7 x 10@xmath48 @xmath49 ( kkm0.2em s@xmath7)@xmath7 , and is similar to the conversion factor that has been derived for the galaxy ( between @xmath9 2 - 3 x 10@xmath48 @xmath49 ( kkm0.2em s@xmath7)@xmath7 ; @xcite and references therein ) . the masses ( @xmath50 ) computed from the @xmath1co integrated intensities were calculated by adopting the conversion factor derived from the w51 data , and include a multiplicative factor of 1.36 to incorporate the mass contribution from heavier elements . the results summarized in table [ tbl-1 ] indicate that the w51 molecular cloud has a mean diameter of @xmath997 pc and a mass cloud and several other velocity components that we did not associate with the w51 molecular cloud . ] of @xmath910@xmath51 . the similarity in the two mass estimates indicate the gravitational potential energy is approximately equal to the kinetic energy , and that self gravity must play a critical role in the evolution of the w51 molecular cloud . the 68km0.2em s@xmath7 cloud is 136 pc in length but only 22 pc wide over most of the minor axis . such an obvious departure from spherical symmetry renders the virial mass estimate suspect , and so the mass of the 68km0.2em s@xmath7 cloud was estimated only using the @xmath1co conversion factor . the mass obtained , @xmath910@xmath52 , is an order of magnitude less than that of the w51 cloud . a comparison of the w51 cloud properties with the size and mass spectrum of molecular clouds in the galaxy ( e.g. @xcite ; @xcite ) shows that the w51 molecular cloud is one the largest gmcs in the galactic disk . among the @xmath95000 molecular clouds with diameters in excess of @xmath922 pc ( and corresponding masses @xmath910@xmath52 ; i.e. gmcs ) , the w51 gmc is in the upper 1% of the clouds by size and the upper 510% by mass . for the 68km0.2em s@xmath7 cloud , although its mass is typical of a relatively low mass gmc , a distinguishing feature is its shape . the ratio of the major to minor axis of the 68km0.2em s@xmath7 cloud is @xmath96 . in the cloud catalog by solomon et al . ( 1987 ) , 85% of the clouds have an aspect ratio less than 2 , and only one object has an aspect ratio greater than observed for the 68km0.2em s@xmath7 cloud . clearly an elongated shape over such a large length scale is unusual in the molecular interstellar material and may indicate that the 68km0.2em s@xmath7 cloud is a transient structure originating from a relatively recent dynamical event . now that the individual molecular clouds in the w51 region have been identified , their relationship to the massive star forming sites can be explored . three images of the w51 region are shown in figure [ fig6 ] : a map of the integrated @xmath1co(10 ) intensity generated from gaussian fits with mean velocities between 56 and 71km0.2em s@xmath7 ( i.e. the w51 and 68km0.2em s@xmath7 clouds ) , an image of the 60 emission from the iras sky survey atlas , and a @xmath1321 cm radio continuum image ( @xcite ) . the bulk of the 60 emission is elongated parallel to , but slightly below , the galactic plane , and is coincident with bright radio continuum emission . of the sources labeled in the @xmath1321 cm continuum map , w51c has predominantly a non thermal continuum spectrum and is thought to be a supernova remnant ( @xcite ) , while the other sources have thermal spectra and are compact hii regions . the radio continuum sources g49.4 - 0.3 and g49.5 - 0.4 are classically referred to as w51a ( @xcite ) , with the g49.5 - 0.4 region containing the infrared source w51irs1 ( @xcite ) and the @xmath36 masers w51north , w51south , and w51main ( @xcite ) . sources g48.9 - 0.3 , g49.1 - 0.4 , and g49.2 - 0.4 are collectively known as w51b ( @xcite ) . the observed radio continuum fluxes ( @xcite ) and far infrared luminosities ( @xmath9 10@xmath53 ; @xcite ; @xcite ) imply the presence of one or more o stars in each of these regions . figure [ fig6 ] shows that many of the radio continuum sources have corresponding peaks in the molecular line and far infrared images . in addition to the spatial coincidences , these radio continuum sources have recombination line velocities ( @xcite ) similar to the velocity components identified from the @xmath1co observations . the g49.5 - 0.4 hii region has a recombination line velocity of 59km0.2em s@xmath7 and is spatially coincident with the strong @xmath1co emission associated with the w51 molecular cloud . the recombination line velocities toward g48.9 - 0.3 , g49.1 - 0.4 , and g49.2 - 0.4 are 66 , 72 , and 66km0.2em s@xmath7 respectively and are located along the 68km0.2em s@xmath7 molecular cloud . finally , the g49.4 - 0.3 hii region has a recombination line velocity of 53km0.2em s@xmath7 and is coincident with the bright @xmath1co and @xmath2co emission from the 53km0.2em s@xmath7 molecular cloud . ( note that the molecular gas associated with this hii region is outside the velocity range that defines the w51 gmc . ) hi ( @xcite and h@xmath54co ( @xcite ) spectra toward this source exhibit absorption features at velocities of 63 - 65km0.2em s@xmath7 , indicating that g49.4 - 0.3 must be located behind the w51 gmc . however , these observations do not indicate whether this source is situated just beyond w51 and hence is physically related to the hii region complex , or if it is a distant , unrelated background massive star forming site . the molecular line maps presented here provide no compelling reason for ( or against ) such an association . to search for any additional massive star forming sites in the w51 molecular cloud , the iras point source catalog was examined for objects that have at least two `` high '' quality detections among the four iras bands and a rising spectral energy distribution toward longer wavelengths . these criteria were designed to select a reliable sample of objects with far - infrared colors characteristic of embedded star forming regions ( @xcite ) . of the 40 iras point sources within the mapped region that meet these criteria , the seven brightest objects at 25 are located along the interface between the w51 and the 68km0.2em s@xmath7 clouds . visual inspection of iras 60 image in figure [ fig6 ] confirms that the brightest sources are located along this ridge . assuming that the other point sources are located at the distance of the w51 molecular cloud ( although many of them are almost certainly foreground objects ) , point sources found away from this interface have far - infrared luminosities in the four iras bands less than 40,000 @xmath55 and inferred spectral types later than zams b0.5 ( @xcite ) . thus embedded o type stars in the w51 region currently are confined to the southern extreme of the molecular cloud . the extreme star formation characteristics of the w51 region raises the question as to whether the star formation activity stems from unusual _ global _ properties in the w51 molecular cloud , or from unusual conditions found _ local _ to the massive star forming sites . these possibilities can be explored by comparing the w51 cloud properties with other star forming regions that have also been extensively mapped in @xmath1co and @xmath2co . in particular , we shall compare the w51 and 68km0.2em s@xmath7 clouds with the molecular clouds associated with the hii regions sh 140 , sh 155 , sh 235 , sh 247 , sh 252 , and sh 255 which have been mapped with the same receiver and telescope used for the w51 observations ( @xcite ; @xcite ) . the embedded high mass stellar content in this comparison sample is generally limited to a single early b / late o type star and is in stark contrast to the cluster of o stars forming in the w51 molecular cloud . the clouds in the comparison sample have diameters of @xmath92055 pc and masses of @xmath9@xmath56 to @xmath57 . thus the w51 molecular cloud is @xmath152.5 times larger and @xmath1510 times more massive than these objects . note that the molecular clouds associated with sh 247 , sh 252 , and sh 255 are distinct regions within the gem ob1 cloud complex , which has a total mass of @xmath58 and a diameter of @xmath9150 pc ( @xcite ) . while the spatial size of the gem ob1 complex is larger than the w51 cloud , the w51 cloud exhibits a continuous structure @xmath9100 pc in size as opposed to the fragmentary appearance of the gem ob1 complex . the 68km0.2em s@xmath7 cloud is also more massive than the objects in the comparison sample and is significantly more elongated than any of the clouds considered here . thus the w51 and 68km0.2em s@xmath7 clouds are at the extreme in terms of cloud masses and sizes compared to objects in this sample . with the possible exception of the 68km0.2em s@xmath7 cloud , these clouds are similar though in that they appear to be gravitationally bound . the size and mass of a cloud are not necessarily the key parameters that control the star formation activity within the cloud . intuitively , one might expect that the amount of matter above a critical density to be the critical variable for otherwise similar clouds . thus if the massive star formation activity in w51 has resulted from the large scale collapse of the cloud , one would expect the volume and column density densities to be larger than found in a typical cloud . as an indirect measure of the h@xmath11 column densities , figure [ fig7 ] shows histograms of the observed @xmath2co integrated intensities for each of the clouds in our sample . the lowest integrated intensity shown for any cloud is 3 kkm0.2em s@xmath7 since that is approximately the highest 3@xmath59 detection limit among the various @xmath2co surveys . comparison of the @xmath1co and @xmath2co intensities suggests that the @xmath2co emission is optically thin if the @xmath1co and @xmath2co excitation temperatures are equal as assumed in the lte analysis ( see @xcite ) . therefore , the distribution of @xmath2co integrated intensities should accurately trace the h@xmath11 column density distributions as along as the @xmath2co abundance is roughly constant within a cloud . the @xmath2co integrated intensities in figure [ fig7 ] can be converted to h@xmath11 column densities for an assumed @xmath2co / h@xmath11 abundance of 1.5 x 10@xmath47 ( @xcite ) with the formula @xmath60 for an excitation temperature of @xmath61 = 10 k , the 3 kkm0.2em s@xmath7 integrated intensity limit imposed for figure [ fig7 ] corresponds to an h@xmath11 column density of 1.5 x 10@xmath62 @xmath49 , or an visual extinction of @xmath91.5@xmath63 ( @xcite ) . variations in the @xmath2co abundance and excitation conditions will obviously effect the absolute conversion from @xmath2co integrated intensities to h@xmath11 column densities , and the comparisons here are intended to search for large differences ( factors of several or more ) in the typical column densities in these clouds . figure [ fig7 ] shows that the distributions of @xmath2co integrated intensities peak near the detection limit of 3 kkm0.2em s@xmath7 for each cloud , with a long tail toward the higher integrated intensities . the tail of these distributions correspond to the high column density regions and are often associated with star forming sites . the mean @xmath2co integrated intensity among the clouds varies between 4.9 and 9.7 kkm0.2em s@xmath7 , with the w51 cloud containing the third highest mean intensity and the 68km0.2em s@xmath7 cloud the highest . the high values found for the 68km0.2em s@xmath7 cloud may be a result of an unusual viewing angle , as either this cloud is a sheet of gas observed edge on or a long , narrow filament . these results imply that most of the mass in each cloud is contained in lines of sight with column densities corresponding to less than a few magnitudes of visual extinction . thus the w51 gmc is similar to other clouds in that the diffuse envelope contains more mass than the high column density cores . while the w51 cloud contains a higher column density on average than the other clouds , this can attributed to the fact that it is more than twice as large as some clouds in the sample . indeed , assuming that the @xmath64 w51 cloud is distributed in a sphere of diameter of 97 pc ( see table [ tbl-1 ] ) , the average h@xmath11 volume density is 40 @xmath65 , comparable to the volume density inferred in nearby molecular clouds ( @xcite ; @xcite ) . this suggests that the entire w51 molecular is probably not in an advanced stage of collapse , and that the intense star formation activity in w51 likely results from forces acting on a localized region . in retrospect , this is perhaps not surprising given that the massive star forming regions in w51 are located at the edge of the cloud and not in the center as expected if , for example , the entire cloud was systematically collapsing . contrary to the global properties , the gas properties local to the w51 massive star forming regions do appear to be unusual compared to the molecular clouds in the solar neighborhood . submillimeter continuum observations have shown that the core containing the g49.5 - 0.4 hii region contains @xmath9 10@xmath52 of gas with a mean h@xmath11 volume density of @xmath9 5 x 10@xmath66 @xmath65 over a 3 pc radius region ( @xcite ) . by contrast , submillimeter observations indicate that the most massive cores in nearby molecular clouds typically have masses @xmath923 orders of magnitude less than that of the g49.5 - 0.4 core ( e.g. @xcite ; @xcite ; @xcite ; @xcite ) . while the g49.5 - 0.4 core does not necessarily contain higher gas densities , it does contain more gas at the densities needed to form stars . the above discussion suggests that the key to understanding the massive star formation activity in the w51 complex is determining the forces that acted on a localized region within the cloud . the molecular line maps presented here allow us to speculate on what these forces may be . as shown in figures [ fig3 ] and [ fig4 ] , the diffuse emission from the w51 gmc truncates at the location of the 68km0.2em s@xmath7 cloud for velocities @xmath15 63km0.2em s@xmath7 . this morphology was used in [ analysis ] to argue that the w51 and 68km0.2em s@xmath7 clouds are at a common distance since such an interface is unlikely to result from the chance superposition of unrelated molecular clouds . one way such an interface could form is if the 68km0.2em s@xmath7 cloud has collided into the w51 gmc ( see also @xcite ; @xcite ) . molecular gas does extend below the 68km0.2em s@xmath7 cloud at lower velocities , however . in this picture , given the three dimensional structure of the w51 gmc , this material has not crossed the path of the 68km0.2em s@xmath7 cloud . both h@xmath11co and hi spectra toward the g49.5 - 0.4 hii region ( @xcite ; @xcite ; @xcite ) contain absorption lines at velocities @xmath15 65km0.2em s@xmath7 , and relative to the line of sight , the 68km0.2em s@xmath7 cloud must be located in front of the w51 gmc . since the velocity difference between the two clouds ( at least 5 km0.2em s@xmath7 ) is larger than the sound speed , a shock front will form that will compress the molecular gas and possibly induce star formation ( @xcite ) . the qualitative model of two colliding clouds accounts for several properties of the w51 region . first , one would expect that star formation should occur preferentially along the interface region . the iras image in figure [ fig6 ] shows that this is indeed the case for massive stars , as the northern half of the w51 gmc appears devoid of embedded o type stars despite containing most of the cloud mass . the cloud collision model would also suggest that the massive star forming regions along the collision interface should have a common age . while the ages of these stars are not known , the lifetime of the mid o type stars found in the w51 region ( @xcite ; see also @xcite ) places an upper limit to the stellar ages of @xmath95 myr ( @xcite ) . the actual ages may be considerably less since these o stars are still in the compact hii region phase , which has a lifetime of less than 1 myr ( @xcite ; @xcite ) . further support for a cloud cloud collision model comes from considering the projected distance ( 73 pc ) between the two furthest separated star forming regions along the w51/68km0.2em s@xmath7 cloud interface . the sound travel time across this distance is nearly two orders of magnitude larger than the o star lifetime . therefore , these star forming regions must have been created by either a single event operating along the entire southern edge of the cloud or from up to 4 separate , but nearly simultaneous , events . given the scarcity of embedded o stars in the galaxy , the cloud collision model would provide a natural explanation for simultaneous star formation along the ridge . ultimately , the suggested collision between the w51 and 68km0.2em s@xmath7 clouds may be related to a spiral density wave . the anomalous gas velocities in the w51 region have long been attributed to streaming motions in the sagittarius spiral arm ( @xcite ; @xcite ) . the associated spiral density wave may have indirectly led to the massive star formation activity in the w51 region by enhancing the number density of clouds and increasing the probability of a cloud - cloud collision . further , a spiral wave shock , if present ( see @xcite ) , will compress and flatten any clouds ( @xcite ) . indeed , such a shock could account for the highly elongated shape of the 68km0.2em s@xmath7 cloud . the w51 gmc , however , remains roughly circular in shape , and globally its evolution is likely still dominated by self gravity . finally , we briefly consider the implications of these results for other star forming regions . while the w51 star forming region exceeds all nearby embedded star forming sites in terms of the number of o stars , bolometric luminosity , and dense core mass , w51 itself is dwarfed by some star forming regions in nearby galaxies . most notably , the 30 doradus region in the large magellanic cloud contains an order magnitude more o stars than w51 ( @xcite ) . at larger distances , many interacting galaxies contain even yet more vigorous massive star forming regions that may be dense enough to represent young globular clusters ( @xcite ) . most of the molecular gas appears to have been dispersed in these clusters already , and in any event , the distances to these systems precludes any detailed studies of the natal clouds . the most significant piece of information afforded by the w51 molecular maps is that despite containing one of the most massive dense cores known in our galaxy , most of the mass in the w51 molecular cloud is not currently forming massive stars . thus it is easy to imagine that the increase in the number of cloud cloud collisions that presumably results in interacting galaxies may lead to a larger number of star forming regions throughout a single cloud or more intense star formation activity within a small region . in fact , the mass within the w51 molecular cloud is comparable to that in globular clusters , and it may not require the conglomeration of many giant molecular clouds to form these stellar systems , but the large scale collapse of a single gmc . we have mapped a 1.39 x 1.33 region toward the w51 hii region complex at 45 - 47 resolution and 50 sampling in the j=10 transitions of @xmath1co and @xmath2co . from these data we have identified the major molecular clouds and have associated these clouds with the massive embedded star forming sites in the w51 region . we find that : \(1 ) the two most prominent clouds in the w51 region are the 58 - 6063km0.2em s@xmath7 cloud ( defined as the w51 gmc ) and the 68km0.2em s@xmath7 cloud . the w51 gmc is associated with the brightest @xmath136 cm continuum source in w51 ( g49.5 - 0.4 ) , and the 68km0.2em s@xmath7 cloud contains the hii regions g48.9 - 0.3 , g49.1 - 0.4 , and g49.2 - 0.4 . published absorption line spectra ( @xcite ; @xcite ) indicate that the fifth bright hii region in the area , g49.4 - 0.3 , must be located behind the w51 gmc , but it remains unclear whether or not it is physically associated with the other star forming regions . \(2 ) the mass of the w51 gmc and the 68km0.2em s@xmath7 clouds are @xmath9 1.2@xmath310@xmath51 and @xmath67 respectively . the w51 molecular cloud is roughly circular in shape with a mean diameter of @xmath997 pc and appears to be gravitationally bound . compared to the @xmath95000 gmcs in the galactic disk , w51 is among the top 1% by size and the top 510% in terms of cloud mass . the 68km0.2em s@xmath7 cloud is an elongated filament of @xmath9 136pc@xmath322pc in size . the 6:1 aspect ratio of the major and minor axis in the 68km0.2em s@xmath7 are rare in the galaxy over such a large size scale ( @xcite ) , and suggests that this molecular cloud may represent a transient feature . \(3 ) the properties of the w51 and 68km0.2em s@xmath7 clouds are compared with nearby clouds that have been studied in a similar manner but contain lower levels of massive star formation activity . while the w51 cloud is larger and more massive than nearby clouds , the mean h@xmath11 column density is not unusual given the large size , and the mean h@xmath11 volume density is comparable . the w51 gmc is similar to other clouds in that most of the molecular mass is contained in a diffuse molecular envelop that is not forming massive stars . the 68km0.2em s@xmath7 cloud contains the largest mean column density among the clouds studied here , but this may be a result of an unusual viewing angle for this elongated cloud . we suggest that much of the star formation activity in the w51 region has not resulted from global collapse of the w51 cloud , but from forces acting on localized regions within the cloud . \(4 ) we speculate that much of the massive star formation activity in w51 has resulted from a collision between the w51 and 68km0.2em s@xmath7 molecular clouds . this conjecture can explain the string of embedded o stars that are spread out for 70 pc along the interface between the w5168km0.2em s@xmath7 clouds , and why massive star formation is currently confined to the southern ridge of the w51 gmc . we would like to thank mark heyer for completing the @xmath2co map of w51 and bon chul koo for making available his @xmath1321 cm continuum image of w51 . jmc acknowledges support from the james clerk maxwell telescope fellowship . dbs was supported in part by nasa grant nagw-3938 . the five college radio astronomy observatory is operated with support from nsf grant 9420159 . arnal , e. m. , & goss , w. m. 1985 , , 145 , 369 bachiller , r. & cernicharo , j. 1986 , , 166 , 283 bally , j. , stark , a. a. , wilson , r. w. , & langer , w. d. 1987 , , 312 , l45 bieging , j. 1975 , in lecture notes in 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we present 45 - 47 angular resolution maps at 50@xmath0 sampling of the @xmath1co and @xmath2co j=1 - 0 emission toward a 1.39 @xmath3 1.33 region in the w51 hii region complex . these data permit the spatial and kinematic separation of several spectral features observed along the line of sight to w51 , and establish the presence of a massive ( @xmath4 ) , large ( @xmath5 ) giant molecular cloud ( gmc ) , defined as the w51 gmc , centered at ( @xmath6km0.2em s@xmath7 ) . a second massive ( @xmath8 ) , elongated ( 136 pc @xmath3 22 pc ) molecular cloud is found at velocities of @xmath968km0.2em s@xmath7 along the southern edge of the w51 gmc . of the five radio continuum sources that classically define the w51 region , the brightest source at @xmath10 cm ( g49.5 - 0.4 ) is spatially and kinematically coincident with the w51 gmc and three ( g48.9 - 0.3 , g49.1 - 0.4 , and g49.2 - 0.4 ) are associated with the 68km0.2em s@xmath7 cloud . published absorption line spectra indicate that the fifth prominent continuum source ( g49.4 - 0.3 ) is located behind the w51 molecular cloud . the w51 gmc is among the upper 1% of clouds in the galactic disk by size and the upper 510% by mass . while the w51 gmc is larger and more massive than any nearby molecular cloud , the average h@xmath11 column density is not unusual given its size and the mean h@xmath11 volume density is comparable to that in nearby clouds . the w51 gmc is also similar to other clouds in that most of the molecular mass is contained in a diffuse envelope that is not currently forming massive stars . we speculate that much of the massive star formation activity in this region has resulted from a collision between the 68km0.2em s@xmath7 cloud and the w51 gmc . 10000 10000

You are an expert at summarizing long articles. Proceed to summarize the following text: first experimental studies of electron transport through single molecules attached to metal contacts by using a scanning tunnelling microscope ( stm ) or mechanically controlled break junction techniques @xcite , also triggered considerable theoretical activity in this field since the beginning of the new millennium . the theoretical framework most widely used in this context is a non - equilibrium green s function ( negf ) formalism @xcite , where coherent electron transmission according to landauer s theory is assumed . the conductance of a molecular junction can then be described in dependence on the incoming electron s energy @xmath1 in terms of the transmission probability @xmath2 , which within negf is defined as @xmath3,\ ] ] where the self energy matrices @xmath4 and @xmath5 contain the coupling of the molecule to the left and right electrodes and @xmath6 and @xmath7 are the retarded and advanced green s functions ( gf ) of the ( extended ) molecule . the negf formalism has been implemented in a variety of codes , where in combination with density functional theory ( dft ) it allows for a first principles treatment of the conductance of single molecule junctions @xcite@xmath8@xcite . the usefulness of such calculations , however , still relies on simple models for interpreting them in terms of quantum chemical concepts such as atomic orbitals ( aos ) or molecular orbitals ( mos ) in order to achieve a qualitative understanding of the observed electron transport characteristics in the context of our general knowledge of the electronic properties of molecules or what is regarded as chemical intuition . in principle , electron transmission can be viewed as a particular manifestation of the more general phenomenon of electronic communication through a molecule , where a green s function describes the propagation of a perturbation and is a measure of the connected paths made available by the bonding pattern of the molecule @xcite . the zeroth green s function @xmath9 can be used to rewrite the expression for @xmath2 given in eq . [ negf ] under the assumption that the central molecule is coupled to both metal contacts only via a single ao labeled @xmath10 and @xmath11 on the respective side , because then each matrix @xmath12 contains only one non - zero element , @xmath13 and @xmath14 , respectively and therefore a single term remains from the trace resulting in @xmath15 by evaluating the relevant matrix elements of @xmath9 , chemical understanding of the general properties of molecules can arguably be complemented by studying their transmission properties . for example , the low - bias conductance through benzene is orders of magnitude lower when it is contacted at positions which are meta ( _ m _ ) with respect to each other when compared with ortho ( _ o _ ) or para ( _ p _ ) @xcite@xmath8@xcite . this result comes as no surprize for a chemist even without any prior exposure to the theory of molecular conductivity , who knows that the influence of a substituent in a benzene ring is `` felt '' , in both electrophilic and nucleophilic substitution reactions , in o- and p - position to it , while m - positions `` do not communicate '' . this knowledge might be referred to as chemical intuition but is actually based on rules stemming from resonance theory within the valence bond ( vb ) framework , where the relation of electronic communication to the topology of mos is not self - evident but should be contained in @xmath9 . in their classical 1947 - 1950 series of papers coulson and co - workers attempted to put the one - electron green s function ( without using the term at the time ) at the heart of chemical theory @xcite@xmath8@xcite . they demonstrated how starting from a hckel hamiltonian in ao representation basic molecular characteristics such as mo energies , atomic charges , bond orders and response coefficients can be derived directly from the secular determinant without referring to explicit mo vectors , where the relation of this work to electron transport phenomena has been commented on very recently @xcite@xmath8@xcite . although it was correctly claimed by datta @xcite amongst others that in a single molecule junction the conductance is defined not only by the central molecule but rather by the entire system including the metal contacts , the individual contributions of the components are separable in eq . therefore , for the purpose of device design the molecular contribution can be optimized independently from the coupling to the metal contacts , a notion which has been recently confirmed in a joint theoretical and experimental work by manrique et al . @xcite . in this study it was shown that molecules and even their fragments contribute well defined and transferable factors to electron transport as a crucial observation for the investigation of destructive quantum interference ( dqi ) effects , a phenomenon which has been the topic of a tremendous number of recent articles , where for a rather complete bibliography we refer to ref . @xcite . such dqi effects when occurring in the transmission close to the fermi energy @xmath16 can be used for data storage @xcite , inducing thermoelectricity @xcite or the design of logic circuits @xcite . simple models have been proposed for their analysis , which were derived from tight - binding ( tb ) or topological hckel theory and validated by dft calculations : one of them which we refer to as `` the graphical ao scheme '' in the following has been derived specifically for the prediction of dqi and is based on a graphical analysis of the connectivity matrix of atomic orbitals ( aos ) @xcite@xmath8@xcite , while the other interprets the efficiency of transmission in a broader sense in terms of the signs and amplitudes of molecular orbitals on the atomic sites directly connected to the electrodes @xcite@xmath8@xcite , and the analysis is sometimes limited to a `` frontier orbital approximation '' where only the highest occupied ( homo ) and the lowest unoccupied mo ( lumo ) are taken into account . the aim of this article is to reconcile the predictions from these two conceptually different approaches for an interpretation and analysis of the molecular green s function . it is expected that by focusing on either an ao or a mo representation of the same quantum - mechanical problem one should obtain the same results . their reconcilation is akin to that of the vb and mo theories in the earlier days of quantum chemistry by van vleck et al . in 1935 @xcite . but while vb and mo approaches become variationally equivalent for the ground state only in the limit of full configuration interaction , for electron transport within a single particle framework the representations of the molecular green s function in the ao and mo bases are already strictly equivalent on a single determinant level . from eq . [ trans ] it can be seen that it is both necessary and sufficient to evaluate the purely molecular quantity @xmath9 for estimating whether the transmission will be finite or zero at any given energy @xmath1 . the derivation of both `` the graphical ao scheme '' and the mo based scheme mentioned above start from this observation . within a frontier orbital approximation , however , only the homo and lumo are taken into accound instead of all mos contained in @xmath9 and this approximation then limits the range of applicability of the mo based scheme to that of the coulson - rushbrooke pairing theorem @xcite as we explain in detail in the next section . if the contributions of all mos and not only the frontier orbitals to @xmath9 in eq . [ trans ] are correctly accounted for on a quantitative level , however , dqi can be analyzed from a mo perspective leading to equivalent results as the graphical ao scheme from refs . @xcite@xmath8@xcite for all conjugated @xmath0 systems both alternant and non - alternant , with and without hetero - atoms and regardless of which subset of sites the contact atoms belong to , which is the main message of our article . the paper is organized as follows : in the next section we shortly review the graphical ao scheme and highlight its relation to eq . [ trans ] . here we also explain on the basis of the pairing theorem that dqi effects entering @xmath9 can in general only be quantitatively described and understood in terms of the onsite energies of all mos and their respective amplitudes at the contacted atomic sites . furthermore , we clarify the connection of such a mo centered analysis scheme to larsson s formula , which has been originally proposed for the definition of an effective coupling from the mo contributions to the transfer integral in a marcus theory description of electron hopping @xcite@xmath8@xcite but later on also used for the analysis of coherent electron transport in single molecule junctions @xcite . in the following section we provide computational studies for a variety of test systems in order to substantiate our claim that it is possible to gain understanding of dqi effects in accurate terms for any conjugated @xmath0 system without the limitations of applicability facing the original frontier mo rules . for all the molecular systems in our article numerical calculations on a dft level exist in the literature and most of them have also been studied experimentally . since the focus of our work is on topological properties of @xmath9 , for the calculations we present here topological hckel hamiltonians are used in combination with negf . in this numerical chapter we also present the respective predictions from the graphical ao scheme for all systems as a reference and demonstrate their equivalence to the results obtained from an analysis of mo contributions , where the convergence with respect to the number of mos included plays a prominent role . in the final chapter we provide a summary . the zeroth green s function @xmath9 in eq . [ trans ] describes the propagation of a tunnelling charge between the atomic sites @xmath10 and @xmath11 mediated by all molecular orbitals ( mos ) which in the weak coupling limit can be formulated as @xcite @xmath17^{-1},\ ] ] where @xmath18 is the molecular hamiltonian , i a unity matrix and @xmath19 an infinitesimal imaginary term introduced in order to avoid divergence of @xmath9 at the eigenvalues of @xmath18 . since @xmath9 is obtained from the inversion of the hamiltonian matrix @xmath18 , which is defined in an ao representation , one can analyze the properties of @xmath9 from the ratio of one of the minors of @xmath18 and its determinant @xcite@xmath8@xcite where @xmath9 is only equal to zero when the respective minor , as defined by the connection of the leads to two particular atomic sites @xmath10 and @xmath11 on the molecule , is also zero . in this way the graphical ao scheme for the prediction of dqi effects was derived , which has been formulated as the following set of rules : dqi occurs at @xmath20 if it is impossible to connect the two atomic sites @xmath10 and @xmath11 in a molecular topology , i.e. the only two sites with a direct coupling to the leads , by a continuous chain of paths , and at the same time fulfill the conditions ( i ) two sites can be connected by a path if they are nearest neighbors and ( ii ) at all atomic sites in the molecule other than @xmath10 and @xmath11 , there is one incoming and one outgoing path . in other words , for a finite zero - bias conductance all aos of the molecular topology have to be either traversed within a continuous chain of paths from @xmath10 to @xmath11 or be part of a closed loop in the topology , where the latter can be a double line due to the pairing of connected orbitals or a triangle or any larger loop @xcite . we will demonstrate in the following section how to apply these rules for any given molecular topology in praxis . in a later extension of this scheme it has been clarified that such defined paths can also cancel each other out in special cases and that therefore a sign has to be attributed to them @xcite . it has to be noted that these `` paths '' are just mathematical terms coming from forming the minor of @xmath18 and should not be interpreted as the physical path of an electron moving through the molecule . this graphical ao scheme has the advantage that it allows for the prediction of dqi without any numerical calculations being required simply by a visual assessment of the chemical structure of the central molecule in the junction . the scheme has been designed for molecules with a conjugated @xmath0 system , because it is only @xmath0 electrons which are taken into account in the topological hckel hamiltonian it was derived from . in praxis this is not really a limitation , since potential functional molecules of interest are usually conjugated systems , where @xmath0-transmission is dominant . in order to allow for a simple analytical treatment , the derivation of the scheme also originally assumed sites with identical onsite energies and couplings to each other @xcite . this assumption was later lifted in an attempt to generalize the method now also allowing for hetero atoms in the molecular structure but this came at the price of increased mathematical complexity @xcite . another assumption was that the only energy @xmath1 , where @xmath2 is of interest is the fermi energy because it defines the zero - bias conductance and therefore the rules only apply at @xmath20 . this latter assumption is rather delicate considering that in the model hamiltonian the graphical rules were derived from the onsite energy of carbon sites were artificially set to @xmath16 . quite surprisingly , it was found in praxis that this rather crude approximation did not seem to limit the predictive qualities of the model even for cases where hetero atoms such as oxygen were involved in the molecular structures under investigation @xcite as long as the fermi energy defined by the metal leads lies within the homo - lumo gap of the molecule when energy levels are aligned @xcite . in order to gain a mo perspective of @xmath9 instead of an ao one , @xmath18 has to be looked at in its diagonalized form as @xmath21 , where @xmath22 is the diagonal matrix of mo eigenenergies and @xmath23 is the matrix of the coefficients for the expansion of all mos as a linear combination of all aos in the molecule . inserting this definition of @xmath18 into eq . [ zero ] gives @xmath24 which is the spectral representation of @xmath9 in a hckel ao basis with @xmath25 the coefficient of the l(r)-th ao in the m - th mo in a sum that runs over all @xmath26 occupied and unoccupied mos , which result from the coupling of the aos defining the basis vectors for @xmath18 . it should be stressed that eq . [ spectral ] is exact for any hamiltonian with an orthogonal ao basis and that this spectral representation of @xmath9 served as the starting point for the formulation of the molecular orbital rules for efficient transmission by yoshizawa and co - workers @xcite@xmath8@xcite . for the special case of alternant hydrocarbons ( ah ) , which are molecules with a conjugated @xmath0 system where carbon atoms can be divided into two subsets , `` starred and unstarred '' , such that the atoms of one subset are bonded only to those from the other , the coulson - rushbrooke pairing theorem @xcite applies which states that ( i ) the @xmath0 electron energy levels are symmetrically distributed about the zero energy level ( which is assumed to be @xmath16 in single molecule junctions ) and ( ii ) that each occupied mo obtained from diagonalizing the corresponding hamiltonian in an orthogonal ao basis with an energy @xmath27 has its mirror image in the unoccupied region with the energy @xmath28 , which regarding its shape differs only in the sign of all ao coefficients of one subset . in the following we focus on molecules with an even number of mos , which we can then group in eq . [ spectral ] into pairs of the contributions from the mos whose energies are linked by the symmetry relation it defines , i.e ( h , l ) , ( h-1,l+1), ... ,(h-(n/2 - 1),(l+(n/2 - 1 ) ) with h = homo , l = lumo and @xmath29 . we can then redefine @xmath30 as the sum of these pairs , which in the following we will refer to as coulson - rushbrooke or cr pairs : @xmath31 the pairing theorem now predicts for aos @xmath10 and @xmath11 on carbon atoms of the same subset that @xmath32 because either @xmath33 and simultaneously @xmath34 , or @xmath35 and simultaneously @xmath36 , as all the coefficients in only one subset change their sign when comparing an occupied with its mirrored unoccupied level . therefore , the terms in every cr pair of eq . [ coulson ] cancel exactly at @xmath16 for this case and dqi occurs as a result as has also been observed in refs . @xcite . if on the other hand the contact aos @xmath10 and @xmath11 belong to carbon atoms from different subsets , then @xmath37 because either @xmath33 and @xmath36 , or @xmath35 and @xmath34 . for this case , the contributions coming from the two individual parts of each cr pair of mos ( h - k , l+k ) including the homo and the lumo always add up constructively at @xmath16 in eq . [ coulson ] . although any individual cr pair contribution is therefore nonvanishing , it is important to stress that destructive interference is still possible between cr pairs , as each of them can contribute either a positive or a negative term to @xmath9 . the pairing theorem , however , does not provide the means for an assessment of prediction of such inter - pair interference . the general conclusion from the pairing theorem is therefore that dqi will always occur for the electron transport through junctions containing alternant hydrocarbons when carbon atoms of the same subset are contacted , which is already sufficient to account for the low conductance of a variety of systems such as polyenes with contact atoms of the same parity , meta - contacted benzene or generic cross - conjugated molecules , where these cases can readily be identified from their chemical structure without any deeper analysis of the shapes and signs of their frontier mos . on the other hand , for alternant hydrocarbons contacted on carbons belonging to different subsets , i.e. where one contact atom is starred and the other one unstarred or for non - alternant hydrocarbons or for conjugated @xmath0 systems containing hetero atoms , the pairing theorem can neither predict nor rule out dqi . in the literature these two cases are sometimes distinguished in terms of `` easy zeros '' ( the same subset contacted ) and `` hard zeros '' ( different subsets contacted ) @xcite or linked to the occurence of an odd or even number of zeroes in @xmath2 @xcite . but for the purpose of our article the important distinction is that for even - membered alternant hydrocarbons contacted at sites of the same subset dqi will always occur , while for all other cases dqi can not be predicted without numerical calculations from a mo perspective . we note that our discussion above only refers to alternant hydrocarbons with an even number of mos and therefore also an even number of carbon sites . this is the general case for stable alternant hydrocarbons . when the total number of mos is odd , which implies the existence of a non - bonding mo at the fermi energy with non - vanishing contributions from only one subset follows from the pairing theorem , which then allows for a conduction peak instead of a dqi induced minimum at @xmath16 when the contacted atoms belong to the subset contributing to this non - bonding mo . we now turn our attention to the molecular orbital rules derived by yoshizawa and co - workers @xcite@xmath8@xcite , where the starting point was also the spectral representation of @xmath9 given in eq . [ spectral ] . these rules are amongst the earliest formulated providing a link between the complex phenomenon of dqi in electron transmission and the standard output of quantum chemical calculations , in this case the sign of the amplitudes of mos . within a frontier orbital approximation they also become particularly simple to apply because then the entire sum in eq . [ coulson ] is dominated by only one cr pair , namely the contribution to @xmath30 coming from the homo and the lumo , and then the remaining pairs can all be neglected because their large energetic distance @xmath38 to @xmath16 results in large denominators in the respective terms , thereby making them numerically negligible . from this assumption , it can be concluded that the transport through a single molecule would be effective , i.e. dqi would be absent , when on the two contact atoms to the two leads i ) the sign of the product of the mo expansion coefficients in the homo ( @xmath39 ) is different from that in the lumo ( @xmath40 ) and ii ) all four involved amplitudes @xmath41 , @xmath42 , @xmath43 and @xmath44 are of significant magnitude . if these conditions are not fulfilled , then `` inefficient '' transmission due to at least a partial cancellation of the contributions from the homo and lumo was predicted which was not formulated as necessarily the zero transmission which is typical for dqi in a rigid sense . such a frontier orbital approximation , however , only delivers correct results for the prediction of dqi whre the cr pairing theorem @xcite is applicable . if the atoms contacted by the two electrodes belong to the same subset ( either starred or unstarred ) of carbon atoms in an even - membered ah , the cancellation of the contributions from the homo and the lumo to @xmath30 is a reliable indicator of dqi not necessarily because they are dominant , but because it represents the cancellation of also the contributions within all other cr pairs entering eq . [ coulson ] . this is the reason why dqi can be understood in this case in terms of the frontier orbitals alone . for all other cases all mos in the system need to be considered . if an alternant hydrocarbon is contacted at atomic sites belonging to different subsets , i.e one being starred and one being unstarred according to the cr framework , then although contributions from the homo and lumo can only interfere constructively the tails related to lower lying occupied and higher lying unoccupied mos might still cancel out with those of the frontier orbitals at @xmath16 and cause dqi . for non - alternant hydrocarbons and organic molecules containing hetero atoms , it turns out to be equally insufficient to limit the analysis to just one or even two cr pairs of mo contributions . in the next section we will provide a range of numerical examples justifying this assertion . a somewhat simplified form of eq . [ spectral ] has been known for decades as larsson s formula in a different but related context , where it was used for the definition of the transfer integral mediated by a selected set of mos in a marcus theory description of electron hopping @xcite@xmath8@xcite . more recently , it has been realized @xcite that the same formula can be also employed to define an approximation for @xmath2 in coherent eletron tunnelling as @xmath45 with @xmath46 being an energy dependent effective coupling containing the contributions from all mos of a molecular bridge and defined by @xmath47 here @xmath48 , @xmath49 and @xmath50 are the eigenenergy , and the respective couplings to the left and right contact of the @xmath51th mo and @xmath1 is the kinetic energy of a transferred electron . it is easy to see by direct comparison that the effective coupling @xmath46 in eq . [ gamma ] is very much related to the zeroth green s function in eq . [ spectral ] . there are only two differences between the two equations . first , the mo amplitudes @xmath25 have been replaced by couplings @xmath49 and @xmath50 , which describe the overlap of each mo with a respective contact ao on the two leads . since in the negf - tb description we employ for our numerical studies in the following section the contact ao on the leads is always the same orbital , this difference amounts to just the same constant factor for all the mo terms of the sum in eq . [ gamma ] and is therefore irrelevant for our study where we just set this value to 1 . the second difference between the two expressions , namely the dropping of the infinitesimal term @xmath19 just means that @xmath46 diverges at the eigenergies of all mos . in principle this deficiency can be repaired by introducing an energy dependent normalisation factor as has been derived from the more general theory in ref . @xcite by sautet et al . @xcite under the very limiting condition of @xmath52 , which is relevant when the focus is on the analysis of stm measurements . since the qualitative behaviour of @xmath53 does not deviate from that of @xmath2 obtained from negf - tb in all systems investigated in this article , we avoide such a normalisation factor as an unnecessary complication . in the following section we just truncate @xmath53 as obtained from eq . [ gamma ] at the poles and scale it with the arbitrary constant of @xmath54 for the purpose of its graphical presentation in the related figures . while the poles of @xmath30 in eq . [ spectral ] or @xmath46 in eq . [ gamma ] define the peaks in @xmath2 when these quantities are squared and each of these peaks can be identified with the electron transmission through one individual mo in the absence of degeneracies , the non - resonant transmission for energies between the peaks contains contributions from all mos @xmath51 with the respective couplings @xmath49 and @xmath50 and these contributions can interfere constructively or destructively in dependence on the energy @xmath1 . in the sums of eqns . [ spectral ] and [ gamma ] the sign of the contribution from any mo is determined by i ) the numerator of its corresponding term in the summation as defined by the product of amplitudes at the contact site or product of couplings to the metal leads and ii ) its denominator which depends on whether the energy @xmath1 for which @xmath2 is evaluated lies below or above the onsite energy of the mo in question . for ahs contacted via carbon atoms belonging to the same subset , dqi at the center of the homo - lumo gap can be directly concluded from the cr pairing theorem by using eq . [ coulson ] . for all other cases , neither the pairing theorem applies nor can reliable predictions on dqi be obtained within a frontier orbital approximation , because the occurrence or absence of dqi for any given value of @xmath1 seems to depend on a fine balance of cumulative contributions with different signs from all mos , where a quantitative description of the decay of their respective tails is crucial as we will demonstrate in the next section . all the case studies in this article are based on simple models both in ao and mo representations as derived from the topological properties of the molecular hamiltonian @xmath18 in a tight binding approximation with all onsite energies for the carbon aos contributing to the @xmath0 system set to zero , i.e. to the origin of energy assumed to represent @xmath16 , and the couplings between them to the resonance integral @xmath55 from hckel theory which also defines the unit of the energy axis . as an appropriate numerical benchmark for our conclusions we therefore present negf - tb calculations which have been conducted within the atomic simulation environment ( ase ) @xcite with a coupling of @xmath55 between the molecular topology and the semi infinite carbon chains used for the electrodes . sulfur atoms have been given an onsite energy of 1.11 @xmath55 and c - s bonds a coupling value of 0.69 @xmath55 @xcite in order to account for the effect of the hetero atom . since the molecules we investigate here all have been chosen due to the recent interest they attracted , we also refer to the relevant literature for each system in order to show that our conclusions harmonize with the results from more realistic negf - dft calculations or experimental conductance measurements . as a first example for our arguments in the last section we consider butadiene contacted at different sites ( fig . [ fig1 ] ) . since this molecule with a conjugated @xmath0 system is an even - membered ah , its even- and odd - numbered atoms belong to different starred / unstarred subsets and therefore dqi can be predicted for the ( 1,3- ) connection within a frontier orbital approximation in agreement with the pairing theorem as outlined in section ii b because carbon sites belonging to the same subset are contacted ( fig . [ fig1]a ) . for all other possible connections , sites belonging to different subsets are contacted , and therefore constructive interference of the contributions from the homo and lumo alone is found according to the pairing theorem . the graphical ao scheme ( fig . [ fig1]b ) as well as negf - tb calculations for @xmath2 and their estimates as @xmath53 from eq . [ gamma ] after a diagonalisation of @xmath18 where for both all four mos have been properly accounted for ( fig.[fig1]c ) find dqi not only for the 1,3- but also for the 2,3-connection . within the graphical ao scheme dqi is predicted if it is not possible to form a continuous line between the two contact sites and have all aos which are not on this line grouped up in pairs or as part of a closed loop . the single sites which are not crossed or grouped up are marked by green dots for the sake of clarity , which also allows for the simple correspondence that dqi occurs where green dots are unavoidable . the mo based scheme within a frontier orbital approximation on the other side also correctly predicts dqi for 1,3-positioning of the contacts , but not for the 2,3-connection . while the amplitudes of both the homo and lumo are low on sites @xmath56 and @xmath57 ( fig . [ fig1]d ) , this justifies a reduced conductance but not a cancelling out to zero which is characteristic for dqi and found for the 2,3-connection for @xmath2 . in fig . [ fig1]e we plot @xmath53 when only the contributions from the homo and the lumo enter the expression for @xmath46 in eq . [ gamma ] ( green lines ) . there it can be seen that indeed zero transmission is found also considering only the two frontier orbitals for the 1,3-connection in agreement with the predictions from the pairing theorem , while the 2,3-case shows a reduced but finite conductance when compared to 1,2- and 1,4-positions of the contacts . we also plot @xmath53 from the contributions of the homo-1 and lumo+1 alone ( blue lines ) and find that only for the ( 2,3- ) connection they cross those of the frontier orbitals at @xmath16 . since the corresponding sum of terms entering eq . [ gamma ] for the two pairs have different signs at their crossing point , they cancel each other out and lead to zero transmission . this is probably the simplest example to contrast a case of dqi , which can be predicted within a frontier orbital approximation with one that is beyond its range of applicability . and @xmath11 and with `` stars '' related to the pairing theorem and involving only the @xmath0 electrons for a ) benzene connected in ortho- ( _ o_- ) , meta- ( _ m_- ) and para ( _ p_- ) positions , b ) dinaphthylethene ( dne ) and c ) dithienylethene ( dte ) in their open and closed forms , respectively , where for the latter also the alternant hydrocarbon analogs obtained by removing the s sites are shown . in d ) the chemical structure of azulene is shown and the positions for connecting metal contacts are numbered for making reference to them in the text . ] in figure [ fig2 ] we show the other molecular systems which we investigate as case studies in this article and also the corresponding tb next neighbor connectivity , which provides the basis for all our negf - tb computations as well as the application of the pairing theorem and the graphical ao scheme in the following . we note that only unsaturated carbon atoms are part of the @xmath0 system and that it is only those which need to be considered in a tb framework . for benzene ( fig . [ fig2]a ) , it is established knowledge @xcite@xmath8@xcite that the conductance is finite for an _ p_-connected pair of contacts but dqi occurs at @xmath16 for a _ m_-connection , which is also consistent with the chemical understanding of communication through an aromatic ring . in figure [ fig2]a we illustrate that these findings can be also understood in the context of the pairing theorem ( fig . [ fig2]b ) because only for the m - connection two `` starred '' carbon sites are contacted in this example of an even - membered ah meaning that each cr pair will provide a contribution of exactly zero in eq . [ coulson ] . another type of systems where dqi plays an important role are molecular switches @xcite based on `` conducting '' and `` insulating '' isomers that can be transformed into each other in a highly reversible photochemical reaction . we will consider here one family of such switches , namely diarylethenes @xcite , and in particular the homocyclic dinaphthylethene ( dne ) ( fig . [ fig2]b ) and the heterocyclic dithienylethene ( dte ) ( fig . [ fig2]c ) . for both systems the closed isomer is much better conducting than the open isomer in a molecular junction which has been demonstrated experimentally @xcite and confirmed theoretically @xcite , where the formation ( or breaking ) of a single bond distinguishes one from the other in structural terms . negf - dft calculations can be found e.g. in ref . @xcite and ref . @xcite for dne and dte , respectively . the molecular orbital rules by yoshizawa and co - workers have also been applied for both systems , where although their application in a narrow sense would have suggested constructive interference for the `` closed '' ( conducting ) and `` open '' ( insulating ) form for dne , the differences in conductance between the two forms found with negf - dft has been attributed to the larger orbital amplitudes for the `` closed '' form @xcite . as for the butadiene example we discussed above , this argument explains quantitative differences in the conductance but not the qualitative difference defined by the occurrence or absence of an interference minimum . for dte on the other side , the contributions from the homo-1 and lumo+1 had to be added to those from the frontier orbitals in order to reach better agreement with experimental findings @xcite . as can be seen from their respective tb next neighbor connectivity in figs . [ fig2]b and c the electrodes are attached to carbon atoms which belong to different subsets of the starring scheme for the open forms of both dne and dte . as a consequence of the mo symmetry properties following from the pairing theorem therefore constructive qi has to be found within all cr pairs of mos defined by eq . [ coulson ] including the homo and the lumo , and hence dqi can only occur due to cancellation of terms between different cr pairs which can not be assessed by using a frontier orbital approximation , and this is the reason why we included these systems in the present study . dte is heteorocyclic , which means that we need to include an onsite energy for the sulfur atoms differing from those of the carbon sites and also a value for the c - s coupling in the negf - tb calculations in the following as specified in the computational details . in order to have a reference point for the application of the graphical ao scheme also for dte , we also introduce a structure with the s atoms of dte deleted in fig . [ fig2]c , which transforms it into an even - membered ah , where in the next section we will compare the transmission functions of both dte forms with and without sulfur sites . as the final system for our study we chose azulene ( fig . [ fig2]d ) , which is a non - alternant hydrocarbon and therefore it is not possible to divide its carbon sites into `` starred '' and `` unstarred '' subsets in relation to the pairing theorem or derive any conclusions regarding destructive or constructive qi within a frontier orbital approximation . this system is also of interest because it has been wrongly claimed in a joint experimental and theoretical study that for azulene the graphical ao scheme also fails in its predictions at least when the electrodes contact the sites @xmath58 and @xmath57 in fig . [ fig2]d @xcite . this claim has later been refuted @xcite , where it was shown that the predictions of the graphical ao scheme were also correct for azulene with @xmath58,@xmath57-contacts when closed loops of aos and not only pairs of them are considered which always has been a central aspect of the scheme @xcite . negf - dft calculations for azulene containing compounds with different electrode contact sites including the @xmath58,@xmath57- and @xmath59,@xmath60-cases can be e.g. found in ref . @xcite and ref . @xcite . , @xmath57- and @xmath59,@xmath60-positions . dqi is predicted if it is not possible to form a continuous line between the two contact sites and have all aos which are not on this line grouped up in pairs or as part of a closed loop . the single sites which are not crossed or grouped up are marked by green dots for the sake of clarity . ] in fig . [ fig3 ] , @xmath2 as calculated from negf - tb for all systems introduced in fig . [ fig2 ] is shown in the left panels for each label and compared with @xmath53 in the right panels , which was obtained from eq . [ gamma ] with mo onsite energies and amplitudes resulting from a diagonalisation of @xmath18 in the same tb setup , where for both quantities the curves for the systems exhibiting dqi at @xmath16 are shown in red and the others in black . apart from the units , which are in multiples of the conductance quantum @xmath61 for @xmath2 and chosen arbitrarily for @xmath53 the agreement in all cases is excellent , which fully justifies to investigate the absence or occurrence of dqi solely in terms of the contributions entering eq . [ gamma ] . from both @xmath2 and @xmath53 the m - connection is correctly identified as the only one with dqi for benzene ( fig . [ fig3]a ) and zero conductance found only for the open form of dne ( fig . [ fig3]b ) , a result which needs the inclusion of all mos and not just the frontier orbitals @xcite as we will further argue below . also for dte ( fig . [ fig3]c ) only the open form exhibits a transmission zero at @xmath16 for the analog alternant hydrocarbons ( right two panels ) , which is shifted to higher energies if the s atoms are included in the calculations ( left two panels ) but still lowers the conductance at the fermi energy in its vicinity quite substantially even then . for azulene ( fig . [ fig3]d ) there are qi minima across the energy spectrum for the two investigated junctions which differ in their respective contact sites . but while contacts in the @xmath59,@xmath60-positions ( red lines ) result in zero conductance , the minima for the @xmath58,@xmath57-connected system are not only lying above @xmath16 but are also so narrow that they do not seem to have an impact on @xmath62 . we note that all these features we summarized here are in good qualitative agreement with the respective negf - dft calculations in the literature we referred to in the last section when introducing the respective molecular structures above . in fig . [ fig4 ] we demonstrate the application of the graphical ao scheme @xcite for all systems in fig . [ fig2 ] without hetero atoms and its predictions for dqi identify the cases with @xmath63 in the calculations shown in fig . [ fig3 ] without a single failure , regardless of whether the molecular topology belongs to an alternant or non - alternant hydrocarbon or which subset of carbon atoms in relation to the `` starring '' scheme the electrodes are connected to . in principle , it would also be possible to account for the presence of hetero atoms within the scheme as it has been done elsewhere @xcite for a treatment of dte containing its sulfur sites but this would come at the price of diminishing the scheme s simplicity and would not provide any important arguments for the present discussion . this ao scheme considers all orbitals in an ao representation and relies on the structure of the entire hamiltonian thereby strongly reflecting the respective molecular topology . this is in contrast to any frontier orbital approximation within a mo based scheme where by definition all but two mos are disregarded . the pairing theorem justifies this omission for the specific case , where each cr pair defined by the respective equal distance of its parts to @xmath16 cancels out individually for symmetry reasons . for the other cases where interference is constructive within each pair dqi can still occur due to cancellation between pairs this is why all mo contributions are significant in this latter case as we will further explore below . independently of the frontier orbital approximation as we discuss it in this article , there is the common conception in studies of the conductance of single molecule junctions that @xmath62 is dominated by the mos close to @xmath16 since the tails of the peaks further apart decay rapidly and their contributions can therefore safely be disregarded @xcite . this assumption is also motivated by the fact that the respective distance of each mo to the fermi energy enters its respective term in eqns . [ spectral ] and [ gamma ] explicitly in the denominator and its increase can therefore be expected to reduce the terms significance . there are two reasons why such preconceptions should be questioned regarding their validity in general : i ) while it is true that the denominator of a term in eqns . [ spectral ] and [ gamma ] increases for high values of @xmath64 , this effect might be outweighed by large couplings or large mo amplitudes at the contact sites ; ii ) the distinction between the occurrence and absence of dqi is often one between an exact value of zero ( at least in the framework of tb where only @xmath0 electrons are considered ) and a rather small number which appears to be bigger than it actually is due to the logarithmic plotting of @xmath2 . in fig . [ fig5 ] we increase the number of cr pairs of mos included in the sum of eq . [ gamma ] for the calculation of @xmath53 stepwise for the systems in fig . [ fig2 ] which exhibit dqi close to the fermi level but where this can not be predicted within a frontier orbital approximation . here we first consider only the homo and the lumo ( label @xmath58 ) , then the homo , the lumo , the homo-1 and the lumo+1 ( label @xmath56 ) and so on where only the red curve with the highest label includes the contributions from all cr pairs of mos corresponding to the respective molecular topology . quite contrarily to what might be expected , for the open forms of dne ( fig . [ fig5]a ) and dte ( fig . [ fig5]b and c ) the convergence of @xmath53 with the number of included pairs is not smooth but oscillating because the contributions of cr pairs of mos enter with alternating signs . even for the label with only the cr pair with the energies most remote from @xmath16 missing , i.e. the labels @xmath65 , @xmath66 and @xmath59 for dne , dte without and with s , respectively , the conductance is still far from zero on a logarithmic scale . for the label @xmath57 with the three cr pairs of mos closest to the fermi energy included it even has a magnitude comparable to that of the conducting closed form of the respective switch , where for dte the convergence behavior seems to be rather unaffected by the presence or absence of the s atoms . this analysis strongly indicates that in order to capture dqi effects for a particular molecular topology correctly really all mos belonging to its @xmath0 system need to be properly accounted for in order to achieve a reliable theoretical description . even for the non - alternant hydrocarbon azulene contacted in ( @xmath59,@xmath60- ) positions ( fig . [ fig5]d ) , where no destructive but also no constructive interference within each pair can be indicated directly from the pairing theorem , the contributions from the frontier orbitals , i.e. the homo and lumo , alone ( label @xmath58 ) do not result in any dqi feature close to @xmath16 . the inclusion of the second cr pair of mos ( label @xmath56 ) produces this feature but it then again needs the contributions from all cr pairs to position it energetically exactly at @xmath16 . , where @xmath46 is taken as the sum over all five occupied ( blue curve ) or only the three occupied mos closest to @xmath16 ( magenta curve ) in eq . [ gamma ] for the open form of the ah analog of dte with the s atoms removed . ] another property that arises from the pairing theorem is that in the case of dqi for even - membered ahs connected at sites belonging to different subsets , the contributions from all occupied and all unoccupied mos to @xmath46 in eq . [ gamma ] must each cancel out individually at @xmath16 . this is because in those cases the contribution from each half of a cr pair is equal to the other half in both sign and magnitude at @xmath16 , which means that it is then sufficient to consider either all occupied or all unoccupied mos alone . making use of this knowledge , in fig . [ fig6 ] we plot @xmath53 from the sum over the occupied states in eqn . [ gamma ] , where we compare taking all five occupied mos for the open form of the ah analog of dte with the s atoms removed ( blue curve ) with the case where the lowest lying two mos have been excluded from the summation ( magenta curve ) . as can be seen from the figure , cutting out the lowest lying two mos does not make any difference in the energy regions of the peaks of the three higher lying mos , because the transmission in the vicinity of a peak is always largely dominated by the contribution of the one mo it is related to , but crucially decides whether dqi occurs at the fermi level or not . this enforces the main message of our article that in order to identify dqi reliably from a mo perspective the contribution of all mos need to be taken into account and not just a selected few of them . this finding has also high importance for the analysis of negf - dft results , where a cut coupling approach is routinely used to describe dqi in terms of a few mos close to @xmath16 only @xcite . in summary , we showed that dqi in the electron transport of single molecule junctions can be reliably discussed from a mo perspective if the contributions from all mos are accounted for and not only those from mos close to the fermi level . this applies in general and not only for even - 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destructive quantum interference ( dqi ) in single molecule electronics is a purely quantum mechanical effect and entirely defined by inherent properties of the molecule in the junction such as its structure and symmetry . this definition of dqi by molecular properties alone suggests its relation to other more general concepts in chemistry as well as the possibility of deriving simple models for its understanding and molecular device design . recently , two such models have gained wide spread attention , where one was a graphical scheme based on visually inspecting the connectivity of carbon sites in conjugated @xmath0 systems in an atomic orbital ( ao ) basis and the other one put the emphasis on the amplitudes and signs of the frontier molecular orbitals ( mos ) . there have been discussions on the range of applicability for these schemes , but ultimately conclusions from topological molecular hamiltonians should not depend on whether they are drawn from an ao or a mo representation , as long as all the orbitals are taken into account . in this article we clarify the relation between both models in terms of the zeroth order green s function and compare their predictions for a variety of systems . from this comparison we conclude that for a correct description of dqi from a mo perspective it is necessary to include the contributions from all mos rather than just those from the frontier orbitals . the cases where dqi effects can be successfully predicted within a frontier orbital approximation we show to be limited to alternant even - membered hydrocarbons , as a a direct consequence of the coulson - rushbrooke pairing theorem in quantum chemistry . _ .. you have a program , for god s sake use it , play with it , do a calculation on any small problem related to your problem . let the calculations teach you . they are so easy ! shall we stop teasing one another about mo and vb ? _ ... quantum chemistry has given us two wonderful tools to reason about chemistry , and denying any one of them would impoverish our ability to reason .

You are an expert at summarizing long articles. Proceed to summarize the following text: we consider the grassmannian @xmath0 , the set of two - planes inside @xmath1 . a plane @xmath2 is given by two linearly independent vectors or by any two linear combinations of them that are independent , so @xmath3 there is a transitive action of @xmath4 ( or @xmath5 ) on @xmath0 . @xmath6 if we consider a particular point @xmath7 @xmath8 @xmath9 we notice that the conformal group of space time , @xmath10 , has spin group @xmath11 . its complexification , @xmath12 , has spin group @xmath13 . how to extract the minkowski space from @xmath0 ? notice that since the two vectors are independent , @xmath14 at least one of the @xmath15 determinants in this matrix is @xmath16 . the space is covered by the atlas @xmath17 @xmath18 is the _ big cell _ , and using the @xmath19 freedom , a plane in @xmath18 can be represented by @xmath20 with the entries of @xmath21 totally arbitrary . so @xmath22 , and it is a good candidate for the minkowski space . what about the group action ? @xmath18 is left invariant by the _ lower parabolic _ subgroup of @xmath23 , @xmath24 and it acts on @xmath21 as @xmath25 the group is @xmath26 so it is the poincar group where instead of the lorentz group we have put its double cover . @xmath21 belongs to the _ twistor space _ associated to spacetime . using the pauli matrices , we can revert to the spacetime notation and obtain the standard action of the poincar group on minkowski spacetime @xmath27 also the spacetime metric has an interpretation in the twistor formalism , @xmath28 quantization of spacetime means to deform the commutative algebra of functions ( can be polynomials or smooth functions ) to a non commutative algebra . other properties that we want to consider in the quantum setting have to be first defined in the algebraic formalism and then quantized. this is the case of the group actions . the respective algebras are @xmath29/ ( \det g-1),\qquad a , b=1,\dots , 4.\ ] ] @xmath30/ ( \det x\cdot \det y -1 ) , \qquad i , j=1,2,\quad a , b=3,4.\ ] ] @xmath31.\ ] ] the group law is expressed in terms of a _ coproduct _ @xmath32with the property @xmath33 the action on the minkowski space is a _ coaction _ @xmath34 in refs . @xcite one substitutes the group @xmath13 by @xmath35 in the twistor construction . all the scheme of coaction and big cell can be repeated in the quantum case , which gives a quantization for the minkowski space as a big cell inside a quantum conformal space ( a quantum grassmannian ) . we just state the result : the quantum minkowski space is a quantum matrix algebra with the rows interchanged . the correspondence @xmath36 is given in terms of the respective generators : @xmath37 this means that the commutation relations among the quantum generators are the following @xmath38 what happened to the groups ? they have become _ quantum groups _ with a non commutative product and a coproduct that is _ the same _ than the one we had before . this means that the group law has not changed , nor the coaction on the grassmannian and the minkowski space . the only change is that all these varieties have become non commutative . it is a remarkable property of matrix quantum groups that the coproduct is compatible with both , the commutative product and the non commutative one . in the quantum version @xmath39 , the sets of generators @xmath40 , @xmath41 and @xmath42 are separately isomorphic to @xmath15 matrix algebras , but while @xmath40 and @xmath41 commute among them , @xmath43 does not commute with the rest of the generators . a quantum matrix algebra is an algebra over @xmath44 $ ] , where @xmath45 is a parameter . moreover , as a module over @xmath46 , it is a free module , which means that it has a basis . it is well known that there is at least one _ ordering _ among the generators such that the standard monomials associated to this ordering are a basis of the quantum matrix algebra . ( this is a non trivial property ) . the ordering is the following @xmath47 and then there is an isomorphism ( _ ordering rule _ or _ quantization map _ ) between @xmath48 and @xmath49 $ ] : @xmath50@ > q_{\mathrm{m}}>>{\mathcal{o}}_q({\mathrm{m}})\\t_{41}^a t_{42}^b t_{31}^c t_{32}^d @ > > > { \hat{t}}_{41}^a { \hat{t}}_{42}^b { \hat{t}}_{31}^c { \hat{t}}_{32}^d\end{cd}.\ ] ] with the quantization map we can pull back the non commutative product to @xmath49 $ ] . this defines a _ star product _ , @xmath51,\ ] ] which can be computed _ explicitly _ @xmath52 @xmath53 are numerical factors defined recursively . we recover the semiclassical interpretation of the algebra being an algebra of functions , but with a star product . the previous formula for the star product is nice and compact , but can only be computed on polynomials . can we extend it to smooth functions ? not obvious . we prove that there exists a ( unique ) differential star product that coincides with the one given above on polynomials . change of the parameter : @xmath54 . we expand in powers of @xmath55 so we obtain a star product of the form @xmath56 with @xmath57 at each order , we have contributions from each of the terms with different @xmath58 @xmath59 we want to write @xmath60 as a bidifferential operator . but this is not trivial because all the dependence in the exponents should cancel . @xmath61 antisymmetrizing and changing variables we obtain the poisson bracket @xmath62 notice that if the @xmath40 s are real , then the poisson bracket is pure imaginary . also , it is quadratic . we have computed explicitly up to the order @xmath63 , but the expression is already too big to display it here . we looked for an argument to show it at arbitrary order . this can be done by careful inspection . the proof that it is differential at each order is rather technical and we do not reproduce it here @xcite but having the explicit formula for the polynomials is essential to apply the argument . an example of the possible difficulty : suppose that we want to reproduce @xmath64 as the result of applying a differential operator to @xmath65 . we have several choices , @xmath66 but none of them is both , independent on the exponent @xmath67 and polynomial in the variable . so the right combination of coefficients should appear in order to cancel the factors that appear when differentiating . for example , if the result were @xmath68 , then we have a differential operator @xmath69 since we have recovered the interpretation of functions for the non commutative algebra , we can try to express the coaction as an action of this space of functions . remember that , formally , for the generators the coaction is the same than in the commutative algebra . we just need to pull it back to the star product algebra . we define the transformed variables ( no translations are considered here ) @xmath70 @xmath71 one just has to expand the star products in the right hand side . up to order @xmath55 we have computed it in terms of a differential operator , @xmath72 and the coefficients are polynomials of order 6 in the variables @xmath73 . let @xmath74 be a commutative algebra over @xmath75 . an involution @xmath76 of @xmath74 is an antilinear map satisfying , for @xmath77 and @xmath78 @xmath79 let us consider the set of fixed points of @xmath76 , @xmath80 it is easy to see that this is a real algebra whose complexification is @xmath74 . @xmath81 is a _ real form _ of @xmath74 . _ classical minkowski space _ : @xmath82 the combinations @xmath83 are fixed points of the involution . one has @xmath84.\ ] ] _ classical euclidean space _ : @xmath85 the commbinations @xmath86 are fixed points of @xmath87 , and as before , @xmath88.\ ] ] formally , the same expressions on the generators as in the classical case are involutions in the quantum algebra . a few things change : * checking with the commutation relations they are antiautomorphisms , this is @xmath89 * this discards the interpretation of the real form of the non commutative algebra as the set of fixed points of the involution . * when pulling back to the star product algebra , the poisson bracket is purely imaginary . finally one finds also the corresponding involutions in the group , @xmath90 it is not difficult to realize that in the minkowskian case the real form of the lorentz group ( corresponding to the generators @xmath40 and @xmath41 ) is @xmath91 and in the euclidean case is @xmath92 . d. cervantes wants to thank the departament de fsica terica , universitat de valncia for the hospitality during the elaboration of this work .

we present a deformation of the minkowski space as embedded into the conformal space ( in the formalism of twistors ) based in the quantum versions of the corresponding kinematic groups . we compute explicitly the star product , whose poisson bracket is quadratic . we show that the star product although defined on the polynomials can be extended differentiably . finally we compute the eucliden and minkowskian real forms of the deformation .

You are an expert at summarizing long articles. Proceed to summarize the following text: the purpose of this paper is to introduce and analyze two exactly solvable discrete time versions of @xmath0-tasep . before introducing them we recall the continuous time poisson @xmath0-tasep which was previously studied in @xcite . let us fix some notation used throughout . for @xmath2 we denote the state of @xmath0-tasep with @xmath3 particles as @xmath4 , where we have fixed a virtual particle at infinity by setting @xmath5 . the gap between particle @xmath6 and @xmath7 is denoted @xmath8 . we define and provide some background for the continuous time poisson @xmath0-tasep . all of the results of this paper will be stated simultaneously for this process as well as its two discrete time variants . the @xmath3-particle continuous time poisson @xmath0-tasep with particle rate parameters @xmath9 is an interacting particle system @xmath10 in which for each @xmath11 , @xmath12 moves to @xmath13 at exponential rate @xmath14 ( which vanishes when @xmath15 ) , where @xmath0 is a parameter in @xmath16 . here we assume that these exponentially distributed jumping events are all independent , and we note that since they occur in continuous time , no two jumps occur simultaneously , almost surely . also note that the evolution of @xmath17 only depends on those particles with lower indices than @xmath6 . step initial condition is defined as setting @xmath18 for @xmath11 . [ zerorem ] the continuous time poisson @xmath0-tasep and the recognition of its exact solvability comes from the work of @xcite on macdonald processes . it was soon after studied in its own right in @xcite via the type of many body system approach methods which we develop herein for the two discrete time variants of the process . recently @xcite showed how @xmath0-tasep arises from considerations involving a @xmath0-deformed generalization of the dual rsk correspondence and @xcite showed how it arises from a certain class of nearest - neighbor stochastic dynamics on gelfand - tsetlin patterns . it would be very interesting to extend those works or the method of @xcite to include the ( more general ) discrete time @xmath0-taseps of the present paper as well . the evolution of @xmath19 for @xmath11 ( for the continuous time poisson @xmath0-tasep ) is given by a totally asymmetric zero range process with site dependent jump rate @xmath20 and infinite sink and source at the boundary . a variant of this zero range process can be seen as corresponding to a particular representation of the @xmath0-boson hamiltonian considered in @xcite . in @xcite , integrability ( in the form of @xmath21 and @xmath22 matrices satisfying yang - baxter relations ) for this @xmath0-boson hamiltonian was shown , though the connection of that to the exact solvability of @xmath0-tasep discussed below remains to be understood . a stationary ( infinite lattice ) variant of the above gap zero range process was also discussed immediately after theorem 2.9 of @xcite and an @xmath23 upper and lower bound on the variance of the stationary current is established therein . poisson @xmath0-tasep has an interpretation as a model for traffic on a one - lane ( discrete ) road @xmath24 in which the rate at which cars jump forward is modulated by the distance to the next car ( as well as a car dependent rate parameter @xmath25 ) . as the distance goes to zero , the jump rate goes to zero , and as the distance grows , the jump rate approaches @xmath25 . step initial condition corresponds to an initially jammed configuration . when @xmath26 , continuous time poisson @xmath0-tasep becomes the well - studied model of continuous time tasep ( and likewise the discrete time versions of @xmath0-tasep we introduce become known discrete time version of tasep ) . the configuration of particles in tasep for a fixed time @xmath27 can be described as a determinantal point process in which all correlation functions are given by minors of a single correlation kernel ( cf . @xcite and references therein ) . in other terminology , tasep is free - fermion , or related to schur processes , or non - interacting line ensembles . given that structure there exists a clean path to fredholm determinant formulas for marginal distributions , which in turn allow readily for asymptotic analysis ( cf . @xcite and references therein ) . the present work , where @xmath28 , does not appear to be determinantal , hence new ideas are necessary to study and extract asymptotic distributional information about the processes considered . in recent years there has been a flurry of activity surrounding the analysis of non - determinantal yet still exactly solvable stochastic interacting particle systems @xcite . in @xcite it was explained how , instead of the determinantal structure of correlation functions , these non - determinantal systems are exactly solvable due to the existence of concise and exact formulas for expectations of large classes of observables of the particle systems ( in fact , in many cases large enough classes to uniquely characterize the fixed time distribution of the particle systems ) . a natural question about @xmath0-tasep is to compute the distribution of @xmath29 ( which is the location of particle @xmath30 at time @xmath27 ) . there are ( presently ) two approaches to compute this distribution . the first is through the theory of macdonald processes @xcite , and the second is through the many body system ( or duality ) approach @xcite . the first approach is entirely based on the integrable properties of macdonald polynomials ( see section [ macsec ] for a brief review on the relationship of the two approaches ) , and these properties naturally lead to the discovery of @xmath0-tasep as well as the computation of nested contour integral formulas for expectations of certain observables of the process . in particular , for step initial condition this shows that ( see also @xcite or theorem [ momthm ] below ) for @xmath31 , @xmath32= \frac{(-1)^k q^{k(k-1)/2}}{(2\pi \iota)^k } \int \cdots \int \prod_{1\leq a < b\leq k } \frac{z_a - z_b}{z_a - qz_b } \prod_{j=1}^{k } \left(\prod_{i=1}^{n_j } \frac{a_i}{a_i - z_j}\right ) e^{(q-1)tz_j } \frac{dz_j}{z_j},\ ] ] where the contour of integration for @xmath33 contains @xmath34 as well as @xmath35 , but not zero ( see figure [ circontours ] for an example of such a contour when all @xmath36 ) . the above formula is given in theorem [ momthm ] along with parallel formulas for the discrete time versions . since the observables @xmath37 are all in @xmath16 , their moment problem is well - posed . this means that the above formulas uniquely characterize the joint distribution of @xmath0-tasep at time @xmath27 ( provided it is started from step initial condition at time zero ) . it remains a challenge to extract manageable ( from the perspective of large @xmath27 and @xmath30 asymptotics ) formulas for joint distributions , though it is understood how to derive a fredholm determinant formula which characterizes the one - point distribution of @xmath29 . this is described in theorem [ distthm ] . the second approach to solving the continuous time poisson @xmath0-tasep is based solely on the observation that the dynamics of the particle system implies that the expectations on the left - hand side of ( [ exp1 ] ) satisfy certain closed systems of coupled odes , which we call _ many body systems_. these systems have unique solutions and it is easy to check that the nested contour integral formulas on the right - hand side of ( [ exp1 ] ) satisfy these systems ( and hence are equal to the expectations on the left - hand side of ( [ exp1 ] ) ) . this second approach is more direct ( it avoids explicit mention of macdonald polynomials ) and appears to be more general ( it applies to asep @xcite , which is not known to fit into the theory of macdonald processes ) . on the other hand , it requires non - trivial input of a suitable interacting particle system , the correct observables to study and some inspiration in guessing a closed form solution to the associated many body system . for all of the versions of @xmath0-tasep the fact that these expectations satisfy the many body system can be seen immediately from the macdonald process perspective as a consequence of certain commutation relations involving macdonald difference operators ( see section [ macsec ] ) . it is this second approach which we employ in this paper to study the two exactly solvable discrete time @xmath0-taseps we now introduce . before we proceed , let us fix some additional notation . we use @xmath38 to denote particle rate parameters , and @xmath39 and @xmath40 to denote ( discrete ) time dependent jump parameters . the indicator function of an event @xmath41 is written @xmath42 . the @xmath0-pochhammer symbol is defined as @xmath43 finally , we define a variant of the weyl chamber as @xmath44 the two discrete time versions of @xmath0-tasep which we introduce and study were not fabricated out of thin air , but rather came from further investigations into dynamics related to macdonald processes which may be detailed in a future work ( cf . @xcite for some related developments ) . all three versions of @xmath0-tasep are illustrated in figure [ qtaseps ] . -tasep , discrete time geometric @xmath0-tasep with parallel update , and discrete time bernoulli @xmath0-tasep with right - to - left sequential update . ] we will make use of the following @xmath0-deformation of the truncated geometric distribution . for @xmath45 and @xmath46 the distribution @xmath47 is defined as @xmath48 for @xmath49 we extend the definition so that @xmath50 [ sumone ] for @xmath51 and @xmath46 , @xmath52 for all @xmath53 and @xmath54 the positivity is immediate as @xmath55 . that these @xmath47 sum to one can be seen inductively ( in @xmath56 ) by using the recursion ( see ( 10.0.3 ) in @xcite ) for the @xmath0-binomial coefficients ( which occur in the definition of @xmath47 as the fraction ) . we may now introduce the first discrete time version of @xmath0-tasep . we work in the greatest generality for which this is exactly solvable . for simplicity , a reader may want to think of all parameters @xmath36 and all @xmath57 . the @xmath3-particle discrete time geometric @xmath0-tasep with particle rate parameters @xmath38 and time dependent jump parameters @xmath58 is an interacting particle system @xmath10 in discrete time @xmath59 . at time @xmath60 the system stochastically evolves from its state at time @xmath27 according to the following _ parallel _ update rule : for each @xmath11 , @xmath61 where the update is independent for each @xmath6 and @xmath27 . since we observe the convention of @xmath62 , in the above update rule @xmath63 . we assume throughout the paper at the @xmath25 and @xmath64 are such that @xmath65 for all @xmath6 and @xmath27 , otherwise the process is not well - defined . these dynamics mean that at time @xmath60 each particle @xmath66 hops to a random location in the set @xmath67 with hop length distributed according to independent random variables with distribution @xmath68 . these dynamics clearly preserve the order of particles . when @xmath26 the jump probabilities become @xmath69 this corresponds with a geometric jump of rate @xmath70 in which all of the weight given to jumps @xmath71 are collapsed onto the weight of @xmath72 . in other words , in parallel each particle attempts a geometric jump to the right subject to blocking by the previous location of the next particle . this process was studied previously in @xcite as a marginal of dynamics on gelfand - tsetlin patterns which preserve the class of schur processes . the @xmath3-particle discrete time bernoulli @xmath0-tasep with particle rate parameters @xmath38 and time dependent jump parameters @xmath73 is an interacting particle system @xmath10 in discrete time @xmath59 . at time @xmath60 the system stochastically evolves from its state at time @xmath27 according to the following _ sequential _ update rules : first update @xmath74 according to @xmath75 \frac{a_1\beta_{t+1}}{1+a_1\beta_{t+1 } } & \textrm{for } b=1,\\[.5em ] 0 & \textrm{otherwise}.\end{cases}\ ] ] then , for @xmath76 through @xmath3 ( in that order ) if @xmath77 ( i.e. , the previous jump occurred ) then update @xmath78 \frac{a_i\beta_{t+1}}{1+a_i\beta_{t+1 } } & \textrm{for } b=1,\\[.5em ] 0 & \textrm{otherwise}.\end{cases}\ ] ] if @xmath79 ( i.e. , the previous jump did not occur ) then update @xmath80 ( 1-q^{{\ensuremath{\mathrm{gap}_{i}}}(t)})\frac{a_i\beta_{t+1}}{1+a_i\beta_{t+1 } } & \textrm{for } b=1,\\[.5em ] 0 & \textrm{otherwise}.\end{cases}\ ] ] the updates are all independently distributed . these dynamics mean that @xmath81 jumps right by one with probability @xmath82 and otherwise stays put . sequentially , if @xmath83 jumped , then @xmath66 jumps right by one with probability @xmath84 , otherwise if @xmath83 stayed put , then @xmath66 jumps right by one with probability @xmath85 . these dynamics clearly preserve the order of particles . when @xmath26 the update rule becomes the following : at time @xmath60 , particle @xmath81 jumps to the right by 1 with probability @xmath82 and stays put otherwise ; then sequentially for @xmath86 through @xmath87 , the particle @xmath66 jumps to the right by 1 with probability @xmath84 if the destination site is unoccupied , otherwise it stays put . to be clear , if @xmath88 and @xmath89 ( neighbors ) and @xmath81 jumps right so that @xmath90 , then the rightward jump for @xmath86 is available and @xmath91 . this sequential discrete time tasep was studied previously in @xcite as a marginal of dynamics on gelfand - tsetlin patterns which preserve the class of schur processes . as @xmath92 and @xmath93 go to zero and time is suitably rescaled , the discrete time @xmath0-taseps both converge to the continuous time poisson version . that is to say ( taking all @xmath94 and @xmath95 ) that with @xmath96 and @xmath97 geometric @xmath0-tasep @xmath10 converges as a process to continuous time poisson @xmath0-tasep with @xmath98 representing time . the same result holds for bernoulli @xmath0-tasep with @xmath99 and @xmath100 . as @xmath101 , the dynamics of the continuous time poisson @xmath0-tasep converge ( under appropriate scaling ) to the solution of the semi - discrete stochastic heat equation ( equivalently the oconnell - yor directed polymer @xcite ) see theorem 4.1.26 of @xcite or proposition 6.2 of @xcite . this semi - discrete stochastic heat equation converges under diffusive space / time scaling and weak noise scaling to the continuum stochastic heat equation @xcite . it is also shown in @xcite that the continuous time poisson @xmath0-tasep has a direct limit to the solution to the continuum stochastic heat equation . presumably a similar sequence of limits for the discrete time @xmath0-tasep dynamics should exist . it would be interesting to investigate whether there exist other degenerations of these discrete time @xmath0-taseps besides those just mentioned . one could , for instance , keep the @xmath92 or @xmath93 parameters fixed and scaling @xmath101 . at least for the geometric case , one expects to make some contact with the work of @xcite . the three versions of @xmath0-tasep share the following three surprising properties , which amount to their exact solvability : 1 . the expectations of observables @xmath102 for @xmath103 evolve according to closed systems of coupled odes ( in the continuous time case ) or coupled difference equations ( in the discrete time case ) . we call these systems _ true evolution equations_. 2 . the true evolution equations are almost constant coefficient or separable , except for effects which arise when some of the @xmath104 are close together . rather than trying to solve the true evolution equation directly , one can look for solutions to the free evolution equation ( i.e. , the constant coefficient and separable system neglecting the boundary effects ) on the larger space @xmath105 which have the right initial data when restricted to @xmath106 and which satisfy certain boundary conditions when some of the @xmath104 are all equal . the restriction of such a solution to @xmath106 will coincide with the solution to the true evolution equation . generally , for every possible combination of clusters of @xmath107 ( i.e. , strings of equal coordinates ) there will be additional boundary conditions which must be satisfied . it turns out that for the three versions of @xmath0-tasep we consider , it suffices to consider only @xmath1 two - body boundary conditions ( which ends up being the same for all three systems ) corresponding to when @xmath108 . that the two - body boundary conditions imply all many body boundary conditions is the hallmark of _ integrability _ in the language of ( quantum ) many body systems . this type of reduction goes back to the 1931 work of on diagonalizing the heisenberg spin chain @xcite . our reduction , under the scaling limit to the continuum stochastic heat equation mentioned above , cincides with that for the quantum delta bose gas ( also known as the lieb - liniger model @xcite ) . 3 . a general class of solutions to the free evolution equation exists since it is separable and constant coefficient . it is not immediately clear how to combine solutions from this class so as to additionally satisfy the @xmath1 two - body boundary conditions and the initial data . however , for the initial data corresponding to step initial condition , it is also possible to ( easily ) check that _ nested contour integral formulas _ ( such as in ( [ exp1 ] ) ) solve the free evolution equation and satisfy the @xmath1 two - body boundary conditions . one may speculate that these formulas should be related to the _ bethe ansatz _ for diagonalizing this system ( cf . @xcite for such a relationship for a limiting continuum version of these many body problems ) . from this it would be possible to produce solutions corresponding to general initial conditions . to our knowledge the bethe ansatz has not been worked out sufficiently in this context ( see however related work of @xcite ) . in section [ results ] we record the consequences of the above mentioned properties . the rest of the paper is devoted to proving these results along the lines of the three steps given above . in particular , in section [ truesec ] we prove that the expectations of the above mentioned observables do satisfy explicit true evolution equations . in section [ freesec ] we demonstrate the integrability of these systems by reducing consideration to free evolution equations with @xmath1 boundary conditions . in section [ checkingformulas ] we check that for initial data corresponding to step initial conditions this system is indeed solvable via a nested contour integral formula . in section [ endsec ] we also briefly remark on the relationship between the true evolution equations and markov process duality as well as certain commutation relations involving macdonald difference operators . ab was partially supported by the nsf grant dms-1056390 . ic was partially supported by the nsf through dms-1208998 as well as by microsoft research through the schramm memorial fellowship , and by a clay mathematics institute research fellowship . we state in parallel our main results for all three variants of @xmath0-tasep . to facilitate this we use @xmath10 for each process and instead use different expectation symbols to denote running the different dynamics . [ momthm ] fix @xmath28 . consider : 1 . continuous time @xmath0-tasep with particle rate parameters @xmath38 . let @xmath109 represent the expectation operator for this process started from step initial condition . define @xmath110 2 . discrete time geometric @xmath0-tasep with particle rate parameters @xmath38 and time dependent jump parameters @xmath58 . let @xmath111 represent the expectation operator for this process started from step initial condition . define @xmath112 3 . discrete time bernoulli @xmath0-tasep with particle rate parameters @xmath38 and time dependent jump parameters @xmath73 . let @xmath113 represent the expectation operator for this process started from step initial condition . define @xmath114 fix @xmath115 , then for all @xmath116 ( i.e. , @xmath117 ) and @xmath118 , @xmath119 = \frac{(-1)^k q^{\frac{k(k-1)}{2}}}{(2\pi \iota)^k } \int \cdots \int \prod_{1\leq a < b\leq k } \frac{z_a - z_b}{z_a - qz_b } \prod_{j=1}^{k } \left(\prod_{i=1}^{n_j}\frac{a_i}{a_i - z_j}\right ) \frac{f^{\ell}(qz_j)}{f^{\ell}(z_j ) } \frac{dz_j}{z_j},\ ] ] where the contour of integration for @xmath33 contains @xmath34 , and @xmath35 but not 0 or any poles of @xmath120 . and @xmath36 . ] the remaining sections of this article are concerned with proving the above result ( see in particular section [ checkingformulas ] ) . [ arborder ] consider running an arbitrary combination of the three types of @xmath0-tasep with general parameters : run continuous time poisson @xmath0-tasep for time @xmath121 and discrete time geometric @xmath0-tasep with @xmath122 and discrete time bernoulli @xmath0-tasep with @xmath123 . the methods we develop herein imply the following results . the terminal state of the particle system after running combinations of the three dynamics does not depend on the order in which they were run ( in fact , one can take turns between running the three different dynamics ) . moreover , letting @xmath10 represent the state of the process after having run these three dynamics we find that for all @xmath115 and @xmath116 , @xmath124 & = & \frac{(-1)^k q^{\frac{k(k-1)}{2}}}{(2\pi \iota)^k } \int \cdots \int \prod_{1\leq a < b\leq k } \frac{z_a - z_b}{z_a - qz_b } \\ & & \times\,\prod_{j=1}^{k } \left(\prod_{i=1}^{n_j } \frac{a_i}{a_i - z_j}\right ) e^{(q-1)\gamma z_j } \prod_{m\geq 1 } ( 1-\alpha_m z_j)\frac{1+q\beta_m z_j}{1+\beta_m z_j } \frac{dz_j}{z_j},\end{aligned}\ ] ] where the contour of integration for @xmath33 contains @xmath34 , and @xmath35 but not 0 or any other poles of the integrand . the above expression is just a product of @xmath120 over @xmath118 . the above nested contour integral formulas arise naturally in the study of macdonald processes ( cf . proposition 3.1.5 of @xcite and corollary 4.7 of @xcite ) and the existence of the associated integrable quantum many body systems is actually a consequence of this connection ( see section [ macsec ] for more details ) . [ distrem ] the expectations considered above in theorem [ momthm ] uniquely characterize the fixed time joint distribution of @xmath125 , and hence also that of @xmath10 . this is because the @xmath126 are deterministically in @xmath16 , and thus their moments uniquely identify their distributions @xcite . in principle it should be possible to extract formulas for any fixed time expectations of @xmath10 from the explicit formulas in theorem [ momthm ] . it would be desirable to find moment generating functions ( which characterize the joint distributions of @xmath10 ) for which there are sufficiently compact formulas to allow for asymptotics . presently it is only known how to do this for the one - point distribution of @xmath29 . as an application of the above moment formulas it is possible to prove the following fredholm determinant formula for the @xmath0-laplace transform of @xmath37 which in turn characterize the distribution of @xmath29 via a simple inversion formula ( see remark [ qinversion ] ) . the procedure for going from the nested contour integral formulas above to the fredholm determinant formula is developed in @xcite and the below result follows immediately from this procedure . for simplicity of contours , we assume here that all @xmath36 ( see remark [ notasrem ] ) . [ distthm ] fix @xmath28 and particle rate parameters @xmath36 . consider the three versions of @xmath0-tasep corresponding with @xmath118 ( in the notation of theorem [ momthm ] ) . then for all @xmath127 @xmath128 = \det(i + k^{\ell}_{\zeta})\ ] ] where @xmath129 is the fredholm determinant of @xmath130 for @xmath131 a small positively oriented circle containing 1 . the operator @xmath132 is defined in terms of its integral kernel @xmath133 with @xmath134 where the function @xmath135 is defined as in theorem [ momthm ] . the following second formula also holds : @xmath136 = \frac{\det(i + \zeta \tilde{k}^{\ell})}{(\zeta;q)_{\infty}}\ ] ] where @xmath137 is the fredholm determinant of @xmath138 times the operator @xmath139 for @xmath140 a positively oriented circle containing 0 and 1 ( and no poles of @xmath135 ) . the operator @xmath141 is defined in terms of its integral kernel @xmath142 where the function @xmath135 is defined as in theorem [ momthm ] . we refer to the first fredholm determinant ( [ mellinbarnes ] ) as mellin - barnes type , and the second fredholm determinant ( [ cauchy ] ) as cauchy type . these mellin - barnes type and cauchy type fredholm determinant formulas were previously proved in the form of theorems 3.2.10 and 3.2.16 ( respectively ) of @xcite . however , those results are phrased in terms of macdonald processes ( in which @xmath143 corresponds with @xmath144 $ ] ) . the general scheme which leads to these results is recorded in sections 3.1 and 3.2 of @xcite and shows how to go from the nested contour integral formulas for @xmath145 $ ] to the ( respective ) mellin - barnes and cauchy type fredholm determinant formulas in the theorem . in the application of propositions 3.6 and 3.10 of @xcite it is necessary to check certain technical conditions on the function @xmath146 to make sure that the formal manipulations which lead to the fredholm determinants are , in fact , numerical identities . these conditions are checked in the proofs of theorems 3.2.10 and 3.2.16 ( respectively ) of @xcite so we do not repeat them . [ notasrem ] the only modification of theorem [ distthm ] for general @xmath25 is that in the first fredholm determinant , @xmath147 and consequently some care must be taken in specifying the appropriate choice of contours for the @xmath148 and @xmath149 integration ( so that the @xmath148 contour contains the @xmath150 while avoiding other poles ) . see for example , theorem 3.2.11 of @xcite , or theorem 4.13 of @xcite . likewise , in the second fredholm determinant formula , the term @xmath151 in defining @xmath152 is replaced by @xmath153 and the contour @xmath140 is replaced by a suitable one which contains 0 and all @xmath150 . [ qinversion ] the transform from the probability distribution of @xmath37 to the expectation on the left - hand sides of ( [ mellinbarnes ] ) and ( [ cauchy ] ) is called a @xmath0-laplace transform . just as the usual laplace transform can be inverted via a single contour integral in the spectral variable , so too can we recover the distribution of @xmath154 or @xmath155 from its transform . to state this inversion , write @xmath156.\ ] ] then it follows from proposition 3.1.1 of @xcite ( see also @xcite ) that @xmath157 where @xmath158 is a positively oriented circle which encircles @xmath159 . there are a number of interesting limit theorem results which should be accessible via asymptotic analysis of the fredholm determinants in theorem [ distthm ] . the mellin - barnes type fredholm determinant seems to be most easily analyzed asymptotically . presently , the only asymptotics related to @xmath0-tasep which have been worked out correspond with first taking the limit for @xmath30 fixed and @xmath101 in which @xmath0-tasep converges @xcite to the semi - discrete stochastic heat equation @xcite . at that level @xcite have proved gue tracy - widom limit theorems ( see also @xcite for a related discrete stochastic heat equation @xcite ) . there remain a number of limit theorems which should be provable from the fredholm determinant formulas above , such as the gue tracy - widom limit theorems for the three variants of @xmath0-tasep . it is possible that in proving such results slightly different choices of contours ( as in @xcite ) will be necessary . the first surprising property of q - tasep is that the expectations of the observables @xmath102 evolve according to closed systems of coupled odes ( in the continuous time case ) or coupled difference equations ( in the discrete time case ) . we call these systems the _ true evolution equations_. this property is not a generic property of interacting particle systems , but rather something quite special to these systems and these choices of observables . in order to state the true evolution equations it will be useful to have an alternative notation for @xmath116 wherein @xmath160 counts the number of @xmath161 ( i.e. , the size of the cluster of @xmath162 s equal to @xmath6 ) . more precisely , define @xmath163 to each @xmath116 associate @xmath164 via @xmath165 , and to each @xmath166 associate @xmath167 as the unique @xmath107 such that @xmath168 . thus @xmath169 lists the multiplicities of each number in @xmath170 in @xmath107 and @xmath171 associates to such multiplicities an ordered list in @xmath106 . for instance if @xmath172 then @xmath173 . we say that @xmath107 has @xmath174 clusters if @xmath175 . for the present example @xmath172 has three clusters . for a vector @xmath176 and a vector @xmath177 with @xmath178 define @xmath179 both of these map @xmath180 into another element of @xmath181 . the following combinatorial coefficients are used in defining the true evolution equation . for @xmath182 , @xmath28 , @xmath183 and integers @xmath184 define @xmath185 with the convention that @xmath186 for @xmath187 or @xmath188 . we record the following easily checkable properties of the coefficients @xmath189 which will prove useful : @xmath190 for @xmath191 , @xmath11 and @xmath192 define the following difference operators @xmath193 [ zeroremark ] if @xmath194 for @xmath180 with @xmath195 , then it follows that @xmath196 moreover , for such an @xmath146 , we have that @xmath197 with @xmath198 . [ truedef ] we say that @xmath199 solves the ( rate parameter @xmath34 ) continuous time poisson @xmath0-tasep / discrete time geometric @xmath0-tasep ( with time dependent jump parameters @xmath200 ) / discrete time bernoulli @xmath0-tasep ( with time dependent jump parameters @xmath201 ) _ true evolution equation _ with initial data @xmath202 if 1 . for all @xmath192 and @xmath203 , @xmath204 2 . for all @xmath192 such that @xmath195 , @xmath205 for all @xmath203 ; 3 . for all @xmath192 , @xmath206 . [ existuniq ] the above respective true evolution equations have unique solutions . the operators @xmath207 and @xmath208 map the space of functions @xmath209 onto itself . in fact , these operators restrict to mapping the space of functions @xmath210 onto itself . on account of this , the above true evolution equations restrict to a collection of closed systems of linear odes ( for the continuous time case ) or difference equations ( for the two discrete time cases ) , indexed by @xmath211 . it suffices , therefore , to prove existence and uniqueness when restricted to the system corresponding to each @xmath115 . for fixed @xmath211 , our system of linear differential or difference equations is also triangular in the following sense . define a partial ordering on @xmath181 so that @xmath212 if for all @xmath213 , @xmath214 . then triangularity means for the continuous time case that @xmath215 depends only upon @xmath216 for @xmath217 , and for the discrete time cases that @xmath218 depends only upon @xmath216 for @xmath217 . this is easily seen from the definition of the true evolution equations . finally , for each @xmath115 , the associated closed , triangular system of linear odes / difference equations is also finite . one account of this , one can apply standard methods ( such as in @xcite for the continuous time case or linear algebra for the discrete time cases ) to conclude the existence and uniqueness of solutions . we may now state the main result of this section , which is that expectations of certain observables of the three variants of @xmath0-tasep solve the above ( respective ) true evolution equations . [ truethm ] consider the ( rate parameter @xmath34 ) continuous time poisson @xmath0-tasep / discrete time geometric @xmath0-tasep ( with time dependent jump parameters @xmath39 ) / discrete time bernoulli @xmath0-tasep ( with time dependent jump parameters @xmath40 ) started from an arbitrary initial condition @xmath219 . then for any @xmath115 and @xmath116 , @xmath220 = h(t;\vec{y}(\vec{n}))\ ] ] where @xmath199 solves the ( respective ) true evolution equation with initial data @xmath221.\ ] ] due to lemma [ existuniq ] it suffices to show that the left - hand side of ( [ truelhs ] ) satisfies the true evolution equation ( in each of the three cases ) . in what follows let @xmath222 . it is then convenient to rewrite the left - hand side of ( [ truelhs ] ) as @xmath223= { \ensuremath{\mathbb{e}}}\left[\prod_{i=0}^{n } q^{(x_i(t)+i)y_i}\right]\ ] ] with the convention that when @xmath195 , the above is zero , and when @xmath224 , the above product starts at @xmath225 . from this one readily sees that condition ( 2 ) of the true evolution equation is satisfied . condition ( 3 ) is immediate as well . we prove condition ( 1 ) for each of the three cases . in fact , the poisson case was previously proved in @xcite , though we include it here as well for completeness . * poisson case * : let @xmath21 denote the generator of continuous time poisson @xmath0-tasep . then it follows that @xmath226 & = & l { \ensuremath{\mathbb{e}}}\left [ \prod_{i=0}^{n } q^{(x_i(t)+i)y_i}\right]= { \ensuremath{\mathbb{e}}}\left[l \prod_{i=0}^{n } q^{(x_i(t)+i)y_i}\right]\\ & = & { \ensuremath{\mathbb{e}}}\left[\sum_{j=1}^{n } a_j(1-q^{x_{j-1}(t)-x_{j}(t)-1})\big(q^{(x_j(t)+1+j)y_j}-q^{(x_j(t)+j)y_j}\big ) \prod_{\substack{i=0\\i\neq j}}^{n } q^{(x_i(t)+i)y_i}\right]\\ & = & { \ensuremath{\mathbb{e}}}\left[\sum_{j=1}^{n } a_j(1-q^{x_{j-1}(t)-x_{j}(t)-1})(q^{y_j}-1 ) \prod_{i=0}^{n } q^{(x_i(t)+i)y_i}\right]\\ & = & { \ensuremath{\mathbb{e}}}\left[\sum_{j=1}^{n } { \ensuremath{\mathcal{l}^{(a_j)}_{j } } } \prod_{i=0}^{n } q^{(x_i(t)+i)y_i}\right ] = \sum_{j=1}^{n } { \ensuremath{\mathcal{l}^{(a_j)}_{j } } } { \ensuremath{\mathbb{e}}}\left[\prod_{i=0}^{n } q^{(x_i(t)+i)y_i}\right].\end{aligned}\ ] ] this shows that the expectation in question does satisfy the true evolution equation . * geometric case * : we show the following stronger statement . let @xmath227 be the sigma - field generated by the random variables @xmath228 . then , as @xmath227 measurable random variables @xmath229 = { \ensuremath{\mathcal{a}^{(-a_1 \alpha_{t+1})}_{1 } } } \cdots { \ensuremath{\mathcal{a}^{(-a_n \alpha_{t+1})}_{n } } } \prod_{i=0}^{n } q^{(x_i(t)+i)y_i}.\ ] ] taking expectations of both sides ( with respect to the random variables @xmath10 ) recovers the desired result . in what follows we write @xmath92 instead of @xmath230 . in order to show the above result , observe that it follows immediately from the geometric dynamics that @xmath231 = \prod_{i=1}^{n } \left(\sum_{j=0}^{{\ensuremath{\mathrm{gap}_{i}}}(t ) } { \ensuremath{\mathbf{p}_{{\ensuremath{\mathrm{gap}_{i}}}(t ) , a_i \alpha}}}(j ) q^{jy_i}\right ) \prod_{i=0}^{n } q^{(x_i(t)+i)y_i}.\ ] ] [ cqm ] for all @xmath232 and @xmath191 @xmath233 and likewise @xmath234 applying this lemma we find @xmath235 with the convention @xmath236 . the second line follows from the definition of @xmath237 . thus , we have reached a formula for the left - hand side of ( [ geolhs ] ) which , in light of equation ( [ rhsexp ] ) matches the right - hand side of ( [ geolhs ] ) . we will prove equation ( [ lemmalhs ] ) . the @xmath49 case is in the same spirit so we leave it out . taking @xmath56 fixed , denote the left - hand side of ( [ lemmalhs ] ) as @xmath238 . we may apply lemma [ sumone ] with the parameter @xmath92 from its statement taken to be @xmath239 . this yields @xmath240 from the definition of @xmath241 this equality can be rewritten as @xmath242 we may expand the product @xmath243 as @xmath244 where @xmath245 is the degree @xmath246 elementary symmetric polynomial ( see e.g. section i.2 of @xcite ) . plugging this into ( [ secondset ] ) and rearranging the summations of @xmath53 and @xmath246 yields @xmath247 rewriting in terms of the @xmath238 , we find @xmath248 note also that ( see e.g. section i.3 , exercise 1 of @xcite ) @xmath249 the relation ( [ aboverelation ] ) on the @xmath238 uniquely characterizes them , hence it suffices to check that the right - hand side of ( [ lemmalhs ] ) also satisfies the relation when substituted for the @xmath238 . this amounts to @xmath250 in order to check that the above relation holds we can gather all coefficients associated with @xmath251 and check that for @xmath252 the coefficients combine to equal 1 , and for @xmath253 they combine to 0 . letting @xmath254 , we must therefore check that @xmath255 using the definition of the @xmath256 coefficients the above relation reduces to @xmath257 from corollary 10.2.2(c ) of @xcite we find that @xmath258 from which ( [ aboverelation2 ] ) immediately follows . * bernoulli case * : we will show the following stronger statement . for @xmath259 and @xmath203 , let @xmath260 the sigma - field generated by the random variables @xmath261 . then , as @xmath227 measurable random variables @xmath262 = { \ensuremath{\mathcal{a}^{(qa_1\beta_{t+1})}_{1 } } } \cdots { \ensuremath{\mathcal{a}^{(qa_n\beta_{t+1})}_{n } } } \prod_{i=0}^{n } q^{(x_i(t+1)+i)y_i}.\ ] ] taking expectations of both sides ( with respect to the random variables @xmath10 ) recovers the desired result . in what follows we write @xmath93 instead of @xmath263 . the following lemma will quickly yield a proof of ( [ berncond ] ) . [ bernlem ] for @xmath264 , @xmath265\qquad\qquad\qquad\qquad&\\ = \sum_{s_i=0}^{y_i } c_{qa_i\beta } ( y_i , s_i ) q^{(x_i(t)+i)(y_i - s_i)}q^{(x_{i-1}(t)+i-1)s_i},\qquad\qquad\qquad\qquad\qquad&\end{aligned}\ ] ] and for @xmath225 @xmath266 = c_{qa_1\beta } ( y_1,0 ) q^{(x_1(t)+1)y_1}.\ ] ] let us first consider the @xmath225 case . given the knowledge of @xmath267 , the dynamics of bernoulli @xmath0-tasep implies that with probability @xmath268 , we have that @xmath269 and with probability @xmath270 , we have that @xmath271 . this implies that @xmath272 & = & c_{a_1\beta}(y_1,0 ) q^{(x_1(t)+1)y_1}\left(\frac{a_1\beta}{1+a_1\beta } q^{y_1 } + \frac{1}{1+a_1\beta}\right ) \\ & = & c_{qa_1\beta } ( y_1,0 ) q^{(x_1(t)+1)y_1},\end{aligned}\ ] ] as desired . here we used the third relation of ( [ cid ] ) with @xmath273 to reach the above conclusion . now consider the case when @xmath264 . let @xmath274 represent the indicator function for the event that @xmath275 ( i.e. , particle @xmath7 jumped at time @xmath60 ) . note that this is measurable with respect to @xmath276 . thus , by virtue of the bernoulli dynamics , we have that for @xmath277 @xmath278\\ & = i_{i-1 } q^{(x_i(t)+i)r } q^{(x_{i-1}(t)+1+i-1)s } \left(\frac{a_i\beta}{1+a_i\beta } q^r + \frac{1}{1+a_i\beta}\right)\\ & \quad + ( 1-i_{i-1 } ) q^{(x_i(t)+i)r } q^{(x_{i-1}(t)+i-1)s } \left((1-q^{x_{i-1}(t)-x_i(t)-1})\frac{a_i\beta}{1+a_i\beta } q^r + \frac{1+a_i\beta q^{x_{i-1}(t)-x_i(t)-1}}{1+a_i\beta}\right)\\ & = q^{(x_i(t)+i)r } q^{(x_{i-1}(t)+i-1)s } \frac{1}{1+a_i\beta } \left(i_{i-1 } a_i\beta q^{r+s } + i_{i-1 } q^s + ( 1-i_{i-1 } ) q^r + ( 1-i_{i-1})\right)\\ & \quad + q^{(x_i(t)+i)(r-1 ) } q^{(x_{i-1}(t)+i-1)(s+1 ) } \frac{1}{1+a_i\beta } \left(a_i\beta(1-i_{i-1})(1-q^r)\right).\end{aligned}\ ] ] using the above , we may now compute @xmath279 \\ & = \sum_{s_i=0}^{y_i } q^{(x_i(t)+i)(y_i - s_i)}q^{(x_{i-1}(t)+i-1)s_i } \\ & \qquad\qquad\times \ , \bigg(c_{a_i\beta}(y_i , s_i)\frac{1}{1+a_i\beta}\left(i_{i-1}a_i\beta q^{y_i } + i_{i-1 } q^{s_i } + ( 1-i_{i-1})a_i\beta q^{y_i - s_i } + ( 1-i_{i-1})\right ) \\ & \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+ c_{a_i\beta}(y_i , s_i-1 ) \frac{a_i\beta}{1+a_i\beta}(1-i_{i-1})(1-q^{y_i - s_i+1})\bigg).\end{aligned}\ ] ] observe that we may rewrite the above factor ( in large parentheses ) as @xmath280 this last equality came from two facts . the left - hand side has two terms . the first is equal to @xmath281 by the third relation of ( [ cid ] ) . the second term ( with @xmath274 ) is zero by the first relation of ( [ cid ] ) . as a consequence of the above calculation we find that @xmath282\\ & = \sum_{s_i=0}^{y_i } c_{qa_i\beta}(y_i , s_i ) q^{(x_i(t)+i)(y_i - s_i)}q^{(x_{i-1}(t)+i-1)s_i},\end{aligned}\ ] ] as desired to complete the proof of the lemma . in order to conclude the proof of the bernoulli case we use conditional expectations to rewrite the left - hand side of ( [ berncond ] ) with @xmath224 as @xmath283 \big\vert \sigma^n_t,\sigma^{n-2}_{t+1}\big]\cdots \big\vert \sigma^{n}_t , \sigma^{1}_{t+1 } \big]\big\vert \sigma^{n}_t\big]\\ & = { \ensuremath{\mathcal{a}^{(a_1\beta)}_{1 } } } \cdots { \ensuremath{\mathcal{a}^{(a_{n-1}\beta)}_{n-1 } } } { \ensuremath{\mathbb{e}}}\big[q^{(x_1(t+1)+1)y_1 } { \ensuremath{\mathbb{e}}}\big [ q^{(x_2(t+1)+2)y_2 } \cdots { \ensuremath{\mathbb{e}}}\big [ q^{(x_{n-1}(t+1)+n-1)y_{n-1 } } \\ & \quad\times\,\sum_{s_n=0}^{y_n } c_{a_n\beta}(y_n , s_n ) { \ensuremath{\mathbb{e}}}\big [ q^{(x_n(t+1)+n)(y_n - s_n)}q^{(x_{n-1}(t+1)+n-1)s_n } \big\vert \sigma^n_t,\sigma^{n-1}_{t+1}\big]\big\vert \sigma^n_t,\sigma^{n-2}_{t+1}\big]\cdots \big\vert \sigma^{n}_t , \sigma^{1}_{t+1 } \big]\big\vert \sigma^{n}_t\big]\\ & = { \ensuremath{\mathcal{a}^{(a_1\beta)}_{1 } } } \cdots { \ensuremath{\mathcal{a}^{(a_{n-1}\beta)}_{n-1 } } } { \ensuremath{\mathbb{e}}}\big[q^{(x_1(t+1)+1)y_1 } { \ensuremath{\mathbb{e}}}\big [ q^{(x_2(t+1)+2)y_2 } \cdots { \ensuremath{\mathbb{e}}}\big [ q^{(x_{n-1}(t+1)+n-1)y_{n-1 } } \big\vert \sigma^n_t,\sigma^{n-2}_{t+1}\big]\cdots \big\vert \sigma^{n}_t , \sigma^{1}_{t+1 } \big]\big\vert \sigma^{n}_t\big ] \\ & \quad\times\,\sum_{s_n=0}^{y_n } c_{qa_n\beta } ( y_n , s_n ) q^{(x_n(t)+n)(y_n - s_n)}q^{(x_{n-1}(t)+n-1)s_n}\end{aligned}\ ] ] where the first equality was from definition and the second equality followed from applying lemma [ bernlem ] with @xmath284 . we may continue to use lemma [ bernlem ] to reduce the expression above , ultimately leading to @xmath285 with the convention @xmath236 . on account of remark [ zeroremark ] we may now recognize that the right - hand side above , is equal to the right - hand side of ( [ berncond ] ) , as desired to complete the proof of the bernoulli case . it is not a priori clear how to approach the problem of solving the true evolution equations of theorem [ truethm ] in closed form . away from the boundary of @xmath106 ( i.e. , when all of the elements of @xmath107 are different , or equivalently all @xmath286 ) the true evolution equation is constant coefficient and separable . this constant coefficient , separable equation can be extended to all of @xmath287 and is what we will call the _ free evolution equation_. the true and free evolution equations do not match near the boundary of @xmath106 when there is clustering in @xmath107 . the following idea can be traced to bethe s 1931 solution to the heisenburg spin chain @xcite and can be thought of as a generalization of the method of images or reflection principle . bethe s idea is to try to rewrite the true evolution equation as the restriction of the free evolution equation subject to certain boundary conditions which , if satisfied , would imply that the restriction matched the true evolution equation . generally the free evolution equation may be defined on a larger space ( here @xmath287 ) than is physically meaningful ( here @xmath106 ) and the initial data need only be imposed on the physically relevant portion of the space . if such a solution satisfying the free evolution equation , boundary conditions and initial data exists , then its restriction to the physically relevant space will necessarily solve the true evolution equation for the right initial data . this existence of solutions is not assured . if the system in consideration is @xmath211-dimensional , then generically there could be boundary conditions corresponding to all possible compositions of @xmath288 . if in fact only the @xmath1 nearest neighbor boundary conditions are necessary , then the system is called _ integrable _ in the language of quantum many body systems . this is the case for the true evolution equations of poisson , geometric and bernoulli @xmath0-tasep . for the poisson @xmath0-tasep this reduction was observed in proposition 2.7 of @xcite . one should note that it is not a priori clear that the boundary conditions for the discrete time @xmath0-taseps with time varying parameters @xmath64 or @xmath289 would not depend on @xmath27 . in fact , it turns out that this @xmath27 dependence of the true evolution equation plays no role in the form of the boundary condition . in fact , between all three versions of @xmath0-tasep , the exact same boundary condition arise . for a function @xmath290 define the operators @xmath291 and @xmath292 via @xmath293 for a function @xmath294 , let @xmath295_i$ ] and @xmath296_i$ ] act on the @xmath297 coordinate of @xmath146 . the following theorem relates the true evolution equations for @xmath0-tasep to free evolution equations with @xmath1 boundary conditions . [ freethm ] if @xmath298 solves : 1 . for all @xmath299 and @xmath203 , @xmath300_i(t;\vec{n } ) \qquad\qquad\qquad\quad\ , \textrm{(poisson)},\\ u(t+1;\vec{n } ) & = & [ { \ensuremath{\nabla_{-a_{n_1 } \alpha_{t+1 } } } } ] _ 1 \cdots [ { \ensuremath{\nabla_{-a_{n_k } \alpha_{t+1}}}}]_k u(t;\vec{n } ) \qquad\textrm{(geometric)},\\[.2em ] \,[{\ensuremath{\nabla _ { a_{n_1}\beta_{t+1 } } } } ] _ 1 \cdots [ { \ensuremath{\nabla_{a_{n_k } \beta_{t+1}}}}]_k u(t+1;\vec{n } ) & = & [ { \ensuremath{\nabla_{q a_{n_1}\beta_{t+1}}}}]_1 \cdots [ { \ensuremath{\nabla_{q a_{n_k}\beta_{t+1}}}}]_k u(t;\vec{n})\qquad\ , \textrm { ( bernoulli)};\end{aligned}\ ] ] 2 . for all @xmath301 such that for some @xmath302 , @xmath303 , @xmath304_i - q [ \nabla]_{i+1}\big ) u(t;\vec{n})=0;\ ] ] 3 . for all @xmath299 such that @xmath305 , @xmath306 for all @xmath203 ; 4 . for all @xmath116 , @xmath307 ; then for all @xmath176 , @xmath308 where @xmath309 is the solution to the true evolution equation for the ( rate parameter @xmath34 ) continuous time poisson @xmath0-tasep / discrete time geometric @xmath0-tasep ( with time dependent jump parameters @xmath39 ) / discrete time bernoulli @xmath0-tasep ( with time dependent jump parameters @xmath73 ) , started from initial data @xmath202 . it is immediate from the third and fourth hypotheses of theorem [ freethm ] that conditions ( 2 ) and ( 3 ) of definition [ truedef ] are satisfied . it suffices to check that condition ( 1 ) of the true evolution equation is satisfied by @xmath310 . we show this first for the poisson case ( which previously appeared in the proof of proposition 2.7 of @xcite ) . we then deal simultaneously with the geometric and bernoulli cases by using lemma [ dblem ] below . in what follows we assume that @xmath311 and hence @xmath312 . * poisson case : * recall that the size of the cluster of elements of @xmath107 equal to @xmath6 is given by @xmath160 . consider the cluster of elements equal to @xmath3 : @xmath313 . every other cluster works similarly to what we now describe and though it is possible that the cluster we study is empty , we may repeat the below calculation for any other cluster . in order to prove the true evolution equation it suffices ( by summing over all clusters ) to show that @xmath314_i u(t;\vec{n } ) = a_n ( 1-q^{y_n } ) \nabla_{y_n } u(t;\vec{n } ) = { \ensuremath{\mathcal{l}^{(a_n)}_{n}}}u(t;\vec{y}),\ ] ] where the second equality follows immediately from the definition of @xmath315 . to see the first equality we use the boundary condition for the free evolution equation which implies that for @xmath316 , @xmath296_i u(t;\vec{n } ) = q^{y_n - i}[\nabla]_{y_n } u(t;\vec{n})$ ] . summing over @xmath6 gives the desired equality . * geometric and bernoulli cases : * we use lemma [ dblem ] below to prove the equivalence of the free and true evolution equations for both the geometric and bernoulli cases . however , we first provide a lemma which quickly leads to the proof of lemma [ dblem ] . [ prevlem ] fix @xmath317 and @xmath191 . assume that a function @xmath318 is such that if @xmath108 for any @xmath319 then @xmath320_i - q[\nabla]_{i+1})f(\vec{n } ) = 0 $ ] . then @xmath321_1 \cdots [ { \ensuremath{\nabla_{a}}}]_y f(n,\ldots , n ) = \sum_{s=0}^{y } c_{a}(y , s ) f(\underbrace{n,\ldots , n}_{y - s } , \underbrace{n-1,\ldots , n-1}_{s}).\ ] ] we prove this by induction in @xmath322 . for @xmath323 observe that by definition of @xmath324 $ ] , @xmath325_1 f(n ) & = & ( 1+a)f(n ) - a f(n-1)\\ & = & c_a(1,0 ) f(n ) + c_a(1,1 ) f(n-1),\end{aligned}\ ] ] since @xmath326 and @xmath327 . for @xmath328 assume that we have proved the lemma for @xmath329 . thus we have @xmath330_1 [ { \ensuremath{\nabla_{a}}}]_2 \cdots [ { \ensuremath{\nabla_{a}}}]_y f(n,\ldots , n ) & = & [ { \ensuremath{\nabla_{a}}}]_1 \sum_{s=0}^{y-1 } c_{a}(y-1,s ) f(n,\underbrace{n,\ldots , n}_{y-1-s } , \underbrace{n-1,\ldots , n-1}_{s})\\ & = & \sum_{s=0}^{y-1 } c_{a}(y-1,s ) ( 1+a ) f(n,\underbrace{n,\ldots , n}_{y-1-s } , \underbrace{n-1,\ldots , n-1}_{s})\\ \nonumber & & + \sum_{s=0}^{y-1 } c_{a}(y-1,s ) ( -a ) f(n-1,\underbrace{n,\ldots , n}_{y-1-s } , \underbrace{n-1,\ldots , n-1}_{s}).\end{aligned}\ ] ] using the relation @xmath320_i - q[\nabla]_{i+1})f(\vec{n } ) = 0 $ ] for @xmath107 such that @xmath108 , we see that @xmath331 plugging this into ( [ aboveplug ] ) we arrive at @xmath332 grouping the coefficients of @xmath333 in the above summations and using the second relationship in ( [ cid ] ) we recover ( [ desiredf ] ) and hence complete the inductive step . [ dblem ] fix @xmath334 and @xmath38 . assume that a function @xmath318 is such that if @xmath108 for any @xmath319 then @xmath320_i - q[\nabla]_{i+1})f(\vec{n } ) = 0 $ ] . then for all @xmath31 , @xmath335_1 \cdots [ { \ensuremath{\nabla_{a_{n_k}}}}]_k f(n_1,\ldots , n_k ) = \sum_{s_1=0}^{y_1 } c_{a_1}(y_1,s_1 ) \cdots \sum_{s_n=0}^{y_n } c_{a_n}(y_n , s_n ) f(\vec{n}(\vec{y}^{\,\vec{s}})).\ ] ] observe that @xmath107 can be split into clusters of equal variables so that @xmath336 and @xmath337 and so on ( recall that @xmath222 gives the sizes of these clusters ) . due to the ordering of @xmath107 , the result we seek to prove follows readily from applying lemma [ prevlem ] to each cluster of @xmath107 starting with the cluster of @xmath104 which equal 1 , and ending with the cluster of @xmath104 equal to @xmath3 . let us now complete the proof of theorem [ freethm ] . we may apply lemma [ dblem ] ( given below ) to the solution @xmath298 of the geometric / bernoulli free evolution equation ( the boundary condition on @xmath338 implies the hypotheses of the lemma are met ) . since @xmath306 for @xmath305 , it follows that the summation over @xmath339 in the outcome of the application of the lemma should be removed and only the @xmath198 term remain ( that is , because all terms involving @xmath340 are necessarily zero ) . comparing the outcome to the ( respective ) true evolution equation of definition [ truedef ] one finds that they match . the following is an immediate corollary of the combination of theorems [ freethm ] and [ truethm ] . [ freethmcor ] consider the ( rate parameter @xmath34 ) continuous time poisson @xmath0-tasep / discrete time geometric @xmath0-tasep ( with time dependent jump parameters @xmath39 ) / discrete time bernoulli @xmath0-tasep ( with time dependent jump parameters @xmath40 ) started from an arbitrary initial condition @xmath219 . then for any @xmath115 and @xmath31 , @xmath341 = u(t;\vec{n})\ ] ] where @xmath298 solves the ( respective ) free evolution equation with @xmath1 boundary conditions started from initial data @xmath342.\ ] ] we now conclude the proof of theorem [ momthm ] by showing that the right - hand side of ( [ momthmeq ] ) solves the free evolution equations with @xmath1 boundary conditions which is given in theorem [ freethm ] . corollary [ freethmcor ] then immediately implies theorem [ momthm ] . for ease of readability let us recall the right - hand side of ( [ momthmeq ] ) and denote it as @xmath343 there are four hypotheses to check in theorem [ freethm ] . the first hypothesis to check is that @xmath344 solves the free evolution equation corresponding with the choice of @xmath118 . since the free evolution equation is linear and separable , it suffices to check that for arbitrary @xmath345 and @xmath30 , @xmath346 satisfies the @xmath347 version of the respective free evolution equations . * poisson case : * we must check that @xmath348 the @xmath349 brings out a factor @xmath350 on the right - hand side , while the @xmath292 brings out a factor @xmath351 . comparing both sides we find equality . * geometric case : * we must check that @xmath352 the @xmath353 brings out a factor @xmath354 which simplifies to @xmath355 , and hence we find equality . * bernoulli case : * we must check that @xmath356 the @xmath357 on the left - hand side brings out a factor @xmath358 which simplifies to @xmath359 , while the @xmath360 on the right - hand side brings out a factor @xmath361 which simplifies to @xmath362 . comparing both sides we find equality . the second hypothesis to check is that the boundary condition is satisfied . without loss of generality assume @xmath363 ( general @xmath108 is identical ) . we wish to show that @xmath364_1-q[\nabla]_2\right ) m(t;\vec{n } ) = 0.\ ] ] applying the operator @xmath296_1-q[\nabla]_2 $ ] to the integrand of @xmath344 brings out a factor of @xmath365 . this new factor cancels the denominator @xmath366 . on account of this we may freely ( without encountering any poles ) deform the contours for @xmath367 and @xmath368 to coincide . hence we may write @xmath364_1-q[\nabla]_2\right ) m(t;\vec{n } ) = \int \int ( z_1-z_2 ) g(z_1)g(z_2)dz_1 dz_2\ ] ] where the function @xmath369 involves the integrals in @xmath370 . since the two contours are identical , this integral is clearly zero , as desired . the third hypothesis to check is that for @xmath305 , @xmath371 . this follows from simple residue calculus since when @xmath305 there is no pole at @xmath372 and since this was the only pole contained by the @xmath373 contour , by cauchy s theorem the @xmath373 integral equals 0 . the final hypothesis to check is the initial data . since we are dealing with step initial condition @xmath18 , it follows that @xmath374 and hence we must show that @xmath375 . this follows from residue calculus as well . expand the @xmath367 contour to infinity . since @xmath376 , @xmath377 and so the only pole in @xmath367 is encountered at @xmath378 ( @xmath379 is not a pole because of the decay coming from @xmath380 ) . because we pass the pole at 0 from the outside , the contribution of the residue is @xmath381 times the same integral , but with @xmath1 variables . repeating this procedure yields the desired result . this completes the proof of theorem [ momthm ] . at least for the poisson and geometric @xmath0-tasep , the true evolution equations are closely related to markov process dualities . this was understood for poisson @xmath0-tasep in @xcite . two markov processes @xmath382 and @xmath383 ( with state spaces @xmath384 and @xmath385 respectively ) are said to be dual with respect to a function @xmath386 if for all @xmath387 , @xmath388 and @xmath203 , @xmath389 = { \ensuremath{\mathbb{e}}}^{y } \left[h(x;y(t))\right]\ ] ] where @xmath390 and @xmath391 refer to the expectations of the respective markov chains @xmath382 and @xmath383 started from @xmath392 and @xmath393 . let us recall the duality for the poisson @xmath0-tasep . let @xmath394 be the totally asymmetric zero range process with state space @xmath395 from ( [ yn ] ) in which the rate at which a particle moves from site @xmath6 to @xmath7 ( for @xmath396 ) is given by @xmath397 . notice that this is the same jump rate as in remark [ zerorem ] though presently there are no sources or sinks . let @xmath10 be the continuous time poisson @xmath0-tasep with @xmath3 particles . then it is easily checked that @xmath10 and @xmath394 are dual with respect to @xmath398 where for @xmath224 the product is over @xmath399 and for @xmath195 , the product is taken to be zero ( due to the virtual particle @xmath400 ) . the true evolution equation follows immediately from this duality since @xmath401 = \frac{d}{dt } { \ensuremath{\mathbb{e}}}^y\left[h(\vec{x};\vec{y}(t))\right ] = \sum_{i=1}^{n } { \ensuremath{\mathcal{l}^{(a)}_{i } } } \left[h(\vec{x};\vec{y}(t))\right].\ ] ] here , the first equality is from duality and the second one is from the kolmogorov forward equation and the fact that @xmath402 is the generator of the dual zero range process . it is also possible to relate the discrete time geometric @xmath0-tasep to a discrete time totally asymmetric zero range process @xmath394 with state space @xmath395 . in parallel , the state of @xmath394 is updated to that of @xmath403 via the following procedure which occurs at each site @xmath396 . at time @xmath27 there are @xmath404 particles lined up above site @xmath6 . start with the bottom - most particle and with probability @xmath70 move the particle to site @xmath7 at time @xmath60 . then proceed sequentially by considering the second particle from bottom , and then the third and so on . for the @xmath405 particle from bottom , with probability @xmath406 move it to site @xmath7 at time @xmath60 . here @xmath407 counts the number of particles in site @xmath6 which are below the @xmath405 particle and which have been tagged to move to site @xmath7 at time @xmath60 . the update rule is sequential within each site @xmath6 , but parallel amongst different sites . this duality can be proved using similar considerations as in the above proof of the geometric true evolution equation . since it is not necessary for the purposes of this paper , we leave this just as a remark and do not present this proof . ( ascending ) macdonald processes ( see section 2 and 3 of @xcite for more details ) are measures on triangular arrays of interlacing nonnegative integers @xmath408 we write @xmath409 for the @xmath405 level of the triangle , and note that @xmath409 is a partition of length at most @xmath53 . the measure on @xmath410 is specified by @xmath38 , @xmath411 , @xmath412 , @xmath121 and two parameters @xmath413 as follows : @xmath414 and the marginal of this measure on a single level @xmath415 is given by @xmath416 in the above , @xmath417 and @xmath418 are macdonald symmetric functions ( indexed by partitions or skew partitions ) and @xmath419 is a macdonald positive specialization ( i.e. , a homomorphism from the algebra of symmetric functions to @xmath420 that sends macdonald symmetric functions to elements of @xmath421 ) specified via the generating function @xmath422 for such a specialization and positive @xmath25 , the numerator in the definition of @xmath423 above is non - negative for all triangular arrays @xmath410 , and as long as the normalizing constant @xmath424 is finite , the above expression describes a probability measure . for a partition @xmath425 of length @xmath3 , the macdonald polynomial @xmath426 is an eigenfunction for the first macdonald difference operator @xmath427 ( where @xmath428 ) with eigenvalue @xmath429 notice that when @xmath376 the eigenvalue is simply @xmath430 . we will assume henceforth that @xmath376 ( though everything except the commutation relations holds for general @xmath27 ) . the normalizing constant is given by @xmath431 it was observed in @xcite that by linearity of @xmath432 and by the above eigenvalue relation , for any @xmath433 @xmath434.\ ] ] a generalization of this was provided in @xcite , showing that if @xmath435 represents the first difference operator acting on @xmath436 , then for any @xmath437 @xmath438.\ ] ] since the @xmath439 and @xmath440 are explicit and suitably nice , it is possible to encode the above repeated application of difference operators onto @xmath440 in terms of a nested contour integral formula ( one recovers the difference operators by expansion of the formula into residues ) . it is shown in @xcite that for @xmath441 with @xmath442 , and @xmath443 , @xmath444 = \frac{(-1)^k q^{\frac{k(k-1)}{2}}}{(2\pi \iota)^k } \int \cdots \int \prod_{1\leq a < b\leq k } \frac{z_a - z_b}{z_a - qz_b } \prod_{j=1}^{k } \left(\prod_{i=1}^{n_j } \frac{a_i}{a_i - z_j}\right ) \frac{\pi(qz_j;\rho)}{\pi(z_j;\rho ) } \frac{dz_j}{z_j}.\ ] ] where the contour of integration for @xmath33 contains @xmath34 , and @xmath35 but not 0 or any other poles of the integrand . plugging in the formula for @xmath440 ( given above in terms of @xmath92 s , @xmath93 s and @xmath445 ) it is immediately clear that @xmath446 = { \ensuremath{\mathbb{e}}}\left[\prod_{i=1}^{n } q^{(x_i(t)+i)y_i } \right],\ ] ] where the expression on the right - hand side corresponds to the version of @xmath0-tasep considered in remark [ arborder ] . therefore , since these moments characterize the joint law of @xmath447 and @xmath448 , it follows that these two sets are equal in distribution . there are two immediate questions this equality prompts . first is whether there is a way to embed the various @xmath0-tasep dynamics into a markov chain on interlacing triangular arrays so that this equality becomes natural . second is how the true evolution equation ( which the right - hand side of the above equality satisfies ) can be seen directly from the left - hand side as a consequence of macdonald difference operator relations ( without any reference to @xmath0-taseps ) . we answer both of these questions for continuous time poisson @xmath0-tasep first , and then indicate some extensions of the answer to the second question for the two discrete time versions as well . there exist markov dynamics on the space of interlacing triangular arrays which preserve the class of macdonald processes . one example of such a dynamic was introduced and studied in sections 2 and 3 of @xcite ( see @xcite for other such dynamics ) . when the parameter @xmath376 the continuous time dynamics on the triangular arrays becomes quite simple ( cf . section 3.3 of @xcite ) . each @xmath449 attempts to increase its value by one according to independent exponentially distributed jumping times with rate given by @xmath450 where terms involving indices outside of the triangular array are omitted . the simplest macdonald process with which to initiate @xmath410 is specified by taking all @xmath451 and corresponds with taking @xmath452 . then if the above dynamics are run for time @xmath27 ( not to be confused with the macdonald parameter @xmath27 which has been fixed to be zero ) then @xmath453 is distributed according to a macdonald process specified by taking all @xmath454 but @xmath455 . the above dynamics have the property that the evolution of @xmath456 is marginally markovian ( with respect to its own filtration ) . in fact , calling @xmath457 we readily observe that @xmath12 evolves according to continuous time poisson @xmath0-tasep with particle jump parameters @xmath34 . it was through this line of reasoning that poisson @xmath0-tasep was first introduced and through the above macdonald difference operator considerations that the formulas for moments of @xmath458 were initially calculated . in light of equation ( [ lambdax ] ) we are led to the second question , of how the true evolution equation for poisson @xmath0-tasep can be seen directly from the language of macdonald difference operators . in particular , we would like to show that @xmath459\ ] ] solves @xmath460 we claim that the fact that @xmath199 solves the true evolution equation can be seen as a consequence of a commutation relation for macdonald first difference operators , as well as the fact that by the above discussion @xmath461 1 . for any @xmath464 and any @xmath465 , @xmath466 = ( 1-q^k ) x_n ( d_{n-1}-d_n ) ( d_n)^{k-1}.\ ] ] 2 . for any @xmath464 , @xmath465 and @xmath46 , @xmath467 3 . for any @xmath464 , @xmath465 and @xmath468 , @xmath469 we first remark that in the statement of the lemma , @xmath470 , @xmath471 and @xmath472 should all be treated as operators which act by multiplication . thus , we seek to prove the above identities as operators applied to general functions @xmath473 . by polarization it suffices to prove these identities when applied to functions of the form @xmath474 . we prove part 1 of the lemma and simply note that analogous considerations ( or comparison to the treatment of the geometric and bernoulli many - body systems earlier in this paper ) suffice to prove parts 2 and 3 . we can rewrite the desired identity we wish to prove as ( adding harmless @xmath475 factors to both sides ) @xmath476 for @xmath474 . this may be expressed in terms of nested contour integrals . the left - hand side becomes @xmath477 while the right - hand side becomes @xmath478 in both sets of integrals we assume that the @xmath33 contour can be chosen so as to contain @xmath479 for @xmath480 as well as @xmath481 , but not 0 or any poles of @xmath482 . observe that due to the choice of nested contours , it follows that for @xmath483 and any symmetric function @xmath484 ( which does not introduce new poles into the below integrand ) @xmath485 using this , we find that the term @xmath486 in the integrand of ( [ exp1s ] ) can be turned into @xmath487 which exactly matches the term in the integrand of ( [ exp2s ] ) and thus proves the lemma . to conclude , let us explain how the commutation relations imply the true evolution equation . observe that @xmath488 where @xmath489 , and on the right - hand side we assign @xmath490 after applying all of the operators ( we suppress writing this assignment above and in what follows ) . the commutation relation allows us to pull the @xmath491 in the second term from the right side of the expression to the left side . in the first application of ( [ commute ] ) we replace @xmath492 and find @xmath493 repeating this @xmath3 times we find @xmath494 the right - hand side of which is readily matched to that of the true evolution equation .

we introduce two new exactly solvable ( stochastic ) interacting particle systems which are discrete time versions of @xmath0-tasep . we call these geometric and bernoulli discrete time @xmath0-tasep . we obtain concise formulas for expectations of a large enough class of observables of the systems to completely characterize their fixed time distributions when started from step initial condition . we then extract fredholm determinant formulas for the marginal distribution of the location of any given particle . underlying this work is the fact that these expectations solve closed systems of difference equations which can be rewritten as free evolution equations with @xmath1 two - body boundary conditions discrete @xmath0-deformed versions of the quantum delta bose gas . these can be solved via a nested contour integral ansatz . the same solutions also arise in the study of macdonald processes , and we show how the systems of equations our expectations solve are equivalent to certain commutation relations involving the macdonald first difference operator . [ section ] [ section ] [ theorem]conjecture [ theorem]lemma [ theorem]proposition [ theorem]corollary [ theorem]claim [ theorem]critical point derivation [ theorem]experimental result # 1 # 1#1 # 1 [ theorem]remark [ theorem]example [ theorem]definition [ theorem]definitions

You are an expert at summarizing long articles. Proceed to summarize the following text: typical wavelength of gravitational waves from astrophysical compact objects such as bh(black hole)-bh binaries is in some cases very long so that wave optics must be used instead of geometrical optics when we discuss gravitational lensing . more precisely , if the wavelength becomes comparable or longer than the schwarzschild radius of the lens object , the diffraction effect becomes important and as a result the magnification factor approaches unity @xcite . mainly due to the possibility that the wave effects could be observed by future gravitational wave observations , several authors @xcite have studied wave effects in gravitational lensing in recent years . in most of the works which studied gravitational lensing phenomenon in the framework of wave optics , isolated and normal astronomical objects such as galaxies are concerned as lens objects . recently yamamoto and tsunoda@xcite studied wave effects in gravitational lensing by an infinite straight cosmic string . the metric around a cosmic string is completely different from that around a usual massive object . cosmic strings generically arise as solitons in a grand unified theory and could be produced in the early universe as a result of symmetry breaking phase transition@xcite . if symmetry breaking occurred after inflation , the strings might survive until the present universe . recently , cosmic strings attract a renewed interest partly because a variant of their formation mechanism was proposed in the context of the brane inflation scenario@xcite . in this scenario inflation is driven by the attractive force between parallel d - branes and parallel anti d - branes in a higher dimensional spacetime . when those brane - anti - brane pairs collide and annihilate at the end of inflation , lower - dimensional d - branes , which behave like monopoles , cosmic strings or domain walls from the view point of four - dimensional observers , are formed generically @xcite . for some time , cosmic string was a candidate for the seed of structure formation of our universe , but this possibility was ruled out by the measurements of the spectrum of cosmic microwave background ( cmb ) anisotropies@xcite . the current upper bound on the dimensionless string tension @xmath4 is around @xmath5 , which comes from the observations of cmb@xcite and/or the pulsar timing @xcite . although cosmic string can not occupy dominant fraction of the energy density of the universe , its non - negligible population is still allowed observationally@xcite . in fact , sazhin et al.@xcite reported that csl-1 , which is a double image of elliptical galaxies with angular separation @xmath6 , could be the first case of the gravitational lensing by a cosmic string with @xmath7 . we study in detail wave effects in the gravitational lensing by an infinite straight cosmic string . in ref . @xcite , wave propagation around a cosmic string was studied but they put the waveform around the string by hand . compared with that corresponding to the geometrical optics . we show that the importance of the diffraction effects are determined by the combination of three parameters , @xmath4 , the distance from the string to the observer and the wavelength and that the relative amplitude of the diffracted wave can be @xmath8 for realistic astrophysical situations . ] their prescription is correct only in the limit of geometrical optics , which breaks down when the wavelength becomes longer than a certain characteristic length . in this paper , we present exact solutions of the ( scalar ) wave equation in a spacetime with a cosmic string . we analytically show that our solutions reduce to the results of the geometrical optics in the short wavelength limit . we derive a simple analytic formula of the leading order corrections to the geometrical optics due to the finite wavelength effects and also an expression for the long wavelength limit . interference caused by the lensing remains due to the diffraction effects even when only a single image can be seen in the geometrical optics . this fact increases the lensing probability by cosmic strings . this paper is organized as follows . in section ii , we construct a solution of the wave equation on a background spacetime with an infinite straight cosmic string in the case that a source of the wave is located infinitely far . an extension to the case in which a point source is located at a finite distance is given in appendix b. in section iii , we study properties of the solution obtained in sec . ii in detail . in section iv , we focus on compact binaries as the sources of gravitational waves and discuss the possible effects due to finiteness of the lifetime and the frequency evolution of the binaries on the detection of the gravitational waves which pass near a cosmic string . we also give a rough estimate for the event rate of the lensing of gravitational waves from ns - ns mergers assuming decigo / bbo . section v is devoted to summary . a solution of einstein equations around an infinite straight cosmic string to first order in @xmath9 is given by @xcite @xmath10 where @xmath11 is a cylindrical coordinate(@xmath12 ) and @xmath13 is the deficit angle around the cosmic string . spatial part of the above metric describes the euclidean space with a wedge of angular size @xmath14 removed . due to the deficit angle around a string , double images of the source are observed with an angular separation @xmath15 when a source is located behind the string in the limit of geometrical optics . in general for a wave with a finite wavelength , some interference pattern appears . an exact solution of einstein equations around a finite thickness string has been already obtained @xcite , but we use the metric ( [ 1.1 ] ) as a background since the string thickness is negligibly small compared with the einstein radius , @xmath16 , where @xmath17 is the distance from the observer to the string . throughout the paper , we consider waves of a massless scalar field instead of gravitational waves for simplicity , but the wave equations are essentially the same in these two cases . an extension to the cosmological setup is straightforwardly done by adding an overall scale factor . in that case the time coordinate @xmath18 is to be understood as the conformal time . the wave equation remains unchanged if we consider a conformally coupled field , but it is modified for the other cases due to curvature scattering . the correction due to curvature scattering of the friedmann universe is suppressed by the square of the ratio between the wavelength and the hubble length , which can be neglected in any situations of our interest . our goal of this section is to construct a solution of the wave equation which corresponds to a plane wave injected perpendicularly to and scattered by the cosmic string . this situation occurs if the distance between the source and the string is infinitely large . in order to construct such a solution , we introduce a monochromatic source uniformly extended in the @xmath19-direction and localized in @xmath20 plane , @xmath21 where @xmath22 is the frequency and we have introduced @xmath23 , a constant independent of @xmath24 , to adjust the overall normalization when we later take the limit @xmath25 . the factor @xmath26 appears because @xmath27-coordinate used in the metric ( [ 1.1 ] ) differs from the usual angle @xmath28 here we consider a uniformly extended source instead of a point source since the former is easier to handle . when the limit @xmath25 is taken , the answers are identical in these two cases . the case with a point - like source at a finite distance is more complicated . this case is treated in appendix b. now the wave equation that we are to solve is @xmath29 since @xmath30 satisfies the same equation ( [ 1.3 ] ) as @xmath31 does , @xmath31 is even in @xmath27 . thus , it can be expanded as @xmath32 from eqs . ( [ 1.3 ] ) and ( [ 1.4 ] ) , the equations for @xmath33 are @xmath34 where @xmath35 and @xmath36 . the solution of eq . ( [ 1.5 ] ) except for @xmath37 is a linear combination of bessel function and hankel function . we impose that the wave @xmath38 is regular at @xmath39 and pure out - going at infinity . further , imposing that the wave is continuous at @xmath37 , @xmath33 becomes @xmath40 where @xmath41 is the heaviside step function . substituting eq . ( [ 1.6 ] ) into eq . ( [ 1.5 ] ) , the normalization factor @xmath42 is determined as @xmath43^{-1}\cr & = & { br_0 \epsilon_m(-1)^m\over 4i(1-\delta ) } , \label{1.7}\end{aligned}\ ] ] where @xmath44 denotes a differentiation with respect to the argument . from eqs . ( [ 1.6 ] ) and ( [ 1.7 ] ) with the aid of the asymptotic formulae of the bessel and hankel functions , @xmath31 for @xmath45 can be written as @xmath46 we determine the overall normalization of the source amplitude @xmath23 , independently of @xmath4 , so that eq . ( [ 1.8 ] ) becomes a plane wave @xmath47 when @xmath48 . this condition leads to @xmath49 . then , finally @xmath38 becomes @xmath50 the solution ( [ 1.9 ] ) describes the waveform propagating around a cosmic string . but it is not easy to understand the behavior of the solution because it is given by a series . in fact , it takes much time to perform the summation in eq . ( [ 1.9 ] ) numerically for a realistic value of tension of the string , say , @xmath51 because of slow convergence of the series . in particular it is not manifest whether the amplification of the solution in the short wavelength limit coincides with the one which is obtained by the geometrical optics approximation . therefore it will be quite useful if one can derive a simpler analytic expression . here we reduce the formula by assuming that the distance between the string and the observer is much larger than the wave length , @xmath52 which is valid in almost all interesting cases . using an integral representation of the bessel function , @xmath53 where the contour of the integral @xmath54 is such as shown in fig . [ contour ] , eq . ( [ 1.9 ] ) can be written as @xmath55 when @xmath18 is in the segment of the integration contour @xmath54 along the imaginary axis , the summation over @xmath56 does not converge because the absolute value of each term in the summation is all unity . in order to make the series to converge , we need to think that the integration contour @xmath54 is not exactly on the imaginary axis but @xmath18 always has a positive real part . for bookkeeping purpose , we multiply each term in the sum by a factor @xmath57 ( @xmath58 is an infinitesimally small positive real number ) . then eq . ( [ 3.2 ] ) becomes @xmath59 where @xmath60 is defined by @xmath61 with @xmath62 and @xmath63 . now we find that all we need to evaluate is @xmath60 in order to obtain an approximate formula for @xmath64 . this integral will not be expressed by simple known functions in general , but the integration can be performed by using the method of steepest descent in the limit @xmath65 . the integrand of eq . ( [ 3.3 ] ) has two saddle points located at @xmath66 and @xmath67 in the vicinity of the integration contour @xmath54 . we should also notice that the integrand has a pole at @xmath68 , which is also infinitesimally close to the contour of the integral @xmath54 . this pole is located near the saddle point at @xmath69 as far as @xmath24 and @xmath27 are small . hence the treatment of the saddle point at @xmath69 is much more delicate than that of the saddle point at @xmath70 . we only discuss the saddle point at @xmath69 , then the case at @xmath70 is a trivial extension . when @xmath71 or @xmath72 , which corresponds to shaded regions in fig . [ contour ] , @xmath73 diverges in the limit @xmath74 . if @xmath75 , the pole at @xmath68 is in the bottom - left unshaded region . in this case we can not deform the contour to the direction of the steepest descent at @xmath69 without crossing the pole at @xmath68 . the deformed contour which is convenient to apply the method of the steepest descent is such that is shown as @xmath76 in fig . [ contour ] . when we deform the integration contour from @xmath54 to @xmath77 , there arises an additional contribution corresponding to the residue at @xmath78 when @xmath75 . on the other hand , if @xmath79 , the pole is in the top - left shaded region . in this case , we can deform the contour of the integral to the direction of the steepest descent without crossing the pole @xmath80 . hence no additional term arises . from these observations , we find that it is necessary to evaluate the integral ( [ 3.3 ] ) separately depending on the signature of @xmath81 . though the calculation itself can be done straightforwardly , it is somewhat complicated because the saddle point and the pole are close to each other . when the pole is located inside the region around the saddle point that contributes dominantly to the integral , a simple gaussian integral does not give a good approximation . detailed discussions about this point are given in appendix a. here we only quote the final result which keeps terms up to @xmath82 , @xmath83 where @xmath84 , \label{3.3b1}\\ \sigma(\theta ) & : = & { \rm sign } ( { \alpha}(\theta ) ) , \label{3.3c}\end{aligned}\ ] ] and @xmath85 we are mostly interested in the cases with @xmath86 . then , we have @xmath87 , and therefore @xmath88 reduces to @xmath89 . the second term in eq . ( [ 3.3bb ] ) is the contribution from the integral around the saddle point at @xmath69 along the contour @xmath90 . this term is not manifestly suppressed by @xmath91 . as far as @xmath89 is fixed , this term does not vanish in the limit @xmath92 . of course , if we fix @xmath24 and @xmath27 first , and take the limit @xmath92 , the argument of the error function goes to @xmath93 and the function itself vanishes . however , @xmath89 vanishes at @xmath94 . hence even for a very large value of @xmath95 there is always a region of @xmath27 in which this second term can not be neglected . however , for @xmath27 in such a region , @xmath89 can not be very large . therefore , we can safely drop the second term in the exponent . on the other hand , the last term in eq . ( [ 3.3bb ] ) , which is the contribution from the saddle point at @xmath70 , is always suppressed by @xmath91 . hence , this term does not give any significant contribution for @xmath65 . the first term in eq . ( [ 3.31 ] ) can be dropped in the same manner for @xmath65 . keeping only the terms which possibly remain in the limit @xmath92 , we finally obtain @xmath96 for illustrative purpose , we compared the estimate given in eq . ( [ 3.3bb ] ) with the exact solution eq . ( [ 1.9 ] ) in fig . [ approx ] . they agree quite well at @xmath65 . the deficit angle and the observer s direction are chosen to be @xmath97 and @xmath98 , respectively . , @xmath76 and @xmath99 , respectively . @xmath100 are the saddle points of @xmath73 . ] geometrical optics limit corresponds to the limit @xmath74 with @xmath24 and @xmath27 fixed . in this limit @xmath89 also goes to @xmath93 , and hence the error function in eq . ( [ 3.3b ] ) vanishes . hence the waveform in the geometrical optics limit , which we denote as @xmath101 , becomes @xmath102 where @xmath103 is defined by eq . ( [ 1.25 ] ) . since @xmath38 and hence @xmath101 are even in @xmath27 , it is sufficient to consider the case with @xmath104 . in fig . [ configuration ] , the configuration of the source , the lens and the observer is drawn in the coordinates in which the deficit angle @xmath105 is manifest , i.e. , the wedge aob is removed from the spacetime . both points a and b indicate the location of the source . the lines oa and ob are to be identified . the angle made by these two lines is the deficit angle . the locations of the string and the observer are represented by o and p , respectively . in our current setup the distance between o and a ( @xmath106 ) is taken to be infinite . when @xmath107 , only the source a can be seen from the observer . this corresponds to the fact that only the first term remains for @xmath108 in eq . ( [ go1 ] ) . for @xmath107 , we have @xmath109 this is a plane wave whose traveling direction is @xmath110 , which is the direction of @xmath111 in fig . [ configuration ] in the limit @xmath112 . for @xmath113 , @xmath101 is @xmath114 this is the superposition of two plane waves whose traveling directions are different by the deficit angle @xmath105 . hence amplification of the images and interference occur for @xmath113 as expected . as we shall explain below , eq . ( [ go1 ] ) coincides with the one derived under the geometrical optics . in geometrical optics , wave form is given by @xcite @xmath115 , \label{go6}\ ] ] where @xmath116 represents a two - dimensional vector on the lens plane and @xmath117 represents the summation of time of flight of the light ray from the source to the point @xmath118 on the lens plane and that from the point @xmath118 to the observer . @xmath119 is a stationary point of @xmath117 , and @xmath120 when @xmath119 is a minimum , saddle and maximum point of @xmath117 , respectively . the amplitude ratio @xmath121 is written as @xmath122 , \label{go7}\ ] ] where @xmath123 in eq . ( [ go7 ] ) is the deflection potential @xcite which is the integral of the gravitational potential of the lens along the trajectory between the source and the observer . ( [ go6 ] ) represents that the wave form is obtained by taking the sum of the amplitude ratio @xmath124 of each images with the phase factor @xmath125 . if the lens is the straight string , the spacetime is locally flat everywhere except for right on the string . this means that the deflection potential @xmath126 is zero and hence the amplitude ratio is unity for all images @xcite and the trajectory where the time of flight @xmath127 takes the extremal value is a geodesic in the conical space , and @xmath127 of any geodesic takes minimum , which means @xmath128 . there are two geodesics if the observer is in the shaded region in fig . [ configuration ] . the time of flight along the trajectory ap is @xmath129 where @xmath130 . the time of flight along the trajectory bp is obtained by just replacing @xmath103 with @xmath131 . hence , substituting ( [ go8 ] ) into ( [ go6 ] ) , we find that the waveform in the geometrical optics is the same as eq . ( [ go3 ] ) except for an overall phase @xmath132 . this factor has been already absorbed in the choice of the normalization factor @xmath23 in our formula ( [ 1.9 ] ) . we define the amplification factor @xmath133 where @xmath134 is the unlensed waveform . using eq . ( [ go3 ] ) , the amplification factor of @xmath101 for @xmath135 is given by @xmath136 where we have assumed @xmath103 and @xmath24 are small and dropped terms higher than quadratic order . it might be more suggestive to rewrite the above formula into @xmath137 where @xmath138 . the distance from a node to the next of when the observer is moved in @xmath139-direction is @xmath140 , where @xmath3 is a wavelength . this oscillation is seen in the right panel of fig . [ approx ] . in the previous subsection , we have derived the waveform in the limit @xmath141 which corresponds to the geometrical optics approximation . here we expand the waveform ( [ 3.3b ] ) to the lowest order in @xmath142 . this includes the leading order corrections to the geometrical optics approximation due to the finite wavelength effects . for the same reason as we explained in the previous subsection , we assume that @xmath24 and @xmath103 are small . using the asymptotic formula for the error function eq . ( [ ap.5a ] ) , the leading order correction due to the finite wavelength , which we denote as @xmath143 , is obtained as @xmath144 as is expected , the correction blows up for @xmath145 , where @xmath81 or @xmath146 vanishes , irrespectively of the value of @xmath95 . in such cases , we have to evaluate the error function directly , going back to eq . ( [ 3.3b ] ) . the expression on the first line in eq.([qgo1 ] ) manifestly depends only on @xmath147 aside from the common phase factor @xmath148 . this feature remains true even if we consider a small value of @xmath147 . this can be seen by rewriting eq . ( [ 3.3b ] ) as @xmath149 the common phase @xmath148 does not affect the absolute magnitude of the wave . except for this unimportant overall phase , the waveform is completely determined by @xmath150 . the geometrical meaning of these parameters @xmath150 is the ratio of two length scales defined on the lens plane . to explain this , let us take the picture that a wave is composed of a superposition of waves which go through various points on the lens plane . in the geometrical optics limit the paths passing through stationary points of @xmath127 , which we call the image points , contribute to the waveform . the first length scale is @xmath151 which is defined as the separation between an image point and the string on the lens plane . in this picture we expect that paths whose pathlength is longer or shorter than the value at an image point by about one wavelength will not give a significant contribution because of the phase cancellation . namely , only the paths which pass within a certain radius from an image point need to be taken into account . then such a radius will be given by @xmath152 , which we call fresnel radius . namely , a wave with a finite wavelength can be recognized as an extended beam whose transverse size is given by @xmath153 . the ratio of these two scales gives @xmath150 : @xmath154 when @xmath155 , i.e. , @xmath156 , the beam width is smaller than the separation . in this case the beam image is not shadowed by the string , and therefore the geometrical optics becomes a good approximation . when @xmath157 , i.e. , @xmath158 we can not see the whole image of the beam , truncated at the location of the string . then the diffraction effect becomes important . the ratio of the beam image eclipsed by the string determines the phase shift and the amplification of the wave coming from each image . if we substitute @xmath159 as a typical value , we obtain a rough criterion that the diffraction effect becomes important when @xmath160 or @xmath161 in terms of @xmath95 . the same logic applies for a usual compact lens object . in this case the fresnel radius does not change but the typical separation of the image from the lens is given by the einstein radius @xmath162 , where @xmath163 is the mass of the lens . then the ratio between @xmath164 and @xmath153 is given by @xmath165 , which leads to the usual criterion that the diffraction effect becomes important when @xmath166@xcite . from the above formula ( [ qgo1 ] ) , we can read that the leading order corrections scales like @xmath167 . this dependencies on @xmath3 and @xmath168 differ from the cases that the lens is composed of a normal localized object , in which the leading order correction due to the finite wavelength is @xmath169 @xcite . the condition for the diffraction effect to be important ( [ qgo4 ] ) can be also derived directly from eq . ( [ qgo1 ] ) . in order that the current expansion is a good approximation , @xmath170 must be smaller than @xmath101 . this requires that @xmath171 , which is identical to ( [ qgo4 ] ) . we plot the absolute value of the amplification factor under the quasi - geometrical optics approximation as dashed line in fig . [ quasi1 ] . we find that the quasi - geometrical optics approximation is a good approximation for @xmath172 . for @xmath173 , the quasi - geometrical optics approximation gives a larger amplification factor than the exact one . in the quasi - geometrical optics approximation , we find from eqs . ( [ go3 ] ) and ( [ qgo1 ] ) the absolute value of the amplification factor for @xmath174 is @xmath175^{1/2}. \label{qgo5}\ ] ] from this expression , we find that the position of the first peak of the amplification factor lies at @xmath176 , which can be also verified from fig . [ quasi1 ] . for @xmath173 the present approximation is not valid , but we know that the amplification factor should converge to unity in the limit @xmath177 , where @xmath153 is much larger than @xmath178 . for @xmath98 . black line and dashed one correspond to eq . ( [ 3.3b ] ) and the quasi - geometrical optics approximation , respectively . the string tension is chosen to be @xmath179 . ] we show in fig . [ quasi2 ] the absolute value of the amplification factor as a function of @xmath103 for four cases of @xmath95 around @xmath180 . top left , top right , bottom left and bottom right panels correspond to @xmath181 and @xmath182 , respectively . black curves are plots for eq . ( [ 3.3b ] ) and the dotted ones are plots for the quasi - geometrical optics approximation . as is expected , the error of the quasi - geometrical optics approximation becomes very large near @xmath183 , where @xmath89 vanishes . as the value of @xmath95 increases , the angular region in which the quasi - geometrical optics breaks down is reduced . interestingly , the absolute value of the amplification factor deviates from unity even for @xmath184 which is not observed in the geometrical optics limit . this is a consequence of diffraction of waves , the amplitude of oscillation of the interference pattern becomes smaller as @xmath27 becomes larger , which is a typical diffraction pattern formed when a wave passes through a single slit . the broadening of the interference pattern due to the diffraction effect means that the observers even in the region @xmath185 can detect signatures of the presence of a cosmic string . but the deviation of the amplification from unity outside the wedge @xmath186 is rather small except for the special case @xmath187 : for @xmath188 the magnification is inefficient and for @xmath189 the magnification itself does not occur . hence the increase of the event rates of lensing by cosmic strings compared with the estimate under the geometrical optics approximation could be important only when the relation @xmath187 is satisfied . if we take @xmath190 and @xmath191 which is in the frequency band of lisa(laser interferometer space antenna)@xcite , we find that the typical value of @xmath4 is @xmath192 . so far , we have considered the stringy source rather than a point source . extension to a point source can be done in a similar manner to the case of the stringy source and is treated in appendix b. the result is @xmath193 where @xmath194 is the distance between the source and the observer . @xmath195 , which is defined by eq . ( [ b1 ] ) , is related to @xmath196 as @xmath197 { \big|}_{\xi \to x}+(\theta \to -\theta).\ ] ] hence @xmath38 for the point source is similar to that for the stringy source . in particular , assuming that @xmath198 , and keeping terms which could remain for @xmath199 , we have @xmath200 we have derived an approximate waveform ( [ qgo1.5 ] ) which is valid in the wave zone from the exact solution of the wave equation eq . ( [ 1.9 ] ) . here we show that eq . ( [ qgo1.5 ] ) can be obtained by a more intuitive and simpler method . in the path integral formalism @xcite , the wave form is given by the sum of the amplitude @xmath201 for all possible paths which connect the source and the observer . here @xmath202 is the time of flight along the path @xmath203 . if the cosmic string resides between the source and the observer , the wave form will be given by the sum of two terms one of which is obtained by the path integral over the paths which pass through the upper side of the string ( @xmath204 ) in fig . [ configuration ] , and the other through the lower side of it ( @xmath205 ) . the waveform coming from the former contribution will be given by @xmath206 where @xmath207 is a point on the lens plane specified by @xmath208 . one can determine the normalization constant @xmath209 by a little more detailed analysis , but we do not pursue it further here . by integrating eq . ( [ sd1 ] ) , we recover the first term in eq . ( [ qgo1.5 ] ) . for completeness , we consider the case in which the wavelength is longer than the distance from the string @xmath210 . in this limit , the first few terms in eq . ( [ 1.9 ] ) dominate , and we find @xmath211 in particular , for @xmath212 eq . ( [ 3.17 ] ) becomes @xmath26 which is larger than unity . this differs from the cases of gravitational lensing by a normal compact object , where the amplification becomes unity in the long wavelength limit . the reason why the amplification differs from unity even in the long wavelength limit is that the space has a deficit angle and hence the structure at the spatial infinity is different from the usual euclidean space . waves with very long wavelengths do not feel the local structure of string . however , uniform amplification of waves should occur as a result of total energy flux conservation because the area of the asymptotic region at a constant distance from the source is reduced due to the deficit angle . in this sense such modes feel the existence of a string . in this section , we consider compact binaries as sources of gravitational waves . gravitational waves from compact binaries are clean in the sense that the waves are almost monochromatic : the time scale for the frequency to change is much longer than the orbital period of the binary except for the phase just before plunge . hence interference between two waves coming from both sides of the cosmic string could be observed by future detectors . since each compact binary has a finite lifetime , lensing events can be classified roughly into two cases . if the difference between the times of flight along two geodesics is larger than the lifetime of the binary , we will observe two independent waves separately at different times . on the other hand , if the time delay is shorter than the lifetime , what we observe is the superposition of two waves . the remaining lifetime of the binary @xmath213 when the period of the gravitational waves measured by an observer is @xmath214 is estimated as @xmath215 where @xmath216 is the mass ratio of the binary ( @xmath217 ) , @xmath163 is the mass of the more massive star in the binary and @xmath218 is the source redshift . the time delay @xmath219 is @xmath220 taking the typical values of parameters as @xmath221 and @xmath222 , the condition @xmath223 gives the upper bound on the mass @xmath163 , @xmath224 the time scale for the orbital frequency of the binary to change is the same order as @xmath213 . hence the condition @xmath223 implies that the frequencies of two waves are almost the same . the left and right panels in fig . [ lisa - decigo ] which correspond to different frequencies of gravitational waves show the region where the condition eq . ( [ com3 ] ) is satisfied for three different values of string parameter @xmath24 . the shaded area represent the parameter region beyond the detector s sensitivities . in the left and right panels we assumed , respectively , that the threshold value for detection in strain amplitude for lisa and decigo(decihertz interferometer gravitational wave observatory)@xcite / bbo(big bang observer)@xcite , which are given by @xmath225 and @xmath226 . we find that both cases @xmath223 and @xmath227 can occur both for lisa and bbo / decigo . we can easily extend our waveform ( [ 1.9 ] ) to the case that the frequency of the source changes in time . let us write the source as @xmath228 . the fourier transformation of @xmath229 is defined by @xmath230 denoting the solution @xmath231 for a monochromatic source obtained in the previous sections by @xmath232 , @xmath38 can be written as @xmath233 where we assumed that @xmath229 is real . substituting eq . ( [ qgo7 ] ) to the above expression , we have @xmath234 eq . ( [ com4.1 ] ) is a general formula which applies to any time dependent source . here we consider the special case in which @xmath229 takes the form @xmath235 with @xmath236 where @xmath237 and @xmath238 are assumed . this represents a quasi - monochromatic source with its frequency slowly changing . then @xmath239 is @xmath240 substituting eqs . ( [ qgo8 ] ) and ( [ com4.3 ] ) into eq . ( [ com4.1 ] ) , and using the method of the steepest descent , we have @xmath241 with @xmath242 this represents a superposition of two waves coming from both sides of the string whose arrival times differ by @xmath243 . in the preceding subsections , we study the waveforms observed in two cases with @xmath244 and @xmath245 . as we have explained in the preceding subsection , what we observe is a superposition of two waves in this case . because the relative phase difference of these waves slowly increases or decreases in time due to the frequency change of the binary source and the optical path difference between two geodesics , we will observe the beat if the amplitude of the integrated relative phase difference over observation time is larger than @xmath8 . the condition that the beat is observed can be derived as follows . if we denote the total observation period by @xmath246 , then from eq . ( [ com4.4 ] ) the integrated relative phase difference is @xmath247 , where both @xmath168 and @xmath248 are assumed to be @xmath249 . hence we can observe the beat if @xmath250 because @xmath213 is roughly the same as the time scale for the frequency of the binary to change , i.e. @xmath251 , eq . ( [ com10 ] ) can be written as @xmath252 if @xmath246 is fixed , e.g. @xmath253 for lisa , eq . ( [ com12 ] ) is written as an lower bound on @xmath163 . for @xmath254 and @xmath255 , eq . ( [ com12 ] ) becomes @xmath256 we show in fig . [ lisa2 ] the region where eq . ( [ com13 ] ) is satisfied for lisa with @xmath254 . we find that if @xmath257 which is about one order of magnitude below the current upper bound , lisa will detect the beat of gravitational waves for all observable ranges in @xmath258 space as long as @xmath259 . ) is satisfied . the frequency of the gravitational waves is assumed to be @xmath260.,width=302 ] if @xmath261 , we observe the waveform of either the first term or the second one in eq . ( [ com4.4 ] ) at a given time . we show in fig . [ oneside ] the amplification of the wave corresponding to the first term in eq . ( [ com4.4 ] ) as a function of @xmath262 normalized by @xmath263 , which is nothing but @xmath264 in the case discussed in sec.[sec : behaviorsofsolution ] . we find that the amplification approaches zero more slowly for @xmath265 and oscillates around unity for @xmath266 and the angular size in which non - trivial oscillations due to the diffraction effect can be observed is given by @xmath263 . since @xmath267 in the present case , we have @xmath268 . therefore this angular size of oscillation is much smaller than @xmath269 . hence it will be very difficult to detect a lensing event in which this diffraction effect is relevant . in this section , we estimate the detection rate of the gravitational lensing caused by cosmic strings for planned gravitational wave detectors such as lisa , decigo and bbo . it is well known that string network obeys the scaling solution where the appearance of the string network at any time looks alike if it is scaled by the horizon size . there are a few dozen strings spread crossing the horizon volume and a number of string loops @xcite . since the horizon scale increases in the comoving coordinates as time goes , the number of strings increase if there is no interaction between them . however , since strings are typically moving at a relativistic speed , they frequently intersect with each other . as a result reconnection between strings occurs , reducing the number of long strings which extend over the horizon scale . during the process of reduction of the number of long strings a large number of string loops are formed , but they shrink and decay via gravitational radiation . due to the balance of two effects , the number of long strings in a horizon volume remains almost constant in time . the reconnection probability @xmath270 is essentially @xmath271 for gauge theory solitons @xcite because reconnection allows the flux inside the string to take an energetically favorable shortcut . for f - strings , the reconnection is a quantum process and its probability is roughly estimated as @xmath272 , where @xmath273 is the string coupling and is predicted in @xcite that @xmath274 for d - strings , the reconnection probability might be @xmath275 @xcite . if the reconnection probability is less than @xmath271 , the number of long strings is expected to be @xmath276 times larger than that in the case with @xmath277 . therefore it is expected that in the context of cosmic strings motivated by superstring theory the number of long strings in a horizon volume can be @xmath278 or more . to estimate the event rate for the gravitational lensing , here we consider a compact binary ( such as binary neutron stars and/or black holes ) as a source of gravitational waves . there are large uncertainties about the event rate of mbh ( massive black hole ) merger detected by lisa or decigo / bbo . several authors @xcite employed a model in which mbh mergers are associated with the mergers of host dark matter halos to estimate the event rate of mbh - mbh mergers . in this model , the event rate is dominated by halos with the minimum mass @xmath279 above which halos have a central mbh and some scenario predict that the event rate could reach @xmath280 events / yr . for decigo / bbo , the binary neuron stars will be observed @xmath281 events / yr . the probability of lensing for a single source by an infinite straight cosmic string both at cosmological distances is @xmath282 eq . ( [ es3 ] ) is derived under the geometrical optics approximation . in section iii , we found that the signal of lensing by cosmic strings ( the interference pattern of gravitational waves at detectors ) extends over an angular scales larger than the deficit angle @xmath283 when the diffraction effect is marginally important . this is a well known fact for the gravitational lensing by usual stellar objects @xcite . as we estimated in section iii , the critical distance @xmath284 below which the diffraction effect becomes important is @xmath285 therefore the probability of lensing by cosmic strings may be enhanced due to the diffraction effect for @xmath286 at lisa band ( @xmath287sec ) and for @xmath288 at decigo / bbo band ( @xmath289sec ) . assuming the prospective values of the parameters that determines the rate of lensing events @xmath290 , we obtain @xmath291 where @xmath292(@xmath293 ) denotes the numerical factor arising from the enhancement of the lensing probability due to the diffraction effect . @xmath294 is almost upper bound on the total event rate of neutron star mergers detectable by decigo / bbo . if the event rate is even higher , the number of events becomes comparable to or larger than the number of frequency bins . then we will not be able to distinguish each event , and undistinguishable signals become confusion noise . in the case of lisa , this bound on @xmath295 is even lower . unfortunately , a large number of lensing events by cosmic strings can be expected only for marginally large @xmath296 with a small reconnection probability @xmath270 . finally we briefly comment on the validity of the assumption that most of cosmic strings can be treated as straight ones in studying gravitational lensing by them . in geometrical optics approximation , only light paths which satisfy the fermat s principle contribute to the amplification factor . if we take into account the finiteness of the wavelength , the trajectories whose optical path differences are less than a few times of its wavelength will dominantly contribute to the amplification factor . in terms of the distance on the lens plane ( @xmath208-plane in fig . [ configuration ] ) , the optical paths within @xmath297 from the intersection of the geodesic will give a dominant part of the amplification factor . in the standard literature , the typical size of small - scale structure of a long string is given by the gravitational back - reaction scale @xmath298 , where @xmath18 is a cosmic time @xcite . but this is not an established argument and some recent studies suggest that the smallest size of the wiggles could be much smaller than @xmath299 @xcite . if we assume here that the smallest size of the wiggles is @xmath299 , then the condition that the straight string approximation is good is @xmath300 . substituting the appropriate values of the parameters , it gives the condition , @xmath301 hence approximating a cosmic string by a straight one is good for wide range of possible values of the parameters . we have constructed a solution of the klein - gordon equation for a massless scalar field in the flat spacetime with a deficit angle @xmath13 caused by an infinite straight cosmic string . we showed analytically that the solution in the short wavelength limit reduces to the geometrical optics limit . we have also derived the correction to the amplification factor obtained in the geometrical optics approximation due to the finite wavelength effect and the expression in the long wavelength limit . the waveform is characterized by a ratio of two different length scales . one length scale @xmath178 is defined as the separation between the image position on the lens plane in the geometrical optics and the string . we have two @xmath178 since there are two images corresponding to which side of the string the ray travels . ( when the image can not be seen directly , we assign a negative number to @xmath178 . ) the other length scale @xmath153 , which is called fresnel radius , is the geometrical mean of the wavelength and the typical separation among the source , the lens and the observer . the waveform is characterized by the ratios between @xmath178 and @xmath153 . if @xmath302 , the diffraction effect becomes important and the interference patterns are formed . even when the image in the geometrical optics is not directly seen by the observer , the interference patterns remain . in contrast , in the geometrical optics magnification and interference occur only when the observer can see two images which travel both sides of the string . namely , the angular range where lensing signals exist is broadened by the diffraction effect . this broadening may increase the lensing probability by an order of magnitude compared with that estimated by using the geometrical optics when the distance to the source is around the critical distance @xmath284 given in eq . ( [ es4 ] ) . we finally estimated the rate of lensing events which can be detected by lisa and decigo / bbo assuming bh - bh or ns - ns mergers as a source of gravitational waves . for possible values of the parameters that determines the event rate such as string reconnection rate , string tension and the event rate of the unlensed mergers , the lensing event rate could reach several per yr . thanks kunihito ioka , takashi nakamura and hiroyuki tashiro for useful comments . this work is supported in part by grant - in - aid for scientific research , nos . 14047212 and 16740141 , and by that for the 21st century coe `` center for diversity and universality in physics '' at kyoto university , both from the ministry of education , culture , sports , science and technology of japan . here we derive a formula eq . ( [ 3.3b ] ) from the integral representation of the solution eqs . ( [ 3.31 ] ) and ( [ 3.3 ] ) . as we explained in the sec.iii , we have to calculate the integral for @xmath303 and @xmath304 separately . in this case , there is no contributions from the pole @xmath80 . since @xmath305 , the integral @xmath306 is dominated from the two regions @xmath307 , where @xmath308 is the saddle points of @xmath73 . let us first calculate the integral around @xmath309 . we can not apply the method of steepest descent where the denominator of the integrated function is replaced with the value at @xmath310 because the pole @xmath80 of the integrand can lie in the region @xmath311 and the denominator is no longer constant around @xmath309 . fortunately the integral can be approximated written by the special function which can be evaluated easily . we first do the transformation of variable such that @xmath312 ( u : real number ) which corresponds to the deformation of the contour of the integral from @xmath54 to @xmath76 as shown in fig . [ contour ] . expanding @xmath73 around @xmath309 to second order in @xmath313 and the denominator of the integral to the first order in @xmath313 gives the integral @xmath314 where @xmath315 is defined by eq . ( [ 3.3b1 ] ) . hence we need to evaluate the integral @xmath316 where @xmath58 was introduced to remember that the imaginary part of @xmath317 is positive when @xmath318 . this integral is given by an error function as @xmath319 this can be derived by solving a differential equation @xmath320 which follows from the definition of @xmath321 , with the boundary condition that @xmath322 . asymptotic formulas for the error function are @xmath323 using eq . ( [ ap.3 ] ) , the integral eq . ( [ ap.2 ] ) becomes @xmath324 next let us calculate the integral eq . ( [ ap.1 ] ) around @xmath325 . since the pole @xmath80 is far from @xmath326 , we can approximate the denominator of the integrated function as a constant and apply the usual saddle point method . this gives @xmath327 the sum of eqs . ( [ ap.6 ] ) and ( [ ap.7 ] ) gives @xmath328 for @xmath304 . in this case , there is a contribution from the pole @xmath80 . hence the integral is divided into the integral around the pole and the one whose circuit of integration is @xmath76 . the integral around the pole gives @xmath329 the integral around @xmath330 along the trajectory @xmath76 is the same as for @xmath304 and is given by eq . ( [ ap.2 ] ) . the only difference is the signature of @xmath331 . by changing the integration variable from @xmath313 to @xmath332 , @xmath88 is replaced with @xmath333 and the overall signature flips . as a result we find that the integration along @xmath77 gives @xmath334 integral eq . ( [ ap.1 ] ) around @xmath326 is also given by eq . ( [ ap.7 ] ) . combining the results of subsections @xmath335 and @xmath336 , adding the similar terms @xmath337 , and also using the asymptotic form of @xmath338 , we have eq . ( [ 3.3b ] ) . here we consider a point source at a finite distance . for a point source , @xmath339 where @xmath340 . we consider a solution written in the form of the following expansion , @xmath341 the solution for @xmath342 is the same as @xmath33 in ( [ 1.6 ] ) but @xmath22 contained in @xmath95 and @xmath343 are here replaced with @xmath344 , and @xmath345 first we compute @xmath346 for @xmath347 . as in the case of bessel function , we also use the integral representation for hankel function @xmath348 here the integration is to be performed along the path @xmath349 presented in fig . [ contour ] . using the above formula and ( [ 3.1 ] ) , we have @xmath350 we introduce a new variable @xmath351 . under the assumption that @xmath352 , the integration over @xmath203 is dominated by the contribution around @xmath353 . hence , the integration contour for @xmath18 is unaltered even if we change the integration variable from @xmath18 to @xmath354 . after this change of the variable , we have @xmath355 , \cr & & \end{aligned}\ ] ] where @xmath356 we expand the exponent around a zero of its derivative . the derivative vanishes at @xmath357 , and @xmath358 is given by @xmath359 taylor expansion of @xmath360 around @xmath357 becomes @xmath361 we truncate this expansion at the quadratic order because the higher order terms are suppressed by @xmath362 or @xmath363 . performing gaussian integral , we obtain @xmath364 further , we expand @xmath365 around an approximate stationary point at @xmath366 . then we have @xmath367 again we truncate this expansion at the quadratic order for the same reason as before . then one finds that @xmath368 is approximately given by @xmath369 where @xmath370 and @xmath371 the function @xmath372 is almost identical to @xmath31 discussed in sec.iii , except that @xmath148 and other @xmath95 are replaced with @xmath373 and @xmath374 , respectively . finally , we perform the integration over @xmath375 . from ( [ b1 ] ) , we have @xmath376 since @xmath377 , we can invoke the saddle point method again to perform @xmath375-integral when @xmath378 is large . evaluating the contribution from the saddle point at @xmath379 with @xmath380 , we obtain @xmath381 the calculation for @xmath382 can be done in a completely parallel way , and the final result becomes identical to the case with @xmath347 . s. deguchi and w. d. watson , astrophys . j. * 307 * , 30 ( 1986 ) . r. takahashi and t. nakamura , astrophys . j. * 595 * , 1039 ( 2003 ) [ arxiv : astro - 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we present exact solutions of the massless klein - gordon equation in a spacetime in which an infinite straight cosmic string resides . the first solution represents a plane wave entering perpendicular to the string direction . we also present and analyze a solution with a static point - like source . in the short wavelength limit these solutions approach the results obtained by using the geometrical optics approximation : magnification occurs if the observer lies in front of the string within a strip of angular width @xmath0 , where @xmath1 is the string tension . we find that when the distance from the observer to the string is less than @xmath2 , where @xmath3 is the wave length , the magnification is significantly reduced compared with the estimate based on the geometrical optics due to the diffraction effect . for gravitational waves from neutron star(ns)-ns mergers the several lensing events per year may be detected by decigo / bbo .

You are an expert at summarizing long articles. Proceed to summarize the following text: * theoretical framework for a minimal signaling circuit . * to obtain the central results , we start with an example which illustrates the efficacy of the wk theory , and suggests a way to a more detailed , realistic model of the enzymatic push - pull loop . consider a small portion of a signaling pathway [ fig . [ f1]c ] , involving two chemical species : one with time - varying population @xmath0 ( the `` input '' ) , and another one with population @xmath1 ( the `` output '' ) whose production depends on @xmath0 . these could be , for example , the active , phosphorylated forms of two kinases within a signaling cascade , with @xmath2 downstream of @xmath3 . the upstream part of the pathway contributes an effective production rate @xmath4 for species @xmath3 , which in general can be time - dependent , though for now we will make @xmath4 constant . the output @xmath2 is produced by a reaction , @xmath5 , with a rate @xmath6 that depends on the input . the species are deactivated with respective rates @xmath7 and @xmath8 , mimicking the role of the phosphatases . the input @xmath0 will vary over a characteristic time scale @xmath9 , fluctuating around the mean @xmath10 . the output deactivation rate sets the response time scale @xmath11 over which @xmath1 can react to changes in the input . the dynamical equations , within a continuum , chemical langevin ( cl ) description @xcite , are given by : @xmath12 where the additive noise contribution @xmath13 , with @xmath14 and @xmath15 denoting the mean of population @xmath16 . the function @xmath17 is gaussian white noise with correlation @xmath18 . the @xmath19 brackets denote an average over the ensemble of all possible noise realizations . for small deviations @xmath20 from the mean populations @xmath15 , eq . can be solved using a linear approximation , where we expand the rate function to first order , @xmath21 , with coefficients @xmath22 , @xmath23 . ( we will return later to the issues of nonlinearity and discrete populations . ) the result is : @xmath24,\nonumber\end{aligned}\ ] ] where in the second line we have introduced an arbitrary scaling factor @xmath25 ( to be defined below ) inside the brackets , and divided through by @xmath26 outside the brackets . the solution for @xmath27 has the structure of a linear noise filter equation : @xmath28 , with @xmath29 . in this analogy , we have a signal @xmath30 together with a noise term @xmath31 forming a corrupted signal @xmath32 . the output @xmath33 is produced by convolving @xmath32 with a linear filter kernel @xmath34 . as a consequence of causality , the integrals in eq . run over @xmath35 , so the filtered output @xmath36 at any time @xmath37 depends only on @xmath38 from the past . the utility of mapping the push - pull system onto a noise filter comes from the application of wk theory , which is designed to solve a key optimization problem : out of all possible causal , linear filters @xmath39 , what is the optimal function @xmath40 that minimizes the differences between the output @xmath36 and input @xmath41 time series . in our example , this means having @xmath27 reproduce as accurately as possible the scaled input signal @xmath42 . specifically , we would like to minimize the relative mean - squared error @xmath43 . for a particular @xmath44 and @xmath27 , the value of @xmath45 is smallest when @xmath46 , which we will use to define the gain @xmath26 . in this case @xmath45 reduces to @xmath47 . the great achievement of wiener @xcite and kolmogorov @xcite was to show that @xmath48 satisfies the following wiener - hopf equation : @xmath49 where @xmath50 is the correlation between points in time series @xmath51 and @xmath52 , assumed to depend only on the time difference @xmath37 . given @xmath53 and @xmath54 , which are properties of the signal @xmath41 and noise @xmath55 , it is possible to solve eq . for @xmath48 . the corresponding minimum value of the error @xmath45 is : @xmath56 the solution of the wiener - hopf equation requires the following correlation functions , which can be derived from eq . : @xmath57 , @xmath58 , and @xmath59 , where the parameter @xmath60 . plugging these into eq . , we can solve for the optimal filter function by assuming a generic ansatz @xmath61 , finding the unknown coefficients @xmath62 and rate constants @xmath63 by comparing the left and right sides of the equation . in our case , a single exponential ( @xmath64 ) is sufficient to exactly satisfy eq . ( see details in appendix [ a : wk ] ) , and we get @xmath65 . the conditions for achieving wk optimality , @xmath66 , are then : @xmath67 from eq . the minimum relative error is : @xmath68 the fidelity between output and input is described through a single dimensionless optimality control parameter , @xmath69 . it can be broken up into two multiplicative factors , reflecting two physical contributions : @xmath70 . the first term , @xmath71 , is a burst factor , measuring the mean number of output molecules produced per input molecule during the active lifetime of the input molecule . the second term , @xmath72 , is a sensitivity factor , reflecting the local response of the production function @xmath73 near @xmath74 ( controlled by the slope @xmath75 ) relative to the production rate per input molecule @xmath76 . note that @xmath77 only if @xmath73 is globally nonlinear , since physical production functions satisfy @xmath78 for all @xmath79 . if @xmath73 is perfectly linear , @xmath80 , then @xmath81 , and @xmath82 . thus the limit of efficient noise suppression , @xmath83 , where @xmath84 becomes small , can be achieved by making the burst factor @xmath85 and/or enhancing the sensitivity @xmath86 , at the cost of introducing nonlinear effects ( discussed in detail below ) . for optimality to be realized , we additionally need an appropriate separation of scales [ eq . ] between the characteristic time of variations in the input signal , @xmath9 , and the response time of the output , @xmath11 . the latter should be faster by a factor of @xmath87 . the scaling @xmath88 for large @xmath69 is the same as the burst factor scaling of the target population variance in biochemical negative feedback networks intended to maintain homeostasis and suppress fluctuations @xcite . the slow @xmath89 decay in both cases , compared to the more typical scaling of variance with @xmath90 ( inversely proportional to the number of signaling molecules produced ) reflects the same underlying physical challenge : the difficulty of suppressing or filtering noise in stochastic reaction networks . the error @xmath45 defined above is based on the instantaneous difference between the input @xmath41 and output @xmath36 time series . one of the powerful features of the wk formalism is that it naturally extends error minimization to cases where the goal is extrapolating the future signal , where we seek to minimize the difference between @xmath36 and @xmath91 for some @xmath92 @xcite . given the time delays inherent in many biological responses , particularly where feedback is involved , such predictive noise filtering has significant applications @xcite , which we will explore in subsequent work . for now , we confine ourselves to the instantaneous error , which is sufficient to treat the kinase - phosphatase push - pull loop . we also note that there is no unique measure of signal fidelity . besides @xmath45 , one can optimize the mutual information between the output and input species in the cascade @xcite . for example , in the two - component cascade with nonlinear regulation , considered below , a spectral expansion of the master equation allows for efficient numerical optimization of the system parameters for particular forms of the rate function , maximizing the mutual information @xcite . * effects of nonlinearity and discrete populations . * for the subclass of gaussian - distributed signal @xmath41 and noise @xmath55 time series ( as is the case within the cl picture ) , the wk filter derived above , based on the linearization of the cl , is optimal among all possible linear or nonlinear filters @xcite . if the system fluctuates around a single stable state , and the copy numbers of the species are large enough that their poisson distributions converge to gaussians ( mean populations @xmath93 ) , the signal and noise are usually approximately gaussian . however , the rate function @xmath73 will never be perfectly linear in practice , and thus one needs to consider how nonlinearities in @xmath73 will affect the minimal @xmath45 . in addition , the discrete nature of population changes , which becomes important at lower copy numbers , has to be explicitly taken into consideration . surprisingly , the wk result of eq . can be generalized even to cases where the linear , continuum assumptions underlying wk theory no longer hold . starting from the exact master equation , valid for discrete populations and arbitrary @xmath73 , we have rigorously solved the general optimization problem for the error @xmath45 between output and input using the principles of umbral calculus @xcite . the detailed proof is in appendix [ a : non ] , but the main results are as follows . any function @xmath73 can be expanded in terms of a set of polynomials @xmath94 as @xmath95 . the @xmath94 are polynomials of degree @xmath96 , given by @xmath97 and the coefficients @xmath98 are related to moments of @xmath73 , @xmath99 . the average is taken with respect to the poisson distribution @xmath100 . the first two polynomials are @xmath101 and @xmath102 , giving @xmath103 and @xmath104 . remarkably , the relative error @xmath45 has an exact analytical form in terms of the @xmath98 , @xmath105^{-1}.\ ] ] this expression is bounded from below by @xmath106 where @xmath107 . the equality is only reached when @xmath108 and @xmath73 has an optimal linear form , @xmath109 , with all @xmath110 for @xmath111 . in this optimal case , @xmath112 and @xmath113 , and hence @xmath114 , @xmath115 from eq . . a : numerical optimization results for the hill production function @xmath73 that minimizes relative error @xmath45 between input and output , with each color corresponding to different values of the parameter @xmath116 ( see text for other parameters ) . the input probability distribution @xmath117 is superimposed in black ( the height scale is arbitrary ) . b : for each value of @xmath118 from panel a , circles show the minimal @xmath45 . the lower bound @xmath119 [ eq . ] is drawn as a blue curve . c : analogous to panel b , but showing the ratio @xmath120 at which the minimum @xmath45 is achieved . the blue curve shows the wk prediction for this ratio , @xmath121 . ] making @xmath122 large , for example by increasing @xmath123 , is desirable for better signal transduction , but with a caveat . we can keep @xmath45 near @xmath119 even for a globally nonlinear @xmath73 so long as @xmath73 remains approximately linear in the vicinity of the mean @xmath74 , and the nonlinear corrections @xmath98 for @xmath124 are negligible . large @xmath123 can be achieved through a highly sigmoidal input - output response , known as ultrasensitivity , which is biologically realizable in certain regimes of signaling cascades @xcite . however , our theory predicts that as @xmath73 goes to the extreme limit of a step - like profile around @xmath74 , @xmath45 should become significantly higher than @xmath119 , and the benefits of ultrasensitivity vanish . the reason for this is that letting @xmath123 become arbitrarily large ( making the step sharper ) necessarily implies that @xmath73 eventually deviates substantially from @xmath125 . we know that any physically sensible @xmath73 satisfies the constraint @xmath78 for @xmath126 . if @xmath127 and @xmath128 for @xmath124 , the function @xmath73 would be negative for @xmath129 , violating the physical constraint . hence the coefficients @xmath98 for @xmath124 must be non - negligible when @xmath123 is sufficiently large , leading to @xmath130 . we can illustrate this result numerically for @xmath73 that have the form of a hill function , @xmath131 , defined by the three parameters @xmath132 , @xmath133 , and @xmath134 . this represents a typical sigmoidal behavior in biochemical systems , with a small production rate for @xmath135 switching over to a saturation level @xmath132 for @xmath136 . we performed a numerical minimization of @xmath45 ( evaluated using eq . ) over the parameter space , at fixed @xmath4 , @xmath7 , @xmath137 , and @xmath118 . using eq . is numerically extremely efficient , since the coefficients @xmath98 typically decay quite rapidly , allowing the infinite sum to converge after a small ( @xmath138 ) number of terms . fixing @xmath137 and @xmath118 is equivalent to specifying the first two moments of @xmath73 , which in turn defines a curve in the three - dimensional parameter space of @xmath132 , @xmath133 , and @xmath134 . after numerically solving for this curve , the minimization procedure consists of searching along the curve ( and varying the free system parameter @xmath8 ) to find the parameter set that yields the smallest @xmath45 . [ f2]a shows optimization results for @xmath139 s@xmath140 , @xmath141 s@xmath140 , @xmath142 s@xmath140 , and varying @xmath118 , with the optimal hill function @xmath73 ( the one with smallest @xmath45 ) at each @xmath118 drawn in a different color . the corresponding minimal values of @xmath45 are shown in fig . [ f2]b as circles in the same colors , with @xmath119 using eq . drawn as a blue curve for comparison . larger values of @xmath118 have optimal @xmath73 profiles that are increasingly step - like , with steeper slopes near @xmath74 . for the range @xmath143 the maximum slope ( @xmath144 ) is still small enough that @xmath73 remains approximately linear across the entire @xmath3 range where @xmath117 is non - negligible ( the distribution is superimposed in fig . [ f2]a ) . hence minimal @xmath45 values are very close to @xmath119 , decreasing with @xmath118 . the ratios @xmath120 at which these minimal @xmath45 values occur , shown in fig . [ f2]c , are nearly equal to the predicted value @xmath145 ( blue curve ) . we can estimate that this near - optimality will persist up to @xmath146 , since that is roughly the slope of an @xmath73 that rises from zero near the left edge of @xmath147 ( at @xmath148 ) to a value of @xmath137 at @xmath149 . for @xmath150 , or equivalently @xmath151 , the nonlinearity of @xmath73 becomes appreciable around @xmath74 , distorting the output signal and leading to minimal @xmath45 noticeably larger than @xmath119 , and actually increasing with @xmath118 . thus moving towards the ultrasensitive limit @xmath152 is initially beneficial for noise filtering , but only up to a point : @xmath73 does not have to be globally linear , but local linearity of @xmath73 near @xmath74 , which can be satisfied readily , is best for accurate signal transduction . * enzymatic push - pull loop can act as an optimal wk filter . * the system considered so far is the simplest realization of a signaling circuit , in the sense that it involves only two species , related through a single phenomenological production function , @xmath73 . in reality , an enzymatic push - pull loop involves intermediates complexes of the substrate with the kinase or phosphatase whose binding , unbinding , and catalytic reactions all contribute to the stochastic nature of signal transmission . can the wk theoretical framework be used to describe optimality in this complicated context ? let us consider a more microscopic model of the loop reaction network [ fig . the active kinase is either free ( @xmath153 ) or bound to substrate ( @xmath154 ) . the input @xmath3 is defined as the total active kinase population @xmath155 . upstream modules control kinase activation and deactivation , described by rates @xmath4 and @xmath156 respectively . the kinase can phosphorylate the substrate , converting it from inactive ( @xmath157 ) to active ( @xmath158 ) form . analogously , in the reverse direction , free phosphatases ( @xmath159 ) form complexes with the active substrate ( @xmath160 ) , which lead to dephosphorylation , returning the substrate to inactive form . the output @xmath2 is the total active substrate population @xmath161 . the reactions for substrate modification , with corresponding rate constants , are : @xmath162{\kappa_\text{b } } s_k \xrightarrow{\kappa_\text{r } } k + s^\ast\\ p + s^\ast & \xrightleftharpoons[\rho_\text{u}]{\rho_\text{b } } s^\ast_p \xrightarrow{\rho_\text{r } } p + s. \end{split}\ ] ] we chose representative rate values based on a model of the map kinase cascade @xcite ( all units are in s@xmath140 ) : @xmath163 , @xmath164 , @xmath165 , @xmath166 , @xmath167 , @xmath168 . the rate @xmath156 in the model controls the characteristic time scale over which the input signal varies . we let @xmath169 s@xmath140 , which sets this scale to minutes . mean free substrate and phosphatase populations ( which together with the rates determine all equilibrium population values ) are in the ranges : @xmath170 , @xmath171 molecules / cell . we simulated the dynamics of this system numerically using kinetic monte carlo ( kmc ) @xcite , with sample input and output trajectories shown in fig . [ f3]d - f for @xmath172 and three values of @xmath173 . as the free phosphatase population is varied , we see different degrees of signal fidelity , with the closest match between @xmath27 and @xmath174 for the intermediate case in fig . are we seeing behavior similar to an optimal wk filter ? as detailed in appendix [ a : map ] , we can approximately map the phosphorylation cycle to a noise filter using the same method as in our first example : starting from the full dynamical equations in the linear cl approximation , we derive the correlation functions required to solve the wiener - hopf relation , eq . . the effective parameters resulting from the mapping are : @xmath175 where @xmath176 , @xmath177 , @xmath178 , @xmath179 , @xmath180 , and @xmath181 . . is valid in the regime @xmath182 , with corrections of order @xmath183 and @xmath184 shown in appendix [ a : map ] . such a mapping allows us to use wk results in eqs . and to predict the conditions for optimality and the minimal possible @xmath45 . [ f3]a - b show the left ( solid lines ) and right - hand ( dashed lines ) sides of both conditions in eq . as a function of @xmath173 for @xmath185 ( @xmath172 ) . the @xmath173 value at the intersections , where the conditions are fulfilled , is marked by a diamond . [ f3]c shows that exactly at this value @xmath45 achieves a minimum , given by @xmath84 from eq . ( dashed line ) . the cl approximation ( solid curves ) and kmc simulations ( circles ) are in excellent agreement . thus , the phosphorylation cycle can indeed be tuned to behave like an optimal wk noise filter , even for a realistic signaling model . in light of the mapping in eq . , we can now understand the behavior of the trajectories in fig . [ f3]d - f , which correspond to @xmath185 . in panel d , where @xmath186 , we have @xmath187 [ fig . [ f3]a ] , and the output @xmath27 becomes excessively smooth , since it can not respond quickly enough to changes in the input signal @xmath174 . the corresponding power spectral density ( psd ) of the output , shown in panel g , is smaller at high frequencies compared to the psd of the input . in panel f , we have the opposite situation of @xmath188 at @xmath189 . the output response is too rapid , generating additional noise that obscures the signal . in this case the output psd ( panel i ) has an extra high frequency contribution relative to the input psd . panel e represents the optimal intermediate @xmath190 , where @xmath191 and the wk conditions are fulfilled . the input and output psds ( panel h ) are similar at all frequencies . the minimum of @xmath45 in fig . [ f3]c is shallow , meaning that near - optimal filtering persists even when the phosphatase population is not precisely tuned to the wk condition . for @xmath173 values that vary nearly five - fold between @xmath192 , the error @xmath45 remains within 5% of the minimum value @xmath84 . another aspect of the filter s robustness can be highlighted by perturbing the enzymatic parameters @xmath193 , @xmath194 , @xmath195 , @xmath196 , @xmath197 , and @xmath198 . if we randomly vary all these parameters within a range between 0.1 and 10 times the values listed above after eq . , and calculate the resulting conditions for wk optimality [ eq . ] for each new parameter set , we obtain the results in fig . [ f4 ] . for a given @xmath173 , the shaded intervals in the figure correspond to the 68% confidence intervals on the input kinase frequency scale @xmath156 and the mean substrate population @xmath199 at optimality . thus , for a broad range of biologically relevant enzymatic parameters , we get a sense of how the populations of @xmath173 and @xmath199 must complement each other , and an associated time scale @xmath200 reflecting how quickly the input signal can vary and still be accurately transduced . from the trends in fig . [ f4 ] , we see that to get the system to respond to more rapidly varying signals , we need larger populations of @xmath173 and @xmath199 . as a concrete example , for the hyperosmolar glycerol ( hog ) signaling pathway in yeast , discussed further in the next section , kinase substrates have cell copy numbers of between @xmath201 and @xmath202 , while the ptp and ptc phosphatases that have been identified as targeting the pathway are present in cell copy numbers between @xmath203 and @xmath204 @xcite . using these population scales as a rough guide for @xmath199 and @xmath173 ( ignoring complications like multiple phosphorylation steps and sharing of phosphatases between different pathways ) we see from fig . [ f4 ] that the corresponding @xmath205 s@xmath140 . this range of optimal time scales is consistent with the experimental observation that the hog pathway can faithfully transduce osmolyte signals at frequencies @xmath206 s@xmath140 @xcite . and the characteristic frequency scale @xmath156 over which the active kinase input signal varies . this is at wk optimality [ eq . ] , using the mapping of eq . and the parameter values @xmath193 , @xmath194 , @xmath195 , @xmath196 , @xmath197 , @xmath198 listed in the text after eq . . the shaded region between the dashed curves shows the 68% confidence interval for achieving wk optimality , resulting from randomly perturbing all the parameter values so that they can be up to ten - fold smaller or larger . b : analogous to panel a , but showing the relation between @xmath173 and substrate population @xmath199 at wk optimality . ] ( solid blue ) and @xmath1 ( solid purple ) in a @xmath207 system driven by an oscillatory upstream flux @xmath208 ( see text for parameters ) . dashed lines are local means @xmath209 and @xmath210 . b : for trajectories in a , the deviations from local means , @xmath211 ( blue ) and @xmath212 ( purple ) . c - e : results calculated from kmc for a system with @xmath207 and oscillatory @xmath208 at varying driving periods @xmath213 . @xmath9 is marked by a vertical dashed line . c : the minimum errors @xmath45 ( circles ) and @xmath214 ( squares ) . @xmath84 is marked by a horizontal dashed line . d : the minimum local coefficient of variation @xmath215 . e : the mean phosphatase population values @xmath173 at which the minima shown in panels c and d are achieved ( @xmath45 : circles , @xmath214 : squares , @xmath216 : crosses ) . the @xmath173 value for wk optimality is marked by a horizontal dashed line . ] * noise filtration in a push - pull loop driven by oscillatory input . * remarkably , since eq . is independent of @xmath4 , the system can serve as an optimal filter for a range of @xmath4 values , so long as the condition @xmath217 is satisfied . this regime , involving saturated kinases and unsaturated phosphatases , has been previously identified as a candidate for efficient signal transmission by gomez - uribe _ et al . _ @xcite . to check the filter operation with varying upstream flux , we used a time - dependent @xmath208 , driving the system with oscillatory input . this is motivated by microfluidic experimental setups @xcite , where the hog pathway of yeast was probed by exposing the cells to periodic osmolyte pulses . in the experiments , the input signal is the extracellular osmolyte concentration , and the output is the degree to which the activated kinase hog1 localizes in the nucleus , where it initiates a transcriptional response to the osmolar shock . though the biochemical network relating the output to input consists of a complex series of enzymatic push - pull loops , the overall behavior was quantified through response functions in terms of input signal frequency @xcite , related to the fourier transforms of the input - output correlation functions . such correlation functions are the basic ingredients in assessing filter optimality in the wk theory . here , we will focus only on a single push - pull loop , and use input at varying frequencies to determine whether @xmath84 remains a meaningful constraint on filter performance even for non - stationary signals . in fig . [ f5]a we show a sample @xmath0 and @xmath1 kmc trajectory at optimality for @xmath218 , with @xmath219 s@xmath140 , @xmath220 , and @xmath221 s. the input has two characteristic time scales , @xmath213 and @xmath222 s. for @xmath223 , we define relative error in terms of deviations from local , time - dependent means : @xmath214 defined using @xmath224 and @xmath225 [ fig . [ f5]b ] , where @xmath226 , @xmath227 are shown as dashed curves in fig . [ f5]c shows kmc results for minimum @xmath45 and minimum @xmath214 as a function of @xmath213 for a system tuned to optimality with @xmath207 . the values of @xmath173 at which these minima are achieved are shown in fig . [ f5]e . at @xmath228 we find @xmath229 , since both the input and output have time to adjust to the slowly varying local means . in fact , the minimum @xmath214 approaches @xmath84 for @xmath223 , as optimality is unaffected by the slow oscillation in @xmath208 . the @xmath173 where the minimum @xmath214 occurs also approaches the value predicted by wk theory ( fig . the filter transduces the signal with high fidelity . in the opposite limit of small @xmath230 , the rapidly varying @xmath208 essentially averages out , since neither the input nor the output have time to respond to the sharp changes in @xmath208 . thus the system sees an effective constant flux @xmath231 . here @xmath45 , the error estimate with respect to the global mean , is more relevant than @xmath214 . in this regime the minimum @xmath232 , @xmath45 approaches @xmath84 for @xmath233 , and the @xmath173 value where @xmath45 is minimized agrees with the wk prediction . the two regimes in system behavior , with a changeover at the time scale @xmath9 , reflect the fact that the enzymatic loop acts an effective low - pass filter @xcite : it can accurately transmit the low frequency component of @xmath208 , but integrates over the high - frequency portion above a certain bandwidth . the overall bandwidth of a cascade of push - pull loops has been experimentally characterized for the yeast hog pathway , yielding a value of @xmath234 s@xmath140 @xcite . using this as a rough estimate of the bandwidth scale @xmath7 in individual loops , we could expect to see a changeover between the two regimes depending on whether the driving frequency is much slower or faster than @xmath235 . regardless of the magnitude of the driving frequency , both @xmath45 and @xmath214 always remain greater than @xmath84 , so the latter remains a bound on noise filter efficiency even for dynamic input . more generally , the low - pass filtering property of the enzymatic loop can be fine - tuned to optimize other signal transmission characteristics besides @xmath45 and @xmath214 . these two errors are minimized when the output fluctuations ( @xmath236 or @xmath237 ) closely follow the scaled input fluctuations ( @xmath238 or @xmath239 ) . however one could imagine biological scenarios where the desired outcome was a smoothed output that mirrored the oscillatory driving signal . in other words we could demand that @xmath1 , as shown for example in fig . [ f5]a ( purple trajectory ) , deviates minimally from the oscillatory local mean @xmath240 ( superimposed dashed line ) . in this case , the natural quantity to minimize would be a local coefficient of variation , @xmath241 . from the oscillatory kmc simulations described above , we calculate @xmath216 , and find that it can be made small in the slow oscillation regime @xmath242 , as shown in fig . [ f5]d , which plots the minimum @xmath216 as a function of @xmath213 for @xmath243 s. from fig . [ f5]e , which shows the @xmath173 values at which the minimum @xmath216 occurs ( crosses ) , we see that in the large @xmath213 limit this @xmath173 value is smaller than the wk prediction . this makes sense , since as we know from the case of a constant driving function ( @xmath244 ) , illustrated in fig . [ f3]d , keeping @xmath173 below the wk optimum smooths the output . for systems more complex than the enzymatic loop , smoothed output ( homeostasis around a constant mean , or tracking of a driven , time - varying local mean ) can be enhanced by introducing some negative feedback mechanism from the output back to the input @xcite . for such negative feedback systems it turns out there exists a mapping onto a different wk filter @xcite . we have demonstrated the usefulness of a generalized wk filter theory as a way of characterizing signal fidelity in an enzymatic push - pull loop . this basic motif of biological signal transduction can effectively realize an optimal wk noise filter . through a novel analytical approach , we have generalized wk ideas beyond their original linear context , thus providing fidelity bounds in strongly nonlinear cases , including ultrasensitive production and oscillatory input driving . even for a complex kinase - phosphatase reaction network with multiple intermediates , the theory predicts the conditions for accurate signal transduction , yielding a bound on the error in terms of a single dimensionless optimality control parameter @xmath69 . the results highlight how physics and engineering concepts can be use to understand how biology robustly tunes push - pull loops to optimality by setting the copy numbers of phosphatase and substrate molecules . we can relate the wide range of cellular signaling protein copy numbers observed experimentally to optimal time scales on which the cell can accurately transduce the signal , and thus yield an effective physiological response . since our approach is formulated in terms of correlation functions of signal and noise , quantities readily accessible from both theory and simulation , the current work can be generalized to other complex signaling networks . the ultimate goal is to give insights into the design principles underlying the large , intertwined biochemical pathways that determine how the cell can process and respond to diverse sources of external stimuli . this work was supported by a grant from the national science foundation ( che13 - 61946 ) . given the correlation functions , @xmath245 we would like to find the optimal filter function @xmath246 that satisfies the wiener - hopf equation , @xmath247 since @xmath248 and @xmath249 consist of exponential terms and dirac delta functions , a reasonable ansatz for @xmath246 is a sum of @xmath250 exponentials , @xmath251 , with parameters @xmath62 , @xmath63 , @xmath252 . plugging this into eq . , along with the correlation functions from eq . , and carrying out the integral , we find @xmath253 , \quad t>0 . \end{split}\ ] ] comparing the left - hand and right - hand sides of eq . , we see that the coefficients of the linearly independent exponential terms on both sides must match , giving @xmath254 equations : @xmath250 coefficients of @xmath255 , plus one for @xmath256 . since there are @xmath257 unknown parameters in the ansatz , the only value of @xmath250 that gives a closed set of equations is @xmath64 . with this choice of @xmath250 , the resulting two equations are @xmath258 the only physically sensible solution of eq . for @xmath259 and @xmath260 ( where @xmath261 as @xmath262 ) is @xmath263 thus the optimal filter is @xmath264 to obtain results for the general signal pathway model , where we assume neither linearity of the production function @xmath73 or a continuum description , we start with an exact equation for the stationary joint distribution @xmath265 of the input and output . using this , we will derive expressions for various moments of the distribution which enter into the relative mean - squared error @xmath266 from the master equation , @xmath265 satisfies @xmath267\\ & \qquad + f \left [ { \cal p}(i-1,o)-{\cal p}(i , o)\right]\\ & \qquad + \gamma_o \left[(o+1){\cal p}(i , o+1)-o { \cal p}(i , o)\right]\\ & \qquad + r(i)\left[{\cal p}(i , o-1)-{\cal p}(i , o)\right]=0 . \end{split}\ ] ] let us define a generating function @xmath268 . by multiplying eq . by @xmath269 and then summing over @xmath2 , we can derive the following equation for @xmath270 , @xmath271 + f \left [ h_{i-1}(z)-h_i(z)\right]\\ & \qquad+ \gamma_o ( 1-z)h^\prime_i(z ) + r(i)(z-1)h_i(z)=0 . \end{split}\ ] ] plugging in @xmath272 , eq . can be solved for @xmath273 , the marginal probability distribution of the input . the result is @xmath274 , the poisson distribution . this implies that the first and second input moments are given by @xmath275 moments involving the output @xmath2 can be obtained by manipulation of eq . . taking its first derivative with respect to @xmath276 , and then setting @xmath272 , we find @xmath277 + f \left [ h^\prime_{i-1}(1)-h_i^\prime(1)\right]\\ & \qquad - \gamma_o h^\prime_i(1 ) + r(i)h_i(1)=0 . \end{split}\ ] ] similarly , taking the second derivative of eq . with respect to @xmath276 , and setting @xmath272 , gives @xmath278 + f \left [ h^{\prime\prime}_{i-1}(1)-h_i^{\prime\prime}(1)\right]\\ & \qquad- 2\gamma_o h^{\prime\prime}_i(1 ) + 2 r(i)h^\prime_i(1)=0 . \end{split}\ ] ] from the definition of the generating function , @xmath279 and @xmath280 . summing eqs . and over all @xmath3 yields the following moment relations , @xmath281 evaluating @xmath282 involves finding the mean of @xmath73 over the known input distribution @xmath283 . however , finding @xmath284 involves the unknown distribution @xmath285 . moreover , the last remaining moment in eq . for the mean - squared error , @xmath286 , can also be expressed in terms of this distribution , @xmath287 . thus it is crucial to have additional information about @xmath285 . we know that @xmath285 satisfies eq . , and let us assume an ansatz for @xmath285 of the form @xmath288 for some function @xmath289 . plugging this into eq . , and using the fact that @xmath290 is the poisson distribution , we find @xmath291=0,\ ] ] where @xmath292 is an operator acting on @xmath289 , defined as @xmath293 here @xmath294 is the finite difference operator , which acts on a function @xmath295 as @xmath296 . thus the function @xmath289 which solves eq . is @xmath297 , where the operator @xmath298 . thus @xmath299 , and @xmath300 note that the terms on the right - hand sides inside the @xmath301 brackets are solely functions of @xmath3 , and hence the averages depend on @xmath117 . plugging eqs . , , and into eq . gives an expression for the relative error , @xmath302 } , \\ { \cal m}[r(i ) ] & \equiv \langle r(i ) { \cal l } r(i ) \rangle - \langle r(i ) \rangle^2 . \end{split}\ ] ] to make further progress on the evaluation of @xmath45 , it would be helpful to express @xmath73 in terms of eigenfunctions of @xmath292 ( which would also be eigenfunctions of @xmath303 ) . to do this , we employ a set of techniques known as umbral calculus @xcite , which starts with the observation that the function @xmath73 can be expanded in a newton series ( the finite difference analogue of the taylor series ) , @xmath304 where @xmath305 is the @xmath306th falling factorial of @xmath3 ( with @xmath307 ) . the newton series expansion exists assuming @xmath73 fulfills certain analyticity and growth conditions @xcite , which are satisfied for all physically realistic production functions . finite difference operators acting on @xmath308 result in linear combinations of falling factorials . in particular , @xmath309 and @xmath310 . thus the operator @xmath292 acting on @xmath308 gives @xmath311.\ ] ] if we consider functions like @xmath73 as vectors in the basis of falling factorials @xmath312 , with components @xmath313 , then from eq . the operator @xmath292 is a simple bidiagonal matrix in this basis , with elements @xmath314 the eigenvalues of @xmath315 of @xmath292 , labeled by @xmath316 in decreasing order , are just the diagonal matrix components , @xmath317 . the corresponding eigenfunctions are @xmath318 the @xmath96th eigenfunction @xmath94 is a polynomial in @xmath3 of degree @xmath96 , with the first few given by @xmath319 the eigenfunctions @xmath94 are mathematically related to expansions of the master equation through alternative approaches , for example the spectral method of refs . @xcite . in fact , @xmath320 , where @xmath321 is the mixed product defined in eq . a8 of ref . @xcite ( with @xmath74 substituted for the rate parameter @xmath322 ) . since eq . can be inverted to express @xmath308 in terms of the eigenfunctions , @xmath323 we can write @xmath73 as in terms of the eigenfunctions by plugging eq . into eq . , @xmath324 where we have used the property that @xmath325 for @xmath326 . the operator @xmath327 acting on @xmath73 is then @xmath328 since the quantities in eq . for @xmath45 involve averages with respect to @xmath117 , it is useful to derive the first and second moments of the eigenfunctions . from the fact that the falling factorials have very simple averages in the poisson distribution , @xmath329 , we find using eq . that @xmath330 . this implies that @xmath331 . to find @xmath332 , we start from the chu - vandermonde identity @xcite , the umbral analogue of the binomial theorem , @xmath333 for @xmath334 and @xmath335 this gives @xmath336 where we have used the fact that @xmath337 . multiplying both sides by @xmath338 , we find @xmath339 the second equality is based on the relation @xmath340 , which follows from the definition of the falling factorial . taking the average of both sides of eq . yields @xmath341 an alternative expression for @xmath342 can be derived by substituting the eigenfunction expansion of eq . for both @xmath338 and @xmath308 , @xmath343 comparing the right - hand sides of eqs . and we see that @xmath344 . together with eqs . and this allows us to calculate @xmath345 & = \langle r(i ) { \cal l } r(i ) \rangle - \langle r(i ) \rangle^2\\ & = \sum_{n^\prime=0}^\infty \sum_{n=0}^\infty \sigma_{n^\prime}\sigma_n \frac{\gamma_o}{\gamma_o+n\gamma_i } \langle v_{n^\prime}(i ) v_n(i)\rangle - \sigma_0 ^ 2\\ & = \sum_{n=1}^\infty \sigma_n^2 \frac{\gamma_o n ! \bar{i}^n}{\gamma_o+n\gamma_i}. \end{split}\ ] ] using the fact that @xmath346 , we can similarly evaluate @xmath347 -\bar{i } \sigma_0\\ & = \frac{\gamma_o \bar{i } \sigma_1}{\gamma_o+\gamma_i}. \end{split}\ ] ] plugging eqs . and into eq . , we obtain our final expression for the relative error , @xmath348^{-1}.\ ] ] this expression can be readily calculated numerically for any given @xmath73 , as was done in the main text for the family of hill function production rates . to facilitate evaluation , we express the coefficients @xmath98 as moments with respect to the poisson distribution @xmath117 in the following manner , using the expansion of eq . , @xmath349 from the definition of @xmath94 in eq . , the coefficients @xmath98 can be written @xmath350 using eq . the @xmath98 can be numerically calculated for any @xmath73 . the sum in eq . converges quickly because the @xmath98 decrease rapidly with @xmath96 , so typically only @xmath98 for @xmath351 are needed to get accurate results for @xmath45 . the expression in eq . also allows us to determine under what conditions the relative error @xmath45 becomes minimal . for this to occur we need @xmath352 , since otherwise @xmath45 takes its maximum value of 1 . the sum within the brackets in eq . is composed of only non - negative terms , and @xmath45 is smallest when this sum is minimal . this can be achieved by setting @xmath353 for all @xmath124 . thus @xmath45 is bounded from below by @xmath354^{-1},\ ] ] where the equality is only reached when @xmath73 has an optimal linear form , @xmath355 . the right - hand side of eq . is minimized with respect to @xmath8 when @xmath356 , with @xmath357 . at this optimal @xmath8 , the inequality in eq . the full set of reactions for the enzymatic push - pull loop is given by @xmath359{f } k\\ k + s & \xrightleftharpoons[\kappa_\text{u}]{\kappa_\text{b } } s_k \xrightarrow{\kappa_\text{r } } k + s^\ast,\\ p + s^\ast & \xrightleftharpoons[\rho_\text{u}]{\rho_\text{b } } s^\ast_p \xrightarrow{\rho_\text{r } } p + s. \end{split}\ ] ] the corresponding steady - states populations are @xmath360 where @xmath176 , @xmath177 , @xmath178 , @xmath179 , @xmath180 , and @xmath181 . for the system in eq . , the associated set of chemical langevin equations is @xmath361 where the equations on the last line come from the assumptions that the total populations of free / bound phosphatase ( @xmath362 ) and free / bound substrate in all its forms ( @xmath363 ) remain constant . the noise terms @xmath364 , where the @xmath365 are gaussian white noise functions with correlations @xmath366 . the constants @xmath367 are the power spectra of the noise terms , given by @xmath368 we are interested in how the kinase input signal @xmath369 is transduced into the active substrate output @xmath370 , and in particular whether the system can be approximately mapped onto a wk noise filter of the form given in the main text ( eq . 2 ) . ( recall that @xmath371 for any time series @xmath372 . ) since the wk description hinges on the form of the correlation functions of input and output , we will need to calculate such correlations for the dynamical equations in eq . . after linearizing these equations , it will be easier to work in fourier space , where the fourier - transformed correlation functions correspond to power spectra : @xmath373 for a given @xmath374 . hence it will useful , before proceeding further , to recast main text eq . 2 , the time - domain noise filter , as a fourier - space relation in terms of the power spectra . the result is @xmath375 . \end{split}\ ] ] our goal in this section is to show that @xmath376 and @xmath377 calculated for the enzymatic push - pull loop in eq . have the approximate form of eq . , with effective values for @xmath7 , @xmath8 , @xmath378 , and @xmath69 expressed in terms of the loop reaction rate parameters . the equilibrium populations @xmath379 and @xmath380 scale with @xmath74 as @xmath381 and @xmath382 . similarly , @xmath383 and @xmath384 . each deviation from the mean@xmath385 , @xmath386 , @xmath387 , and @xmath388we will explicitly divide into a component that scales with @xmath389 or @xmath236 like the mean population ( the `` slowly '' varying component ) , and the remainder ( the `` quickly '' varying component , denoted with subscript @xmath390 ) : @xmath391 we can interpret eq . as defining a change of variables from the set @xmath385 , @xmath386 , @xmath387 , and @xmath388 to the set @xmath236 , @xmath392 , @xmath389 , @xmath393 . the nomenclature of slow and quick components comes from the fact that if the enzymatic reaction rates ( @xmath394 , @xmath395 , @xmath396 , @xmath397 ) are made extremely rapid , the characteristic time scales for the @xmath398 and @xmath392 fluctuations become so small that the quick components can be neglected , since there would be nearly instantaneous equilibration between the free and bound enzyme populations . in general , however , we can not assume this limiting case always holds , so we will take into account both the slow and quick components in our analysis . where @xmath400 denotes the fourier transform of @xmath372 . eq . can be solved analytically for @xmath401 , @xmath402 , @xmath403 , @xmath404 , though for simplicity we will not write out the full solutions , since these would take up too much space . rather we will sketch out the basic approach to calculating and approximating the associated power spectra . the structure of the solutions to eq . , for example @xmath403 , is a linear combination of the the noise functions , @xmath405 , with coefficients @xmath406 . the corresponding power spectrum is @xmath407 , with @xmath367 given by eq . . the function @xmath408 can be written out in the form of a rational function with even powers of @xmath409 in the numerator and denominator , @xmath410 where @xmath411 and @xmath412 are coefficients independent of @xmath409 , and @xmath413 , @xmath414 for the case of @xmath415 . in order to simplify eq . further , we will make two assumptions : ( i ) the characteristic time scale over which the input signal varies , @xmath200 , is much longer than the characteristic time scales of the enzymatic reactions , @xmath416 and @xmath417 , where @xmath16 denotes the various subscripts @xmath418 , @xmath419 , and @xmath420 . for the parameters in the main text , @xmath421 , while @xmath416 , @xmath422 . this the physically interesting regime , since we can expect the system to efficiently transduce signals that vary more slowly than the intrinsic reactions that carry out the transduction . limiting our focus to frequencies @xmath423 , @xmath417 , it turns out that the higher order powers of @xmath409 in both the numerator and denominator of eq . are negligible , and the power spectrum can be approximated by @xmath424 ( ii ) we assume that the system is in the regime where @xmath425 . thus we will expand the coefficients @xmath426 and @xmath427 in eq . up to first order in @xmath183 and @xmath184 , resulting in a @xmath428 that has the form of eq . . namely , @xmath429 and @xmath430 , where the effective @xmath431 is given by @xmath432 31ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop ( ) @noop _ _ ( , )

cellular signaling involves the transmission of environmental information through cascades of stochastic biochemical reactions , inevitably introducing noise that compromises signal fidelity . each stage of the cascade often takes the form of a kinase - phosphatase push - pull network , a basic unit of signaling pathways whose malfunction is linked with a host of cancers . we show this ubiquitous enzymatic network motif effectively behaves as a wiener - kolmogorov ( wk ) optimal noise filter . using concepts from umbral calculus , we generalize the linear wk theory , originally introduced in the context of communication and control engineering , to take nonlinear signal transduction and discrete molecule populations into account . this allows us to derive rigorous constraints for efficient noise reduction in this biochemical system . our mathematical formalism yields bounds on filter performance in cases important to cellular function like ultrasensitive response to stimuli . we highlight features of the system relevant for optimizing filter efficiency , encoded in a single , measurable , dimensionless parameter . our theory , which describes noise control in a large class of signal transduction networks , is also useful both for the design of synthetic biochemical signaling pathways , and the manipulation of pathways through experimental probes like oscillatory input . extracting signals from time series corrupted by noise is a challenge in a number of seemingly unrelated areas . minimizing the effects of noise is a critical consideration in designing communication and navigation systems , and analyzing data in diverse fields like medical and astronomical imaging . more recently , a number of studies have focused on how biological circuits , comprised of chemical signaling pathways mediated by genes , proteins , and rna , cope with noise @xcite . one of the key discoveries in the past decade is that the naturally occurring systems that control all aspects of cellular processes undergo substantial stochastic fluctuations both in their expression levels and activities . noise may even have a functional role @xcite , providing coordination between multiple interacting chemical partners in typical circuits . because of the variety of ways noise influences cellular functions , it is important to develop a practical and general theoretical framework for describing how biological systems cope with and control the inevitable presence of noise arising from stochastic fluctuations . in the context of communication theory , the optimal noise - reduction filter , discovered independently by wiener @xcite and kolmogorov @xcite in the 1940 s , inaugurated the modern era of signal processing , providing the first general solution to the problem of extracting useful information from corrupted signals . we show that this classic result of wartime mathematics , developed to guide radar - assisted anti - aircraft guns , yields insights into the efficiency limits of generic biochemical signaling networks . dealing with noise in biological signal transduction is at first glance even more daunting than in engineered systems . in order to survive , cells must process information about their external environment @xcite , which is transmitted and amplified from stimulated receptors on the cell surface through elaborate pathways of post - translational covalent modifications of proteins . a typical example is phosphorylation by protein kinases of target proteins , which then become activated to modify targets further downstream . signaling occurs through cascades involving multiple stages of such activation [ fig . [ f1]a ] . since each enzymatic reaction is stochastic , noise inevitably propagates through the cascade , potentially corrupting the signal @xcite . our work focuses on a basic signaling circuit : a `` push - pull loop '' where a substrate is activated by one enzyme ( i.e. phosphorylation by a kinase ) and deactivated by another ( i.e. dephosphorylation by a phosphatase ) @xcite [ fig . [ f1]b ] . since cascades have a modular structure , formed through many such loops in series and parallel , understanding the stochastic properties at the single loop level is a prerequisite to addressing the complex behavior of entire pathways @xcite . the push - pull loop can act like an amplifier , taking the input signal the time - varying population of kinase and approximately reproducing it at larger amplitude through the output the population of active , phosphorylated substrate @xcite . depending on the parameters , small changes in the input can be translated into large ( but noise - corrupted ) output variations . the amplification is essential for sensitive response to external stimuli , but it must also preserve signal content to be useful for downstream processes . thus , the signaling circuit , despite operating in a noisy environment , needs to maintain a high fidelity between output and amplified input . from a design perspective , the natural question that arises is what are the general constraints on filter efficiency ? are there rigorous bounds , which depend only on certain collective features of the underlying biochemical network architecture ? discovering such bounds is important both to explain the metabolic costs of noise suppression in biological systems @xcite , and also for bioengineering purposes . in particular , for constructing synthetic signaling networks , we would like to make the most efficient communication pathway with a limited set of resources ( free energy costs ) . to answer these questions , using the enzymatic push - pull loop as an example , we introduce a new mathematical framework , inspired by the wiener - kolmogorov ( wk ) theory for optimal noise filtration . the original wk theory has restrictions that make it of limited utility in the biological context it assumes that the input and output are continuous variables describing stationary stochastic processes . more critically , the filter is assumed to be linear . exploiting the power of exact analytical techniques based on umbral calculus @xcite , we overcome these limitations , thus generalizing the wk approach . this crucial theoretical development enables us to provide a rigorous solution to the filter optimization problem , taking into account discrete populations and nonlinearity . we can thus understand constraints in biologically significant regimes of the push - pull loop behavior , for example highly nonlinear , `` ultrasensitive '' amplification @xcite . our theory predicts that optimality can be realized by tuning phosphatase levels , which we verified through simulations of a microscopic model of the loop reaction network , including cases where the system is driven by an oscillatory input @xcite , which is relevant to recent experimental probes @xcite . the optimality is robust , with the filter operating at near - optimal levels even when the wk conditions are only approximately fulfilled , over a broad range of realistic parameter values . although illustrated using a push - pull loop , the theory is applicable to a large class of signaling networks , including more complex features such as negative feedback or multi - site phosphorylation of substrates .

You are an expert at summarizing long articles. Proceed to summarize the following text: in newtonian theory a self - gravitating incompressible fluid body rotating at a moderate velocity around a fixed axis with respect to some inertial frame takes the shape of a maclaurin ellipsoid , which is axisymmetric with respect to the rotation axis . for a higher rotation rate , namely when the ratio of kinetic to gravitational potential energy @xmath6 is larger than @xmath7 , another figure of equilibrium exists : that of a jacobi ellipsoid , which is triaxial and rotates around its smallest axis @xcite . actually the jacobi ellipsoid is a preferred figure of equilibrium , since at fixed mass and angular momentum , it has a lower total energy @xmath8 than a maclaurin ellipsoid , due to its greater moment of inertia @xmath9 with respect to the rotation axis . indeed , at fixed angular momentum @xmath10 , the kinetic energy @xmath11 is a decreasing function of @xmath9 , and for large values of @xmath10 , this decrease overcomes the effect of the gravitational potential energy @xmath12 , which increases with @xmath9 . therefore , provided some mechanism acts for dissipating energy while preserving angular momentum ( for instance viscosity ) , a maclaurin ellipsoid with @xmath13 will break its axial symmetry and migrate toward a jacobi ellipsoid @xcite . this is the secular `` bar mode '' instability of rigidly rapidly rotating bodies . the qualifier _ secular _ reflects the necessity of some dissipative mechanism to lower the energy , the instability growth rate being controlled by the dissipation time scale , the maclaurin spheroids are subject to another instability , which is on the contrary _ dynamical _ , i.e. it develops independently of any dissipative mechanism and on a dynamical time scale ( one rotation period ) . ] . as shown by christodoulou et al . @xcite , the jacobi - like bar mode instability appears only if the fluid circulation is not conserved . if on the contrary , the circulation is conserved ( as in inviscid fluids submitted only to potential forces ) , but not the angular momentum , it is the dedekind - like instability which develops instead . the famous chandrasekhar - friedman - schutz ( cfs ) instability ( see @xcite for a review ) belongs to this category . the jacobi - like bar mode instability , applied to neutron stars , is particularly relevant to gravitational wave astrophysics . indeed a jacobi ellipsoid has a time varying mass quadrupole moment with respect to any inertial frame , and therefore emits gravitational radiation , unlike a maclaurin spheroid . for a rapidly rotating neutron star , the typical frequency of gravitational waves ( twice the rotation frequency ) falls in the bandwidth of the interferometric detectors ligo and virgo currently under construction . neutron stars being highly relativistic objects , the classical critical value @xmath14 , established for incompressible newtonian bodies , can not a priori be applied to them . the aim of the present article is thus to investigate the effect of general relativity on the secular bar mode instability of homogeneous incompressible bodies . we do not discuss compressible fluids here . it has been shown that compressibility has little effect on the triaxial instability @xcite . chandrasekhar @xcite has examined the first order post - newtonian ( pn ) corrections to the maclaurin and jacobi ellipsoids , by means of the tensor virial formalism . this work has been revisited recently by taniguchi @xcite . however , these authors have not computed the location of the maclaurin - jacobi bifurcation point at the 1-pn level . this has been done only recently by shapiro & zane @xcite and di girolamo & vietri @xcite . on the numerical side , bonazzola , frieben & gourgoulhon @xcite have investigated the secular bar mode instability of rigidly rotating compressible stars in general relativity . in the newtonian limit , they recover the classical result of james @xcite ( see also @xcite ) , namely that , for a polytropic equation of state , the adiabatic index must be larger than @xmath15 for the bifurcation point to occur before the mass shedding limit ( keplerian frequency ) . in the relativistic regime , they have shown that general relativistic effects stabilize rotating stars against the viscosity driven triaxial instability . in particular , they have found that @xmath16 is an increasing function of the stellar compactness , reaching @xmath17 for a typical neutron star compaction parameter . this stabilizing tendency of general relativity has been confirmed by the pn study of shapiro & zane @xcite and di girolamo & vietri @xcite mentioned above . note that this behavior contrasts with the cfs instability , which is strengthened by general relativity @xcite . in this paper , we improve the numerical technique over that used by bonazzola et al . @xcite by introducing surface fitted coordinates , which enable us to treat the density discontinuity at the surface of incompressible bodies . indeed the technique used in refs . @xcite did not permit to compute any incompressible model . in particular , it was not possible to compare the numerical results in the newtonian limit with the classical maclaurin - jacobi bifurcation point . we shall perform such a comparison here . the very good agreement obtained ( relative discrepancy @xmath18 ) provides very strong support for the method we use for locating the bifurcation point and which is essentially the same as that presented in ref . @xcite . the plan of the paper is as follows . the analytical formulation of the problem , including the approximations we introduce , is presented in sec . [ s : basic ] . section [ s : num ] then describes the numerical technique we employ , as well as the various tests passed by the numerical code . the numerical results are presented in sec . [ s : cal ] , as well as a detailed comparison with the pn studies @xcite and @xcite . finally , sec . [ s : summ ] provides some summary of our work . let us consider a rotating star that is steadily increasing its rotation rate , e.g. by accretion in a binary system . before the triaxial instability sets in , the spacetime generated by the rotating star can be considered as _ stationary _ and _ axisymmetric _ , which means that there exist two killing vector fields , @xmath19 and @xmath20 , such that @xmath19 is timelike ( at least far from the star ) and @xmath20 is spacelike and its orbits are closed curves . when the axisymmetry of the star is broken , the stationarity of spacetime is also broken . in newtonian theory , there is no inertial frame in which a rotating triaxial object appears stationary , i.e. does not depend upon the time . it can be stationary only in a corotating frame , which is not inertial , so that the stationarity is broken in this sense . the stationarity in the corotating frame can be expressed geometrically by stating that the newtonian spacetime possesses a one - parameter symmetry group , whose integral curves are helices . a generator of this symmetry group thus has the form @xmath21 where @xmath22 and @xmath23 are respectively the time and azimuthal coordinates associated with an inertial observer , and @xmath24 is the angular velocity with respect to the inertial frame . in general relativity , a rotating triaxial system can not be stationary , even in the corotating frame , as it radiates away gravitational waves and therefore loses energy and angular momentum . however , at the very point of the symmetry breaking , no gravitational wave has yet been emitted . for sufficiently small deviations from axisymmetry , we may neglect the gravitational radiation . therefore we shall assume that the spacetime has a helical symmetry , as in the newtonian case . the ( suitably normalized ) associated symmetry generator @xmath25 is then a killing vector , which can be written in the form ( [ e : helical ] ) in weak - field regions ( spacelike infinity ) . note that spacetimes with helical symmetry have been also used for describing binary systems with circular orbits @xcite . we model the stellar matter by a perfect fluid , for which the stress - energy tensor takes the form @xmath26 , where @xmath27 is the fluid 4-velocity , @xmath28 the fluid proper energy density , @xmath29 the fluid pressure and @xmath30 the spacetime metric tensor . a rigid motion is defined in relativity by the vanishing of the expansion tensor @xmath31 of the 4-velocity @xmath27 . in presence of the killing vector @xmath25 , this can be realized by requiring the colinearity of @xmath27 and @xmath25 ( supposing that the fluid occupies only the region where @xmath25 is timelike ) : @xmath32 where @xmath33 is a scalar field related to the norm of @xmath25 by the normalization of the 4-velocity @xmath34 . for a perfect fluid at zero temperature , the momentum - energy conservation equation @xmath35 can be recast as @xcite @xmath36 @xmath37 where @xmath38 is the proper baryon number density and @xmath39 is the momentum 1-form @xmath40 , @xmath41 being the fluid specific enthalpy : @xmath42 , where @xmath43 is some mean baryon mass . in eq . ( [ e : canon ] ) , @xmath44 denotes the exterior derivative of @xmath39 , the so - called _ vorticity 2-form _ @xcite . for the rigid motion we are considering , the baryon number conservation equation ( [ e : conserv ] ) is automatically satisfied , thanks to the colinearity of @xmath27 with the symmetry generator @xmath25 . moreover , the equation of motion ( [ e : canon ] ) can be reduced to a first integral . indeed cartan s identity applied to the lie derivative of the 1-form @xmath39 along the vector field @xmath25 leads to @xmath45 where the second equality holds thanks to the helical symmetry : @xmath46 . inserting relation ( [ e : rigid ] ) into the equation of fluid motion ( [ e : canon ] ) shows that the first term in eq . ( [ e : cartan ] ) vanishes identically , so that one gets the first integral of motion @xcite @xmath47 in the axisymmetric and stationary case , where @xmath25 is a linear combination of the two killing vectors [ eq . ( [ e : ell_axi ] ) below ] , one recovers the classical expression @xcite . the first integral ( [ e : int_prem_rigid ] ) can be re - expressed in terms of the 3 + 1 formalism of general relativity . let us introduce the spacetime foliation by the @xmath48 hypersurfaces @xmath49 and the associated future - directed unit vector field @xmath50 everywhere normal to @xmath49 . @xmath50 is the 4-velocity of the so - called _ eulerian observer _ ( also called _ zamo _ or _ locally non - rotating observer _ ) . we have the following orthogonal split of the fluid 4-velocity : @xmath51 with the lorentz factor @xmath52 the spacelike vector @xmath53 is the fluid 3-velocity as measured by the eulerian observer and the second equality in the above equation results from the normalization of the 4-velocity @xmath27 . similarly , we have an orthogonal split of the helical killing vector : @xmath54 where @xmath55 is the lapse function , governing the proper time evolution between two neighboring hypersurfaces @xmath49 . from relation ( [ e : rigid ] ) and the two orthogonal decompositions ( [ e : u_ortho ] ) and ( [ e : ell_ortho ] ) , we get @xmath56 , so that the ( logarithm of the ) first integral ( [ e : int_prem_rigid ] ) can be written @xmath57 with @xmath58 in the non - relativistic limit , @xmath59 tends toward the classical specific enthalpy ( excluding the rest - mass energy ) of the fluid , whereas @xmath60 tends toward the newtonian gravitational potential . therefore , in the newtonian limit , where @xmath61 and @xmath62 , eq . ( [ e : int_prem ] ) reduces to the classical first integral of motion @xmath63 when the rigidly rotating star is still stationary and axisymmetric , it is well known that a coordinate system @xmath64 can be chosen so that the metric takes the papapetrou form ( see e.g. @xcite and references therein ) @xmath65 where @xmath55 , @xmath66 , @xmath67 and @xmath68 are four functions of @xmath69 . the coordinate vectors @xmath70 and @xmath71 are then the two killing vectors @xmath19 and @xmath20 mentioned in sec . [ s : helic ] . the helical killing vector @xmath25 is expressible as @xmath72 with the form ( [ e : metric_axi ] ) , the einstein equations reduce to a set of four coupled elliptic equations ( see e.g. @xcite for the precise form ) . when the triaxial instability sets in , we shall consider that the metric is a perturbation of ( [ e : metric_axi ] ) , which we write as @xcite @xmath73 where @xmath55 , @xmath74 , @xmath75 , @xmath66 , @xmath67 and @xmath68 are six functions of @xmath76 , with @xmath77 note that @xmath70 and @xmath71 are no longer killing vectors , only @xmath25 remains killing . the coordinate system @xmath78 is adapted to the killing vector @xmath25 and @xmath22 is an ignorable coordinate in this system . we call @xmath79 the _ corotating _ coordinate system , and @xmath80 the _ nonrotating _ one . @xmath55 is the lapse function already introduced in eq . ( [ e : ell_ortho ] ) . @xmath81 is ( minus . ] ) the _ shift vector _ of the nonrotating coordinate system . in the triaxial case , there is no equivalent of the papapetrou theorem @xcite which , in absence of convective motions ( in the meridional planes ) , allows one to set to zero all the off - diagonal components of the metric tensor of axisymmetric and stationary spacetimes , except for @xmath82 . so , in principle , all the metric components should be non - vanishing in eq . ( [ e : metric_3d ] ) . however , we retained only @xmath83 and @xmath84 as the extra non zero components with respect to the axisymmetric case . we did so as an approximation in order to simplify the writing of einstein equations . this approximation can be justified by the following remarks : ( i ) the metric element ( [ e : metric_3d ] ) is exact in the stationary axisymmetric case , ( ii ) it encompasses the first order pn metric . this means that the neglected terms are of second order pn and moreover vanish for stationary axisymmetric rotating stars . we consider these terms to be negligible in our study of the verge of the non - axisymmetric instability . the einstein equation leads to the following set of partial differential equations : @xmath85 \ ; = \ ; 16 \pi na^2 b p \ , r \sin\theta \label{e : einstein3 } \\ & & \delta_2 \ , \zeta \ ; = \ ; 8\pi a^2 \ , [ p + ( e+p)u_i u^i ] + \frac{3}{2 } a^2 k_{ij } k^{ij } - \overline\nabla_i \nu \overline\nabla^i \nu \ , \label{e : einstein4}\end{aligned}\ ] ] where the following notations have been introduced [ see also eq . ( [ e : hnu_def ] ) ] @xmath86 and @xmath87 denotes the covariant derivative with respect to the flat 3-metric @xmath88 , @xmath89 the corresponding laplacian and @xmath90 is the laplacian in the 2-dimensional space spanned by @xmath69 : @xmath91 in eqs . ( [ e : einstein1])-([e : einstein4 ] ) , @xmath92 denotes the spatial components of the fluid 3-velocity @xmath53 [ cf . ( [ e : u_ortho ] ) ] . they can be expressed as @xmath93 and @xmath94 is deduced from @xmath53 by means of eq . ( [ e : gamma ] ) : @xmath95 - b^2 r^2 \sin^2\theta ( u^\varphi)^2 \right ) ^{-1/2 } \ .\ ] ] we have also introduced the energy density @xmath96 as measured by the eulerian observer : @xmath97 another quantity not yet defined and which appears in eqs . ( [ e : einstein1])-([e : einstein4 ] ) is the extrinsic curvature tensor @xmath98 of the hypersurface @xmath49 . the above equations must be supplemented by an equation of state ( eos ) , i.e. a relation between @xmath59 , which appears in the first integral ( [ e : int_prem ] ) , and @xmath28 and @xmath29 which appear in the source terms of the einstein equations ( [ e : einstein1])-([e : einstein4 ] ) , via eq . ( [ e : ener_euler ] ) . for the incompressible matter we are considering , we choose this eos to be @xmath99 where @xmath100 is a constant , representing the constant proper energy density of the fluid . the basic idea is to solve the equations presented in sec . [ s : basic ] by means of an iterative scheme to get an axisymmetric stationary solution , and then to introduce some triaxial perturbation and resume the iterative scheme . if the perturbation is damped ( resp . grows ) as long as the iteration proceeds , the equilibrium configuration will be declared stable ( resp . unstable ) . this procedure has been used already in the works @xcite . we shall prove here , by comparison with the analytical result , that it correctly locates the secular instability point along the maclaurin sequence . we solve the system of non - linear elliptic equations ( [ e : einstein1])-([e : einstein4 ] ) by means of the multi - domain spectral method presented in ref . the nice feature of this method is that it introduces surface - fitted coordinates @xmath101 , so that the density discontinuity at the stellar surface is exactly located at the boundary between two domains . in this way , all the fields are @xmath102 functions in each domain . this avoids any spurious oscillations ( gibbs phenomenon ) and results in a very high precision . as discussed in sec . [ s : present ] , this technique constitutes the major improvement with respect to the numerical method used in ref . another difference with ref . @xcite is that we employ the technique presented in ref . @xcite to solve the vector elliptic equation ( [ e : einstein2 ] ) for the shift vector . in particular , we solve for the cartesian components of the shift vector , whereas @xcite solved for the spherical components . a weak point of the surface - fitted coordinate technique of ref . @xcite is that the transformation from the computational coordinates @xmath101 to the physical ones @xmath103 becomes singular when the ratio of polar to equatorial radius @xmath104 is lower than @xmath105 . this prevents us from computing very flattened configurations , but fortunately this causes no trouble in reaching the maclaurin - jacobi bifurcation point , which is at @xmath106 . the iterative procedure is as follows . first one must set a value of the central enthalpy @xmath107 and the rotation frequency @xmath24 , in order to pick out a unique rotating axisymmetric configuration . the iteration is then started from very crude values : flat spacetime and spherical stellar shape . we solve the einstein equations ( [ e : einstein1])-([e : einstein4 ] ) with the spherical energy density and pressure distribution in their right - hand side . we then plug the obtained value of @xmath60 in the first integral of motion ( [ e : int_prem ] ) , along with @xmath107 , to get the enthalpy field @xmath59 . the zero of this field defines the new stellar surface to which we adapt the computational coordinates @xmath101 ( see @xcite for details ) . from @xmath59 we compute new values of the energy density and pressure according to the eos ( [ e : eos_e])-([e : eos_p ] ) . these values are then put on the right - hand side of the einstein equations ( [ e : einstein1])-([e : einstein4 ] ) and a new iteration begins . usually the rotation velocity is set to zero for the ten first steps . it is then switched on , either integrally or ( for very rapidly rotating configurations ) gradually . the convergence of the procedure is monitored by computing the relative difference @xmath108 between the enthalpy fields at two successive steps ( long - dashed line in fig . [ f : evol_q ] ) . the iterative procedure is stopped when @xmath108 goes below a certain threshold , typically @xmath109 , or for high precision computations @xmath110 . evolution during the iterative procedure of the convergence indicator @xmath108 ( long - dashed line ) , the triaxial perturbation @xmath111 ( short - dashed line ) and its growth rate @xmath112 ( solid line ) . the left figure corresponds to a stable configuration with respect to nonaxisymmetric perturbation , the right one an unstable one . @xmath108 and @xmath111 are depicted in logarithmic units , while @xmath112 is multiplied by 10 for better readability . the discontinuity at step 10 is due to the switch of the rotation and that at step 25 to the switch of the triaxial perturbation.,title="fig:",height=226 ] evolution during the iterative procedure of the convergence indicator @xmath108 ( long - dashed line ) , the triaxial perturbation @xmath111 ( short - dashed line ) and its growth rate @xmath112 ( solid line ) . the left figure corresponds to a stable configuration with respect to nonaxisymmetric perturbation , the right one an unstable one . @xmath108 and @xmath111 are depicted in logarithmic units , while @xmath112 is multiplied by 10 for better readability . the discontinuity at step 10 is due to the switch of the rotation and that at step 25 to the switch of the triaxial perturbation.,title="fig:",height=226 ] isocontours of the enthalpy @xmath59 in the equatorial plane at steps 20 , 150 , 180 and 200 for the unstable model presented in fig . [ f : evol_q ] . dashed lines denote negative values of @xmath59 , which correspond to the exterior of the star . the thick solid line denotes the stellar surface.,title="fig:",height=170 ] isocontours of the enthalpy @xmath59 in the equatorial plane at steps 20 , 150 , 180 and 200 for the unstable model presented in fig . [ f : evol_q ] . dashed lines denote negative values of @xmath59 , which correspond to the exterior of the star . the thick solid line denotes the stellar surface.,title="fig:",height=170 ] isocontours of the enthalpy @xmath59 in the equatorial plane at steps 20 , 150 , 180 and 200 for the unstable model presented in fig . [ f : evol_q ] . dashed lines denote negative values of @xmath59 , which correspond to the exterior of the star . the thick solid line denotes the stellar surface.,title="fig:",height=170 ] isocontours of the enthalpy @xmath59 in the equatorial plane at steps 20 , 150 , 180 and 200 for the unstable model presented in fig . [ f : evol_q ] . dashed lines denote negative values of @xmath59 , which correspond to the exterior of the star . the thick solid line denotes the stellar surface.,title="fig:",height=170 ] after this convergence has been achieved , we switch on a triaxial @xmath113 perturbation by modifying the metric potential @xmath60 according to @xmath114 where @xmath115 is a small constant , typically @xmath116 to @xmath117 . then we continue the iteration procedure as described above , without any further modification of the equations . at each step , we evaluate the quantity @xmath118 where @xmath119 and @xmath120 are the @xmath113 coefficients of the fourier expansion of the @xmath23 part of @xmath60 : @xmath121 and the @xmath122 in eq . ( [ e : max_triax ] ) is taken over all the @xmath123 and @xmath124 coefficients . @xmath111 is used to monitor the evolution of the triaxial perturbation : if @xmath125 as the iteration proceeds , we conclude that the perturbation decays and the axisymmetric configuration is a stable one . in the vicinity of the marginally stable configuration , the decay or growth of the perturbation turns out to be pretty small . in order to facilitate the diagnostic , we monitor instead the _ relative _ growth rate of the @xmath113 part , defined as @xmath126 after some transitory regime , @xmath112 turned out to be constant . if @xmath127 ( resp . @xmath128 ) we conclude that the configuration is unstable ( resp . the behaviors of @xmath108 , @xmath111 and @xmath112 in two typical computations are shown in fig . [ f : evol_q ] , whereas fig . [ f : bar ] depicts the development of the triaxial instability . note that the above method does not require to specify some value of viscosity . whatever this value , the effect of viscosity is simulated by the rigid rotation profile that we impose at each step in the iterative procedure via eq . ( [ e : u_i ] ) . if we consider the iteration as mimicking some time evolution , this means that the time elapsed between two successive steps has been long enough for the actual viscosity to have rigidified the fluid flow . consequently , a quantity that we can not get by our method is the instability time scale . as recalled in the introduction , this time scale depends upon the actual value of viscosity , but not the instability itself . the numerical code implementing the above procedure has been constructed upon the c++ library lorene @xcite . numerical computations have been performed on sgi origin200 as well as linux pc workstations . three domains have been used , the innermost one corresponding to the interior of the star , and the outermost one extending to spacelike infinity by means of a suitable compactification ( see @xcite for details ) . the number of spectral coefficients used in each domain is @xmath129 . the corresponding memory requirement is 40 mb and a typical cpu time ( e.g. corresponding to the first 60 steps of fig . [ f : evol_q ] , which are sufficient to conclude about the stability of the configuration ) is 3 min on an intel pentium iv 1.5 ghz processor . numerous tests have been performed to assess the validity of the method and the accuracy of the numerical code . we present here successively tests for axisymmetric configurations in general relativity and tests about the determination of the triaxial instability point in the newtonian limit . tests regarding the triaxial instability in the relativistic case are deferred to sec . [ s : resu_inst ] , where we present a detailed comparison with analytical ( post - newtonian ) results . the multi - domain spectral technique with surface - fitted coordinates has already been tested in the newtonian regime , giving a rapidly rotating maclaurin ellipsoid with a relative error of the order of @xmath110 @xcite . it has been also shown in ref . @xcite that the error is evanescent , i.e. that it decays exponentially with the number of spectral coefficients , which is typical of spectral methods . the accuracy of the computed relativistic axisymmetric models is estimated using two general relativistic virial identities grv2 @xcite and grv3 @xcite . these two virial error indicators are integral identities which must be satisfied by any solution of the einstein equations and which are not imposed during the numerical procedure ( see ref . @xcite for the computation of grv2 and grv3 ) . grv3 is a generalization to general relativity of the classical virial theorem . we have obtained values of grv2 and grv3 of the order of @xmath110 in newtonian regime and @xmath130 for high rotation and high compaction parameter . the virial errors are at least one order of magnitude better than obtained in the previous code used for calculating rapidly rotating strange stars . .[t : ansorg ] comparison with the numerical results of ansorg et al . @xcite for a highly relativistic stationary axisymmetric model @xmath131 ( @xmath132 ) and @xmath133 , with compaction parameter @xmath134 . meaning of the symbols [ in geometrized units ( @xmath135 ) ] are as follows : dimensionless angular velocity @xmath136 , gravitational mass @xmath137 , baryon mass @xmath138 , circumferential equatorial radius @xmath139 , angular momentum @xmath140 . @xmath141 , @xmath142 , and @xmath143 are the redshifts in the polar direction , in the equatorial plane along the direction of motion and opposite to that direction respectively . [ cols="<,<,<",options="header " , ] some important quantities at the viscosity driven instability points for different values of compaction parameter are reported in table [ t : instab ] . we are using two kinds of compaction parameters : @xmath144 already defined and the _ proper compaction parameter _ @xmath145 , where @xmath146 is the circumferential radius , i.e. the length of the equator ( as given by the metric ) divided by @xmath147 of the actual configuration ( unlike @xmath148 which is the circumferential radius of the non rotating star having the same baryon mass ) . our results are shown as thick lines in figs . [ f : omeg_ecc_ours ] , [ f : omeg_tsw_ours ] and [ f : ecc_comp ] . according to our analysis the critical value of the eccentricity very weakly depends on the compaction parameter and for @xmath149 is by only @xmath5 larger than newtonian value of the onset of the secular bar mode instability . the critical value of the ratio of the kinetic energy to the absolute value of the gravitational potential energy @xmath150 for compaction parameter as high as 0.275 is by @xmath151 ( table [ t : instab ] ) higher than the newtonian value . the dependence of @xmath1 on the compactness can be very well approximated by the function @xmath152 the comparison between our relativistic calculations of the viscosity driven instability and the corresponding two different pn calculations derived by @xcite and by @xcite is shown in fig [ f : omeg_ecc ] , [ f : omeg_tsw ] and [ f : tsw_comp ] . shapiro and zane @xcite employ an energy variational principle to determine equilibrium shape and stability of homogeneous triaxial ellipsoids . the method used by them is valid for arbitrary rotation rate , but only for constant density bodies . di girolamo and vietri @xcite determined the value of the eccentricity and @xmath153 at the instability onset point using landau s theory of second - order phase transitions . this method is the extension to pn regime of that used by bertin and radicati @xcite for the newtonian treatment of bar mode instability and valid for any equation of state . considering the dependence of @xmath153 with respect to the eccentricity ( fig . [ f : omeg_ecc ] ) we found good agreement between our results and the pn ones of @xcite . as can be seen , both calculations show much weaker influence of relativistic effects on location of the instability onset point than shapiro & zane pn calculations . figure [ f : tsw_comp ] shows the dependence of the critical eccentricity and the critical value of @xmath6 on the compaction parameter @xmath144 . the authors of @xcite use a proper `` conformal compaction parameter '' @xmath154 , where @xmath155 , @xmath156 , @xmath157 , @xmath158 being the ellipsoid semiaxes . in order to make a comparison between our relativistic calculations and those of @xcite we find the @xmath144 corresponding to their @xmath159 using the formula @xmath160 ( see @xcite , expression [ 4 ] , p. 422 ) . this formula is valid in the spherical limit , but the difference between the proper `` conformal compaction parameter '' and the `` spherical conformal compaction parameter '' is at most 3 % . according to our analysis the critical value of the eccentricity depends very weakly on the compaction parameter ( solid line ) ; this is in agreement with the pn calculations by di girolamo & vietri @xcite . the relative differences between our and their pn calculations are less than @xmath161 . a much stronger weakening of the bar mode instability by general relativity is suggested by the pn study of shapiro & zane @xcite , who find that @xmath162 ( dashed line in the left panel of fig . [ f : tsw_comp ] ) could be as large as 0.94 at @xmath163 . this discrepancy may be ascribed mainly to the ellipsoidal approximation for the deformation and equilibrium shape . we refer to the article @xcite for a detailed explanation of discrepancies between the two pn calculations of instability points . the right panel of fig . [ f : tsw_comp ] presents the comparison between @xcite and our calculation of the critical @xmath6 as a function of the compaction parameter @xmath144 . we found that for the compaction parameter as high as 0.05 ; 0.15 ; 0.25 @xmath1 is only by @xmath164 @xmath165 higher than newtonian value , while according to ref . @xcite , the increase is @xmath166 respectively . according to our study relativistic effects weaken the jacobi - like bar mode instability ( solid line in figs . [ f : omeg_ecc_ours ] , [ f : omeg_tsw_ours ] , [ f : ecc_comp ] and [ f : tsw_comp ] ) , but the stabilizing effect is not very strong . this is in agreement with the pn calculations @xcite . eccentricity ( left ) and ratio of the kinetic energy to the absolute value of the gravitational potential energy ( right ) at the onset of the secular bar mode instability , versus the compaction parameter @xmath145.,title="fig:",height=226 ] eccentricity ( left ) and ratio of the kinetic energy to the absolute value of the gravitational potential energy ( right ) at the onset of the secular bar mode instability , versus the compaction parameter @xmath145.,title="fig:",height=226 ] eccentricity ( left ) and ratio of the kinetic energy to the absolute value of the gravitational potential energy ( right ) at the onset of the secular bar mode instability , versus the compaction parameter @xmath144 . our results are denoted by the solid line . the dashed line and the dash - dotted line correspond to the pn calculations by @xcite and @xcite respectively.,title="fig:",height=226 ] eccentricity ( left ) and ratio of the kinetic energy to the absolute value of the gravitational potential energy ( right ) at the onset of the secular bar mode instability , versus the compaction parameter @xmath144 . our results are denoted by the solid line . the dashed line and the dash - dotted line correspond to the pn calculations by @xcite and @xcite respectively.,title="fig:",height=226 ] triaxial instabilities of rotating compact stars can play an important role as emission mechanisms of gravitational waves in the frequency range of the forthcoming interferometric detectors . a rapidly rotating neutron star can spontaneously break its axial symmetry if the ratio of the rotational kinetic energy to the absolute value of the gravitational potential energy @xmath6 exceeds some critical value . we have investigated the effects of general relativity upon the nonaxisymmetric viscosity - driven bar mode instability of incompressible , uniformly rotating stars . this is the relativistic analog of the newtonian maclaurin - jacobi bifurcation point . our method of finding the instability point is similar to that used in ref . @xcite for compressible fluid stars . the main improvement with respect to this work regards the numerical technique . we have indeed introduced surface - fitted coordinates , which enable us to treat the strongly discontinuous density profile at the surface of incompressible bodies . this avoids any gibbs - like phenomenon and results in a very high precision , as demonstrated by comparison with the analytical result for the newtonian maclaurin - jacobi bifurcation point , that our code has retrieved with a relative error of @xmath167 . according to our results , general relativity weaken the jacobi - like bar mode instability : the values of @xmath168 , eccentricity and @xmath153 increase at the onset of instability above the newtonian values . this general tendency is in agreement with pn analytical results @xcite for rigidly rotating incompressible bodies and with the numerical calculations of @xcite for relativistic polytropes . however we found that the stabilizing effect of general relativity is much weaker than that obtained in the pn treatment of @xcite . the critical value of the ratio of the kinetic energy to the absolute value of the gravitational potential energy @xmath1 for a compaction parameter as high as 0.275 is only @xmath151 larger than the newtonian value , whereas it has been found @xmath169 larger by @xcite . according to our analysis the critical value of the eccentricity very weakly depends on the compaction parameter and for a compaction parameter as high as 0.275 is only @xmath5 ( @xmath170 according to @xcite ) larger than the newtonian value of the onset of the secular bar mode instability . regarding the dependence of @xmath171 and @xmath162 with respect to compactness , we found very good agreement between our result and the recent pn ones of @xcite , the relative differences being lower than @xmath161 . we are grateful to nick stergioulas and tristano di girolamo for helpful discussions and to brandon carter for reading the manuscript . we also thank stuart shapiro and silvia zane for providing tables of their results and tristano di girolamo and mario vietri for providing their results prior to publication . this work has been funded by the following grants : kbn grants 5p03d01721 ; the greek - polish joint research and technology program epan - m.43/2013555 and the eu program `` improving the human research potential and the socio - economic knowledge base '' ( research training network contract hprn - ct-2000 - 00137 ) .

we perform some numerical study of the secular triaxial instability of rigidly rotating homogeneous fluid bodies in general relativity . in the newtonian limit , this instability arises at the bifurcation point between the maclaurin and jacobi sequences . it can be driven in astrophysical systems by viscous dissipation . we locate the onset of instability along several constant baryon mass sequences of uniformly rotating axisymmetric bodies for compaction parameter @xmath0 . we find that general relativity weakens the jacobi like bar mode instability , but the stabilizing effect is not very strong . according to our analysis the critical value of the ratio of the kinetic energy to the absolute value of the gravitational potential energy @xmath1 for compaction parameter as high as @xmath2 is only @xmath3 higher than the newtonian value . the critical value of the eccentricity depends very weakly on the degree of relativity and for @xmath4 is only @xmath5 larger than the newtonian value at the onset for the secular bar mode instability . we compare our numerical results with recent analytical investigations based on the post - newtonian expansion .

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Proceed to summarize the following text: we can study the effect of electromagnetic fields on fluids only if we know the stress induced due to the fields in the fluids . despite its importance , this topic is glossed over in most works on the otherwise well - established subjects of fluid mechanics and classical electrodynamics . the resultant force and torque acting on the body as a whole are calculated but not the density of body force which affects flow and deformation of materials . helmholtz and korteweg first calculated the body force density in a newtonian dielectric fluid in the presence of an electric field , in the late nineteenth century . however , their analysis was criticized by larmor , livens , einstein and laub , who favoured a different expression proposed by lord kelvin . it was later on shown that the two formulations are not contradictory when used to calculate the force on the body as whole and that they can be viewed as equivalent if we interpret the pressure terms appropriately . we refer to bobbio s treatise @xcite for a detailed account of the controversy , the experimental tests of the formulas and their eventual reconciliation . the few published works on the topic like the text books of landau and lifshitz @xcite , panofsky and phillips @xcite and even bobbio @xcite treat fluids and elastic solids separately . further , they restrict themselves to electrically and magnetically linear materials alone . in this paper , we develop an expression for stress due to external electromagnetic fields for materials with simultaneous fluid and elastic properties and which may have non - linear electric or magnetic properties . our analysis is thus able to cater to dielectric viscoelastic fluids and ferro - fluids as well . we also extend rosensweig s treatment @xcite , by allowing ferro - fluids to have elastic properties . let us first see why the problem of finding stress due to electric or magnetic fields inside materials is a subtle one while that of calculating forces on torques on the body as a whole is so straightforward . the standard approach in generalizing a collection of discrete charges @xmath0 to a continuous charge distribution is to replace the charges themselves with a suitable density function @xmath1 and sums by integrals . thus , the expression for force @xmath2 , ( @xmath3 is the electric field at the location of the charge @xmath0 . ) on a body on @xmath4 discrete charges in an electric field @xmath5 , is replaced with @xmath6 , when the body is treated as a continuum of charge , the integral being over the volume of the body . the integral can be written as @xmath7 where @xmath8 is the force density in the body due to an external electric field . it can be shown that @xcite that the same expression for force density is valid even inside the body . if instead , the body were made up of discrete dipoles instead of free charges , then the force on the body as a whole would be written as @xcite @xmath9 where @xmath10 is the dipole moment of the @xmath11th point dipole and @xmath3 is the electric field at its position . if the body is now approximated as a continuous distribution of dipoles with polarization @xmath12 , then the force on the whole body is written as @xmath13 while this is a correct expression for force on the body as a whole , it is not valid if applied to a volume element inside the material . in other words , @xmath14 is not a correct expression for density of force in a continuous distribution of dipoles although @xmath15 is the density of force in the analogous situation for monopoles . we shall now examine why it is so . consider two bodies @xmath16 and @xmath17 that are composed of charges and dipoles respectively . ( the subscripts of quantities indicate their composition . ) let @xmath18 and @xmath19 be volume elements of @xmath16 and @xmath17 respectively . the volume elements are small compared to dimensions of the body but big enough to have a large number of charges or dipoles in them . the forces @xmath20 and @xmath21 on @xmath18 and @xmath19 respectively due to the surrounding body are @xmath22 where @xmath4 is the number of charges or dipoles inside the volume element under consideration . in both these expressions , @xmath3 is the macroscopic electric field at the position of @xmath11th charge or dipole . it is the average value of the microscopic electric field @xmath23 at that location . that is @xmath24 , where @xmath25 denotes the spatial average of the enclosed quantity . the microscopic field @xmath23 can be written as @xmath26 where @xmath27 is the microscopic field due to the charges or dipole outside the volume element and @xmath28 is the field due to charges or dipoles inside the volume element other than the @xmath11th charge or dipole . for the volume element @xmath18 of point charges , @xmath29 where @xmath30 is the microscopic electric field at the position of @xmath11th charge due to @xmath31th charge inside @xmath18 . therefore , @xmath32 newton s third law makes the second sum on the right hand side of the above equation zero . @xmath20 is thus due to charges outside @xmath18 alone for which the standard approach of replacing sum by integral and discrete charge by charge density is valid . therefore , @xmath15 continues to be the volume force density inside the body . if the same analysis were to be done for the volume element @xmath19 of point dipoles , it can be shown that the contribution of dipoles inside @xmath19 is not zero . in fact , the contribution depends on the shape of @xmath19 @xcite . that is the reason why @xmath14 , also called kelvin s formula , is not a valid form for force density in a dielectric material . we would have got the same results for a continuous distribution of magnetic monopoles , if they had existed , and magnetic dipoles . that is @xmath33 is not the correct form of force density of a volume element in a material with magnetization @xmath34 in a magnetic field @xmath35 . the goal of this paper is to develop an expression for stress inside a material with both viscous and elastic properties in the presence of an external electric or magnetic field , allowing the materials to have non - linear electric and magnetic properties . we demonstrate that by making some fairly general assumptions about thermodynamic potentials , it is possible to develop a theory of stresses for materials with fluid and elastic properties . we check the correctness of our results by showing that they reduce to the expressions developed in earlier works when the material is a classical fluid or solid . to our knowledge , there is no theory of electromagnetic stresses in general continua with simultaneous fluid and elastic properties . since we are using techniques of equilibrium thermodynamics for our analysis , we will not be able to get results related to dissipative phenomena like viscosity . deriving an expression for viscosity for even a simple case of a gas requires full machinery of kinetic theory @xcite . developing a theory of electro and magneto viscous effects is a much harder problem and we shall not attempt to solve it in this paper . we begin our analysis in section ( [ sec : thermod ] ) by reviewing expressions for the thermodynamic free energy of continua in electric and magnetic fields . after pointing out the relation between stress and free energy in section ( [ sec : dielectric ] ) , we obtain a general relation for stress in a dielectric material in presence of an electric field . we check its correctness by showing that it reduces to known expressions for stress in newtonian fluids and elastic solids . the framework for deriving electric stress is useful for deriving magnetic stress in materials that are not permanently magnetized . section ( [ sec : magnetic ] ) mentions the expression for stress in a continuum in presence of a static magnetic field . we then point out the assumptions in derivations of ( [ sec : dielectric ] ) and ( [ sec : magnetic ] ) that render the expressions of stress unsuitable for ferro - fluids and propose the one that takes into account the permanent magnetization of ferro particles . we derive expressions for ponderomotive forces in section ( [ sec : ponder ] ) from the expressions for stress obtained in previous sections . most of our analysis rests on framework scattered in the classic works of landau and lifshitz on electrodynamics @xcite and elasticity @xcite generalizing it for continua of arbitrary nature . electromagnetic fields alter thermodynamics of materials only if they are able to penetrate in their bulk . conducting materials have plenty of free charges to shield their interiors from external static electric fields . therefore , the effect of external static electric fields are restricted to their surface alone , in the form of surface stresses . the situation in dielectrics is different - a paucity of free charges allows an external static field to penetrate throughout its interior polarizing its molecules . the external field has to do work to polarize a dielectric . this is akin to work done by an external agency in deforming a body . the same argument applies to a body exposed to a magnetic field . unlike static electric fields that are shielded in conductors , magnetic fields always penetrate in bodies , magnetizing them . the nature of the response depends on whether a body is diamagnetic , paramagnetic or ferromagnetic . in all the cases , magnetic fields have to do work to magnetize them and therefore the thermodynamics of continua is always affected by a magnetic field . we shall develop thermodynamic relations for materials exposed to static electromagnetic fields in this section . at a molecular level , electric and magnetic fields deform matter for which the fields have to do work . the material and the field together form a thermodynamic system . the work done on it is of the form @xmath36 where @xmath37 is an intensive quantity and @xmath38 a related extensive quantity denotes the possibly inexact differential of a quantity @xmath37 . ] . in the case of a dielectric material in a static electric field , the intensive quantity is the electric field @xmath5 and the extensive quantity is the total dipole moment @xmath39 , @xmath12 being the polarization and @xmath40 being the volume of the material . in the case of a material getting magnetized , the intensive quantity is the magnetizing field @xmath35 and the extensive quantity is the total magnetic moment @xmath41 , @xmath34 being the polarization and @xmath40 being the volume of the material . the corresponding work amounts are @xmath42 and @xmath43 respectively . we added a subscript @xmath44 because this is only one portion of the work . the other portion of the work is required to increment the fields themselves to achieve a change in polarization or magnetization . they are @xmath45 and @xmath46 respectively , where @xmath47 is the permittivity of free space and @xmath48 is the permeability of free space respectively and that of a magnetic field is @xmath49 . ] therefore , the total work needed to polarize and magnetize a material , at constant volume , are @xmath50 we have derived these relations for linear materials . we will now show that they are true for any material . imagine a dielectric immersed in an electric field . let the electric field be because of a charge density @xmath51 . let the electric field be increased slightly by changing the charge density by an amount @xmath52 . work done to accomplish this change is @xmath53 where @xmath54 is the electric potential . since @xmath55 , @xmath56 if the charge density is localized then the volume of integration can be taken as large as we like . we do so and also convert the first integral on the right hand side to a surface integral . the first term then makes a vanishingly small contribution to the total and the work done in polarizing a material can be written as @xmath57 let a material be magnetized by immersing it in a magnetic field . the magnetic field can be assumed to be created because of a current density @xmath58 . let the magnetic field be increased slightly by changing the current density . we further assume that the rate of increase of current is so small that @xmath59 at all stages . the source of current has to do an additional work while increasing the amount of current density in order to overcome the opposition of the induced electromotive force . if @xmath60 is the induced emf , then the sources will have to do an additional work at the rate @xmath61 , where @xmath62 is the magnetic flux and the dot over head denotes total time derivative . the amount of work needed is @xmath63 . if @xmath64 is the cross sectional area of the current , then @xmath65 but @xmath66 , therefore , @xmath67 since we assumed the current to be increased at an infinitesimally slow rate , there are no displacement currents and @xmath68 . @xmath69 where we have used the vector identity @xmath70 . we once again assume that the current density is localized and therefore converting the second integral on the right hand side of equation ( [ thermod : e6 ] ) into a surface integral results in an infinitesimally small quantity . the work done in magnetizing a material is therefore , @xmath71 a change in the helmholtz free energy of a system is equal to the work done by the system in an isothermal process , which in turn is related to stresses in the continuum . we will show how stress is related to the helmholtz free energy . let us consider the example of an ideal gas . the change in its helmholtz free energy , is given by @xmath72 . using the first and the second laws of thermodynamics we have @xmath73 . therefore , @xmath74 , which under isothermal conditions means @xmath75 . in this simple system , @xmath76 is the isotropic portion of the stress and @xmath77 is related to the isotropic strain . thus , we can get @xmath76 is we know change in helmholtz free energy and volume . therefore , a first step toward getting an expression for stress is to find the helmholtz free energy . under isothermal conditions , the first law of thermodynamics is @xmath78 where @xmath79 is the total free energy , @xmath80 is the absolute temperature , @xmath81 is the total entropy and @xmath82 is the work done on the system . first law of thermodynamics for polarizable and magnetizable media is @xmath83 where @xmath84 is the mechanical work done on the system . if @xmath85 , @xmath86 and @xmath87 are internal energy , mechanical work and entropy of the media _ per unit volume _ , first law of thermodynamics for polarizable and magnetizable media is @xmath88 the mechanical work done on a material is @xmath89 where @xmath90 is the stress tensor and @xmath91 is the strain tensor in the medium . further , with this substitution , all quantities in equations ( [ fe:4 ] ) and ( [ fe:5 ] ) become exact differentials allowing us to replace @xmath92 with @xmath93 . if @xmath94 is the helmholtz free energy per unit volume , @xmath95 these relations give change in helmholtz free energy in terms of change in @xmath96 and @xmath97 . the @xmath96 field s source is free charges alone while the @xmath97 field s source is all currents . in an experiment , we can control the total charge and free currents . therefore , it is convenient to express free energy in terms of @xmath5 , whose source is all charges - free and bound , and @xmath35 , whose source is free currents . we therefore introduce associated helmholtz free energy function @xmath98 for polarizable media as @xmath99 and for magnetizable media as @xmath100 . equations ( [ fe:6 ] ) and ( [ fe:7 ] ) therefore become @xmath101 if @xmath102 is the deviatoric stress , @xmath103 , where @xmath76 is the hydrostatic pressure . therefore we have , @xmath104 the quantity @xmath105 is the dilatation of the material during deformation . therefore the thermodynamic potential @xmath98 of a polarizable ( magnetizable ) medium is thus , a function of @xmath80 , @xmath40 , @xmath91 and @xmath5(@xmath35 ) . equivalently , it can be considered a function of @xmath80 , @xmath106 , @xmath91 and @xmath5(@xmath35 ) , where @xmath106 is the mass density of the medium . we will now calculate the stress tensor in a polarizable medium . we consider a small portion of the material and find out the work done by the portion in a deformation in presence of an electric field . the portion is small enough to approximate the field to be uniform throughout its extent . we emphasize that through this assumption we are not ruling out non - uniform fields but only insisting that the portion be small enough to ignore variations in it . since a sufficiently small portion of a material can be considered to be plane , the volume element under consideration can be assumed to be in form of a rectangular slab of height @xmath107 . let it be subjected to a virtual displacement @xmath108 which need not be parallel to the normal @xmath109 to the surface . the virtual work done by the medium per unit area in this deformation is @xmath110 , where @xmath111 is the stress _ on _ the portion . if @xmath90 is the stress _ due to _ the portion _ on _ the medium , then @xmath112 . therefore , the virtual work done by the medium on the portion is @xmath113 . further , since both @xmath111 and @xmath90 are symmetric , the virtual work can also be written as @xmath114 . the change in helmholtz free energy during the deformation is @xmath115 per unit surface area . if we assume the deformation to be isothermal , @xmath116 change in height of the slab is @xmath117 [ dielectric : f1 ] the geometry of the problem is described in figure 2 . for an isothermal variation @xmath118 we depart from the convention in thermodynamics , to indicate variables held constant as subscripts to partial derivatives , to make our equations appear neater . we shall also use the traditional notation for partial derivatives . we will now get expressions for each term on the right hand side of equation ( [ dielectric:3 ] ) . 1 . if @xmath119 is the helmholtz free energy in absence of electric field , @xmath120 , where @xmath121 is the permittivity tensor . permittivity is known to be a function of mass density of a material , the dependence being given by clausius - mossotti relation@xcite . electric field is _ usually _ independent of mass density of the material . however , that is not so if the material has a pronounced density stratification like a fluid heated from above . if @xmath122 and @xmath123 are two elements of such a fluid , at the top and bottom respectively , both having identical volume then @xmath122 will have less number of dipoles than @xmath123 . the electric field inside them , due to matter within their confines too will differ . we point out that although divergence of @xmath5 depends only on the density of free charges , @xmath5 itself is produced by all multipoles . therefore , @xmath124 the last term in equation ( [ dielectric:4 ] ) is absent if the material has a uniform temperature . it is not included in the prior works of bobbio@xcite and landau and lifshitz@xcite . if the @xmath125 ( or @xmath126 ) axis is assumed to be along the normal and the deformation is uniform , the displacement of a layer of the volume element can be described as @xmath127 where @xmath125 is the vertical distance from the lower surface . since @xmath108 is fixed , @xmath128 and @xmath129 since the strain tensor is always symmetric , @xmath130 the electric field does not depend on strain but permittivity does . this is because , deformation may change the anisotropy of the material , which determines its permittivity . likewise , permittivity and strain tensors do not depend on electric field ] . therefore they can be pulled out of the integral and @xmath131 3 . from equation ( [ fe:8 ] ) , @xmath132 therefore , the last term is just @xmath133 . using equations ( [ dielectric:4 ] ) , ( [ dielectric:9 ] ) and ( [ dielectric:10 ] ) in ( [ dielectric:3 ] ) , we get @xmath134 substituting ( [ dielectric:11 ] ) and ( [ dielectric:2 ] ) in ( [ dielectric:1 ] ) we get @xmath135 we have gathered terms independent of electric field in the first curly bracket of equation ( [ dielectric:12 ] ) , keeping the contribution of electric field to stress in the rest . we still have to find out the expressions for @xmath136 and @xmath137 . a change in density of a layer depends on the change it its height ( or thickness ) , therefore , @xmath138 or , @xmath139 we will now estimate change in electric field due to deformation . consider a volume element at a point @xmath140 . let it undergo a deformation by @xmath141 . as a result , matter that used to be at @xmath142 now appears at @xmath140 . in a virtual homogeneous deformation , every volume element carries its potential as the material deforms . therefore , the change in potential at @xmath140 is @xmath143 . since @xmath144 ( see equation ( [ dielectric:5 ] ) ) , @xmath145 since @xmath146 , @xmath147 we have used the assumption that the region is small enough to have almost uniform electric field and therefore it can be pulled out of the gradient operator . equation ( [ dielectric:12 ] ) therefore becomes @xmath148 stress in a polarized viscoelastic material at rest is therefore , @xmath149 we can simplify equation ( [ dielectric:16a ] ) by writing the terms in the first curly bracket as familiar thermodynamic quantities . if @xmath150 is the total helmholtz free energy of the substance in absence of electric field and @xmath119 is the helmholtz free energy per unit volume then @xmath151 , where @xmath152 is the volume of the substance , @xmath153 the mass and @xmath106 the density . maxwell relation for pressure in terms of total helmholtz free energy is @xmath154 similarly , the dependence of @xmath119 on strain tensor @xmath155 can be written as @xmath156@xcite , where we have retained only the deviatoric of the strain tensor because the isotropic part is already accounted in hydrostatic pressure of equation ( [ dielectric:16a ] ) . the constant @xmath157 is the shear modulus of the substance . therefore , @xmath158 equation ( [ dielectric:16a ] ) can therefore be written as @xmath159 we will now look at some special cases of ( [ dielectric:16 ] ) , 1 . if there is no matter , terms with pressure , density and strain tensor will not be present . further @xmath160 and equation ( [ dielectric:16 ] ) becomes the maxwell stress tensor for electric field in vacuum . @xmath161 we emphasize that the general expression for stress in a material exposed to static electric field reduces to maxwell stress tensor only when we ignore all material properties . if there is no electric field , all terms in the second and third curly bracket of ( [ dielectric:16 ] ) vanish . further , if the medium is a fluid without elastic properties , @xmath119 , will not depend on @xmath155 and the stress will be @xmath162 thus the stress in a fluid without elastic properties is purely hydrostatic . we do not see viscous terms in ( [ dielectric:17 ] ) because viscosity is a dissipative effect while @xmath163 is obtained from helmholtz free energy which has information only about energy than can be extracted as work . if the material were a solid and if there are no electric fields as well , the stress is @xmath164 it is customary to write the first term of equation ( [ dielectric:18 ] ) in terms of @xmath165 , the bulk modulus as @xmath166 4 . for a fluid dielectric with isotropic permittivity tensor , @xmath119 is independent of @xmath155 and @xmath167 . if the fluid has a uniform density , equation ( [ dielectric:16 ] ) then becomes @xmath168 this expression matches the one obtained in @xcite , after converting to gaussian units , and after accounting for the difference in the interpretation of stress tensor . landau and lifshitz s stress tensor is @xmath169 . 5 . for a fluid dielectric with isotropic permittivity tensor and in which the electric field depends on density equation ( [ dielectric:16 ] ) then becomes @xmath170 6 . for a solid dielectric we can assume that @xmath119 , @xmath5 and @xmath171 are independent of @xmath106 . equation ( [ dielectric:16 ] ) now becomes @xmath172 if the solid is isotropic and remains to be so after application of electric field , @xmath167 and ( [ dielectric:20 ] ) simplifies to @xmath173 where @xmath174 is the part of stress tensor that exists even in absence of electric field . this expression matches with the one in @xcite if one converts it to gaussian units , assumes the constitutive relation @xmath175 and takes into account that their stress tensor is @xmath169 . 7 . for a viscoelastic liquid that is also a linear dielectric with uniform density , @xmath176 in order to calculate stress in a magnetic fluid , we continue to use the physical set up used in section ( [ sec : dielectric ] ) of a small slab of viscoelastic liquid subjected to magnetic field . if there are no conduction and displacement currents , ampere s law becomes @xmath177 , making the @xmath35 field conservative . it can then be treated like the electrostatic field of section ( [ sec : dielectric ] ) . in order to extend the analysis of section ( [ sec : dielectric ] ) to magnetic fluids , we need an additional assumption of magnetic permeability being independent of @xmath35 . although the first assumption , of no conduction and displacement currents , is valid in the case of ferro - viscoelastic fluids , the second assumption of field - independent permeability is not . therefore , this analysis is valid only for the single - valued , linear section of the @xmath97 versus @xmath35 curve of ferro - viscoelastic liquid , giving @xmath178 where @xmath179 is the magnetic permeability tensor . we have omitted the term accounting for dependence of @xmath35 on mass density @xmath106 because we are not aware of a situation where it may happen . the expressions derived in sections ( [ sec : dielectric ] ) and ( [ sec : magnetic ] ) are valid only if permittivity and permeability are independent of electric and magnetic fields respectively . ferro - fluids are colloids of permanently magnetized particles . as the applied magnetic field increases from zero , an increasing number of sub - domain magnetic particles align themselves parallel to the field opposing the random thermal motion leading to a magnetization that increased in a non - linear manner . the magnetic susceptibility and therefore permeability depend on the field . it can not be pulled out of the integral sign . equation ( [ magnetic:1 ] ) should be written as @xmath180 if the elastic effects are negligible , equation ( [ ferro:1 ] ) reduces to @xmath181 if @xmath182 , as is normally for ferro - fluids @xcite , @xmath183 since the applied magnetic field is independent of density , @xmath184 where to get the last equation we have used the relation @xmath185 and the fact that @xmath35 does not depend on @xmath106 . if @xmath186 is the specific volume , that is @xmath187 , equation ( [ ferro:5 ] ) can be written as @xmath188 further , @xmath189 and @xmath185 imply , @xmath190 using equations ( [ ferro:5 ] ) and ( [ ferro:6 ] ) in equation ( [ ferro:3 ] ) , we get @xmath191dh_r\right\}\delta_{ij } - h_i b_j\ ] ] this is same as rosensweig s @xcite equation ( 4.28 ) except that he calculates @xmath111 , which is related to our stress tensor as @xmath169 . we do not know of electric analogues of ferro fluids ( electro - rheological fluids are analogues of magneto - rheological fluids , not ferro fluids ) . however , there are permanently polarized solids , called ferro - electrics . for such materials , the stress is @xmath192 the old term `` ponderable media '' means media that have weight . ponderomotive force is the one that cause motion or deformation in a ponderable medium . in contemporary terms , it is the density of body force in a material . it is related to the stress tensor @xmath90 as @xmath193 we mention a few familiar special cases of this equation for fluids of various kinds . 1 . for incompressible , newtonian fluids the stress tensor is given by ( [ dielectric:17 ] ) and the force density is @xmath194 note that the force density does not include the dissipative component due to viscosity . 2 . for an incompressible , newtonian , dielectric fluid in presence of static electric field , assuming that the electric field inside the fluid is independent of density , the stress tensor is given by equation ( [ dielectric:19 ] ) and the ponderomotive force is @xmath195 where @xmath196 is the density of free charges in the fluid and @xmath197 is its relative permittivity . in deriving equation ( [ ponder:3 ] ) we used gauss law @xmath198 and the fact that we are dealing with an electrostatic field ( @xmath199 ) , for which @xmath200 . in an ideal , dielectric fluid @xmath201 and @xmath202 the relative permittivity is a function of temperature and the term @xmath203 is significant in a single - phase fluid only if there is a temperature gradient . the third term in equation ( [ ponder:4 ] ) is called the electro - striction term and it is present only when the electric field or @xmath204 or both are non - uniform . the derivative of the relative permittivity with respect to mass density is calculated using the clausius - mossotti relation @xcite , @xcite . 3 . continuing with the same fluid as above but now having a situation in which the electric field is a function of mass density @xmath106 , we have an additional term in equation ( [ ponder:4 ] ) given by @xmath205 we come across such a situation when there is a strong temperature gradient in the fluid resulting in a gradient of dielectric constant @xmath197 . since the electric field depends on @xmath197 and @xmath197 depends on mass density through the clausius - mossotti relation , the electric field is a function of mass density and we have to consider this additional term . we hasten to add that it not necessary for there to be a temperature gradient to have such a situation , a gradient of electric permittivity suffices to give rise to such a situation . the derivation for force density in an incompressible , newtonian , diamagnetic or paramagnetic fluid in presence of a static magnetic field is similar except that we use the maxwell s equations @xmath206 and @xmath207 . we also assume the auxiliary magnetic field , @xmath35 , inside the fluid is independent of density . we get @xmath208 the term @xmath209 is the lorentz force term and it is zero if the fluid is not conducting . @xmath210 is the relative permeability of the fluid . the fourth term in equation ( [ ponder:5 ] ) is called the magneto - striction force . it is present only if the magnetic field or @xmath211 or both are non - uniform . the derivative of relative permeability with respect to mass density is calculated using the magnetic analog of the clausius - mossotti relation @xcite . several forms of body force density , all equivalent to each other , can be derived for ferro - fluids from equations ( [ ferro:7 ] ) and ( [ ponder:1 ] ) . we refer to @xcite for more details . [ ponder : i5 ] if the material is dielectric and viscoelastic , we assume that the permittivity depends on the strain . even though the material was isotropic before applying electric field , it may turn anisotropic as its molecules get polarized and align with the field . the scalar permittivity is then replaced with a second order permittivity tensor @xmath171 . following landau and lifshitz s treatment of solid dielectrics @xcite , we assume that the permittivity tensor is a linear function of the strain tensor and write it as @xmath212 where @xmath213 and @xmath214 are constants indicating rate of change of permittivity with strain . we call them @xmath213 and @xmath214 to differentiate them from @xmath215 and @xmath216 used to describe behavior of solid dielectric @xcite . if we assume the material to be incompressible , @xmath217 and @xmath218 for an incompressible material , equation ( [ dielectric:16c ] ) becomes , @xmath219 since @xmath213 and shear modulus @xmath157 are constants , they do not survive in the expression for @xmath220 . the expression for ponderomotive force in a dielectric , viscoelastic fluid is same as that for a dielectric , newtonian fluid . [ ponder : i6 ] the same conclusion follows for a viscoelastic fluid subjected to a magnetic field if we assume that @xmath221 , @xmath222 and @xmath223 being constants , when a fluid is magnetized . time - varying electric fields can penetrate conductors up to a few skin depths that depends on the frequency of the fields and physical parameters of the material like its ohmic conductivity or magnetic permeability . the general problem of response of materials to time - varying fields is quite complicated . however , the results in this paper can be applied for slowly varying fields , that is , the ones that do not significantly radiate . for such fields , the time varying terms of maxwell equations can be ignored . whether a time varying field can be considered quasi - static or not depends on the linear dimension of the materials involved . if @xmath224 is the angular frequency of the fields , the wavelength of corresponding electromagnetic wave is @xmath225 , @xmath226 being the velocity of light in vacuum . if the linear dimension @xmath227 of the materials is much lesser than @xmath228 , for any element @xmath229 of the path of current , there is another within @xmath227 that carries same current in the opposite direction , effectively canceling the effect of current . for power line frequencies , the value of @xmath227 is a few hundred miles and even for low frequency radio waves , with @xmath230 hz , @xmath227 is of the order of @xmath231 m. thus the _ slowly - varying fields _ or _ quasi - static _ approximation @xcite is valid for frequencies up to that of radio waves and our results can be applied under those conditions . 1 . proof of equation ( [ dielectric:8 ] ) . @xmath232 interchange the indices in the first term , @xmath233 since @xmath91 is a symmetric tensor , @xmath234 99 bobbio , s. , electrodynamics of materials : forces stresses , and energies in solids and fluids , academic , new york , chapter 4 , ( 2002 ) . landau l. d. and lifshitz e. m. , electrodynamics of continuous media , pergamon , oxford , ( 1960 ) . panofsky w. k. h. and phillips m. , classical electricity and magnetism , 2nd edition , dover publications inc . , new york , ( 1962 ) . rosensweig r. e. , ferrohydrodynamics , dover publications , mineola , new york , ( 1997 ) . reitz , j. r. , milford , f. j. and christy , r. w. , foundations of electromagnetic theory , 3rd edition , narosa publishing house , new delhi , ( 1990 ) . jackson , j. d. , classical electrodynamics , 3rd edition , john wiley & sons inc , new york , ( 1999 ) . chapman , s. and cowling , t. g. , mathematical theory of non - uniform gases , 3rd edition , cambridge university press , cambridge , ( 1970 ) . landau l. d. and lifshitz e. m. , theory of elasticity , pergamon , oxford , ( 1960 ) . joshi amey , radhakrishna m.c . and rudraiah n , rayleigh - taylor instability in dielectric fluids , physics of fluids , volume 22 , issue 6 , ( 2010 ) .

a clear understanding of body force densities due to external electromagnetic fields is necessary to study flow and deformation of materials exposed to the fields . in this paper , we derive an expression for stress in continua with viscous and elastic properties in presence of external , static electric or magnetic fluids . our derivation follows from fundamental thermodynamic principles . we demonstrate the soundness of our results by showing that they reduce to known expressions for newtonian fluids and elastic solids . we point out the extra care to be taken while applying these techniques to permanently polarized or magnetized materials and derive an expression for stress in a ferro - fluid . lastly , we derive expressions for ponderomotive forces in several situations of interest to fluid dynamics and rheology .

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Proceed to summarize the following text: suppose @xmath6 particles are placed on vertices of the @xmath0-path , with no site multiply occupied . the _ biased exclusion process _ is the markov chain @xmath7 with transitions as follows : * choose uniformly among the @xmath8 edges of the path , * if both vertices of the selected edge are either occupied or unoccupied , do nothing , * if there is exactly one particle on the edge , place it on the right vertex with probability @xmath9 and on the left with probability @xmath10 . the canonical case is when @xmath0 is even and @xmath11 . this defines a reversible ergodic markov chain , which has a unique stationary distribution @xmath12 . it is natural to ask about its _ mixing time _ , @xmath13 we write @xmath14 for @xmath15 . when @xmath16 , proved @xmath17 n^3 \log ( n/{\varepsilon } ) \,,\ ] ] and conjectured that the lower bound is sharp . recently , lacoin answered this , proving that the process has a _ cutoff _ , i.e. @xmath18 it is worth observing that the eigenfunction lower bound method introduced in wilson turns out to be widely applicable , giving sharp lower bounds for many models . when @xmath19 , the mixing time was first studied by , who proved @xmath20 . a simpler path coupling proof was given by . ( this proof is repeated here as the upper bound in theorem [ thm : mixbe ] . ) the purpose of this paper is to understand the mixing behavior when the bias may depend on @xmath0 and in particular when @xmath21 as @xmath3 . we show that in all cases , there is a _ pre - cutoff _ , meaning that there are universal constants @xmath22 so that @xmath23 we find that , depending on the rate at which @xmath24 , the mixing time interpolates between the unbiased and constant bias cases . below summarizes our results . we write @xmath25 to mean that there exist constant @xmath26 , not depending on @xmath27 , so that @xmath28 . [ thm : main ] consider the @xmath27-biased exclusion process on @xmath29 with @xmath6 particles . we assume that @xmath30 . if @xmath31 , then @xmath32 if @xmath33 , then @xmath34 if @xmath35 , then @xmath36 we provide more precise estimates on @xmath37 in proposition [ prop : lb1 ] , proposition [ prop : lb2 ] , and theorem [ thm : mixbe ] . in particular , the lower bound in follows from proposition [ prop : lb1 ] , the lower bound in follows from proposition [ prop : lb2 ] , and the lower bound in follows from proposition [ prop : lbu ] . the upper bounds in and follow from theorem [ thm : mixbe ] , and the upper bound in follows from proposition [ prop : nearun ] . since the behavior of the individual particles remains diffusive in the @xmath38 regime , it is not surprising that the mixing time has the same order as the unbiased process in this case . the change of the functional form of the mixing time at @xmath39 is a more unexpected transition . a path coupling gives useful upper bounds for @xmath40 . when @xmath41 is small , we use a simple coupling adapted from a coupling for ( unbiased ) random adjacent transpositions given in . in the unbiased case , @xmath6 coupled unbiased random walks must hit zero . the bias introduced when @xmath41 is small does nt overwhelm the diffusive motion , so the same idea works . for lower bounds , when @xmath42 , we use wilson s method ( introduced in ) . thus we need the eigenfunction corresponding to the second eigenvalue , which we explicitly compute . when @xmath43 , we follow the left - most particle , and show it needs at least order @xmath44 moves to mix . the organization of the paper is as follows . after giving definitions in section [ sec : defn ] , in section [ sec : spectrallb ] we compute the eigenfunction needed for wilson s method , and provide the corresponding lower bounds . in particular , the lower bounds in theorem [ thm : main ] ( i ) and ( ii ) are given in propositions [ prop : lb1 ] and [ prop : lb2 ] , respectively . we give the two upper bounds in section [ sec : ub ] : the upper bound in is given in proposition [ prop : nearun ] , and the other upper bounds in theorem [ thm : main ] are all immediate from theorem [ thm : mixbe ] . we conclude with the single particle lower bound needed for theorem [ thm : main ] ( iii ) in section [ sec : lbu ] . it will sometime be convenient to use a bijection of the state - space @xmath45 of the particle process to the space of nearest - neighbor paths of length @xmath0 which begin at @xmath2 and have exactly @xmath6 _ up _ increments and @xmath46 _ down _ increments . for a particle configuration @xmath47 , let @xmath48 be defined by @xmath49 , and @xmath50 so occupied sites correspond to increments and vacant sites correspond to decrements of the path . see figure [ fig : exclmov ] for an illustration . . node @xmath51 of the path is updated in configuration @xmath52 to obtain @xmath53 . this corresponds to exchanging the particle at vertex @xmath51 with the hole at vertex @xmath54 . [ fig : exclmov ] ] the dynamics on the path are as follows : pick among the @xmath8 interval vertices of the path . if the path is a local extremum , refresh it with a local maximum with probability @xmath55 , and a local minimum with probability @xmath56 . if the chosen vertex is not an extremum , do nothing . see again figure [ fig : exclmov ] for an illustration of a transition , and figure [ fig : biasedtrans ] for the possible transitions from a particular path . ] it will be convenient to move back and forth from the particle description and the path description , and we will freely do so . here we set @xmath57 ; our assumption is always that @xmath58 . [ prop : eigenfunction ] let @xmath59 . the function @xmath60 , defined for the path @xmath61 as @xmath62 is the second eigenfunction for the biased exclusion process , with eigenvalue @xmath63 we let @xmath64 ; note our convention is @xmath65 . for a path @xmath61 and vertex @xmath66 , let @xmath67 be the number of up - edges before @xmath68 . we have @xmath69 and @xmath70 . define @xmath71 for @xmath72 . let @xmath73 be the path obtained by applying an update to @xmath61 at internal vertex @xmath68 . then @xmath74 = q g^\star_{h}(i-1))+ p g^\star_{h}(i+1 ) \,.\ ] ] consider the case where @xmath68 is a local extremum in @xmath61 . if the path at @xmath68 is refreshed to a local maximum , then @xmath75 , while if the path is refreshed to a local minimum , then @xmath76 . therefore , @xmath77 = q \theta^{i - ( f_h(i-1 ) + 1 ) } + p \theta^{i - ( f_h(i+1 ) -1 ) } = q g^\star_h(i-1 ) + p g^\star_h(i+1 ) \,.\ ] ] in the case where @xmath78 , the update at @xmath68 must leave the path unchanged . in this case , @xmath79 and @xmath80 . therefore , @xmath81 \,.\ ] ] finally , suppose @xmath82 ; again , the update at @xmath68 does not change the path . since @xmath83 in this case , @xmath84 for any constant @xmath85 , the function @xmath86 also satisfies @xmath87 = q g_h(i-1 ) + p g_h(i+1 ) \,.\ ] ] define @xmath88 and let @xmath89 define @xmath90 let @xmath91 satisfy @xmath92 that is , @xmath93 is the eigenfunction for the @xmath94 random walk on @xmath95 with absorbing states @xmath2 and @xmath0 . a direct verification shows that @xmath96 is a solution . note that @xmath97 \theta^{-1/2 } q \sin(\pi / n ) \\ & \quad + [ \theta^{n - k } - c ] \theta^{-n/2}\theta^{1/2}p \sin(\pi - \pi / n ) \nonumber \\ & = \sqrt{pq } \sin(\pi / n)[1 + \theta^{n/2-k } - c[1 + \theta^{-n/2 } ] ] \nonumber \\ & = 0 \ , . \nonumber \end{aligned}\ ] ] define @xmath98 let @xmath99 be the configuration obtained after one step of the chain when started from @xmath61 ; as before let @xmath100 be the update given that internal vertex @xmath52 is selected for an update . @xmath101 & = \sum_{x=1}^{n-1 } { { \mathbf e}}_h [ g_{\tilde{h}}(x ) ] \phi(x)\\ & = \sum_{x=1}^{n-1 } \bigr [ \bigl(1 - \frac{1}{n-1}\bigr ) g_h(x ) + \frac{1}{n-1}{{\mathbf e}}_h [ g_{\tilde{h}^{(x ) } } ] \bigr ] \phi(x ) \\ & = \bigl ( 1 - \frac{1}{n-1 } \bigr ) \phi(h ) + \frac{1}{n-1}\sum_{x=1}^{n-1 } [ q g_h(x-1 ) + p g_h(x+1 ) ] \phi(x ) \end{aligned}\ ] ] the sum on the right equals @xmath102 + [ g_h(0 ) \phi(1 ) q + g_h(n ) \phi(n-1 ) p ] \\ = \lambda \sum_{x=1}^{n-1 } g_h(x ) \phi(x ) = \lambda \phi(h ) \ , , \end{gathered}\ ] ] by . therefore , @xmath103 = \bigl ( 1 - \frac{1 - \lambda}{n-1 } \bigr ) \phi(h)\ ] ] note that @xmath104 for @xmath105 , and @xmath106 is increasing in @xmath61 , so @xmath60 is increasing . an increasing eigenfunction always corresponds to the second eigenvalue , so it must be the one with largest ( non unity ) eigenvalue . the second largest eigenvalue equals @xmath63 note that @xmath107 , so we have @xmath108 \theta^{-x/2 } \sin(\pi x / n ) \\ & = \sum_{x=1}^{n-1 } \bigl [ \alpha^{h(x ) } - \theta^{(n - x)/2 } \frac{1 + \theta^{n/2-k}}{1 + \theta^{n/2 } } \bigr ] \sin(\pi x / n ) \\ & = \sum_{x=1}^{n-1 } \alpha^{h(x ) } \sin(\pi x / n ) - \xi(n , k,\alpha)\ , . \end{aligned}\ ] ] let @xmath109 since @xmath110 does not depend on @xmath61 , and the eigenfunction @xmath60 must be orthogonal to the constants , it follows that @xmath111 . since @xmath112 , @xmath113 to apply wilson s lower bound , we need to bound @xmath114 from below , and @xmath115 from above . define @xmath116 for @xmath117 defined in , @xmath118 using that @xmath119 , we pair together the terms at @xmath52 and @xmath120 in so that @xmath121 the first sum simplifies to @xmath122 and the second to @xmath123 [ lem : efrb ] let @xmath117 be as in , and for a path @xmath61 , let @xmath99 be one step of the exclusion chain started from @xmath61 . let @xmath124 be the spectral gap . define @xmath125 if @xmath126 , then @xmath127 \log n \,.\ ] ] fix @xmath128 . from , @xmath129 if @xmath99 is obtained by a single update to @xmath61 at @xmath52 , the @xmath130 , and @xmath131 thus , if @xmath132 , then @xmath133 letting @xmath134 so that @xmath135 , equations and show that @xmath136 ^ 2 \,.\ ] ] the spectral gap @xmath137 satisfies @xmath138 suppose that @xmath139 . then from and we have @xmath140\log n \,.\ ] ] if @xmath141 , where @xmath142 , then @xmath143 ^ 2 & \zeta > 0 \\ c_0 \left ( \frac{3\rho^2}{4 } \right)^2 & \zeta = 0 \end{cases } \,.\ ] ] the right - hand side is bounded below for @xmath142 , so we conclude that @xmath127 \log n \,.\ ] ] [ prop : lb1 ] if @xmath144 where @xmath145 , then @xmath146 \bigl(\log n + \log[(1-{\varepsilon})/{\varepsilon } ) ] \bigr ) \,.\ ] ] from , the spectral gap @xmath137 satisfies @xmath147 \,.\ ] ] using lemma [ lem : efrb ] in ( see also theorem 13.5 of for a discussion ) yields @xmath148 \label{eq : wilsonlb } \\ & = \frac{n^3 } { ( \pi^2 + \zeta^2)}[1 + o(1)]\bigl(\log n + \log[(1-{\varepsilon})/{\varepsilon } ] \bigr)\ , , \nonumber \end{aligned}\ ] ] which yields . note that this matches the lower bound in theorem 4 of wilson ( 2004 ) for the symmetric exclusion when @xmath149 . [ prop : lb2 ] if @xmath150 but @xmath151 , then @xmath152(\log n + \log[(1-{\varepsilon})/{\varepsilon } ] ) \,.\ ] ] this again follows from , and lemma [ lem : efrb ] . [ prop : nearun ] there exists a constant @xmath153 such that if @xmath154 , then @xmath155 we now define a markov chain @xmath156 so that * @xmath157 and @xmath158 are _ labelled _ @xmath6-particle configurations , * if the labels are erased , @xmath159 and @xmath160 each are biased exclusion processes . we say a labelled particle is _ coupled _ at time @xmath161 if it occupies the same vertex in both @xmath157 and @xmath158 . we now describe a move of this chain from state @xmath162 : pick an edge @xmath163 among the @xmath8 edges uniformly at random . we consider several cases . * _ both @xmath164 and @xmath165 have no particles on @xmath163 . _ the chain remains at @xmath162 . * _ one of @xmath166 contains two particles on @xmath163 , and one of @xmath166 contains one particle on @xmath163 . _ suppose , without loss of generality , that @xmath164 contains one particle on @xmath163 . toss a @xmath56-coin to determine where the particle is placed in @xmath164 . if the single particle on @xmath163 in @xmath164 is coupled , or has the same label as one of the particles on @xmath163 in @xmath165 , arrange the two particles on @xmath163 in @xmath165 to preserve or facilitate the coupling . otherwise , toss a fair coin to determine the placement of the two particles in @xmath165 . * _ both @xmath164 and @xmath165 have two particles on @xmath163_. toss a fair coin to determine the placement of the two particles on @xmath163 in @xmath164 . place the particles in @xmath165 on @xmath163 to preserve or facilitate any couplings ; if no coupling is possible , toss a fair coin to determine the particle placement on @xmath163 . the distance @xmath167 between particle @xmath68 in @xmath164 and particle @xmath68 in @xmath165 performs a delayed nearest - neighbor walk , with possible bias @xmath27 at each move ( sometimes the bias is to the right , sometimes to the left ) . the probability it moves is at least @xmath168 . we can thus couple it to a random walk @xmath169 with constant upward bias @xmath27 so that @xmath170 until @xmath167 hits zero . consider the biased random walk @xmath169 on @xmath171 with positive bias @xmath27 , holding probability @xmath172 , and @xmath173 ; if @xmath174 then @xmath175 we have @xmath176 where @xmath177 . by the central limit theorem , since @xmath178 , there is a constant @xmath179 such that , for @xmath0 large enough , @xmath180 thus by taking @xmath153 large enough , @xmath181 if we run @xmath182 blocks of @xmath183 moves , then we have @xmath184 setting @xmath185 , @xmath186 if @xmath187 , then @xmath188 , and @xmath189 for @xmath0 large enough . and @xmath53 . [ fig : walkneighbors ] ] we consider configurations @xmath52 and @xmath53 to be adjacent if @xmath53 can be obtained from @xmath52 by taking a particle and moving it to an adjacent unoccupied site . in the path representation , moving a particle to the right corresponds to changing a local maximum ( i.e. , an `` up - down '' ) to a local minimum ( i.e. a `` down - up '' ) . moving a particle to the left changes a local minimum to a local maximum . see figure [ fig : exclmov ] , where @xmath190 . [ thm : mixbe ] consider the biased exclusion process with bias @xmath191 on the segment of length @xmath0 and with @xmath6 particles . set @xmath192 . for @xmath193 , if @xmath0 is large enough , then @xmath194 \right ] \,.\ ] ] in particular , if @xmath195 , then @xmath196 , so @xmath197 -2 \log \beta + o(\beta ) \bigr ] \,.\ ] ] note that whenever @xmath198 for constants @xmath153 and @xmath199 , the ratio of the upper and lower bounds is bounded . thus there is a pre cut - off for this chain in this regime . for @xmath200 , define the distance between two configurations @xmath52 and @xmath53 which differ by a single transition to be @xmath201 where @xmath61 is the height of the midpoint of the diamond that is removed or added . ( see figure [ fig : walkneighbors ] . ) note that @xmath58 and @xmath202 guarantee that @xmath203 , so we can use path coupling see , e.g. , theorem 14.6 of . we again let @xmath204 denote the path metric on @xmath205 corresponding to @xmath206 . we couple from a pair of initial configurations @xmath52 and @xmath53 which differ at a single vertex @xmath207 as follows : choose the same vertex in both configurations , and propose a local maximum with probability @xmath208 and a local minimum with probability @xmath56 . for both @xmath52 and @xmath53 , if the current vertex @xmath207 is a local extremum , refresh it with the proposed extremum ; otherwise , remain at the current state . let @xmath209 be the state after one step of this coupling . there are several cases to consider . the first case is shown in figure [ fig : walkneighbors ] . let @xmath52 be the upper configuration , and @xmath53 the lower . here the edge between @xmath210 and @xmath211 is `` up '' , while the edge between @xmath212 and @xmath213 is `` down '' , in both @xmath52 and @xmath53 . if @xmath207 is selected , the distance decreases by @xmath214 . if either @xmath211 or @xmath212 is selected , and a local minimum is selected , then the lower configuration @xmath53 is changed , while the upper configuration @xmath52 remains unchanged . thus the distance increases by @xmath215 in that case . we conclude that @xmath216 - \rho(x , y ) & = -\frac{1}{n-1}\alpha^{h+n - k } + \frac{2}{n-1}p \alpha^{h+n - k -1 } \nonumber \\ & = \frac{\alpha^{h+n - k}}{n-1 } \left ( \frac{2p}{\alpha } - 1 \right ) = \frac{\alpha^{h+n - k}}{n-1}\left ( 2\sqrt{pq } - 1 \right ) \ , . \label{eq : aepc1 } \end{aligned}\ ] ] in the case where @xmath52 and @xmath53 at @xmath217 are as in the right panel of figure [ fig : walkneighbors ] , we obtain @xmath216 - \rho(x , y ) & = -\frac{1}{n-1}\alpha^{h+n - k } + \frac{2}{n-1}(1-p ) \alpha^{h+n+1 } \nonumber \\ & = \frac{\alpha^{h+n - k}}{n-1}\left ( 2\alpha ( 1-p ) - 1 \right ) = \frac{\alpha^{h+n - k}}{n-1}\left(2 \sqrt{pq } - 1 \right ) \label{eq : aepc2 } \end{aligned}\ ] ] ( we create an additional disagreement at height @xmath218 if either @xmath211 or @xmath212 is selected and a local maximum is proposed ; the top configuration can accept the proposal , while the bottom one rejects it . ) since @xmath219 , we have @xmath220 , and both and reduce to @xmath221 - \rho(x , y ) = - \frac{\alpha^{h+n - k}}{n-1 } \delta\,.\ ] ] now consider the case on the left of figure [ fig : moreconfigs ] . we have @xmath216 - \rho(x , y ) & = -\frac{1}{n-1}\alpha^{h+n - k } + \frac{1}{n-1}q \alpha^{h+n - k+1 } + \frac{1}{n-1}p \alpha^{h+n - k-1 } \nonumber \\ & = \frac{\alpha^{h+n - k}}{n-1}\left ( q\alpha + \frac{p}{\alpha } - 1 \right ) \nonumber \\ & = -\frac{\alpha^{h+n - k}}{n-1 } \delta \ , , \end{aligned}\ ] ] which gives again the same expected decrease as . ( in this case , a local max proposed at @xmath211 will be accepted only by the top configuration , and a local min proposed at @xmath212 will be accepted only by the bottom configuration . ) the case on the right of figure [ fig : moreconfigs ] is the same . thus , holds in all cases . that is , since @xmath222 , @xmath223 = \rho(x , y)\left(1 - \frac{\delta}{n-1 } \right ) \leq \rho(x , y ) e^{-\frac{\delta}{n-1 } } \,.\ ] ] the diameter of the state - space is the distance from the configuration with @xmath6 `` up '' edges followed by @xmath46 `` down '' edges to the configuration with @xmath46 `` down edges '' followed by @xmath6 `` up '' edges . to move from the former to the latter , first flip the top - most maxima , next the subsequent two maxima , continuing down @xmath224 levels . at level @xmath225 , there are @xmath225 maxima to flip . each of the next @xmath226 levels will have @xmath6 maxima to flip . the number of maxima in the last @xmath224 levels decrease by a unit at each depth . thus , the distance travelled equals @xmath227 since @xmath228 , corollary 14.7 of gives @xmath229 \right ] \,.\ ] ] note that @xmath196 as @xmath24 , so @xmath230 -2 \log \beta + o(\beta ) \right ] \,.\ ] ] in particular , if @xmath231 , then @xmath232 , which is the same order as the mixing time in the symmetric case . ] [ prop : lbu ] suppose that @xmath150 . for any @xmath193 and @xmath233 , if @xmath0 is large enough , then @xmath234 we use the particle description here . the stationary distribution is given by @xmath235 where @xmath236 are the locations of the @xmath6 particles in the configuration @xmath52 , and @xmath237 is a normalizing constant . to see this , if @xmath238 is obtained from @xmath52 by moving a particle from @xmath225 to @xmath239 , then @xmath240 let @xmath241 be the location of the left - most particle of the configuration @xmath52 , and let @xmath242 be the location of the right - most unoccupied site of the configuration @xmath52 . let @xmath243 and consider the transformation @xmath244 which takes the particle at @xmath225 and moves it to @xmath206 . note that @xmath245 is one - to - one on @xmath246 . we have @xmath247 so @xmath248 letting @xmath249 , we have @xmath250 we consider now starting from a configuration @xmath251 with @xmath252 . the trajectory of the left - most particle , @xmath253 , can be coupled with a delayed biased nearest - neighbor walk @xmath169 on @xmath171 , with @xmath254 and such that @xmath255 , as long as @xmath256 . the holding probability for @xmath169 equals @xmath257 . by the gambler s ruin , the chance @xmath258 ever reaches @xmath259 is bounded above by @xmath260 therefore . @xmath261 by chebyshev s inequality ( recalling @xmath254 ) , @xmath262 taking @xmath263 and @xmath264 shows that @xmath265 as long as @xmath266 . combining with shows that @xmath267 we conclude that as long as @xmath266 , @xmath268 as @xmath3 , whence @xmath269 for sufficiently large @xmath0 . we thank perla sousi and nayantara bhatnagar for helpful comments on an earlier version of this paper .

we analyze the mixing behavior of the biased exclusion process on a path of length @xmath0 as the bias @xmath1 tends to @xmath2 as @xmath3 . we show that the sequence of chains has a pre - cutoff , and interpolates between the unbiased exclusion and the process with constant bias . as the bias increases , the mixing time undergoes two phase transitions : one when @xmath1 is of order @xmath4 , and the other when @xmath1 is order @xmath5 .

You are an expert at summarizing long articles. Proceed to summarize the following text: the so - called acceleration theorem for wave - packet motion in periodic potentials , formulated already in 1928 by bloch , @xcite has proven to be of outstanding value to solid - state physics for understanding the dynamics of bloch electrons within a semiclassical picture . @xcite in its most - often used variant , this theorem states that if we consider an electronic wave packet in a spatially periodic lattice , which is centered in @xmath0 space around some wave vector @xmath1 , and if an external electric field @xmath2 is applied under single - band conditions , then this center wave vector evolves in time according to @xmath3 , with @xmath4 being the electronic charge . perhaps its best - known application is the explanation of bloch oscillations of particles exposed to a homogeneous , constant force , @xcite which we recapitulate here in the simplest guise : take a particle in a one - dimensional tight - binding energy band @xmath5 , where @xmath6 is the band width and @xmath7 denotes the lattice period . assume that the particle s wave packet is centered around @xmath8 initially and subjected to a homogeneous force of strength @xmath9 . then the acceleration theorem , now taking the form @xmath10 tells us @xmath11 , so that the packet moves through @xmath0 space at a constant rate . @xcite according to another classic work by jones and zener , @xcite the particle s group velocity @xmath12 in real space is determined , quite generally , by the derivative of @xmath13 with respect to @xmath0 when evaluated at the moving center @xmath14 , @xmath15 in our case , this relation immediately gives @xmath16 implying that the particle s response to the constant force is an oscillating motion with the bloch frequency @xcite @xmath17 . this elementary example , to which we will come back later in sec . [ sec : s_4 ] , strikingly illustrates the power of this type of approach . but an obvious restriction stems from the necessity to remain within the scope of the single - band approximation ; the above acceleration theorem ( [ eq : oat ] ) is put out of action when several bloch bands are substantially coupled by the external force . nonetheless , in the present work we demonstrate that there exists a generalization of the acceleration theorem which can be applied even under conditions of strong interband transitions . specifically , we consider situations in which a bloch particle is subjected to a strong oscillating force which possibly induces pronounced transitions between the unperturbed energy bands . by abandoning the customary crystal - momentum representation @xcite and introducing an alternative floquet representation instead , we show that the effect of an additional force then is well captured by another acceleration theorem which closely mimics the spirit of the original . we obtain two major results : the floquet analog ( [ eq : gat ] ) of bloch s acceleration theorem ( [ eq : oat ] ) , and the floquet analog ( [ eq : fgv ] ) of the jones - zener expression ( [ eq : ogv ] ) for the group velocity . these findings are particularly useful for control applications , when a strong oscillating field `` dresses '' the lattice and thus significantly alters its band structure , while a second , comparatively weak homogeneous force is employed to effectuate controlled population transfer . we first outline the formal mathematical arguments in secs . [ sec : s_2 ] and [ sec : s_3 ] , and then we give two applications of topical interest , discussing `` super '' bloch oscillations in sec . [ sec : s_4 ] and coherently controlled interband population transfer in sec . [ sec : s_5 ] . although we restrict ourselves here for notational simplicity to one - dimensional lattices , our results can be carried over to general , higher - dimensional settings . we consider a particle of mass @xmath18 moving in a one - dimensional lattice potential @xmath19 with spatial period @xmath7 under the influence of a homogeneous , time - dependent force @xmath20 , as described by the hamiltonian @xmath21 subjecting the particle s wave function @xmath22 to the unitary transformation @xmath23 the new function @xmath24 obeys the schrdinger equation @xmath25 with the transformed hamiltonian @xmath26 now let us further assume that the force @xmath20 is periodic in time with period @xmath27 , such that its one - cycle integral either vanishes or equals an integer multiple of @xmath28 times the reciprocal lattice wave number @xmath29 : @xmath30 for example , this is accomplished by a monochromatic oscillating force with an additional static bias , @xmath31 provided the latter satisfies the condition @xmath32 . then the floquet theorem guarantees that the time - dependent schrdinger equation ( [ eq : tse ] ) admits a complete set of spatiotemporal bloch waves , @xcite that is , of solutions of the form @xmath33 with spatially _ and _ temporally periodic functions @xmath34 as usual , @xmath35 is the band index and @xmath0 a wave number ; @xmath36 thus is the quasienergy dispersion relation for the @xmath35th band . if @xmath37 in eq . ( [ eq : res ] ) , the existence of these solutions is obvious , because then @xmath38 , so that the wave functions ( [ eq : stb ] ) generalize the customary bloch waves @xcite for particles in spatially periodic lattice potentials by also accounting for the temporal periodicity of the driving force . when @xmath39 , so that @xmath40 itself is not periodic in time , spatiotemporal bloch waves ( [ eq : stb ] ) emerge nonetheless because @xmath0 is projected to the first quasimomentum brillouin zone , as first discussed by zak . @xcite in any case , the quasienergies @xmath36 may depend in a complicated manner on the parameters of the driving force , and the wave functions @xmath41 pertaining to a single quasienergy band may be nontrivial mixtures of several unperturbed energy bands . for later use , we observe that their spatial parts @xmath42 obey the quasienergy eigenvalue equation @xmath43 as follows immediately when plugging the solutions ( [ eq : stb ] ) into the schrdinger equation ( [ eq : tse ] ) . throughout , we adopt the standard normalization @xmath44 an arbitrary wave packet @xmath24 may now be expanded with respect to these spatiotemporal bloch waves , and written in the form @xmath45 with @xmath46 denoting the fundamental brillouin zone . the expansion coefficients @xmath47 depend on the way the system has been prepared and on the way the driving force has been turned on , whereas the basis functions @xmath48 and their quasienergies @xmath36 are given by the eigenvalue equation ( [ eq : qee ] ) and obviously are independent of such details . clearly , one has @xmath49 so that the populations @xmath50 remain constant in time . this expansion ( [ eq : flr ] ) , referred to as the floquet representation of the wave packet , is formally reminiscent of its customary crystal - momentum representation , that is , of an expansion with respect to the bloch states of the unperturbed potential @xmath51 which underlies the standard acceleration theorem . @xcite there are , however , substantial differences which become most clear when considering a wave packet occupying a single quasienergy band , @xmath52 here and in the following , we omit the band index @xmath35 for ease of notation . now this wave packet ( [ eq : sbp ] ) may describe , for instance , the dynamics in a situation where two unperturbed energy bands are resonantly coupled by the driving force @xmath20 ; consequently , in a crystal - momentum representation one would have to account for rabi - type oscillations between these two bands by coefficients which quantify the oscillating band populations . in the floquet respresentation , on the other hand , the rabi oscillations are already incorporated into the basis states ( [ eq : stb ] ) , so that one merely encounters single quasienergy band dynamics , with the remaining time evolution of @xmath53 simply given by eq . ( [ eq : dgt ] ) . thus , although the external force effectuates transitions between the unperturbed bloch bands , there are no inter - quasienergy band transitions ; @xmath54 remains constant in time . second , even in a situation where @xmath20 does not couple different energy bands , the wave packet s center @xmath14 evolves according to the standard acceleration theorem @xmath55 in the crystal - momentum representation , whereas in the floquet representation the moment @xmath56 obviously stays constant in time . in short , an expansion of the wave packet with respect to the spatiotemporal bloch waves ( [ eq : stb ] ) implies constant coefficients , and hence constant occupation probabilities , if the external force @xmath20 adheres to the specification ( [ eq : res ] ) . this formal shift of the dynamics from the occupation numbers to the basis states which is implied by the floquet representation now allows for a clear and physically transparent description of the additional effects which emerge when the external force does _ not _ obey eq . ( [ eq : res ] ) ; these effects are captured by the generalized acceleration theorem exposed in the following . we take a wave packet occupying a single quasienergy band and stipulate that in addition to the possibly strong driving force @xmath20 there is a second homogeneous force @xmath57 which we denote as the _ probe force _ ; this is assumed to be sufficiently weak so that it does not introduce transitions among different quasienergy bands . to be precise , the total hamiltonian now reads @xmath58 where the time - periodic force @xmath20 is resonant in the sense of eq . ( [ eq : res ] ) and thus creates a basis of spatiotemporal bloch waves ( [ eq : stb ] ) , whereas the probe force @xmath57 also is spatially homogeneous , but not necessarily periodic in time . after performing the unitary transformation ( [ eq : unt ] ) , we obtain the hamiltonian in the form @xmath59 with @xmath40 given by eq . ( [ eq : uph ] ) . moreover , we start from an initial wave packet of the form ( [ eq : sbp ] ) . because of the additional probe force @xmath57 , the time evolution of @xmath53 is no longer given by eq . ( [ eq : dgt ] ) ; the aim now is to find an effective hamiltonian @xmath60 which governs the resulting dynamics of @xmath53 , under the proposition that this remains restricted to the single , initially occupied quasienergy band . exploiting the normalization ( [ eq : nrm ] ) , we have @xmath61 this gives @xmath62 \varphi_k \right)^ * \psi \nonumber\\ & & - \sqrt{\frac{a}{2\pi } } { f_{\rm p}}\int \ ! { { \rm d}}x \ , \varphi_k^ * x \psi \ ; , \end{aligned}\ ] ] having suppressed the arguments @xmath63 and @xmath64 for better legibility ; all integrals here are taken over the entire lattice . in the first term on the right - hand side of this equation we exploit the quasienergy eigenvalue equation , eq . ( [ eq : qee ] ) , yielding @xmath65 . for rewriting the second term we use @xmath66 which is obtained by taking the derivative of the complex conjugate to eq . ( [ eq : dfp ] ) with respect to @xmath0 , and leads to @xmath67 for making the final step , we have resubstituted the expression ( [ eq : sbp ] ) for @xmath68 and have made use of the identity @xmath69 with the scalar product @xmath70 being given by an integral over a single lattice period . note that @xmath71 as an immediate consequence of eq . ( [ eq : nrm ] ) , which implies @xmath72 so that @xmath73 is purely imaginary . collecting all the pieces , we obtain the desired evolution equation @xmath74 with the effective hamiltonian for the floquet representation , @xmath75 from this expression we deduce the generalized acceleration theorem , that is , the acceleration theorem for the floquet representation : since the moment ( [ eq : fra ] ) obeys the the equation @xmath76 \rangle\ ] ] and the commutator appearing here on the right - hand side is easily evaluated , @xmath77 = { f_{\rm p}}$ ] , we are directly led to @xmath78 this is the central result of the present work ; its analogy to the standard acceleration theorem ( [ eq : oat ] ) for the crystal - momentum representation is evident . observe that there is an intuitively clear reason for the appearance of the term proportional to @xmath73 in the effective hamiltonian ( [ eq : efh ] ) : the twofold periodic parts @xmath79 of the spatiotemporal bloch waves are obtained by solving the eigenvalue equation ( [ eq : qee ] ) . this is done for each wave number @xmath0 separately , so that one is free to bestow upon each eigensolution an arbitrary phase factor @xmath80 . on the other hand , the evolution equation ( [ eq : eeg ] ) for the wave function @xmath53 in the floquet representation naturally establishes a `` connection '' between those different eigensolutions @xcite and therefore requires information about the gauge function @xmath81 ; this is provided by the expression @xmath73 . note further that when multiplying eq . ( [ eq : eeg ] ) by @xmath82 and subtracting the complex conjugate of the resulting equation , this piece drops out , and one is left with @xmath83 thus , @xmath84 does not depend on @xmath0 and @xmath64 separately , but rather on the combination @xmath85 , so that the distribution @xmath53 moves through the floquet @xmath0 space without change of shape , again in precise analogy to the classic behavior . @xcite but we reemphasize that this seemingly simple dynamics might be unrecognizable in the usual crystal - momentum representation , because the system might undergo violent transitions between different energy bands when monitored in a basis of time - independent bloch waves . as the introductory example has shown , the standard ( crystal - momentum ) acceleration theorem develops its main power in combination with the jones - zener expression ( [ eq : ogv ] ) for the wave packet s group velocity in real space , and the question naturally arises whether there exists a similar connection in the floquet representation . obviously , one can establish a relation corresponding to eq . ( [ eq : ogv ] ) by applying a stationary - phase argument to the expansion ( [ eq : flr ] ) , but here we follow an alternative line of reasoning which may be found particularly enlightening . considering a well - localized wave packet @xmath22 in the original frame of reference to which the hamiltonian operators ( [ eq : hor ] ) and ( [ eq : hot ] ) pertain , that packet s group velocity is given by @xmath86 on the other hand , exploiting the operator identity @xmath87 the eigenvalue equation ( [ eq : qee ] ) transforms into the even more basic eigenvalue equation @xmath88 for the periodic core pieces @xmath89 of the spatiotemporal bloch waves ( [ eq : stb ] ) , invoking the parametrically @xmath0-dependent operator @xmath90 this eigenvalue problem can efficiently be implemented for numerical calculations . @xcite it also manifestly contains the origin of the condition ( [ eq : res ] ) imposed on the oscillating force @xmath20 , since @xmath0 is reduced to fall within @xmath91 . most importantly , this eigenvalue problem ( [ eq : epu ] ) poses itself in an extended hilbert space made up of functions @xmath89 which are periodic in both space and time , in accordance with eq . ( [ eq : dpu ] ) . consequently , `` time '' has to be regarded as a coordinate in this extended hilbert space and therefore needs to be integrated over when forming a scalar product , just like any spatial coordinate . thus , the natural scalar product in this extended hilbert space is given by @xcite @xmath92 with @xmath93 denoting the standard scalar product in the original , physical hilbert space , as already employed in eqs . ( [ eq : def ] ) and ( [ eq : nru ] ) . it follows that the quasienergies @xmath36 can be written as diagonal elements of the matrix of the quasienergy operator , @xmath94 inviting us to make use of an analog of the hellmann - feynman theorem : @xcite @xmath95 in the final step we have undone the shift ( [ eq : udo ] ) ; the wave functions @xmath96 then denote the functions which are obtained from the spatiotemporal bloch waves ( [ eq : stb ] ) by inverting the transformation ( [ eq : unt ] ) . comparison of eqs . ( [ eq : vgd ] ) and ( [ eq : vge ] ) , keeping in mind the definition ( [ eq : scp ] ) , now yields the desired relation : supposing that @xmath97 were made up from a single spatiotemporal bloch wave labeled by @xmath35 and @xmath98 , say , one would obtain the formal identity @xmath99 but this is not what we want , because an individual spatiotemporal bloch wave is uniformly extended over the lattice and thus does not correspond to a `` group '' which propagates in space . rather , we require a wave packet ( [ eq : sbp ] ) which is reasonably well centered in the floquet @xmath0 space , with a center @xmath100 given by eq . ( [ eq : fra ] ) . then we have @xmath101 to good accuracy , so that the cycle - averaged group velocity of the floquet wave packet is given by the derivative of its quasienergy dispersion relation , evaluated at its center @xmath100 . again , this floquet relation ( [ eq : fgv ] ) closely mimics its historic crystal - momentum antecessor , given by eq . ( [ eq : ogv ] ) . in contrast to the equation of motion ( [ eq : gat ] ) for @xmath100 itself , which holds exactly within a single quasienergy band setting , this relation ( [ eq : fgv ] ) is an approximation which holds the better , the narrower the packet s floquet @xmath0 space distribution . although it seems self - evident , it might be worthwhile to stress that the argument required to evaluate the derivative ( [ eq : fgv ] ) is floquet @xmath100 , not crystal momentum @xmath102 . the phenomenon termed `` super '' bloch oscillations @xcite arises when a bloch particle is subjected to both a static ( dc ) and an oscillating ( ac ) force , such that an integer multiple of the ac frequency is only slightly detuned from the bloch frequency associated with the dc component of the force . @xcite although the effect itself appears almost trivial from the mathematical point of view , we nonetheless dwell on this at some length , because it provides a particularly instructive example for juxtaposing the familiar crystal - momentum representation to the floquet representation introduced in sec . [ sec : s_2 ] and for demonstrating in detail how they match . to be definite , we consider the total force to be of the form @xmath103 \ ; , \label{eq : tto}\ ] ] where @xmath104 denotes the heaviside function , so that both the dc and the ac component of the force are turned on instantaneously and simultaneously at @xmath105 ; that moment @xmath105 thus determines the relative phase between the bloch oscillations caused by the dc component and the driving oscillations of the ac component . the basic assumptions now are that _ ( i ) _ we are given an initial wave packet which occupies a single energy band , being centered around @xmath106 at the moment @xmath107 in the crystal - momentum representation , and that _ ( ii ) _ interband transitions remain negligible for @xmath108 , despite the action of the force @xmath20 . we then encounter single - band dynamics which are fully covered by the `` old '' acceleration theorem @xmath109 , giving @xmath110 \nonumber \end{aligned}\ ] ] for @xmath108 . as an archetypal example we now take a tight - binding cosine energy dispersion relation for the band considered , @xmath111 parametrized as in the introduction . utilizing eq . ( [ eq : ogv ] ) , one then finds the packet s group velocity : @xmath112 this expression describes super bloch oscillations if we assume further that the dc component of the force is almost resonant in the sense of eq . ( [ eq : res ] ) . we therefore decompose this component according to @xmath113 where @xmath114 with some nonzero integer @xmath115 as previously in eq . ( [ eq : int ] ) , so that @xmath116 equals the bloch frequency @xmath117 , while @xmath118 is quite small compared to @xmath119 . we then have @xmath120 with frequency detuning @xmath121 , so that the group velocity ( [ eq : igv ] ) takes the form @xmath122 having introduced the scaled driving amplitude @xmath123 and a constant phase @xmath124 which accounts for the initial conditions . because @xmath125 according to our specifications , the contribution @xmath126 to the argument of @xmath127 does not vary appreciably during one single cycle @xmath128 of the ac component . thus , when averaging the instantaneous group velocity over one such cycle , this `` slow '' time dependence may be ignored , meaning that @xmath126 may be considered as constant when taking the average . @xcite invoking the jacobi - anger indentity in the guise @xmath129 where @xmath130 denote the bessel functions of the first kind , one immediately obtains @xmath131 according to the above reasoning , here the `` fast '' time dependence is integrated out , but the slow dependence on @xmath126 remains . @xcite integrating , this yields the cycle - averaged drift motion of the packet , that is , its position @xmath132 without the fast ac quiver , @xmath133 with a suitably chosen origin of the @xmath63 axis . this result finally clarifies what is `` super '' with these dynamics : because the residual force @xmath118 is quite small , the amplitude of this oscillation ( [ eq : sbo ] ) can be fairly large ; indeed , in a corresponding experiment with weakly interacting bose - einstein condensates in driven optical lattices haller _ et al . _ have observed giant center - of - mass oscillations with displacements across hundreds of lattice sites . @xcite as far as the phenomenon itself is concerned there is nothing more to add ; because one requires single bloch - band dynamics right from the outset , a floquet treatment is not necessary . nevertheless the floquet approach is of its own intrinsic value even here , since it provides a somewhat different view which , in contrast to the above crystal - momentum calculation , is capable of some generalization . the floquet analysis starts from the spatiotemporal bloch waves and their quasienergies . in a single - band setting with an external homogeneous force , these are exceptionally easy to obtain : writing the bloch waves of the undriven lattice in the form @xmath134 where @xmath135 denotes a wannier function localized around the @xmath136th lattice site , @xcite the so - called houston functions @xcite @xmath137 are solutions to the time - dependent schrdinger equation in the original frame , for arbitrary @xmath20 , provided the `` moving wave numbers '' @xmath138 are given by @xmath139 always assuming the viability of the single - band approximation . @xcite taking a force of the particular form ( [ eq : int ] ) with exactly resonant @xmath119 obeying @xmath32 , we have @xmath140 this implies that both the exponentials @xmath141 and @xmath142 are @xmath27 periodic in time , with @xmath128 , whereas the integral over @xmath142 is not , because the fourier expansion of @xmath142 contains a zero mode , so that its integral contains a linearly growing contribution . but this observation reveals that the `` accelerated bloch waves '' ( [ eq : hou ] ) with resonant time - periodic forcing ( [ eq : int ] ) are precisely the required spatiotemporal bloch waves in the original frame , with their quasienergies being determined by the zero mode : @xmath143 the remarkable fact that the quasienergy bands collapse , _ i.e. _ , become flat when @xmath144 is such that @xmath145 , indicates that an oscillating force can effectively shut down the tunneling contact between neighboring wells ; this `` coherent destruction of tunneling '' is a generic feature of driven single - band systems . @xcite a bit of reflection then shows that the core pieces @xmath79 of the spatiotemporal bloch waves , that is , the solutions to the eigenvalue equation ( [ eq : epu ] ) , are given by @xmath146 \right ) \ ; . \end{aligned}\ ] ] although this has not been particularly emphasized , the above construction makes sure that any spatiotemporal bloch wave ( [ eq : hou ] ) is labeled by the same wave number @xmath0 as the ordinary bloch wave to which it reduces when the external force vanishes . @xcite otherwise , there is nothing particular about the choice @xmath147 for the lower bound of integration in eq . ( [ eq : qkt ] ) for @xmath138 : in contrast to eq . ( [ eq : tto ] ) , where @xmath107 has been singled out as the moment when the force is turned on , and which thus designates an initial - value problem for a particular wave packet , the solution of the eigenvalue problem ( [ eq : epu ] ) for the entire spatiotemporal bloch basis requires a force @xmath20 which is perfectly periodic in time ; the resulting expression for @xmath138 thus holds for both @xmath148 and @xmath149 . also note that it would be meaningless to include some additional constant phase into the argument of the ac component of the force ( [ eq : int ] ) : because this expression holds for all times @xmath64 , such a phase would merely amount to a shift of the origin of the time coordinate and thus is as irrelevant for the calculation of the quasienergy dispersion relation as would be a shift of the origin of the spatial coordinate system for the calculation of a crystal s energy band structure . knowing the quasienergy dispersion relation ( [ eq : qed ] ) , the machinery established in sec . [ sec : s_3 ] can be set to work : according to eq . ( [ eq : fgv ] ) , the cycle - averaged group velocity of a floquet wave packet ( [ eq : sbp ] ) is given by @xmath150 if we now turn back to the specific forcing ( [ eq : tto ] ) , and thus consider exactly the same initial - value problem as in the previous crystal - momentum exercise , we can make operational use of the decomposition ( [ eq : dec ] ) of the dc force : its resonant part @xmath119 has already been incorporated into the spatiotemporal bloch waves ( [ eq : hou ] ) , which means that it has already been accounted for in `` dressing '' the lattice and changing its original energy dispersion @xmath13 to the quasienergy dispersion @xmath151 . therefore , it is only the small residual part @xmath118 which enters into the equation of motion for @xmath100 , that is , into the generalized acceleration theorem ( [ eq : gat ] ) ; this part @xmath118 thus constitutes a particular , time - independent example of a probe force @xmath57 as considered in sec . [ sec : s_3 ] . we now have @xmath152 giving @xmath153 all that remains to be done now is to express the initial floquet center @xmath154 in terms of the initial wave packet s center @xmath106 , which had been specified in the crystal - momentum representation . but this is an easy task , comparing the original bloch waves ( [ eq : blo ] ) to their spatiotemporal descendents ( [ eq : hou ] ) : at the moment @xmath105 when the force ( [ eq : tto ] ) is turned on , @xmath106 coincides with @xmath155 for one particular @xmath0 ; this evidently is the desired @xmath154 . the equality identifying @xmath154 thus is @xmath156 which , written out in full detail , reads @xmath157 using this to eliminate @xmath154 from eq . ( [ eq : fkt ] ) , we arrive at @xmath158 with precisely the same phase @xmath159 as already defined in eq . ( [ eq : phi ] ) . inserting this argument ( [ eq : arg ] ) into the cycle - averaged group velocity ( [ eq : cag ] ) , and comparing with the previous expression ( [ eq : cac ] ) , one confirms that the result of the floquet analysis fully coincides with that of the more customary crystal - momentum calculation . the necessity to painstakingly distinguish between crystal momentum @xmath102 and floquet @xmath100 at all stages may appear a bit mind - boggling ; if this is not done with sufficient care , one might overlook a contribution to @xmath159 . @xcite but if respected properly , the mathematical structure of the floquet picture unerringly leads to the correct answer . if one strips the above reasoning to the bare essentials , that is , if one starts from the quasienergy dispersion relation ( [ eq : qed ] ) , takes its derivative to obtain the formal expression ( [ eq : cag ] ) for the cycle - averaged group velocity , and then inserts the solution to the equation of motion ( [ eq : par ] ) dictated by the generalized acceleration theorem in order to compute the group velocity of the wave packet actually considered , one sees that this procedure exactly parallels the explanation of the usual bloch oscillations , as reviewed in the introduction . thus , super bloch oscillations may be seen as ordinary bloch oscillations arising in response to a weak probe force @xmath118 , but occurring in a spatiotemporal lattice , as created by dressing the original lattice through application of the strong force ( [ eq : int ] ) . one might finally wish to get away from the particular , instantaneous onset of the forcing assumed in eq . ( [ eq : tto ] ) : the dc and the ac component might not be switched on simultaneously , or not abruptly , possibly involving two different turn - on functions for the two components . in any case , at some moment @xmath105 the final amplitudes will have been reached , so that the previous analysis goes through unaltered for @xmath108 , if one only interprets @xmath106 correctly : this would no longer indicate the crystal - momentum wave number around which the initial wave packet had been prepared , but rather that to which the latter had been shifted during the turn - on phase . expressed differently , the phase @xmath159 in eqs . ( [ eq : cac ] ) and ( [ eq : sbo ] ) depends significantly on the precise turn - on protocol : not surprisingly , the way the external force has been turned on in the past crucially affects the coherent wave - packet motion after the turn - on is over . aside from its aesthetic value , the floquet picture offers at least one further benefit : bloch oscillations in dressed lattices may also occur under conditions such that the quasienergy bands are mixtures of several unperturbed energy bands , disabling a crystal - momentum treatment . a floquet analysis , on the other hand , would merely require one to replace the single - band quasienergies ( [ eq : qed ] ) by the actual ones and then again invoke the generalized acceleration theorem ( [ eq : gat ] ) , similar to the examples worked out in the next section . a field of major current interest in which the floquet picture may find possibly groundbreaking applications concerns ultracold atoms , or weakly interacting bose - einstein condensates , in time - periodically driven optical lattices . @xcite as opposed to ordinary crystalline matter exposed to high - power laser fields , such systems offer the advantage that one can apply even nonperturbatively strong driving forces without inducing unwanted inhomogeneities , as caused by polarization effects or domain formation . @xcite the issue at stake here is not merely redoing well - known condensed - matter physics in another setting , and thus selling old wine in new skins , but rather finding genuinely new ways of coherently controlling mesoscopic matter waves , such that target states are created which have not been accessible before , and are manipulated according to some prescribed protocol . here we point out that the generalized acceleration theorem ( [ eq : gat ] ) may be a valuable tool in this quest . a standard one - dimensional ( 1d ) optical lattice is described by a cosine potential @xmath160 where @xmath161 is the wave number of the two counterpropagating laser beams generating the lattice . @xcite its depth @xmath162 is measured in multiples of the single - photon recoil energy @xmath163 for orientation , if one traps @xmath164rb atoms in a lattice with @xmath161 corresponding to the wavelength @xmath165 nm , as in a recent experiment by zenesini _ et al . _ , @xcite one finds @xmath166 ev ; typical optical lattices are a few recoil energies deep . figure [ fig : f_1 ] shows quasienergy spectra for such a 1d cosine lattice ( [ eq : ola ] ) with depth @xmath167 under pure ac forcing , that is , for @xmath168 not containing a dc component , with driving frequency @xmath169 . under the laboratory conditions specified above ( @xmath164rb at @xmath165 nm ) , this corresponds to @xmath170 khz . figure [ fig : f_1](a ) results when the scaled driving amplitude ( [ eq : sda ] ) is set to zero ; this subfigure therefore is obtained by projecting the lowest three unperturbed energy bands to the fundamental quasienergy brillouin zone , which extends from @xmath171 to @xmath172 on the ordinate . figure [ fig : f_1](b ) displays the quasienergy band structure for the moderate driving strength @xmath173 ; here avoided crossings show up which generally indicate multiphotonlike resonances . @xcite figure [ fig : f_1](c ) then reveals pronounced ac stark shifts ( that is , shifts of the quasienergies against the zone - projected original energies ) for @xmath174 , corresponding to truly strong forcing . we now turn from the quasienergy spectrum to an exemplary initial - value problem : at @xmath147 we prepare an initial wave packet ( [ eq : sbp ] ) in the lowest bloch band @xmath175 with a gaussian momentum distribution , @xmath176 centered around @xmath177 with width @xmath178 , and subject it to a pulse , @xmath179 starting at @xmath147 and ending at @xmath180 , endowed with a smooth , squared - sine envelope function : @xmath181 we again set @xmath169 , as in fig . [ fig : f_1 ] ; adjust the pulse length to 50 cycles ; @xmath182 , and fix the maximum driving amplitude @xmath183 such that @xmath184 , corresponding to the conditions reached in fig . [ fig : f_1](c ) . we then monitor the resulting wave - packet dynamics both in the basis of the unperturbed energy bands and in the bases provided by the instantaneous spatiotemporal bloch waves , that is , in the family of floquet bases which are obtained when the driving amplitude is kept fixed at any value @xmath185 reached during the pulse . figure [ fig : f_2 ] displays the results : the jagged lines in the main frame show the occupation probabilities of the lowest three unperturbed bloch bands @xmath186 , and 3 ; in the middle of the pulse the band @xmath187 contains even more population than the band @xmath188 . on the other hand , the horizontal line at the top depicts the occupation of the instantaneous floquet band emerging from the lowest bloch band : this floquet band contains practically _ all _ the population during the entire pulse , which means that the wave function adjusts itself adiabatically to the changing morphology of its quasienergy band , @xcite as previously sketched in fig . [ fig : f_1 ] , when the driving amplitude @xmath189 is first increased and then decreased back to zero . to quantify the precise degree of adiabatic following , the inset in fig . [ fig : f_2 ] shows the variation of the floquet band population on a much finer scale . observe that the final adiabaticity defect is on the order of merely @xmath190 , even though the driving amplitude reaches its fairly high maximum strength within no more than 25 cycles . with respect to the concepts developed in sec . [ sec : s_2 ] , fig . [ fig : f_2 ] strikingly demonstrates the advantages of the floquet picture over the traditional crystal - momentum representation for the situation considered . if there were an additional probe force , its effect would have to be tediously disentangled from the fast oscillations of the bloch band populations . when the same dynamics are seen from the floquet viewpoint , essentially `` nothing '' happens , because practically all inter - bloch - band transitions are already accounted for by continuously adapting the floquet basis , so that the action of a probe force would stand out most clearly . although , of course , the crystal - momentum representation is mathematically equivalent to the floquet picture , there is no question which one is preferable here . note also that fig . [ fig : f_2 ] answers one further pertinent question : how do we prepare a wave packet which occupies merely a single quasienergy band , although it is undergoing rapid transitions between several bloch bands at the same time ? the recipe for achieving this is simple : start with a wave packet occupying a single bloch band and switch on the driving force smoothly thereby enabling adiabatic following . at this point an important issue needs to be stressed : the concept of adiabatic following , or parallel transport in a differential - geometric language , usually is applied to energy eigenstates ; @xcite in the context of optical lattices this has been exploited , e.g. , by fratalocchi and assanto for studying nonlinear adiabatic evolution and emission of coherent bloch waves . @xcite in contrast , here we consider adiabatic following of explicitly time - dependent _ quasi_energy eigenstates , that is , of solutions to the quasienergy eigenvalue equation ( [ eq : qee ] ) ; this is what allows us to separate the fast , oscillating time dependence of the driving force from the slow , parametric time dependence of its envelope . having learned these lessons , we now set the generalized acceleration theorem ( [ eq : gat ] ) to work . suppose that we are prompted to empty the ground - state energy band . starting again from an initial wave packet ( [ eq : ini ] ) , we then may proceed as follows : first we smoothly turn on an ac force which dresses the lattice , creating avoided quasienergy crossings initially not `` seen '' by the adiabatically following packet . for instance , we may wish to utilize the avoided crossings showing up in fig . [ fig : f_1](b ) . to this end , we again take an ac force with frequency @xmath169 and fix its scaled driving amplitude at the plateau value @xmath173 . this dressing force is switched on during 25 cycles with half a squared - sine envelope , maintained at maximum amplitude for 50 further cycles , and switched off again for another 25 cycles , as sketched in fig . [ fig : f_3](a ) . if this were all we did , the wave packet would simply undergo adiabatic evolution and finally restore its initial condition , as previously observed in fig . [ fig : f_2 ] . instead , once the maximum dressing amplitude has been reached , we now apply an additional weak probe force @xmath57 in order to exploit eq . ( [ eq : gat ] ) for moving the packet away from the brillouin zone center , driving it over the avoided crossings that have opened up in fig . [ fig : f_1](b ) . this probe force is implemented in the form of two smooth , squared - sine shaped dc pulses , one acting during the plateau of the dressing pulse , the other acting with reversed sign after the dressing pulse is over , as drawn in fig . [ fig : f_3](a ) . the maximum strength of the probe force here is only 2.5% of that of the dressing force ; for better visibility , the probe force is magnified in fig . [ fig : f_3](a ) by a factor of 10 . it is now almost obvious how to describe the response of the wave packet within the floquet picture : the initial state ( [ eq : ini ] ) first is adiabatically shifted into a single quasienergy - band packet during the turn - on of the dressing force . in contrast to a crystal - momentum representation , all dressing - induced fast oscillations are taken out of the dynamics of @xmath191 in the floquet representation , as shown in fig . [ fig : f_3](b ) . when the first probe pulse acts at constant dressing amplitude , it forces the wave packet over the avoided crossing seen in fig . [ fig : f_1](b ) , so that the packet undergoes zener - type transitions to `` higher '' quasienergy bands , @xcite splitting into individual subpackets associated with the different quasienergy bands involved . when the dressing force is switched off , each of these subpackets moves adiabatically on its own quasienergy surface , finally reaching the continuously connected bloch bands . the second , reversed probe pulse , applied after the dressing pulse is over , then acts in accordance with bloch s original acceleration theorem ( [ eq : oat ] ) , shifting the various subpackets back to the brillouin zone center . in the scenario displayed in fig . [ fig : f_3 ] , the lowest band is almost entirely depopulated by the probe - induced zener transitions , so that only a marginal fraction of the initial packet returns , as depicted in fig . [ fig : f_3](c ) . thus , the main part of the initial packet has been placed in higher bloch bands , as intended . we have also checked by explicit calculation that without the comparatively weak probe pulses the returning wave packet would be almost identical to the initial one . the above example of our `` dressing and probing '' strategy immediately lends itself to a host of further modifications and extensions . to give but one further instance , if the probe pulse is still weaker , such that the wave packet does not pass over the avoided - crossing regime , but rather stops there , the zener transitions are incomplete , so that a signifcant part of the initial state is recovered when the process is over . this is elaborated in fig . [ fig : f_4 ] with the same dressing force as above , but now the maximum strength of the probe force amounts to only 1.7% of that of the dressing force . the final subpacket still occupying the lowest bloch band then is no longer centered around @xmath192 , implying that this subpacket will move over the lattice . in a sense , the left wing of the initial wave packet has been cut out , so that fig . [ fig : f_4 ] may be regarded as a particular paradigm of `` wave - packet surgery . '' summarizing our line of reasoning , we have introduced in sec . [ sec : s_2 ] a representation of wave packets of quantum particles in spatially periodic lattices subjected to homogeneous , time - periodic forcing which is based on an expansion with respect to spatiotemporal bloch waves and reduces to the standard crystal - momentum representation when the forcing is turned off . it embodies forcing - induced oscillations into the basis , so that only the actually relevant dynamics remain to be dealt with . within this floquet representation one encounters many features already familiar from solid - state physics in time - independent lattice potentials , but here their scope is different . as a prominent example , the generalized acceleration theorem derived in sec . [ sec : s_3 ] takes the same form as its historic antecessor formulated by bloch , @xcite but applies to single quasienergy band dynamics , which can be drastically different from single energy band behavior . there are further features which can be carried over from the crystal - momentum representation to the floquet picture and acquire a modified meaning there , such as the expression for the group velocity of a wave packet or zener transitions among different bands . the super bloch oscillations considered in sec . [ sec : s_4 ] provide a mainly pedagogical example which can be worked out in full detail analytically . here the floquet picture can not exert its full strength , because one assumes _ a priori _ that the driving force does not induce transitions from the initially occupied energy band to other ones , so that the historic acceleration theorem remains capable of describing the entire dynamics . the floquet approach leads to exactly the same result , but implies a different viewpoint , separating the dc component of the force into one part which is resonant with the ac component , and together with the latter dresses the lattice , creating a quasienergy band ; the remaining residual part of the dc force then probes this new quasienergy band , rather than the original unperturbed energy band . this theme of `` dressing and probing '' also prompts far - reaching strategies for achieving coherently controlled interband population transfer and even more . two basic examples for this have been discussed in sec . [ sec : s_5 ] , but the possibilities obviously extend much farther . utilizing the generalized acceleration theorem , an initial wave packet may by split coherently into two components at an avoided quasienergy band crossing in a dressed lattice , and the lattice may then be redressed ( that is , exposed to an ac force with different parameters ) such that another quasienergy band structure is generated , possibly involving avoided crossings which affect only one of the daughter wave packets created in the first step , but not the other . moreover , daughter wave packets can be made to move , possibly into different directions , and to interfere with other wavelets having been manipulated separately before in distant parts of the lattice . this vision apparently will be hard to realize with traditional solids , but it has come into immediate reach in current laboratory experiments with weakly interacting bose - einstein condensates in driven optical lattices . seen against this background , the generalized acceleration theorem almost provides a blueprint for a wave - packet processor .

a representation is put forward for wave functions of quantum particles in periodic lattice potentials subjected to homogeneous time - periodic forcing , based on an expansion with respect to bloch - like states which embody both the spatial and the temporal periodicity . it is shown that there exists a generalization of bloch s famous acceleration theorem which grows out of this representation , and captures the effect of a weak probe force applied in addition to a strong dressing force . taken together , these elements point at a `` dressing and probing '' strategy for coherent wave - packet manipulation , which could be implemented in present experiments with optical lattices .

You are an expert at summarizing long articles. Proceed to summarize the following text: the exponentially growing number of known extrasolar planets now enables statistical analyses to probe their formation mechanism . two theoretical frameworks have been proposed to account for the formation of gas giant planets : the slow and gradual core accretion model @xcite , and the fast and abrupt disk fragmentation model @xcite . the debate regarding their relative importance is still ongoing . both mechanisms may contribute to planet formation , depending on the initial conditions in any given protoplanetary disk ( * ? ? ? * and references therein ) . by and large , our understanding of the planet formation process is focused on the case of a single star+disk system . yet , roughly half of all solar - type field stars , and an even higher proportion of pre - main sequence ( pms ) stars , possess a stellar companion ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? since the disk and multiplicity phenomena are associated with similar ranges of distances from the central star , the dynamical influence of a companion on a disk may be dramatic . theory and observations agree that stellar companions can open large gaps in disks , or truncate them to much smaller radii than they would otherwise have ( e.g. , * ? ? ? * ; * ? ? ? the consequences for planet formation are still uncertain , however . observations of protoplanetary disks among pms stars have revealed that tight binaries generally show substantially reduced ( sub)millimeter thermal emission @xcite as well as a much rarer presence of small dust grains in regions a few au from either component @xcite . both trends can be qualitatively accounted for by companion - induced disk truncation , which can simultaneously reduce the disk s total mass , outer radius and viscous timescale . these observational facts have generally been interpreted as evidence that binaries tighter than @xmath0au are much less likely to support gas giant planet formation . however , follow - up imaging surveys have identified some 50 planet - host stars that possess at least one stellar companion ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? in particular , it it is worth noting that about 20% of all known planets in binary systems have a stellar companion within less 100au , so that planet formation in such an environment can not be considered a rare occurrence . in this _ letter _ , i review several key statistical properties of pms and field binary systems that provide insight on the planet formation process ( sections[sec : ci ] and [ sec : end ] ) . i then discuss the implications for the main mechanisms of planet formation in binary systems as a function of their projected separation ( section[sec : implic ] ) . in this study , i only consider binaries in the 51400au separation range , for which current pms multiplicity surveys are reasonably complete . the tightest binary system known to host a planet has a 19au separation . stellar companions beyond 1400au are not expected to have much influence on planet formation . in order to draw a broad and homogeneous view of the initial conditions for planet formation , i compiled a sample of 107 pms binaries for which deep ( sub)millimeter continuum observations and/or near- to mid - infrared colors are available in the literature . the ( sub)millimeter data are taken from the work of @xcite ; for almost all targets , a 1@xmath1 sensitivity of 15mjy or better at 850@xmath2 m and/or 1.3 mm is achieved . the median projected separation in this sample is 92au . i also defined a comparison sample of 222 pms stars for which no companion has ever been detected . i focus here on the taurus and ophiuchus star forming regions , the only ones for which high - resolution multiplicity , photometric and millimeter surveys have a high completeness rate . the two clouds contribute an almost equal number of binaries to the sample . furthermore , both regions have similar stellar age distributions ( median age around 1myr , ophiuchus being probably slighter younger on average than taurus ) and their mass function fully samples the 0.11.5@xmath3 range ( e.g. , * ? ? ? * ; * ? ? ? finally , taurus represents an instance of distributed star formation , while ophiuchus is a more clustered environment . these two clouds therefore offer a global view of the early stages of planet formation among solar - type and lower - mass stars . i first address the question of the presence of dust in the planet - forming region , namely the innermost few au around each component , within binary systems . to probe the presence of an optically thick dusty inner disk , i used near- to mid - infrared colors . i selected the following standard thresholds to conclude that a circumstellar disk is present : @xmath4-[8.0 ] \ge 0.8$]mag , @xmath5mag , @xmath6mag , @xmath7 ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? about 80% of the pms binaries considered here have _ spitzer_/irac colors , which are used whenever available . @xcite have demonstrated that tighter binaries have a much lower probability of hosting circumstellar dust . the same effect is observed here in a somewhat smaller sample . the median separation of binaries with an inner disk in this sample is about 100au , whereas that of disk - free binaries is 40au . the simplest interpretation of this trend is that disks in tight binaries are dissipated much faster than in wide systems ( * ? ? ? * kraus et al . , in prep . ) . to extend upon this previous analysis , i used the two - sided fischer exact test to determine the probability that wide and tight binaries have a different proportion of diskless systems , using a sliding threshold to split the sample . as shown in figure[fig : proba ] , the difference is significant at the 2@xmath1 level or higher for a wide range of threshold separations . in particular , this analysis reveals that _ the observed reduced disk lifetime in binaries only applies to systems that are tighter than about 100au_. on the other hand , there is no statistical difference between binaries wider than 100au and single stars . while near- and mid - infrared emission best traces the presence of dust within a few au of star , only long - wavelength flux measurements can probe the total dust mass of protoplanetary disks ( e.g. , * ? ? ? * ) . from the sample defined above , i selected those objects which show evidence of an optically thick inner disk ( as defined above ) and have been observed in the ( sub)millimeter . the median separation in this subsample of 44 binaries is 130au . while the 850@xmath2 m survey of ophiuchus is not yet as complete as that of taurus , the existing 1.3 mm observations of pms stars are generally less sensitive to cold dust . since using both wavelengths yield similar conclusions but with lower significance for the 1.3 mm one , i focus here on 850@xmath2 m measurements . as has long been known , tight binaries have a different distribution of submillimeter fluxes than wide ones , with a much lower median flux ( 13mjy vs 50mjy at 850@xmath2 m using a 100au separation threshold ) and only very few high - flux systems @xcite . i compared the distributions of 850@xmath2 m fluxes for tight and wide binaries defined by the same sliding threshold as above using the conservative survival analysis peto - pentrice generalized wilcoxon test to account for upper limits . i find that wide and tight binaries are different at the 2@xmath1 level or higher if the separation threshold is in the 75 - 300au range ( see figure[fig : proba ] ) . i therefore conclude that _ binaries with a projected separation smaller than 300au have a substantially reduced submillimeter flux_. on the other hand , the distribution of 850@xmath2 m fluxes for wide binaries is indistinguishable from that of single stars . in past studies , it has been assumed that a reduced ( sub)millimeter flux necessarily implies a reduced total dust mass independently of the disk properties ( for instance , see the prescription used by * ? ? ? while this is true in general , it is unclear whether this assumption is valid for severely truncated disks for which optical depth effects may become important . the model constructed by @xcite seems to support this hypothesis , but these authors assumed that tight binaries are always surrounded by a massive circumbinary structure , which we now know is rare . to revisit this issue , i have computed a grid of radiative transfer models using the mcfost code @xcite to compute the 850@xmath2 m flux of a disk with a typical @xmath8 surface density profile , an 0.1au inner radius and a flaring power law @xmath9 . emission from the central star is modeled as a 4000k , 2@xmath10 photosphere and a distance of 140pc is assumed . the dust is assumed to be made of astronomical silicates with a @xmath11 power law size distribution ranging from 0.03@xmath2 m to 1 mm . the only variables in the model are the disk outer radius , @xmath12 , and the total dust mass , @xmath13 . figure[fig : diskmass ] demonstrates that the proportionality between total dust mass and submillimeter flux observed for large disks breaks down for @xmath14au as the disk becomes optically thick to its own emission . disk truncation by an outer stellar component is dependent on the orbital parameters and mass ratio of the binary system @xcite . it is therefore not possible to uniquely associate a binary separation with a tidally - set value of @xmath12 . the ratio between these quantities is typically in the broad 2.55 range . systems whose separation is less than 100au are therefore expected to possess disks whose outer radius is 40au or less . in this configuration , total disk masses of at least @xmath15 are necessary to produce 850@xmath2 m fluxes as low as @xmath1630mjy . in the sample studied here , about a third ( 6 out of 19 ) of all binaries that are tighter than 100au and possess an inner disk have an 850@xmath2 m flux that is higher than 30mjy . therefore , _ a significant fraction of the circumstellar disks in tight binaries ( @xmath17au ) are massive enough to potentially form gas giant planets , despite their much lower ( sub)millimeter fluxes . _ les us turn our attention to mature planetary systems . as of this writing , there are 38 exoplanets that are in a system with at least 2 stellar components ( using 1400au as the upper limit for binary separation ) , including 5 systems with a stellar companions within 25au ( * ? ? ? * and references therein ) . most of these planet - bearing stars are of solar type . the overall detection rates of gas giant planets in binary systems and in single stars are undistinguishable @xcite . there is marginal evidence that planets in binaries tighter than @xmath0au may be somewhat less frequent than one would assume based on the frequency of planets in wider binaries ( by 6.0@xmath182.7% , * ? ? ? however , the small sample size , adverse selection biases and incompleteness of current multiplicity surveys are such that it is premature to reach definitive conclusions . in any case , we can use this sample to test whether the separation of the stellar binary has any influence on planet properties . despite an earlier claim for a distinct period - mass distribution @xcite , @xcite have shown that there is essentially no difference in the properties of planetary systems around single stars and in binary systems . however , a previously unrecognized trend is evident in figure[fig : planets ] . while planets covering two orders of magnitude in mass can be found in wide binaries ( as around single stars ) , systems tighter than @xmath0au appear to host only high - mass , @xmath19 , planets . to quantify this effect , i used the two - sided fischer exact test to determine whether close and wide binaries ( with the usual sliding threshold ) have different proportion of high- and low - mass planets . i used 1.6@xmath20 , the median for all planets known to date , to separate low- from high - mass planets . figure[fig : proba ] confirms that _ binary systems tighter than about 100au produce a distribution of planets that is strongly biased towards the highest masses . _ this conclusion is significant at the 3@xmath1 level . it is important to test whether this trend is not a mere consequence of a selection bias , as a close stellar companion can alter the detectability of a planet - induced radial velocity signal . to evaluate this possibility , i build on the `` uniform detectability '' sample defined by @xcite which contains all stars for which close - in planets as low mass as 0.3@xmath20 ( well below the apparent cut - off in mass for planets in tight binaries ) , as well as 1@xmath20 planets on a 4yr orbit , could be detected . in the current sample of binary planet hosts , the proportions of stars that belong to the uniform detectability sample among binaries tighter and wider than 100au are indistinguishable ( 5/9 and 24/40 , respectively ) . i therefore conclude that the trend discussed above is unlikely to be the consequence of a selection bias or of observational limitations . an indirect signpost of planet formation is the debris disk phenomenon . in these systems , small dust grains are produced via the collisions of large solid bodies @xcite . @xcite observed 69 a- and f - type known binaries with _ spitzer _ and found debris disks in systems spanning 6 decades in separation . they further suggested that intermediate separation ( 330au ) binaries are substantially less likely to host a debris disks than either tighter or wider systems , although the formal significance of this difference is marginal at best . no such trend was found by @xcite , who included 24 a- through m - type binaries in their own _ spitzer _ survey . this latter survey focused on targets that are more similar in mass to exoplanet hosts and the pms population discussed in the previous section . i used the two - sided fischer test to determine whether the occurrence of debris disks is indeed different in tight and wide binaries , using the same sliding threshold as above ( see figure[fig : proba ] ) . there is no significant difference for any value of the threshold in the sample from @xcite , nor in a combined sample that also includes systems from @xcite . the combined sample contains 52 binaries in the 51400au range , with a median separation of 50au , an increase of 15 sources from the sole sample of @xcite . in addition , the occurrence rates of debris disks in binary systems and single stars are very similar @xcite . in other words , _ any 0.52@xmath3 star , irrespective of the presence of a companion ( within the 51400au range studied here ) , may experience the early phases of planet formation up to the planetesimal stage_. this analysis has revealed a clear dichotomy between tight and wide binaries . systems with separation @xmath21au are indistinguishable from single stars as far as the initial conditions and end product of planet formation are concerned . the only caveat to this statement is the possibility of mild disk truncation in 100 - 300au systems , but most disks in these systems retain a mass reservoir that is sufficient to build up gas giant planets . on the other hand , planet formation in binaries with separations @xmath22au is characterized by a much shorter clearing timescale for the protoplanetary disks and a strong bias towards high - mass planets . despite these differences , planetesimals and mature planetary systems appear to form at roughly the same frequency as around other stars . furthermore , while protoplanetary disks are more compact in tight systems because of truncation , a significant fraction of them possess large mass reservoirs ( at least several times @xmath20 ) . taken together , _ these results suggest that planet formation in binaries tighter than 100au proceeds through a different , but not much less frequent , mechanism compared to wide binaries and single stars . _ the shorter disk lifetime in tight binaries makes it extremely difficult to form gas giant planets through the core accretion model , especially if the final planets are particularly massive . rather , this combination of observed trends supports an abrupt process to form planets in tight binaries , such as the disk fragmentation model . indeed , this mechanism can be extremely efficient in the case of a compact , massive protoplanetary disk which is naturally prone to gravitational instability . furthermore , gravitational perturbations induced by a close stellar companion can trigger the instability even though the disk itself is not unstable to its own gravity @xcite . on the other hand , considering the long survival timescale and slim chances of gravitational instabilities , disks located within wide binaries and around single stars are good candidates to form planets via the core accretion model in their inner regions ( e.g. , * ? ? ? while a violent process is most likely responsible for the formation of planets in tight binaries , it is however unclear whether all planets in wide binaries form through a single mechanism . indeed , it is also conceivable that high - mass planets ( @xmath23 ) mostly form via disk fragmentation , while lower mass planets are preferentially the result of core accretion . this scenario would naturally alleviate the difficulty of the core accretion model to form the highest - mass planets in less than a few myr . this hypothesis has the additional advantage that it could also apply to tight binaries . indeed , since a stellar companion located within less than 100au dramatically shortens the disk lifetime , core accretion is essentially prevented from occurring , accounting for the absence of low - mass ( @xmath24 ) gas giant planets in tight binaries . planetesimals can presumably form in either scenario , accounting for the observations regarding the debris disks phenomenon . in summary , it remains to be determined whether the trends discussed here indicate an actual dichotomy between the main planet formation theories or a mere change of the relative importance of the two models as a function of the location of the stellar companion . improving the statistical significance of the various trends discussed here and determining the exact properties of disks within tight pms binaries will help shed further light these two possibilities . i am grateful to silvia alencar and jane gregorio - hetem for organizing and inviting me to `` special session 7 '' at the iau 27th general assembly held in rio de janeiro , where this work was first presented , as well as to anne eggenberger , deepak raghavan , david rodriguez and peter plavchan for invaluable input regarding exoplanets and debris disks . the work presented here has been funded in part by the agence nationale de la recherche through contract anr-07-blan-0221 . andrews , s. m. , & williams , j. p. 2005 , , 631 , 1134 andrews , s. m. , & williams , j. p. 2007 , , 671 , 1800 artymowicz , p. , & lubow , s. h. 1994 , , 421 , 651 beckwith , s. v. w. , sargent , a. i. , chini , r. s. , & guesten , r. 1990 , , 99 , 924 boley , a. c. 2009 , , 695 , l53 bonavita , m. , & desidera , s. 2007 , , 468 , 721 bontemps , s. , et al . 2001 , , 372 , 173 boss , a. p. 2006 , , 641 , 1148 cieza , l. a. , et al . 2009 , , 696 , l84 chauvin , g. , lagrange , a .- m . , udry , s. , fusco , t. , galland , f. , naef , d. , beuzit , j .- l . , & mayor , m. 2006 , , 456 , 1165 duchne , g. , et al . , 2010 , apj , submitted duchne , g. , delgado - 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in this article , i examine several observational trends regarding protoplanetary disks , debris disks and exoplanets in binary systems in an attempt to constrain the physical mechanisms of planet formation in such a context . binaries wider than about 100au are indistinguishable from single stars in all aspects . binaries in the 5100au range , on the other hand , are associated with shorter - lived but ( at least in some cases ) equally massive disks . furthermore , they form planetesimals and mature planetary systems at a similar rate as wider binaries and single stars , albeit with the peculiarity that they predominantly produce high - mass planets . i posit that the location of a stellar companion influences the relative importance of the core accretion and disk fragmentation planet formation processes , with the latter mechanism being predominant in binaries tighter than 100au .

You are an expert at summarizing long articles. Proceed to summarize the following text: organic charge - transfer ( ct ) crystals made up by @xmath1 electron - donor ( d ) and electron acceptor ( a ) molecules often exhibit a typical stack structure , with d and a molecules alternating along one direction.@xcite the quasi - one - dimensional electronic structure is stabilized by the ct interaction between d and a , so that the ground state average charge on the molecular sites , or degree of ionicity , @xmath2 , assumes values between 0 and 1 . crystals characterized by @xmath3 0.5 are _ conventionally _ classified as quasi - neutral ( n ) , as opposed to the quasi - ionic ( i ) ones , with @xmath4 0.5 . as discussed for the prototypical system of tetrathiafulvalene - chloranil ( ttf - ca),@xcite a few ct salts have n - i and peierls transition , in which @xmath2 changes rapidly and the regular stack dimerizes , yielding a potentially ferroelectric ground state.@xcite n - i transitions are valence instabilities implying a _ collective _ ct between d and a sites , and as such are accompanied by many intriguing phenomena , such as dielectric constant anomalies , current - induced resistance switching , relaxor ferroelectricity , and so on.@xcite the isostructural series formed by 4,4-dimethyltetrathiafulvalene ( dmttf ) with substituted cas , in which one or more chlorine atom is replaced by a bromine atom , is particularly interesting . in this case , in fact , the transition temperature and related anomalies can be lowered towards zero by chemical or physical pressure , attaining the conditions of a quantum phase transition.@xcite albeit several aspects of the n - i transition in br substituted dmttf - ca family are worth further studies , the motivation of the present work is far more limited , as we want first of all clarify the mechanism of the transition in the pristine compound , dmttf - ca . despite intensive studies,@xcite the transition still presents controversial aspects . through visible reflectance spectra of single crystals and absorption spectra of the powders , aoki@xcite suggested that by lowering the temperature below 65 k , dmttf - ca does not undergo a full n - i transition , but forms a phase in which both n ( @xmath5 ) and i ( @xmath6 ) species are present . the structural investigation as a function of temperature@xcite put in evidence a fundamental aspect of the transition , only implicit in aoki s work:@xcite at 65 k the unit cell doubles along the _ c _ axis ( _ a _ is the stack axis ) . the order parameter of the transition , which is second - order , is the cell doubling coupled with the dimerization.@xcite so above 65 k the cell contains one stack , and at 40 k contains two stacks , both dimerized , and inequivalent ( space group @xmath7 ) . from the bond distances , @xmath2 is estimated at 0.3 and 0.7 - 0.8 for the two stacks , respectively.@xcite in this view , and considering that the two stacks are dimerized in anti - phase , at low temperature dmttf - ca has a _ ferrielectric _ ground state . however , the above scenario has been questioned.@xcite polarized single crystal infrared ( ir ) reflectance measurements suggests that n and i stacks do not cohexist . only one ionicity is observed , changing continuously from about 0.25 at room temperature to about 0.48 at 10 k , the maximum slope in the @xmath8 occurring around 65 k. the crystal structure at 14 k indicates a @xmath9 space group , with two equivalent , dimerized stacks in the unit cell , and _ anti - ferroelectric _ ground state.@xcite according to this picture , the mechanism of dmttf - ca phase transition is very similar to the other n - i transitions.@xcite the madelung energy change yields an appreciable change of @xmath2 ( about 0.1 ) within a few degrees of temperature , accompanied by a stack dimerization . the cell doubling appears to be a secondary aspect , whereas the most important feature is the continuous variation of @xmath2 , as opposed for instance to the discontinuous , first order transition of ttf - ca.@xcite some questions remain however unanswered in the above picture.@xcite the transition displays a continuous ionicity change with @xmath10 , and consequently one would expect huge anomalies at the transition , whereas for instance the dielectric constant increase at @xmath11 is less than in the case of ttf - ca.@xcite furthermore , what is the driving force of the transition ? in ttf - ca , the n - i transition is attributed to the increase of madelung energy by the lattice contraction.@xcite if it is so also for dmttf - ca , what is the role of cell doubling ? finally , although @xmath7 and @xmath9 space groups are sometimes difficult to disentangle by x - ray diffraction , the issue of the different published structures is not solved , both exhibiting good confidence factors in the refinement process @xcite in order to clarify these open questions , and to understand the mechanism of the phase transition in dmttf - ca , we have decided to collect and re - analyze complete polarized ir and raman spectra of dmttf - ca single crystals , along the same lines followed for ttf - ca.@xcite indeed , a careful analysis can give information about @xmath2 , stack dimerization , and the peierls mode(s ) inducing it . vibrational spectra give information about the _ local _ structure , and from this point of view are complementary to the x - ray analysis , which probes long range order . we shall show that dmttf - ca transition can hardly be classified as a n - i transition , the most important aspect being the stack dimerization and cell doubling . we shall also offer some clues about the origin of the discrepancies in the two x - ray determinations.@xcite dmttf - ca single crystals have been prepared as previously described.@xcite the ir spectra ( 600 - 8000 ) have obtained with a bruker ifs66 ftir spectrometer , equipped with a590 microscope . raman spectra have been recorded with a renishaw 1000 micro - spectrometer . the excitation of raman has been achieved with a lexel krypton laser ( @xmath12 = 647.1 nm ) , backscattering geometry , with less than 1 mw power to avoid sample heating . a pre - monochromator has been used for the low - frequency spectra ( below 200 ) . for the high frequency raman spectra we report only the spectra obtained for incident and scattered light both polarized perpendicularly to the stack axis , ( @xmath13 ) in the conventional notation . in this arrangment the in - plane molecular modes , notably the totally symmetric ones , are more clearly visible . the spectral resolution of ir and raman spectra is 2 . temperatures down to 10 k have been reached with a ars closed - circle cryostat , fitted under the ir and raman microscopes . the temperature reading on the cold finger has been tested and considered accurate to @xmath142 k for the raman and ir reflectance measurement , where silver paste has been used to glue the sample to the cold finger . for ir absorption the temperature reading is far less accurate , due to the unperfect thermal contact between the sample and the kbr window on the cold finger . temperature reading corrections have been applied based on the comparison with the reflectivity data . dmttf - ca reflectivity has been normalized to that of an al mirror , without further corrections . therefore the reflectance values are not absolute , and relative values can be compared with confidence only within each low - temperature run . we consider reflectance values of the spectra below 20 k not reliable in any case , because the deposition of an unknown contaminant on the dmttf - ca surface introduces high noise above @xmath0 2000 . the first question we address is that of the ionicity as a function of temperature . to such aim , we have collected both ir reflectance and absorbance spectra , with polarization perpendicular to the stack axis . the two types of spectra allow us to ascertain whether probing the surface or the bulk yields the same result . unfortunately , we were unable to obtain crystals sufficiently thin to avoid saturation of the most intense absorption bands , so the information provided by the two types of spectra are complementary . [ fig : irperp ] shows some examples of spectra as a function of temperature in the frequency range 1500 - 1700 . the two structures at 1649 and 1539 ( at 150 k ) are assigned to the @xmath15 and @xmath16 modes of the ca moiety , corresponding to the c = o and c = c antisymmetric stretching vibrations , respectively.@xcite the c = o stretching mode is the most sensitive to the molecular charge , so it has been almost invariably used to estimate the ionicity . however , recent investigations on ca and ca@xmath17 molecular vibrations have shown that also the c = c mode should be a good @xmath2 indicator.@xcite therefore we shall use c = o @xmath15 as a primary @xmath2 indicator , and c = c @xmath16 mode as secondary , internal consistency probe . as the c = o stretching mode saturates in absorption ( fig . [ fig : irperp ] , bottom panel ) , we have performed the usual kramers - kronig transformation of the reflectance spectra . from the frequency reading of the c = o @xmath15 mode we have estimated the ionicity by the usual relationship : @xmath18/\delta_{ion}$ ] where @xmath19 is the c = o stretching frequency of the neutral molecule and @xmath20 is the ionization frequency shift.@xcite the resulting @xmath8 is reported in the top panel of fig . [ fig : ionint ] . the top panel of fig . [ fig : ionint ] is rather similar to the corresponding one of ref . . indeed , our spectra ( fig . [ fig : irperp ] ) do not show the onset of a strong band around 1580 below 65 k , that according to aoki _ _ et al.__@xcite signals the presence of @xmath21 species . following ref . , we attribute the band around 1580 to an activated @xmath22 mode , present in the spectra polarized parallel to the stack ( see section [ sec : dimerization ] ) . however , our data show two significant differences compared with horiuchi _ results.@xcite first of all , our @xmath8 curve is consistently shifted downward by about 0.05 @xmath2 units with respect to the corresponding one of ref . . as a consequence , the maximum ionicity at the lowest temperature is well below the n - i borderline , namely 0.43 instead of 0.48 . we believe the discrepancy is due to the extrapolation involved in the kramers - kronig transformation from reflectance to conductivity . we have indeed verified that different extrapolation procedures may yield different frequencies , and we have chosen extrapolation parameters giving conductivity spectra with frequencies matching those read in absorption ( bottom panel of fig . [ fig : irperp ] ) . our datum is also in agreement with what reported by aoki,@xcite who assigned a @xmath2 value of @xmath23 to the n species at 20 k ( although , as stated above , we do not see the insurgence of bands due to i species ) . the second important difference of our results compared with those of horiuchi _ et al . _ @xcite is that just below the phase transition temperature , between 62 and 54 k , the c = o stretching mode shows a clear doublet structure ( fig . [ fig : irperp ] , top panel ) , suggesting the presence of two slightly differently charged molecular species . the indication is confirmed by the band due to the c = c stretching mode , which also shows a doublet structure , clearly seen in absorption ( bottom panel of fig . [ fig : irperp ] ) . the frequency ( and ionicity ) difference is small , but clearly visible irrespectively of the direction of temperature change , and reproducible in different runs . actually , a hint of a doublet structure is visible also in ref . spectra , but it was interpreted as a band broadening . in fig . [ fig : irperp ] the dashed area indicates the temperature interval of this coexistence . . middle panel : normalized intensity of ir vibronic bands , connected to the stack dimerization amplitude . bottom panel : intensity of x - ray reflections signaling the cell doubling ( from ref . ) . the vertical dashed line marks the critical temperature @xmath24 65 k. ] to summarize , our results present a valence instability scenario different from both the previously reported ones.@xcite the ionicity change appears to be continuous across the phase transition . the crystal remains neutral ( @xmath25 at 20 k ) , therefore excluding the simple term of n - i transition : it is better to refer to it as a valence instability . finally , in a temperature interval of less than 10 k below 65 k there is coexistence of two species with slightly different molecular ionicity , both on the neutral side , with @xmath26 0.36 and 0.38 . it is well known that ir spectra polarized along the stack are sensible to the symmetry breaking associated with stack distortions . in fact , the loss of inversion center on the molecular units makes the raman - active totally - symmetric ( @xmath22 ) molecular modes also ir active , with huge intensity due to their coupling with the ct electronic transition ( ir `` vibronic bands'').@xcite in addition , it has been recently shown that the ir spectra polarized parallel to the stack , associated with raman , also yield information about pre - transitional phenomena , like the softening of the phonons inducing the stack distortion.@xcite to investigate these aspects of the dmttf - ca phase transition , we have collected the ir reflection spectra polarized parallel to the stack axis . [ fig : reflpar ] shows some examples of the spectra at different temperatures across the phase transition . [ fig : reflpar ] shows the insurgence of strong features below @xmath11 . the spectra are identical to those of ref . , and we follow a similar analysis . we focus the attention on the two main features around 1400 and 980 , associated with the @xmath27 mode of the ttf skeleton in the dmttf moiety , and to the @xmath28 mode of ca , respectively.@xcite their intensity @xmath29 as a function of @xmath10 is extracted by fitting the reflectance spectra with a drude - lorentz oscillator model , where the dielectric constant is given by : @xmath30 in this equation @xmath31 is the high - frequency dielectric constant , and @xmath32 is the line width . now , @xmath29 is related to @xmath33 , the dimerization amplitude.@xcite the relationship is not of direct proportionality , but in any case @xmath29 gives an indication of the increase of the dimerization amplitude at the transition.@xcite the @xmath34 for the two modes , normalized at the lowest temperature value ( 20 k ) , are reported in the middle panel of fig . [ fig : ionint ] for temperatures below 65 k. indeed , around the critical temperature one can not disentangle the contribution to the intensity coming from the combination modes , as discussed in detail in the next section . in the same panel we report , for comparison , the normalized intensity relevant to the dmttf @xmath22 mode , as given in ref . for the same temperature range . the middle panel of fig . [ fig : ionint ] shows that the ir oscillator strengths , related to the stack dimerization amplitude @xmath33,@xcite display a behavior typical of an order parameter relevant to a second - order phase transition . it is instructive from this point of view to compare the present data with the intensity of the x - ray diffraction spots related to the cell doubling , as reported in ref . and shown in the bottom panel of fig . [ fig : ionint ] . the two sets of data exhibit the same behavior , demonstrating that the cell doubling , as detected by x - ray , and stack dimerization amplitude , as detected by ir , occur at the same time , representing two inseparable aspects of dmttf - ca phase transition . on the other hand , the comparison of the three panels in fig . [ fig : ionint ] puts in evidence that the valence instability actually _ follows _ the structural modification . dimerization and cell doubling start at @xmath24 65 k , whereas the rapid increase in @xmath2 occurs slightly below , and implies the simultaneous presence of species with two slightly different ionicities in a @xmath10 interval of about 10 degrees . if dmttf - ca phase transition is displacive , it should imply the occurrence of soft phonon(s ) yielding the stack dimerization and the cell doubling . in a simplified but effective view , we can think of the phase transition as due to just one critical phonon , with wavevector @xmath35 . such a phonon belongs to a phonon branch that corresponds to stack dimerization along the @xmath36 crystal axis , and at the zone - center is optically active . the driving force of the transition is then provided by the peierls mechanism , which couples the zone - center dimerization mode , i.e. , the peierls mode , with the ct electronic structure.@xcite the electron - phonon causes the softening of the peierls mode , eventually laeding to stack dimerization . however , in the proximity of the phase transition , where interstack interactions are more effective,@xcite the peierls mode evolves to a stack dimerization out - of - phase in nearest - neighbors cells , when it softens yielding the cell doubling along the crystallographic direction @xmath37 . of course , in the complicated phonon structure of a molecular crystal like dmttf - ca , the peierls mode may result from the superposition ( mixing ) of several phonons , all directed along the stack . a spectroscopic investigation of dmttf - ca , along the lines already developed for ttf - ca,@xcite should yield the identification of these phonons or of the resulting `` effective '' peierls mode . phonons coupled to the ct electrons along the chain are most likely inter - molecular , or lattice , phonons.@xcite we start by classifying the lattice phonons and their raman and ir activity by adopting the rigid molecule approximation . this approximation is known to be not fully valid for ttf - ca,@xcite but is the only reasonable starting point in the lack of explicit calculations of dmttf - ca phonon dynamics . then , in the high temperature ( ht ) phase ( @xmath38)@xcite we expect 9 opticallly active lattice modes , @xmath39 . the raman active @xmath40 modes can be described as molecular librations ( @xmath41 ) , and are decoupled from the ct electrons . coupling is instead possible for the ir active @xmath42 phonons , which indeed correspond to translations ( @xmath43 ) . there are no symmetry constraint about the direction of molecular displacements , so we may have some component of all the three @xmath42 phonons along the stack axis , contributing to the peierls mode . in the low - temperature ( lt ) phase it is not clear if the two stacks inside the unit cell are inequivalent , with space group @xmath7,@xcite or equivalent , with space group @xmath9.@xcite since in any case the inequivalence is small , for the spectral predictions we find more convenient to use the centro - symmetric description . the center of inversion is between the two stacks in the unit cell , then we expect 21 optically active modes , @xmath44 and @xmath45 . therefore , the phonons modulating the ct integral are ir active in both ht and lt phases , whereas the cell doubling in the lt phase makes raman active 6 translational phonons , which correspond to the coupled in - phase displacements of the two chains . direct investigation of peierls coupled modes in the far - ir ( 5 - 200 ) is not an easy task.@xcite however , in the case of ttf - ca it has been shown@xcite that useful information can be obtained from the comparison between raman and ir spectra polarized parallel to the stack , in the frequency region of molecular vibrations . we have then performed the kramers - kronig transformation of the reflectivity data of section [ sec : dimerization ] ( fig . [ fig : reflpar ] ) , obtaining the optical conductivity spectra which are compared with raman in fig . [ fig : ramanir ] . and ca @xmath27 bands and the corresponding ir sidebands of fig . [ fig : ramanir ] as a function of temperature . bottom panel : temperature evolution of the most intense low - frequency raman bands . the vertical dashed line marks the critical temperature . ] we again focus attention on the @xmath28 mode of the ttf skeleton in the dmttf moiety and to the @xmath28 mode of ca , which correspond to the two most prominent raman bands of fig . [ fig : ramanir ] , located around 1400 and 980 , respectively.@xcite the ir spectra above 80 k exhibit pairs of absorptions ( `` side - bands '' ) , symmetrically located above and below the just mentioned raman bands . the side - bands are quite naturally interpreted as sum and difference combination bands between the corresponding @xmath22 mode and a lattice phonon.@xcite by lowering temperature the side - bands approach each other , and around 80 k they start to coalesce and to overlap to the central raman band . on the other hand , below the transition temperature ( for instance , 60 k in fig . [ fig : ramanir ] ) there is coincidence between raman and ir bands . as discussed in section [ sec : dimerization ] , both are indeed due to the same @xmath22 molecular vibration , active in both type of spectra due to the symmetry breaking connected to the stack dimerization . analysis of the side - bands therefore gives information on the peierls mode in the ht phase.@xcite in the top panel of fig . [ fig : peierls ] the frequency difference between the dmttf @xmath27 raman band and the corresponding ir side - bands is plotted as a function of temperature . we also plot the frequency semi - difference between the side - bands associated with both dmttf and ca @xmath27 modes . [ fig : peierls ] shows that the data points coincide within experimental error , supporting the idea that _ the same _ lattice mode is involved in the combination , and clearly indicates a soft mode behaviour . this softening suggests that this lattice phonon is indeed the peierls mode , or to be precise , the `` effective '' peierls mode , resulting from the superposition of several modes coupled to the ct . we can not follow the frequency evolution down to the transition temperature , since below 80 - 75 k it becomes impossible to separate the contribution of the two side - bands , letting aside the interference from the fluctuations occurring near the phase transition . polarization ; dashed line ( @xmath46 ) polarization ( see text ) . ] we now turn attention to dmttf - ca lt phase , where the peierls mode(s ) are active in raman in addition to ir . we have then measured the raman spectra in the low - frequency region in order to identify the possible soft phonons . low - frequency raman also gives indications about the cell doubling , given the difference in the number of phonons present in the two phases . an example of the low - frequency raman spectra above and below the phase transition is shown fig . [ fig : rlattice ] . we report the spectra with two polarizations . in one , both incident and scattered light are polarized perpendicularly to the stack axis ( @xmath13 spectra ) . in the other , the polarization of the incident light is rotated parallel to the stack axis , the polarization of the scattered light being kept perpendicular to it ( @xmath46 spectra ) . the list of the observed bands in both polarizations is reported in table i. the number of observed phonon modes is less than predicted by the selection rules , but in any case the phonons detected in the ht phase are about a half of those detected at lt , an obvious consequence of cell doubling . table i. dmttf - ca raman active lattice + modes in the lt and ht phases . [ cols="^,^,^,^,^",options="header " , ] the temperature dependence of the raman frequencies are shown in the bottom panel of fig . [ fig : peierls ] . one immediately notice the usual softening for all the modes as we increase the temperature , due to lattice expansion . however , in the lt phase the softening of some phonons is more pronounced close to the critical temperature . the frequency lowering is not as large as in the ht phase ( top panel of fig . [ fig : peierls ] ) , but is certainly present . the relative weakness of the effect compared to the ht phase can be explained considering that in the ht phase we observe the softening of an `` effective '' peierls mode , superposition of several phonons all coupled to the ct electrons . in the lt phase , on the other hand , the softening is distributed on several modes , and the phonon description and mixing changes as we approach the phase transition . the effect is clearly seen in fig . [ fig : peierls ] , where at about 40 k there is a case of avoided crossing of two phonons located around 70 . in addition , we have to keep in mind that the dmttf - ca transition is second order , but can not be considered a strictly one - dimensional peierls transition , as the transition implies a change in the number of stacks per unit cell . the actual phase transition mechanism is a complex one , as shown by the fact that just below @xmath11 there is coexistence of two slightly different degrees of ionicity on the molecular sites ( section [ sec : valence ] ) . this finding might be explained in terms of an inequivalence of the two stacks inside the unit cell in proximity of @xmath11 . the present work does not allow us to draw definitive conclusions about the equivalence / inequivalence of the two dmttf - ca stacks inside the unit cell of the lt phase ( anti - ferroelectric or ferrielectric arrangement ) . as just discussed , the presence of two slightly different degrees of ionicity may imply that just after the phase transition we have a temperature interval ( @xmath47 k ) in which the two stacks are inequivalent with the @xmath7 structure,@xcite , followed by a definitive structural rearrangement yielding to the 14 k @xmath9 structure.@xcite this picture would support the @xmath7 structural determination,@xcite collected at 40 k below our coexistence @xmath10 region , only assuming that the @xmath7 structure refers to a non - equilibrium phase . such `` frozening '' of a metastable phase may be a consequence of a too fast sample cooling , a case not uncommon in organic solid state , although most of the times refers to some disordered , glassy phase.@xcite at this point we wish to underline that inequivalence of the stacks does not necessarily imply an appreciably different degree of ionicity . dft calculations made for the @xmath7 structure at 40 k indeed found practically identical @xmath2 for the two stacks with different dimerization amplitudes.@xcite we may then have a scenario with two ( slightly ) inequivalent stacks , but practically identical @xmath2 . ir spectra polarized parallel to the stack ( fig . [ fig : reflpar ] ) of course can not disentangle dimerization amplitudes on different stacks . optical spectroscopy selection rules , on the other hand , are based on the factor group ( unit cell group ) , therefore reflecting the translational long - range order of the crystal.@xcite from this perspective , two findings are in favor of inequivalent stacks , down to at least 20 k. the first fact refers to the raman - ir coincidence observed for the @xmath22 molecular modes in the lt phase ( fig . [ fig : ramanir ] ) . if the two stacks are equivalent , and connected by an inversion center , each @xmath22 mode of one stack would be coupled in - phase and out - of - phase with the same mode on the other stack . the in - phase mode is raman active , and the out - of - phase one ir active , therefore we should not observe precise frequency coincidence , the difference being related to the strength of inter - stack interaction . unfortunately , our data are not conclusive in this respect , due to the frequency uncertainties associated with the kramers - kronig transformation.@xcite the other experimental observation that can be explained in terms of inequivalent stacks is the doubling of localized electronic transitions below 65 k.@xcite this experimental observation was the first one that induced aoki _ _ et al.__@xcite to suggest the coexistence of neutral and ionic species , but since then it has been almost forgotten . et al._,@xcite as well as the present measurements , exclude such coexistence , and the only explanation we can think of the doubling is in term of ordinary davydov splitting.@xcite however , the two components of the davydov splitting can be both optically active only in the lack of inversion center relating the two stacks , otherwise the _ gerade _ component is inactive . only the replica of structural measurements and/or of the refinement process starting from the two different hypothesis will definitely settle the question of equivalence - inequivalence of dmttf - ca stacks . on the other hand , this question is not particularly relevant as far as the mechanism of dmttf - ca phase transition is concerned . the present analysis departs from the previous ones@xcite only in some seemingly marginal details , but actually the resulting picture of the phase transition is completely different . first of all , we have ascertained that the phase transition implies only a limited change of @xmath2 , dmttf - ca remaining on the _ neutral _ side down to the lowest temperature . furthermore , the major charge rearrangement _ follows _ , by a few degrees k , the onset of cell doubling and stack dimerization . the latter finding can be well appreciated from fig . [ fig : ionint ] , where the inflection of the @xmath8 curve occurs around 61 k , rather than at 65 k for the symmetry breaking . close scrutiny of fig . 5 of ref . conveys the same information . therefore the transition can hardly be termed n - i , since cell doubling and stack dimerization clearly constitute the driving force of the transition . indeed , we can envision a scenario in which the dimerization and cell doubling lead to a better molecular packing , with an increase in the madelung energy and consequent small , continuous change in @xmath2 . the presence of slightly differently charged molecular species before the dimerization / cell doubling has reached completion ( fig . [ fig : ionint ] ) fits quite naturally into this picture . disentangling the contribution of the cell doubling from that of the stack dimerization is a useless endeavor . in any case , our measurements have clearly evidenced the presence of an effective soft mode along the chain ( fig . [ fig : peierls ] , top panel ) , so a peierls - like mechanism is certainly at work in the precursor regime of the phase transition . x - ray diffuse scattering also reveals the importance of electron - phonon coupling along the chain , and of one - dimensional correlations . it has been interpreted in terms of lattice relaxed exciton strings ( lr - ct ) rather than in terms of a soft mode . the phase transition is then regarded more as a disorder - order transition ( ordering of lr - ct exciton strings along and across the chains ) , and not as a displacive one , with progressive uniform softening of the peierls mode up to the final chain dimerization . it would be interesting to re - analyze the x - ray diffuse scattering data to examine whether and to what extent they are compatible with the soft - mode picture . the present results of course only evidence the soft - mode mechanism , and the presence of lr - ct exciton strings can not be excluded , in particular close to @xmath11 , in the region where our data can not be unambiguously interpreted . the lr - ct exciton string picture has been invoked mainly to account for the dielectric constant anomaly at @xmath11 , attributed to the progressive ordering of the a para - electric phase before reaching anti - ferroelectric ( or ferrielectric ) ordering.@xcite on the other hand , it has been shown that the peierls mechanism is also able to _ quantitatively _ explain the experimental increase of the dielectric constant at @xmath11 , interpreted as due to charge oscillations induced by the peierls mode.@xcite we underline in this respect that although the dimerization of the stack in the i phase has been often attributed to a spin - peierls mechanism,@xcite the electronic degrees of freedom are involved as well , particularly in proximity of the n - i borderline.@xcite in addition , the dimerization transition may occur also on the n side , provided the electron - phonon interaction is strong enough . this is just the present case , and correspondingly we have an increase of the dielectric constant less important than in the case of ttf - ca,@xcite as predicted by the calculations.@xcite in summary , the present interpretation stresses the importance of the lattice instability over that of charge instability in dmttf - ca and related compounds . we gratefully thank n. karl for providing the dmttf - ca crystals . the reflectivity data have been fitted by the freely available reffit program ( optics.unige.ch/alexey/reffitt.html ) . we thank a. painelli for many useful discussions . work in italy supported by the `` ministero dell universit e ricerca '' ( mur ) , through firb - rbne01p4jf and prin2004033197_002 . torrance , j.e . vasquez , j.j . mayerle , v.y . lee , phys . lett . * 46*,253 ( 1981 ) ; j.b . torrance , a. girlando , j.j . mayerle , j.i . crowley , v.y . lee , p. batail , s.j . lapaca , phys . lett . * 47 * , 1747 ( 1981 ) .

we report a detailed spectroscopic investigation of temperature - induced valence and structural instability of the mixed - stack organic charge - transfer ( ct ) crystal 4,4-dimethyltetrathiafulvalene - chloranil ( dmttf - ca ) . dmttf - ca is a derivative of tetrathiafulvalene - chloranil ( ttf - ca ) , the first ct crystal exhibiting the neutral - ionic transition by lowering temperature . we confirm that dmttf - ca undergoes a continuous variation of the ionicity on going from room temperature down to @xmath0 20 k , but remains on the neutral side throughout . the stack dimerization and cell doubling , occurring at 65 k , appear to be the driving forces of the transition and of the valence instability . in a small temperature interval just below the phase transition we detect the coexistence of molecular species with slightly different ionicities . the peierls mode(s ) precursors of the stack dimerization are identified .

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Proceed to summarize the following text: planetary nebulae ( pne ) are a key pathway in the evolution of low to intermediate mass stars , and their central stars are the immediate precursors of white dwarfs . studies of pn central stars ( herein cspne ) are motivated by : the desire to understand the origin of the rich variety of pn morphologies ; to establish the mass - loss process via fast winds driven by radiation pressure by spectral lines ; and to secure fundamental stellar parameters that can test post - agb stellar evolution models . time - series spectroscopy is an important diagnostic tool in developing our understanding of cspne . recently , far - uv and uv datasets have revealed signatures of large - scale wind structures and evidence for modulated temporal behaviour that may provide a handle on the central star rotation rates ( e.g. prinja et al . 2012a , 2012b ) . similarities between the wind properties of h - rich cspne and those of massive population i ob stars ( which also have line - driven winds ) suggest that instabilities in variable fast winds may result in shock heated gas which emits x - rays in the central cavities of pne ( e.g. guerrero 2006 ; kastner et al . 2012 ) . in the optical waveband , time - series data are requisite for establishing systematic radial velocity shifts in cspne absorption lines . de marco et al . ( 2004 ) have for example conducted a radial velocity survey of 11 cspne to provide constraints on the binary properties of the parent agb population and thus the extent to which binarity may play a causal role in shaping non - spherical nebulae . in this paper we present time - series optical spectra of the central star of ngc 2392 ( eskimo nebula ) . our study is motivated by several interesting characteristics , discrepancies and scenarios for this pn : ( i ) the central star of ngc 2392 exhibits high he , n and low c , o abundances suggesting that the photosphere has been processed ( mndez et al . a possible scenario is that the abnormal central star abundances are due to a common - envelope evolutionary phase thus implying a close binary companion ; ( ii ) danehkar et al . ( 2011 ) employ photoionization models of high excitation pn emission lines to argue that ngc 2392 has a hot white dwarf ( @xmath0 1 m@xmath2 ) companion ; ( iii ) detailed kinematic modelling of the ( eskimo ) nebula by garcia - diaz et al . ( 2012 ) supports a near - pole orientation , complex nebula morphology with multiple kinematic components , and an evolution path that may invoke a common - envelope binary ; ( iv ) the extended and point x - ray emission from ngc 2392 ( e.g. kastner et al . 2012 ) is not entirely consistent with the predicted thermal energy converted from the kinetic energy of the fast wind . additional coronal energy from a binary companion may explain the observed high x - ray temperatures . despite all the implications of the above studies , there is no definitive evidence so far of a binary nucleus in ngc 2392 , and the time - variable and geometric characteristics of its fast wind are not established . in this study we present the analysis of high - resolution optical time - series datasets secured over two epochs in 2006 and 2010 using the 3.6 m eso and canada - france - hawaii ( cfht ) observatories . our goal is to investigate for the first time @xmath0 hourly changes in the fast wind of ngc 2392 and fluctuations close to the surface of the central star . we characterise here evidence for evolving structure in the outflow and indications of radial velocity changes in deep - seated absorption lines . a log of the time - series spectra of the central star of ngc 2392 is given in table 1 . thirteen spectra were secured over 3 consecutive nights in 2010 march using the espadons echelle spectrograph ( manset & donati , 2003 ; donati et al . 1997 ) on the 3.6 m cfht at mauna kea , hawaii . the continuum signal - to - noise ratio ( s / n ) of an individual spectrum is @xmath0 100 for 30 min exposures , with a spectral resolution , r , @xmath0 68000 . the data were reduced using the standard cfht pipeline upena . the cfht data are complemented in this study by 18 spectra obtained at the eso la silla 3.6 m telescope using the harps echelle spectrograph ( mayor et al . , 2003 ) . the eso observations were carried out by us during 2006 march over 3 consecutive nights ( table 1 ) . typical individual harps spectra have s / n @xmath0 40 ( for 30 min exposures ) and r @xmath0 110,000 . the eso automatic online pipeline was used for homogeneous reduction . lll observatory & mjd range ( days ) & no . of spectra + eso harps & 53818.002 @xmath1 53818.109 & 6 + & 53818.995 @xmath1 53819.101 & 6 + & 53820.002 @xmath1 53820.108 & 6 + + cfht espadons & 55256.358 @xmath1 55256.447 & 5 + & 55257.271 @xmath1 55257.337 & 4 + & 55258.270 @xmath1 55258.336 & 4 + all the spectra were normalised by fitting a low - order polynomial through continuum windows . the line profiles discussed here have been corrected for a radial velocity of 82 km s@xmath3 , measured in weak absorption lines . the fundamental central star parameters adopted in this study are listed in table 2 . .ngc 2392 adopted central star parameters . [ cols="<,<,<",options="header " , ] there are no archival fuv or uv _ time - series _ datasets available for ngc 2392 that permit an investigation of systematic variability on time - scales comparable to the wind flow time ( i.e. @xmath0 hours ) . patriarchi & perinotto ( 1995 ) show sparse @xmath4 spectra separated by @xmath0 7 years which suggest some fluctuations in the civ and nv lines , and guerrero & de marco ( 2013 ) comment on the two _ fuse _ spectra available for the central star . the optical time - series data of the nuclei of ngc 2392 that we present here provide access to the fast wind via excited transitions arising from hei and heii . ( note that the h@xmath5 and h@xmath6 lines are heavily affected by nebular emission . ) the he lines are primarily the result of recombination from a higher ionization stage followed by a radiative de - excitation . since recombination is sensitive to the square of wind density ( @xmath7 ) , the optical lines form in the densest region of the wind ( in contrast to the uv lines ) . furthermore the hei @xmath85876 ( 2@xmath9p@xmath10 @xmath1 3@xmath9d ) transition is a potential exception if large populations are acquired in its lower level such that it effectively becomes the ground level and consequently behaves as a scattering or resonance line . we find that the stellar lines in the optical spectra of ngc 2392 are undoubtedly variable within @xmath0 hours , with peak - to - peak amplitude changes of @xmath0 5% in hei @xmath85876 and heii @xmath84686 . the corresponding equivalent width and standard deviations for the combined eso and cfht datasets are @xmath0 0.32 s.d . 0.12 and @xmath0 @xmath13.8 s.d . 0.6 . the line profile variability is quantified in fig . 1 where we employ the temporal variance spectrum ( tvs ) method ( e.g. fullerton et al . 1996 ) to estimate the statistical significance of the changes , having accounted for differences in the data quality at spectrum and pixel levels . the variability level is higher in heii @xmath84686 than hei @xmath85876 or hei @xmath84472 ; note that numerous other stellar lines are also weakly variable in the night - to - night data . significant variability extends to @xmath0 @xmath1180 km s@xmath3 ( i.e. @xmath0 0.45 @xmath11 ) in the eso ( 2006 ) and cfht ( 2010 ) spectra of heii @xmath84686 , but the redward extend of these changes is lower in 2010 compared to 2006 . for both epochs the hei @xmath85876 weak p cygni profile varies between @xmath0 @xmath1150 km s@xmath3 to 100 km s@xmath3 . an important feature of the tvs in fig . 1 is that it is _ double - peaked _ in the heii and hei lines for the eso and cfht datasets . typically the peaks in the tvs are separated by @xmath0 100 to 120 km s@xmath3 . the amplitude of the tvs peaks are not systematically stronger or weaker in either the blue or red sides . for line profile variability dominated by clumps we may expect small - scale structure to be distributed evenly over the whole inner wind thus presenting a broadly symmetrical single - peak tvs . one possibility for the double - peak tvs is that it results from radial velocity shifts perhaps due to a binary nature in ngc 2392 . we return to the issue of radial velocity changes in sect . the systematics of the heii @xmath84686 and hei @xmath85876 profile changes are examined further in figs . 2 and 3 where the data are displayed in two - dimensional velocity@xmath1time dynamic spectra ( images ) . the intensity levels of individual spectra are determined as _ differences _ with respect to the average spectrum for each observing night , and represented by a colour scale from minimum ( black ) to maximum ( white ) intensity cut level . the features seen in figs . 2 and 3 therefore represent pseudo - absorption and pseudo - emission features relative to the mean . the fast wind in ngc 2392 is undoubtedly variable on very short time - scales and can switch between an overall pseudo - emission to pseudo - absorption pattern in the dynamic spectra in less than 30 minutes ( i.e. time lengths comparable to the characteristic wind flow time , @xmath0 @xmath12 ) . the evidence from figs . 2 and 3 is that episodes of strong blue and red changes in heii @xmath84686 are mimicked in hei @xmath85876 at the same velocities . for example enhanced redward emission in heii @xmath84686 is accompanied by an increase in the ( weak ) p cygni emission component in hei @xmath85876 . a substantial low velocity blueward emission increase in heii @xmath84686 is ` matched ' by an increase in the p cygni absorption strength in hei @xmath85876 . the overall impression from our optical time - series is that the fast wind in ngc 2392 is stochastically variable over @xmath0 hours as opposed to revealing coherent modulated behaviour . we do not for example see signs of blueward and/or redward migrating features during each night , nor is there evidence for much slower evolving ( large - scale ) structures in the wind that persist over @xmath0 3 nights . 5411 , hei @xmath85876 and heii @xmath84686 line profiles taken @xmath0 1 day apart . the left- and right - hand panels show examples from the eso and cfht data , respectively . ] the overall line profile morphology can change in complex manners from night - to - night . we show in fig . 4 corresponding pairs of hei @xmath85876 , heii @xmath85411 and heii @xmath84686 spectra . each pair is separated by @xmath0 1 day . the left - hand panel in fig . 4 shows a case where as the total hei @xmath85876 absorption increases across all velocities , in the corresponding heii @xmath84686 pair the overall emission decreases . for this same pair , the changes in heii @xmath85411 are however clearly _ asymmetric _ toward redward velocities . in a further twist , the right - hand panels in fig . 4 show a case where the hei @xmath85876 line profile transforms from an absorption profile to a clear p cygni profile with redward emission . in this case the corresponding heii @xmath84686 pair exhibit almost no change in total equivalent width but the peak emission shifts blueward by @xmath0 30 km s@xmath3 . but , once more , the heii @xmath85411 profile changes are almost entirely at redward velocities . to gain further insights into the implications of these overall profile changes , we examined model predictions using the unified non - lte , line - blanketed model atmosphere code cmfgen ( e.g. hillier & miller 1998 ) . briefly , cmfgen solves the non - lte radiation transfer problem assuming a chemically homogeneous , spherically symmetric , steady - state outflow . each model is defined by the stellar radius , the luminosity , the mass - loss rate , the wind terminal velocity ( @xmath11 ) , the stellar mass and by the abundances of the species included in the calculations . the code does not solve for the hydrodynamical structure , hence the velocity field has to be defined using the output of a plane - parallel model ( tlusty in this case ; see hubeny & lanz , 1995 ) to define the pseudo - static photosphere , connected just below the sonic point to a beta - type velocity law to describe the wind regime . we calculated a substantial grid of cmfgen model atmosphere spectra and explored sequences of different mass - loss and clumping . the mass - loss rate has been incremented between @xmath0 7 @xmath13 10@xmath14 m@xmath2 yr@xmath3 to @xmath0 1.6 @xmath13 10@xmath15 m@xmath2 yr@xmath3 . we also adjusted the clumping factor , @xmath16 , which measures the over - density inside the clumps with respect to the average wind density . finally , the velocity at which optically thin clumping starts ( @xmath17 ) near the stellar surface was adjusted between 10 km s@xmath3 to 40 km s@xmath3 , using an exponential law defined by the parameters @xmath16 and @xmath17 . in each sequence all but one of the above parameters are kept fixed . the model sequences are shown in fig . 5 for heii @xmath84686 , heii @xmath85411 and hei @xmath85876 . comparisons between this grid with variable mass - loss rates and wind clumping sequences lead us to conclude that the absorption versus emission strength changes seen in the cfht and eso data in the left - hand panels in fig . 4 may qualitatively be explained by ( global ) mass - loss differences , but not the asymmetric behaviour evident in heii @xmath85411 . furthermore the relatively large emission line shift seen in the right - hand panels in fig . 4 _ can not _ be explained by line - synthesis model predictions with a spherically homogeneous wind . ordinarily the search for radial velocity shifts in symmetric photospheric absorption lines would be a relatively straightforward exercise in hot stars . trends in velocity changes may then betray for example the causal role of stellar binarity or multiplicity . the situation in ngc 2392 is more complex however for two key reasons ; ( i ) we have demonstrated in sect . 3 that the nucleus of ngc 2392 drives a highly variable fast wind , and the origin of these changes are deep - seated and close to the stellar surface . this means that a large selection of the he i , he ii and metal absorption lines in the optical spectra may be disturbed by the effects of a variable wind ; ( ii ) in their analysis of the 3-d and kinematic structure of the nebula , garcia - diaz et al . ( 2012 ) present evidence that the inner nebular shell has an almost pole - on orientation , such that the inclination angle with respect to the line - of - sight is only @xmath18 . assuming that the central star has the same orientation , the geometry is obviously not favourable for the detection of binary induced radial velocity motion of the nucleus . taking up this challenge nevertheless , we have examined weak , relatively symmetrical stellar absorption lines in ngc 2392 for evidence of radial velocity shifts . the cmfgen line - synthesis models ( sect . 3 ) suggest that most of the optical lines in ngc 2392 are potentially affected by mass - loss and wind clumping changing . our grid of models points to the niv @xmath86380.8 absorption line as a rare example of a primarily photospheric absorption line , suitable for a study of subtle radial velocity fluctuations . inspection of the time - series data suggests that the niv line _ does _ in fact shift in central velocity by @xmath0 10 to 15 km s@xmath3 in the eso and cfht datasets , over hourly timescales . pairs of niv profiles separated by @xmath0 2 hours are shown in fig . 6 where the central minimum and blue and red wings are shifted by @xmath0 15 km s@xmath3 . a particularly useful consistency check is provided by h@xmath19 , since the caiih @xmath83968.5 interstellar / circumsystem lines provide an excellent nearby fiducial for the accuracy of the wavelength scale close to h@xmath19 . we note in fig . 6 that the velocity behaviour seen in niv is systematically mimicked in h@xmath19 , thus firming confidence in the notion that there is a genuine radial velocity shift in ngc 2392 . the niv line profiles were fitted with gaussian model profiles with least squares to estimate the central absorption velocities . for a typical internal fitting error of @xmath20 1 km s@xmath3 , the mean and median absolute deviation ( mad ) of the eso and cfht data are 6.4 ; mad @xmath0 3.7 km s@xmath3 and 3.9 ; mad @xmath0 3.4 km s@xmath3 . the time - series data are unfortunately not intensive enough over a sufficiently extensive time - scale to allow for a confident search for periodic behaviour in the recorded radial velocities . with this caveat , fourier power spectra were calculated independently for the cfht and harps measurements and the results are shown in fig . 7 ( upper panel ) . there are undoubtedly spurious features and aliases throughout the power spectrum , but it is interesting to note that in both datasets the maximum power occurs at very similar frequencies i.e. @xmath0 8.1 d@xmath3 . 6380.8 line profiles secured @xmath0 2 hours apart in cfht and eso spectra . simultaneous velocity changes are also evident h@xmath19 , where the sharp interstellar feature of caiih @xmath83968.5 is an excellent wavelength scale fiducial . ] for a slightly more sophisticated treatment we also employed the clean algorithm ( roberts et al . 1987 ) which is more suited to the temporal analysis of unequally spaced finite data samples . using a gain of 0.5 and 100 iterations , the spectral window function was iteratively subtracted in the fourier domain from the raw power spectrum shown in the upper panel in fig . the resultant ` cleaned ' power spectrum is shown in the bottom panel in fig . 7 , where the strongest peak frequency is at @xmath0 8.13 d@xmath3 , _ for both datasets_. we estimate an uncertainty of at least @xmath0 10% in this frequency based on the half - width at half - maximum of the main peak in the window function . these results provide tentative evidence for a period of @xmath0 0.123-day in the niv radial velocity changes , with a semi - amplitude of @xmath0 10 km s@xmath3 . figure 8 shows the niv central velocities phased on 0.123-day . ( in each case phase 0 is arbitrarily set to the timing of the first observation . ) the results in fig . 8 suggest some coherency on the 0.123-day period _ independently _ for the eso ( 2006 ) and cfht ( 2010 ) datasets , though there is relatively larger scatter ( in the very small velocity displacements ) at some phases . ultimately a more intensive , very high - resolution time - series dataset of the central star is needed to confirm the significance of this signal . 6380.8 stellar radial velocity changes in ngc 2392 . the cfht ( solid black ) and eso ( dotted red ) power spectra are shown in the upper panels , and corresponding cleaned versions are shown in the lower panel . in _ both _ datasets a peak frequency of @xmath0 8.1 d@xmath3 is apparent . ( power is in arbitrary units . ) ] 6380.8 absorption line phased on the @xmath0 0.123-d period . the eso harps and cfht espadons data are separated by @xmath0 4 years . pairs of representative fitting error bars are shown in each panel . ] we have provided evidence and demonstrated that in two independently secured optical spectroscopic time - series separated by @xmath0 3 years , both datasets reveal for the central star of ngc 2392 ; ( i ) stochastic variations in the fast wind - formed recombination lines on timescales down to @xmath0 30 min . , ( ii ) changes in the overall morphology of the hei and heii line profiles that can not be accounted for by 1-d line - synthesis predictions for a spherically homogeneous wind , ( iii ) radial velocity shifts of semi - amplitude @xmath0 10 km s@xmath3 in niv @xmath86380.8 ( and tentatively in h@xmath19 ) that show maximum fourier power spectra signal at @xmath0 0.123-d in eso ( 2006 ) and cfht ( 2010 ) data . our detection of radial velocity motion of the central star in ngc 2392 is obviously tentative . cross - correlation with the majority of stellar absorption lines in the optical range is not fruitful since most of features are ` contaminated ' by the imprints of stellar wind variability which we have established here . in advance of more definitive optical spectroscopy and photometry being secured , we can only speculate on the potential binary components in ngc 2392 : for an assumed circular orbit and line - of - sight inclination @xmath0 @xmath21 ( e.g. garcia - diaz et al . 2012 ) ; semi - amplitude of 10 km s@xmath3 and period = 0.123-d ( figs . 8 and 9 ) ; assumed ( primary ) central star mass = 0.6 m@xmath2 , the implied dynamical mass of the secondary in circular keplerian orbit is @xmath0 0.1 m@xmath2 . such a low - mass companion would for example correspond to a late m - dwarf of @xmath22 @xmath0 2000 - 3000k . the morphologies of the fuv and uv lines are complex in ngc 2392 and provide additional signatures for an asymmetric geometry . the cmfgen line - synthesis models ( sect . 3 ) do not provide consistent matches to doppler widths , and absorption and emission strengths across all the uv ion stages observed in the fast wind . a selection of uv wind line profiles in ngc 2392 is presented in fig . 9 , ranging from ovi @xmath81031.9 and svi @xmath8944.5 , to pv @xmath81118.0 and nv @xmath81238.8 , and civ @xmath81548.2 and mgii @xmath82795.5 . ( the data have been retrieved from the _ fuse _ and _ iue _ archives . ) there are some key points to note in fig 9 : ( i ) all the wind lines get weaker with increasing outflow velocity . it may be that the wind plasma is shifting to a very high ionization state ( beyond ovi ) as it travels to larger radii . alternatively , the line shapes in fig . 9 could be an indication that the fast wind of ngc 2392 is moving out of the line - of - sight , somewhat as may be expected for a polar , high - latitude wind in an asymmetric geometry ; ( ii ) the low excitation mgii line is very narrow ( @xmath0 100 km s@xmath3 ; as is siiii @xmath81206.5 ) and consistent with a low - velocity equatorial wind ( see e.g. bjorkman et al . 1994 ; massa 1995 ) ; ( iii ) the presence of pv most likely indicates that the wind is optically very thick , since phosphorous has a low cosmic abundance and this line would otherwise not be so clearly detected . however civ and nv are weak at intermediate velocities ( @xmath0 200 to 300 km s@xmath3 ) . an optically very thick wind that causes weak absorption can arise in a scenario where the wind is not covering the entire stellar disk , as may be expected from a polar wind . we conclude that uv lines provide evidence for an asymmetric , two - component outflow in ngc 2392 , where high - speed high - ionization gas forms preferentially in the polar region . slower , low ionization material is then confined primarily to a cooler equatorial component of the outflow . 1031.9 , svi @xmath8944.5 , pv @xmath81118.0 , nv @xmath81238.8 , civ @xmath81548.2 , mgii @xmath82795.5 . based on observations obtained at the canada - france - hawaii telescope ( cfht ) which is operated by the national research council of canada , the institut national des sciences de lunivers of the centre national de la recherche scientique of france , and the university of hawaii , and on observations collected at the european southern observatory , la silla ( programme i d eso 076.d-0207(a ) . we thank derck massa for discussions about the fast wind of ngc 2392 . we acknowledge the helpful comments of the referee . bjorkman , j , e . , ignance , r. , tripp , t.m . , cassinelli , j.p . 1994 , apj , 435 , 416 danehkar , a. , frew , d.j . , parker , q.a . , de marco , o. 2011 , in interacting binaries to exoplanets : essential modeling tools , iau symp . 282 , eds . m. t. richards and i. hubeny , p. 470 donati , j .- f . , semel , m. , carter , b.d . , rees , d.e . , collier cameron , a. 1997 , mnras , 291 , 658 fullerton , a.w . , gies , d.r . , bolton , c.t 1996 , apjs , 103 , 475 garcia - diaz , m.t . , lpez , j.a . , steffen , w. , richer , m.g . 2012 , apj , 761 , 172 guerrero , m.a . , de marco , o. 2013 , a&a , 553 , 126 guerrero , m.a . 2006 , in planetary nebulae in our galaxy and beyond , proceedings of the international astronomical union , symposium 234 . barlow and r.h . mndez , cambridge university press , 2006 . , p.153 herald , j.e . , bianchi , l. 2011 , mnras , 417 , 2440 hillier , d.j . , miller , d. , 1998 , apj , 496 , 407 hubeny , i. , lanz , t. 1995 , apj , 439 , 875 kaschinski , c.b . , pauldrach , a.w . a. , hoffmann , t.l . , 2012 , a&a , 542 , 45 kastner , j.h . , montez , r. , balick , b. 2012 , aj , 144 , 58 manset , n. , donati , j .- f . , 2003 , in polarimetry in astronomy , proceedings of the spie . s. fineschi , 4843 , p. 425 massa , d.l . 1995 , apj , 438 , 376 mayor , m. et al . 2003 , the messenger 114 , 20 mndez , r.h . 1991 , in evolution of stars : the photospheric abundance connection , iau symp . 145 , eds . g. michaud and a.v . tutukov , kluwer academic publishers , dordrecht , p.375 mndez , r.h . , urbaneja , m.a . , kudritzki , r .- p . , prinja , r.k . 2012 , in planetary nebulae : an eye to the future , iau symp . 283 , eds . a. manchado , l. stanghellini and d. schoenberner , cambridge university press , p. 436 de marco , o. , bond , h.e . , harmer , d. , fleming , a.j . 2004 , apj , 602 , 93 patriarchi , p. , perinotto , m. 1995 , a&s , 110 , 353 prinja , r.k . , massa , d.l . , urbaneja , m.a . , kudritzki , r .- 2012a , mnras , 422 , 3142 prinja , r.k . , massa , d.l . , cantiello , m. 2012b , apj , 759 , l28 roberts , d.h . , lehr , j. , dreher , j.w . 1987 , aj , 93 , 968

we report on high - resolution optical time - series spectroscopy of the central star of the ` eskimo ' planetary nebula ngc 2392 . datasets were secured with the eso 2.3 m in 2006 march and cfht 3.6 m in 2010 march to diagnose the fast wind and photospheric properties of the central star . the hei and heii recombination lines reveal evidence for clumping and temporal structures in the fast wind that are erratically variable on timescales down to @xmath0 30 min . ( i.e. comparable to the characteristic wind flow time ) . we highlight changes in the overall morphology of the wind lines that can not plausibly be explained by line - synthesis model predictions with a spherically homogeneous wind . additionally we present evidence that the uv line profile morphologies support the notion of a high - speed , high - ionization polar wind in ngc 2392 . analyses of deep - seated , near - photospheric absorption lines reveals evidence for low - amplitude radial velocity shifts . fourier analysis points tentatively to a @xmath0 0.12-d modulation in the radial velocities , independently evident in the eso and cfht data . we conclude that the overall spectroscopic properties support the notion of a ( high inclination ) binary nucleus in ngc 2392 and an asymmetric fast wind . [ firstpage ] stars : mass - loss @xmath1 stars : evolution @xmath1 stars : individual : ngc 2392 @xmath1 optical : stars

You are an expert at summarizing long articles. Proceed to summarize the following text: in search for new fundamental structures in theoretical physics and mechanics the physicists are increasingly turning their attention to algebraic structures which are based on ternary multiplication law ( more generally on @xmath0-ary multiplication law ) . in 1973 y. nambu proposed a generalization of hamiltonian mechanics where he replaced a canonically conjugate pair of variables by a triple of canonical variables and the usual poisson bracket by a ternary operation ( nambu bracket ) @xcite . a geometric formalism for this generalization of hamiltonian mechanics based on a notion of nambu bracket of order @xmath0 , the fundamental identity and a concept of nambu - poisson manifold was developed by l. takhtajan in @xcite . an important part of this geometric formalism is a notion of an @xmath0-lie algebra which was also studied by v. t. filippov in @xcite . j. arnlind , a. makhlouf and s. silvestrov constructed an @xmath0-lie algebra by means of a binary lie algebra endowed with an analog of a trace @xcite and this @xmath0-lie algebra was called an @xmath0-lie algebra induced by lie algebra . they also studied the cohomologies of @xmath0-lie algebra induced by a lie algebra and found the relation of these cohomologies to the cohomology of initial binary lie algebra . in this paper we propose a notion of super @xmath0-lie algebra and follow an approach of j. arnlind , a. makhlouf and s. silvestrov to construct a super @xmath0-lie algebra with the help of a binary super lie algebra which is equipped with an analog of a supertrace , and this super @xmath0-lie algebra is called the super @xmath0-lie algebra induced by a super lie algebra . we apply this approach to the super lie algebra of clifford algebra ( with even number of generators ) with the matrix representation given by a supermodule of spinors and for any even integer @xmath1 we construct a super 3-lie algebra . in this section we remind the definition of @xmath0-lie algebra . let @xmath3 be a field of real or complex numbers , @xmath4 be a vector space over @xmath5 , and @xmath6 . a vector space @xmath7 is said to be an @xmath0-lie algebra if @xmath7 is endowed with a multilinear skew - symmetric mapping @xmath8:{\mathfrak g}^n\to { \mathfrak g}$ ] which satisfies the identity @xmath9 = \nonumber\\ & & \qquad\quad\sum_{k=1}^{n}[x_1,x_2,\ldots , x_{k-1},[y_1,y_2,\ldots , y_{n-1},x_{k}],x_{k+1},\ldots , x_n ] , \label{identity for n - lie algebra}\end{aligned}\ ] ] where @xmath10 . particularly if @xmath11 then the above definition yields the definition of a ( binary ) lie algebra and the identity ( [ identity for n - lie algebra ] ) takes on the form of jacobi identity . let @xmath12 be a ( binary ) lie algebra with a lie bracket @xmath13:{\frak h}\times{\frak h}\to { \frak h}$ ] , and @xmath14 be an @xmath5-multilinear skew - symmetric @xmath15-form which satisfies the condition @xmath16,x_{m+1},\ldots , x_k)=0,\ ] ] where @xmath17 are elements of @xmath12 . particularly if @xmath18 then a form @xmath19 satisfies @xmath20)=0 $ ] for any @xmath21 . let @xmath22 be an integer , @xmath23 be a permutation of the integers @xmath24 such that @xmath25 , @xmath26 and @xmath27 be the parity of @xmath28 . define @xmath29=\sum_{\sigma } ( -1)^{|\sigma|}\phi(x_{i_1},x_{i_2},\ldots , x_{i_k})[x_{i_{k+1}},x_{i_{k+2 } } ] . \label{n - bracket for lie algebra}\ ] ] then @xmath12 endowed with ( [ n - bracket for lie algebra ] ) is the @xmath0-lie algebra , where @xmath30 . it is worth noting that in the particular case of @xmath18 this theorem yields the 3-lie algebra with ternary lie bracket @xmath31=\phi(x)[y , z]+\phi(y)[z , x]+\phi(z)[x , y ] , \label{ternary lie bracket}\ ] ] and a form @xmath19 which for any @xmath21 satisfies @xmath20)=0 $ ] can be viewed as an analog of a trace . the 3-lie algebras of this kind induced by a binary lie algebra are introduced and studied @xcite . in this section we give a definition of a super @xmath0-lie algebra and prove that one can construct a super @xmath0-lie algebra by means of a super trace . let @xmath32 be a finite - dimensional super vector space , where @xmath33 is the subspace of even elements and @xmath34 is the subspace of odd elements . if @xmath35 is a homogeneous element then its degree will be denoted by @xmath36 , where @xmath37 and @xmath38 . let @xmath39 . for any @xmath40 , where @xmath41 are homogeneous elements , and for every integer @xmath42 we define @xmath43 let @xmath44 be the super vector space of endomorphisms of a super vector space @xmath45 . the composition of two endomorphisms @xmath46 determines the structure of superalgebra in @xmath44 , and the graded binary commutator @xmath47=a\circ b-(-1)^{|a||b|}$ ] induces the structure of super lie algebra in @xmath44 . the supertrace of an endomorphism @xmath48 can be defined by @xmath49 for any endomorphisms @xmath50 it holds @xmath51)=0 $ ] . given @xmath40 and a permutation @xmath52 of the integers @xmath53 we assign to them the element @xmath54 defined by @xmath55 let @xmath56 be an element of permutation @xmath28 . assume that @xmath57 are the elements of permutation @xmath28 which precede an element @xmath56 in @xmath28 and greater than @xmath56 . obviously @xmath58 is the number of inversions of an element @xmath56 in @xmath28 . define the integer @xmath59 by @xmath60 a super vector space @xmath61 is said to be a super @xmath0-lie algebra if it is endowed with an @xmath0-ary bracket @xmath62 : { \frak g}\times { \frak g}\times\ldots\times{\frak g}($]n@xmath63 which satisfies the following conditions : 1 . @xmath0-ary bracket is a multilinear mapping and the degree of @xmath0-ary bracket of @xmath0 homogeneous elements is equal to the sum of degrees of these elements , i.e. for any homogeneous element @xmath64 it holds @xmath65|=|{\bf x}|_n,\ ] ] 2 . @xmath0-ary bracket is a graded skew - symmetric multilinear mapping , i.e. for any homogeneous elements @xmath66 and for any @xmath67 it holds @xmath68=-(-1)^{|x_k||x_{k+1}|}[x_1,x_2,\ldots , x_{k+1},x_{k},\ldots , x_n],\ ] ] 3 . @xmath0-ary bracket satisfies the identity @xmath9 = \nonumber\\ & & \;\;\sum_{k=1}^{n}(-1)^{|{\bf x}|_{k-1}|{\bf y}|_{n-1}}[x_1,x_2,\ldots , x_{k-1},[y_1,y_2,\ldots , y_{n-1},x_{k}],x_{k+1},\ldots , x_n],\nonumber\end{aligned}\ ] ] where @xmath69 are homogeneous elements . let @xmath44 be the super vector space of endomorphisms of a super vector space @xmath45 and @xmath70 be a sequence of @xmath0 endomorphisms of @xmath45 . define @xmath71=\sum_{\sigma}(-1)^{|\sigma|+|\bf{a}_\sigma|}a_{i_1}\circ a_{i_2}\circ\ldots\circ a_{i_n } , \label{n - bracket for endomorphisms}\ ] ] where @xmath52 is a permutation of integers @xmath53 and @xmath27 is the parity of this permutation . then the super vector space @xmath44 endowed with the @xmath0-ary bracket ( [ n - bracket for endomorphisms ] ) is the super @xmath0-lie algebra . a representation of a super @xmath0-lie algebra @xmath4 is a linear mapping @xmath72 , where @xmath45 is a super vector space ( a representation space of @xmath4 ) , which satisfies : 1 . for any homogeneous element @xmath73 its image @xmath74 in @xmath44 is the homogeneous element and @xmath75 , 2 . for any elements @xmath66 it holds @xmath76)=[\rho(x_1),\rho(x_2),\ldots,\rho(x_n)].\ ] ] let @xmath77 be a super @xmath0-lie algebra with @xmath0-ary bracket @xmath8 $ ] , and @xmath72 be a representation of @xmath4 . for any homogeneous element @xmath78 of @xmath4 we define the @xmath79-ary bracket by @xmath80_\rho & = & \sum_{k=1}^{n+1}(-1)^{k-1}(-1)^{|x_k||{\bf x}|_{k-1}}\mbox{str}(\rho(x_k))\nonumber\\ & & \qquad\qquad\qquad\quad\times [ x_1,x_2,\ldots,\hat x_k,\ldots , x_{n+1 } ] , \label{n - bracket with supertrace}\end{aligned}\ ] ] where the hat over an element @xmath81 means that this element is omitted . a super @xmath0-lie algebra @xmath4 equipped with the @xmath79-ary bracket ( [ n - bracket with supertrace ] ) is the super @xmath79-ary lie algebra . [ theorem for n+1 lie algebra with super trace ] a clifford algebra is a unital associative algebra , which can be equipped with an @xmath82-graded structure , and it provides a well known example of a superalgebra . taking the graded commutator of two elements of this superalgebra one can consider it as the super lie algebra . in this section we consider a supermodule over clifford algebra with even number of generators , and this supermodule provides us with a representation of mentioned above super lie algebra . making use of a supertrace of this representation we construct a super 3-lie algebra with the help of the formula ( [ n - bracket with supertrace ] ) . we remind that clifford algebra @xmath83 is the unital associative algebra over @xmath84 generated by @xmath85 which obey the relations @xmath86 where @xmath87 is the unit element of clifford algebra . let @xmath88 be the set of integers from 1 to @xmath0 . if @xmath89 is a subset of @xmath90 , i.e. @xmath91 where @xmath92 , then one can associate to this subset @xmath89 the monomial @xmath93 . if @xmath94 one defines @xmath95 . the number of elements of a subset @xmath89 will be denoted by @xmath96 . it is obvious that the vector space of clifford algebra @xmath83 is spanned by the monomials @xmath97 , where @xmath98 . hence the dimension of this vector space is @xmath99 and any element @xmath100 can be expressed in terms of these monomials as @xmath101 where @xmath102 is a complex number . it is easy to see that one can endow a clifford algebra @xmath83 with the @xmath103-graded structure by assigning the degree @xmath104 to monomial @xmath97 . then a clifford algebra @xmath83 can be considered as the superalgebra since for any two monomials it holds @xmath105 . let @xmath89 be a subset of @xmath88 , and @xmath97 be a symbol associated to @xmath89 . let @xmath83 be the vector space spanned by the symbols @xmath97 . define the degree of @xmath97 by @xmath106 , where @xmath96 is the number of elements of @xmath89 , and the product of @xmath107 by @xmath108 where @xmath109 , @xmath110 is the number of elements of @xmath89 which are greater than @xmath111 , and @xmath112 is the symmetric difference of two subsets . then @xmath83 is the unital associative superalgebra , where the unit element @xmath87 is @xmath113 . this theorem can be proved by means of the properties of symmetric difference of two subsets . we remind a reader that the symmetric difference is commutative @xmath114 , associative @xmath115 and @xmath116 . the latter shows that @xmath117 is the unit element of this superalgebra . the symmetric difference also satisfies @xmath118 . hence @xmath83 is the superalgebra . the superalgebra @xmath83 can be considered as the super lie algebra if for any two homogeneous elements @xmath119 of this superalgebra one introduces the graded commutator @xmath120=xy-(-1)^{|x||y|}yx$ ] and extends it by linearity to a whole superalgebra @xmath83 . we will denote this super lie algebra by @xmath121 . then @xmath122 are the generators of this super lie algebra @xmath121 , and its structure is entirely determined by the graded commutators of @xmath97 . then for any two generators @xmath107 we have @xmath123=f(i , j)\;\gamma_{i\delta j } , \label{binary commutators}\ ] ] where @xmath124 is the integer - valued function of two subsets of @xmath90 defined by @xmath125 it is easy to verify that the degree of graded commutator is consistent with the degrees of generators , i.e. @xmath126=|\gamma_i|+|\gamma_j|.$ ] indeed the function @xmath127 satisfies @xmath128 and @xmath129 hence @xmath126=-(-1)^{|i||j|}[\gamma_j,\gamma_i]$ ] which shows that the relation ( [ binary commutators ] ) is consistent with the symmetries of graded commutator . it is obvious that if the intersection of subsets @xmath130 contains an even number of elements then @xmath131 , and the graded commutator of @xmath107 is trivial . particularly if at least one of two subsets @xmath130 is the empty set then @xmath131 . thus any graded commutator ( [ binary commutators ] ) containing @xmath87 is trivial . as an example , consider the super lie algebra @xmath132 . its underlying vector space is 4-dimensional and @xmath132 is generated by two even degree generators @xmath133 and two odd degree generators @xmath134 the non - trivial relations of this super lie algebra are given by @xmath135=[\gamma_2,\gamma_2]=2\,e,\ ; [ \gamma_1,\gamma_{12}]=2\,\gamma_2,\ ; [ \gamma_2,\gamma_{12}]=-2\,\gamma_1.\end{aligned}\ ] ] now we assume that @xmath136 is an even integer . the super lie algebra @xmath121 has a matrix representation which can be described as follows . fix @xmath11 and identify the generators @xmath137 with the pauli matrices @xmath138 , i.e. @xmath139 then @xmath140 where @xmath141 let @xmath142 be the 2-dimensional complex super vector space @xmath143 with the odd degree operators ( [ pauli matrices 1,2 ] ) , where the @xmath82-graded structure of @xmath142 is determined by @xmath144 . then @xmath145 , and @xmath142 can be considered as a supermodule over the superalgebra @xmath146 . let @xmath147 . then @xmath148 can be viewed as a supermodule over the @xmath149-fold tensor product of @xmath146 , which can be identified with @xmath83 by identifying @xmath137 in the @xmath150th factor with @xmath151 in @xmath83 . this @xmath83-supermodule @xmath148 is called the supermodule of spinors @xcite . hence we have the matrix representation for the clifford algebra @xmath83 , and this matrix representation or supermodule of spinors allows one to consider the supertrace , and it can be proved @xcite that @xmath152 now we have the super lie algebra @xmath153 with the graded commutator defined in ( [ binary commutators ] ) and its matrix representation based on the supermodule of spinors . hence we can construct a super 3-lie algebra by making use of graded ternary commutator ( [ n - bracket with supertrace ] ) . applying the formula ( [ n - bracket with supertrace ] ) we define the graded ternary commutator for any triple @xmath154 of elements of basis for @xmath153 by @xmath155 = \mbox{str}(\gamma_i)\,[\gamma_j,\gamma_k]-(-1)^{|i||j|}\mbox{str}(\gamma_j)\ , [ \gamma_i,\gamma_k]\nonumber\\ & & \qquad\qquad\qquad\qquad\qquad\qquad\quad + ( -1)^{|k|(|i|+|j|)}\mbox{str}(\gamma_k)\,[\gamma_i,\gamma_j ] , \label{ternary graded commutator with super trace}\end{aligned}\ ] ] where the binary graded commutator at the right - hand side of this formula is defined by ( [ binary commutators ] ) . according to theorem [ theorem for n+1 lie algebra with super trace ] the vector space spanned by @xmath156 and equipped with the ternary graded commutator ( [ ternary graded commutator with super trace ] ) is the super 3-lie algebra which will be denoted by @xmath157 . making use of ( [ binary commutators ] ) we can write the expression at the right - hand side of the above formula in the form @xmath155 = f(j , k ) \mbox{str}(\gamma_i)\,\gamma_{j\delta k}-(-1)^{|i||j|}f(i , k)\mbox{str}(\gamma_j)\ , \gamma_{i\delta k}\nonumber\\ & & \quad\qquad\qquad\qquad\qquad\qquad\qquad\quad + ( -1)^{|k|(|i|+|j|)}f(i , j)\mbox{str}(\gamma_k)\,\gamma_{i\delta j}.\nonumber\end{aligned}\ ] ] from the formula for supertrace ( [ super trace ] ) it follows immediately that the above graded ternary commutator is trivial if none of subsets @xmath158 is equal to @xmath159 . similarly this graded ternary commutator is also trivial if all three subsets @xmath160 are equal to @xmath90 , i.e. @xmath161 , or two of them are equal to @xmath159 . the graded ternary commutators of the generators @xmath162 of the super 3-lie algebra @xmath157 are given by @xmath163=\left\ { \begin{array}{ll } ( 2i)^mf(i , j)\gamma_{i\delta j } & \text{if } i\neq { \mathcal n } , j\neq { \mathcal n } , k={\mathcal n},\\ 0 & \text{in all other cases } . \end{array } \right . \label{proposition}\ ] ] the author gratefully acknowledges that this work was financially supported by the estonian science foundation under the research grant etf9328 and this work was also financially supported by institutional research funding iut20 - 57 of the estonian ministry of education and research . 1 j. arnlind , a. kitouni , a. makhlouf , s. silvestrov , _ structure and cohomology of 3-lie algebras induced by lie algebras _ , arxiv : 1312.7599v . j. arnlind , a. makhlouf , s. silvestrov , _ construction of @xmath0-lie algebras and @xmath0-ary hom - nambu - lie algebras _ , j. math . * 52 * ( 2011 ) , 123502 , 13 pp . v. t. filippov , _ @xmath0-lie algebras _ , siberian math . j. * 26 * ( 1985 ) , 879 891 . y. nambu , _ generalized hamiltonian dynamics _ , phys . d ( 3 ) 7 ( 1973 ) , 2405 2412 . v. mathai , d. quillen , _ superconnections , thom classes , and equivariant differential forms _ , topology * 25 * , 1 ( 1986 ) , 85110 . l. takhtajan , _ on foundation of the generalized nambu mechanics _ , commun.math.phys . 160 ( 1994 ) , 295 - 316

we propose a notion of a super @xmath0-lie algebra and construct a super @xmath0-lie algebra with the help of a given binary super lie algebra which is equipped with an analog of a supertrace . we apply this approach to the super lie algebra of a clifford algebra with even number of generators and making use of a matrix representation of this super lie algebra given by a supermodule of spinors we construct a series of super 3-lie algebras labeled by positive even integers .

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Proceed to summarize the following text: experimental studies of neutrino oscillations have provided us with compelling evidence that neutrinos have masses and lepton flavors mix . among various theoretical models , the famous seesaw mechanism @xcite provides us with a very natural description of why the masses of the three known neutrinos are so small compared to the masses of the other standard model ( sm ) fermions . in the simplest type - i seesaw model , heavy right - handed neutrinos with a mass scale @xmath0 are introduced in addition to the sm particle content . in order to stabilize the masses of the light neutrinos around the sub - ev scale , @xmath1 is naturally expected , if the dirac mass @xmath2 between the left- and right - handed neutrinos is comparable with the mass of the top quark . the testability of conventional seesaw models is therefore questionable . furthermore , the heavy right - handed neutrinos potentially contribute to the hierarchy problem through loop corrections to the higgs potential , unless a supersymmetric framework is considered . the large hadron collider ( lhc ) will soon start to probe tev scale physics , and the question of whether we can find hints on the neutrino mass generation mechanism at the lhc or not is relevant and interesting . there are several indications that new physics will show up at the tev scale , in particular theories that are able to stabilize the higgs mass and to solve the gauge hierarchy problem . the geometric mean of the planck mass and the @xmath3 k background temperature also suggests that 1 tev is the maximum mass that any cosmologically stable perturbatively coupled elementary particle can have , otherwise the density of the universe exceeds its critical value @xcite . within the seesaw framework , for the purpose of lowering the seesaw scale without spoiling the naturalness criterion , some underlying symmetry preserving the lepton number , @xmath4 , is usually incorporated . for example , in the type - i seesaw with more than one heavy right - handed neutrino , contributions to the light - neutrino masses from different right - handed neutrinos may cancel each other due to the symmetry , which results in massless left - handed neutrinos after integrating out the heavy degrees of freedom from the theory @xcite . such a low - scale fermionic seesaw mechanism may not be able to stabilize the masses of the light neutrinos , since loop corrections may be unacceptably large . a possible way to avoid this problem of the type - i seesaw model is given by the inverse seesaw model , which contains a majorana insertion used to reduce the @xmath5 scale @xcite . in the type - ii seesaw model , extending the sm with an @xmath6 triplet higgs scalar @xcite , the coupling between the triplet and the sm higgs scalar breaks lepton number explicitly and is expected to be very small . thus , the masses of the light neutrinos are suppressed through the approximate symmetry . in general , the canonical leptogenesis mechanism @xcite , which provides a very attractive description of the origin of the observed baryon asymmetry of the universe , does not work for the low - scale seesaw mechanisms unless severe fine - tuning is invoked @xcite . in this paper , we employ the alternative framework of extra spacetime dimensions , where the fundamental grand unified scale and the planck scale are lowered in a natural way @xcite . we work exclusively within the context of flat extra dimensions . in our higher - dimensional seesaw model , a truncating scale restoring the renormalizability of the theory plays the role of breaking @xmath5 , so that the masses of the light neutrinos are suppressed , while the lower kaluza klein ( kk ) states can be searched for at the lhc . significant low - energy non - unitary leptonic mixing , due to integrating out the heavy kk states , could give observable phenomena in future neutrino oscillation experiments , such as a neutrino factory @xcite . in addition , resonant leptogenesis could possibly be achieved in this model . for earlier studies of the generation of small neutrino masses in the context of extra dimensions , see for example refs . a study of unitarity violation in scenarios with bulk gauge singlet neutrinos was performed in ref . an alternative higher - dimensional seesaw model was investigated in ref . @xcite . the remaining parts of the paper are organized as follows : first , in sec . [ sec : introduction ] , we present the general formalism of our model . then , in sec . [ sec : nu ] , we show explicitly how sizable non - unitarity effects emerge in the leptonic flavor mixing . section [ sec : lhc ] is devoted to the collider signatures and the discovery potential of the heavy kk modes at the lhc . we comment on the origin of baryon number asymmetry in our model in sec . [ sec : leptogenesis ] . finally , a summary and our conclusions are given in sec . [ sec : summary ] . we consider a brane world theory with a five - dimensional bulk , where the sm particles are confined to the brane . we also introduce three sm singlet fermions @xmath7 ( @xmath8 ) @xcite . being singlets , they are not restricted to the brane and can propagate in the extra spacetime dimensions . the action responsible for the neutrino masses is given by @xmath9 \nonumber \\ & & + \int_{y=0 } { \rm d}^4 x \left ( - \frac{1}{\sqrt{m_s } } \overline{\nu_{\rm l } } \hat m^c \psi - \frac{1}{\sqrt{m_s } } \overline{\nu^c_{\rm l } } \hat m \psi + { \rm h.c.}\right),\end{aligned}\ ] ] where @xmath10 is the coordinate along the extra compactified dimension and @xmath11 denotes the mass scale of the higher - dimensional theory . note that , although @xmath12 is defined in the same way as in four dimensions , it does not represent the charge conjugate of @xmath13 in five dimensions @xcite , and hence , the term @xmath14 is not a majorana mass term . however , in the four - dimensional theory , it leads to effective majorana mass terms for the kk modes of @xmath13 . due to the freedom in the choice of basis for the singlet fermion fields , one can always apply a unitary transformation in flavor space in order to diagonalize @xmath0 . without loss of generality , we will therefore work in a basis in which @xmath15 is real and diagonal . the dirac masses @xmath16 and @xmath17 could be generated by couplings of the bulk neutrinos to a brane - localized higgs boson receiving a vacuum expectation value . we decompose the spinors of the bulk singlet fermions into two two - component objects : @xmath18 , where @xmath19 . since the extra dimension is compactified on the @xmath20 orbifold , the kk modes of @xmath21 and @xmath22 are four - dimensional weyl spinors . we take @xmath21 to be even under the @xmath23 transformation @xmath24 , while @xmath25 is taken to be odd . thus , in eq . , the @xmath17 term corresponding to the coupling between @xmath26 and @xmath25 is not allowed . the kk expansions of @xmath21 and @xmath25 are given by @xmath27 in general , an extra - dimensional model must be viewed as an effective theory , since it is non - renormalizable . this means that the kk towers are expected not to be infinite , but truncated after a finite number of levels . the nature of this cutoff depends on the specific ultraviolet ( uv ) completion of the model , which is not known . here , we impose a truncation of the kk towers at a maximum kk index @xmath28 . a cutoff of this kind arises , for example , in deconstructed models of extra dimensions @xcite . in general , other kinds of truncation schemes are possible , but the one that we consider has the virtue of giving rise to a mechanism for generating small neutrino masses from the tops of the kk towers , as will be discussed below . inserting the above expansion into eq . and integrating over the compactified extra dimension , we arrive at the following form for the four - dimensional action @xmath29 \right . \nonumber \\ & & \phantom{\int d^4 x}-\left . { \rm i } \left(\nu_{\rm l}^t \sigma^2 m_{\rm d } \xi^{(0 ) } + \sqrt{2 } \sum^n_{n=1 } \nu_{\rm l}^t \sigma^2 m_{\rm d } \xi^{(n ) } + { \rm h.c . } \right ) \right\},\end{aligned}\ ] ] where , written in block - form , the mass matrix @xmath30 for the kk modes at the @xmath31th level takes the form @xmath32 the dirac mass term is then given by @xmath33 . for the purpose of simplicity in the following discussion , we define the linear combinations @xmath34 for @xmath35 . the full mass matrix in the basis @xmath36 then reads @xmath37 the scale of @xmath0 is not governed by the electroweak symmetry breaking , and hence , one can expect that @xmath38 holds . then , by approximately solving the eigenvalue equation of the matrix in eq . with respect to the small ratio @xmath39 , the light - neutrino mass matrix is found to be @xmath40 in refs . @xcite , the limit @xmath41 is considered , and the light - neutrino mass matrix is then given by @xmath42 the masses of the light neutrinos are suppressed only if @xmath43 in the denominator of eq . is very large . therefore , a severe fine - tuning between @xmath44 and @xmath45 has to be invoked , which appears quite unnatural . however , bare majorana masses of the form @xmath46 , where @xmath47 is an odd integer , emerge naturally from the sherk schwarz decomposition in string theory as a requirement of topological constraints , and hence , such relations do not suffer any fine - tuning problems ( see detailed discussions in ref . @xcite ) . with our chosen cutoff scheme , together with the above condition on @xmath48 , lepton number violation will be induced only at the top of the kk tower , as we will see shortly . there could , of course , be other lepton number violating processes at some intermediate point , but we choose to treat the simple scenario where the cutoff is the only source . one can easily prove that , in the simplest case @xmath49 , the light - neutrino mass matrix is given by @xmath50 instead of a large mass scale @xmath0 for the singlet fermions , the light - neutrino masses are suppressed by the large cutoff scale @xmath51 . we consider the interesting case where the scale of the uv completion is much larger than the scale of the extra dimension @xmath52 and the singlet fermion masses , _ i.e. _ , we assume @xmath53 to hold . in this limit , the neutrino mass matrix is simply given by @xmath54 , _ i.e. _ , the scale of the neutrino masses is determined by a high - energy scale associated with the fundamental theory underlying the effective extra - dimensional model . as for the heavy kk modes , from eq . , the masses of the @xmath31th excited kk modes are given by @xmath55 as we will discuss later , this implies that @xmath56 and @xmath57 ( as well as @xmath58 and @xmath59 ) form dirac pairs . thus , lepton number can be assigned to these pairs and the lepton number violating effects , such as neutrino masses , can only arise from the unpaired @xmath60 at the top of the kk tower . in order to compute the effective low - energy leptonic mixing , we first consider the light - neutrino mass matrix . generally , @xmath61 is a complex symmetric matrix , and can be diagonalized by means of a unitary matrix @xmath62 as @xmath63 where @xmath64 , with @xmath65 being the masses of the light neutrinos . note that , similarly to the ordinary fermionic seesaw mechanism , the light neutrinos mix with the heavy kk modes . thus , @xmath62 is not the exact leptonic mixing matrix entering into neutrino oscillations , even if one works in a basis where the charged - lepton mass matrix is diagonal . to see this point clearly , we can fully diagonalize eq . and then write down the neutrino flavor eigenstates in terms of the mass eigenstates @xmath66,\end{aligned}\ ] ] where @xmath67 denotes the mass eigenstates of the light neutrinos , and @xmath68 is the upper - left @xmath69 sub - matrix of the complete mixing matrix containing the light neutrinos as well as the full kk tower for the singlet fermions . furthermore , we have introduced the quantities @xmath70 which represent the mixing between the light neutrinos and the kk modes . the charged - current lagrangian in mass basis can be rewritten as @xmath71w^-_\mu + { \rm h.c.},\end{aligned}\ ] ] where @xmath72 is the @xmath6 coupling constant . due to the existence of the kk modes , the light - neutrino mixing matrix is no longer unitary . to a very good precision , we have @xmath73 assuming that @xmath74 , eq . can be approximated by @xmath75 compared to the conventional parametrization of non - unitarity effects @xmath76 @xcite , where @xmath77 is a hermitian matrix , we thus obtain @xmath78 an interesting feature of eq . arises immediately : the non - unitarity effects are dominated only by the combination @xmath79 . as a rough estimate , if we keep @xmath52 at the tev scale and @xmath80 , @xmath81 can be naturally expected . another typical feature is that , if @xmath82 holds , then both the neutrino mixing and the non - unitarity effects are determined by a single dirac mass matrix @xmath2 . therefore , in such a realistic low - scale extra - dimensional model , the non - unitarity effects are strongly correlated with the neutrino mixing matrix and the radius of the extra spacetime dimension . in our numerical computations , we adopt a convenient parametrization @xcite , and rewrite @xmath2 as @xmath83 with @xmath84 being an arbitrary complex orthogonal matrix . with this parametrization , eq . takes the form @xmath85 the present bounds at 90 % c.l . on the non - unitarity parameters are given by @xcite @xmath86 where the most severe constraint is that on the @xmath87 element , coming from the @xmath88 decay . however , in the case that @xmath0 lies below the electroweak scale , but above a few gev , the @xmath88 constraint is lost due to the restoration of the glashow iliopoulos maiani ( gim ) mechanism @xcite , and a less stringent bound of @xmath89 should be used . apart from resulting in non - unitarity effects in neutrino mixing , the heavy singlet fermions in the bulk will also contribute to the lepton flavor violating ( lfv ) decays of charged leptons , _ e.g. _ , @xmath88 and @xmath90 , through the loop exchange of kk modes @xcite . different from the standard type - i seesaw mechanism , the corresponding branching ratios are not dramatically suppressed by the light - neutrino masses , but only driven down by the factor @xmath91 defined in eq . . thus , appreciable lfv rates could be obtained . as shown in eq . , the heavy singlets @xmath59 , @xmath56 , and @xmath92 couple to the gauge sector of the sm , and thus , if kinematically accessible , they could be produced at hadron colliders . for a quantitative discussion , we now restrict ourselves to the simplest case @xmath93 . note that @xmath59 and @xmath58 are two - component majorana fields with equal masses but opposite cp parities @xcite . thus , they are equivalent to a single dirac field @xmath94 with @xmath95 $ ] , @xmath96 $ ] , and mass @xmath97 . similarly , @xmath98 can be combined with @xmath99 , and hence , forms a higher kk dirac mode with @xmath100 $ ] and mass @xmath101 . as a general result of the mass degeneracy , all the kk modes are paired together except for the highest mode @xmath60 with mass @xmath102 . actually , @xmath60 is now the sole source of lepton number violation , and thus , gives rise to the masses of the light neutrinos , which can also be seen from eq . . the structure of the singlet dirac and majorana fermions is schematically depicted in fig . [ fig : dirac ] . illustration of the construction of dirac particles from pairs of modes in the kk tower . two heavy kk majorana modes with equal masses , but opposite cp parities , can be grouped together , as shown with double lines , in order to form a dirac particle . in the case @xmath103 ( left column ) , the heaviest mode @xmath60 is left , while for the case @xmath104 ( right column ) , there are three modes left : @xmath105 , @xmath106 , and @xmath60.,width=604 ] the weak interaction lagrangian for the heavy states can now be rewritten as @xmath107w^-_\mu + { \rm h.c . } , \\ { \cal l}_{\rm nc } & = & \frac{g}{2 \cos \theta_{\rm w } } { \nu_{m\rm l}^\dagger } \bar{\sigma}^\mu v^\dagger \left [ \sqrt{2 } \sum^{n-1}_{n=0 } k^{(n ) } p^{(n)}_l + k^{(n ) } y^{(n ) } \right ] z_\mu+ { \rm h.c . } , \\ { \cal l}_{h } & = & \frac{-{\rm i } g}{\sqrt{2}m_{w } } { \nu_{m\rm l}^t } \sigma^2 v^t m_d \left [ \sqrt{2 } \sum^{n-1}_{n=0 } p^{(n)}_l + y^{(n ) } \right ] h + { \rm h.c.},\end{aligned}\ ] ] where @xmath108 denotes the weak mixing angle and @xmath109 is the mass of the @xmath110 boson . in the case @xmath111 ( where @xmath112 denotes the higgs mass ) , the heavy kk modes decay in the channels @xmath113 , @xmath114 , and @xmath115 . the corresponding partial decay widths are given by @xcite @xmath116 where @xmath117 and @xmath118 denote that masses of @xmath119 and @xmath120 , respectively . since the lower kk modes are dirac particles , and lepton number breaking occurs only at the top of the kk towers , we focus our attention on lepton number conserving channels mediated by the lightest kk modes . for example , an interesting channel is the production of three charged leptons and missing energy @xcite , _ i.e. _ , @xmath121 , which is depicted in fig . [ fig : lhc ] . feynman diagrams for the potentially interesting lhc signatures with three charged leptons and missing energy in the model under consideration.,width=302 ] another possible process is the pair production of charged leptons with different flavor and zero missing energy , _ i.e. _ , @xmath122 . however , it is difficult to make significant observations in this channel at the lhc , due to the large sm background @xcite . an analysis of the collider signatures of an extra - dimensional model similar to the one that we consider was performed in ref . it was found that the most promising channel for that model is three leptons and large missing energy . since taus are difficult to detect , due to their short lifetime , only electrons and muons in the final state were considered . the signals were combined into two classes , the @xmath123 signal , given by the sum of the @xmath124 and @xmath125 signals , where @xmath126 and @xmath127 denote both leptons and antileptons of the indicated flavors , and the @xmath128 signal , given by the sum of the @xmath129 and @xmath130 signals . for the case of normal neutrino mass hierarchy ( @xmath131 ) , it was found that the @xmath123 combination gives the most promising signal . in order to reduce the sm background , which mainly comes from decays of @xmath119 bosons , the following kinematic cuts , taken from ref . @xcite , were adopted : i ) the two like - sign leptons must each have a transverse momentum larger than 30 gev and ii ) the invariant masses from the two opposite - sign lepton pairs must each be separated from the mass of the @xmath119 boson by at least 10 gev . only the effects of the lowest kk level were considered , as it was concluded that the contributions from higher modes would be more than one order of magnitude smaller . we have calculated the @xmath123 as well as the @xmath128 signals for our model . the results , using an integrated luminosity of @xmath132 , are shown in fig . [ fig : lhcsignals ] . we have considered the normal neutrino mass hierarchy ( @xmath131 ) as well as the inverted hierarchy ( @xmath133 ) , and for each case , we have chosen the mass of the lightest neutrino to be equal to zero or @xmath134 ev , corresponding to the hierarchical or nearly degenerate neutrino mass spectrum , respectively . for the neutrino oscillation parameters , we have used the best - fit values from ref . @xcite , _ i.e. _ , @xmath135 , @xmath136 , @xmath137 , @xmath138 , and @xmath139 . we have put the dirac cp - violating phase to zero . for each case , we have set the value of the cutoff scale in order to maximize the signal , while respecting the non - unitarity bounds given in eq . . like ref . @xcite , we have only taken the lightest kk modes of the singlet fermions into account . the signals are dominated by the on - shell production of the internal gauge bosons and sterile fermions . since @xmath140 , on - shell production of the gauge bosons is not possible if @xmath141 , and in that case , the signals are suppressed by the off - shell propagators . hence , we have chosen @xmath142 as the lower bound in our figures . in the case that the lightest neutrino is massless , the @xmath123 signal is stronger than the @xmath128 signal by approximately one order of magnitude for the normal hierarchy , while the opposite is true for the inverted hierarchy . in the case of a nearly degenerate mass spectrum , _ i.e. _ , that the lightest neutrino has a non - zero mass equal to 0.1 ev , the two signals are almost identical , especially in the inverted hierarchy case . since the expected background , after the kinematic cuts have been imposed , is of the order of 100 events @xcite and none of the signals is stronger than @xmath143 events , we conclude that , for our model , the non - unitarity bounds are strong enough to rule out the part of the parameter space that could possibly be probed by the lhc . the expected number of events for the @xmath123 and @xmath128 signals at the lhc as functions of the inverse radius @xmath45 , for an integrated luminosity of @xmath144 . note that the masses of the lightest singlet fermions are equal to @xmath145 . for @xmath141 , on - shell production of the internal gauge bosons is not possible , and the signal is suppressed . the values of the neutrino oscillation parameters are given in the main text . left panel : normal neutrino mass hierarchy . right panel : inverted neutrino mass hierarchy.,title="fig : " ] the expected number of events for the @xmath123 and @xmath128 signals at the lhc as functions of the inverse radius @xmath45 , for an integrated luminosity of @xmath144 . note that the masses of the lightest singlet fermions are equal to @xmath145 . for @xmath141 , on - shell production of the internal gauge bosons is not possible , and the signal is suppressed . the values of the neutrino oscillation parameters are given in the main text . left panel : normal neutrino mass hierarchy . right panel : inverted neutrino mass hierarchy.,title="fig : " ] baryogenesis via leptogenesis is one of the main candidates for being the theory appropriately describing the production of a baryon asymmetry in the early universe , which is measured to be @xmath146 @xcite . in its most basic form , leptogenesis occurs in a type - i seesaw scenario , where a net lepton asymmetry is produced through the out - of - equilibrium decay of the heavy neutrinos and then partially converted to a baryon asymmetry through sphaleron processes . the sakharov conditions @xcite are fulfilled by the decays occurring out of equilibrium , the loop level cp - violation of the decays through complex yukawa couplings , and the baryon number violation of the sphalerons , respectively . usually , the net lepton number is produced by the decays of the lightest singlet fermions , since asymmetries produced by the heavier neutrinos will be washed out . however , in our scenario , the tower of dirac fermions can be given definite lepton number assignments and lepton number violation only occurs at the top of the tower through the unpaired @xmath147-states , which could take on the role of the singlet fermions in the basic scenario . it is important to note that for @xmath93 ( @xmath148 ) , there will be no net lepton number violation , since all of the three unpaired states will be degenerate in mass . however , if the @xmath47 are different , _ e.g. _ , @xmath149 and @xmath150 , @xmath151 , then @xmath152 ( see fig . [ fig : dirac ] ) will be the unique lightest majorana state and a net lepton asymmetry could be produced . since the mass splitting of @xmath52 between the @xmath147-states is expected to be very small compared to the masses , the model would have to be treated within the framework of resonant leptogenesis @xcite . furthermore , to accurately examine the prospects for leptogenesis in this model , one would have to properly take into account the effects of the dirac tower . even if the dirac fermions in the tower preserve lepton number , they do not participate in the sphaleron processes , since they are sm singlets , which could hide some part of the produced lepton number from the sphalerons if all dirac fermions do not decay before sphaleron processes become inactive . thus , a detailed analysis , which is beyond the scope of this paper , would be required to properly analyze the prospects for leptogenesis in this model . for earlier studies of leptogenesis in extra dimensions , see for example refs . in this work , we have studied a possible mechanism for generating small neutrino masses in the context of extra dimensions . in the model that we consider , the sm particles are confined to a four - dimensional brane , while three sm singlet fermions are allowed to propagate in an extra dimension , compactified on the @xmath153 orbifold . since extra - dimensional models are generally non - renormalizable , and can only be considered as effective theories , the kk expansions of the higher - dimensional fields are expected to be truncated at some cutoff scale . we have imposed a cut on the kk number , truncating the towers at @xmath28 . in the case that the bulk majorana mass term for the singlet fermions has the form @xmath154 , where @xmath155 is an odd integer , the kk modes of the singlet fermions pair to form dirac fermions . such a form for a majorana mass is motivated by , for example , the scherk schwarz mechanism . due to the truncation of the kk towers , a number of unpaired majorana fermions remain at the top of each kk tower , and these are the only sources of lepton number violation in this model . if the cutoff scale is large , small masses for the left - handed neutrinos are naturally generated . due to mixing between the light neutrinos and the kk modes of the singlet fermions , large non - unitarity effects can be induced . since the masses of the light neutrinos are generated by the top of each tower , these non - unitarity effects are not suppressed by the light - neutrino masses . current bounds on the non - unitarity parameters have constrained the parameter space of the model . finally , we have considered the prospects of observing the effects of the lowest kk modes of the singlet fermions at the lhc . in particular , we have considered the three leptons and large missing energy signal , which has previously been found to be promising for a similar model . we have found that , in contrast to the previous results in the literature , the potential of discovering such models at the lhc is actually pessimistic . in particular , the parts of the parameter space that could be probed at the lhc are ruled out by the bounds imposed by the stringent constraints on the effective low - energy leptonic mixing . however , the non - unitarity effects in neutrino oscillations could be observable at future neutrino factory experiments . therefore , future long baseline neutrino oscillation experiments could play a very complementary role in searching for hints of extra dimensions . we would like to thank steve blanchet for useful discussions . we acknowledge the hospitality and support from the nordita scientific program `` astroparticle physics a pathfinder to new physics '' , march 30 april 30 , 2009 during which parts of this study was performed . this work was supported by the european community through the european commission marie curie actions framework programme 7 intra - european fellowship : neutrino evolution [ m.b . ] , the royal swedish academy of sciences ( kva ) [ t.o . ] , the gran gustafsson foundation [ h.z . ] , and the swedish research council ( vetenskapsrdet ) , contract no . 621 - 2008 - 4210 [ t.o . ] .

we study the generation of small neutrino masses in an extra - dimensional model , where singlet fermions are allowed to propagate in the extra dimension , while the standard model particles are confined to a brane . motivated by the fact that extra - dimensional models are non - renormalizable , we truncate the kaluza klein towers at a maximal kaluza klein number . this truncation , together with the structure of the bulk majorana mass term , motivated by the sherk schwarz mechanism , implies that the kaluza klein modes of the singlet fermions pair to form dirac fermions , except for a number of unpaired majorana fermions at the top of each tower . these heavy majorana fermions are the only sources of lepton number breaking in the model , and similarly to the type - i seesaw mechanism , they naturally generate small masses for the left - handed neutrinos . the lower kaluza klein modes mix with the light neutrinos , and the mixing effects are not suppressed with respect to the light - neutrino masses . compared to conventional fermionic seesaw models , such mixing can be more significant . we study the signals of this model at the large hadron collider , and find that the current low - energy bounds on the non - unitarity of the leptonic mixing matrix are strong enough to exclude an observation .

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Proceed to summarize the following text: the recent revival of interest in the properties of two - band ( or , more generally , multiband ) superconductors has been largely stimulated by the discovery of superconductivity in mgb@xmath0 ( ref . ) . other candidates for multiband superconductivity include nickel borocarbides ( ref . ) , nbse@xmath0 ( ref . ) , cecoin@xmath1 ( ref . ) , and also the iron - based high - temperature superconductors , see ref . for a review . these discoveries have shown that multiband superconductivity , which is characterized by a significant difference in the order parameter magnitudes and/or phases in different bands , might be a much more common phenomenon than it was previously thought . one important class of multiband superconductors is noncentrosymmetric compounds with a strong spin - orbit ( so ) coupling of conduction electrons with the lattice . since the discovery of superconductivity in cept@xmath2si ( ref . ) , the list of noncentrosymmetric superconductors has grown to include dozens of materials , such as uir ( ref . ) , cerhsi@xmath2 ( ref . ) , ceirsi@xmath2 ( ref . ) , y@xmath0c@xmath2 ( ref . ) , li@xmath0(pd@xmath3,pt@xmath4)@xmath2b ( ref . ) , and many others . in noncentrosymmetric crystals , the bloch bands are split by the so coupling and the cooper pairing of electrons from different bands is suppressed , resulting in an effectively two - band picture of superconductivity . both the spin structure of the bands and the momentum - space topology of the bloch wavefunctions are nontrivial , which brings about a number of novel properties , such as the magnetoelectric effect in the superconducting state,@xcite topologically protected gapless boundary modes and quantum spin hall effect,@xcite and the anomalous de haas - van alphen and hall effects in the normal state.@xcite a comprehensive review of the recent developments in the field can be found in ref . . one of the most striking features of noncentrosymmetric superconductors is the existence of various unusual nonuniform superconducting states in the presence of a magnetic field @xmath5 ( refs . ) , or even without any field ( ref . ) . the best studied example is the helical states , with the order parameter proportional to @xmath6 , where @xmath7 is linear in @xmath5 , which originate from the deformation of the bloch bands by the field , see ref . for a review . the helical states , as well as their nonlinear modifications , such as `` multiple-@xmath7 '' states , have been previously studied only in some limiting cases , either using a bardeen - cooper - schrieffer ( bcs ) model with a spin - singlet attraction@xcite or assuming that superconductivity appears only in one of the bands.@xcite in both cases , the order parameter has one component . the purpose of this paper is to develop a phenomenological theory of nonuniform states in noncentrosymmetric superconductors in the general case , fully taking into account the two - component nature of superconductivity in these materials . generalization of the bcs theory to the case of two spin - degenerate bands was originally introduced in ref . . subsequent work has shown that many properties of multiband superconductors differ qualitatively from the single - band case , with the most spectacular features associated with the additional degrees of freedom the relative phases of the pair condensates in different bands . for example , if the condensate phases have different windings around a vortex core , than the vortex will carry a fractional magnetic flux.@xcite in addition to exotic vortices , there is another type of topological defects specific to two - band superconductivity , namely phase solitons , in which the relative phase exhibits a kink - like variation by @xmath8 between its asymptotic mean - field values.@xcite increasing the number of bands opens up even more intriguing possibilities . for instance , superconducting states that break time - reversal symmetry can exist in `` frustrated '' three - band systems,@xcite with domain walls separating degenerate ground states.@xcite it has been proposed that the phase solitons in two - band superconductors can be dynamically created in nonequilibrium current - carrying states,@xcite or by the proximity effect with a conventional @xmath9-wave superconductor,@xcite but experimental observation of these effects has remained a challenge . in this paper , we show how the elusive phase solitons can be spontaneously formed in noncentrosymmetric superconductors in a sufficiently strong magnetic field . we focus on two - dimensional ( 2d ) noncentrosymmetric superconductors in a parallel field . these systems can be realized experimentally at an interface between two different non - superconducting materials,@xcite near a doped surface of an insulating crystal,@xcite or near the surface of a topological insulator.@xcite in all these systems the mirror symmetry between two half - spaces separated by an interface is explicitly broken . we use the two - band ginzburg - landau ( gl ) formalism@xcite modified to include first - order gradient terms ( the lifshitz invariants ) to catch the effects specific to noncentrosymmetric superconductors . the paper is organized as follows : in sec . [ sec : two - band picture ] we develop a two - band description of noncentrosymmetric superconductors . in sec . [ sec : helical state ] , the helical instability in a weak magnetic field is considered and the quasiparticle spectrum in the helical state is calculated . in sec . [ sec : phase solitons ] we discuss the phase solitons and soliton lattices in a strong field . throughout the paper we use the units in which @xmath10 . in order to see how nondegenerate bands are formed in a noncentrosymmetric crystal with the so coupling , we start with the following hamiltonian of noninteracting electrons : @xmath11a^\dagger_{\bk s}a_{\bk s'}.\ ] ] here @xmath12 are spin indices , @xmath13 are the pauli matrices , @xmath14 is the band dispersion without the so coupling , and the sum over @xmath15 is restricted to the first brillouin zone . the so coupling of electrons with the crystal lattice is described by @xmath16 , which satisfies @xmath17 . the momentum dependence of the so coupling is dictated by the requirement that it must be invariant under the crystal symmetry operations , in the following sense : if @xmath18 is any operation from the point group @xmath19 of the crystal , then @xmath20 . the complete list of representative expressions for @xmath16 for all noncentrosymmetric point groups can be found in ref . . for instance , in the least symmetric case of a triclinic crystal , i.e. if @xmath21 , the direction of @xmath22 is not related to the crystallographic axes and we have @xmath23 , where the coefficients @xmath24 form a real @xmath25 matrix . in a cubic crystal with @xmath26 , which describes the point symmetry of li@xmath0(pd@xmath3,pt@xmath4)@xmath2b , the simplest form compatible with all symmetry requirements is @xmath27 , where @xmath28 is a constant . expressions become more complicated if the point group contains improper elements . for example , for the tetragonal group @xmath29 , which is relevant for cept@xmath2si , the so coupling is given by @xmath30 . since we focus on the purely 2d case , we can set @xmath31 . then the so coupling takes the rashba form : @xmath32 which was originally introduced to describe the effects of the absence of mirror symmetry in semiconductor quantum wells.@xcite diagonalizing eq . ( [ h - free ] ) we arrive at the following band dispersion functions : @xmath33 where @xmath34 is called helicity . therefore , the bands are nondegenerate ( except the lines or points where @xmath35 ) , invariant with respect to all operations from @xmath19 , and also even in @xmath15 . the last property is a consequence of time reversal symmetry . indeed , the bloch states @xmath36 and @xmath37 belong to @xmath15 and @xmath38 , respectively , and have the same energy . here @xmath39 is the time reversal operation for spin-@xmath40 particles and @xmath41 is the complex conjugation . since the bands are nondegenerate , one can write @xmath42 , where @xmath43 is a nontrivial phase factor.@xcite the latter is given by @xmath44 for the model described by eq . ( [ h - free ] ) . let us now introduce an external magnetic field @xmath5 . we consider only a 2d superconductor , with the field parallel to its plane , therefore the vector potential and the orbital effects of the field can be neglected . adding the zeeman interaction @xmath45 to eq . ( [ h - free ] ) , where @xmath46 is the bohr magneton , and diagonalizing the resulting hamiltonian we obtain the following energy eigenvalues : @xmath47 . assuming that the zeeman energy is small compared with the so coupling and expanding these eigenvalues to the first order in @xmath5 , we have@xcite @xmath48c^\dagger_{\bk\lambda}c_{\bk\lambda}.\ ] ] therefore , the bands are asymmetrically deformed by the field and no longer even in @xmath15 . this allows for the cooper pairing with a nonzero centre - of - mass momentum to occur , leading to a variety of field - induced nonuniform superconducting states , which are discussed in the subsequent sections . we assume the following hierarchy of the energy scales : @xmath49 , where @xmath50 is the superconducting critical temperature , @xmath51 is the energy cutoff of the pairing interaction , and @xmath52 is the fermi energy , which is a good assumption in realistic noncentrosymmetric superconductors.@xcite then the so - split bands are sufficiently separated for the pairs to form independently in each band , and it is natural to use the basis of the helicity band eigenstates to introduce an attractive interaction between electrons in the cooper channel . following the standard bcs ideology , we assume that the pairing interaction is only effective when the quasiparticle momenta are close to the fermi surfaces and , therefore , the interband pairing , i.e. the formation of the cooper pairs of electrons with opposite helicities , is suppressed . this leads us to the pairing hamiltonian @xmath53 where @xmath54 is the system volume and @xmath55 is the pairing interaction function . the latter has the form @xmath56 where @xmath57 can be represented as a bilinear combination of the basis functions of irreducible representations of @xmath19 and , therefore , is amenable to the usual symmetry analysis.@xcite note that the dependence of the pairing interaction on the center - of - mass momentum of the pairs is neglected , which is legitimate since @xmath58 is small compared to the fermi momenta in the two bands . the simplest model corresponds to isotropic basis functions of the unit representation , with @xmath59 the strength of intraband pairing is described by the coupling constants @xmath60 and @xmath61 , while that of interband pairing by @xmath62 . in a uniform superconducting state , using the mean - field approximation to decouple the interaction in eq . ( [ h int reduced ] ) we arrive at the following expression : @xmath63,\ ] ] where @xmath64 is the gap function in the @xmath65th band ( we have omitted a @xmath66-number term on the right - hand side of the last equation ) . in the isotropic pairing model , see eq . ( [ isotropic - pairing ] ) , the gap functions have the form @xmath67 where @xmath68 play the role of the superconducting order parameters . the model defined by eqs . ( [ h_0 ] ) and ( [ h int reduced ] ) is formally similar to the two - band bcs theory,@xcite whose applications to mgb@xmath0 , iron - based superconductors , and other systems have been extensively studied recently . the order parameter is represented by two complex functions @xmath69 and @xmath70 . an obvious difference is that in our case the two bands are nondegenerate and the gap functions contain the phase factors @xmath71 . however , these phase factors do not affect the bulk observable quantities which are determined by the quasiparticle excitations , such as spin susceptibility and electronic specific heat , nor do they enter the gl free energy , which is expressed in terms of @xmath69 and @xmath70 . the order parameters in the helicity band representation are related to the singlet ( @xmath72 ) and `` protected '' triplet ( @xmath73 ) components in the spin representation as follows : @xmath74 and @xmath75 ( ref . ) . in particular , in the case of a bcs - like point attraction all the coupling constants in eq . ( [ isotropic - pairing ] ) are the same : @xmath76 , and we have @xmath77 , i.e. the pairing is isotropic singlet . in the opposite case of a `` single - band '' model , either @xmath69 or @xmath70 is zero . in general , however , both components are nonzero and different , and can also depend on coordinates . in the spin representation this translates into an order parameter having both singlet and triplet components . nonuniform superconducting states are most efficiently treated using the gl formalism . the phenomenological gl functional of a noncentrosymmetric superconductor can be obtained in the standard fashion by expanding the free energy ( or , more precisely , the difference between the free energies in the superconducting and normal states ) in powers of @xmath69 and @xmath70 and their gradients and keeping all terms allowed by symmetry . in the case of a 2d noncentrosymmetric superconductor in a parallel magnetic field @xmath5 we have the following expression for the gl free energy density : @xmath78 the intraband contributions are given by @xmath79+l_\lambda h^2|\eta_\lambda|^2,\ ] ] while @xmath80 is the band - mixing term describing the `` josephson coupling '' of the two order parameters , which originates in the interband pairing terms in eq . ( [ h int reduced ] ) . the free energy ( [ f_gl ] ) yields the following expression for the supercurrent : @xmath81 where @xmath82 is the absolute value of the electron charge . the first three terms in eq . ( [ f - pm ] ) are the usual gl uniform and gradient terms . the temperature dependence enters through the coefficients @xmath83 , which have the form @xmath84 , where @xmath85 and @xmath86 is the transition temperature the @xmath65th band would have at zero field in the absence of any interband coupling . the fourth term , sometimes called the lifshitz invariant , is linear both in gradients and in the magnetic field , and is specific to noncentrosymmetric superconductors . its origin can be traced to the deformation of the helicity bands by the field , see eq . ( [ h_0 ] ) . microscopic derivation of the lifshitz invariant can be found in ref . . it is the lifshitz invariants that are responsible for unusual nonuniform states created by the field , see secs . [ sec : helical state ] and [ sec : phase solitons ] below . the last term in eq . ( [ f - pm ] ) describes the paramagnetic suppression of the critical temperature , which is due to the change in the paramagnetic susceptibility in the superconducting state compared with the normal state . microscopic theory yields expressions for the coefficients in the gl functional in terms of the fermi - surface averages of the order parameter , the fermi velocity , and the so coupling direction @xmath87 , see ref . . assuming isotropic bands , one has the following order - of - magnitude estimates : @xmath88 where @xmath89 is the fermi - level dos and @xmath90 is the fermi velocity in the @xmath65th band , while @xmath91 is the superconducting critical temperature at zero field . one can recover the single - band and the singlet bcs limits from the general gl functional ( [ f_gl ] ) as follows . the single - band case corresponds to the absence of the interband coupling , i.e. @xmath92 . assuming @xmath93 , superconductivity appears only in the `` @xmath94 '' band , i.e. @xmath95 , at least near the critical temperature . in the bcs case , @xmath96 , and one obtains a one - component gl free energy in terms of @xmath97 , with the lifshits invariant the critical temperature , @xmath98 , of the second - order superconducting phase transition is obtained by solving the linearized gl equations . assuming @xmath99 and anticipating a possible helical instability modulated perpendicularly to the field , we seek the order parameters in the form @xmath100 . then from the free energy ( [ f_gl ] ) it follows that @xmath101 where @xmath102 . setting the determinant of the matrix on the left - hand side to zero , one arrives at the following expression for the critical temperature as a function of the magnetic field and the helical modulation wavevector : @xmath103 the actual transition temperature @xmath98 and the modulation wavevector are obtained by maximizing the above expression with respect to @xmath104 . at zero field the maximum critical temperature is achieved in the uniform state , i.e. at @xmath105 , and is given by @xmath106 at a small but nonzero field , we seek the wavevector in the form @xmath107 . expanding eq . ( [ t_c - hq ] ) in powers of @xmath108 and maximizing with respect to @xmath104 , we obtain : @xmath109 which corresponds to @xmath110 here @xmath111 and @xmath112 thus we see that , similarly to the single - component helical state,@xcite the superconducting order parameter is nonuniform , with the modulation wavevector linearly proportional to the field . the suppression of the critical temperature is quadratic in @xmath108 , which is typical of a paramagnetic pair breaking . according to eq . ( [ t_c - h ] ) , the pair breaking is weakened in the presence of the helical instability . we would like to stress that , in contrast to the larkin - ovchinnikov - fulde - ferrell nonuniform state,@xcite which only exists in paramagnetically - limited superconductors at sufficiently strong magnetic fields , the helical state appears at an arbitrarily weak field . the last term on the right - hand side of eq . ( [ f - pm ] ) , which describes the paramagnetic suppression of superconductivity , is crucial for maintaining the stability of the system at @xmath113 . if this term were not included , the linear in gradient terms would result in the critical temperature unphysically increasing as a function of the field . to avoid this , one has to assume that @xmath114 . despite the fact that the order parameter in the helical state is proportional to @xmath115 , the supercurrent is equal to zero . indeed , eq . ( [ supercurrent ] ) yields the following expression : @xmath116 where @xmath117 , and @xmath118 . substituting eq . ( [ q - h ] ) , we obtain : @xmath119 here we used the ratio of the order parameters at zero field , @xmath120 , which follows from eq . ( [ linear - gl ] ) . that the current must vanish can also be understood using the following simple argument . uniform supercurrent is obtained by differentiating the total free energy with respect to a uniform vector potential : @xmath121 , where @xmath66 is the speed of light . due to the gauge invariance , we have @xmath122 and , therefore , @xmath123 , because the free energy has a minimum at the equilibrium value of the helical modulation . although the above calculations apply in the case of a 2d isotropic noncentrosymmetric superconductor , they can be straightforwardly extended to an arbitrary in - plane symmetry . in the general case , the lifshitz invariant is bilinear in both the order parameter gradients and the magnetic field and eq . ( [ f - pm ] ) is replaced by @xmath124 where @xmath125 , and the einstein summation convention is assumed . the matrices @xmath126 and @xmath127 are symmetric , while @xmath128 is neither symmetric nor antisymmetric , in general . the isotropic case is recovered when @xmath129 , @xmath130 , and @xmath131 . in the least symmetric case of @xmath21 all elements of the matrices @xmath126 , @xmath128 , and @xmath127 are nonzero . we seek the order parameter in the form @xmath132 and obtain for the critical temperature the same expression as eq . ( [ t_c - hq ] ) . the only difference is that @xmath133 are now given by @xmath134 maximizing the critical temperature with respect to @xmath7 , we obtain : @xmath135 and @xmath136h_ih_j,\ ] ] where @xmath137 therefore , the critical temperature is suppressed by the field , quadratically in @xmath5 , but the helical modulation wavevector is no longer perpendicular to the field , in general . the supercurrent is equal to zero , for the same reason as explained above . the helical states can be observed , for instance , in tunneling experiments , which measure the density of states ( dos ) of quasiparticle excitations . in this subsection we calculate the quasiparticle spectrum in the state with @xmath138 , by solving the bogoliubov - de gennes ( bdg ) equations independently in each band . derivation of the bdg equations for a noncentrosymmetric superconductor is presented in appendix [ sec : bdg - helical ] . the helical state order parameter in the momentum representation is given by @xmath139 , and the bdg hamiltonian , see eq . ( [ h - bdg - transformed ] ) , takes the following form : @xmath140 where @xmath141 . to make the above hamiltonian diagonal in @xmath15-space , we perform a unitary transformation @xmath142 , where @xmath143 and @xmath144 is the position operator in @xmath15-space . one can see that the exponentials in @xmath145 act as finite displacement operators in momentum space , because @xmath146 . after the unitary transformation , the bdg hamiltonian becomes @xmath147 in a weak field , the helical modulation wavevector is small and one can expand the last expression in powers of @xmath7 and @xmath5 , with the following result : @xmath148 where @xmath149 describes the field - induced deformation of the fermi surface and @xmath150 is the band velocity of quasiparticles . now we are in a position to calculate the quasiparticle dos in the the @xmath65th band : @xmath151,\ ] ] where @xmath152 are the eigenvalues of the hamiltonian ( [ h - bdg - weak - field ] ) . near the fermi surface , one can integrate eq . ( [ dos ] ) over @xmath153 and obtain : @xmath154 ^ 2-|\eta_\lambda|^2}}\right\rangle_{fs},\ ] ] where the angular brackets denote the average over the @xmath65th fermi surface , restricted by the condition @xmath155 . since the dos satisfies the bdg electron - hole symmetry , @xmath156 , we focus only on the upper half of the spectrum , i.e. on @xmath157 . to make analytical progress we assume a cylindrical fermi surface with the rashba so coupling , see eq . ( [ rashba - soc ] ) , and @xmath99 , with @xmath158 . then , @xmath159 it follows from eqs . ( [ as ] ) and ( [ coeff - estimates ] ) that both the zeeman and `` helical '' contributions to @xmath160 , described by the first and second terms in @xmath161 , respectively , are of the same order . from eq . ( [ dos - final ] ) we obtain the following expression for the dos : @xmath162 where @xmath163 the angular integration in here is restricted by the condition @xmath164 . we calculated the last integral numerically , with the result presented in fig . [ fig : dos ] . this plot reveals two prominent features . first , the gap in the quasiparticle spectrum is given by @xmath165 , i.e. it is smaller in the helical state than in the uniform state . second , the inverse - square - root dos singularity , which is a hallmark of the bcs theory , is replaced by a much weaker singularity at @xmath166 . a straightforward analytical evaluation of eq . ( [ ixy ] ) gives @xmath167 immediately above the gap edge , and @xmath168 near the peak . tunneling experiments probe the total dos , which is given by @xmath169 and sketched in fig . [ fig : dos - total ] . th band , for @xmath170 . the gap edge is shifted to @xmath171 , while the logarithmic singularity occurs at @xmath172.,width=226 ] in this section we show that a soliton - like texture is spontaneously formed in the superconducting state above a certain critical magnetic field . phase solitons are the simplest topological defects that can exist in a two - band superconductor . in a phase soliton the relative phase of the two order parameters exhibits a kink - like variation of @xmath8 , approaching its mean - field value at infinity.@xcite according to sec . [ sec : two - band picture ] , noncentrosymmetric superconductors can be viewed as two - band systems and , therefore , are expected to support phase solitons . we assume the in - plane isotropy and set @xmath99 . to make analytical progress , we work at zero temperature and employ the london approximation , in which the gap magnitudes are constant . considering a planar texture perpendicular to the @xmath173 axis , the order parameter components are given by @xmath174 . the supercurrent in this state is given by eq . ( [ current - x ] ) . the current conservation implies that @xmath175 , where the value of the constant is set by external sources and can be assumed to be zero . the condition @xmath123 allows one to express the gradients of @xmath176 and @xmath177 in terms of the gradient of the relative phase @xmath178 : @xmath179 where @xmath180 it follows from eq . ( [ phi - grads ] ) that one can have two very different physical situations , depending on whether the phases @xmath176 and @xmath177 are locked together , with @xmath181 taking a constant value throughout the system , or they are allowed to vary independently , leading to a spatially - nonuniform @xmath181 . while the former case corresponds to the helical states considered in sec . [ sec : helical state ] , in the present section we focus on the latter possibility . substituting eq . ( [ phi - grads ] ) into eq . ( [ f_gl ] ) , we obtain : @xmath182 . here the ellipsis denote the terms which depend only on @xmath183 and @xmath184 , while the contributions containing the relative phase have the form @xmath185 where @xmath186 and @xmath187 according to eq . ( [ coeff - estimates ] ) , we have the following estimate : @xmath188 . without loss of generality we assume that @xmath189 . , in the two - band helical state.,width=226 ] for concreteness we consider only the case @xmath190 ( interband attraction ) , which means that if the relative phase is constant then it is equal to zero . the results for @xmath191 are obtained by shifting the whole phase texture by @xmath192 . for the free energy difference , @xmath193 , between the state with a nonuniform @xmath194 and the state with @xmath195 we obtain : @xmath196 where @xmath197 . variational minimization of the last expression yields the following nonlinear differential equation for the relative phase : @xmath198 in addition to the uniform solution @xmath195 , this equation also has various nonuniform ones corresponding to phase solitons or soliton lattices . for example , a single soliton connects @xmath195 at @xmath199 and @xmath200 at @xmath201 and has the following explicit form : @xmath202 $ ] , where @xmath203 is the soliton width . while the last term in eq . ( [ delta f ] ) is a full derivative and does not contribute to the equation of motion , it does affect the free energy . in fact it is easy to see that this term is responsible for a phase transition in the system , which is controlled by the external field . qualitatively , the parameter @xmath204 , which is proportional to the magnetic field , provides a bias favoring a nonzero average gradient of the relative phase . when this bias becomes large enough to overcome the energy cost of creating the solitons , the latter will be spontaneously formed in the system . denoting the single - soliton energy by @xmath205 and assuming that the density of solitons @xmath206 is low , i.e. the spacing between them is much greater than @xmath207 , one can expand the difference between the total free energies of the system with and without solitons in powers of @xmath206 : @xmath208 where @xmath209 is the length of the system in the @xmath173 direction . the neglected terms , denoted by the ellipsis , take into account the interaction between the solitons . one can see that at @xmath210 , where @xmath211 the leading term in eq . ( [ delta - f - n_s ] ) becomes negative , resulting in the proliferation of phase solitons . the transition between the state with a uniform relative phase and the soliton lattice state is mathematically similar to the commensurate - incommensurate transition of noble gas atoms adsorbed on a periodic substrate.@xcite banishing the technical details of the solution to appendix [ sec : sl - transition ] , here we present only the results . the critical magnetic field above which the phase soliton lattice is formed is given by @xmath212 see eqs . ( [ h_s - v_0 ] ) and ( [ h - def ] ) . note that the critical field of the superconducting transition in 2d noncentrosymmetric systems diverges at @xmath213 , see refs . and , therefore , the soliton instability of the superconducting state is always realized at strong enough fields . according to eq . ( [ l near h_s ] ) , the spacing between solitons grows logarithmically at @xmath214 : @xmath215 the relative phase of the order parameter components in the soliton lattice at @xmath216 is shown in fig . [ fig : theta ] . it follows from eq . ( [ phi - grads ] ) that at low fields , @xmath217 , when the relative phase is uniform , the order parameters in the two bands are still nonuniform and given by @xmath218 . thus the two - band helical state studied in sec . [ sec : helical state ] is recovered . , between the two bands , at high magnetic fields , @xmath216 ( @xmath219 is the soliton lattice period).,width=226 ] we have developed a general theory of the helical instability in two - dimensional noncentrosymmetric superconductors , taking into account the two - component nature of the order parameter in these systems . we have found that the paramagnetic pair breaking is weakened in the presence of the helical modulation . the quasiparticle dos in the helical state is significantly different from that in a uniform state , showing field - dependent gap edges and logarithmic singularities , which could be probed in tunneling expreriments . we have also found a novel type of field - induced nonuniform superconducting state , namely the lattice of phase solitons . unlike the previously studied nonuniform states in noncentrosymmetric superconductors , the phase solitons appear only in the two - component model . in contrast to the phase solitons in centrosymmetric two - band superconductors , which are difficult to create , the soliton instability predicted in this paper is always present at sufficiently strong magnetic fields . the transition into the soliton state takes place when the bias provided by the lifshitz invariants in the gl free energy , which are unique to noncentrosymmetric systems , overcomes the energy cost of creating a soliton . this phase transition should show up as a feature in the high - field low - temperature portion of the phase diagram . one can also expect that the spatial inhomogeneity of the order parameters will result in a qualitative modification of the quasiparticle spectrum , similar to the soliton bound states in centrosymmetric two - band superconductors,@xcite which could be studied by tunneling . this and other issues , such as the fate of the soliton state at finite temperatures or in the presence of the orbital effects of magnetic field , will be studied elsewhere . this work was supported by a discovery grant from the natural sciences and engineering research council of canada . the bdg equations , which determine the quasiparticle spectrum in an arbitrary nonuniform superconducting state , can be derived using the standard mean - field approach of the bcs theory , with some modifications pertinent to noncentrosymmetric superconductors . the starting point is the hamiltonian @xmath220 , where @xmath221 is the noninteracting part , with @xmath141 being the band dispersion function deformed by the magnetic field , see eq . ( [ h_0 ] ) , and @xmath222 is the pairing interaction given by eq . ( [ h int reduced ] ) . decoupling the latter in the mean - field approximation , we obtain : @xmath223\nonumber\\ -\frac{1}{2{\cal v}}\sum_{\bk\bk'\bq}\sum_{\lambda\lambda'}\delta^*_\lambda(\bk,\bq)v^{-1}_{\lambda\lambda'}(\bk,\bk')\delta_{\lambda'}(\bk',\bq).\end{aligned}\ ] ] here @xmath64 is the gap function in the @xmath65th band , which satisfies the self - consistency equation @xmath224 and @xmath225 in the second term in eq . ( [ h_int - mf ] ) should be understood as the inverse matrix both in @xmath15- and @xmath65-spaces . separating the phase factors @xmath71 introduced in sec . [ sec : two - band picture ] , we have @xmath226 and , therefore , @xmath227 , where the @xmath15-dependence of @xmath228 is determined by the basis functions of an irreducible representation of the point group . assuming an isotropic pairing , which corresponds to the unit representation , we have @xmath229 and @xmath230 two complex functions @xmath69 and @xmath70 comprise the order parameter of our noncentrosymmetric superconductor . introducing the nambu operators in each band , @xmath231 , one can write the fermionic part of the mean - field hamiltonian in the following form : @xmath232 where @xmath233 are the matrix elements of the bdg hamiltonian in momentum representation . thus the spectrum of the bogoliubov quasiparticles can be found independently in each band . the phase factors in the off - diagonal elements of the hamiltonian ( [ h - bdg ] ) can be removed by a unitary transformation : @xmath234 where @xmath235 the matrix elements of the transformed hamiltonian in momentum space have the following form : @xmath236 we see that the phase factors @xmath71 do not affect the quasiparticle spectrum in a nonuniform superconducting state ( as long as there is no impurities or external fields in the diagonal elements of the bdg hamiltonian ) . in a uniform state , the order parameter has the form @xmath237 , and the diagonalization of eq . ( [ h - bdg - transformed ] ) yields @xmath238 for the energy of an excitation with wavevector @xmath15 . it is straightforward to check that eq . ( [ theta - eq ] ) has the following integral of motion : @xmath239 this admits a simple mechanical analogy : interpreting @xmath181 as a coordinate and @xmath173 as a time variable , @xmath240 has the meaning of the total energy of a pendulum of mass @xmath241 in the potential @xmath242 . if @xmath243 , then the pendulum oscillates near one of the minima of the potential , which corresponds to a periodic modulation of the relative phase in real space . the case @xmath244 corresponds to the pendulum completing just one full rotation from @xmath245 to @xmath8 , or to a single phase soliton connecting @xmath195 at @xmath199 and @xmath200 at @xmath201 . if @xmath246 then the pendulum has enough energy to complete an infinite number of full rotations , which corresponds to a soliton lattice . it follows from eq . ( [ int - of - motion ] ) that @xmath247 which implicitly determines @xmath181 as a function of @xmath173 . from the last expression we can immediately obtain the soliton lattice period as a function of @xmath240 : @xmath248 focusing on the vicinity of the transition at @xmath244 we have @xmath249 with logarithmic accuracy at @xmath250 . the next step is to relate @xmath251 and @xmath219 to the magnetic field @xmath204 . using eq . ( [ delta f ] ) and the fact that the phase winding per soliton is equal to @xmath8 , we obtain the following expression for the free energy density : @xmath252 where @xmath253\ ] ] is the free energy per one cell of the soliton lattice . since the lattice period diverges at the transition , we obtain from eq . ( [ delta f over l ] ) the following expression for the critical field : @xmath254 . introducing the single - soliton energy @xmath255 , we recover eq . ( [ h_s ] ) . in order to calculate @xmath256 and @xmath205 , we use the integral of motion , eq . ( [ int - of - motion ] ) , and obtain : @xmath257\nonumber\\ & = & \int_0^{2\pi}d\theta\sqrt{2{\cal e}+2v_0(1-\cos\theta)}-{\cal e}\ell.\end{aligned}\ ] ] the second line follows after changing the variable , @xmath258 , in the first one and using the expression for @xmath259 from eq . ( [ x vs theta ] ) . putting @xmath244 , we have @xmath260 . the lattice period is determined by minimizing the free energy , eq . ( [ delta f over l ] ) , with respect to @xmath219 . in this way , we obtain : @xmath261 it follows from eqs . ( [ epsilon_1 ] ) and ( [ sl - period - gen ] ) that @xmath262 . substituting this and eq . ( [ epsilon_1 ] ) into eq . ( [ epsilon - derivative ] ) , we have @xmath263 this last equation implicitly determines @xmath251 and , therefore , the lattice period , as functions of the field . the critical field of the soliton transition is given by @xmath264 m. a. tanatar , j. paglione , s. nakatsuji , d. g. hawthorn , e. boaknin , r. w. hill , f. ronning , m. sutherland , l. taillefer , c. petrovic , p. c. canfield , and z. fisk , phys . lett . * 95 * , 067002 ( 2005 ) . d. r. tilley , proc . . soc . * 84 * , 573 ( 1964 ) ; m. e. zhitomirsky and v .- h . dao , phys . b * 69 * , 054508 ( 2004 ) ; m. silaev and e. babaev , phys . rev . b * 85 * , 134514 ( 2012 ) ; e. babaev and m. silaev , phys . rev . b * 86 * , 016501 ( 2012 ) . the zeeman interaction is not diagonal in the helicity basis , leading to the interband transitions caused by the magnetic field . the effect of these transitions on the helicity band quasiparticles is quadratic in @xmath108 and is neglected if the zeeman energy is much smaller than the so band splitting .

we show how the two - band nature of superconductivity in noncentrosymmetric compounds leads to a variety of novel nonuniform superconducting states induced by a magnetic field . at low fields , a two - band helical state is realized , with a distinctly non - bcs quasiparticle spectrum . at high fields , the superconducting state becomes unstable towards the formation of a lattice of topological phase solitons .

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Proceed to summarize the following text: it is widely believed , as suggested by a host of independent cosmological and astrophysical observations , that the most part of the matter in the universe is not visible , revealing its existence only through gravitational effects . in particular , data on the rotational curves of galaxies indicate that the galactic visible parts are surrounded by approximately spherical dark halos which extend up to several times the size of the luminous components . the best candidates to provide dark matter in galaxies are weakly interacting massive particles ( wimp ) . several wimp direct detection experiments are operating @xcite , with the goal of measuring the nuclear recoil energy ( in the kev range ) expected to be deposited in solid , liquid or gaseous targets by the scattering of the non - relativistic dark halo wimps . unfortunately , expected rates are small and the exponential decay of the wimp recoil spectrum resembles that of the background at low energies . however , a specific signature can be exploited in order to disentangle a wimp signal from the background : the annual modulation of the rate @xcite . this effect , expected to be of the order of a few per cent , is induced by the rotation of the earth around the sun . due to its smallness , the annual modulation signature requires large mass detectors with high statistics in order to overcome background fluctuations and be unambiguously detected . the annual modulation effect has been experimentally investigated by the dama collaboration , which has indeed reported a positive evidence by using a 100 kg sodium iodide detector @xcite . one of the most important sources of uncertainty in the calculation of wimp direct detection rates is the modeling of the velocity distribution function ( df ) of the particles populating the dark halo . in the literature , a simple isothermal sphere model is usually adopted , i.e. a wimp gas described by an isotropic maxwellian with r.m.s velocity of the order of 300 km s@xmath0 this leads to a sinusoidal time dependence of the expected signal with maximum ( or minimum ) around june 2@xmath1 , i.e. with the same ( or opposite ) phase as the relative velocity between the earth and the halo rest frame . however , the actual form of the wimp velocity df is unknown , and many different models , alternative to the isothermal sphere , are compatible with observations @xcite . the goal of the present letter is to show that the main features of the annual modulation effect ( the sinusoidal dependence with time , the occurrence of maxima and minima during the year and , under some circumstances , the one year period ) may be affected by anisotropies in the velocity df . the most relevant effect is a distortion of the sinusoidal time behaviour at low recoil energies . these energies , though below the current detector thresholds , might be reached in the future . the observation of the effects discussed in this letter could provide informations on the phase space distribution of our galactic halo , especially on the degree of its anisotropy . due to the rotation of the disk around the galactic center , the solar system moves through the wimp halo , assumed to be at rest in the galactic rest frame . in the following , we will assume a right handed system of orthogonal coordinates : the @xmath2 axis in the galactic plane , pointing radially outward ; the @xmath3 axis in the galactic plane , pointing in the direction of the disk rotation ; the @xmath4 axis directed upward , perpendicular to the galactic plane . notice that our system differs from standard `` galactic coordinates '' by the different choice of the @xmath2 axis , which for us is directed outward . the relative velocity between the wimp halo and the detector is given by the earth velocity @xmath5 , as seen in the galactic rest frame . it is the sum of three components : the galactic rotational velocity @xmath6 km s@xmath0 ( we will assume : @xmath7 km s@xmath0 ) , the sun proper motion @xmath8 km s@xmath0@xcite and the earth orbital motion @xmath9@xcite : @xmath10 \label{eq : vearthx } , \\ v^e_y & = & v^g_y + v^s_y + u^e(\lambda ) \cos\beta_y \cos[\omega ( t - t_y ) ] \label{eq : vearthy } , \\ v^e_z & = & v^g_z + v^s_z + u^e(\lambda ) \cos\beta_z \cos[\omega ( t - t_z ) ] \label{eq : vearthz},\end{aligned}\ ] ] where @xmath11 is the ecliptic longitude , which is function of time . we can express @xmath11 as @xcite : @xmath12 where @xmath13 and @xmath14 and @xmath15 denotes the time expressed in days relative to ut noon on december 31 . in eqs.([eq : vearthx][eq : vearthz ] ) @xmath16 $ ] is the modulus of the earth rotational velocity , which slightly changes with time due to the small ellipticity @xmath17 of the earth orbit ( @xmath18 km s@xmath0 , @xmath19 and @xmath20 ) @xcite . in eqs.([eq : vearthx][eq : vearthz ] ) the @xmath21 denote the ecliptic latitudes and the @xmath22 are the phases of the three velocity components : @xmath23 , @xmath24 , @xmath25 and @xmath26 day , @xmath27 day , @xmath28 day . the angular velocity has a period of 1 year , and is given by : @xmath29 . with the numbers given above , the modulus of the earth velocity changes in time as : @xmath30 $ ] ( in km s@xmath0 ) , where @xmath31 days , i.e. june 2@xmath32 . notice the slight offset between @xmath33 and @xmath34 , due to the composition of the velocity components . as a good approximation , @xmath35 is usually taken as : @xmath36 $ ] . the direct detection differential rate @xmath37 is proportional to the integral : @xmath38 where @xmath39 and @xmath40 are the wimp velocity df and the wimp velocity , respectively , in the earth s rest frame ; @xmath41 is the minimum value of @xmath40 for a given recoil energy @xmath42 , wimp mass @xmath43 and nuclear target mass @xmath44 and is given by : @xmath45 . by indicating with @xmath46 and @xmath47 the wimp velocity df and the wimp velocity in the galactic reference frame , the following transformations hold : @xmath48 which imply that @xmath49 , and so @xmath37 , develops a time dependence induced by @xmath50 . time dependence of @xmath49 for an isotropic isothermal sphere . the different curves refer to values of @xmath41 ranging from 0 ( lower ) to 400 ( upper ) km s@xmath0 , in steps of 20 km s@xmath0 ( the dashed curves correspond to @xmath51 , 200 and 300 km s@xmath0 ) . the vertical dashed line denotes @xmath52 days . ] in the case of the isothermal sphere model the df is given by a truncated isotropic maxwellian which depends only on @xmath53 : @xmath54 where @xmath55 is the wimp r.m.s . velocity , given by : @xmath56 . it is clear that , through the change of reference frame of eq . ( [ eq : transformation ] ) , @xmath49 depends on time only through @xmath35 . since the relative change of @xmath35 during the year is of the order of a few per cent , we can approximate @xmath49 with its first order expansion in the small parameter @xmath57 , around its mean value @xmath58 : @xmath59 $ ] . the well known result is then obtained that the wimp rate has a sinusoidal time dependence with the same phase ( @xmath60 june 2@xmath1 ) as @xmath35 , for all values of @xmath41 . this is shown in fig.[fig : ivmin_isotropic ] , where @xmath49 is plotted as a function of time for various values of @xmath41 . the simplest generalization of the isothermal sphere model is given by a triaxial system described by a multivariate gaussian : @xmath61 where @xmath62 is the normalization constant . for @xmath63 , then @xmath64^{-1/2}$ ] . the usual isothermal sphere is the spherical limit of eq.([eq : triaxialgs ] ) , obtained with : @xmath65 . in order to discuss the effect of anisotropy at fixed wimp mean kinetic energy , we will fix @xmath66 as in the isothermal case ( @xmath56 ) and discuss our results in terms of the two independent parameters : @xmath67 and @xmath68 . at variance with the isothermal sphere , now @xmath49 depends in general on all the three components of @xmath69 , and not simply on @xmath35 . we can write : @xmath70 , \label{eq : triaxiales}\ ] ] where we have defined the reduced ( dimensionless ) variables : @xmath71 and @xmath72 $ ] ( @xmath73 ) . in the isotropic case ( isothermal sphere ) one has @xmath74 , @xmath75 and @xmath76 . the presence of the order one parameter @xmath77 and of the small oscillation amplitudes @xmath78 allows the taylor expansion of @xmath49 in terms of @xmath79 parameters . a straightforward calculation shows that the conclusions of the previous section are recovered . the same as in fig . [ fig : ivmin_isotropic ] , for an anisotropic model with @xmath80 and @xmath81 . ] on the contrary , allowing anisotropies such that @xmath82 and/or @xmath83 , the values of the parameters @xmath84 and @xmath85 are enhanced , and the time dependence of @xmath86 and @xmath87 in @xmath49 may become important . an example of this situation is shown in fig . [ fig : ivmin_tangential ] , where , the time evolution of @xmath49 is plotted for @xmath88 and @xmath89 ( i.e. : @xmath90 km s@xmath0 . ) this choice of @xmath91 , @xmath92 , which refers to a tangential anisotropy , corresponds to triaxial models discussed , for instance , in refs . @xcite . a distortion of the curves of fig . [ fig : ivmin_tangential ] , as compared to the familiar sinusoidal time dependence , appears : this effect may be explained by the fact that now @xmath93 , a taylor expansion of the type used in the isothermal sphere case breaks down and a full numerical calculation of the integral of eq.([eq : ivmin ] ) is required . the final result is not sinusoidal . this peculiar behaviour is more pronounced at low values of @xmath41 ( i.e. low recoil energies ) namely for @xmath94 80 km s@xmath0 , for which the distortion is strong and the maxima ( in absolute value ) of the rate are shifted as compared to the standard case @xcite . for larger values of @xmath41 the distortion is less pronounced , and it dies away when @xmath95 km s@xmath0 . this may be explained by the fact that , as @xmath41 grows , the integral of eq.([eq : ivmin ] ) becomes less sensitive to the parameter @xmath96 since it gets increasingly dominated by wimps with velocities along the @xmath3 axis , which is the one along which the boost due to the galactic rotation is directed . we notice that for values of @xmath41 around ( @xmath97 ) km s@xmath0 a distortion is present , but the amplitude of the modulation is suppressed and therefore difficult to detect . the same as in fig . [ fig : ivmin_isotropic ] , for an anisotropic model with @xmath98 and @xmath99 . the curves refer to : @xmath100 ( lower ) , 180 , 190 , 200 ( dashed ) , 210 , 220 , 230 , 240 , 250 ( upper ) km s@xmath0 . ] as a second example , in fig.[fig : ivmin_radial ] we plot @xmath49 as a function of time for the case @xmath101 , @xmath102 ( i.e. : @xmath103 km s@xmath0 . this situation is representative of a radial anisotropy . in this case , we have further enhanced the contribution of @xmath104 over @xmath85 , so that one should expect to draw the same conclusions as in the isothermal sphere case , with the usual sinusoidal time dependence of the rate and a phase close to @xmath105 . this is indeed the case , except for a narrow interval of values of @xmath41 around @xmath106 210 km s@xmath0 . in this range , @xmath49 develops two maxima because , for that particular choice of @xmath41 , there is an exact cancellation in the first term of the expansion in @xmath107 , so that the term @xmath108 proportional to @xmath109 $ ] sets in . this particular cancellation of the first term in the taylor expansion of @xmath49 happens also for the isothermal sphere model , but in that case the size of the quadratic term is strongly suppressed because @xmath110 is much smaller . in a wimp direct detection experiment this effect would show up in a very peculiar way : a halving of the modulation period of the rate in a narrow range of recoil energies . note , however , that in order to have some realistic chance to detect this effect , it should show up in one of the experimental energy bins just above threshold , where the highest signal / background ratio is usually attained . values of @xmath41 as a function of the quenched nuclear recoil energies @xmath111 , for : nai scintillators @xcite , ge ionization detectors @xcite and te bolometers @xcite . for each panel , the different curves refer to wimp masses of : 20 gev ( solid ) , 50 gev ( dotted ) , 100 gev ( dashed ) , 200 gev ( long dashed ) and 1 tev ( dot dashed ) . the dashed vertical lines denote the current energy thresholds . ] in order to establish a link between our discussion and wimp direct detection experiments @xcite , in fig . [ fig : vmin ] we plot @xmath41 as a function of the quenched nuclear recoil energy @xmath112 ( @xmath113 is the quenching factor ) for the target nuclei : na , i , ge , te and for different wimp masses . the vertical dashed lines show current energy thresholds achieved by each type of detector . [ fig : vmin ] shows that values of @xmath94 80 km s@xmath0 , i.e. sufficiently low to observe a sizeable distortion effect as the one discussed for tangential anisotropy , correspond to wimp recoil energies below the threshold of present direct detection experiments , and that the effect would be more easily detected at higher wimp masses . however , a foreseeable lowering of the threshold , down to 0.51 kev , could be enough to observe the distortion . on the other hand , for radial anisotropy , we can conclude that the recoil energy corresponding to a halving of the modulation period can actually coincide to the experimental thresholds within the reach of present day detectors for 20 gev @xmath114 100 gev , depending on the particular target nucleus . in this respect , we note that the properties of the annual modulation effect observed by the dama / nai experiment @xcite ( a one year period sinusoidal behaviour in the 26 kev energy bins @xcite ) implies that the dama / nai experiment is already able to set constraint on strong radial anisotropies . in the present letter we have shown that the main features of the annual modulation of the signal of wimp direct searches , i.e. the sinusoidal dependence of the rate with time , the position of its maxima and minima during the year and even the period , may be affected by relaxing the isothermal sphere hypothesis in the description of the wimp velocity phase space . we have considered a multivariate gaussian and found that different situations may occur , depending on the pattern of anisotropy : tangential anisotropies induce a departure at low energies from the usual sinusoidal time dependence , along with a shift in the position of the maximum of the signal during the year , while radial anisotropies may produce a halving of the modulation period in a particular energy bin . the former effect turns out to be relevant at low recoil energies , actually below the threshold of present day experiments , while the latter should be already within the reach of current detectors . in particular , the properties of the annual modulation effect observed by the dama / nai experiment @xcite may already indicate that strong radial anisotropies are excluded .

we show that the main features of the annual modulation of the signal expected in a wimp direct detection experiment , i.e. its sinusoidal dependence with time , the occurrence of its maxima and minima during the year and ( under some circumstances ) even the one year period , may be affected by relaxing the isothermal sphere hypothesis in the description of the wimp velocity phase space . the most relevant effect is a distortion of the time behaviour at low recoil energies for anisotropic galactic halos . while some of these effects turn out to be relevant at recoil energies below the current detector thresholds , some others could already be measurable , although some degree of tuning between the wimp mass and the experimental parameters would be required . either the observation or non observation of these effects could provide clues on the phase space distribution of our galactic halo .

You are an expert at summarizing long articles. Proceed to summarize the following text: since the invention of neutrino beams at accelerators and the consequent discovery of the two flavors of neutrinos@xcite , the reactions @xmath3 and @xmath4 , which are the dominant reactions of muon and electron neutrinos with energies from @xmath5 mev to @xmath6 gev , have played an important role in studies of neutrino flavor . these charged - current quasi - elastic ( ccqe ) interactions are important not only because they identify the flavor of the neutrino through the production of the lepton in the final state , but also because the two body kinematics permit a measurement of the neutrino energy from only the observation of the final state lepton . accelerator neutrino experiments like t2k@xcite , nova@xcite and a number of proposed experiments seek to make precision measurements of the neutrino flavor oscillations @xmath0 or @xmath7 in order to determine the mass hierarchy of neutrinos and to search for cp violation in neutrino oscillations . uncertainties on differences between these cross - sections for muon and electron neutrinos will contribute to experimental uncertainties in these flavor oscillation measurements . interactions of the charged - current with fundamental fermions , such as @xmath8 , have no uncertainties in the differences between the reactions for muon and electron neutrino interactions because the weak interaction is experimentally observed to be flavor universal . in particular , the effect of the final state lepton mass in this two body reaction of fundamental fermions can be unambiguously calculated . one such calculable difference occurs because of radiative corrections to the tree - level ccqe process . radiative corrections from a particle of mass @xmath9 in a process with momentum transfer @xmath10 are of order @xmath11 , which implies a significant difference due to the mass of the final state lepton@xcite . although this effect is calculable , it is not accounted for in neutrino interaction generators used in recent analysis of experimental data , such as genie@xcite , neut@xcite and nuance@xcite . there are , however , cross - section differences due to lepton mass which can not be calculated from first principles with current theoretical tools . the presence of the target quarks inside a strongly bound nucleon lead to a series of _ a priori _ unknown form factors in the nucleon level description of the reaction , e.g. , @xmath1 . it is the uncertainties on these form factors combined with the alteration of the kinematics due to lepton mass that leads to uncertainties , and that is the focus of the results of this paper . there is also a modification of the reaction cross - sections due to the effects of the nucleus in which the target nucleons are bound . the model incorporated in genie@xcite , neut@xcite and nuance@xcite is a relativistic fermi gas model@xcite which provides a distribution of nucleon kinematics inside the nucleus . a more sophisticated description from a nuclear spectral function model@xcite is implemented in the nuwro generator@xcite . we do not consider the effect of the nucleus in this work , although it may be important in the relative weighting of nucleon kinematics at low energy . however , this work provides a blueprint for studying the effect of the final state lepton mass in different nuclear models . the cross section for quasi - elastic scattering of neutrinos at energies relevant for oscillation experiments may be calculated from the fermi theory of weak interactions with the effective lagrangian , @xmath12 where @xmath13 is the fermi constant and the @xmath14 are the leptonic and hadronic currents . the form of the leptonic current is specified by the theory to be @xmath15 because the leptons are fundamental fermions . however , as mentioned above the hadronic current for quasi - elastic scattering depends on unknown form factors of the nucleons . the hadronic current can be decomposed into vector and axial components , @xmath16 @xmath17 contains three terms related to the vector form factors @xmath18 , @xmath19 and @xmath20 , and @xmath21 contains three terms related to the axial form factors @xmath22 , @xmath23 and @xmath24 . a description the the bilinear covariant structure of the currents is given in several standard texts and review papers@xcite . we follow most closely the notation of ref . . from the effective lagrangian of eq [ eq : efflagranian ] and currents above in eqs . [ eq : leptoniccurrent ] and [ eq : hadroniccurrent ] , the quasi - elastic cross section on free nucleons is : @xmath25 \nonumber\\ & & \times\frac{m^2 g_f^2 \cos^2 \theta_{c}}{8 \pi e_{\nu}^2}\end{aligned}\ ] ] where the invariant @xmath26 and @xmath27 is the four momentum transfer from the leptonic to hadronic system , @xmath28 is the mass of the nucleon , @xmath29 is the cabibbo angle , and @xmath30 is the neutrino energy in the lab . the combination of mandelstam invariants @xmath31 and @xmath32 can be written as , @xmath33 where @xmath9 is the mass of the final state lepton . the functions a(@xmath34 ) , b(@xmath34 ) and c(@xmath34 ) depend on the nucleon form factors and @xmath35 , the difference between the anomalous magnetic moment of the proton and the neutron : @xmath36 , \\ % \end{split } % \end{equation } % \begin{equation } \label{eq : bfunc } % \begin{split } b(q^2 ) % & & = & \frac{q^2}{m^2 } re f_a^ * \left ( f_v^1 + \xi f_v^2\right ) % & - \frac{m^2}{m^2 } re \left [ \left ( f_v^1 -\frac{q^2}{4 m^2 } \xi f_v^2 \right ) ^ * f_v^3 \right . % & \left . - \left ( f_a - \frac{q^2 f_p}{2 m^2 } \right)^ * f_a^3 \right ] { \rm\textstyle and}\\ % \end{split } % \end{equation } % \begin{equation } \label{eq : cfunc } c(q^2 ) & = & \frac{1}{4 } \left ( \vert f_a \vert ^2 + \vert f_v^1 \vert ^2 + \frac{q^2}{m^2 } \left| \frac{\xi f_v^2}{2 } \right| ^2 + \frac{q^2}{m^2 } \vert f_a^3 \vert ^2 \right ) . % \end{equation}\end{aligned}\ ] ] note that the form factors themselves are functions of @xmath34 in eqs . [ eq : afunc][eq : cfunc ] . @xmath18 and @xmath19 are the vector and @xmath22 and @xmath37 the axial form factors of the first class currents . first class currents conserve both time and charge symmetry . in addition , first class vector currents commute with the g - parity operator while first class axial currents anti - commute with it@xcite . the terms associated with @xmath18 and @xmath22 are considered the leading terms in the hadron current since they have no dependence on the four - momentum transfer , excepting that of the form factors , and they are not suppressed by powers of the final state lepton mass as @xmath37 is . vector elastic form factors are precisely known at @xmath38 from the nucleon electric charges and magnetic moments and are precisely measured over a wide range of @xmath34 in charged - lepton elastic scattering from protons and deuterium . the axial nucleon form factor at zero @xmath34 is precisely measured in studies of neutron beta decay . at higher @xmath34 , much of our knowledge of the axial form factors comes from muon neutrino quasi - elastic scattering measurements . for @xmath39 ( gev / c)@xmath40 , the vector form factors and the axial form factors are observed to follow a dipole form , that is @xmath41 where the constant @xmath42 is often expressed as a mass of the same order of magnitude as the mass of the nucleon . at high @xmath34 the vector form factors do not follow the dipole structure@xcite . the neutrino scattering data contains conflicting results among different experiments@xcite which limit our ability to effectively use that information to constrain the axial form factor . pion electroproduction experiments@xcite have also measured the axial form factor at @xmath43 0.2 ( gev / c)@xmath40 . the form factor @xmath37 is determined from pcac which , under minimal assumptions , states that@xcite : @xmath44 where @xmath45 is the renormalized field operator that creates the @xmath46 and @xmath42 is a constant which may be computed at @xmath38 . pcac gives the following relation between @xmath37 and the pion nucleon form factor , @xmath47 , @xmath48 where @xmath49 is the pion mass . the goldberger - treiman relation@xcite predicts @xmath50 where @xmath51 is the pion decay constant . under the assumption that the goldberger - treiman relation holds for all values of @xmath34 , then @xmath37 is @xmath52 this is the relationship that is used in all modern neutrino generators@xcite . @xmath20 and @xmath23 are form factors associated with the second class current ( scc ) . the existence of such currents requires charge or time symmetry violation , and current measurements show the size of these violations to be small . additionally a nonzero @xmath20 term would violate conservation of the vector current ( cvc ) . both @xmath53 and @xmath54 can be limited experimentally in studies of beta decay . almost all current analyses of neutrino data assume that the sccs are zero . the vector sccs only enter the cross - section in terms suppressed by @xmath55 , but there are unsuppressed terms involving the axial scc form factor . in this section , we will study the dependence of the muon - electron cross - section differences as a function of @xmath56 and @xmath34 . differences arise due to the variation of kinematic limits due to the final state lepton mass , different radiative corrections to the tree level process and uncertainties in nucleon form factors . equations [ eq : afunc ] and [ eq : bfunc ] contain explicitly the dependence of the ccqe cross - section in terms of the form factors . lepton mass , @xmath9 , enters in both @xmath57 and @xmath58 and these terms involve all the form factors discussed above . note that @xmath37 and @xmath20 _ only _ appear in terms multiplied by @xmath55 and therefore are negligible in the electron neutrino cross - section , but not in the muon neutrino cross - section . as metrics , we define the fractional differences between the muon and electron neutrino ccqe cross - sections : @xmath59 the integrals over @xmath34 in eqs . [ eq : diff ] and [ eq : intdiff ] are taken within the kinematic limits of each process , and those limits depend on lepton mass as discussed in the next section . another useful metric is the difference between a cross - section in a model with a varied assumption from that of a reference model . our reference model derives @xmath60 and @xmath61 from the electric and magnetic vector sachs form factors which follow the dipole form of eq . [ eq : dipole ] with @xmath62 ( gev / c)@xmath40 , and it assumes @xmath22 is a dipole with @xmath63 ( gev / c)@xmath40 . the reference model uses the derived @xmath37 from eq . [ eqn : fp ] , and assumes that @xmath64 at all @xmath34 . we then define : @xmath65 where @xmath66 is the reference model for @xmath67 or its anti - neutrino analogue and @xmath68 is the model to be compared to the reference . large differences between the electron and muon neutrino quasi - elastic cross - sections exist at low neutrino energies from the presence of different kinematic limits due to the final state lepton mass and due to the presence of the pseudoscalar form factor , @xmath37 , derived from pcac and the goldberger - treiman relation . these differences are typically accounted for in modern neutrino interaction generators . there are also significant differences due to radiative corrections , particularly in diagrams that involve photon radiation attached to the outgoing lepton leg which are proportional to @xmath69 . these differences are calculable , but are typically not included in neutrino interaction generators employed by neutrino oscillation experiments . if our estimate of these differences , of order @xmath70 , is confirmed by more complete analyses , then this is a correction that needs to be included as it is comparable to the size of current systematic uncertainties at accelerator experiments@xcite . modifications of the assumed @xmath37 from pcac and the goldberger - treiman relation and the effect of the form factors @xmath20 and @xmath23 corresponding to second class vector and axial currents , respectively , are not included in neutrino interaction generators . a summary of the possible size of these effects , as we have estimated them , is shown in fig . [ fig : summary ] . these differences , particularly from the second class vector currents , may be significant for current@xcite and future@xcite neutrino oscillation experiments which seek precision measurements of @xmath71 and its anti - neutrino counterpart at low neutrino energies . previous work@xcite has demonstrated sensitivity to these second class currents in neutrino and anti - neutrino quasi - elastic muon neutrino scattering , and future work with more recent data@xcite and newly analyzed data@xcite may help to further limit uncertainties on possible second class currents . the suggestion for this work came out of conversations with alain blondel about systematics in future oscillation experiments and we thank him for inspiring this work . we are grateful to ashok das , tamar friedmann and tom mcelmurry for their clear and patient explanations of the bilinear covariant structure of weak interactions . we thank arie bodek for a helpful discussion of available tests of the cvc hypothesis . we thank gabriel perdue and geralyn zeller for helpful comments on a draft of this manuscript . we are grateful to bill marciano for his helpful insights into the radiative corrections after the initial draft of this paper appeared online . a. de rujula , r. petronzio and a. savoy - navarro , nucl . b * 154 * , 394 ( 1979 ) . c. andreopoulos [ genie collaboration ] , acta phys . b * 40 * , 2461 ( 2009 ) . y. hayato , nucl . suppl . * 112 * , 171 ( 2002 ) . y. hayato , acta phys . polon . b * 40 * , 2477 ( 2009 ) . d. casper , nucl . suppl . * 112 * , 161 ( 2002 ) . smith and e.j . moniz nucl . * b43 * 605 ( 1972 ) . o. benhar , a. fabrocini , s. fantoni and i. sick , nucl . a * 579 * , 493 ( 1994 ) . j. sobczyk , pos nufact * 08 * , 141 ( 2008 ) . r.e marshak , riazuddin and c.p . ryan , _ theory of weak interactions in particle physics _ , wiley - interscience ( 1969 ) . a. bodek , s. avvakumov , r. bradford and h. s. budd , j. phys . conf . ser . * 110 * , 082004 ( 2008 ) . v. lyubushkin _ et al . _ [ nomad collaboration ] , eur . j. c * 63 * , 355 ( 2009 ) . j. l. alcaraz - aunion _ et al . _ [ sciboone collaboration ] , aip conf . proc . * 1189 * , 145 ( 2009 ) . m. dorman [ minos collaboration ] , aip conf . * 1189 * , 133 ( 2009 ) . a. a. aguilar - arevalo _ et al . _ [ miniboone collaboration ] , phys . d * 81 * , 092005 ( 2010 ) . s. choi , v. estenne , g. bardin , n. de botton , g. fournier , p. a. m. guichon , c. marchand and j. marroncle _ et al . _ , phys . lett . * 71 * , 3927 ( 1993 ) . a. liesenfeld _ et al . _ [ a1 collaboration ] , phys . b * 468 * , 20 ( 1999 ) . stephen l. adler , phys . rev . * 137 * , 10221033 ( 1964 ) . m. l. goldberger and s. b.treiman , phys . rev . * 5 * , 1178 - 1184 ( 1958 ) . k. kubodera , j. delorme and m. rho , nucl . phys . * b66 * , 253 - 292 ( 1973 ) . m. oka and k. kubodera , phys . * b90 * 45 ( 1980 ) . k. minamisono _ et al . _ , phys . rev . * c65 * , 015501 ( 2001 ) . k. minamisono _ et al . _ , phys . * c84 * , 055501 ( 2011 ) . d.h . wilkinson , eur . j. * a7 * 307 ( 2000 ) .

accelerator neutrino oscillation experiments seek to make precision measurements of the neutrino flavor oscillations @xmath0 in order to determine the mass hierarchy of neutrinos and to search for cp violation in neutrino oscillations . these experiments are currently performed with beams of muon neutrinos at energies near 1 gev where the charged - current quasi - elastic interactions @xmath1 and @xmath2 dominate the signal reactions . we examine the difference between the quasi - elastic cross - sections for muon and electron neutrinos and anti - neutrinos and estimate the uncertainties on these differences .

You are an expert at summarizing long articles. Proceed to summarize the following text: the @xmath0 model in flat space is a scalar field theory whose configuration space @xmath2 consists of finite energy maps from euclidean @xmath3to the complex projective space @xmath0 , the energy functional being constructed naturally from the riemannian structures of the base and target spaces ( that is , the model is a pure sigma model in the broad sense ) . the requirement of finite energy imposes a boundary condition at spatial infinity , that the field approaches the same constant value , independent of direction in @xmath3 , so that the field may be regarded as a map from the one point compactification @xmath4 to @xmath0 . since @xmath5 also , finite energy configurations are effectively maps @xmath6 , the homotopy theory of which is well understood , and the configuration space is seen to consist of disconnnected sectors @xmath7 labelled by an integer @xmath8 , the `` topological charge '' ( degree ) , @xmath9 each configuration is trapped within its own sector because time evolution is continuous . the lorentz invariant , time - dependent model is not integrable but complete solution of the static problem has been achieved by means of a bogomolnyi argument and the general charge @xmath8 moduli space , the space of charge-@xmath8 static solutions @xmath10 , is known ( that _ all _ static , finite energy solutions of the @xmath0 model saturate the bogomolnyi bound is a non - trivial result @xcite ) . each static solution within the charge-@xmath8 sector has the same energy ( minimum within that sector and proportional to @xmath8 ) , and @xmath11 is parametrized by @xmath12 parameters ( the moduli ) , so such a moduli space may be thought of as the @xmath13-dimensional level bottom of a potential valley defined on the infinite dimensional charge-@xmath8 sector , @xmath7 . low energy _ dynamics _ may be approximated by motion restricted to this valley bottom , a manifold embedded in the full configuration space , and thus inheriting from it a non - trivial metric induced by the kinetic energy functional . the approximate dynamic problem is reduced to the geodesic problem with this metric , and has been investigated by several authors @xcite . in the unit - charge sector one here encounters a difficulty : certain components of the metric are singular and the approximation is ill defined . for example , unit - charge static solutions are localized lumps of energy with arbitrary spatial scale , so one of the six moduli of @xmath14 is a scale parameter . motion which changes this parameter is impeded by infinite inertia in the geodesic approximation , a result in conflict with numerical evidence which suggests that lumps collapse under scaling perturbation @xcite . this problem should not be present in the model defined on a compact two dimensional physical space . the obvious choice is the @xmath15-sphere because the homotopic partition of the configuration space carries through unchanged . also , @xmath16 with the standard metric is conformally equivalent to euclidean @xmath17 , and the static @xmath0 model energy functional is conformally invariant , so the whole flat space static analysis is still valid and all the moduli spaces are known . however , the kinetic energy functional _ does _ change and induces a new , well defined metric on the unit - charge moduli space . by means of the isometry group derived from the spatial and internal symmetries of the full field theory we can place restrictions on the possible structure of this metric , greatly simplifying its evaluation . the geodesic problem is still too complicated to be solved analytically in general , but by identifying totally geodesic submanifolds , it is possible to obtain the qualitative features of a number of interesting solutions . in particular , the possibilities for lumps travelling around the sphere are found to be unexpectedly varied . the @xmath0 model on the @xmath15-sphere is defined by the lagrangian @xmath19=\int_{s^{2}}\ , ds\ , \frac{\partial_{\mu}w\partial_{\nu}\bar{w}}{(1+|w|^{2})^{2}}\ , \eta^{\mu\nu}\ ] ] where @xmath20 is a complex valued field , @xmath21 is the invariant @xmath16 measure and @xmath22 are the components of the inverse of the lorentzian metric @xmath23 on r(time)@xmath24(space ) , @xmath25 being the natural metric on @xmath16 . although the language of the @xmath0 model is analytically convenient , the homotopic classification and physical meaning of the field configurations are more easily visualized if we exploit the well known equivalence to the @xmath26 sigma model @xcite . in the latter , the scalar field is a three dimensional isovector constrained to have unit length with respect to the euclidean @xmath27 norm ( @xmath28 ) , that is , the target space is the 2-sphere of unit radius with its natural metric , which we will denote @xmath29 for clarity . ( the suffix refers to `` isospace '' in analogy with the internal space of nuclear physics models . ) the @xmath0 field @xmath20 is then thought of as the stereographic image of in the equatorial plane , projected from the north pole , @xmath30 . explicitly , @xmath31 and @xmath32 then @xmath19\equiv l_{\sigma}[{\mbox{\boldmath $ \phi$}}]=\frac{1}{4}\int_{s^{2}}\ , ds\ , \partial_{\mu}{\mbox{\boldmath $ \phi$}}\cdot\partial_{\nu}{\mbox{\boldmath $ \phi$}}\ , \eta^{\mu\nu}\ ] ] the familiar @xmath26 sigma model lagrangian . a @xmath20 configuration , then , may be visualized as a distribution of unit length arrows over the surface of the physical 2-sphere @xmath33 . each smooth map @xmath33@xmath34@xmath29 falls into one of a discrete infinity of disjoint homotopy classes , each class associated with a unique integer which may be thought of as the topological degree of the map ( see , for example @xcite ) , so homotopic partition of the configuration space is built in to the model from the start . we also choose stereographic coordinates @xmath35 on @xmath33 , in terms of which , @xmath36 where @xmath37 , @xmath35 takes all values in @xmath3 and @xmath38 , @xmath39 , @xmath40 . the radius of @xmath33 has been normalized to unity . the invariant measure is , @xmath41 and so , @xmath19 = \int\ , dx\ , dy\ , \frac{1}{(1+|w|^{2})^{2}}\left ( \frac{|\dot{w}|^{2}}{(1+r^{2})^{2}}-\left|\frac{\partial w}{\partial x}\right|^{2 } -\left|\frac{\partial w}{\partial y}\right|^{2 } \right).\ ] ] we identify kinetic energy , @xmath42 = \int\ , \frac{dx\ , dy}{(1+r^{2})^{2}}\frac{|\dot{w}|^{2}}{(1+|w|^{2})^{2}}\ ] ] and potential energy @xmath43 = \int\ , dx\ , dy\ , \frac{1}{(1+|w|^{2})^{2}}\left ( \left|\frac{\partial w}{\partial x}\right|^{2}+ \left|\frac{\partial w}{\partial y}\right|^{2 } \right).\ ] ] note that the potential energy is identical to that for flat space by virtue of the conformal invariance of the static model ( stereographic projection is a conformal transformation ) . thus the familiar bogomolnyi argument @xcite follows immediately and ( @xmath44 run over @xmath45 and @xmath46 represents the @xmath27 vector product in space ) : @xmath47 , \nonumber \\ \rightarrow v[w ] & = & \frac{1}{4}\int\ , dx\ , dy\ , \partial_{i}{\mbox{\boldmath $ \phi$}}\cdot\partial_{i}{\mbox{\boldmath $ \phi$}}\nonumber \\ & \geq & \frac{1}{2}\left|\int\ , dx\ , dy\ , \left(\frac{\partial{\mbox{\boldmath $ \phi$}}}{\partial x}\times\frac{\partial{\mbox{\boldmath $ \phi$}}}{\partial y}\right)\cdot{\mbox{\boldmath $ \phi$}}\right| = 2\pi|n|,\end{aligned}\ ] ] where @xmath20 is in the degree @xmath8 homotopy class , equality holding if and only if @xmath48 which , on substitution of ( [ eq : phidef ] ) becomes the cauchy - riemann condition for @xmath20 to be an analytic function of @xmath49 ( upper sign ) or @xmath50 ( lower sign ) . the former ( latter ) case corresponds to static solutions of positive ( negative ) degree , and if @xmath20 is single valued with finite degree @xmath8 , then it must be a rational map of degree @xmath8 in @xmath51 if @xmath52 or in @xmath53 if @xmath54 . we shall deal with the unit charge moduli space , consisting of all rational maps of degree 1 in @xmath51 . since the configuration space and moduli spaces of the flat space and spherical space models are diffeomorphic , we shall use the same notation ( @xmath55 etc . ) in both cases . the simplest static unit - charge solution is @xmath56 which we shall call the symmetric hedgehog because its field points radially outwards at all points on @xmath33 . its energy density is uniformly distributed , so it is not really a lump . since the static model is conformally invariant , any configuration obtained from this by a mbius transformation must be another point on the moduli space . in fact the orbit of @xmath57 under the mbius group _ is _ the space of degree 1 rational maps , each map being generated by one and only one group element . thus we may identify the moduli space with the parameter space of the mbius group . there is a well known matrix representation of mbius transformations @xcite which we denote thus : @xmath58 where @xmath59 so that @xmath60 . the last condition ensures the invertibilty of the transformation and fixes the degree of @xmath20 at 1 . the mbius group product becomes matrix multiplication , @xmath61 where the left hand side means @xmath62 in obvious notation . all matrices differing by a constant factor yield the same configuration , and @xmath60 so when we divide by this scaling equivalence we can choose a unimodular matrix as the representative for each equivalence class . there are two such matrices possible for each distinct configuration because if @xmath63 is unimodular , so is @xmath64 . thus @xmath65 is a double cover of the moduli space , which we recover by dividing out the equivalence @xmath66 : the moduli space is @xmath67 . coincidentally , @xmath65 is also a double cover of the proper orthochronous lorentz group . the statement that any lorentz transformation may be formed by a unique composition of a boost then a rotation ( or _ vice versa _ ) translates to the existence , for all @xmath68 , of @xmath69 and @xmath70 , a positive definite , unimodular , hermitian @xmath71 matrix ( call this set @xmath72 ) , satisfying @xmath73 both @xmath74 and @xmath70 being unique @xcite . it follows that the space @xmath65 is locally a product of @xmath75 ( the group manifold of @xmath76 ) and @xmath27 ( the parameter space of @xmath72 ) , a result which generalizes globally , @xmath77 . we may choose local coordinates on @xmath65 by defining the standard euler angles @xmath78 on @xmath75 , @xmath79 and expanding @xmath70 in terms of pauli matrices , @xmath80 @xmath81 being chosen to ensure the unimodular and postive definite properties : @xmath82 the 3-vector ( modulus @xmath83 ) takes all values in @xmath27 , while @xmath84 $ ] , @xmath85 $ ] and @xmath86 $ ] . these ranges allow @xmath63 to take all values in the double cover @xmath65 . in analyzing the structure of the metric , it is convenient to work with @xmath65 , checking that the metric is single valued under the identification of @xmath63 with @xmath64 . the true moduli space @xmath67 is charted by the same coordinates but with @xmath87 lying in the reduced range @xmath88 $ ] , for @xmath74 is then restricted to the `` upper half '' of @xmath75 . the chart has a coordinate singularity at @xmath89 and at @xmath90 . the explicit connexion between a point in @xmath14 and the corresponding static solution will be made in section 5 , below . field dynamics of the @xmath0 model may be visualized as the dynamics of a point particle with `` position '' @xmath91 moving in an infinite - dimensional configuration space . a solution @xmath92 of the field equations is thought of as a trajectory in this space , motion on which is determined by metric @xmath93 $ ] and potential @xmath94 $ ] . in the unit - charge sector , the bogomolnyi argument shows that there is a six - dimensional subspace on which the potential achieves its topological minimum value of @xmath95 , and that any perturbation departing from this subspace must involve increasing @xmath96 . if a configuration sitting at the bottom of this potential valley is given a small velocity tangential to it then we expect the ensuing time - evolved field to stay close to the valley bottom , for departure from it entails climbing up the valley walls . in the geodesic approximation @xcite we restrict motion to the valley bottom , assuming that orthogonal modes are insignificant . thus , at all times @xmath92 is a solution of the _ static model _ , but we allow the moduli @xmath97 , denoted collectively by @xmath98 , to vary with time in accordance with the inherited action principle . so , @xmath99 and the lagrangian is @xmath100 defining the induced metric , @xmath101 and ignoring the irrelevant constant , the lagrangian is recast as that of a free particle moving on a riemannian manifold with metric @xmath102 : @xmath103 the equations of motion are the geodesic equations . in principle all we need do is evaluate the integrals of ( [ eq : metricdef ] ) , but these are 21 functions of 6 variables so as it stands this is intractable in practice . it is profitable to take a more circumspect approach , using symmetries of the model to place restrictions on the structure of @xmath102 . consider the rotation group @xmath104 acting on @xmath33 and @xmath29 . the former is the group of spatial rotations ( under which @xmath20 , or equivalently , transforms as a scalar ) while the latter is the group of global internal rotations ( henceforth called `` isorotations '' ) of the field of which @xmath20 is the stereographic image . any such transformation @xmath105 leaves invariant ( a ) the topological charge , so @xmath105 is a bijection @xmath106 , ( b ) the potential energy , so within @xmath7 static solutions are mapped to other static solutions , @xmath107 , and ( c ) the kinetic energy , which induces the metric on @xmath11 . hence @xmath105 is an isometry of @xmath108 . the @xmath76 subgroup of the mbius group s double cover , @xmath65 , acting via the operation @xmath109 defined by equation ( [ eq : dotdef ] ) is the double cover of the group of rotations of the 2-sphere @xcite considered as operations on the projective plane ( spatial or internal , ie acting on @xmath49 or @xmath20 ) . thus we find that ( @xmath110 ) , @xmath111 produces an isorotation of the configuration @xmath112 , while @xmath113 produces a spatial rotation , both isometries of the induced metric . the action of the isorotation on the moduli space is simple : @xmath114 the isometry takes the @xmath76 left multiplication action on @xmath115 while leaving the @xmath27 moduli unchanged . using a technique standard in the analysis of isometries in general relativity @xcite , we change from the coordinate basis on @xmath116 , to a non - coordinate basis , in this case the left - invariant 1-forms of the lie group @xmath76 . these may be found by expanding the left - invariant 1-form @xmath117 in terms of a convenient basis of the lie algebra @xmath118 , for example @xmath119 . explicitly , @xmath120 where @xmath121 if we evaluate the metric at one particular point on @xmath115 , for all possible @xmath122 , we can obtain the metric at all other points on @xmath115 because @xmath115 is the isorotation orbit of our base point , and isorotation is an isometry , so the metric must remain constant ( for each ) over the entire orbit . `` constant '' means unchanging when considered as a geometric object , not that the components with respect to the original coordinate basis are constant , because the basis vectors themselves transform non - trivially . the basis of ( [ eq : sigma ] ) _ is _ invariant however , so the metric must be of the form @xmath123 where @xmath124 and each of the component functions is independent of @xmath78 . let us now consider the spatial rotations : @xmath125 the latter in terms of coordinates is @xmath126 the action of conjugation of the hermitian , traceless matrix @xmath127 by a unitary matrix @xmath128 is well known @xcite it is equivalent to a @xmath104 rotation of : @xmath129 where @xmath130 with components @xmath131 . the action on the left - invariant 1-forms is similar . under @xmath132 , @xmath133 where @xmath134 is the same @xmath104 matrix defined above . thus both and transform as 3-vectors under spatial rotations and as scalars under isorotations . the metric must be invariant under spatial rotations also , so the task is to construct from , @xmath135 and the most general possible @xmath136 tensor which is scalar under these rotations . this is @xmath137 @xmath138@xmath139 being 7 unknown functions of @xmath140 only . the metric may be restricted still further on consideration of a discrete isometry . the kinetic energy is invariant under the discrete `` parity '' transformations @xmath141 and @xmath142 . however , neither is an isometry of the moduli space because each reverses the sign of the topological charge , mapping lumps to anti - lumps . the composite transformation @xmath143 _ is _ an isometry . using the configuration of ( [ eq : dotdef ] ) , @xmath144 in terms of the moduli , @xmath145 is the transformation , @xmath146 this isometry removes two of the terms in ( [ eq : metric ] ) because under it , @xmath147 and @xmath148 so that @xmath149 . the remaining five functions of @xmath150 are evaluated by choosing convenient orientations for , positions on @xmath115 and tangent vectors ( velocities ) , then calculating the kinetic energy and comparing with ( [ eq : metric ] ) . repeating this four times it is possible to extract the following ( see figure 1 ) : @xmath151 \\ \nonumber c & = & \frac{\pi}{2}-2\pi s_{1}(\chi ) \\ \nonumber d & = & \frac{\pi}{\lambda^{2}}\left[6s_{1}(\chi)-\frac{1}{2}\right ] \\ \nonumber g & \equiv & a\end{aligned}\ ] ] where , @xmath152 \\ \nonumber s_{2}(\chi ) & = & \frac{\chi}{(\chi^{2}-1)^{3}}\left[\chi^{4}-2\chi^{2}\log\chi^{2}-1\right ] . \end{aligned}\ ] ] note that @xmath153 is a strictly increasing function of @xmath150 , and that @xmath154 . there appear to be divergences of the functions @xmath138@xmath155 at @xmath156 , but these are in fact removable singularities , so all the limits of vanishing @xmath150 exist . although @xmath157 and @xmath155 are negative it is straightforward to show that this metric is positive definite , as of course it must be . the veracity of the statement @xmath158 is established by explicit calculation , there being no obvious symmetry argument in its favour . in summary then , the metric is @xmath159 before discussing geodesics of the metric ( [ eq : g ] ) we must describe the connexion between a point on the moduli space @xmath160 and its corresponding field configuration . consider first the 3-dimensional submanifold defined by @xmath161 , parametrized by @xmath162 . any point in this subspace may be written as @xmath163 where @xmath164 , @xmath165 and @xmath166 . the lump represented by @xmath167 is @xmath168 this is a distorted hedgehog with the arrows pulled towards the north pole . the larger @xmath150 is , so the larger @xmath153 is and the greater is the distortion . although it is usual to define the position of a lump as the position of maximum energy density , we shall refer to this as a lump of sharpness @xmath150 located at @xmath169 , the antipodal point to the energy density peak which occurs where the arrows are stretched apart . obviously the motion of any point is trivially mirrored by its antipodal image , so this terminology makes sense . the lump represented by is @xmath170\odot z \\ \nonumber & = & [ r^{\dagger}({\mbox{\boldmath $ \lambda$}}'\cdot{\mbox{\boldmath $ \tau$}})]\odot(r\odot z)=r^{\dagger}\odot[w'(r\odot z)].\end{aligned}\ ] ] this configuration is formed by first performing a spatial rotation taking the arrow at the old point @xmath51 and placing it at the new point @xmath171 without changing its orientation then performing the inverse isorotation . the result looks like the arrows have been fixed to @xmath33 which has then been rotated by @xmath134 which , as defined , has the action on @xmath16 equivalent to @xmath172 , _ not _ @xmath128 , acting on @xmath0 via @xmath109 . that is , if we define @xmath173 to be stereographic projection , @xmath174@xmath0 so that @xmath175 , then @xmath176 . so the lump at the north pole is shifted to @xmath177 . all other points on the moduli space are on the isorotation orbit of this submanifold , and isorotation , while changing the internal orientation of the lump , does not move the lump around on physical space . thus we can always interpret @xmath178 as the lump s position , and @xmath150 as parametrizing its sharpness . the symmetric hedgehog has @xmath156 , and large @xmath150 lumps have taller , narrower energy density peaks than small @xmath150 lumps . one way of attacking the geodesic problem is to reduce its dimension by identifying totally geodesic submanifolds , that is , choosing initial value problems whose solution is simplified by some symmetry . the easiest method for identifying such submanifolds is to find fixed point sets of discrete groups of isometries . any isometry maps geodesics to geodesics , so if there were a geodesic starting off in the fixed point set of the isometry and subsequently deviating from it , this would be mapped under the isometry to another geodesic , identical to the first throughout its length in the fixed point set , but deviating from the set in a different direction . this violates the uniqueness of solutions of ordinary differential equations , so no such geodesic may exist . if the initial data are a point on the fixed point set and a velocity tangential to it , then the geodesic must remain on the fixed point set for all subsequent time . examining ( [ eq : g ] ) we see that @xmath179 is an isometry . its fixed point set is @xmath115 , the isorotation orbit of the symmetric hedgehog , on which the metric is @xmath180 the kinetic energy is the rotational energy of a totally symmetric rigid body , moment of inertia @xmath181 . the solutions are just isorotations of the symmetric hedgehog at constant frequency about some fixed axis . in this case isorotation is equivalent to spatial rotation because @xmath182 . a less trivial geodesic submanifold is the fixed point set of the parity transformation described above , @xmath145 . this is a 3-dimensional manifold , the product of the plane @xmath183 in @xmath27 with the circle @xmath184 , \beta=\gamma=0\}\cup\{\alpha\in[0,\pi ] , \beta=\gamma=\pi\}$ ] in @xmath115 . the circle is more conveniently parametrized if we temporarily allow @xmath185 the domain @xmath88 $ ] , for it is then @xmath186 , \beta=\gamma=0\}$ ] . this space contains lumps of arbitrary sharpness located on a great circle through the poles of @xmath33 , each lump having an internal phase , so certain of its geodesics may be candidates for `` travelling lumps . '' introducing spherical polar coordinates for , @xmath187 the plane @xmath183 is parametrized by @xmath188 where @xmath189 $ ] , again gluing two semicircles together and extending the domain of @xmath190 to cover the whole circle in one go . the metric on this geodesic submanifold is @xmath191 so the kinetic energy is @xmath192\ ] ] where we have used the cyclicity of @xmath190 and @xmath185 to eliminate @xmath193 and @xmath194 in favour of their constant , canonically conjugate momenta , @xmath195 note that constant @xmath196 ( @xmath197 ) does _ not _ imply constant @xmath193 ( @xmath194 ) , nor does @xmath198 ( @xmath199 ) imply @xmath200 ( @xmath201 ) . this system can be visualized as a point particle of position dependent mass @xmath202 moving in a potential . it is the form of the potential which determines the broad qualitative features of its behaviour : @xmath203 as can be seen from figure 2 while @xmath204 is monotonically decreasing , @xmath205 is monotonically increasing . this allows the possibility of potential minima where the forces @xmath206 outwards ( in the sense of increasing @xmath150 ) and @xmath207 inwards are in stable equilibrium . it certainly is _ not _ possible if @xmath208 , for then @xmath209 as a whole is monotonically decreasing . this region in the @xmath210 plane is shown shaded in figure 3 . whatever the initial conditions on @xmath150 , the lump always moves towards infinite @xmath150 without passing through @xmath156 ( which would correspond to the lump swapping hemispheres ) , reaching the singularity @xmath211 , an infinitely tall , sharp spike , in finite time . thus @xmath108 is geodesically incomplete . this result follows from the rapid vanishing of the inertia to sharpening , @xmath212 , in the large @xmath150 limit ( see figure 4 ) . for example , consider the simple case @xmath213 and let @xmath214 and @xmath215 be strictly positive . it is easily seen that @xmath216 , the time taken to reach the singular spike is proportional to the following integral : @xmath217 the integrand is finite over the integration range ( even if @xmath218 ) , so if @xmath216 diverges it can only be due to the large @xmath150 behaviour . but the integrand vanishes like @xmath219 at large @xmath150 , fast enough to ensure convergence . the inclusion of repulsive potentials can only make matters worse , so this singular behaviour extends to the rest of the shaded area . in the unshaded region , one can define the positive constant @xmath220 such that @xmath221 then the forms of the functions @xmath222 and @xmath223 ( see figure 5 ) suggest that for each @xmath150 , there is one ( and only one ) value of @xmath224 ( call it @xmath225 ) for which @xmath226 has a minimum at @xmath150 . the equilibrium condition is @xmath227 , so @xmath228 inverting the definition of @xmath224 we find that there are two distinct values of @xmath229 for each @xmath225 . if @xmath229 takes one of these and @xmath230 then @xmath150 will not subsequently change and hence @xmath194 and @xmath193 will also remain constant , allowing the lump to travel around a great circle on @xmath33with constant speed and shape while undergoing constant frequency isorotation . the two values are @xmath231 substituting ( [ eq : pdefs ] ) we can find the corresponding pair of stable ratios @xmath232 as functions of @xmath150 , @xmath233 ( see figure 6 ) . thus , for any lump sharpness @xmath150 and travel speed @xmath193 there are two possible isorotation frequencies @xmath194 which allow stable , uniform travel and these two stability `` branches '' never coincide . it is interesting to note that @xmath234 meaning that very tall , sharp lumps can travel uniformly with @xmath235 . motion with constant @xmath150 and @xmath236 is simply constant speed spatial rotation carrying the lump around a great circle . so when the extent of the lump s structure is negligible relative to the radius of curvature of @xmath33 , it can travel in analogous fashion to a flat - space @xmath0 lump @xcite . since @xmath237 takes all positive values , whatever value @xmath224 takes there is an equilibrium @xmath150 . if @xmath214 is near this value , then ( assuming @xmath238 is not too large ) the shape of the lump will oscillate periodically about the preferred sharpness , and its speed of travel round the sphere will vary with the same period . if @xmath238 _ is _ too large , or the lump is initially much too spread out for its @xmath224 , then it will escape to the singular spike in finite time . let us examine the concrete example @xmath239 . figure 7 shows the potential @xmath226 with its minimum and the lump travel speed @xmath193 as functions of @xmath150 . we imagine a particle of position dependent mass moving in this potential and for simplicity take @xmath230 . clearly , if we release the particle with @xmath240 , it will move off to infinity , the last case mentioned above . but if @xmath241 , oscillatory motion ensues . even here there are two qualitatively different cases , because @xmath242 has an absolute minimum at @xmath243 , a turning point which is only reached if @xmath244 ( or @xmath245 ) . if @xmath246 then the speed of travel oscillates in simple phase with the lump sharpness , going from fast , spread - out lump to slow , sharp lump and back again . but if @xmath247 or @xmath245 the speed undergoes an extra wobble during the middle of the sharpness cycle , speeding up then slowing down again as it passes through its maximum sharpness . this case corresponds to lumps whose shape oscillates more acutely . other interesting geodesic submanifolds are generated by computing the fixed point sets @xmath248 of the isometries @xmath249 , simultaneous isorotation and spatial rotation by @xmath250 about the @xmath251 and @xmath252 axes respectively : @xmath253 thus if @xmath254 must point along the @xmath252-axis . on @xmath255 , @xmath256 where @xmath257 $ ] , whereas if @xmath258 then @xmath259 where @xmath260 . it follows that also points along the @xmath252-axis , independent of @xmath251 . the @xmath261 submanifolds are the images of @xmath255 under @xmath262 isorotations about the three axes , so it suffices to solve the geodesic problem on @xmath255 geodesics on @xmath263 , are then obtained by acting with the appropriate isometry . the choice of @xmath252 does nt matter , and we choose to study the cylinder @xmath264 consisting of lumps of every sharpness located at the north ( south ) pole if @xmath265 ( @xmath266 ) , arbitrarily rotated about the north - south axis . note that @xmath267=0 $ ] on @xmath268 so `` isorotated '' and `` spatially rotated '' mean the same thing in this case . the kinetic energy on @xmath268 is @xmath269\ ] ] where once again @xmath270 is the momentum conjugate to @xmath271 , @xmath272 and is constant by virtue of the cyclicity of @xmath271 . this looks like a particle in one dimension moving in a potential @xmath273 with postion dependent mass . from the potential ( figure 8) we see that all motion is oscillatory and that @xmath274 periodically changes sign . thus a lump set spinning about its own axis will spread out , its rotation slowing , until it is uniformly spread over the sphere , whereupon it will shrink to its mirror image in the opposite hemisphere , regaining its original spin speed as it does so . the process then reverses and the lump `` bounces '' between antipodal points indefinitely . defining the new coordinate @xmath275 which takes values in a finite open interval @xmath276 symmetric about @xmath277 , the metric on @xmath268 becomes @xmath278 where @xmath279 . since @xmath280 , the manifold may be embedded as a surface of revolution in @xmath27 and geodesics on it can be visualized directly . figure 9 is a sketch of the embedded surface , which is of finite length and sausage - shaped with its ends pinched to infinitely sharp spikes , the tips of which are the points @xmath281 and so are missing . the coordinates @xmath282 are geodesic orthogonal coordinates : a curve of constant @xmath271 is a geodesic along the length of the cylinder , lying in a plane containing the cylinder s axis , parametrized by arc length @xmath283 , while a curve of constant @xmath283 is a circle of radius @xmath284 , lying in a plane orthogonal to the cylinder s axis . two such curves always intersect at right angles . the spinning geodesics described above wind around the cylinder , never reaching the ends ( this would violate conservation of `` angular momentum '' ) but winding back and forth between two circles @xmath285 , @xmath286 which they touch tangentially . the angle @xmath287 at which the geodesic intersects the circle @xmath277 determines @xmath288 . when @xmath289 the geodesic stays on the circle , @xmath290 ( a spinning symmetric hedgehog ) and @xmath291 monotonically increases , tending to the supremum @xmath292 as @xmath287 tends to @xmath293 ( @xmath294 is an irrotational geodesic between antipodal singular spikes ) . note that the geodesic incompleteness already mentioned appears again , this time characterized by the finite length of the cylinder and the missing points @xmath295 ( @xmath281 ) . the behaviour of isolated topological solitons in flat space is generally rather trivial , whereas , as we have seen , despite the homogeneity of @xmath16 , the motion of a single lump on the sphere is surprisingly complicated . it does travel on great circles , but while doing so its shape may oscillate in phase with its speed , whose periodic variation is of one of two types depending on the violence of the shape oscillations , or it may collapse to an infinitely tall , thin spike in finite time . a lump sent spinning about its own axis spreads out then re - forms in the opposite hemisphere , endlessly commuting between antipodal points . the infinities in the unit - charge metric in flat space can be attributed to the lumps polynomial tail - off : the kinetic energy needed to rigidly spin or scale - deform a lump diverges because such motions involve changing the field at spatial infinity . the @xmath0 model on any compact space should be free of this problem because the kinetic energy , being an integral over a space of finite volume , must be finite provided the kinetic energy density is non - singular . conversely , one would expect the singularity to persist in the model defined on hyperbolic space . the flat - space @xmath0 model can be made more `` physical '' by adding a @xmath296-dimensional version of the skyrme term to stabilize against lump collapse , and a potential to stabilize against spread . the bogomolnyi bound remains valid but unsaturable . the potential is somewhat arbitrary , but one interesting possibilty @xcite gives a mass to small amplitude travelling waves of the field , termed pions in analogy with the skyrme model , and gives the lump an exponential rather than polynomial tail . this allows the lump to rotate , a problem if one attempts a collective coordinate approximation to low - energy dynamics along the lines recently proposed in @xcite . the idea is to restrict the field to the `` bogomonyi regime '' moduli space ( in this the space of static @xmath0 solutions ) , introducing a potential and a perturbed metric ( in @xcite but not @xcite ) to account for the new interactions , which are assumed to be weak . there seems little hope of perturbing the singular flat - space metric such that rotations become possible , but the problem does not arise on the sphere . the geodesic approximation could be used to investigate the interaction of two lumps moving on @xmath16 . right angle scattering in head on collisions emerges naturally from the geodesic approximation of many flat - space models as a consequence of the classical indistinguishability of topological solitons . it would be interesting to see if there is some analogous behaviour on the sphere . however , evaluating the two - lump metric could be difficult since the action of the isometry group on the charge-2 moduli space is far less accessible than in the present case . even in flat space @xcite , the scattering problem is sufficiently complicated to require considerable numerical effort . * acknowledgments : * i would like to thank richard ward , who suggested this work , and bernd schroers for many useful discussions . i also acknowledge the financial support of the uk particle physics and astronomy research council .

low - energy dynamics in the unit - charge sector of the @xmath0 model on spherical space ( space - time @xmath1 ) is treated in the approximation of geodesic motion on the moduli space of static solutions , a six - dimensional manifold with non - trivial topology and metric . the structure of the induced metric is restricted by consideration of the isometry group inherited from global symmetries of the full field theory . evaluation of the metric is then reduced to finding five functions of one coordinate , which may be done explicitly . some totally geodesic submanifolds are found and the qualitative features of motion on these described . epsf = -0.4 in = -0.2 in = 6.6 in

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Proceed to summarize the following text: cylindrical semiconductor nanostructures bridge between quasi-1d systems at small diameters and quasi-2d in the opposite limit , thus extending the wealth of physics and applications of low - dimensional solid - state systems . the controlled growth of semiconductor quantum tubes ( qts ) with diameters in the 10 - 100 nm range has been recently demonstrated through several techniques , including multi - layer overgrowth of nanowires@xcite and strain - induced bending of a planar heterostructure.@xcite in addition to qts with a solid semiconductor core , it is possible to grow _ hollow _ qts , where the charge carriers are confined in a thin semiconductor shell , encompassed by a barrier material which is only a few nm thick.@xcite large surface - to - volume ratios and the possibility of various functionalizations on both the internal and external surfaces make the latter systems particularly interesting for applications.@xcite although experiments concerning the optical properties of these systems are still limited , advancements in the optical quality of the samples point to a rapid increase of these investigations.@xcite the excitonic properties of semiconductor qts are particularly interesting with respect to conventional semiconductor quantum wires , where excitons are confined in the core of the nanostructure.@xcite on the one hand , due to the combined effect of the qt curvature and of the quasi-2d confinement of carriers in the cylindrical shell , excitonic binding energies might be substantially stronger than in bulk , even for large diameter qts . on the other hand , a dielectric medium outside the shell of the qt may result in a dielectric confinement of the electric field felt by the optically excited electron - hole pairs , in most cases enhancing their excitonic binding energy . since the dielectric interface is spatially separated from the carriers , which are confined deep inside the shell , excitonic binding and sensitivity to the medium might be strongly enhanced without spoiling the optical properties of the electronic system,@xcite analogously to core - shell nanowires.@xcite the screening provided by the dielectric environment can be varied in a broad range.@xcite the tunability of the dielectric constant in the core of the qt , obtained , _ e.g. _ , by oxidation,@xcite can further increase such effects . present work on qts theoretically considered magnetic states,@xcite reported experimental evidence of the aharonov bohm effect,@xcite and treated optical properties,@xcite but the influence of the dielectric dismatch between the nanostructure and the environment has been studied so far only for conventional quantum wires@xcite and freestanding nanowires.@xcite here we will consider also a dielectric mismatch between the core and the shell , which will lead to a considerable change in the electron - hole interaction , as shown in fig . [ fig : potential ] . hereafter we investigate the excitonic binding and oscillator strength in hollow and filled qts for different geometries and dielectric configurations . besides increasing due to the reduced screening , the excitonic binding strongly depends on the qt diameter and on the dielectric medium . the paper is organized as follows . in section [ sec : model ] we outline the theoretical model , which includes the exact solution of the poisson equation and the diagonalization of the electron - hole hamiltonian within the envelope - function approximation . in sections iii and iv we report our results and draw the conclusions , respectively . the system we consider consists of an infinite tube with cylindrical symmetry@xcite ( see fig . [ fig : tube_scheme ] ) . for simplicity we assume that the motion in the radial direction is frozen , and that charge carriers are radially confined in a @xmath0-like well at a distance @xmath1 from the tube axis . this electronic layer is buried in the middle of a coaxial cylindrical shell of thickness @xmath2 with dielectric constant @xmath3 , while the core and the environmet have in general different dielectric constants , @xmath4 and @xmath5 , respectively . since the shell is a semiconductor material , typically @xmath6.@xcite the invariance under translations along , and rotations around the tube axis warrants the separation of the center of mass and relative coordinates . the motion of the wannier exciton@xcite in the relative degrees of freedom is determined by the envelope - function hamiltonian @xmath7 - v ( x , y),\ ] ] expressed in units of the effective hartree @xmath8@xmath9 , with @xmath10 the reduced electron - hole mass . the relative coordinates around the circumference ( @xmath11 ) and along the tube axis ( @xmath12 , see fig . [ fig : tube_scheme ] ) , are in units of the effective bohr radius , a@xmath13 . the effective coulomb interaction potential @xmath14 between the confined electron and hole depends parametrically on the dielectric constants ( @xmath15 ) and on the tube geometry through @xmath16 and @xmath1 . in cylindrical coordinates , the potential ( scaled with @xmath8 ) generated by a charge at @xmath17 reads @xmath18 g_{m,\alpha}(k , \rho , \rho'),\end{aligned}\ ] ] where @xmath19 indicates whether the position of the test - charge @xmath20 is in the core , shell or environment region , respectively , and @xmath21 is the solution of the radial poisson equation in that region ( see the appendix for further details ) . the interaction @xmath22 in eq . coincides with @xmath23 , with @xmath24 . as shown in the appendix , @xmath25 \left[\tilde{b}_m^>+c_m^>\right ] \times \nonumber \\ & & i_m(kr ) k_m(kr),\end{aligned}\ ] ] where @xmath26 are the bessel functions of the first and second kind ; the coefficients @xmath27 are given in eqs . and in terms of @xmath28 and their derivatives . to illustrate how the electron - hole interaction is influenced by the dielectric environment , we shown in fig . [ fig : potential ] the potential @xmath22 for i ) a _ filled _ qt , with a core of the same material as the shell , immersed in a substance with a low - dielectric constant ( @xmath29 ) , and ii ) a _ hollow _ qt , with the same low - dielectric constant substance inside and outside the shell ( @xmath30 . for comparison , we also show the dielectrically homogeneous case ( @xmath31 ) , where the @xmath22 reduces to the usual coulomb potential @xmath32 . figure [ fig : potential](a ) shows the interaction _ along _ the qt @xmath33 , while fig . [ fig : potential](b ) shows the interaction _ around _ the cylinder @xmath34 . the coulomb interaction for the hollow and filled cases is for all distances stronger than in the homogeneous case , since the _ average _ dielectric constant of the system is smaller , and the electric field is not screened outside and , for the hollow case , also inside the qt . the interaction in the filled and hollow case is substantially different only for distances smaller or comparable to the bohr radius , with the interaction in the hollow case being stronger . for larger distances ( inset of fig . [ fig : potential](a ) ) , on the other hand , the non - trivial influence of the dielectric mismatch between the core and the shell leads to crossing of the potentials for hollow and filled qts , before both converge to the same value . between an electron - hole pair confined to a cylindrical surface with diameter @xmath35 a@xmath36 , buried in a shell with thickness @xmath37 a@xmath38 with dielectric mismatch , as follows . homogeneous case : @xmath39 . filled case : @xmath40 . hollow case : @xmath41 . ( a ) interaction along the qt , @xmath42 . ( b ) interaction around the qt , @xmath43 . inset in ( a ) : @xmath44 in a larger range of @xmath45 . ] a convenient basis set to represent the exciton wavefunction is obtained by multiplying eigenfunctions of the linear momentum operator along @xmath12 ( @xmath46 ) and of the angular momentum operator along the tube axis ( @xmath47 ) . imposing periodic born - von karman boundary conditions@xcite along @xmath12 , with period @xmath48 sufficiently larger than the effective bohr radius of the material , results in : @xmath49 with @xmath50 . the wavefunction thus reads @xmath51 where @xmath52 and @xmath53 indicates the @xmath54th exciton state . the coefficients @xmath55 are obtained from the schrdinger equation in the above basis : @xmath56 \delta_{n , n'}\delta_{p , p'}\\ -u_{{n'\!,p'\atop{\!\!n , p}}}\bigg \ } { \ensuremath{c _ { n'\!,\,p'}^{j}}}=e_j { \ensuremath{c _ { n , p}^{j}}}.\end{gathered}\ ] ] the diagonal term in the first line represents the kinetic energy , whereas the matrix elements of the electron - hole coulomb interaction term are given by @xmath57 in order to reduce the dimension of the hamiltonian matrix we introduce a cutoff energy @xmath58 , set the maximum number of plane waves @xmath59 and choose the maximum number of orbital modes @xmath60 in eq . as the nearest integer to @xmath61 the hamiltonian matrix is block diagonalized using a symmetrized basis set . in particular , we consider linear combinations of the above basis functions that are even or odd with respect to the inversion of the relative coordinates @xmath62 and @xmath12 , which is the equivalent of inverting the absolute coordinates , since the corresponding inversion operators @xmath63 commute with the relative motion hamiltonian @xmath64 = 0 & , & [ h , \pi_y ] = 0.\end{aligned}\ ] ] the resulting energy @xmath65 is obtained with respect to the energy minimum of the conduction band . therefore the binding energy of the exciton ground state is @xmath66 . in the presence of a photon gauge field the electron - hole pair recombines emitting a photon of energy @xmath67 . the recombination rate is related to the dimensionless oscillator strength @xmath68 , which in the dipole approximation reads@xcite @xmath69 here @xmath70 is the envelope function of the exciton ground state given by eq . , @xmath71 the momentum of the center of mass , @xmath72 is the energy gap between valence and conduction band and @xmath73 , where @xmath74 is the energy associated with kane s matrix elements.@xcite in the following we investigate the excitonic properties of qts made of the direct gap materials , inas , gaas and inp , and two different dielectric configurations : _ filled _ qts , with a core of the same material of the shell ( @xmath75 ) , and _ hollow _ qts , with the core of the same material as the environment ( @xmath76 ) . we consider qts with diameters in the @xmath77 nm range and a constant shell thickness @xmath78 nm , comparable to state - of - the - art samples.@xcite material parameters used in the calculations are listed in table [ tab : parameters ] . [ cols="^,^,^ " , ] in order to further investigate the effect of the coulomb interaction between the carriers , we plot the squared modulus of the ground state excitonic wave function for three different dielectric environments , for a inas tube of diameters @xmath79 nm ( fig . [ fig : wave_20 nm ] ) and @xmath80 nm ( fig . [ fig : wave_100 nm ] ) , for both hollow ( top ) and filled ( bottom ) tubes . these are two relevant cases , since the former exhibits a geometrical confinement caused by the small circumference , whereas the latter falls fully in the 2d regime without any confinement . as shown in fig . [ fig : wave_20 nm ] , the wave function for small tubes is distributed over all the circumference , best visible in the homogeneous case ( @xmath81 ) . reducing the screening by diminishing @xmath5 affects the wavefunctions only weakly , leading to a slightly increased localization , both of the hollow ( upper panels ) as well as of the filled case ( lower panels ) . on the other hand , both dielectric configurations lead to similar wavefunctions , reflected in the energies reported in fig . [ fig : ene_vs_epsilon ] , too . for large tubes of fig . [ fig : wave_100 nm ] the wavefunction is no more distributed over all the circumference , but well localized . therefore the curvature of the tube has no effects on the exciton for larger diameters , making it fully 2d . again , changing the dielectric configuration , by diminishing @xmath5 as well as by going from hollow ( top panels ) to filled ( bottom panels ) tubes , is changing the wavefunctions only marginally , while the respective energies are very sensitive to it ( see fig . [ fig : ene_vs_epsilon ] ) . therefore , while the diameter has a definite influence on the dimensionality of the excitonic states , changing the dielectric configuration amounts to modulating the mean screening with nearly no effects on the wavefunction , shifting only the energy . we have studied theoretically the excitonic properties of semiconductor qts , focusing on the influence of their dielectric environment and its interplay with structural parameters . we find that , due to the strong increase of the electron - hole interaction and ensuing very large excitonic binding which is possible in these structures , the spectral properties of excitonic absorption are strongly dependent on geometrical parameters and dielectric environment , with energies well below the energies of the dielectrically homogenous case which is always in the 2d regime for typical parameters . calculations have been performed for inas , gaas and inp . the low gap material inas shows a peculiar behavior , since in the investigated systems the exciton binding energy is a substantial fraction of the gap . the very large binding energies , their tunability in a wide range , and the large sensitivity of the excitonic response to the dielectric medium , point to perspective applications of these systems . we thank financial support from the italian minister for university and research through firb rbin04ey74 and cineca iniziativa calcolo parallelo 2009 . the inner radius @xmath82 and the outer radius @xmath83 divide the space into three regions : core ( @xmath84 ) , shell ( @xmath85 ) and environment ( @xmath86 ) with dielectric constants @xmath87 , @xmath88 , @xmath89 , respectively ( see fig . [ fig : tube_scheme ] ) . the electrostatic potential at point @xmath90 induced by an electron localized in the shell , _ i.e. _ with @xmath91 , screened by @xmath3 has to obey the poisson equation in cylindrical coordinates ( with charge @xmath92 : @xmath93 here @xmath94 indicates one of the three possible regions of the test charge : core ( @xmath95 ) , shell ( @xmath96 ) or environment ( @xmath97 ) . is solved by the _ ansatz _ @xmath98 where @xmath99 is the solution of the radial poisson equation in each region @xmath100 @xmath101 and can be written as a linear combination of the solutions of the homogeneous laplace equation , _ i.e. _ modified bessel functions of the first kind , @xmath102 , and the second kind , @xmath103 , with the following properties:@xcite where @xmath108 are no green s functions , but solutions of the ( homogeneous ) laplace equation , from which we construct the solution @xmath109 of eq . in the following . we define @xmath110 $ ] and @xmath111 $ ] . matching components of fields @xmath112 and @xmath113 at the interfaces is equivalent to @xcite to determine the last two unknowns we use the symmetry of the green s function @xmath109 with respect to the exchange of @xmath115 and @xmath116 making,@xcite @xmath117 and normalization @xmath118 defining @xmath119 and @xmath120 , and the quantities in particular , for two charges localized at the same distance @xmath24 from the center @xmath125 \times \nonumber\\ & \left [ b_m^ > i_m(kr ) + c_m^ > k_m(kr ) \right ] \nonumber \\ = & \frac{4 \pi}{\epsilon_s } \left [ \tilde{b}_m^ < + \tilde{c}_m^<\right ] \left[\tilde{b}_m^>+c_m^>\right ] \times \nonumber \\ & i_m(kr ) k_m(kr),\end{aligned}\ ] ] hence , taking eq . in the special case @xmath24 , gives the coulomb potential for two particles localized in the shell on a cylindrical surface of radius @xmath1 @xmath127 \left[\tilde{b}_m^>+c_m^>\right ] \times \nonumber \\ & i_m(kr ) k_m(kr ) \cos(k(z - z ' ) ) dk.\end{aligned}\ ] ] for @xmath128 this reduces to the usual form @xmath129 in cylindrical coordinates,@xcite while for @xmath130 eq . reproduces the result of ref . . note that @xmath22 is scalable , since all arguments in eq . are products of lengths and momenta and thus dimensionless , only the measure @xmath131 of the integral is reciprocal in length . the latter one scales with the effective bohr length a@xmath132 nm and therefore @xmath22 itself with the effective hartree @xmath8@xmath133 ev .

we study theoretically the optical properties of quantum tubes , one - dimensional semiconductor nanostructures where electrons and holes are confined to a cylindrical shell . in these structures , which bridge between 2d and 1d systems , the electron - hole interaction may be modulated by a dielectric substance outside the quantum tube and possibly inside its core . we use the exact green s function for the appropriate dielectric configuration and exact diagonalization of the electron - hole interaction within an effective mass description to predict the evolution of the exciton binding energy and oscillator strength . contrary to the homogeneous case , in dielectrically modulated tubes the exciton binding is a function of the tube diameter and can be tuned to a large extent by structure design and proper choice of the dielectric media .

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Proceed to summarize the following text: fu orionis systems ( hereafter fuors ) , named after their prototype object , are low - mass ( sun - like ) protostars that undergo a rapid accretion episode in the innermost region of the circumstellar disc ( see @xcite for a review ) . the gravitational energy released in such an event leads to an emission outburst ( with a rise time of @xmath2yr and a duration of @xmath3yr ) , during which the disc is more luminous than the central protostar by a factor of @xmath4 . the inferred mass accretion rate during the outburst is @xmath5 , much higher than typical accretion rates during the quiescent phase . statistical arguments , first advanced by @xcite , indicate that such outbursts occur , on average , ten or more times during the protostellar lifetime . there is evidence that the outbursts are more frequent during the early ( the so - called class 0 and class i ) evolutionary phases and peter out as the mass accretion rate declines and the protostar enters the visible ( class ii , or classical t tauri ) phase . the picture that has emerged from the observations and their interpretation is that most of the mass that ends up in the protostar is transferred from the disc during such outbursts ( e.g. @xcite ; see , however , @xcite for a different viewpoint ) . if our understanding is correct , the fu orionis phenomenon represents a key element of the star formation process . fuor outbursts are accompanied by strong winds of maximum line - of - sight speeds @xmath6 ( e.g. * ? ? ? * ) , whose inferred mass outflow rates @xmath7 can reach @xmath8 ( e.g. @xcite ; @xcite , hereafter chk93 ) . if most of the mass accreted through the disc is indeed processed through outbursts of this type then most of the mass and momentum ejected over the protostellar lifetime and hence most of the impact that protostellar outflows may have on their environment ( e.g. in contributing to the dynamical support of the parent cloud against gravitational collapse and to the regulation of the mass inflow to the centre ) will be associated with these eruptions . it is also likely that the repeated , powerful ejections have a strong influence on the properties and appearance of the large - scale jets that emanate from these protostars ( e.g. * ? ? ? * ) . it is therefore important to understand the nature and origin of these outflows . neither thermal nor radiative acceleration is likely to be important in fuor winds , which leaves magnetic driving as the most promising mechanism . this conclusion is supported by a zeeman - signature least - square deconvolution measurement in the prototype object fu ori , which was interpreted as indicating the presence of a @xmath9 poloidal magnetic field on scales of @xmath10au @xcite . one possible scenario is that the outflows represent a centrifugally driven wind that is launched along magnetic field lines that thread the disc and are sufficiently inclined ( at an angle @xmath11 for a keplerian rotation law ) to the disc surface ( e.g. * ? ? ? these field lines could correspond to the interstellar magnetic field that threads the natal molecular cloud core and is dragged in by the accretion flow , although an origin in a disc dynamo is also conceivable . in this picture , the outflow need not be launched from the immediate vicinity of the central star . alternatively , the wind , while still comprising material removed from the accretion disc , could be driven along stellar magnetic field lines . in view of the fact that the massive accretion flow during an fuor outburst is expected to compress the stellar magnetosphere to an equatorial radius @xmath12 not much larger than the stellar radius @xmath13 , the wind in this scenario necessarily originates close to the stellar surface . one version of the latter scenario , proposed by @xcite , corresponds to the x - celerator model presented in @xcite . in this picture , the accretion flow spins up the outer layers of the star to breakup , resulting in a magnetocentrifugal wind being driven along a narrow bundle of opened - up field lines that emerge from an ` x - point ' at the stellar equator . in the x - wind model described in @xcite , the ` x - point ' is associated more generally with the corotation radius @xmath14 ( the radius where the disc angular speed equals the stellar angular speed ) , which in quiescent protostars is typically a few stellar radii away from the stellar surface . chk93 and @xcite presented arguments in favour of the ` disc field ' interpretation of fuor outflows and against a ` stellar field ' scenario . they demonstrated that increasingly stronger photospheric lines observed in fu ori become progressively more blueshifted even as their two absorption components ( attributed to the disc rotation ) move closer together in wavelength , and pointed out that this is precisely the behaviour expected in a disc - driven wind . in addition , based on the evidence that most of the optical continuum in fuors is emitted by an extended disc and on the fact that typical stellar winds accelerate on scales comparable to the stellar radius , they contended that a stellar field - driven wind launched from the stellar surface could not reproduce the observations . in particular , they argued that the strong , but only moderately blueshifted , intermediate - strength lines detected in fu ori could not originate in such a wind because , by the time such an outflow covered a significant fraction of the continuum emission region in the disc ( necessary for producing strong absorption ) , it would have already attained a high velocity ( and would therefore exhibit strongly blueshifted lines ) . another potential problem that they cited involves the large rotational broadening that could be expected from a source rotating at breakup ( as envisioned in the x - celerator picture ) . recent axisymmetric and 3d numerical simulations of disc accretion on to stellar magnetospheres ( @xcite , hereafter r09 ) have revealed features that resemble the x - wind configuration proposed by @xcite but that are nevertheless different on several counts . specifically , it was found that such systems drive conical disc winds along stellar field lines that are bunched up by the accretion flow . however , even though these winds also originate in the inner disc , their launching region is not confined to the immediate vicinity of the corotation radius , as hypothesized in the x - wind scenario . furthermore , the conical winds are driven by the pressure gradient of the azimuthal magnetic field component ( wound up by the differential rotation between the disc and the star ) rather than centrifugally , and they have a smaller opening angle and a narrower lateral extent than x - winds . interestingly , even though r09 only presented results for model parameters appropriate to protostars with comparatively low ( @xmath15 ) accretion rates , the outflows produced in their simulations exhibited several properties that could potentially mitigate the aforementioned arguments against stellar field - driven outflow models for fuors . in particular , it was found that the acceleration of a conical wind is more extended than in a typical ( hydrodynamic ) stellar outflow and that its rotation speed generally decreases along the flow , in contrast with the initial behaviour of a centrifugally driven wind . these findings provide a strong motivation for reevaluating the viability of the ` stellar field ' class of wind models for fuors . more recent investigations , employing larger simulation regions and higher accretion rates , have begun to extend the results of r09 . one notable finding of this new work , analysed in @xcite , is that the collimation of conical winds increases with distance from the origin and that they can eventually become fully collimated . in this paper we focus on simulations that we performed for parameters that are relevant to fuors . our goal is to verify that conical winds are still produced under these circumstances and to examine whether they could potentially account for the inferred properties of fuor outflows . in section [ sec : model ] we provide analytic estimates that are used to guide our simulations and we summarize our numerical scheme . in section [ sec : results ] we present representative results and derive the physical properties of the simulated flows . we discuss the implications of this study for fuors in section [ sec : analysis ] and give our conclusions in section [ sec : conclude ] , where we also outline steps toward further progress . the fuor phenomenon has been convincingly argued to represent an enhanced accretion episode in a protostellar accretion disc , most likely associated with an instability that arises from a mismatch between the mass accretion rates in the inner and outer disc regions ( e.g. @xcite , @xcite and references therein ) . accordingly , we set up a numerical model that simulates a non - steady disc accretion ` burst ' on to a magnetized star . we first discuss some basic scaling relations that allow us to choose the appropriate model parameters , and then briefly describe our numerical model . our model is based on the assumption that the stellar magnetic field can effectively diffuse into the inner region of the disc , allowing the bulk of the inflowing disc material to be channelled on to the stellar surface along closed magnetic field lines ( e.g. * ? ? ? * ; * ? ? ? * ) and the remainder to be expelled in an outflow along opened - up field lines ( e.g. * ? ? ? * ; * ? ? ? the disc truncation ( or ` magnetospheric ' ) radius @xmath1 corresponds to the location in the disc where the torque exerted on the disc plasma by the stellar magnetic field becomes large enough to brake the disc keplerian rotation and enforce corotation with the star . for an aligned dipolar field , it is given by @xmath16 where @xmath17 is the stellar mass and @xmath18 is its magnetic dipole moment , @xmath19 is the gravitational constant and the mass accretion rate is measured at @xmath1 @xcite . under stationary conditions , the numerical factor @xmath20 is estimated to be @xmath21 @xcite . when @xmath1 is close to @xmath13 , as in the fuor case , higher order magnetic moments @xmath22 , which produce magnetic field amplitudes @xmath23 , can also be expected to play a role . in this case , equation ( [ eq : r_m ] ) generalizes to @xmath24 @xcite , where , in particular , @xmath25 , 2 and 3 correspond , respectively , to the dipole , quadrupole and octupole field components . for a given dipole component , the incorporation of additional multipole components will tend to increase the value of the disc truncation radius over the estimate ( [ eq : r_m ] ) . however , for the sake of simplicity , we restrict the discussion in the rest of this paper to a purely dipolar field . in choosing our model parameters , we adopt as fiducial values the physical parameters inferred from observations of fu ori . in particular , based on the results given in @xcite , we take @xmath26 and @xmath27 . these authors also estimate , from spectral modeling , that the inner radius @xmath28 of the fu ori disc is @xmath29 . we identify this radius with @xmath1 , which allows us , using equation ( [ eq : r_m ] ) , to infer the value of @xmath30 : @xmath31 where the normalization of the equatorial surface magnetic field @xmath32 is consistent with typical values inferred in quiescent class - i @xcite and class - ii ( e.g. * ? ? ? * ) protostars as well as with the results reported by @xcite for the poloidal field near fu ori . is larger than that of the corresponding unperturbed dipole field on account of the compressional amplification of the stellar field by the accretion flow . ] the contribution of higher order multipole field components , which could become important near the stellar surface , would have the effect of increasing this ratio . ) and setting , for definiteness , @xmath33 , we deduce , using equation ( [ eq : r_m_n ] ) , that @xmath34 and @xmath35 . note in this connection that recent spectropolarimetric observations of the t tauri stars v2129 oph and bp tau ( @xcite , @xcite ; see also @xcite and @xcite ) inferred an octupolar surface field component that is stronger than the dipolar component . ] previous accretion disc models of fuors have generally ignored the role of the magnetic field in truncating the disc and therefore identified @xmath28 , the inner radius of the disc , with the stellar radius ( e.g. * ? ? ? the value @xmath36 inferred in this way is measurably higher than typical values for low - mass protostars ( @xmath37 ) , and several explanations have been advanced to account for the difference . ( a brief summary of this issue is given by @xcite , who favour an interpretation that attributes the larger radius to stellar expansion brought about by the deposition of heat produced by the accreting gas . ) since @xmath30 is generally @xmath38 in the magnetic accretion model ( for example , it is 1.4 in the representative simulation presented in this paper ) , the inferred value of @xmath13 is lower in this case , which reduces the implied difference from the radii of quiescent protostars . we employ the same numerical model as the one described in section 2 of r09 , and the reader is referred to that paper for further details . as explained in section 2.3.1 of r09 , the high - density gas that comprises the disc material enters the simulation region through the disc boundary only after the computation commences , and it subsequently flows inward on account of its viscosity . this numerical setup is thus naturally suited for modelling the evolution of an accretion ` burst ' , which is the focus of the present work . although we performed simulations for a variety of model parameters , we present only one representative case in this paper . we use the same parameters as in the reference simulation shown in r09 , except that we reduce the reference radius @xmath39 from @xmath40 to @xmath41 and change the outer radius of the computational domain from @xmath42 to @xmath43.au , which corresponds to @xmath44 for the inferred value of @xmath13 in our reference simulation . ] the latter change enables us to increase the mass accretion rate on to the central star in our simulation to the level inferred in fuors . the larger size of the computational domain also allows us to get a better handle on the collimation properties of conical winds . , is presented in @xcite . ] the mass accretion rate on to the central object is determined from the simulations using the expression @xmath45 where @xmath46 is the reference mass accretion rate and where @xmath47 and @xmath48 are , respectively , the dimensionless magnetic moment and mass accretion rate parameters . we use @xmath49 as in the reference simulation of r09 and obtain the value of @xmath50 from the numerical calculation . as described in section [ sec : results ] , the final ( quasi - steady ) mass accretion rate on to the central object in our representative simulation corresponds to @xmath51 . we also find that @xmath52 at that stage , which implies @xmath53 ( using our fiducial value for @xmath28 ) . we can then use equation ( [ eq : mdot_sim ] ) to infer the value of @xmath54 for our representative model : @xmath55 the reference and fiducial parameters for our model are summarized in table [ tab : refval ] . ) , we infer @xmath56 . although @xmath20 is expected to be @xmath57 under stationary conditions ( e.g. * ? ? ? * ) , we consider the derived value to be physically consistent , especially in view of the time - dependent nature of the simulation and the expected presence of higher order multipole field components . one could in principle obtain a lower value of @xmath20 by increasing the adopted value of the parameter @xmath58 . ] [ cols="<,<,<,<",options="header " , ] the time evolution of the simulated system depends on the magnitudes of the viscosity and the magnetic diffusivity , which are parametrized by @xmath59 and @xmath60 , respectively . r09 ( see their appendix d ) found that conical winds are established only when @xmath61 . this is consistent with the fact that the dragging of the magnetic field by the accretion flow , which in fuors causes the field compression near the inner boundary of the disc , requires the magnetic prandtl number @xmath62 to be @xmath38 ( e.g. * ? ? ? our simulations employ the values adopted in the reference case of r09 , namely @xmath63 and @xmath64 . recent work on non - steady protostellar accretion - disc models ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) has indicated that @xmath59 must be large enough ( @xmath65 ) for outbursts that resemble those of fuors to be produced . our adopted value of the viscosity parameter is consistent with this requirement . it is noteworthy in this connection that r09 found that the formation of a robust conical outflow also requires @xmath59 ( _ and _ @xmath60 ) to be comparatively large ( @xmath66 ) . the large - scale poloidal structure of the simulated flow is shown in fig . [ fig : fig1 ] at the time when the wind has become fully developed . the most striking feature of the figure is its qualitative similarity to fig . 3 in r09 , which corresponds to a quiescent protostar that accretes at a much lower ( by a factor @xmath67 ) rate . this conclusion is reinforced by an inspection of fig . [ fig : fig2 ] , which shows a close - up view of the region near the star . in both cases a high - density , conical disc wind is launched from the vicinity of the disc truncation radius @xmath28 , and a lower - density , higher - velocity jet component is established in the interior of the cone . the main difference between the two simulations is in the value of @xmath68 : it is @xmath69 in our simulation , as compared with @xmath70 in the r09 reference calculation . this difference is consistent with the expectation from equation ( [ eq : ratio ] ) , which indicates that this ratio scales only as a weak power of the mass accretion rate ( @xmath71 ) . thus , even though the accretion flow is much more powerful in this case , the steep radial scaling of the magnetic pressure exerted by the dipolar field component ( @xmath72 ) insures that the disc is still truncated at a finite radius and does not actually ` crush ' the stellar magnetosphere ( as envisioned , for example , in the x - celerator scenario for fuors ; see @xcite ) . although the disc in the current simulation is truncated very close to the stellar surface , the magnetic field structure in the vicinity of its inner radius is qualitatively very similar to the case where @xmath73 is @xmath38 . in particular , the magnetic field lines that guide the conical wind and the axial jet are open . the opening - up of parts of the initially dipolar stellar field is a consequence of the differential rotation between the star , where the magnetic field is anchored , and the disc , into which the field lines diffuse , and is a generic property of magnetically linked star disc systems ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . as in the lower-@xmath74 case considered in r09 , the conical wind is driven primarily by the pressure gradient of the azimuthal magnetic field component generated by the differential rotation rather than centrifugally . note in this connection that the profile of the azimuthal velocity component ( see panel c in fig . 6 of r09 as well as the right panel of fig . [ fig : fig4 ] below ) initially _ decreases _ along the poloidal field lines , which contrasts with the behaviour of centrifugally driven winds , in which the azimuthal speed @xmath75 at first increases along a field line . the acceleration is quite efficient , and the conical wind reaches outflow speeds @xmath76 ( corresponding to @xmath77 of the keplerian speed at @xmath28 ) at the outer edge of the simulation region . the highest poloidal speed observed in this simulation is attained further up and is associated with the lower - density axial flow . as seen in fig . [ fig : fig1 ] , its value is @xmath78 , which is consistent with the maximum line - of - sight speed measured in fu ori @xcite . however , this value may not be accurate since the low - density region in the vicinity of the axis is susceptible to numerical artifacts . ) , which results in a comparatively weak axial outflow . ] near the base of the flow the wind velocity is dominated by the azimuthal component , which arises from the rotational motion of the disc in the wind - launching region and has a maximum value of @xmath79 , attained at @xmath80 . ( at smaller radii the wind azimuthal velocity decreases on account of the interaction with the magnetosphere , which rotates with the comparatively low angular velocity of the star . ) we note that , even though the wind is launched very close to the stellar surface and has a high initial rotation velocity , it does _ not _ originate in the stellar surface and does _ not _ require the star to rotate at break - up speed which distinguishes it from the outflow envisioned in the x - celerator scenario @xcite . as discussed in r09 , the magnetic force also has a component directed toward the symmetry axis , which acts to collimate the wind . by using the poloidal matter flux distribution , r09 determined that the conical outflow in their reference simulation attained an opening half - angle of @xmath81 . from the corresponding distribution presented in fig . [ fig : fig1 ] , we find that the collimation is even more efficient in the case that we simulate , with the outflow half - angle decreasing to @xmath82 within a radial distance ( projected on the equatorial plane ) of @xmath83 from the stellar surface . in general , a magnetically driven outflow is collimated by a combination of two effects ( e.g. * ? ? ? * ) : the magnetic tension force that acts to balance the magnetic pressure - gradient force in the force - free sub - alfvnic regime , and the hoop stress exerted by the azimuthal magnetic field component in the super - alfvnic flow region . the difference in the collimation properties of the conical wind in our simulation and in the reference simulation of r09 can be attributed to the fact that a higher mass accretion rate in the disc results in a stronger compression of the stellar magnetic field and hence in a larger collimating magnetic tension force in the sub - alfvnic region of the wind . the hoop - stress effect is also stronger in the higher-@xmath74 case on account of the compressional amplification of the field and because the differential rotation that twists the field lines gets stronger ( for a given value of @xmath14 ) as @xmath28 is decreased . a detailed analysis of the collimation properties of a conical wind from a high-@xmath74 disc is given in @xcite . [ fig : fig3 ] shows the evolution of the matter fluxes that are deposited by the accretion flow on to the stellar surface and in the outflow ( with the mass outflow rate evaluated over a spherical surface far enough from the centre ) . it is seen that the mean mass accretion rate increases steadily until it attains a fully developed state ( with @xmath74 corresponding to the value inferred in fu ori ) at a time @xmath84d from the start of the simulation . this time is longer than the @xmath85yr observed rise time of the fu ori outburst ( e.g. * ? ? ? * ) , but the discrepancy is probably in large part just a consequence of the particular choice of initial conditions for our simulation ( see section [ subsec : numerical ] ) . in the fully developed state , the average accretion and outflow rates are related by @xmath86 . this result is consistent with the observational findings in fu ori ( e.g. * ? ? ? the rapid accretion during the fuor outburst can be expected to spin up the star , and it is therefore necessary to check whether our assumption of slow stellar rotation is self - consistent . we calculated the torque on the star at the end of our simulation from the expression @xmath87 , where the reference torque @xmath88 is listed in table [ tab : refval ] and where the dimensionless field and matter contributions @xmath89 and @xmath90 are , in our case , @xmath91 and @xmath92 , respectively . and @xmath93 at @xmath94 at the end of the simulation . this is consistent with the fact that , as in the quiescent case ( see fig . 14 in r09 ) , most of the angular momentum transport occurs through the viscous stress in the disc . ] by dividing the total torque calculated in this way , @xmath95 , into the stellar angular momentum @xmath96 , where we assume uniform rotation with angular velocity @xmath97 and scale the normalized radius of gyration @xmath98 by its value for a polytrope of index 1.5 , we infer a characteristic spin - up time @xmath99 , which is of the order of the typical fuor outburst time . this implies that our assumption of a slow rotator is only marginally consistent . we note , however , that a conical wind - like component may be present in the outflow even if this assumption is violated ( see section [ sec : conclude ] ) . the simulation results presented in section [ sec : results ] indicate that the observed properties of fuor outflows could in principle be explained in terms of the conical wind and axial jet that are driven from the vicinity of the stellar surface in these systems along stellar magnetic field lines that are compressed , twisted , and opened up by the interaction between the initially dipolar field component and the strong accretion flow . in particular , for typical values of the stellar mass , radius and surface magnetic field strength , and of the mass inflow rate at the inner edge of the circumstellar disc , our representative simulation demonstrates that this interaction can produce outflows whose properties ( mass outflow rate , strong rotational velocity component near the base and possibly also the maximum outflow speed ) are consistent with the observations . as was mentioned in section [ sec : intro ] , chk93 and @xcite argued against a stellar magnetic field - driven wind being able to account for the spectral properties of the outflow in fu ori . they envisioned the outflow as originating in the stellar surface and accelerating rapidly along strongly divergent field lines even as its rotation speed ( which initially has the stellar breakup value ) continues to increase . in our picture , the absorption features modelled in the above - mentioned papers would arise in the conical - wind component , which exhibits a spatially more extended acceleration ( along fast collimating magnetic field lines ) and a lower initial rotation speed ( that at first actually decreases along the flow ) than the ` stellar field ' outflow assumed in those papers . although the spatial and kinematic properties of our simulated conical wind are distinct from the semi - analytic disc outflow model presented in chk93 ( which combined a hydrostatic disc atmosphere with a simple representation of a centrifugal wind ) , it is probably qualitatively closer to that model ( which chk93 and hartmann & calvet 1995 argued was consistent with the spectral data for fu ori ) than to their hypothesized stellar wind model . to demonstrate that the conical wind model could account for the behaviour of the absorption lines measured in fu ori would require a determination of the thermal structure of the simulated flow and a calculation of the synthesized spectra of the relevant photospheric lines . an analogous radiative - transfer calculation , addressing the rotationally induced line variability from an accreting t tauri star with a misaligned magnetic dipole , was carried out by @xcite . while a detailed computation of this type is outside the scope of the present paper , we can obtain some indication of the potential promise of this model by calculating the density and velocity profiles as functions of distance from the mid - plane at the location of the optical continuum emission region and comparing the results with those obtained in the disc wind model of chk93 . in view of the clear differences between our numerical model and chk93 s semi - analytic model ( which include the fact that the latter model , in contrast with our simulations , incorporates energy loss by radiative diffusion in the disc atmosphere ) , we can not expect to find a full quantitative correspondence between the two calculations . , @xmath100 and @xmath101 , and presented results for @xmath102 . furthermore , in their spectral fits they assumed a disc inclination angle @xmath103 , whereas more recent fits for this source ( e.g. zhu et al . 2007 ) have used @xmath104 . ] however , we can look for common trends in the respective profiles . we take the optical emission radius @xmath105 to correspond to an effective disc temperature of @xmath106k , and we use the disc model of @xcite , in which @xmath107 $ ] and the maximum effective temperature ( attained at @xmath108 ) is @xmath109k , to deduce @xmath110 . as noted by chk93 , the disc model fits of @xcite similarly imply that roughly 60% of the optical spectrum in fu ori arises from disc annuli between @xmath111 and @xmath112 . chk93 suggested that this region could be adequately represented by their calculated ` disc atmosphere plus wind ' structure at @xmath113 , which they presented as a function of the height @xmath114 above the mid - plane in their table 2 . in view of the different model setups and adopted source parameters , there are several possible choices for the value of the wind - launching radius where we could compare our model results with the ones given in that table . for definiteness , we opt to also use @xmath115 . note , however , that this value corresponds to different radial distances in the two models : @xmath116 for our choice of parameters and @xmath117 for those employed by chk93 . the comparison between the predictions of these two models is presented in fig . [ fig : fig4 ] . in the chk93 model , the poloidal velocity only has a vertical ( @xmath114 ) component and the azimuthal velocity is approximated as being constant with height and set equal to the keplerian speed at the base of the flow ( @xmath118 for their adopted parameters ) . the curves showing @xmath119 and the mass density @xmath120 in that model are labelled by chk and plotted as a function of @xmath114 at @xmath115 using the data in table 2 of chk93 . in view of the conical shape of our simulated wind , we plot the poloidal and azimuthal velocity components as well as the density in the numerical model along a slightly inclined path ( represented by the heavy dash - dotted straight line in fig . [ fig : fig2 ] ) . and [ fig : fig2 ] . ] it is seen that , as expected , the values of corresponding quantities at a given distance from the mid - plane can be significantly different for the two cases . reflects , in part , the fact that our simulated disc is hotter and therefore geometrically thicker than the chk93 model disc . the latter model was chosen to be consistent with the minimum - temperature requirement obtained by @xcite for a viscous disc to generate fuor - type outbursts . the hotter disc produced in our simulation is clearly also consistent with this condition . ] however , we also find that the basic velocity and density structure of the two models is very similar . in both cases , the flow is rotation - dominated as it emerges from the disc but eventually @xmath121 comes to exceed @xmath75 ( which , in turn , does not systematically increase along the flow ) . chk93 estimated that the disc photosphere at @xmath122 occurs roughly where the density drops to @xmath123 and @xmath119 increases to @xmath124 . ( for comparison , the sound speed in the optical emission region is @xmath125 . ) in their model , @xmath119 becomes @xmath126 when the density drops to @xmath127 . in our simulation , @xmath128 increases above @xmath124 also roughly when the density drops to @xmath123 , and @xmath121 comes to exceed @xmath75 when @xmath120 decreases by another two orders of magnitude . this correspondence indicates that the observed dependence of photospheric absorption line profiles in fu ori on the line strength , which was successfully reproduced by the chk3 model ( see also * ? ? ? * ) , is consistent with an origin in a stellar field - driven conical wind . even if more extensive and detailed calculations indicate that the observed behaviour of the photospheric absorption lines in fu ori can not be reproduced by a conical wind model ( because , for example , a wind of this type that is launched very close to the stellar surface collimates too rapidly for its acceleration region to intercept a line of sight to the optical continuum emission region for the inferred disc inclination angle , or if the maximum predicted outflow speed is too low ) , the results of our simulations suggest that a stellar field - driven outflow might still be an important ingredient of a comprehensive model of fuors . in particular , such an outflow could still potentially account for much of the mass and momentum injected into the ambient medium in the course of an outburst and perhaps also for the highest measured velocities in the h@xmath129 , h@xmath130 and na i lines ( e.g. * ? ? ? * ; * ? ? ? * ) even if another outflow component ( in particular , a disc wind driven along non - stellar magnetic field lines , which were not included in our simulations ) gives rise to the observed photospheric lines . this is because a strong conical wind and a fast , low - density jet appear to be generic features of the disc / stellar - field interaction under a wide range of conditions . on the other hand , if it can be demonstrated that these predicted outflow components are , in fact , absent during fuor outbursts , this would indicate that at least one of the underlying key assumptions of the model [ e.g. that the magnetic diffusivity in the inner disc does not exceed the viscosity ( @xmath131 ) but is nevertheless sufficiently large ( @xmath132 ) , or that the star possesses a sufficiently strong dipolar field component ( @xmath133 ) ] is not valid , which would also enhance our physical understanding of these systems . we have presented numerical simulation results that support an interpretation of the powerful winds that accompany fuor outbursts in terms of stellar magnetic field - driven disc outflows . in this picture , the massive accretion flow that gives rise to an observed burst strongly compresses the stellar magnetic field lines , and the resulting magnetic stress truncates the accretion disc very close to the stellar surface . some of the field lines diffuse into the disc and become twisted by the differential rotation between the disc and the star . this twisting , in turn , opens up the field lines , and the pressure gradient associated with the azimuthal magnetic field component along the opened field drives a moderate - velocity , dense conical wind that emanates from the vicinity of the truncation radius as well as a high - velocity , tenuous axial jet ( whose properties , however , are less well determined in the simulation that we described ) . the magnetic field also acts to collimate these outflow components . the conical wind and axial jet appear to be generic features of the interaction between an accretion disc and a predominantly dipolar stellar field in cases where the effective viscosity @xmath59 and magnetic diffusivity @xmath60 satisfy @xmath134 and are both comparatively high ( @xmath66 ) . these features were originally identified in simulations of protostars with low and moderate accretion rates ( r09 ) . the representative simulation presented in this paper verifies that the same type of outflow is produced also when the accretion rate is as high as @xmath135 , the value inferred in the archetypal object fu ori . our simulation implies that the disc truncation radius in this source , which was observationally determined to lie at a radius @xmath136 , corresponds to a distance of @xmath137 from the stellar surface , and that the surface magnetic field is @xmath138 , which is consistent with independent indications . the mass outflow rate in the simulated outburst ( dominated by the conical wind ) is a factor @xmath139 of the mass accretion rate on to the star , and the maximum outflow velocity within the computational domain ( attained in the axial jet ) is @xmath140 ; these agree well with the observationally inferred values for fu ori . an interpretation of fuor winds in terms of an accretion - disc outflow driven along stellar magnetic field lines was previously proposed by @xcite on the basis of the x - celerator model of @xcite . in this picture , the outflow is launched centrifugally from the surface of a star whose outer layers rotate at breakup speeds . this contrasts with the conical - wind scenario , in which the star rotates comparatively slowly and the outflow originates at a finite distance from the star and is driven by the @xmath141 magnetic pressure gradient from the start . @xcite generalized the x - celerator model to the case where the star rotates below breakup , corresponding to the corotation radius @xmath14 exceeding @xmath13 . however , in their generalized ( x - wind ) model , @xmath14 still coincides with the magnetospheric radius @xmath1 , and the nature of the outflow from that region ( the x - point ) is qualitatively similar to that of the x - celerator model . our assumption in this paper that @xmath142 is plausible in view of the fact that a rapidly rotating protostar could be efficiently braked through a magnetic interaction with the disc during the relatively long quiescent phase ( e.g. * ? ? ? * ; * ? ? ? and while such a star would be spun up during the rapid accretion event comprising an fuor outburst ( see section [ sec : results ] ) , this need not result in the surface layers reaching breakup speeds . ( note in this connection that , even in the absence of a large - scale magnetic field coupling the disc and the star , the protostellar surface layers are not expected to reach breakup speeds during an outburst of this type ; e.g. @xcite ) . chk93 and @xcite argued that a stellar magnetic field - driven outflow model of the x - celerator type is inconsistent with the detection of moderately blueshifted , intermediate - strength absorption lines in fu ori , which , they suggested , could be explained in terms of a disc outflow originating at a distance @xmath143 of a few stellar radii and driven along magnetic field lines that are not associated with the star . specifically , they showed that the observed line profiles could be reproduced by a model in which gas launched from a keplerian accretion disc gradually accelerates until the poloidal velocity component comes to exceed the azimuthal velocity component . in this paper we have demonstrated that a conical wind naturally exhibits this behaviour since the outflow also starts with a predominantly azimuthal velocity component and eventually accelerates to @xmath144 . in particular , we showed that the density and poloidal velocity profiles along a ray through the conical shell that intercepts the disc at the distance of the optical emission region closely match the corresponding profiles calculated in the disc - outflow model of chk93 , notwithstanding the different setups ( and even the fiducial parameter values ) employed in the two ( respectively , numerical and semi - analytic ) models . the fact that the azimuthal velocity of the conical wind remains much lower than the breakup speed of the star and that , in contrast with the initial behaviour of @xmath75 in a centrifugally driven wind , it does not increase ( but , rather , decreases ) along the outflow , circumvents another objection that chk93 levelled at the x - celerator scenario . we note in this connection that an outflow component resembling a conical wind , as well as a strong axial jet component , have been found in simulations of the disc magnetosphere interaction in the ` propeller ' regime ( @xmath145 ) . based on the results presented in r09 , we expect such a flow to be more strongly influenced by the centrifugal force and less well collimated for given values of @xmath18 and @xmath74 than the conical wind we considered above . however , r09 also found in the propeller case that the magnetic force remains important in driving the wind and that the azimuthal speed of the wind does not increase along a field line ( see their fig . it is therefore conceivable that an outflow in this regime could also account for the observations , although this remains to be verified by an explicit simulation . while the results presented in this paper are highly suggestive , a more detailed calculation ( involving the thermal and spectral properties of the outflow ) is required to evaluate the contribution of a stellar field - driven disc wind to the absorption - line spectrum in an object like fu ori . it will also be useful to carry out additional simulations in order to further check the dependence of the results on the adopted initial mass and magnetic flux distributions . in particular , our assumption that initially there is no disc ( as compared to coronal ) gas in the simulation region is not realistic , and our current numerical setup also does not account for the possibility that some of the stellar magnetic field may have diffused into the disc before the onset of the outburst ( see * ? ? ? * ) . given the comparatively high value of the disc inclination angle ( @xmath104 ; @xcite ) adopted in recent studies of fu ori , it is conceivable that the conical wind model would not be able to reproduce the absorption - line profiles measured in this object if the source of the optical continuum is indeed a region of size @xmath146 in the disc . in that case a disc outflow driven along a non - stellar magnetic field , as proposed by chk93 , might provide the dominant contribution to the absorption - line spectrum of fu ori , although a ` conical wind plus axial jet ' outflow could potentially still contribute to some of the observed properties of this object . note , however , that if a _ stellar _ wind is also present ( see section 5.2 of r09 ) , it would have a decollimating effect on the conical wind ( e.g. * ? ? ? * ; * ? ? ? * ) that could increase the range of disc radii ` covered ' by the conical outflow . to our knowledge , photospheric line shifts such as those detected in fu ori have so far not been found in any other fuor . although a lower-@xmath147 wind or some other factor ( such as a higher projected azimuthal velocity ) could have prevented a detection in other fuors ( see * ? ? ? * ) , it would clearly be useful to be able to test competing models also in other bursting sources . lower - amplitude , repetitive photometric outbursts have been detected in ex lupi and a few other t tauri stars ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , and they have also been interpreted as enhanced mass accretion events . in a few of these exor sources there is evidence for an accompanying outflow , which , as discussed in r09 , may well ( at least in some cases ) arise in a disc magnetosphere interaction . however , these systems generally do not exhibit absorption - line spectra like fuors , and there is also no indication that their continuum emission is dominated by a disc ; in this regard their appearance is similar to that of quiescent systems . therefore , unless other spectroscopic diagnostics are identified in these sources , fuors will remain the best candidates for probing the acceleration regions of protostellar disc outflows . this research was supported in part by nsf grants ast-0908184 ( ak ) and ast-1008636 ( mmr and rvel ) and by a nasa atp grant nnx10af63 g ( mmr and rvel ) . the authors thank g. v. ustyugova and a. v. koldoba for the development of the code used in the simulations discussed in this paper , and the reviewer for comments that helped to improve the presentation . bastian u. , mundt r. , 1985 , a&a , 144 , 57 blandford r. d. , payne d. g. , 1982 , mnras , 199 , 883 calvet n. , hartmann l. , kenyon s. j. , 1993 , apj , 402 , 623 ( chk93 ) calvet n. , hartmann l. , strom s. e. , 2000 , in mannings v. g. , boss a. p. , russell s. , eds , protostars & planets iv . arizona press , tucson , p. 377 clarke c. j. , lin d. n. c. , pringle j. e. , 1990 , mnras , 242 , 439 croswell k. , hartmann l. , avrett e. h. , 1987 , apj , 312 , 227 donati j. f. , et al . , 2007 , mnras , 380 , 1297 donati j. f. , et al . , 2008 , mnras , 386 , 1234 donati j. f. , paletou f. , bouvier j. , ferreira j. , 2005 , nature , 438 , 466 fendt c. , 2009 , apj , 692 , 346 ghosh p. , lamb f. k. , 1979a , apj , 232 , 259 ghosh p. , lamb f. k. , 1979b , apj , 234 , 296 goodson a. p. , winglee r. m. , 1999 , apj , 524 , 159 hartmann l. , calvet n. , 1995 , aj , 109 , 1846 hartmann l , kenyon s. j. , 1996 , ara&a , 34 , 207 herbig g. h. , 1977 , apj , 217 , 693 herbig g. h. , 1989 , in reipurth , b. , ed , eso workshop on low - mas star formation and pre main - sequence objects . eso , garching , p. 233 herbig g. h. , 2007 , aj , 133 , 2679 herbig g. h. , 2008 , apj , 135 , 637 herbig g. h. , petrov p. p. , duemmler r. , 2003 , apj , 595 , 384 johns - krull c. m. , 2007 , apj , 664 , 975 johns - krull c. m. , greene t. p. , doppmann g. w. , covey k. r. , 2009 , apj , 700 , 1440 kenyon s. j. , hartmann l. , hewett r. , 1988 , apj , 325 , 231 knigl a. , 1991 , apj , 370 , l39 kurosawa r. , romanova m. m. , harries t. j. , 2008 , mnras , 385 , 1931 lii p. , romanova m. m. , lovelace r. v. e. , 2011 , mnras , submitted ( arxiv:1104.4374 ) long m. , romanova m. m. , kulkarni a. k. , donati j. f. , 2011 , mnras , 413 , 1061 long m. , romanova m. m. , lamb f. k. , 2011 , new astronomy , submitted ( arxiv:0911.5455 ) long m. , romanova m. m. , lovelace r. v. e. , 2005 , apj , 634 , 1214 lovelace r. v. e. , romanova m. m. , bisnovatyi - kogan g. s. , 1995 , mnras , 275 , 244 lubow s. h. , papaloizou j. c. b. , pringle , j. e. , 1994 , mnras , 267 , 235 lynden - bell d. , boily c. , 1994 , mnras , 267 , 146 malbet f. , et al . , 2005 , a&a , 437 , 627 meliani z. , casse f. , sauty c. , 2006 , a&a , 460 , 1 popham r. , kenyon s. , hartmann l. , narayan r. , 1993 , apj , 473 , 422 popham r. , narayan r. , hartmann l. , kenyon s. , 1993 , apj , 415 , l127 reipurth b. , 1990 , in mirzoyan , l. v. , pettersen b. r. , tsvetkov m. k. , eds , proc . 137 , kluwer , dordrecht , p. 229 romanova m. m. , long m. , lamb f. k. , kulkarni a. k. , donati j .- f . , 2011 , mnras , 411 , 915 romanova m. m. , ustyugova g. v. , koldoba a. v. , lovelace , r. v. e. , 2009 , mnras , 399 , 1802 ( r09 ) shu , f. h. , lizano s. , ruden s. p. , najita j. , 1988 , apj , 328 , l19 shu , f. najita j. , ostriker e. , wilkin f. , ruden s. , lizano s. , 1994 , apj , 429 , 781 skinner s. l. , sokal k. r. , gdel m. , briggs k. r. , 2009 , apj , 696 , 766 ustyugova g. v. , koldoba a. v. , romanova m. m. , lovelace r. v. e. , 2006 , apj , 646 , 304 uzdensky d. , knigl a. , litwin c. , 2002 , apj , 565 , 1191 van ballegooijen a. a. , 1994 , ssrv , 68 , 299 zhu z. , hartmann l. , calvet n. , hernandez j. , 2007 , apj , 669 , 483 zhu z. , hartmann l. , calvet n. , hernandez j. , tannirkulam a .- j . , dalessio p. , 2008 , apj , 684 , 1281 zhu z. , hartmann l. , gammie , c. , 2009a , apj , 694 , 1045 zhu z. , hartmann l. , gammie c. , book l. c. , simon j. b. , engelhard e. , 2010 , apj , 713 , 1134 zhu z. , hartmann l. , gammie c. , mckinney , j. c. , 2009b , apj , 701 , 620

fu orionis ( fuor ) outbursts are major optical brightening episodes in low - mass protostars that evidently correspond to rapid mass - accretion events in the innermost region of a protostellar disc . the outbursts are accompanied by strong outflows , with the inferred mass outflow rates reaching @xmath0 of the mass inflow rates . shu et al . proposed that the outflows represent accreted disc material that is driven centrifugally from the spun - up surface layers of the protostar by the stellar magnetic field . this model was critiqued by calvet et al . , who argued that it can not reproduce the photospheric absorption - line shifts observed in the prototype object fu ori . calvet et al . proposed that the wind is launched , instead , from the surface of the disc on scales of a few stellar radii by a non - stellar magnetic field . in this paper we present results from numerical simulations of disc accretion on to a slowly rotating star with an aligned magnetic dipole moment that gives rise to a kilogauss - strength surface field . we demonstrate that , for parameters appropriate to fu ori , such a system can develop a strong , collimated disc outflow of the type previously identified by romanova et al . in simulations of protostars with low and moderate accretion rates . at the high accretion rate that characterizes the fuor outburst phase , the radius @xmath1 at which the disc is truncated by the stellar magnetic field moves much closer to the stellar surface , but the basic properties of the outflow , which is launched from the vicinity of @xmath1 along opened - up stellar magnetic field lines , remain the same . these properties are distinct from those of the x - celerator ( or the closely related x - wind ) mechanism proposed by shu et al . in particular , the outflow is driven from the start by the magnetic pressure - gradient force , not centrifugally , and it is more strongly collimated . we show that the simulated outflow can in principle account for the main observed characteristics of fuor winds , including the photospheric line shifts measured in fu ori . a detailed radiative - transfer calculation is , however , required to confirm the latter result . [ firstpage ] accretion , accretion discs mhd stars : formation stars : magnetic field stars : winds , outflows .

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Proceed to summarize the following text: partial or full ring - like structures , which apparently are projections of shells , are often found in the interstellar medium ( ism ) . these dense shells could possibly be driven by one of the numerous sources like , ionising radiation from young star - clusters , early type massive stars , blast - waves from supernovae , or energetic stellar winds . detailed observations of these shells in various bands of the infrared wavelength have also revealed isolated sites of massive star formation ( e.g. deharveng , zavagno & caplan 2005 ) . in the recent past a number of such sites have been reported , for instance in the hii region rcw79 ( zavagno _ et al . _ 2006 ) , and rcw120 ( anderson _ et al . a catalogue of 600 such shells in the galactic disk was drawn up by churchwell _ ( 2006 ) as part of the glimpse survey . the survey showed that a large proportion ( @xmath290 % ) of these shells are thin , i.e. the shell thickness is less than a third of the outer shell radius , and driven primarily by massive stars ( @xmath3 shells ) , see also zavagno _ et al . _ 2010 . it is well - known that relatively high - mass stars emit powerful radiation that ionises gas in the local neighbourhood , and heats it to temperatures typically of the order of 10@xmath4 kelvin . this hot plasma propagates in the interstellar medium ( ism ) at a highly supersonic speed ( typical velocity of expansion in the initial phases is @xmath5 km / s ) , whence it gradually equilibrates to a significantly lower temperature , of order a few thousand kelvin . the expanding , roughly spherical volume of hot plasma , the so called stroemgren sphere , cools primarily via collisional excitation of heavier elements such as carbon , nitrogen , and oxygen , and sputtering of dust - grains while sweeping up a dense shell of gas in the ism ; the familiar snow - plough phase . this shell is confined by two shocks , first , due to the wind driving it , and second , due to the reverse shock resulting from the propagation of the shell in the ism . the stability of such shells has been discussed by numerous authors , e.g. elmegreen & elmegreen ( 1978 ) , larson ( 1985 ) , elmegreen ( 1989 ) , vishniac ( 1983 , 1994 ) , whitworth ( 1994 ) , wuensch & palou@xmath6 ( 2001 ) and anathpindika ( 2010 ) . it has been demonstrated by these and several other authors that , a shock - confined shell is unstable to instabilities arising out of shock - induced turbulence within layers of the shell . turbulence leads to enhanced transfer of momentum in different regions of the shell that makes it unstable to the so called thin shell instability ( tsi ) ( vishniac 1983 , 1994 ) , and raises the effective local sound - speed . the stability of the shell depends on the critical interplay between the gravitational instability ( gi ) and the tsi . a full fledged analytic treatment of the stability of a shock - confined slab is rather complex as has been demonstrated by vishniac ( 1994 ) , for instance , who showed that such slabs were likely to be unstable to the so called non - linear thin shell instability ( ntsi ) . traditionally , stability analysis of shocked shells and/or slabs , for all practical purposes , have been simplified by excising shock dynamics in favour of a simple high - pressure approximation thereby eliminating perturbative effects of the tsi , and its non - linear excursion ( e.g. whitworth _ et al . _ 1994b ) . anathpindika ( 2009 , 2010 ) numerically showed that a shocked slab and/or shell is unstable to the tsi , which , soon after its formation , develops wiggles on its surface and grows non - linearly . ehlerov@xmath7 & palou@xmath6 ( 2002 ) derived the critical density for a shell to become gravitationally unstable . under a sinusoidal approximation for these perturbations , anathpindika ( 2010 ) deduced an expression for the wavenumber of the fastest growing mode . below , we propose to test the validity of this expression for hii shells driven by typical candidate stars listed in table 5.3 of spitzer ( 1978 ) , and deduce a mass function for clumps condensing out of these shells . the case of a shell driven by a supernova blast wave will be considered separately . we shall demonstrate that the mass function so derived is consistent with that reported by fukui _ et al._(1999 ) , yamaguchi _ et al . _ ( 2001 ) , and roslowsky ( 2005 ) for massive clouds . in 2 we shall deduce our set of equations and demonstrative calculations , including the calculation of a mass spectrum for fragments , will be undertaken in 3 . we conclude in 4 . let us consider a typical source of ionising radiation that emits n@xmath8 number of photons per second . if @xmath9 , @xmath10 are the respective number of protons and @xmath11 per unit volume , then @xmath12 is the number of electrons captured per @xmath13 in the ground state , and the flux of photons flowing through a shell of radius , @xmath14 , is- @xmath15 where @xmath16 is the recombination coefficient that excludes electron captures to the ground state , and defined as @xmath17 at a temperature @xmath18 ; @xmath19 is the ionic charge and @xmath20 is the recombination coefficient function corresponding to @xmath16 ( see table 5.2 , spitzer 1978 ) . the ionisation fraction of hydrogen , @xmath21 , has been set equal to unity so that the stroemgren radius is defined as - @xmath22 where the condition of approximate charge neutrality forces @xmath23 . the temporal evolution of the ionised shell can be obtained by rewriting eqn . ( 2 ) as @xmath24,\ ] ] integration of which yields , @xmath25.\ ] ] maximising this equation gives the timescale , @xmath26 , over which the shell of ionising radiation makes a transition from the initial rarefied phase , to the dense phase , which is @xmath27 the radius of the expanding shell in the dense phase is given by @xmath28 ( e.g. shore 2007 ) . for an approximately spherical shell , its average volume density , @xmath29 , is @xmath30 after a time @xmath31 , the shell cools down to an equilibrium temperature , @xmath32 , defined by eqn . ( 8) below . to estimate the temperature , t@xmath33 , we shall first account for the likely heating and cooling mechanisms . a crucial contributor towards heating the shell is the photoionisation of h@xmath34 , the corresponding rate of heating is @xmath35 ( tielens 2005 ) , where @xmath36 @xmath37s@xmath38 , is the recombination cooling coefficient . the cooling rate due to collisional excitation of carbon is @xmath39 ( tielens 2005 ) , where @xmath40 , is the abundance of carbon in the ism . the equilibrium temperature of the gas , @xmath32 , within the shell can be estimated using the condition for thermal equilibrium , @xmath41 , and excess energy will be radiated away so that @xmath42 , i.e. @xmath43 which is roughly 10 k. the temperature , @xmath44 , of the ionised gas is @xmath45 , where @xmath46 k , is the average temperature of the preshock ism and @xmath47 is the mach number of the propagating shock , @xmath48 , @xmath49 . the density of the shocked gas , @xmath50 , @xmath51 is the average mass of atomic hydrogen . anathpindika ( 2010 ) , by performing a perturbative analysis , derived an expression for the fastest growing unstable mode in a shocked shell , and the wavenumber , @xmath52 , of this mode is @xmath53}.\ ] ] here @xmath54 and @xmath55 are respectively the outer , and the inner radii of the shell , and @xmath56 . ( 3 ) it follows that , @xmath57 for a shell of thickness , @xmath58 , @xmath59 , so that @xmath60 , and since @xmath61 ; @xmath62 . for the wavenumber , @xmath52 , calculated using eqn . ( 9 ) above , the corresponding wavelength , @xmath63 so that mass of a clump , @xmath64 , and the number of fragments @xmath65 . @xmath66 @xmath67^{2}\frac{1}{g^{2}\sigma_{s}},\ ] ] and since @xmath68 , @xmath69 the quantity @xmath70 , as will be shown in 3 below , so that @xmath71 calculations for typical o - b stars are deferred for 3 below . we now discuss the stability of a shell driven by blast waves originating from a supernova that injects energy , @xmath72 , into the ism . for a shell having average density @xmath29 , and radius , @xmath73 , @xmath74 integrating eqn . ( 12 ) yields the well - known sedov solution for the shell radius , @xmath75 where @xmath76 is a numerical constant of order unity . ] the post - shock temperature , @xmath44 , can be calculated using the usual hugoniot shock conditions for an ism at a preshock temperature , @xmath46 k , as before . the pressure due to the blast - wave , @xmath77 , under the assumption of approximate isothermality ; all symbols have their usual meaning . the average density of the shell , @xmath78 . using eqn . ( 13 ) above , we can estimate the timescale , @xmath79 , over which the shell is likely to acquire its equilibrium temperature , @xmath32 , defined by eqn . ( 8) above , whence it may also slow down substantially , to a sonic or possibly , even sub - sonic speed , @xmath80 , is the local sound - speed . then at @xmath81 , @xmath82 @xmath83 the surface density of the shell , @xmath84 , is calculated as before and the mass of the shell , @xmath85 , @xmath86 @xmath87 , is the radius of the shell at that epoch ; this expression is also used to calculate the mass of the hii shell . the fastest growing unstable mode , as before , is calculated as before , using eqn . ( 9 ) above . the calculations for a typical supernova remnant ( snr ) are demonstrated in the following section . _ the collect and collapse model : the hii shell _ + for illustrative purposes we consider a typical particle density , @xmath88 . the minimum timescale for radiative cooling of the shell , @xmath26 , calculated using eqn . ( 4 ) is approximately 0.1 myrs for a typical o5 star ; we have adopted physical parameters defined in table 5.3 of spitzer ( 1978 ) . the shell , at this epoch , enters in to the dense phase whence it acquires substantial mass during the snow - plough phase , and undergoes substantial deceleration as can be seen from the characteristics plotted in figs . 1 and 2 . relatively cooler stars of intermediate mass drive weaker ionisation fronts and collect considerably lesser mass . the uppermost characteristic in fig . 1 shows that a typical o5 star may drive a shell that has a typical mass of a few times @xmath89 m@xmath1 . an important parameter associated with the stability of an hii shell is its thickness . by a thin shell we imply , @xmath90 . it can in fact , be easily demonstrated that the ratio of the shell - thickness to radius , may rapidly asymptote to a value less than unity . the average surface density of the shell , @xmath91 which following a little manipulation , leads us to @xmath92 for an hii shell , @xmath93 , and @xmath94 , then for a shell of constant surface density , @xmath95 . for a typical o5 star , this ratio is @xmath96 . the green curves in fig . 3 shows time evolution of the hii shell thickness - to - radius ratio for the set of stars used for demonstrative calculations above . these plots suggest , the driven shell is generally thin irrespective of the spectral type of driving star . this is consistent with that reported by the glimpse survey comprising 600 hii shells ( churchwell _ et al . _ 2006 ) . both , theoretical ( e.g vishniac 1983 , 1994 ) , and numerical ( e.g. anathpindika 2009 , 2010 ) work has demonstrated the susceptibility of thin shells to various shearing instabilities , and particularly , to the thin shell instability ( tsi ) . thin , shock confined shells tend to show a greater proclivity towards the tsi that apparently dominates the classical jeans instability . the thermal jeans mass , @xmath97 , is @xmath98 which is roughly an order of magnitude smaller than the typical mass of a clump , @xmath99 , defined by eqn . ( 11 ) above which suggests , perturbations triggered by dynamical instabilities in driven shells and/or shocked slabs facilitate concentration of material in perturbed regions , the local maximas or minimas . this was explicitly demonstrated by wuensch & palou@xmath6 ( 2001 ) . the clump on gaining sufficient mass condenses out , which is the essence of the collect and collapse model . + _ a supernova shell _ for a demonstrative calculation , let us consider a typical supernova that releases energy , @xmath100 ergs , into the ism that as in the previous case , is assumed to have @xmath101 @xmath13 . the timescale , @xmath79 , defined by eqn . ( 14 ) above , for a shell of average density @xmath102 g @xmath13 is @xmath103 myrs , and the velocity of the shell , @xmath104 , @xmath105 which is @xmath106km / s . the corresponding radius of the shell is then , + @xmath107 kpc . the mass of the shell swept up immediately follows from eqn . ( 15 ) , and @xmath108 m@xmath1 . similarly , the mass of a typical fragment calculated using eqn . ( 11 ) is , @xmath109 m@xmath1 . it might be interesting to calculate the efficiency , @xmath110 with which the initial energy , @xmath72 , is converted into mechanical energy whence matter in the ism is swept up . this efficiency is defined as , @xmath111 which suggests that a large proportion of the initially injected energy is lost in heating the ism . the attenuated sound speed , @xmath112 , defined by eqn . ( 19 ) below accounts for the effects of turbulence generated by various hydrodynamic instabilities which were not directly included in the perturbative analysis that led to eqn . ( 9 ) , and @xmath113}.\ ] ] the glimpse survey of supernovae shells produced a catalogue of 95 snrs ( reach _ et al . _ 2006 ) , none of which have shown unambiguous evidence in favour of the collect and collapse model , although a sample of snrs have shown association with oh masers . however , the source of these masers is unclear and could perhaps be due to sporadic star - formation triggered in molecular clouds over - run by snrs . despite this being the case , one may assert that possible condensation in snrs may take a while , as suggested by the magnitude of @xmath79 , which is roughly two orders of magnitude larger than the timescale over which the hii shell produced fragments . the resulting fragments are therefore likely to be at least as massive as those forming in the former case . the number of clumps , @xmath114 , condensing out of a shell is @xmath115 . then @xmath116 or equivalently , in terms of the wavenumber , @xmath52 , @xmath117 this equation defines the number of clumps in an interval @xmath118 , the integration of which , over the wave number domain yields the mass spectrum for putative clumps . figure 4 shows a typical mass spectrum for clumps condensing out of hii shells , obtained via a monte - carlo integration of eqn . the integration was performed for 10,000 realisations of shell fragmentation , each producing @xmath114 number of clumps . thus , we have @xmath2180,000 clumps for the o5 star and fewer , @xmath119 , for an o9 star . the relatively smaller shell driven by the latter star not only produces fewer fragments but also , those that are comparatively less massive than those in the former case . however , we note that extremely large clumps , @xmath120 , may only be sparingly produced as is evident from the mass spectrum in fig . the mass spectrum for stars of either spectral type is similar and a power - law , @xmath121 , appears to fit the derived spectrum reasonably well in either cases ; @xmath122 , and 1.5 respectively . the spectrum derived here is consistent with the one obtained for large clouds by fukui _ ( 1999 ) , and roslowsky ( 2005 ) . similarly , we also obtain the mass spectrum for fragments condensing out of the snr driven shells . the spectrum for this case is plotted in fig . 5 , and a power - law similar to the one in earlier two cases , fits the spectrum reasonably well . in fact , this latter spectrum is similar , @xmath123 , to that for the fragments resulting from the hii shell driven by an o5 star ; spectrum for the snr was derived for @xmath124 realisations of shell fragmentation , and shows considerable shift towards a higher mass in comparison to the distributions shown in fig . this is probably because the velocity of an hii shell in its dense phase decays more rapidly than that of an snr , which pushes up the attenuated sound - speed , @xmath112 , for the latter that in turn raises the mass , @xmath99 , as seen in fig . star forming clumps as massive as @xmath22500 m@xmath1 have been reported in the hii shell n49 and rcw34 ( e.g. zavagno _ et al . _ 2010 , bik _ et al . a similar treatment of the problem was presented by whitworth _ ( 1994a , b ) who arrived at a minimum clump mass of only a few tens of m@xmath1 , which in the light of present findings , appears somewhat conservative . our claim more massive clumps is supported by recent observations described above . it is well known , the evolution of a cloud is governed by the dynamical effects associated with the complex interplay between self - gravity , and other contributing factors such as turbulence and the magnetic field that support a clump against the former . it is therefore crucial to predict clump masses with reasonably good accuracy ; under - estimation of the mass of putative clumps , for a given radius , which in the present case is roughly equal to the wavelength of the unstable mode , will lower the average density and thus raise the clump - lifetime . an increase in the longevity of clumps will also possibly make events such as clump - clump collisions more probable , than they are known to be . inter - clump collisions , according to simulations , could lead to bursts of star - formation with fewer low - mass stars ( e.g. chapman _ et al . _ 1992 , anathpindika 2009 ) , that will tend to shift the stellar initial mass function towards a top - heavy distribution , away from the widely reported lognormal form . we have examined the stability of thin , dense shells driven by powerful ionising radiation originating from massive , early type stars and/or blast waves from a supernova . our work here shows that fragmentation of these shells is likely to produce large clumps , with masses typically @xmath0 m@xmath1 . this calculated range of fragment masses is consistent with that reported via observations of hii shells , and the fragments so formed could spawn a second generation of stars , which may in turn trigger the next generation of stars in the surrounding ism . the simple calculations discussed above appear to suggest that the ratio of thickness - to - radius for a shell evolves only weakly with time , and that calculated here ( see fig . 3 ; @xmath125 ) , is consistent with the values reported in the glimpse survey for hii shells . similar results are obtained for an snr , which , however , appears to be much thinner compared to the shells driven by ob stars . this could perhaps be the reason why snrs often appear filamentary in the irac bands ( e.g. reach _ et al . simulations studying the evolution of thin shells such those by anathpindika ( 2010 ) have demonstrated the dominance of tsi in shocked shells ; the surface of the shell was also shown to develop ripples , similar to breathing modes on fluid surfaces ( fig . 3 in anathpindika ( 2010 ) ) . while wiggles on the shell surface associated with the tsi though coplanar , are generally orthogonal to its surface , and so in case of a shell in the plane of the sky , can not alone account for the reported filamentary nature however , breathing modes coupled with local magnetic field could perchance explain the occurrence of filaments reported by reach _ ( 2006 ) . _ * acknowledgements * _ the author is supported by a post doctoral fellowship at the indian institute of astrophysics , bangalore , india , and wishes to thank an anonymous referee for a critical review of the original manuscript . useful suggestions from prof . harish bhatt are gratefully acknowledged . anathpindika , s. , 2009 , a&a , 504 , 437 anathpindika , s. , 2010 , mnras , 405 , 1431 anderson , l. , zavagno , a. , rod ' on , j. , _ et al . _ , 2010 , a&a , 518 , l99 bik , a. , puga , e. , waters , l. , _ et al . _ , 2010 , apj , 713 , 883 chapman , s. , pongracic , h. , disney , m. , nelson , a. , turner , j & whitworth , a. , 1992 , nature , 359 , 207 churchwell , e. , povich , m. , allen , d. , _ et al . _ , 2006 , apj , 649 , 759 deharveng , l . , zavagno , a . , caplan , j. , 2005 , a&a , 433 , 565 ehlerov@xmath7 , s & palou@xmath6 , j. , 2002 , mnras , 330 , 1026 elmegreen , b & elmegreen , d. , 1978 , apj , 220 , 1051 elmegreen , b. , 1989 , apj , 344 , 306 fukui , y. , onishi , t. , abe , r. , kawamura , a. , tachihara , k. , yamaguchi , r. , mizuno , a & ogawa , h. , 1999 , pasj , 51 , 751 larson , r. , 1985 , mnras , 214 , 379 reach , w. , rho , j. _ et al . _ , 2006 , apj , 131 , 1479 roslowsky , e. , 2005 , pasj , 117 , 1403 spitzer , l. , 1978 , in _ physical processes in interstellar medium _ , wiley - intersceince . pub . shore , s. , _ astrophysical hydrodynamics _ , wiley - vch , weinheim , 2007 ; p. 141 - 43 tielens , a. , g. , 2005 , _ the physics and chemistry of the ism _ , cambridge university press , uk , p. 54 - 6 vishniac , e. , 1983 , apj , 274 , 152 vishniac , e. , 1994 , apj , 428 , 186 whitworth , a. , bhattal , a. , chapman , s. , disney , m & turner , j. , 1994a , mnras , 268 , 291 whitworth , a. , bhattal , a. , chapman , s. , disney , m & turner , j. , 1994b , a&a , 290 , 421 wuench , r & palou@xmath6 , j. , 2001 , a&a , 374 , 746 yamaguchi , r. , norikazu , m. , onishi , t. , mizuno , a & fukui , y. , 2001 , apj , 553 , l185 zavagno , a. , deharveng , l. , comer ' on , f. , brand , j. , massy , f. , caplan , j & russeil , d. , 2006 , a&a , 171 , 184 zavagno , a. , anderson , l. , russeil , d. , _ et al . _ , 2010 , a&a , 518 , l101

early type massive stars drive thin , dense shells whose edges often show evidence of star - formation . the possibility of fragmentation of these shells , leading to the formation of putative star - forming clumps is examined with the aid of semi - analytic arguments . we also derive a mass - spectrum for clumps condensing out of these shells by performing monte - carlo simulations of the problem . by extending on results from our previous work on the stability of thin , dense shells , we argue that clump - mass estimated by other authors in the past , under a set of simplifying assumptions , are several orders of magnitude smaller than those calculated here . using the expression for the fastest growing unstable mode in a shock - confined shell , we show that fragmentation of a typical shell can produce clumps with a typical mass @xmath0 m@xmath1 . it is likely that such clumps could spawn a second generation of massive and/or intermediate - mass stars which could in turn , trigger the next cycle of star - formation . we suggest that the ratio of shell thickness - to - radius evolves only weakly with time . calculations have been performed for stars of seven spectral types , ranging from b1 to o5 . we separately consider the stability of supernova remnants . star formation , hiiregions , supernova remnants , instabilities

You are an expert at summarizing long articles. Proceed to summarize the following text: the current observations , such as sneia ( supernovae type ia ) , cmb ( cosmic microwave background ) and large scale structure , converge on the fact that a spatially homogeneous and gravitationally repulsive energy component , referred as dark energy , accounts for about @xmath1 % of the energy density of universe . some heuristic models that roughly describe the observable consequences of dark energy were proposed in recent years , a number of them stemming from a certain physics @xcite and the others being purely phenomenological @xcite . dark energy can even behave as a phantom and effectively violate the weak energy condition@xcite . in various cosmological models , fundamental quantities are either geometrical ( if they are constructed from a spacetime geometry directly ) or physical ( if they depend upon physical fields ) . physical quantities are certainly model - dependent , while geometrical quantites are more universal . about thirty years ago , the bouncing cosmological model with torsion was suggested in ref.@xcite , but the torsion was imagined as playing role only at high densities in the early universe . goenner et al . made a general survey of the torsion cosmology @xcite , in which the equations for all the pgt ( poincar gauge theory of gravity ) cases were discussed although they only solved in detail a few particular cases . recently some authors have begun to study torsion as a possible reason of the accelerating universe @xcite . nester and collaborators @xcite consider an accounting for the accelerated universe in term of a riemann - cartan geometry : dynamic scalar torsion . they explore the possibility that the dynamic pgt connection , reflecting the nature of dynamic pgt torsion , provides the accelerating force . with the usual assumptions of homogeneity and isotropy in cosmology and specific cases of the suitable parameters and initial conditions , they find that torsion field could play a role of dark energy . one of the motivation was to avoid singularities in the initial investigations of torsion cosmology @xcite . however , it soon was found that non - linear torsion effects were more likely to produce stronger singularities @xcite . the non - linear effects turn out to play a key role for the outstanding present day mystery : the accelerated universe . in the various pgt , the connection dynamics decomposed into six modes with certain spin and parity : @xmath2 , @xmath3 , @xmath4 . some investigations showed that @xmath4 may well be the only acceptable dynamic pgt torsion modes @xcite . the pseudoscalar mode @xmath5 is naturally driven by the intrinsic spin of elementary fermions , therefore it naturally interacts with such sources . consequently , it is generally thought that axial torsion must be small and have small effects at the late time of cosmological evolution . this is a major reason why one does not focus on this mode at the late time . on the other hand , the scalar mode @xmath6 does not interact in any direct obvious fashion with any known type of matter @xcite , therefore one can imagine it as having significant magnitude and yet not being conspicuously noticed . furthermore , there is a critical non - zero value for the affine scalar curvature since @xmath6 mode can interact indirectly through the non - linear equations . the homogeneity and isotropy of cosmology have received strong confirmation from modern observations , which greatly restrict the possible types of non - vanishing fields . under the assumption of homogeneity and isotropy , @xmath6 mode has only a time component and it can be specified as the gradient of a time - dependent function . therefore , the cosmological models with the scalar mode offer a situation where dynamic torsion may lead to observable effect at late time . we emphasize again that one does not focus on the early universe , where one could indeed expect large effects ( though their signature would have to be separated from other large effects ) , and substitutionally asks about traces of torsion effects at the late time of cosmological evolution @xcite . obviously , the fine - tuning problem is one of the most important issues for the torsion cosmology @xcite . and a good model should limit the fine - tuning as much as possible . the dynamical attractor of the cosmological system has been employed to make the later time behaviors of the model insensitive to the initial condition of the field and thus alleviates the fine - tuning problem @xcite . furthermore , nester et al @xcite have shown that the hubble parameter and @xmath7 have an oscillatory form for the scalar torsion cosmology . the traditional geometrical parameters , i.e. , the hubble parameter @xmath8 and the deceleration parameter @xmath9 , are two elegant choices to describe the expansion state of universe but they can not distinguish various accelerating mechanism uniquely , because a quite number of models may just correspond to the same current values of @xmath10 and @xmath11 . however , sahni , saini , starobinsky and alam @xcite have introduced the statefinder pair @xmath12 : @xmath13 , @xmath14 . it is obviously a natural next step beyond @xmath10 and @xmath11 . fortunately , as is shown in the literatures @xcite , the statefinder parameters which are also geometrical diagnostics , are able to differentiate a series of cosmological models successfully . using the discussion of statefinder parameters in the scalar torsion cosmology , we explain easily why the present field equations modify the expansion of the universe only at late time . if the evolving trajectory of statefinder have a decelerating phase ( @xmath15 ) at early time , then we can understand why the expansion of the universe until @xmath16 remains unchanged in the scalar torsion models . in this paper , we apply the statefinder diagnostics to the torsion cosmology . we find that there are some characteristics of statefinder parameters for the torsion cosmology that can be distinguished from the other cosmological models . the statefinder diagnostics show that the universe naturally has an accelerating expansion at low redshifts ( late time ) and a decelerating expansion at high redshifts ( early time ) . therefore , scalar torsion cosmology can avoid some of the problems which occur in other models . especially , the effect of torsion can make the expansion rate oscillate when torsion parameter @xmath17 or @xmath18 . whether the universe has properties which are easier to explain within the scalar torsion context is a remarkable possibility demanding further exploration . the oscillatory feature of hubble parameter had earlier been reported for the braneworld cosmology @xcite and the quasi - steady state cosmology @xcite . we show that statefinder diagnostic has a direct bearing on the critical points of the dynamical system . one of the most interesting characteristic of the trajectories is that there are loop and curves with the shape of tadpole in the case of the torsion parameter @xmath19 . in this case , we fit the scalar torsion model to current type ia supernova data and find it is consistent with the observations . furthermore , we analyze preliminarily the relevance for realistic observation of the found statefinder parameters . pgt @xcite has been regarded as an interesting alternative to general relativity because of its gauge structure and geometric properties . pgt based on a riemann - cartan geometry , allows for dynamic torsion in addition to curvature . the affine connection of the riemann - cartan geometry is @xmath20 where @xmath21 is the levi - civita connection and @xmath22 is the torsion tensor . meantime , the ricci curvature and scalar curvature can be written as @xmath23 where @xmath24 and @xmath25 are the riemannian ricci curvature and scalar curvature , respectively , and @xmath26 is the covariant derivative with the levi - civita connection ( for a detailed discussion see ref . . theoretical analysis of pgt led us to consider tendentiously dynamic `` scalar mode '' . in this case , the restricted expression of the torsion can be written as @xcite @xmath27\kappa},\label{trestrct}\ ] ] where the vector @xmath28 is the trace of the torsion . then , the ricci curvature and scalar curvature can be expressed as @xmath29 the gravitational lagrangian density for the scalar mode is @xmath30 where @xmath31 and @xmath0 is a torsion parameter . consider that the parameter @xmath32 is associated with quadratic scalar curvature term @xmath33 , so that @xmath32 should be positive @xcite . therefore , the field equations of the scalar mode are @xmath34 where @xmath35 is the source energy - momentum tensor and @xmath36 is the contribution of the scalar torsion mode to the effective total energy - momentum tensor : @xmath37 since current observations favor a flat universe , we will work in the spatially flat robertson - walker metric @xmath38 $ ] , where @xmath39 is the scalar factor . this engenders the riemannian ricci curvature and scalar curvature : @xmath40 where @xmath39 is the scalar factor , and @xmath10 is the hubble parameter . the torsion @xmath41 should also be only time dependent , therefore one can let @xmath42 ( @xmath43 is the torsion field ) and the spatial parts vanish . the corresponding equations of motion in the matter - dominated era are as follows @xmath44 where @xmath45 and the energy density of matter component @xmath46 one can scale the variables and the parameters as @xmath47 where @xmath48 is the present value of hubble parameter and @xmath49 is the planck length . under the transform ( [ scale ] ) , eqs . ( [ dth])-([dtr ] ) remain unchanged . after transform , new variables @xmath50 , @xmath10 , @xmath51 and @xmath52 , and new parameters @xmath53 , @xmath0 , @xmath54 and @xmath32 are all dimensionless . obviously , the newtonian limit requires @xmath55 . for the case of scalar torsion mode , the effective energy - momentum tensor can be represented as @xmath56\,,\label{torpre}\end{aligned}\ ] ] and the off - diagonal terms vanish . the effective energy density @xmath57 which is deduced from general relativity . @xmath58 is an effective pressure , and the effective equation of state is @xmath59 which is induced by the dynamic torsion . , the temporal component of the torsion @xmath51 , the affine scalar curvature @xmath52 and the deceleration parameter @xmath11 as functions of time . we have chosen the parameters @xmath60 , @xmath61 and the initial values @xmath62 , @xmath63 , @xmath64.,title="fig:",width=264 ] , the temporal component of the torsion @xmath51 , the affine scalar curvature @xmath52 and the deceleration parameter @xmath11 as functions of time . we have chosen the parameters @xmath60 , @xmath61 and the initial values @xmath62 , @xmath63 , @xmath64.,title="fig:",width=268 ] in the case of @xmath65 , nester et al showed that the scalar mode can contribute an oscillating aspect to the expansion rate of the universe @xcite . this oscillatory nature can be illustrated in fig . [ hqevol ] where we have chosen @xmath60 , @xmath61 , @xmath62 , @xmath63 , @xmath64 and set the current time @xmath66 . according to scaling ( [ scale ] ) , the present value of the hubble parameter is unity . obviously , @xmath10 is damped - oscillating at late time and @xmath67 is negative today , which means the expansion of the universe is currently accelerating . the value of @xmath11 turns from positive to negative when the time is around @xmath68 , which is the epoch the universe began to accelerate . however , the above result is dependent on the choice of initial data and the values of the parameters . then , the scalar torsion cosmology is unsuited to solving the fine - tuning problem in the case of @xmath65 . in the following sections , we ll investigate the statefinder and give the dynamics analysis for all ranges of the parameter @xmath0 . for the spatially flat @xmath69cdm model the statefinder parameters correspond to a fixed point @xmath70 while @xmath71 for the standard cold dark matter model ( scdm ) containing no radiation . since the torsion cosmology have used the dynamic scalar torsion ( a geometry quantity in the riemann - cartan spacetime ) , the torsion accelerating mechanism is bound to exhibit an essential distinction in contrast with various dark energy models . therefore , its statefinder diagnostic is sure to reveal differential feature . let us now study the torsion cosmological model in detail . using eqs . ( [ dth])-([fieldrho ] ) , we have the deceleration parameter @xmath72(432a_{1}h^{2})^{-1}\,,\label{torsionq } \end{aligned}\ ] ] and the statefinder parameters @xmath73(108a_{1}bh^{3})^{-1}\,,\nonumber\\ \label{torsionr } \end{aligned}\ ] ] and @xmath74}{3bh\left[(6\mu + br)(36h^{2}-24h\phi + 4\phi^{2}-3r)-54\mu r\right]}\ , . \label{torsions } \end{aligned}\ ] ] in the following we will discuss the statefinder for four differential ranges of the torsion parameter @xmath0 , respectively . firstly , we consider the time evolution of the statefinder pairs @xmath12 and @xmath75 in the case of @xmath76 . in fig . [ planecase4 ] , we plot evolution trajectories in the @xmath77 and @xmath78 planes , where we have chosen @xmath79 and @xmath80 . we see easily that cosmic expansion alternates between deceleration and acceleration in the evolving trajectories of @xmath77 plane , and the amplitude becomes larger and larger as increase of time . the trajectories in the @xmath78 plane is quite complicated , so we mark its sequence by the ordinal number . every odd number curve evolves from finite to infinite , but even number curve evolves from infinite to finite . these are quasi - periodic behaviors which corresponds to the numerical solution of ref . noticeably , the trajectories will never pass @xmath69cdm point @xmath70 . secondly , we discuss the time evolution of the statefinder pairs @xmath12 and @xmath75 for the case of @xmath81 . we plot evolving trajectories in fig . [ planecase3 ] , where we have chosen @xmath82 and @xmath83 . we see clearly that the cosmic acceleration is guaranteed by the dynamic scalar torsion in the evolving trajectories of @xmath77 plane , and the curves will converge into @xmath69cdm point . the evolving trajectories go through a climbing - up stage first , then get into a rolling - down stage in the @xmath78 plane . lastly , trajectories tend to @xmath69cdm point @xmath70 . furthermore , the only one forms a loop that starts from @xmath70 then evolves back to @xmath70 , and others show in the shape of tadpole . and @xmath78 planes for the case of torsion parameter @xmath84 , where we choose the parameters @xmath85 and @xmath86 . the arrows show the direction of the time evolution.,title="fig:",width=264 ] and @xmath78 planes for the case of torsion parameter @xmath84 , where we choose the parameters @xmath85 and @xmath86 . the arrows show the direction of the time evolution.,title="fig:",width=264 ] and @xmath78 planes for the case of torsion parameter @xmath87 , where we choose the parameters @xmath88 and @xmath89 . the arrows show the direction of the time evolution.,title="fig:",width=264 ] and @xmath78 planes for the case of torsion parameter @xmath87 , where we choose the parameters @xmath88 and @xmath89 . the arrows show the direction of the time evolution.,title="fig:",width=264 ] and @xmath90 planes for the case of @xmath91 , where we choose @xmath92 and @xmath93 . the arrows show the direction of the time evolution.,title="fig:",width=264 ] and @xmath90 planes for the case of @xmath91 , where we choose @xmath92 and @xmath93 . the arrows show the direction of the time evolution.,title="fig:",width=257 ] and @xmath78 planes for the case of torsion parameter @xmath94 , where we choose the parameters @xmath95 and @xmath86 . the arrows show the direction of the time evolution.,title="fig:",width=264 ] and @xmath78 planes for the case of torsion parameter @xmath94 , where we choose the parameters @xmath95 and @xmath86 . the arrows show the direction of the time evolution.,title="fig:",width=257 ] thirdly , we discuss the time evolution of the trajectories for the case of @xmath96 . we plot evolving trajectories in fig . [ planecase2 ] , where we have chosen @xmath97 and @xmath98 . obviously , the cosmic acceleration can happen since deceleration parameter is negative . @xmath99 , @xmath100 and @xmath101 become larger and larger first , then less and less as the cosmic time increase . finally , we consider the time evolution of the statefinder pairs @xmath12 and @xmath75 in the case of @xmath18 . in fig . [ planecase1 ] , we plot evolving trajectories in the @xmath77 and @xmath78 planes , where we have chosen @xmath102 and @xmath80 . we find easily that the evolving trajectories analogous to the case of @xmath103 except trajectories pass the @xmath69cdm point . to sum up , it is very interesting to see that the scalar torsion naturally provide the accelerating force in the universe for any torsion parameter @xmath0 . however , it is dependent on torsion parameters that there is a decelerating ( @xmath15 ) expansion before an accelerating ( @xmath104 ) expansion . the statefinder diagnostics show that the universe naturally has an accelerating expansion at low redshifts ( late time ) and a decelerating expansion at high redshifts ( early time ) for the cases of @xmath105 and @xmath18 . obviously , scalar torsion cosmology can avoid some of the problems which occur in other models . if we refuse the possibility of non - positivity of the kinetic energy , we will employ normal assumption , i. e. , @xmath17 . in this case , the effect of torsion can make the expansion rate oscillate . with suitable adjustments of the torsion parameters , it is possible to change the quasi - period of the expansion rate as well as its amplitudes . it is worth noting that the true values of the statefinder parameters of the universe should be determined in model - independent way . in principle , @xmath106 can be extracted from some future astronomical observations , especially the snap - type experiment . why there are new features for the statefinder diagnostic of torsion cosmology ? why the torsion parameter @xmath0 is divided into differential ranges by the statefinder answer is very simple . in fact , the statefinder diagnostic has a direct bearing on the attractor of cosmological dynamics . therefore , we will discuss the dynamic analysis in next section . eqs . ( [ dth])-([dtr ] ) is an autonomous system , so we can use the qualitative method of ordinary differential equations . critical points are always exact constant solutions in the context of autonomous dynamical systems . these points are often the extreme points of the orbits and therefore describe the asymptotic behavior . if the solutions interpolate between critical points they can be divided into a heteroclinic orbit and a homoclinic orbit ( a closed loop ) . the heteroclinic orbit connects two different critical points and homoclinic orbit is an orbit connecting a critical point to itself . in the dynamical analysis of cosmology , the heteroclinic orbit is more interesting @xcite . if the numerical calculation is associated with the critical points , then we will find all kinds of heteroclinic orbits . according to equations ( [ dth])-([dtr ] ) , we can obtain the critical points and study the stability of these points . substituting linear perturbations @xmath107 , @xmath108 and @xmath109 near the critical points into three independent equations , to the first orders in the perturbations , gives the evolution of the linear perturbations , from which we could yield three eigenvalues . stability requires the real part of all eigenvalues to be negative . there are five critical points @xmath110 of the system as follows @xmath111 where @xmath112 , @xmath113 and @xmath114 . the corresponding eigenvalues of the critical points ( i)-(v ) are @xmath115 using eq . ( [ criticalpoints ] ) , we find that there is only a critical point @xmath116 in the case of @xmath76 . from eq . ( [ eigenvalues ] ) , the corresponding eigenvalue is @xmath117 , so @xmath118 is an asymptotically stable focus . if we consider the linearized equations , then eqs . ( [ dth])-([dtr ] ) are reduced to @xmath119 + the linearized system ( [ dthphirlinear ] ) has an exact periodic solution @xmath120 where @xmath121 , @xmath122 , @xmath123 and @xmath124 , @xmath125 and @xmath126 are initial values . obviously , @xmath127 is a critical line of center for the linearized eqs . ( [ hrphips ] ) . in other words , there are only exact periodic solutions for the linearized system , but there are quasi - periodic solutions near the focus for the coupled nonlinear equations . this property of quasi - periodic also appears in the statefinder diagnostic with the case of @xmath76 . @l*15@l critical points & property & @xmath128 & stability + ( ` i ` ) & saddle & @xmath129 & unstable + ( ` ii`)&positive attractor&-1&stable + ( ` iii ` ) & negative attractor&-1&unstable + ( ` iv ` ) & saddle&-1&unstable + ( ` v ` ) & saddle&-1&unstable + @l*15@l critical points & property & @xmath128 & stability + ( ` i ` ) & focus & @xmath130 & stable + ( ` ii`)&saddle&-1&unstable + ( ` iii ` ) & saddle&-1&unstable + ( ` iv ` ) & saddle&-1&unstable + ( ` v ` ) & saddle&-1&unstable + in the case of @xmath131 , the critical point ( ii ) is a late time de sitter attractor and ( iii ) is a negative attractor . the properties of the critical points are shown in table [ cripointsa1l0 ] . the de sitter attractor indicates that torsion cosmology is an elegant scheme and the scalar torsion mode is an interesting geometric quantity for physics . in the dynamical analysis of cosmology , the heteroclinic orbit is more interesting . using numerical calculation , we plot the heteroclinic orbit connects the critical point case ( iii ) to case ( ii ) in fig . [ heteroclinicorbit ] . this heteroclinic orbit is just corresponding to the loop in fig . [ planecase3 ] , which is from @xmath69cdm point to @xmath69cdm point . furthermore , the trajectories with the shape of tadpole correspond to saddles . with @xmath132 . the heteroclinic orbit connects the critical points case ( iii ) to case ( ii ) . we take @xmath133.,width=321 ] with @xmath94 . we take @xmath134 and the initial value @xmath135 . @xmath136 is an asymptotically stable focus point.,width=321 ] in the case of @xmath96 , there is only an unstable saddle @xmath116 where the effective equation of state @xmath128 tends to @xmath129 . therefore , the trajectories in fig . [ planecase2 ] show that @xmath99,@xmath100 and @xmath101 become larger and larger , then less and less as time increases . in the case of @xmath18 , the properties of the critical points are shown in table [ cripointsa1l-1 ] . the trajectories correspond to the stable focus ( see fig . 6 ) and unstable saddles with @xmath137 . therefore , the trajectories pass through the @xmath69cdm point . in ref.@xcite , the authors have compared the numerical values of the torsion model with the observational data , in which they fixed the initial values @xmath48 , @xmath138 and @xmath139 , and torsion parameters @xmath0 and @xmath32 . in this section , we fixed the initial value , then fit the torsion parameters to current type ia supernovae data . the scalar torsion cosmology predict a specific form of the hubble parameter @xmath140 as a function of redshifts @xmath141 in terms of two parameters @xmath0 and @xmath32 when we chose initial values . using the relation between @xmath142 and the comoving distance @xmath143 ( where @xmath141 is the redshift of light emission ) @xmath144 and the light ray geodesic equation in a flat universe @xmath145 where @xmath146 is the scale factor . in general , the approach towards determining the expansion history @xmath140 is to assume an arbitrary ansatz for @xmath140 which is not necessarily physically motivated but is specially designed to give a good fit to the data for @xmath147 . given a particular cosmological model for @xmath148 where @xmath149 are model parameters , the maximum likelihood technique can be used to determine the best fit values of parameters as well as the goodness of the fit of the model to the data . the technique can be summarized as follows : the observational data consist of @xmath150 apparent magnitudes @xmath151 and redshifts @xmath152 with their corresponding errors @xmath153 and @xmath154 . these errors are assumed to be gaussian and uncorrelated . each apparent magnitude @xmath155 is related to the corresponding luminosity distance @xmath156 by @xmath157 + 25,\ ] ] where @xmath158 is the absolute magnitude . for the distant sneia , one can directly observe their apparent magnitude @xmath159 and redshift @xmath141 , because the absolute magnitude @xmath158 of them is assumed to be constant , i.e. , the supernovae are standard candles . obviously , the luminosity distance @xmath147 is the ` meeting point ' between the observed apparent magnitude @xmath160 and the theoretical prediction @xmath140 . usually , one define distance modulus @xmath161 and express it in terms of the dimensionless ` hubble - constant free ' luminosity distance @xmath162 defined by@xmath163 as @xmath164 where the zero offset @xmath165 depends on @xmath48 ( or @xmath166 ) as @xmath167 the theoretically predicted value @xmath168 in the context of a given model @xmath169 can be described by @xcite @xmath170 therefore , the best fit values for the torsion parameters ( @xmath171 ) of the model are found by minimizing the quantity @xmath172 ^ 2}{\sigma_i^2}.\ ] ] since the nuisance parameter @xmath165 is model - independent , its value from a specific good fit can be used as consistency test of the data @xcite and one can choose _ a priori _ value of it ( equivalently , the value of dimensionless hubble parameter @xmath166 ) or marginalize over it thus obtaining @xmath173 where @xmath174 ^ 2}{\sigma_i^2},\ ] ] @xmath175}{\sigma_i^2},\ ] ] and @xmath176 in the latter approach , instead of minimizing @xmath177 , one can minimize @xmath178 which is independent of @xmath165 . the eqs . ( [ gradr]-[tort ] ) can be solved explicitly by a series in the form @xmath179,\ ] ] where @xmath180 and @xmath181 using the general relation between hubble parameter @xmath182 and the redshift @xmath141 , @xmath141 can be written as a function of @xmath50 @xmath183}-1,\ ] ] however , the convergence radius of the series ( [ e1 ] ) is @xmath184 , so we can use the expansion directly in the case of the redshift being @xmath185 . by the numerical calculation , we find that @xmath186 corresponds to @xmath187 for the valuses of parameters @xmath139 and @xmath138 in the fig . [ 192clcontours3 ] . for the case of @xmath188 , we should use a direct analytic continuation . weierstrass @xcite had built the whole theory of analytic functions from the concept of power series . given a point @xmath189 ( @xmath190 ) , the function @xmath182 has a taylor expansion @xmath191.\ ] ] where the coefficients @xmath192 is still expressed as eq . ( [ e2 ] ) and @xmath193 can be decided by eq . ( [ e1 ] ) . the new series defines an analytic function @xmath194 which is said to be obtained from @xmath195 by direct analytic continuation . this process can be repeated any number of times . in the general case we have to consider a succession of power series @xmath195 , @xmath194, ... ,@xmath196 , each of which is a direct analytic continuation of the preceding one . by using this method we have the evolution of hubble parameter @xmath182 . furthermore , we have the function @xmath140 from eq . ( [ e3 ] ) . in fact , we need only to consider the case of @xmath197 for the essence supernovae data . we now apply the above described maximum likelihood method using the essence supernovae data which is one of the reliable published data set consisting of 192 sneia ( @xmath198 ) . beside the 162 data points given in table 9 of ref . @xcite , which contains 60 essence sneia , 57 snls sneia and 45 nearby sneia , we add 30 sneia detected at @xmath199 by the hubble space telescope @xcite as in ref.@xcite . in table [ differentvalues ] , we show the best fit of torsion parameters at different initial values of @xmath139 and @xmath138 . @l*15@l @xmath139 & @xmath138 & @xmath0 & @xmath32 + 0.25&0.35 & -0.10&1.44 + 0.20&0.34 & -0.08&1.80 + 0.15&0.34 & -0.06&2.40 + 0.10&0.33 & -0.04&3.60 + in fig . [ 192clcontours3 ] , contours with 68.3% , 95.4% and 99.7% confidence level are plotted , in which we take a marginalization over the model - independent parameter @xmath165 . the best fit as showed in the figure corresponds to @xmath200 and @xmath201 , and the minimum value of @xmath202 . for @xmath203 , one can get @xmath204 and the best fit @xmath205 . therefore , it s easy to know that @xmath206 is consistent at the @xmath207 level with the best fits of scalar torsion cosmology . in fig . [ zu ] , we show a comparison of the essence supernovae data along with the theoretically predicted curves in the context of scalar torsion and @xmath206 . we can see that the scalar torsion model(@xmath208 , @xmath209 , @xmath210 , @xmath211 ) gives a close curve behavior to the one from @xmath206 ( @xmath212 ) . clearly , the allowed ranges of the parameters @xmath0 and @xmath32 favor the case of @xmath213 if we chose @xmath214 and @xmath215 . with the essence supernovae data via the relation between the redshift @xmath141 and the distance modulus @xmath54 . the scalar torsion model(@xmath208 , @xmath209 , @xmath210 , @xmath211 ) gives a close curve behavior to the one from @xmath206 ( @xmath212).,title="fig:",width=264 ] with the essence supernovae data via the relation between the redshift @xmath141 and the distance modulus @xmath54 . the scalar torsion model(@xmath208 , @xmath209 , @xmath210 , @xmath211 ) gives a close curve behavior to the one from @xmath206 ( @xmath212).,title="fig:",width=272 ] and @xmath32 using the essence sneia dataset . here we have assumed @xmath216 , @xmath214 and @xmath215.,width=321 ] we have studied the statefinder diagnostic to the torsion cosmology , in which an accounting for the accelerated universe is considered in term of a riemann - cartan geometry : dynamic scalar torsion . we have shown that statefinder diagnostic has a direct bearing on the critical points . the statefinder diagnostic divides the torsion parameter @xmath0 into four ranges , which is in keeping with the requirement of dynamical analysis . therefore , the statefinder diagnostic can be used to an exceedingly general category of models including several for which the notion of equation of state is not directly applicable . the statefinder diagnostic has the advantage over the dynamical analysis at the simplicity , but the latter can provide more information . the most interesting characteristic of the trajectories is that there is a loop in the case of @xmath217 . this behavior corresponds to the heteroclinic orbit connecting the negative attractor and de sitter attractor . the trajectories with the shape of tadpole show that they pass through the @xmath69cdm fixed point along the time evolution , then the statefinder pairs are going along with a loop and they will pass through the @xmath69cdm fixed point again in the future . it is worth noting that there exists closed loop in the ref . @xcite , but there is no closed loop which contains the @xmath69cdm fixed point . these behaviors indicate that torsion cosmology is an elegant scheme and the scalar torsion mode is an interesting geometric quantity for physics . furthermore , the quasi - periodic feature of trajectories in the cases of @xmath76 or @xmath18 shows that the numerical solutions in ref . @xcite are not periodic , but are quasi - periodic near the focus for the coupled nonlinear equations . we fixed only the initial values , then fitted the torsion parameters to current sneia dataset . we find that the scalar torsion naturally explain the accelerating expansion of the universe for any torsion parameter @xmath0 . however , it is dependent @xmath0 and @xmath32 that there is a decelerating expansion before an accelerating expansion . the statefinder diagnostics show that the universe naturally have an accelerating expansion at late time and a decelerating expansion at early time for the case of @xmath218 and @xmath18 . if we refuse the possibility of non - positivity of the kinetic energy , we have to employ normal assumption ( @xmath17 ) . under this assumption , the effect of torsion can make the expansion rate oscillate . furthermore , with suitable adjustments of the torsion parameters and initial value , it is possible to change the quasi - period of expansion rate as well as its amplitudes . in order to have a quantitative understanding of the scalar torsion cosmology , the matter density @xmath219 , the effective mass density @xmath220 , and the quantity @xmath221 are important . this scenario bears a strong resemblance to the braneworld cosmology in a very different context by sahni , shtanov and viznyuk @xcite . the @xmath222 parameters in the torsion cosmology and in the @xmath69cdm cosmology can nevertheless be quite different . therefore , at high redshift , the torsion cosmology asymptotically expands like a matter - dominated universe with the value of @xmath222 inferred from the observations of the local matter density . at low redshift , the torsion model behaves like @xmath69cdm but with a renormalized value of @xmath223 . the difference between @xmath222 and @xmath223 is dependent on the present value of statefinder parameters @xmath224 . a more detailed estimate , however , lies beyond the scope of the present paper , and we will study it in a future work . finally , @xmath225 and @xmath226 should be extracted from some future astronomical observations in principle , especially the snap - 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we apply the statefinder diagnostic to the torsion cosmology , in which an accounting for the accelerated universe is considered in term of a riemann - cartan geometry : dynamic scalar torsion . we find that there are some typical characteristic of the evolution of statefinder parameters for the torsion cosmology that can be distinguished from the other cosmological models . furthermore , we also show that statefinder diagnostic has a direct bearing on the critical points . the statefinder diagnostic divides the torsion parameter @xmath0 into differential ranges , which is in keeping with the requirement of dynamical analysis . in addition , we fit the scalar torsion model to essence supernovae data and give the best fit values of the model parameters .

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Proceed to summarize the following text: the vast majority ( perhaps all ) of stars are formed in a clustered fashion . however , only a very small percentage of older stars are found in bound clusters . these two observations highlight the importance of clusters in the star - formation process and the significance of cluster disruption . the process of cluster disruption begins soon after , or concurrent with , cluster formation . @xcite found that @xmath3 of stars formed in embedded clusters end up in bound clusters after @xmath4 yr . @xcite and @xcite have shown that at least 20% , but perhaps all , star formation in the merging antennae galaxies is taking place in clusters , the majority of which are likely to become unbound . the case is similar in m51 , with @xmath5 of all young ( @xmath6 myr ) clusters likely to be destroyed within the first 10s of myr of their lives @xcite . on longer timescales , @xcite and @xcite noted a clear lack of older ( @xmath7 few gyr ) open clusters in the solar neighbourhood and @xcite found a strong absence of older clusters in m51 , m33 , smc , and the solar neighbourhood . the lack of old open clusters in the solar neighbourhood is even more striking when compared with the lmc , which contains a significant number of ` blue globular clusters ' with ages well in excess of a gyr ( e.g. @xcite ) . this difference can be understood either as a difference in the formation history of clusters or as a difference in the disruption timescales . this later scenario was suggested by @xcite , who directly compared the age distribution of galactic open clusters and the smc cluster population . he noted that there are @xmath8 times more clusters with an age of 1 gyr in the smc as compared to the solar neighbourhood ( when normalising both populations to an age of @xmath9 yr ) and concluded that disruption mechanisms must be less efficient in the smc . much theoretical work has gone into the later scenario , with both analytic and numerical models of cluster evolution predicting a strong influence of the galactic tidal field on the dissolution of star clusters ( for a recent review see @xcite ) . only recently has there been a large push to understand cluster disruption from an observational standpoint in various external potentials , making explicit comparison with models @xcite . we direct the reader to the review by larsen in these proceedings for a historical look at the observations and theory of cluster disruption . while cluster disruption is a gradual process with several different disruptive agents at work simultaneously , one can distinguish three general phases of cluster mass loss and disruption . as we will see , a large fraction of clusters gets destroyed during the _ primary _ phase . the main phases and corresponding typical timescales of cluster disruption are : _ i ) infant mortality _ ( @xmath0 yr ) , _ ii ) stellar evolution _ ( @xmath1 yr ) and _ iii ) tidal relaxation _ ( @xmath2 yr ) . during all three phases there are additional tidal external perturbations from e.g. giant molecular clouds ( gmcs ) , the galactic disc and spiral arms that heat the cluster and speed up the process of disruption . however , these external perturbations operate on longer timescales for cluster populations and so are most important in phase iii . in fig . [ fig0 ] we schematically illustrate the three phases of disruption and the involved time - scales . note that the number of disruptive agents decreases in time . in this review we will focus on the physics and observations of phase i as well as on recent population studies aimed at understanding phases ii and iii on a statistical basis . for a recent review on the physics of phases ii and iii , we refer the reader to @xcite . before proceeding , it is worthwhile to consider our definition of a cluster . @xcite defines a cluster to be a _ gravitationally bound _ stellar association which will survive for 1020 crossing times . this definition implies that the stars provide enough gravitational potential to bind the cluster and ignores the role of gas in the early evolution of clusters . in this review , we will define a cluster as a collection of gas and stars which was _ initially gravitationally bound_. the reason for this definition will become evident in section [ infantmortality ] recent studies on the populations of young star clusters in m51 @xcite and the antennae galaxies @xcite have shown a large excess of star clusters with ages less than @xmath1010 myr with respect to what would be expected assuming a constant cluster formation rate . the fact that open clusters in the solar neighbourhood display a similar trend @xcite has led to the conclusion that this is a physical effect and not simply that we are observing these galaxies at a special time in their star - formation history . if one adopts this view , then we are forced to conclude that the majority ( between 60 - 90% ) of star clusters become unbound when the remaining gas ( i.e. gas that is left - over from the star formation process ) is expelled . these clusters will survive less than a few crossing times . suppose that a star cluster is formed out of a sphere of gas with an efficiency @xmath11 , where @xmath12 . further suppose that the gas and stars are initially in virial equilibrium . if we define the virial parameter as @xmath13 , with @xmath14 the kinetic energy and @xmath15 the potential energy , virial equilibrium implies @xmath16 . finally , suppose that the remaining gas is removed on a timescale faster than the crossing time of stars in the cluster . in such a scenario the cluster is left in a super - virial state after the gas removal , with @xmath17 , and the star cluster will expand since the binding energy is too low for the stellar velocities . the expanding cluster will reach virial equilibrium after a few crossing times , but only after a ( possibly large ) fraction of the stars have escaped . this process has been shown to remove a significant amount of the stellar mass of a cluster , and if @xmath18 the entire cluster will become unbound on a timescale of 10s of myr @xcite . rapid gas removal of the type discussed above leaves distinct observables . in figure [ fig1 ] we show the surface brightness profiles of three young clusters ( left panels ) as well as two results of @xmath19-body simulations ( right panels ) of clusters including the effects of rapid gas removal . all three young clusters show an excess of light at large radii with respect to the best fitting eff @xcite or @xcite profiles . this is in good agreement with the predictions of the simulations , in which an unbound halo of stars is removed ( although still appearing to be associated with the cluster for 10s of myr ) due to the rapid change of the gravitational potential @xcite . such excess light at large radii has also been found in young clusters in the antennae galaxies @xcite . @xcite show that for values of @xmath11 of 0.1 and 0.6 , clusters will lose 75% and 10% of the stellar mass respectively within the first @xmath20 myr of their lives . thus we see that this is an extremely efficient way to rapidly disperse stars from young clusters into the field . this mechanism provides a natural explanation for the observed diffuse uv light in the field of starburst galaxies @xcite and supports the scenario of these authors that this light is due to rapidly dispersing young clusters . whether or not a cluster survives this phase , and hence more than 1020 crossing times , is largely dependent on the star - formation efficiency of the gmc core in which the cluster formed . thus , two clusters with exactly the same parameters ( radius , mass , metallicity , external potential field , etc ) may experience two radically different evolutionary paths if their star - formation efficiencies are different . @xcite have used the internal dynamical properties of young clusters in order to estimate their @xmath11-values . no clear trend of @xmath11 on cluster ( stellar ) mass or radius was found . -body simulations which include the rapid removal of gas which was left over from a non-100% star - formation efficiency ( right ) . the solid ( red ) and dashed ( blue ) lines are the best fitting eff @xcite and king @xcite profiles respectively . note the excess of light at large radii with respect to the best fitting eff profile in both the observations and models . this excess light is due to an unbound expanding halo of stars caused by the rapid ejection of the remaining gas after the cluster forms . _ hence , excess light at large radii strongly implies that these clusters are not in dynamical equilibrium . _ for details of the modeling and observations see @xcite and @xcite.,height=340 ] even if a cluster survives the gas removal phase , this phase can significantly effect the observed properties of the cluster . hence , deducing the initial properties of a cluster from its current state is not trivial . @xcite have noted the strong effect of residual gas removal on inferring the initial stellar mass of a cluster , while @xcite have refined the mass loss estimates and shown that measurements of the current radii of young clusters may not reflect the initial nor the final value . additionally , @xcite show that this effect can mimic stellar imf variations in young clusters . the clusters that have survived the gas removal phase are subject to disruption phases ii and iii ( [ subsec : phases ] ) as well as tidal effects . disruption due to these effects can be studied on individual clusters , of which the recent observations of the dissolving globular clusters palomar 5 @xcite are probably the most spectacular example . however , much can be learned by approaching this problem from a cluster population point of view . suppose that clusters are formed continuously with a constant cluster formation rate ( a constraint which we can relax later ) . also , we will assume that we know the cluster initial mass function ( here taken to be a power law of the form @xmath21 with @xmath22 ( e.g. @xcite ) and that clusters can be detected down to a known magnitude limit . finally , we will assume that the disruption time of a cluster depends on the cluster mass , such that more massive clusters survive longer ( on average ) than lower mass clusters . for this final assumption we will adopt a function of the form : @xmath23 where @xmath24 is the disruption time of a @xmath25 cluster and @xmath26 @xcite . the beauty of this formulation is that it only has two variables , @xmath24 and @xmath27 , and as we will see , provides extremely good fits to observations . the formulation provided above , when combined with the given assumptions , allows for the parameters @xmath24 and @xmath27 to be found from age and mass distributions of clusters . the first survey using this technique was carried out by @xcite on cluster populations in m51 , m33 , the smc , and the solar neighbourhood . they made a _ sudden disruption _ assumption , meaning that the cluster is in the sample with its initial mass until , when it is disrupted . the somewhat surprising result from this study was that , while @xmath27 had more or less the same value in all environments studied ( @xmath28 ) , @xmath24 varied by over two orders of magnitude , with values of @xmath29 myr in the central regions of m51 to @xmath30 gyr in the smc . the simple _ sudden disruption _ assumption was improved in a more recent model by @xcite , where a gradual loss of cluster mass was implemented . they assumed that the cluster mass decreases exponentially with a time - scale that decreases as the cluster mass decreases . this is done by saying that the mass loss per unit time ( ) relates to as : @xmath31 with @xmath32 from eq . [ eq : ihavebeentypingthisequationtoomuchinmylife ] . this very simple analytical description for cluster mass loss shows remarkably good agreement when compared to the mass loss following from the detailed @xmath19-body simulations of @xcite . in fig . [ dm_lamers ] we show a direct comparison of @xmath33 from the @xmath19-body simulations of clusters with different density profiles and on different orbits ( left ) and the above mentioned analytical model of @xcite ( right ) . in both graphs the time is normalised to and only the mass loss due to stars escaping the cluster is shown , i.e. mass loss due to stellar evolution ( sev ) is not shown . in addition , there is a coupling between the two types of mass loss : if stars loose mass , the cluster will expand and more stars are pushed over the tidal boundary . the simulations of @xcite considered sev , therefore , their results shown in fig . [ dm_lamers ] do include tidal induced by sev . for this reason we can simply add the mass loss due to sev , taken from an ssp model , to eq . [ eq : dm ] . -body simulations of clusters with different number of stars , different concentration and on different orbits ( left ) . the mass loss due to stellar evolution is not shown . in the right panel the analytical model of @xcite is shown.,title="fig:",width=255]-body simulations of clusters with different number of stars , different concentration and on different orbits ( left ) . the mass loss due to stellar evolution is not shown . in the right panel the analytical model of @xcite is shown.,title="fig:",width=255 ] in a series of follow - up works , it has been shown that the similarity of the value of @xmath27 in various environments strongly implies a uniformity in the cluster disruption process , while the varying values of @xmath24 is due to the different tidal field strengths _ ( and gas contents ) _ of the galaxies studied . galaxies with strong tidal fields , as , for example , derived from their rotation curves , having shorter disruption times @xcite . comparison with results of realistic @xmath19-body models performed by @xcite have placed this empirical disruption law on a solid physical footing @xcite . @xcite have also derived a formula for the predicted mass and age distributions of cluster samples that includes both stellar evolution and disruption for any star formation history . @xcite inserted the lamers disruption law into a cluster population synthesis model . this method has two distinct advantages over the earlier formulations . the first is that it removes the requirement of a constant cluster formation rate , and second , it uses the age and mass distributions together to find @xmath27 and @xmath24 . the case of m51 is shown in fig . one first begins by constructing an observed number density grid in age - mass space ( upper - left panel where the shading corresponds to the logarithm of the number of clusters found within that cell ) . then one generates a large number of models with different values of @xmath24 , @xmath27 , ( time dependent ) cluster formation rates , etc . and compares these models with the observed grid . the resultant @xmath34 diagram is shown in the bottom panel of fig . the best fit model is shown in the top right panel of fig . [ fig2 ] . this cluster population synthesis ( cps ) technique , also used in a similar way by @xcite to derive the properties of the cluster population in the galaxy ngc 3627 , holds great promise in disentangling the myriad of effects present in cluster populations . in principle , the dependences of cluster size , galactocentric radius , star - formation efficiency dependent infant mortality rates , or alternative cluster disruption models can be taken into account by this technique . for this technique to be fully exploited one needs large samples of cluster populations with known ages and masses . datasets suitable for these kind of studies are beginning to be collected and released . several face - on spiral galaxies have been imaged in multiple filters with the high resolution / wide field _ hst / acs _ camera ( e.g @xcite for m101 and for m51 ) . and @xmath24 . * bottom : * the @xmath34 diagram in @xmath27-@xmath24 space . the best fitting model is marked with an x , while the accepted fits ( @xmath35 ) are shaded . reproduced from @xcite.,title="fig:",width=226 ] and @xmath24 . * bottom : * the @xmath34 diagram in @xmath27-@xmath24 space . the best fitting model is marked with an x , while the accepted fits ( @xmath35 ) are shaded . reproduced from @xcite.,title="fig:",width=226 ] and @xmath24 . * bottom : * the @xmath34 diagram in @xmath27-@xmath24 space . the best fitting model is marked with an x , while the accepted fits ( @xmath35 ) are shaded . reproduced from @xcite.,title="fig:",width=226 ] it is worth noting possible objections to the lamers disruption law . the first comes from @xcite who find that in the antennae galaxies the number of clusters decreases in time ( @xmath36 ) as @xmath37 , independent of cluster mass . this may be explained if the disruption timescale @xmath24 due to tidal field effects ( e.g. phase ii & iii ) is greater than or similar to the maximum age in the sample . the cluster disruption due to tidal effects would not yet be present in the @xcite sample , instead the decrease in cluster numbers would be the result of infant mortality and the fading of clusters . studies of infant mortality in m51 also suggest that the effect is mass independent @xcite . in fact , if infant mortality was not ( mostly ) independent of cluster mass we would expect the embedded cluster mass function to be significantly different from the optically selected cluster mass function . in fig . [ fig : mf ] we show the dependence of the mass function slope of a multiple age cluster population on the ratio of the @xmath38 and the maximum age of the cluster in the sample ( ) . clusters were created continuously over 1 gyr with an initial mass function of a power - law with index @xmath39 . the lamers disruption law was applied in the same way as in @xcite . the important thing to take away from this figure is that as approaches and exceeds the mass function is less affected by disruption and so it retains its initial form , i.e. the right panel in fig . [ fig : mf ] probably applies to the @xcite sample . ) .,height=151 ] a second observation seemingly contradicting the lamers disruption law is that of @xcite who find an intermediate age ( @xmath40 gyr ) globular cluster in m33 . in m33 , @xcite find a @xmath24 value of @xmath41 myr , implying a disruption time of @xmath42 gyr for a @xmath43 cluster . however , as the authors note , the value of @xmath24 derived by @xcite was presumably of the thin disk of the galaxy , and if the intermediate - age cluster is part of the thick - disk or halo of the galaxy then the expected value of @xmath24 would be significantly larger than that quoted . additionally , it should also be noted that the mass derived by @xcite is the _ present _ mass of the cluster . the cluster presumably started with a much higher mass and disruptive effects have brought this cluster into its current state . if the present mass of the cluster is @xmath44 , then its initial mass ( after infant weight loss ) would have been @xmath45 ( assuming an age of 5 gyr ) , using the value of @xmath24 for m33 given by @xcite . -body models of @xcite ) disruption time of a @xmath46 cluster , @xmath24 , as a function of the mean density @xmath47 in @xmath48pc@xmath49 of the host galaxy . reproduced from @xcite.,width=340 ] as more and more galaxies ( and environments ) have their characteristic disruption timescales measured , it is useful to compare the results to @xmath19-body models in order to check for consistency between the two . this was done in @xcite who compared the @xmath24 values derived for the smc , m33 , m51 and the solar neighbourhood to the @xmath19-body models of @xcite and @xcite which sample a large range in the ambient densities of the host galaxies . their results , shown in fig . [ fig3 ] , are intriguing . while the predicted and observed disruption time of the smc are in excellent agreement , the disruption times of the galaxy , m33 and m51 are observed to be much shorter than predicted by @xmath19-body models . this result is particularly surprising given the fact that the mass loss predictions of a single cluster are in excellent agreement between the lamers empirical description and that given by @xmath19-body models ( @xcite and fig . [ dm_lamers ] ) . = @xmath46 with @xmath50 . * top : * the motion of the gmc is along the x - axis and the line of sight is perpendicular . * bottom : * the motion of the gmc is into the page and the line of sight is along the gmc trajectory . the arrows in the left - hand lower corner of the left - hand panels are parallel to the direction of motion of the gmc . the gmc is shown with grey shades based on the surface density of a gmc with a@xmath51 = 5.8a@xmath52 . the time with respect to the moment of encounter is indicated in each panel of the top row . see @xcite for a description of the methods and parameters used.,width=529 ] thus , we are left to ask , what physical effects are not included in the @xmath19-body models that may be responsible for disrupting clusters ? the @xmath19-body models used in the comparison were carried out in a smooth logarithmic potential which does not realistically represent the thin disk components of disk galaxies . @xcite have attempted to add encounters with giant molecular clouds ( gmcs ) and spiral arm passages to the @xmath19-body models . in fig . [ fig4 ] we show an example of a cluster - gmc encounter ( from @xcite ) . the parameters of this run are for typical open clusters and gmcs in the solar neighbourhood . the top panels show an edge - on view for five different time steps , while the bottom panels show a view along the trajectory of the gmc . encounters with gmcs present the most important external perturbation which cause mass loss of star in clusters . @xcite find that due to encounters with gmcs scales as @xmath53 where @xmath54 for the solar neighbourhood and @xmath55 scales with the surface density of individual gmcs ( @xmath56 ) and the global gmc density ( @xmath57 ) as @xmath58 . the scaling of @xmath55 with @xmath57 implies that it does not matter if the molecular gas is distributed over a large number of low mass clouds or a small number of massive ( giant ) clouds . this makes it easy to estimate from the observed molecular gas density . indeed , for m51 , where the molecular gas density is about an order of magnitude higher than in the solar neighbourhood , a from eq . [ eq : tdis_gmcs ] of 150 myr is predicted . this corresponds well with the value derived from observations of @xmath59 myr @xcite . note that eq . [ eq : tdis_gmcs ] implies a scaling of with the cluster density ( @xmath60 ) . this seems different than the dependence with discussed before . however , there is only a very shallow relation observed between cluster half - mass radius ( ) and , of the form @xmath61 @xcite . with this relation , and eq . [ eq : tdis_gmcs ] , it follows that for external perturbations @xmath62 , i.e. very close to the index of @xmath63 found from observations discussed in [ subsec : appl_pops ] . this suggests that the disruptive effect of the tidal field and additional external perturbations can be added linearly , resulting in a that depends on as @xmath64 . this can explain the large variation found in the @xmath38 value derived from observations and the almost constant @xmath65 @xcite . in [ sec : discussion ] we discuss some of the pitfalls of these results . as seen in the proceeding sections , the observed disruption time of star clusters in the solar neighbourhood is a factor of @xmath66 shorter than predicted by @xmath19-body models . the inclusion of spiral arm passages and gmc encounters into @xmath19-body models is a promising way to bring the predictions into agreement with the observations . this was recently done by @xcite who found excellent agreement after the inclusion gmc encounters and spiral arm passages . they assume that the different mass loss effects ( stellar evolution , tidal field and external perturbations ) can be added linearly . using the mass - radius relation of [ subsec : tdis_external ] and the results from @xcite and @xcite they analytically model the mass loss due to different effects analytically . this is illustrated in the left panel of fig . [ fig5 ] ( from @xcite ) . based on this mass loss description , the age distribution of open clusters in the solar neighbourhood can be predicted ( instead of fitted , as was done hitherto ) . the results are shown in the right panel of fig . [ fig5 ] . cluster due to various disruptive effects . * right : * comparison between the observed age distribution of open clusters ( from ) and the predictions from @xcite for three different maximum masses . , title="fig:",width=255 ] cluster due to various disruptive effects . * right : * comparison between the observed age distribution of open clusters ( from ) and the predictions from @xcite for three different maximum masses . , title="fig:",width=255 ] we showed in [ sec : populations]&[sec : external ] that the simple lamers disruption law can successfully explain the age and mass distribution of young star clusters populations . here we will discuss other observations lending support to the lamers law and some of the standing problems and uncertainties of this scenario which need further attention . @xcite use a variety of studies to look at the cluster population of the lmc . they also find a lack of old clusters ( with respect to what would be expected from a continuous cluster formation rate ) and derive @xmath67 , again in agreement with other galaxies studied by @xcite and @xcite . note that a lower value of @xmath27 is expected to be observed when the typical is of the same order as the oldest clusters in the sample ( fig . [ fig : mf ] ) , as is the case in lmc . outside the local group , the strongly interacting galaxy ngc 6745 has been studied by @xcite who found evidence for mass dependent disruption , with @xmath68 . the rich cluster system of the intermediate - age merger remnant ngc 1316 shows a clear bimodal colour distribution , with the red component presumably being formed during the merger . @xcite showed , using deep _ hst - acs _ images that if one breaks the red component into ` inner ' and ` outer ' regions ( with respect to the galactic centre ) , that the outer region is a continuous power - law while the inner region shows a power - law behavior at the high luminosity end and a flattening at the low luminosity end . the authors interpret this as evidence for mass - dependent cluster disruption , although no attempt was made to find the characteristic disruption timescale or the value of @xmath27 . one standing problem with the lamers disruption law , also present in other studies on disruption , is whether or not an initial power - law cluster initial mass function ( cimf ) can be transformed into a log - normal distribution , which is observed for old globular cluster populations . the lamers law can create such a turnover , however the precise value of the turnover mass should be dependent on the ambient density @xcite , meaning that cluster disruption should be more efficient in the inner regions of a galaxy than in the outer regions . thus , without fine tuning the models ( e.g. having the same disruption time at all radii due to large radially dependent velocity anisotropies ) one would expect a radially dependent turnover peak in the globular cluster mass function , which is not observed . for a more detailed description of this problem , see the review by larsen in these proceedings . additionally , as noted by @xcite the lamers disruption over - predicts the number of low - mass clusters when applied to old globular cluster populations . in [ subsec : tdis_external ] we showed that the scaling of with is a power - law with exponent @xmath63 . this scaling is similar for two - body evaporation in a tidal field with external perturbations , such as shocks by gmcs and spiral arms , and agrees well with the observations . however , there are still some caveats in the theory explaining this , mostly coming from questions regarding the initial conditions of the simulations . 1 . the first caveat stems from the relation between initial mass and radius of the clusters used in the simulations . if we parameterize this relation as @xmath69 @xmath70 , then @xcite use @xmath71 , implying that their clusters fill their tidal radius . however , observations imply that @xmath72 ( with a large scatter ) @xcite , implying that is mostly independent of mass . this shallow relation implies that massive clusters are not filling their tidal radius , which would change the dependence of with @xcite . 2 . in the derivation of @xmath27 for external shocks ( [ subsec : tdis_external ] ) , only clusters in isolation were considered . how would the presence of a tidal field affect this result ? 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we review the theory and observations of star cluster disruption . the three main phases and corresponding typical timescales of cluster disruption are : _ i ) infant mortality _ ( @xmath0 yr ) , _ ii ) stellar evolution _ ( @xmath1 yr ) and _ iii ) tidal relaxation _ ( @xmath2 yr ) . during all three phases there are additional tidal external perturbations from the host galaxy . in this review we focus on the physics and observations of phase i and on population studies of phases ii & iii and external perturbations ( concentrating on cluster - gmc interactions ) . particular attention is given to the successes and short - comings of the lamers cluster disruption law , which has recently been shown to stand on a firm physical footing .

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Proceed to summarize the following text: the origin of the martian methane ( ch@xmath1 ) is still poorly understood . despite the fact that the presence of ch@xmath1 remains under debate ( zahnle et al . 2011 ; zahnle 2015 ) , detections have been claimed at the 1060 parts per billion by volume ( ppbv ) level in mars atmosphere from space and ground - based observations at the end of the 90s and during the following decade ( formisano et al . , 2004 ; krasnopolsky et al . 2004 ; mumma et al . 2009 ; fonti and marzo 2010 ) . recent observations suggest a ch@xmath1 atmospheric abundance of @xmath010 ppbv , and in some cases no or little ch@xmath1 with an upper limit of @xmath07 ppbv in 20092010 , during mars northern spring ( krasnopolsky 2012 ; villanueva et al . more recent _ in situ _ measurements performed by mars science laboratory ( msl ) have evidenced variations in the methane detection at the location of gale crater . despite a background level of methane remaining at 0.69 @xmath2 0.25 ppbv , an elevated level of methane of 7.2 @xmath2 2.1 ppbv was evidenced during a timespan of @xmath06 months ( see table 1 of webster et al . 2015 ) , a range of values comparable to the levels observed remotely during the last decade . because local methane enhancements such as those measured by msl require ch@xmath1 atmospheric lifetimes of less than 1 yr ( lefvre and forget 2009 ) , its release from a subsurface reservoir or an active primary source has widely been discussed in the literature . a plausible explanation is that ch@xmath1 could have been produced either by hydrothermal alteration of basaltic crust ( lyons et al . 2005 ) or by serpentinization of ultramafic rocks producing h@xmath3 and reducing crustal carbon into ch@xmath1 ( oze and sharma 2005 ; atreya et al . 2007 ; chassefire and leblanc 2011 ; chassefire et al . 2013 ; holm et al . this hypothesis is supported by the fact that ultramafic and serpentinized rocks have been observed on mars , in particular in the nili fossae region ( brown et al . 2010 ; ehlmann et al . 2010 ; viviano et al . once formed , methane storage on mars is commonly associated with the presence of hidden clathrate reservoirs . martian clathrates would form an intermediate storage reservoir in the subsurface that regularly releases methane into the atmosphere ( prieto - ballesteros et al . 2006 ; chastain and chevrier 2007 ; thomas et al . 2009 ; gainey and elwood madden 2012 ; herri and chassefire 2012 ; mousis et al . 2013 , 2015 ) . however , because clathrates are more likely thermodynamically stable in the martian subsurface and at depths depending on the soil s porosity ( mousis et al . 2013 ) , their existence has never be proven by remote or _ in situ _ observations . interestingly , it has been recently proposed that halite or regolith could also sequestrate ch@xmath1 on the martian surface ( fries et al . 2015 ; hu et al . 2015 ) , but these mechanisms still need to be thoroughly investigated . here , because of their ability to trap substantial amounts of gases , we suggest that zeolites may form an alternative plausible storage reservoir of methane in the martian subsurface . spectral evidence for the presence of zeolite has been found on the martian surface ( ruff 2004 ; ehlmann et al . 2009 ; carter et al . 2013 ; ehlmann 2014 ) and there is strong geological case arguing for the presence of this aluminosilicate as part of the martian regolith . in sec . 2 , we explain why chabazite , analcime and clinoptilolite are good candidates to account for the widespread occurrence of zeolites on mars . we also provide an estimate of the amount of zeolites potentially existing on the planet . is dedicated to the description of the adsorption properties of chabazite , analcime and clinoptilolite . the amount of methane potentially trapped in these zeolites in martian conditions is estimated in sec . is devoted to discussion . zeolites have been first detected by ruff ( 2004 ) on martian dust using the mars global surveyor ( mgs ) tes spectroscopic observations . fialips et al . ( 2005 ) then suggested that the water - equivalent hydrogen observed by mars odyssey could be partially stored by zeolite minerals present in the first meters in the martian regolith . indeed , dickinson and rosen ( 2003 ) observed up to 18 wt% of authigenic chabazite in frozen soils of antarctica ( equivalent to martian conditions ) . recently , both omega and crism instruments onboard the esa mars express and nasa mars reconnaissance orbiter ( mro ) detected zeolite minerals on the rocky outcrops of several places on mars ( ehlmann et al . 2009 ; carter et al . 2013 , ehlmann 2014 ) . while the first observations on dust and soils suggested a grossly zeolite mineral distribution at mid - latitude , we now have detailed observations revealing the geological / morphological context of zeolite outcrops ( 152 occurrences were detected by carter et al . 2013 ) . for instance , ehlmann et al . ( 2009 ) claimed the identification of pure analcime ( si - al - na form ) in the deposits in and around the central peaks of two 25-km impact craters nearby nili fossae and isidis . these peaks would then reflect post - impact hydrothermal alteration ( osinski and pierazzo 2013 ) . carter et al . ( 2013 ) had a detailed discussion of the issue of their timing of formation and concluded that most hydrous minerals , including zeolites , were formed during the noachian period . however , they also noticed the presence of zeolites in the younger northern lowlands , probably resulting from ice - volcano interaction . in summary , both tes and omega instruments were able to remotely differentiate zeolite spectra from other alteration minerals , namely opal a and saponite formed under similar conditions . such secondary zeolites result from low temperature aqueous alteration by alkaline brines ( or ice ) of volcanic glass included in pyroclastic or volcanic sedimentary rocks and form authigenic cements in volcanoclastic sandstone . note that volcanic ash and tephra , the common contributor to sedimentary material on mars , should be widespread , as explosive volcanism on mars is the rule rather than the exception ( grott et al . however , the resolution of existing infrared spectra remains insufficient to constrain the variety of zeolites that really crystallized on mars . among the possible zeolites , chabazite is a good candidate to account for their widespread occurrence on mars . this mineral is the end product of weathering sequences in a wide range of chemical context ranging from silica - rich to silica - poor volcanic rocks . chabazite typically forms in chemically open systems , in which transports of soluble ions take place efficiently by flowing vadose water or near - surface ground water ( sheppard and hay 2001 ) . on the other hand , in the closed systems in the martian subsurface , more alkali analcime and clinoptilolite should be the major zeolites due to limitation of transports of soluble ions . also , there are several terrestrial locations where nearly pure analcimes form thick bedding ( several tens of meters ) with wide special extent ( hundreds of kilometers ) ( sheppard and gude 1973 ; whateley et al . 1996 ; deer et al . 2004 ) . one can provide an estimate of the amount of zeolites potentially existing on mars . using noachian estimates for the martian crustal thermal flux ( 1220@xmath4c / km ) and thermodynamic data of low - grade metamorphic facies ( @xmath0160220 @xmath4c and from 0 to 3 @xmath510@xmath6 kpa ) , zeolites may be formed at depths ranging from approximately 8 to 1520 km ( e.g. mcsween et al . this estimate is confirmed by the detection of zeolites near central peaks , independently suggesting that those minerals are indeed present at depths of several kilometers in the crust . assuming this depth range ( 815 km ) , it corresponds to a global volume of 10@xmath7 km@xmath8 and a global equivalent layer ( gel ) reaching @xmath07 km of martian zeolites . however , the maps of carter et al . ( 2013 ) show that the area where zeolites ( and all hydrous minerals ) were detected by remote sensing is equivalent to the surface of the 045@xmath4s latitudinal band , i.e. about 35% of the surface of mars . if we do not consider temperature constraints but only different thicknesses ( 0.001 to 10 km ) of a 100% zeolite layer at all 045 @xmath4s latitudes , the total volume and gel are in the @xmath05 @xmath5 10@xmath95 @xmath5 10@xmath10 km@xmath8 and @xmath00.353500 m ranges , respectively . the smallest values may be considered as reasonable estimates ( in the range of @xmath01% of zeolite in crystal clays ; ehlmann et al . 2011 ) , but other geological settings or models can be considered . for instance , the total volume of possible isolated cylindrical zeolite layers located beneath @xmath0150 impact craters ( carter et al . 2013 ) ranging from 5 to 200 km of diameter ( zeolite layer thickness from 0.1 km to unrealistic 20 km ) may reach @xmath010@xmath8 to 10@xmath10 km@xmath8 ( 0.07 to 700 m gel ) . these isolated layers may correspond to zeolite minerals formed by post - impact hydrothermal alteration ( osinski and pierazzo 2013 ) . therefore , it seems that any scenario of zeolite geological generation ( sparse post - impact hydrothermal alteration or crustal global alteration ) can lead to important ranges of volumes / gel . these values , in particular the most optimistic ones , should not be taken as true quantities , but only as starting reasonable estimates . indeed crustal porosity and fluids surely decrease the efficiency of zeolite formation at depths : the low water - to - rock ratio would prevent any alteration while secondary mineralizations fill the pores . in this section , the adsorption selectivity of ch@xmath1 with respect to co@xmath3 is investigated on chabazite , analcime and clinoptilolite . the common chemical formula of a hydrated chabazite is [ ca@xmath11al@xmath12si@xmath13o@xmath14,(h@xmath3o)@xmath15 . there exists several forms of chabazite zeolites that differ in their si / al ratio and the nature of cations ( ba@xmath16 , ca@xmath16 , sr@xmath16 , k@xmath17 , na@xmath17 ) , which counterbalance the electric charges . the framework structure of chabazite is composed of sio@xmath1 and alo@xmath1 tetrahedrons joined by their oxygen atoms . this arrangement forms primary building units interconnected by secondary building units , as shown in fig . [ chab1 ] . the unit cell of chabazite thus contains one large ellipsoidal cavity accessed by six 8-ring windows ( pascale et al . 2002 ) . the chabazite zeolite gets specific adsorption properties for various molecules having a size smaller than the 8-ring apertures , which gives them access to the ellipsoidal cages . owing to the presence of compensation alkali cations , the chabazite is a hydrophilic material . its adsorption capacity of water is around 0.2 cm@xmath8/g at 257 k ( jnchen et al . 2006 ) . this zeolite is also able to adsorb co@xmath3 and ch@xmath1 , two molecules of interest for the martian atmosphere . figure [ chab2 ] shows the adsorption isotherms of co@xmath3 and ch@xmath1 on chabazite at 300 k calculated by monte carlo simulations in the grand canonical ensemble ( gcmc ) ( garcia - perez et al . 2007 ) for pressure ranging below 10@xmath8 kpa . these simulations are in a very good agreement with some experimental data reported in the literature ( watson et al . , 2012 ; jensen et al . 2012 ) . as would expected , owing to the presence of a quadripolar moment in the carbon dioxide molecule , the chabazite adsorbs more co@xmath3 than ch@xmath1 . in spite of this , the amount of methane adsorbed is not insignificant at 300 k and even better at lower temperature . it can reach @xmath02 mol / kg against more than 6 mol / kg in the case of co@xmath3 . this result suggests that the chabazite will selectively adsorb methane and carbon dioxide , with an adsorption in favor of the latter molecule . the adsorption selectivity of methane with respect to carbon dioxide is defined by the relation : @xmath19 where @xmath20 and @xmath21 are the mole fractions of component i in the adsorbed phase and in the gas phase at equilibrium , respectively . @xmath22 can be predicted from the adsorption isotherms of single components by means of the ideal adsorbed solution theory ( ias theory ; myers and prausnitz . when the gas pressure converges towards zero , each single adsorption isotherm exhibits a linear part ( see fig . [ chab2 ] ) , which corresponds to the henry s law region . in this domain , the adsorbed amount of each single component i is proportional to the gas pressure : @xmath23 = @xmath24 . by applying the ias theory to the henry s law region , this allows us to derive the adsorption selectivity from the ratio between the henry constants : @xmath25 the ratio of the adsorbed amounts of ch@xmath1 and co@xmath3 can then be related to the ratio of their partial pressures at equilibrium via the relation : @xmath26 henry constants determined from experimental adsorption isotherms at 300 k are given in table [ table_chab ] with the corresponding selectivities . with the values of the adsorption enthalpies found in the literature , the henry constants can be estimated at any temperature relevant to mars conditions by using the vant hoff relation : @xmath27 once the values of @xmath28 have been determined for ch@xmath1 and co@xmath3 at given temperature , it is possible to derive the corresponding adsorption selectivity from eq . [ eq2 ] . the theoretical chemical formula of analcime is [ naalsi@xmath3o@xmath11],(h@xmath3o ) . the structure of analcime , represented in fig . [ chab1 ] , is very constricted ; the basic sio@xmath1 and alo@xmath1 tetrahedra mutually link to form 4 or 6 membered rings . the maximum diameter of a sphere that can diffuse throughout this structure is @xmath02.4 . because the kinetic diameters of co@xmath3 and ch@xmath1 are 3.3 and 3.8 respectively , these two molecules can not be adsorbed in analcime . only water can be adsorbed in analcime , due to its smaller diameter ( @xmath02.6 ) . the common chemical formula of clinoptilolite is [ m@xmath11al@xmath11si@xmath30o@xmath14,(h@xmath3o)@xmath12 , where m is a compensation cation easily exchangeable which can be na , k , ca , sr , ba and mg according to the source of minerals ( sand et al . clinoptilolite has the same framework structure as heulandite . however it presents a better thermal stability . the porosity is composed of three sets of intersecting channels , all in the same plane ( fig . 1 ) : a channels with 8-membered rings ( aperture 3 @xmath5 7.6 ) , b channels parallel to a channels with 8-membered rings ( aperture 3.3 @xmath5 7.6 ) and c channels quasi perpendicular to the two others with 8-membered rings ( aperture 2.6 @xmath5 4.7 ) ( baerlocher et al . 2007 ) . the microporous volume determined by water adsorption is around 0.16 cm@xmath8/g at 298 k. owing to the presence of compensation cations , clinoptilolite exhibits good adsorption affinity towards water , carbon dioxide and methane which , unike analcime , can enter its micropores despite a pore aperture close to the kinetic diameter of these molecules . as chabazite , this zeolite preferentially adsorbs carbon dioxide compared to methane . at 298 k , some varieties of clinoptilolite can adsorb more than 3.6 mol / kg of co@xmath3 at room temperature under 10 kpa ( breck et al . 1974 ) while only 0.25 mol / kg of ch@xmath1 under the same conditions ( kouvelos et al . 2007 ) . here , the determination of the adsorption selectivity of ch@xmath1 with respect to co@xmath3 in clinoptilolite is calculated following the same approach as for chabazite . adsorption capacities , adsorption enthalpies , henry constants and adsorption selectivities for co@xmath3 and ch@xmath1 have been derived from the experiments of arefi pour et al . in sec . 3 , we have shown that the application of the henry s law allows to extrapolate the amounts of ch@xmath1 and co@xmath3 trapped in chabazite or clinoptilolite at low pressure range . because the current martian surface atmospheric pressure ( 0.6 kpa ) is located in the validity domain of henry s law ( see the example of fig . [ chab2 ] given at 300 k ) , this enables us to investigate the amount of ch@xmath1 that would be potentially trapped in martian chabazite or clinoptilolite in contact with an older martian atmosphere at various temperatures , assuming that the methane abundance was higher than today s value at that time . the adsorption selectivity of methane with respect to carbon dioxide @xmath22 represents the ratio of the ch@xmath1 abundance in chabazite or clinoptilolite to its abundance in the coexisting gas phase at low pressure range . the evolution of @xmath22 as a function of temperature is illustrated by fig . [ chab3 ] in the cases of the two zeolites . with values between 2.5 @xmath5 10@xmath31 and 0.169 ( chabazite ) and between 2.5 @xmath5 10@xmath32 and 0.094 ( clinoptilolite ) in the 150300 k range , we find that the ch@xmath1/co@xmath3 ratio increases with higher temperatures in both zeolites , regardless of the initial ch@xmath1-co@xmath3 gaseous mixture . figure [ chab4 ] represents the evolution of the ch@xmath1/co@xmath3 ratio in the two zeolites as a function of the ch@xmath1/co@xmath3 ratio in the coexisting gas at three temperatures of interest , namely the coldest winter temperature reached in the south pole region ( 150 k ) , and the average night ( 200 k ) and day ( 300 k ) surface temperatures at mid - latitudes . it shows that the ch@xmath1/co@xmath3 ratio must be in the @xmath010@xmath313 @xmath5 10@xmath33 range at 150 k in the coexisting gas phase to give a value in chabazite matching the ch@xmath1 abundance range measured by msl ( in the @xmath00.257.2 @xmath5 10@xmath34 range ) . ch@xmath1/co@xmath3 ratios must also exceed @xmath05 @xmath5 10@xmath35 and 4 @xmath5 10@xmath36 in gas to give values in chabazite higher than those measured by msl at 300 and 200 k , respectively . on the other hand , the ch@xmath1/co@xmath3 ratio in the coexisting gas must be in the @xmath010@xmath373 @xmath510@xmath36 range in clinoptilolite to match the ch@xmath1 abundance range measured by msl at 150 k. interestingly , because @xmath22 is higher in clinoptilolite than in chabazite at temperatures lower than @xmath0270 k , smaller ch@xmath1/co@xmath3 ratios ( @xmath07 @xmath5 10@xmath35 at 300 k and @xmath05 @xmath5 10@xmath37 at 200 k ) are needed to allow this zeolite to match the msl values . comparisons with models depicting the composition of clathrates potentially existing in the martian subsurface show that chabazite or clinoptilolite can be comparable methane sinks ( i.e. , methane trapping from a methane - containing atmosphere on early mars or from an abiotic source in the crust on early / present mars ) at significantly higher temperatures . for example , @xmath38 in chabazite or clinoptilolite becomes greater or equal to 0.1 at temperatures reaching 300 k ( see fig . [ chab3 ] ) , whatever the initial ch@xmath1 atmospheric mole fraction . similar selectivities are achieved in clathrates for ch@xmath1 mole fractions in the 10@xmath3310@xmath39 range but the existence of these structures requires temperatures lower than @xmath0150 k at 0.6 kpa of atmospheric pressure ( mousis et al . 2013 ) , namely the coldest temperature reached during winter in the south pole region . therefore , scenarios advocating a substantial trapping of volatiles in martian clathrates argue that these ices are buried in the soil at sufficient depth , allowing them to be isolated from the atmosphere and remain stable over long time periods ( chastain and chevrier 2007 ; thomas et al . 2009 ; herri and chassefire 2012 ; chassefire et al . 2013 ; mousis et al . this scenario also applies to martian chabazite or clinoptilolite , allowing ch@xmath1 to be extracted either from a potentially methane - rich ancient atmosphere or directly from an abiotic source localized in the crust . indeed , because of their burial in the soil , these zeolites could have preserved the trapped methane over long time periods and create the sporadic releases observed in the atmosphere over the last decade due to impacts , seismic activity or erosion . an alternative possibility would be to assume that zeolites did continuously remain in equilibrium with the martian atmosphere during the course of its evolution . in this case , zeolites would not be able to supply any methane to the atmosphere : because the value of @xmath38 is in the @xmath010@xmath3210@xmath40 range between 200 and 300 k , the amount of ch@xmath1 trapped in chabazite and clinoptilolite would be lower than the measured atmospheric levels . for the same reason , chabazite and clinoptilolite could not act as ch@xmath1 sinks if they remain in contact with the atmosphere . similarly to the proposed trapping scenarios in clathrates , the methane stored in these zeolites could have been produced earlier either via hydrothermal alteration of basaltic crust ( lyons et al . 2005 ) or via serpentinization reactions ( oze and sharma 2005 ; atreya et al . 2007 ; chassefire and leblanc 2011 ; chassefire et al . 2013 ; holm et al . otherwise , alternative methane sinks should be considered on mars . in order to quantitatively test the link between the current time presence of ch@xmath1 in the atmosphere and the possible destabilization of zeolites , we can estimate the total gel of zeolites that must be destabilized each second , assuming an initial quantity of trapped ch@xmath1 . in the following , we make the assumption that chabazite or analcime are the dominant zeolites on mars . if , instead , analcime is the dominant form , then this material can not be at the origin of the atmospheric ch@xmath1 , due to the small size of its porous network . two case studies can be envisaged . in the first case , we assume that ch@xmath1 is trapped in chabazite or clinoptilolite in contact with an ancient martian atmosphere at a surface pressure and temperature of 0.6 kpa and 150 k , respectively . matching the upper msl value at 150 k requires a ch@xmath1/co@xmath3 ratio of 3 @xmath5 10@xmath33 and 3 @xmath5 10@xmath36 in the gas phase released by chabazite and clinoptilolite , respectively . these ratios correspond to amounts of trapped ch@xmath1 of @xmath010@xmath39 mol kg@xmath40 in chabazite and @xmath010@xmath36 mol kg@xmath40 in clinoptilolite ( see fig . [ chab4 ] ) . the total injection flux of ch@xmath1 in the martian atmosphere has been estimated to be 85100 kg s@xmath40 ( mischna et al . 2011 ; holmes et al . 2015 ) . assuming a mean density of 2000 kg m@xmath32 for chabazite and clinoptilolite , this flux corresponds to at least @xmath05.3 @xmath5 10@xmath41 kg s@xmath40 of zeolites and 5 @xmath5 10@xmath42 m yr@xmath40 as gel . a more precise calculation would be based on the realistic value of localized surface flux 10@xmath4310@xmath34 kg m@xmath39 s@xmath40 derived by holmes et al . ( 2015 ) , assuming a source within a homogeneous 5@xmath4x5@xmath4 region . in this case , the corresponding mass of destabilized zeolites would be 6 @xmath5 10@xmath44 kg m@xmath39 s@xmath40 ( chabazite ) and 6 @xmath5 10@xmath45 kg m@xmath39 s@xmath40 ( clinoptilolite ) . these values correspond to localized layers with thicknesses of @xmath010@xmath3210@xmath40 m for chabazite and @xmath0101000 m for clinoptilolite . in the second case , we assume that chabazite or clinoptilolite are directly filled by pure methane produced from an abiotic source localized at depth in the crust . here , given the high lithostatic pressure , the henry s law does not apply anymore and single adsorption temperature isotherms of ch@xmath1 derived from experiments or gcmc computations must be used . as a toy example , we consider the figure [ chab2 ] which shows that at 300 k and 10@xmath8 kpa of gas pressure ( corresponding to a depth of @xmath090 m ) , the amount of ch@xmath1 trapped in chabazite reaches @xmath02 mol kg@xmath40 . assuming a total ch@xmath1 injection flux of 85100 kg s@xmath40 in the martian atmosphere , we find that it corresponds at least to @xmath02650 kg s@xmath40 of chabazite and 2.9 @xmath5 10@xmath37 m yr@xmath40 as gel . using the localized surface flux 10@xmath4310@xmath34 kg m@xmath39 s@xmath40 sourced from a 5@xmath4x5@xmath4 region , the corresponding mass of destabilized chabazite would be 3 @xmath5 10@xmath46 kg m@xmath39 s@xmath40 , representing a localized layer of @xmath05 @xmath5 10@xmath365 @xmath5 10@xmath33 m yr@xmath40 . in the case of clinoptilolite , assuming an amount of trapped ch@xmath1 10 times smaller than the one estimated for chabazite ( see sec . 3.3 ) , the aforementioned values of gel and localized layer would be increased by the same factor . when based on realistic values of the methane flux , our calculations show that the second case appears more plausible than the first because it requires amounts of chabazite or clinoptilolite well below those independently quantified from geological constraints . if the martian methane present in chabazite or clinoptilolite is directly sourced from an abiotic source in the subsurface , the destabilization of a zeolite localized layer of a few millimeters per year at worst may be sufficient to explain the current observations . our study suggests that if the methane outgassing from excavated chabazite or clinoptilolite prevails over any other source on mars , then the presence of these minerals around gale crater could explain the variation of the ch@xmath1 level observed by msl . an interesting follow - up of this work would be to investigate the adsorption / desorption efficiencies of other gases in zeolites . coupling all the data together might lead to predictions of other observable effects in the martian atmosphere that could be used to test the present hypothesis . finally , it is interesting to note that the zeolites adsorption properties depend on their si / al ratios . the smaller is this ratio , the greater are the compensating cations and the hydrophilic and organophilic properties of the zeolites . the adsorption selectivity of ch@xmath1 with respect to co@xmath3 also varies according to this ratio and the nature of the cation . however , in the cases of chabazite and clinoptilolite , there will always be a preferential adsorption of co@xmath3 at the expense of ch@xmath1 , whatever the si / al ratio and the nature of the cation . this preferential adsorption results from the specific interactions induced by the quadrupole moment caused by the presence of pi electrons in co@xmath3 , which is not the case with ch@xmath1 which puts into action non - specific interactions . the work contributed by o.m . was carried out thanks to the support of the a*midex project ( n^o^ anr-11-idex-0001 - 02 ) funded by the `` investissements davenir '' french government program , managed by the french national research agency ( anr ) . we acknowledge support from the `` observatoire des sciences de lunivers '' ( osu ) institut pythas , the `` institut national des sciences de lunivers '' ( insu ) , the `` centre national de la recherche scientifique '' ( cnrs ) and `` centre national detude spatiale '' ( cnes ) and through the `` programme national de plantologie '' and mex / pfs program . we also thank two anonymous reviewers who made very interesting suggestions that helped us improve our first manuscript . arefi pour , a. , sharifnia , s. , neishaborisalehi , r. , ghodrati , m. 2015 . performance evaluation of clinoptilolite and 13x zeolites in co@xmath3 separation from co@xmath3/ch@xmath1 mixture . j. natural gas sci . 26 , 1246 - 1253 . brown , a. j. , hook , s. j. , baldridge , a. m. , crowley , j. k. , bridges , n. t. , thomson , b. j. , marion , g. m. , de souza filho , c. r. , bishop , j. l. 2010 . hydrothermal formation of clay - 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( eds ) natural zeolites : occurrence , properties , applications , reviews in mineral . and geochem . , 45 , 261276 . sheppard , r. a. , gude , a. j. iii 1973 . zeolites and associated authigenic silicate minerals in tuffaceous rocks of the big sandy formation , mohave county , arizona . u. s. geological survey , professional paper 830 , 36 pp . villanueva , g. l. , mumma , m. j. , novak , r. e. , radeva , y. l. , kufl , h. u. , smette , a. , tokunaga , a. , khayat , a. , encrenaz , t. , hartogh , p. 2013 . a sensitive search for organics ( ch@xmath48 , ch@xmath49oh , h@xmath47co , c@xmath47h@xmath50 , c@xmath47h@xmath47 , c@xmath47h@xmath48 ) , hydroperoxyl ( ho@xmath47 ) , nitrogen compounds ( n@xmath47o , nh@xmath49 , hcn ) and chlorine species ( hcl , ch@xmath49cl ) on mars using ground - based high - resolution infrared spectroscopy . icarus 223 , 11 - 27 . viviano , c. e. , moersch , j. e. , mcsween , h. y. 2013 . implications for early hydrothermal environments on mars through the spectral evidence for carbonation and chloritization reactions in the nili fossae region . journal of geophysical research ( planets ) 118 , 1858 - 1872 . watson , g. c. , jensen , n. k. , rufford , t. e. , ida chan , k. , may , e. f. 2012 . volumetric adsorption measurements of n@xmath3 , co@xmath3 , ch@xmath1 , and a co@xmath3 + ch@xmath1 mixture on a natural chabazite from ( 5 to 3000 ) kpa . journal of chemical & engineering data 57 ( 1 ) , 93101 . /co@xmath3 ratio in chabazite and clinoptilolite as a function of its ratio in coexisting gas represented at @xmath52 = 150 , 200 and 300 k and 0.6 kpa of total pressure . the grey area corresponds to the range of ch@xmath1 measurements made so far by msl in the martian atmosphere . bottom : amount of ch@xmath1 trapped chabazite and clinoptilolite as a function of the ch@xmath1/co@xmath3 ratio in coexisting gas calculated at @xmath52 = 150 , 200 and 300 k and 0.6 kpa of total pressure.,width=340 ]

the origin of the martian methane is still poorly understood . a plausible explanation is that methane could have been produced either by hydrothermal alteration of basaltic crust or by serpentinization of ultramafic rocks producing hydrogen and reducing crustal carbon into methane . once formed , methane storage on mars is commonly associated with the presence of hidden clathrate reservoirs . here , we alternatively suggest that chabazite and clinoptilolite , which belong to the family of zeolites , may form a plausible storage reservoir of methane in the martian subsurface . because of the existence of many volcanic terrains , zeolites are expected to be widespread on mars and their global equivalent layer may range up to more than @xmath01 km , according to the most optimistic estimates . if the martian methane present in chabazite and clinoptilolite is directly sourced from an abiotic source in the subsurface , the destabilization of a localized layer of a few millimeters per year may be sufficient to explain the current observations . the sporadic release of methane from these zeolites requires that they also remained isolated from the atmosphere during its evolution . the methane release over the ages could be due to several mechanisms such as impacts , seismic activity or erosion . if the methane outgassing from excavated chabazite and/or clinoptilolite prevails on mars , then the presence of these zeolites around gale crater could explain the variation of methane level observed by mars science laboratory . mars , mars , atmosphere , mars , surface , mineralogy , astrobiology

You are an expert at summarizing long articles. Proceed to summarize the following text: the karlsruhe tritium neutrino experiment katrin @xcite will be the first beta decay experiment attempting to measure the electron neutrino mass with sub - ev precision . presently the experiment is commissioned to start data - taking in 2013/14 and has a projected sensitivity of 0.2 ev ( 90% c.l . ) to the neutrino mass . katrin is the successor of the experiments in mainz @xcite and troitsk @xcite and will be using some of the same techniques as those . for the technical details of katrin see e.g. @xcite . strictly speaking when measuring the electron neutrino mass with @xmath2-decay spectra , what we get is the socalled kinematic neutrino mass . that is , the incoherent sum of neutrino mass eigenvalues weighted by the appropriate entries in the lepton mixing matrix : @xmath3 however , because the mass differences between the active neutrino mass states are known to be smaller than katrins sensitivity the experiment can effectively only see one mass state ( the mass squared differences are @xmath4 ev@xmath5 and @xmath6 ev@xmath5 , respectively @xcite ) . this mass state is sometimes called the electron neutrino mass , but in principle the tritium beta - spectrum could contain the signatures of more than one mass state or of couplings to other particles entirely . in order to be called truly model independent katrin s final data should be analyzed also for alternative scenarios - beyond the minimal extension of the standard model . performing an analysis for non - standard couplings to the electron neutrino adds more parameter space to the @xmath1-function of the experiment . one should therefore consider how an extended analysis should be performed on the katrin output in order to get reliable results . we present here one approach which seems to give several advantages over the standard frequentist analysis . in section 2 we describe our analysis methods before presenting results for a number of cases in section 3 . finally we give some concluding remarks in the last section . let us begin by summarising the procedures for production and analysis of katrin spectra as performed by a toy model monte carlo and analysis code for katrin - like experiments @xcite . this code has previously been used to forecast the experiment s sensitivity to the neutrino mass @xcite . because katrin has an integrating spectrometer ( a consequence of the mac - e ( magnetic adiabatic collimation with electrostatic ) filter technique ) the beta - spectrum must be written as an integral over the electron energy : @xmath7 here @xmath8 is the retarding potential of the spectrometer , @xmath9 is the total number of tritium nuclei in the source , @xmath10 is the measurement time allotted for a given value of the retarding potential and @xmath11 is the experimental response function ( which in turn is a combination of the electron energy loss function of the tritium source and the transmission function of the spectrometer).@xmath12 is the theoretical beta spectrum rate folded with the electronic final state distribution of molecular tritium . a retarding voltage - independent background rate of @xmath13 is now added to eq . ( [ eq : tid ] ) : @xmath14 this gives us the following theoretical expression for a katrin - like spectrum : @xmath15 individual spectra ( to resemble the real measurements ) are built using initial parameters and @xmath16 for the neutrino mass squared endpoint energy , respectively . to the theoretical expression is then added a random component from a gaussian distribution with @xmath17 and @xmath18 : @xmath19 when we want to fit our randomized beta - spectra we have to account for statistical fluctuations by allowing the overall amplitude @xmath20 of the signal as well as the background rate @xmath21 to vary against the theoretical amplitude @xmath22 and background rate @xmath13 . in addition , we allow the neutrino mass squared as well as the endpoint energy @xmath23 to deviate from the initial parameters of the simulation and @xmath16 . @xmath24 combining eq.s ( [ eq : best2 ] ) and ( [ eq : best3 ] ) we finally get katrin s @xmath1-function @xcite : @xmath25 beta - spectra[fig : fig1 ] for a katrin - like parameter set and an assumed value for the neutrino mass of @xmath26 measured with an optimized time distribution over the last 25 ev of the beta - spectrum , e.g. compare to @xcite.,width=597 ] the analysis of the simulated data can be performed with minuit2 which is imbedded in the root - package . this procedure performs a minimization of the @xmath1-function using combinedminimizer . combinedminimizer in turn uses either an evaluation of the covariance matrix or a simplex method to find the best minimum of the @xmath1-function in the parameter space @xcite . one can now do a standard frequentist analysis to find the statistical uncertainty on e.g. the neutrino mass by producing a suitable amount of monte carlo spectra , performing the minimization for each of them and finally inspecting the resulting histograms . an example is shown in figure [ fig : fig1 ] for 12860 spectra produced with @xmath27 0.0 ev . however as previously indicated the minimization approach has a number of drawbacks . for one thing it does not give any information on multiple minima , and it is not well suited for finding shallow minima . furthermore extracting detailed information on correlated parameters is pretty laborious . still the method works just fine for the four well - known free parameters used in a standard katrin analysis - see table [ tab : org ] . but as one adds more parameters the minimization procedure often becomes problematic and rather slow . as an alternative approach we have considered markov chain monte carlo and bayesian inference techniques as in the publicly available cosmomc analysis package for cosmology . typical cosmological models contain @xmath28 8 - 12 parameters and cosmomc is well suited for relatively fast analysis of such multiparameter spaces @xcite . the programme is built for analysis of large cosmological datasets such as cmb data from wmap and supernova surveys but can in principle analyse whatever dataset the user provides - the cosmology can be turned off if it is irrelevant . cosmomc uses bayesian statistics for the analysis . when doing socalled bayesian parameter inference one is interested in knowing the posterior probability , @xmath29 - the probability of the * parameters * , @xmath30 , given the data @xmath31 and the model @xmath32 . the inverse question is for the probability of the * data * , d , given the parameters and the model , @xmath33 - this is simply the likelihood function . with these two probabilities and the well - known bayes theorem , @xmath34 one can write an expression for the posterior probability : @xmath35 here @xmath36 is the likelihood - which can be easily derived from the @xmath1-function is connected to the @xmath1-function as in the following way : @xmath37 . ] . the posterior probability is thus proportional to the likelihood . meanwhile @xmath38 is the socalled prior probability sometimes referred to as the subjective input - it is what we believe we know from theory before even taking the data into account . correspondingly this probability has no dependence on the data . note that we have been using flat priors on all input parameters in this paper . finally @xmath39 , the evidence , is in effect only a parameter - independent normalization constant@xcite . when we want to know best - fit values and confidence levels of specific parameters we can then simply integrate over all the remaining ( nuisance ) parameters . this is called marginalization and the output is called the marginalized probability for the parameter of interest . in addition to this rather convenient production of parameter probability distributions , from the bayesian inference approach , cosmomc gives us another great advantage by using a markov chain monte carlo ( mcmc ) to probe the parameter space . this will provide a very thorough and easy to inspect mapping of the parameter space of interest . the purpose of the mcmc is to probe the whole parameter space in a randomized manner . to achieve this one implements the metropolis hastings algorithm @xcite consisting of three main steps : firstly an initial point , @xmath40 is chosen . secondly a step is proposed in some random direction , after which the new point is evaluated : @xmath41 . here @xmath42 means the iterative point , and @xmath43 is the proposed addition taken from some proposal density . finally the procedure decides whether or not to take the step . the point @xmath44 is accepted if the posterior probability is improved . that is if @xmath45 if the expression above is @xmath46 the step is accepted with some probability @xmath47 ( rejected with probability @xmath48 ) . in this manner we generate a set of points @xmath49 , also called a markov chain . for the number of points , @xmath50 , going to infinity we thus have a representation of the posterior probability . the decision procedure of the metropolis hastings algorithm allows the chain to wander away from any local minima and thus potentially discover other minima ( to a degree determined by the value of r , with temperature , @xmath51 . ] . on the other hand it also guarantees that the parameter space near the minima is very well probed . furthermore one can perform the analysis on a combination of multiple chains - all started at random positions - and get an even better picture of the behavior of the different parameters in the allowed intervals . to get rid of un - physical effects from the random starting points one normally allows for a burn - in i.e. the first part of the markov chain is removed . in our case the burn - in is 50% of the sample size . before running the programme one must carefully choose stepsizes and parameter ranges . several settings in both the cosmomc programme as such and in the parameter files can be tweaked to fit ones purpose . unfortunately it is in principle not possible to determine in any absolute terms whether or not a specific chain has converged @xcite , but various convergence diagnostics have been developed . for instance when analyzing multiple chains a convergence parameter , @xmath52 , defined as the variance of the chain means divided by the mean of the chain variances , can be evaluated . if @xmath53 is less than some chosen small number ( in our case 0.03 ) this information is interpreted as good convergence . when cosmomc has generated the chains we need , the data analysis is performed giving us best - fit values and standard deviations for all the parameters . additionally cosmomc produces a number of useful matlab - files which can be used to produce @xmath54 and @xmath55 plots of the marginalized distributions . inspecting this graphical output allows us to determine if the chains have really converged , whether there are multiple minima and perhaps most importantly it shows parameter correlations right away . if we go through all of this for say a single monte carlo generated beta spectrum we get all the nice advantages mentioned above . but the analysis of that one spectrum would take many hours as compared to minutes or seconds with the minuit2 procedure and we would mostly just have achieved a much slower evaluation of the best fit values for that particular spectrum . however if we in stead use the theoretical beta - spectrum ( which should represent the average of infinitely many measurements or monte carlo realizations ) as our input data - but with monte carlo generated errorbars - our best fit values and standard deviations from cosmomc should correspond to the results of the frequentist approach of building histograms for a very large ( going to infinity ) number of measurements . to recap we implemented our @xmath1-function for katrin - like experiments in cosmomc and simply turned off cosmology . as data - set we have used the theoretical spectra - for any given model - with the monte carlo generated errorbars of the original code . the results will be discussed in the following section . as a first test of our methods we have attempted to reproduce the katrin sensitivity . we thus generated a tritium beta - spectrum using as input so far only four parameters : the electron neutrino mass squared , the endpoint of the beta - spectrum @xmath16 ] , the background count rate @xmath13 and the signal count rate near the beta - spectrum endpoint , @xmath22 - see eq . ( [ eq : best1 ] ) . the input signal count rate can be calculated as a combination of the column density of the source and the magnetic fields and cross sections of the spectrometer and source . in our case the count rate near the endpoint @xmath23 is included in the analysis code via an amplitude factor @xmath22 as in eq . ( [ eq : best3 ] ) . the exact definition and full calculation of this factor @xmath22 is included in appendix a of reference @xcite . given katrin s experimental settings the amplitude has the value @xmath57 hz . we would like to remark , that the value of the endpoint energy @xmath23 needs to be treated as a free parameter to produce realistic fits with respect to fitting of . up to now the @xmath58he - @xmath58h mass difference is known from precision penning trap experiments with 1.2 ev precision @xcite , but already the fits of the experiments at mainz @xcite and troitsk @xcite would have needed a much more precise input value to justify keeping @xmath23 fixed in the fit . .katrin standard analysis parameters . please note , that we show and plot the deviation of the value of the endpoint of the beta - spectrum from the theoretically expected value : @xmath59 ev . [ tab : org ] [ cols="<,^,^,>",options="header " , ] as compared to the case without righthanded couplings and the right figure shows the bias on @xmath60 . this analysis includes the righthanded coupling strength as a free parameter . the bias on the mass is as large as 80% while the values of @xmath60 is up to five times as large as for the standard case ( barring the parameter range below katrins sensitivity where the uncertainty on the mass parameter migrates into the b - dimension in the mcmc ) . [ fig : m_r],title="fig:",width=264 ] as compared to the case without righthanded couplings and the right figure shows the bias on @xmath60 . this analysis includes the righthanded coupling strength as a free parameter . the bias on the mass is as large as 80% while the values of @xmath60 is up to five times as large as for the standard case ( barring the parameter range below katrins sensitivity where the uncertainty on the mass parameter migrates into the b - dimension in the mcmc ) . [ fig : m_r],title="fig:",width=264 ] for the full parameter range of table [ tab : right ] and the right figure is an enlarged version of this plot for @xmath61 ev . in the left figure we see that we get the output values wrong by more than a factor 10 ! this is exacerbated at mass values just above the katrins sensitivity once again demonstrating how the uncertainty on these two ill - determined parameters is redistributed in the parameter space of the markov chain . the right figure shows us that when we look beyond the much larger errorbars around @xmath62 ev the output values still fluctuate with errors of order @xmath63 . [ fig : b_r],title="fig:",width=264 ] for the full parameter range of table [ tab : right ] and the right figure is an enlarged version of this plot for @xmath61 ev . in the left figure we see that we get the output values wrong by more than a factor 10 ! this is exacerbated at mass values just above the katrins sensitivity once again demonstrating how the uncertainty on these two ill - determined parameters is redistributed in the parameter space of the markov chain . the right figure shows us that when we look beyond the much larger errorbars around @xmath62 ev the output values still fluctuate with errors of order @xmath63 . [ fig : b_r],title="fig:",width=264 ] the results shows us firstly that the output mass values fluctuate rather wildly - and in some cases deviate my as much as @xmath64 80 % from the input values as shown in the left panel of figure [ fig : m_r ] . and secondly the statistical uncertainty is up to 5 times larger than in the standard case except in regions where @xmath65 < 0.2 ev as expected from the discussion above . turning to at the output values of the righthanded coupling strength in figure [ fig : b_r ] we get appallingly bad results especially in the @xmath65 = 0.2 ev -region . from the left hand picture of figure [ fig : b_r ] one might get the impression that the output value of b is returned rather nicely for the larger masses . however as shown in the right panel of figure [ fig : b_r ] the relative error is still up to @xmath64 60% in some regions . in conclusion we see from these numerical artifacts that it is extremely difficult to get a good determination of _ both _ the mass and the coupling strength . at least when using fairly large parameter intervals . given stronger limits the situation would no doubt change . but judging from our cosmomc contours the values in some cases will be pressed to the largest allowed parameter values even when the intervals are as broad as here . in other words - tighter parameter values in this case merely amounts to a manual setting of the allowed size of the statistical uncertainties . next we perform the analysis on the same spectra without including the righthanded coupling strength to get an idea of the bias imposed on the neutrino mass in the presence of unaccounted - for righthanded currents . we present our results in figure [ fig : nor ] but here the analysis has been performed ( on the same spectra ) without the inclusion of the righthanded coupling strength . clearly the errors on @xmath65 are much better and for realistic @xmath66-values certainly within acceptable @xmath67-ranges . however we note that @xmath68 is @xmath69 better in the high - b , high - m corners of the right - side plot . this coincides with a turnover of the bias on the mass in the left - side plot . figure [ fig : eor ] shows that this behavior takes place because the @xmath70 parameter is being pushed to the maximally allowed values , which should be avoided . that is the uncertainty on the mass due to the presence of @xmath66 is migrating into the third correlated parameter - the endpoint of the tritium beta spectrum . [ fig : nor],title="fig:",width=264 ] but here the analysis has been performed ( on the same spectra ) without the inclusion of the righthanded coupling strength . clearly the errors on @xmath65 are much better and for realistic @xmath66-values certainly within acceptable @xmath67-ranges . however we note that @xmath68 is @xmath69 better in the high - b , high - m corners of the right - side plot . this coincides with a turnover of the bias on the mass in the left - side plot . figure [ fig : eor ] shows that this behavior takes place because the @xmath70 parameter is being pushed to the maximally allowed values , which should be avoided . that is the uncertainty on the mass due to the presence of @xmath66 is migrating into the third correlated parameter - the endpoint of the tritium beta spectrum . [ fig : nor],title="fig:",width=264 ] as it turns out we get much better results when we remove the @xmath66-dimension from our cosmomc setup - this time the bias on the mass is no larger than around 12 % . we notice however that the statistical error drops steeply for high masses and coupling strengths . inspecting the original cosmomc likelihood contours we see that this is because @xmath70 has been pushed to the edge of the input interval as shown in figure [ fig : eor ] . this also explains why the bias flips in the same parameter - range instead of becoming monotonically larger for maximal coupling strengths . the propagation of the uncertainty on @xmath66 into the @xmath70-dimension is straightforward from the already discussed correlations between the @xmath71 and @xmath70 - parameters . hopefully the upcoming much more precise @xmath58h@xmath58he mass measurements @xcite will be helpful in resolving this issue for the katrin experiment . vs. @xmath72 for the mass range @xmath73 ev to @xmath74 ev ( again going from the upper left corner to the lower right corner ) when the analysis is performed without the inclusion of @xmath66 . the figure on the left used spectra that was produced with @xmath75 while the figure on the right is for @xmath76 . the expected output for @xmath70 is zero , but it is clear to see that in this case the @xmath77 -correlation pushes the uncertainty induced in the mass parameter by the physical presence of @xmath66 into the @xmath70 -parameter instead . [ fig : eor],title="fig:",width=264 ] vs. @xmath72 for the mass range @xmath73 ev to @xmath74 ev ( again going from the upper left corner to the lower right corner ) when the analysis is performed without the inclusion of @xmath66 . the figure on the left used spectra that was produced with @xmath75 while the figure on the right is for @xmath76 . the expected output for @xmath70 is zero , but it is clear to see that in this case the @xmath77 -correlation pushes the uncertainty induced in the mass parameter by the physical presence of @xmath66 into the @xmath70 -parameter instead . [ fig : eor],title="fig:",width=264 ] in conclusion the bias induced on the neutrino mass is now within acceptable bounds and agree well with the results found by bonn _ et al _ @xcite . finally it should be noted that an experiment such as katrin can clearly not be used to put bounds on the size of the righthanded coupling strength at this point . a precise knowledge of the neutrino mass and the tritium beta - spectrum endpoint @xmath23 would be have to be presupposed before measurements of the tritium beta spectrum could be used to determine @xmath66 . our attempt at an analysis of simulated katrin data with various additional parameters has shown the following : for the standard case of analysis with regard to one neutrino mass , the mcmc approach is certainly well suited and gives robust results . the method is very practical when performing analysis for non - standard cases because the cosmomc output lets us inspect the behavior of the parameters and their relation to one another in a straightforward manner . we have used the method to build a sensitivty - plot for a katrin - like experiment , clearly demonstrating the dominating dependence of the sensitivity on the signal countrate . further we have learned that for a suitable mass - squared difference an experiment such as katrin should be able to detect the existence of other neutrino mass states . and finally we have re - evaluated the influence of couplings to righthanded currents in the tritium beta decay and found that ignoring this would maximally induce an error on the neutrino mass of order 10% . in conclusion we find that our bayesian approach to the analysis of the katrin experiment is certainly competitive to a frequentist approach and that it has several advantages over it when using an already well - developed framework such as cosmomc . we acknowledge the use of computing resources from the danish center for scientific computing ( dcsc ) and the grant of bmbf under contract 05a08pm1 . 99 a. osipowicz _ et al . _ [ katrin collaboration ] , + `` katrin : a next generation tritium beta decay experiment with sub - ev sensitivity for the electron neutrino mass , '' + arxiv:0109033 [ hep - ex ] . j. angrik _ et al . _ [ katrin collaboration ] + `` katrin design report 2004 , '' + wissenschaftliche berichte fz karlsruhe 7090 , http://bibliothek.fzk.de/zb/berichte/fzka7090.pdf ch . kraus , b. bornschein , l. bornschein , j. bonn , b. flatt , a. kovalik , b. ostrick , e. w. otten , j. p. schall , th . thmmler and ch . weinheimer , + `` final results from phase ii of the mainz neutrino mass search in tritium @xmath2 decay , '' + eur . j * c 40 * , 447 - 468 ( 2005 ) + arxiv:0412056v2 [ hep - ex ] . s. s. masood , s. nasri , j. schechter , m. a. trtola , j. w. f. valle and c. weinheimer , + `` exact relativistic beta decay endpoint spectrum , '' + phys . c * 76 * 045501 ( 2007 ) + arxiv:0706.0897v1[hep - ph ] . n. christensen , r. meyer , l. knox , and b. luey , + `` bayesian methods for cosmological parameter estimation from cosmic microwave background measurements , '' + arxiv:0103134 [ astro - ph ] . a. sejersen riis , s. hannestad , + `` detecting sterile neutrinos with katrin like experiments , '' + jcap02(2011)011 ( 2011 ) . + arxiv:1008.1495v2 [ astro - ph ] . j. bonn , k. eitel , f. gluck , d. sevilla - sanchez and n. titov , + `` the katrin sensitivity to the neutrino mass and to right - handed currents in beta decay , '' + arxiv:0704.3930 [ hep - ph ] . n. severijns , m. beck and o. naviliat - cuncic , + `` tests of the standard electroweak model in beta decay , '' + rev . * 78 * ( 2006 ) 991 [ arxiv : nucl - ex/0605029 ] . j. hamann , s. hannestad , g. g. raffelt , i. tamborra and y. y. y. wong , + `` cosmology seeking friendship with sterile neutrinos , '' + phys . rev . lett . * 105 * , 181301 ( 2010 ) . + arxiv:1006.5276 . j. hamann , s. hannestad , g. g. raffelt and y. y. y. wong , + `` observational bounds on the cosmic radiation density , '' + jcap * 0708 * , 021 ( 2007 ) . + arxiv:0705.0440 [ astro - ph ] . j. hamann , s. hannestad , j. lesgourgues , c. rampf and y. y. y. wong , + `` cosmological parameters from large scale structure - geometric versus shape information , '' + arxiv:1003.3999 . m. c. gonzalez - garcia , m. maltoni and j. salvado , + `` robust cosmological bounds on neutrinos and their combination with oscillation results , '' + arxiv:1006.3795 . m. maltoni & t. schwetz , + `` sterile neutrino oscillations after first miniboone results , '' + phys . d * 76 * , 093005 ( 2007 ) . s. goswami and w. rodejohann + `` miniboone results and neutrino schemes with two sterile neutrinos : possible mass orderings and observables related to neutrino masses , '' + arxiv:0706.1462v2 [ hep - ph ] . g. j. stephenson , j. t. goldman and b. h. j. mckellar , + `` tritium beta decay , neutrino mass matrices and interactions beyond the standard model , '' + phys . d * 62 * , 093013 ( 2000 ) + arxiv:0006095 [ hep - ph ] .

the katrin ( karlsruhe tritium neutrino ) experiment will be analyzing the tritium beta - spectrum to determine the mass of the neutrino with a sensitivity of @xmath0 ev ( 90% c.l . ) . this approach to a measurement of the absolute value of the neutrino mass relies only on the principle of energy conservation and can in some sense be called model - independent as compared to cosmology and neutrino - less double beta decay . however by model independent we only mean in case of the minimal extension of the standard model . one should therefore also analyse the data for non - standard couplings to e.g. righthanded or sterile neutrinos . as an alternative to the frequentist minimization methods used in the analysis of the earlier experiments in mainz and troitsk we have been investigating markov chain monte carlo ( mcmc ) methods which are very well suited for probing multi - parameter spaces . we found that implementing the katrin @xmath1- function in the cosmomc package - an mcmc code using bayesian parameter inference - solved the task at hand very nicely .

You are an expert at summarizing long articles. Proceed to summarize the following text: it has been recognized that exotic magnetic excitations known as skyrmions may exist @xcite in a two - dimensional electron gas in a strong homogeneous magnetic field ( quantum hall system ) near spin polarized groundstates . these are excitations of a two - dimensional spontaneous ferromagnet , the physics of which is relevant to this system ( despite the presence of a strong magnetic field ) , because of the small land @xmath0-factor in gaas systems ( where most experiments take place ) , which makes the zeeman coupling very small compared to other energy scales ( coulomb interaction , cyclotron energy ) in the problem . skyrmions are spin configurations with a non - trivial winding number ( pontryagin index ) . they were first discussed in the context of four - dimensional field theories @xcite , and were later recognized as states occurring in the non - linear sigma model description of two - dimensional ferromagnets @xcite . for filling factors @xmath1 close to one ( @xmath2 is the number of electrons and @xmath3 the number of magnetic flux quanta penetrating the system ) , these turn out to be the lowest energy quasiparticles under typical experimental circumstances . skyrmions can thus be introduced into the groundstate by adding or removing charge from the system . @xcite . experimentally , the case for the existence of skyrmions in a system close to @xmath4 is quite strong . nmr experiments show a degrading of the spin polarization with deviation of filling factor from one @xcite that is in remarkably good agreement with hartree - fock theory @xcite . the quasiparticle spin measured in transport experiments @xcite are also reasonably well accounted for by hartree - fock calculations @xcite . electromagnetic absorption experiments @xcite further support that doping away from @xmath4 injects skyrmions into the system . in weaker magnetic fields , near filling factor @xmath5 , early experiments @xcite suggested that spin - polarized quasiparticles are lower in energy than skyrmions , so the effects seen near @xmath4 would not be present at higher filling . this is consistent with calculations of skyrmion energies near @xmath5 that include finite thickness corrections @xcite , which indicate that skyrmions will be present only at much smaller zeeman couplings than realized in typical experiments . the size of the skyrmion may be quantified by a number @xmath6 , the difference in the spin component @xmath7 between the skyrmion and the spin - polarized quasiparticle . because of the necessarily small zeeman coupling , stable skyrmions close to @xmath5 have large values of @xmath6 . they also become unstable with respect to spin - polarized quasiparticles at a finite value of @xmath6 . ( for @xmath4 , @xmath8 as the zeeman coupling reaches the maximum value for which the system supports skyrmions ; i.e. , the skyrmion state smoothly goes into the spin - polarized quasiparticle state . ) for a two - dimensional electron gas ( 2deg ) with width of about 2@xmath9 , where @xmath10 is the magnetic length , the minimum @xmath6 expected @xcite is approximately 4 . recently however , nmr experiments @xcite have uncovered evidence that some anomalous degrading of spin polarization @xmath11 occur as one dopes away from @xmath5 at relatively high zeeman couplings . these experiments further indicate that the number of overturned spins per quasiparticle is quite small , @xmath12 . the simple models usually considered @xcite are inconsistent with this , and one is naturally led to inquire as to what other ingredients might change the critical zeeman coupling and smallest @xmath6 observable near @xmath5 . two possible answers are landau level mixing and screening by filled landau levels . it should be noted that these are not distinct effects : screening by filled landau levels occurs because they may admix high ( unoccupied ) landau levels to smooth fluctuations due to external potentials and/or inhomogeneous electron densities in partially filled levels . conversely , the states which may be used for landau level mixing in a partially filled level are limited to those that are not occupied by electrons in other levels , due to pauli exclusion . thus , a correct treatment of either screening or landau level mixing near @xmath5 must include both these effects . in this work , we present a method by which these may be incorporated into the hartree - fock description of skyrmion states . our principal conclusions may be summarized as follows : ( i ) for skyrmions near @xmath4 , landau level mixing tends to lower the quasiparticle energy , although not enough to quantitatively explain the activation energies seen in experiment @xcite . introduction of a finite width of one magnetic length lowers the energy of the skyrmion by approximately 40% , and inclusion of landau level mixing lowers the energy by approximately another 20% for @xmath13 , where @xmath14 is the cyclotron frequency of the electrons , and @xmath15 is the dielectric constant of the host crystal . the resulting quasiparticle gap is approximately a factor of 2 larger than what is found in experiment @xcite . this result agrees qualitatively with that of another study of landau level mixing effects on @xmath4 skyrmions @xcite . ( ii ) for @xmath4 , landau level mixing tends to suppress quasihole - like skyrmions ( i.e. , lowering the maximum zeeman coupling for which they are stable ) , while enhancing the stability of quasielectron ( anti-)skyrmions . ( iii ) for @xmath5 and higher , we find that a sufficiently realistic model of the effective electron - electron interaction , as modified by the finite thickness of the electronic wavefunctions , is necessary to obtain reliable results . use of a simple potential due to zhang and das sarma @xcite grossly overestimates the stability of @xmath5 skyrmions ; more realistic potentials @xcite allow skyrmions only for very small zeeman couplings . ( iv ) screening and landau level mixing for @xmath5 and @xmath16 tend to lower the energy of spin - polarized quasiparticles more than that of skyrmions , making the latter even less stable . ( v ) the results of ref . can not be understood solely on the basis of hartree - fock states for skyrmions . the remainder of this article is organized as follows . in section ii below , we discuss the method used to allow screening and landau level mixing to be included in our calculations . section iii gives details of our results , and we conclude with a summary of our findings in section iv . most previous hartree - fock studies of skyrmions have relied on landau level representation of the single particle states @xcite . we choose instead to construct the wavefunctions in real space . this enables us to include in the model landau level mixing occurring in weak magnetic fields , without having to expand over the large number of landau levels necessary in the former approach . we thus trade the calculational convenience of working with the functions given in closed analytic form ( landau levels ) for a closer description of the single - particle states by representing them on a real - space grid . in this calculation we aim to model all the participating particles , including the ones in the filled levels . our hartree - fock wavefunction is a slater determinant composed of single - particle states which have @xmath17 as a good quantum number @xcite but whose radial form is to be determined self - consistently : @xmath18 e^{im\theta } .\end{aligned}\ ] ] here @xmath19 and @xmath20 are polar coordinates , @xmath21 is the angular momentum quantum number , and @xmath22 labels different states of the same @xmath21 . the sign @xmath23 corresponds to two families of solutions , @xmath24 for antiskyrmion ( or quasielectron spin structured solution ) and @xmath25 for skyrmion ( quasihole ) . in very strong magnetic fields , the functions @xmath26 and @xmath27 take the form expected for landau level states . when the strength of the magnetic field is lowered to bring the ratio of cyclotron and coulomb energy scales close to 1 , the form of the radial part relaxes toward some modified form , as dictated by the interactions in the system . using the trial form of the wavefunction ( [ trial ] ) , the many - body schrdinger equation with the hamiltonian @xmath28 \nonumber\\ & + & \frac { 1}{2 } \sum_{\sigma \sigma ' } \int d^2r d^2r ' \psi_\sigma^\dagger ( \vec r)\psi_{\sigma ' } ( \vec r ' ) v(\vec r - \vec r ' ) \psi_{\sigma'}^\dagger ( \vec r')\psi_\sigma ( \vec r)\end{aligned}\ ] ] ( where @xmath29 denotes spin and @xmath30 the coulomb interaction ) , upon variation with respect to the functions @xmath31 and @xmath32 , gives a system of mean - field single - particle equations : @xmath33 f_{im}(r)\nonumber \\ & -&\int _ { 0}^{\infty } r'\,dr'\sum _ { m'}v^{ex}_{m - m'}(r , r ' ) \rho _ { m'}^{\uparrow \uparrow } ( r',r ) f_{im}(r ' ) \nonumber \\ & -&\int _ { 0}^{\infty } r'\,dr'\sum _ { m'}v^{ex}_{m - m'}(r , r ' ) \rho _ { m'}^{\downarrow \uparrow } ( r',r ) g_{im\pm 1}(r ' ) \nonumber \\ & = & \epsilon_i f_{im}(r)\end{aligned}\ ] ] together with the analogous equation for the function @xmath27 . here is a dictionary of the notation accompanying eq . [ master ] : the operator @xmath34 is given by @xmath35 \nonumber \\ & - & m\frac { \hbar \omega _ c}{2 } + \frac { ( m^*)^2}{4}\frac { \omega _ c^2r^2}{2m^*}\end{aligned}\ ] ] with @xmath36 , @xmath37 the magnitude of the external magnetic field , and @xmath38 the effective mass of the electron . @xmath0 is the land g - factor , and @xmath39 is the bohr magneton . @xmath40 s denote generalized densities : @xmath41,\end{aligned}\ ] ] and @xmath42 is the uniform background density . @xmath43 and @xmath44 are the following integrals of the coulomb potential over the azimuthal variable : @xmath45 and , finally , @xmath46 stands for the single - particle hartree - fock energy . the finite width of the sample is modeled using the form of the in - plane potential due to cooper @xcite : we replace the coulomb interaction @xmath47 in the eq . ( [ potentials ] ) by @xmath48 the symbol @xmath49 denotes the width of system in the direction perpendicular do the plane of the system , and @xmath50 , @xmath51 are coordinates in that direction . to handle the boundaries of the system , we assume the electron states with angular momentum @xmath52 have the ferromagnetic groundstate form ( i.e. landau levels with well defined spin ) . the states with @xmath53 are explicitly included in the calculation . for @xmath6 not too large we find it is sufficient to allow variations of the states with @xmath21 of up to 30 for @xmath54 and up to 50 for @xmath55 . in practice , including boundary electrons from the states with @xmath21 between 31 and 100 ( @xmath54 ) and between 51 and 120 ( @xmath55 ) describes the effect of the system edge with precision matching the rest of the calculation . understanding eq . ( [ master ] ) as a system of coupled eigenproblems , we look for the self - consistent single particle solutions . ( discretization will turn each eigenequation into a matrix diagonalization problem which can be handled using standard methods . ) the results we thus obtain will be largely presented as comparisons between energies of the spin - polarized quasiparticle and energies of the corresponding skyrmion . to assess the energy of the skyrmion in the region of parameters where it is not stable , we add to the hamiltonian a term of the form @xmath56 , @xmath57 being the spin operator , and @xmath58 a tunable parameter . this term favors a state with total spin @xmath59 , but is insensitive to the detailed form of the wavefunctions . this allows the variational scheme to pick out the lowest energy slater determinant of the form given in eq.([trial ] ) within the space of states with the same fixed value of @xmath6 . based on the calculation described in the previous section , we present some of the results the method allows us to obtain ; we focus mainly on the singly charged excitations in the first three landau levels . consideration of higher landau levels is also possible , but computation of the potential lookup tables becomes prohibitively expensive , and , as the results so far indicate , leads to no new insight . in the following we shall take the unit of energy to be @xmath60 , and the unitless zeeman splitting to be @xmath61 , where @xmath62 is the electron charge , @xmath15 is the dielectric constant of the host material , @xmath9 is the magnetic length in the field @xmath37 and @xmath63 is the bohr magneton . in fig . [ lowest ] we show the energy difference between the spin - polarized quasiparticle and the skyrmion of size @xmath6 , @xmath64 . this quantity is a pure interaction energy ( i.e. zeeman energy is not included ) , and represents the energy gained or lost in deforming a spin - polarized quasiparticle into a skyrmion when zeeman coupling is absent . of particular importance is the slope ( negative slopes indicate that skyrmion is stable for some value of @xmath0 ) and the curvature ( concave curves will support small - sized skyrmions ) . for concave curves the largest zeeman splitting that supports skyrmions is the negative of the initial slope of the curve @xcite . for large cyclotron energies our results are essentially identical to those obtained using the single landau level method @xcite . note the quasielectron and quasihole excitations are precisely degenerate in this case , due to particle - hole symmetry . for smaller values of @xmath65 , the two curves split ; surprisingly , the quasihole skyrmion is suppressed by landau level mixing , whereas the quasielectron skyrmion is enhanced . ( the former result is in agreement with ref . ) . = 2.5 in the energy gaps ( fig . [ gap ] ) which result from creation of skyrmion - antiskyrmion pairs when landau level mixing and finite thickness corrections are included are considerably smaller than what is found for two - dimensional layers and no mixing @xcite . however , the resulting energies are still almost a factor of two larger than what is found in experiment @xcite . the discrepancy is likely to be due to disorder . = 2.5 in figs . ( [ first ] ) and ( [ second ] ) present analogous results for @xmath5 and @xmath16 . note the considerably smaller energy scales in these figures , indicating that skyrmions can only be stable ( if ever ) for small values of @xmath66 @xcite . it is apparent that the introduction of landau level mixing and screening destabilizes the skyrmion . evidently , spin - polarized particles are better able to take advantage of the admixture of higher landau levels than skyrmions . = 2.5 in = 2.5 in quasiparticle energies depend on the well width , as illustrated in figs . [ first173 ] and [ first100 ] . as expected @xcite , we find that for narrower wells the difference in energy is less favorable for the skyrmion ( fig . [ first100 ] ) . note that the width used in fig . [ first173 ] is close to an experimentally reported value @xcite . = 2.5 in = 2.5 in it is worth remarking at this juncture that a reasonably realistic model of the electron - electron interaction with finite sample thickness corrections is needed to obtain qualitatively correct results . [ sarma ] shows that the use of a simpler model potential ( zhang and das sarma @xcite ) @xmath67 which is commonly used in studying quantum hall systems ( see for example refs . and ) gives substantially different results then those presented above ( fig . [ first ] ) . = 2.5 in the principal difference between @xmath68 and @xmath69 is the behavior at small r ; the former diverges logarithmically , whereas the latter is regular . other divergent potentials give results consistent with figs . [ first ] and [ second ] ; it is likely that the oversimplified behavior at short distances is responsible for the poor performance of @xmath69 in this problem . based on results in figs . [ lowest ] , [ first ] , and [ second ] we can construct the phase diagrams of skyrmion stability for the filling factors of @xmath70 and @xmath71 . large @xmath72 and small @xmath73 is the region favoring the spin structured excitations . we see that the region `` shrinks '' as one moves to the higher filling factors . also , according to this calculation the breaking of symmetry between the quasihole and quasielectron excitations upon lowering the cyclotron energy is quite spectacular in the lowest landau level , whereas it plays no significant role in the higher ones . = 2.5 in in the paper by song _ et al_. @xcite the reported excitation at the parameter values of @xmath74 , and @xmath75 falls well outside the boundary expected from the hartree - fock calculation . it is tempting to speculate that inclusion of impurity effects can stabilize the skyrmions at values of @xmath0 bigger than allowed in a pure sample . to test this idea we can include a simple model of an impurity in our calculation : a point charge ( impurity ) at a distance @xmath76 above the central plane of the system . it is replaced by an effective non - uniform charge density in the plane producing the same potential , @xmath77 the effective density can be found to be @xmath78 for @xmath79 . results with and without such an impurity are illustrated in fig . [ imp2nu1 ] . as may be seen , the impurity favors spin - polarized quasiparticle over skyrmion . a similar result is expected for a short - range ( e.g. delta - function ) impurity potential . apparently the simplest models of disorder are not likely to explain the results of ref . . it is probable that more complicated impurities ( e.g. multiply charged or magnetic ones ) could stabilize the small - spin skyrmions at @xmath54 . however , in the absence of data indicating such types of disorder in real samples , an investigation of this phenomenon is left for future work . = 2.5 in = 2.5 in in this paper we have presented a real - space method for computing hartree - fock states and energies of two - dimensional systems in magnetic fields , appropriate for systems with circular symmetry in which landau level mixing may be important . the method was applied to compute the effects of landau level mixing and screening on skyrmion states . it was found that in most cases these tend to destabilize skyrmions , with a notable exception occurring for the case of the quasielectron ( antiskyrmion ) around @xmath4 . the calculations indicate that hartree - fock states can not account for the results of ref . ( reporting skyrmions at @xmath5 ) . this is in agreement with earlier studies where landau level mixing and screening were not included . this work was supported by nsf grant nos.dmr98-70681 and phy94 - 07194 , the research corporation , and the center for computational sciences of the university of kentucky . lee and c.l . kane , phys . lett . * 64 * , 1313 ( 1990 ) . s.l.sondhi , a.karlhede , s.a.kivelson , and e.h.rezayi , phys . rev . b * 47 * , 16419 , ( 1993 ) . h.a.fertig , l.brey , r.ct , and a.h.macdonald , phys . b * 50 * , 11018 ( 1994 ) . t.h.r . skyrme , proc . roy . soc . * a262 * , 233 ( 1961 ) .

we present a hartree - fock study that incorporates the effects of landau level mixing and screening due to filled levels into the computation of energies and states of quasiparticles in quantum hall ferromagnets . we use it to construct a phase diagram for skyrmion stability as a function of magnetic field and zeeman coupling strengths . we find that landau level mixing tends to favor spin - polarized quasiparticles , while finite thickness corrections favor skyrmions . our studies show that skyrmion stability in high landau levels is very sensitive to the way in which electron - electron interactions are modified by finite thickness , and indicate that it is crucial to use models with realistic short distance behavior to get qualitatively correct results . we find that recent experimental evidence for skyrmions in higher landau levels can not be explained within our model . 0.08 in

You are an expert at summarizing long articles. Proceed to summarize the following text: the present work deals with the generalized surface quasi - geostrophic equation ( gsqg ) arising in fluid dynamics and which describes the evolution of the potential temperature @xmath1 by the transport equation : @xmath2 here @xmath3 refers to the velocity field , @xmath4 and @xmath5 is a real parameter taken in @xmath60,2[$ ] . the singular operator @xmath7 is of convolution type and defined by , @xmath8 with @xmath9 where @xmath10 stands for the gamma function . this model was proposed by crdoba et al . in @xcite as an interpolation between euler equations and the surface quasi - geostrophic model ( sqg ) corresponding to @xmath11 and @xmath12 , respectively . the sqg equation was used by juckes @xcite and held et al . @xcite to describe the atmosphere circulation near the tropopause . it was also used by lapeyre and klein @xcite to track the ocean dynamics in the upper layers . we note that there is a strong mathematical and physical analogy with the three - dimensional incompressible euler equations ; see @xcite for details . in the last few years there has been a growing interest in the mathematical study of these active scalar equations . special attention has been paid to the local well - posedness of classical solutions which can be performed in various functional spaces . for instance , this was implemented in the framework of sobolev spaces @xcite by using the commutator theory . wether or not these solutions are global in time is an open problem except for euler equations @xmath11 . the second restriction with the gsqg equation concerns the construction of yudovich solutions known to exist globally in time for euler equations @xcite which are not at all clear even locally in time . the main difficulty is due to the velocity which is in general singular and scales below the lipschitz class . nonetheless one can say more about this issue for some special class of concentrated vortices . more precisely , when the initial datum has a vortex patch structure , that is , @xmath13 is the characteristic function of a bounded simply connected smooth domain @xmath14 , then there is a unique local solution in the patch form @xmath15 in this case , the boundary motion of the domain @xmath16 is described by the contour dynamics formulation ; see the papers @xcite . the global persistence of the boundary regularity is only known for @xmath11 according to chemin s result @xcite ; for another proof see the paper of bertozzi and constantin @xcite . notice that for @xmath17 the numerical experiments carried out in @xcite provide strong evidence for the singularity formation in finite time . let us mention that the contour dynamics equation remains locally well - posed when the domain of the initial patch is assumed to be multi - connected meaning that the boundary is composed with finite number of disjoint smooth jordan curves . the main concern of this work is to explore analytically and numerically some special vortex patches called v - states ; they correspond to patches which do not change their shapes during the motion . the emphasis will be put on the v - states subject to uniform rotation around their center of mass , that is , @xmath18 , where @xmath19 stands for the planar rotation with center @xmath20 and angle @xmath21 the parameter @xmath22 is called the angular velocity of the rotating domain . along the chapter we call these structures rotating patches or simply v - states . their existence is of great interest for at least two reasons : first they provide non trivial initial data with global existence , and second this might explain the emergence of some ordered structures in the geophysical flows . this study has been conducted first for the two - dimensional euler equations ( @xmath11 ) a long time ago and a number of analytical and numerical studies are known in the literature . the first result in this setting goes back to kirchhoff @xcite who discovered that an ellipse of semi - axes @xmath23 and @xmath24 rotates uniformly with the angular velocity @xmath25 ; see for instance the references and @xcite . till now this is the only known explicit v - states ; however the existence of implicit examples was established about one century later . in fact , deem and zabusky @xcite gave numerical evidence of the existence of the v - states with @xmath26-fold symmetry for each integer @xmath27 ; remark that the case @xmath28 coincides with kirchhoff s ellipses . to fix the terminology , a planar domain is said @xmath26-fold symmetric if it has the same group invariance of a regular polygon with @xmath26 sides . note that at each frequency @xmath26 these v - states can be seen as a continuous deformation of the disc with respect to to the angular velocity . an analytical proof of this fact was given few years later by burbea in @xcite . his approach consists in writing a stationary problem in the frame of the patch with the conformal mapping of the domain and to look for the non trivial solutions by using the technique of the bifurcation theory . quite recently , in burbea s approach was revisited with more details and explanations . the boundary regularity of the v - states was also studied and it was shown to be of class @xmath29 and convex close to the disc . we mention that explicit vortex solutions similar to the ellipses are discovered in the literature for the incompressible euler equations in the presence of an external shear flow ; see for instance @xcite . a general review about vortex dynamics can be found in the with regard to the existence of the simply connected v - states for the ( gsqg ) it has been discussed very recently in the papers @xcite . in @xcite , it was shown that the ellipses can not rotate for any @xmath30 and to the authors best knowledge no explicit example is known in the literature . lately , in @xcite the last two authors proved the analogous of burbea s result and showed the existence of the @xmath26-folds rotating patches for @xmath310,1[$ ] . in addition , the bifurcation from the unit disc occurs at the angular velocities , @xmath32 where @xmath10 denotes the usual gamma function.the remaining case @xmath33 has been explored and solved by castro , crdoba and gmez - serrano in @xcite . they also show that the v - states are @xmath29 and convex close to the discs . to complete these works we discussed in this work ( which is forwarded in ) some numerical experiments concerning these v - states and their limiting structures when we go to the end of each branch ; new behaviors will be observed compared to the numerical experiments achieved for euler we want in this chapter to learn more about the v - states but with different topological structure compared to the preceding discussion . more precisely , we propose to scrutinize rotating patches with only one hole , also called doubly connected v - states . recall that a patch @xmath34 is said to be doubly connected if the domain @xmath35 , with @xmath36 and @xmath37 being two simply connected bounded domains satisfying @xmath38 this structure is preserved for euler system globally in time but known to be for short time when @xmath00,1[$ ] see @xcite . we notice that compared to the simply connected case the boundaries evolve through extra nonlinear terms coming from the interaction between the boundaries and therefore the existence of the v - states is relatively more complicate to analyze . this problem is not well studied from the analytical point of view and recent progress has been made for euler equations in the papers @xcite . in @xcite , the authers proved the existence of explicit v - states similar to kirchhoff ellipses seems to be out of reach . indeed , it was stated that if one of the boundaries of the v - state is a circle then necessarily the other one should be also a circle . moreover , if the inner curve is an ellipse then there is no rotation at all . another closely related subject is to deal with some vortex magnitude @xmath39 inside the domain @xmath37 and try to find explicit rotating patches . this was done by flierl and polvani @xcite who proved that confocal ellipses rotate uniformly provided some compatibility relations are satisfied between the parameter @xmath39 and the semi - axes of the ellipses . we note that another approach based upon complex analysis tools with a complete discussion can be found now , from the equations we may easily conclude that the annulus is a stationary doubly connected patch , and therefore it rotates with any angular velocity @xmath22 . from this obvious fact , one can wonder wether or not the bifurcation to nontrivial v - states still happens as for the simply connected case . this has been recently investigated in @xcite for euler equations following basically burbea s approach but with more involved calculations . it was shown that for @xmath40 and @xmath26 being an integer satisfying the inequality @xmath41 then there exist two curves of non - annular @xmath26-fold doubly connected patches bifurcating from the annulus @xmath42 at different eigenvalues @xmath43 given explicitly by the formula @xmath44 ^ 2-b^{2m}}.\ ] ] now we come to the main contribution of the current work . we propose to study the doubly connected v - states for the gsqg model when @xmath00,1[$ ] . before stating our result we need to make some notation . we define @xmath45 and @xmath46 where @xmath47 refers to bessel function of the first kind . our result reads as follows . [ main ] let @xmath48 and @xmath490,1[$ ] ; there exists @xmath50 with the following property : + for each @xmath51 there exists two curves of @xmath26-fold doubly connected @xmath52-states that bifurcate from the annulus @xmath53 at the angular velocities @xmath54 with @xmath55 ^ 2 - 4b^2\lambda_{m}^2(b).\end{aligned}\ ] ] [ rmw1 ] 1 . the number @xmath56 is the smallest integer such that @xmath57 this restriction appears in the spectral study of the linearized operator and gives only sufficient condition for the existence of the v - states . 2 . as we shall see later in lemma [ lem1 ] , for @xmath11 we find the result of euler equations established in @xcite and the condition @xmath58 is in accordance with that given by @xmath59 3 . we can check by using the strict monotonicity of @xmath60 that for any @xmath61 @xmath62 consequently the corresponding bifurcating curves generate close to the annulus non trivial clockwise doubly connected v - states . this fact is completely new compared to what we know for euler equations or for the simply connected case where the bifurcation occurs at positive angular velocities . the numerical experiments discussed in section @xmath63 reveal the existence of non radial stationary patches for the generalized quasi - geostrophic equations and it would be very interesting to establish this fact analytically . in a connected subject , we point out that the last author has shown quite recently in @xcite that for euler equations clockwise convex v - states reduce to the discs . now we shall sketch the proof of theorem [ main ] which is mainly based upon the bifurcation theory via crandall - rabinowitz s theorem . the first step is to write down the analytical equations of the boundaries of the v - states . this can be done for example through the conformal parametrization of the domains @xmath36 and @xmath37 : we denote by @xmath64 the conformal mappings possessing the following structure , @xmath65 we assume in addition that the fourier coefficients are real which means that we look only for the v - states which are symmetric with respect to the real axis . moreover using the subordination principle we deduce that @xmath660,1[$ ] ; the parameter @xmath24 coincides with the small radius of the annulus that we slightly perturb . as we shall see later in section [ sec12 ] , the conformal mappings are subject to two coupled nonlinear equations defined as follows : for @xmath67 @xmath68 with @xmath69 in order to apply the bifurcation theory we should understand the structure of the linearized operator around the trivial solution @xmath70 , corresponding to the annulus with radii @xmath24 and @xmath71 , and identify the range of @xmath72 where this operator has a one - dimensional kernel . the computations of the linear operator @xmath73 with @xmath74 in terms of its fourier coefficients are long and tricky . they are connected to the hypergeometric functions @xmath75 simply denoted by @xmath76 throughout this chapter . to find compact formula we use at several steps some algebraic identities described by the contiguous function relations - . similarly to the euler equations @xcite the linearized operator acts as a matrix fourier multiplier . more precisely , for @xmath77 we obtain the formula , @xmath78 where the matrix @xmath79 is given for @xmath80 by @xmath81 therefore the values of @xmath22 associated to non trivial kernels are the solutions of a second - degree polynomial , @xmath82 this can be solved when the discriminant @xmath83 introduced in theorem [ main ] is positive . the computations of the dimension of the kernel are more complicate than the cases raised before in the references @xcite . the matter reduces to count the following discrete set @xmath84 note that in @xcite this set has only one element and therefore the kernel is one - dimensional . this follows from the monotonicity of the `` nonlinear eigenvalues '' sequence @xmath85 which is not very hard to get due to the explicit polynomial structure of the coefficients of the analogous polynomial to . unfortunately , in the current situation this structure is broken because the matrix coefficients of @xmath86 are related to bessel functions . therefore the monotonicity of the eigenvalues is more subtle and will require more refined analysis . this subject will be discussed later with ample details in the subsection [ subsec12 ] . to achieve the spectral study and check the complete assumptions of crandall - rabinowitz s theorem it remains to prove the transversality assumption and check that the image is of co - dimension one . this will be done in section [ sec45 ] in a straightforward way and without serious difficulties . we also mention that the transversality assumption is obtained only when the discriminant @xmath87 meaning that there is no crossing roots for the equation . the proof of the bifurcation will be achieved in section [ sec45 ] . next , we shall make few comments about the statement of the main theorem . @xmath88 for the sqg equation corresponding to @xmath12 the situation is more delicate due to some logarithmic loss . the simply connected case has been achieved recently in @xcite by using hilbert spaces where we take into account this loss . the same approach could lead to the existence of the doubly - connected v - states for the ( sqg ) equation . for the spectral study , the linearized operator can be obtained as a limit of when @xmath5 goes to @xmath89 more precisely , we get @xmath90 where the matrix @xmath79 is given for @xmath80 by @xmath91 with @xmath92 @xmath93 the boundary of the v - states belongs to hlder space @xmath94 . for euler equations corresponding to @xmath11 , we get better result in the simply connected geometry as it was shown the boundary is @xmath29 and convex when the v - states are close to the circle . the proof in this particular case uses in a deep way the algebraic structure of the kernel according to some recurrence formulae . the extension of this result to @xmath00,2[$ ] was done in @xcite . we expect that the latter approach could be also adapted to the model for the doubly connected case . @xmath95 in the setting of the vortex patches the global existence with smooth boundaries is not known for @xmath310,2[$ ] . the simply connected v - states discussed in @xcite offer a first class of global solutions which are periodic in time . we find here a second class of global solutions which are the doubly connected v - states . the remainder of the chapter is organized as follows . in the next section , we shall write down the boundary equations through the conformal parametrization . in we shall introduce and review some background material on the bifurcation theory and gauss hypergeometric functions . in section @xmath96 , we will study the regularity of the nonlinear functionals involved in the boundary equations . in section @xmath97 we conduct the spectral study and formulate the suitable assumptions to get a fredholm operator of zero index . in section @xmath98 we prove theorem [ main ] . finally , the last section will be devoted to some numerical experiments dealing with the simply and doubly connected v - states . * notation . * we need to fix some notation that will be frequently used along this chapter . 1 . we denote by c any positive constant that may change from line to line . 2 . for any positive real numbers @xmath99 and @xmath100 , the notation @xmath101 means that there exists a positive constant @xmath102 independent of @xmath99 and @xmath100 such that @xmath103 . we denote by @xmath104 the unit disc . its boundary , the unit circle , is denoted by @xmath105 . 4 . let @xmath106 be a continuous function . we define its mean value by , @xmath107 where @xmath108 stands for the complex integration . let @xmath109 and @xmath110 be two normed spaces . we denote by @xmath111 the space of all continuous linear maps @xmath112 endowed with its usual strong topology . 6 . for a linear operator @xmath113 we denote by @xmath114 and @xmath115 the kernel and the range of @xmath116 , respectively . 7 . if @xmath110 is a vector space and @xmath117 is a subspace , then @xmath118 denotes the quotient space . before proceeding further with the consideration of the v - states , we shall recall riemann mapping theorem which is one of the most important results in complex analysis . to restate this result we need to recall the definition of _ simply connected _ domains . let @xmath119 denote the riemann sphere . we say that a domain @xmath120 is _ simply connected _ if the set @xmath121 is connected . _ riemann mapping theorem . _ let @xmath104 denote the unit open ball and @xmath122 be a simply connected bounded domain . then there is a unique bi - holomorphic map called also conformal map , @xmath123 taking the form @xmath124 in this theorem the regularity of the boundary has no effect regarding the existence of the conformal mapping but it contributes in the boundary behavior of the conformal mapping , see for instance @xcite . here , we shall recall the following result . _ kellogg - warschawski s theorem . _ it can be found in @xcite or in ( * ? ? ? * theorem 3.6 ) . it asserts that if the conformal map @xmath125 has a continuous extension to @xmath126 which is of with @xmath127 and @xmath128 , then the boundary @xmath129 is a jordan curve of class @xmath130 next , we shall write down the equation governing the boundary of the doubly connected v - states . let @xmath35 be a doubly connected domain , that is , @xmath36 and @xmath37 are two simply connected domains with @xmath131 . denote by @xmath132 and @xmath133 their boundaries , respectively . consider the parametrization by the conformal mapping : @xmath134 satisfying @xmath135 and @xmath136 now assume that @xmath137 is a rotating patch for the model then according to @xcite the boundary equations are given by @xmath138 where @xmath139 denotes a tangent vector to the boundary @xmath140 at the point @xmath141 we shall now rewrite the equations by using the conformal parametrizations @xmath142 and @xmath143 . first remark that for @xmath144 a tangent vector on the boundary @xmath145 at the point @xmath146 is given by @xmath147 inserting this into the equation and using the change of variables @xmath148 give @xmath149 with @xmath150 and @xmath151 . we shall introduce the functionals @xmath152 then equations of the v - states become , @xmath153 now it is easy to ascertain that the annulus is a rotating patch for any @xmath154 . indeed , replacing @xmath142 and @xmath143 in by @xmath155 and @xmath156 , respectively , we get @xmath157 using the change of variables @xmath158 in the two preceding integrals we find @xmath159 note that each integral in the right side is real since , @xmath160,\quad\overline{\mathop{{\fint}}_\mathbb{t}\frac{d\zeta}{\vert 1-a\zeta\vert^\alpha } } & = & -\mathop{{\fint}}_\mathbb{t}\frac{d\overline{\zeta}}{\vert 1-a\overline{\zeta}\vert^\alpha}\\ & = & \mathop{{\fint}}_\mathbb{t}\frac{d\xi}{\vert 1-a\xi\vert^\alpha}\cdot\end{aligned}\ ] ] therefore we obtain , @xmath161 arguing similarly for the second component @xmath162 we get for any @xmath163 @xmath164 which is the desired result . in this section we shall recall in the first part some simple facts about hlder spaces on the unit circle @xmath105 . in the second part we state crandall - rabinowitz s theorem which is a crucial tool in the proof of theorem [ main ] . we shall also recall some important properties of the hypergeometric functions which appear in a natural way in the spectral study of the linearized operator . the last part is devoted to the computations of some integrals used later in the spectral study . throughout this chapter it is more convenient to think of @xmath165-periodic function @xmath166 as a function of the complex variable @xmath167 . to be more precise , let @xmath168 , be a continuous function , then it can be assimilated to a @xmath169 periodic function @xmath170 via the relation @xmath171 hence when @xmath172 is smooth enough we get @xmath173 because @xmath174 and @xmath175 differ only by a smooth factor with modulus one we shall in the sequel work with @xmath174 instead of @xmath175 which appears more suitable in the computations . + moreover , if @xmath172 has real fourier coefficients and is of class @xmath176 then we can easily check that @xmath177 now we shall introduce hlder spaces on the unit circle @xmath105 . let @xmath178 . we denote by @xmath179 the space of continuous functions @xmath172 such that @xmath180 for any integer @xmath181 , the space @xmath182 stands for the set of functions @xmath172 of class @xmath183 whose @xmath184th order derivatives are hlder continuous with exponent @xmath185 . it is equipped with the usual norm , @xmath186 recall that the lipschitz semi - norm is defined by , @xmath187 now we list some classical properties that will be used later at many places . 1 . for @xmath1880,1[$ ] the space @xmath182 is an algebra . 2 . for @xmath189 and @xmath190 we have the convolution law , @xmath191 we intend now to recall crandall - rabinowitz s theorem which is a basic tool of the bifurcation theory and will be useful in the proof of let @xmath192 be a continuous function with @xmath109 and @xmath110 being two banach spaces . assume that @xmath193 for any @xmath194 belonging in a non empty interval @xmath195 whether close to a trivial solution @xmath196 we can find a branch of non trivial solutions of the equation @xmath197 is the main concern of the bifurcation theory . if this happens we say that we have a bifurcation at the point @xmath196 . we shall restrict ourselves here to the classical result of crandall and rabinowitz @xcite . for more general results we refer the reader to the book of kielhfer @xcite . [ c - r ] let @xmath198 be two banach spaces , @xmath52 a neighborhood of @xmath199 in @xmath109 and let @xmath200 with the following properties : 1 . @xmath201 for any @xmath202 . 2 . the partial derivatives @xmath203 , @xmath204 and @xmath205 exist and are continuous . 3 . @xmath206 and @xmath207 are one - dimensional . 4 . _ transversality assumption _ : @xmath208 , where @xmath209 if @xmath210 is any complement of @xmath206 in @xmath109 , then there is a neighborhood @xmath211 of @xmath212 in @xmath213 , an interval @xmath214 , and continuous functions @xmath215 , @xmath216 such that @xmath217 , @xmath218 and @xmath219 we shall give a short introduction on the gauss hypergeometric functions and discuss some of their basic properties . the formulae listed below will be crucial in the computations of the linearized operator associated to the v - states equations . recall that for any real numbers @xmath220 the hypergeometric function @xmath221 is defined on the open unit disc @xmath104 by the power series @xmath222 here , @xmath223 is the pochhammer symbol defined by , @xmath224 it is obvious that @xmath225 for a future use we recall an integral representation of the hypergeometric function , for instance see @xcite . assume that @xmath226 then @xmath227 the function @xmath228 refers to the gamma function which is the analytic continuation to the negative half plane of the usual gamma function defined on the positive by the integral representation : @xmath229 it satisfies the relation @xmath230 from this we deduce the identities @xmath231 provided all the quantities in the right terms are well - defined . later we need the following values , @xmath232 another useful identity is the euler s reflection formula , @xmath233 now we shall introduce the digamma function which is nothing but the logarithmic derivative of the gamma function and often denoted by @xmath234 . it is given by @xmath235 the following identity is classical , @xmath236 when @xmath237 then it can be shown that the hypergeometric series is absolutely convergent on the closed unit disc and one has the expression,@xmath238 the proof can be found in @xcite , now recall the kummer s quadratic transformation @xmath239 next we recall some contiguous function relations of the hypergeometric series , see @xcite . @xmath240 @xmath241 @xmath242 @xmath243 @xmath244 @xmath245 @xmath246 now we close this discussion with recalling bessel function @xmath47 of the first kind of index @xmath127 and review some important identities . it is defined in the full space @xmath247 by the power series @xmath248 the following identity called sonine - schafheitlin s formula will be very useful later . @xmath249 provided that @xmath250 and that the integral is convergent . a detailed proof of this result can be found in @xcite . the main goal of this paragraph is to compute explicitly some integrals that will appear later in the spectral study . [ lem ] let @xmath251 and @xmath127 . then for any @xmath144 we have the following formulae : @xmath252 @xmath253\notag.\end{aligned}\ ] ] @xmath254\notag.\qquad\qquad\end{aligned}\ ] ] @xmath255.\notag\end{aligned}\ ] ] @xmath256.\notag\qquad\quad\end{aligned}\ ] ] to prove the first identity we use successively the change of variables @xmath257 @xmath258 again by the change of variables @xmath259 one gets @xmath260 since @xmath261 then we can use the taylor series @xmath262 consequently , we get @xmath263 we shall now recall the following identity , see for instance @xcite and @xcite , @xmath264 as it was pointed before the gamma function has no real zeros but simple poles located at @xmath265 and therefore the function @xmath266 admits an analytic continuation on @xmath267 apply this formula with @xmath268 and @xmath269 yields , @xmath270 hence , @xmath271 we shall use legendre s duplication formula , @xmath272 which gives @xmath273 therefore using the identity and @xmath274 we find @xmath275 from the elementary fact @xmath276 one deuces @xmath277 by definition of the hypergeometric series we conclude that @xmath278 using kummer s quadratic transformation the last identity becomes @xmath279 this completes the proof of . we intend now to compute the second integral . to this end we use the change of variables as before , @xmath280 where @xmath281 it follows from the identity that @xmath282.\end{aligned}\ ] ] then , in view of the formula one gets @xmath283 replacing @xmath284 by its expression in we conclude that @xmath285.\end{aligned}\ ] ] we shall now compute the integral . we write @xmath286 using the identity , @xmath287 can be rewritten as @xmath288 then , in view of the formula we get @xmath289 to compute @xmath290 we first observe that by the change of variables @xmath291 we find @xmath292 which yields in turn @xmath293 consequently , @xmath294 .\end{aligned}\ ] ] concerning the integral we use a change of variable as before in order to get @xmath295 with @xmath296 observe that @xmath297 then , it follows from the formula that @xmath298.\end{aligned}\ ] ] using once again the identity implies @xmath299 plugging the latter expression of @xmath300 into yields @xmath301.\end{aligned}\ ] ] we shall now move to the computation of the last integral , @xmath302 from the identity we may write @xmath303 therefore by the formula we obtain @xmath304 to compute @xmath305 we write through a change of variables , @xmath306 which implies in view of @xmath307 hence we find @xmath308\end{aligned}\ ] ] and therefore the proof of the lemma is now complete . this section is devoted to the regularity study of the nonlinear functional @xmath309 introduced and which defines the v - states equations . we shall check the regularity assumptions required by crandall - rabinowitz s theorem . the computations are vey heavy and can be done in a straightforward way without new difficulties compared to the simply connected case treated in the paper @xcite . many of the details may be found in that work and will not be reiterated here . therefore for the sake of concise presentation we shall study the new terms involving the interaction between the boundaries . however , regarding the self - induced terms we only recall the results from the paper @xcite . to begin with , we introduce the function spaces that we shall use . we set , @xmath310 with @xmath311 and @xmath312 for @xmath40 , let @xmath52 denote the product @xmath313 , where @xmath314 is the open ball of @xmath109 with center @xmath199 and radius @xmath315 . we note that this choice is done in order to guarantee that @xmath316 and @xmath317 are conformal for @xmath318 and to prevent the intersection between the curves @xmath319 and @xmath320 which represent the boundaries of the v - states . + now recall from the form of the functional @xmath321 , @xmath322 where @xmath323 is defined by @xmath324 we shall rewrite @xmath325 as follows , @xmath326 with @xmath327 and @xmath328 usually with the notation @xmath316 , @xmath329 . + we propose to prove the following result concerning the regularity of @xmath330 [ reg ] the following holds true . 1 . @xmath331 is well - defined . @xmath331 is of class @xmath332 3 . the partial derivative @xmath333 exists and is continuous . notice that the terms @xmath334 appears modulo the sign of @xmath335 in the simply connected case discussed in the paper @xcite and all the computations were done there . therefore we shall restrict ourselves to recalling just the results of those computations : 1 . @xmath336 is well - defined . @xmath336 is of class @xmath176 . moreover the differential @xmath337 is given for @xmath338 by @xmath339\phi_j^\prime(\tau)}{\vert \phi_j(w)-\phi_j(\tau)\vert^{\alpha+2}}d\tau\bigg\}.\end{aligned}\ ] ] in addition , the partial derivative @xmath340 exists and is continuous . it is given by the formula , @xmath341 if we prove the regularity properties for the second part @xmath342 then we can easily deduce that @xmath343 therefore all the regularity assumptions are satisfied for the terms @xmath344 and to complete the proof of the proposition we should check these assumptions for @xmath342 . more precisely , we shall prove that @xmath345 is well - defined and is of class @xmath332 @xmath346 first , we shall prove that for @xmath347 we have @xmath348 . because the space @xmath349 is an algebra the problem reduces to show that for @xmath350 , the function @xmath351 belongs to @xmath349 . this can be deduced easily from the next general result . let @xmath352 and @xmath316 , @xmath329 and define the operator @xmath353 then @xmath354 the proof of this inequality will be done in a straightforward way since as we shall see the kernel is not singular . this is due to the fact that the inner and the outer boundaries do not intersect . indeed , for all @xmath355 we can write@xmath356 the same result remains true if we change @xmath357 by @xmath358 and therefore we get for @xmath350 @xmath359 it follows that @xmath360 which implies that @xmath361 next take @xmath362 . using the inequality gives @xmath363 where we have used in the last estimate the following inequality : for @xmath364 there exists a constant @xmath365 such that @xmath366 finally , using the fact that @xmath367 one can conclude that @xmath368 which is the desired result . now applying to the operator @xmath323 we get @xmath369 to complete the proof of the first point we shall verify that the fourier coefficients of @xmath370 belong to @xmath371 . from the definition of the space @xmath372 in the mapping @xmath373 has real fourier coefficients and thus the fourier coefficients of @xmath374 are real too . since this property is stable under the conjugation and the multiplication the problem reduces to prove that the fourier coefficients of @xmath375 are real . for this last purpose , we take the conjugate and make a change of variables , @xmath376 this proves that the fourier coefficients of the functions @xmath377 are real and the proof of the first part @xmath378 is now achieved . the strategy to get that @xmath379 is of class @xmath176 on @xmath52 consists first in checking the existence of its gteaux derivative . second we show that the gteaux derivative is strongly continuous . this will ensure in the same time the existence of frchet derivative and its continuity . + the gteaux derivative of the function @xmath379 at @xmath380 in the direction @xmath381 is given by the formula @xmath382+\lim_{t\to 0}\frac{1}{t}\big[n_{j}(f_1,f_2+th_2)-n_{j}(f_1,f_2)\big],\end{aligned}\ ] ] where the limits are taken in the strong topology of @xmath110 . thus we shall first prove the existence of these limit in the pointwise sense , that is for every point @xmath383 and after check that these limits exist in the strong topology of @xmath349 . let us first check for each point @xmath384 the existence of @xmath385 as a linear and bounded operator , that is , @xmath386 . with the notation @xmath316 and @xmath317 , one has @xmath387 we shall make use of the following identity : let @xmath388 , @xmath389 , @xmath390 and define the function @xmath391 which is smooth close to zero , then we have @xmath392 combining this formula with few easy computations yield @xmath393 \notag\\ \triangleq&-\frac{\alpha}{2}c_\alpha\big [ & & { \overline{h_j(w)}}{a}_{i}\big(\phi_j\big)(w)+{h_j(w)}{b}_{i}\big(\phi_j\big)(w)\big].\end{aligned}\ ] ] therefore , @xmath394\bigg\}.\end{aligned}\ ] ] using the algebra structure of @xmath349 combined with the estimate , we get @xmath395 it remains to estimate the terms @xmath396 and @xmath397 . for the first one , we get by virtue @xmath398 now let @xmath362 , then @xmath399 few easy computations show that @xmath400 concerning the last term we shall use the following inequality whose proof is classical.@xmath401 hence , we get @xmath402 inserting this in the estimate and using the inequality we find @xmath403 now by plugging the latter estimate into one gets @xmath404 which is the desired result . the estimate of the term @xmath405 can be done in a similar way by observing that @xmath406 consequently , from we deduce that @xmath407 this means that @xmath408 let us now move to the computation of @xmath409 , for @xmath350 when @xmath352 and @xmath410 . from the definition , we obtain the formula @xmath411 some easy computations combined with the relation allow to get @xmath412 it follows that , @xmath413\bigg\}.\ ] ] since @xmath349 is an algebra one finds that @xmath414 the estimate of the term @xmath415 follows immediately from and we get , @xmath416 for the terms @xmath417 and @xmath418 we can proceed similarly as for @xmath419 and we find @xmath420 putting together the preceding estimates yields , @xmath421 this shows that @xmath422 to achieve the existence proof of the gteaux derivatives it remains to check that the convergence in occurs in the strong topology of @xmath423 there are many terms to analyze and they can be treated in a similar way . the computations are straightforward but slightly long and we prefer just to treat a significant term and the remaining ones are quite similar . for example in the first term of the right - hand side of we need to check @xmath424 to simplify the notation we set @xmath425let @xmath426 such that @xmath427 . then by we have @xmath428 it follows from the inequalities and that @xmath429 which implies in turn , @xmath430 therefore we get @xmath431 now for @xmath432 , we write by the mean value theorem @xmath433 observe that @xmath434 can be rewritten in the integral form @xmath435 thus , @xmath436 in view of the formula we readily obtain @xmath437.\end{aligned}\ ] ] using straightforward computations combined with the inequality yield for any @xmath438,$ ] @xmath439 hence we get , @xmath440 this implies according to the estimate that @xmath441 and consequently , @xmath442 this concludes the desired result . the next task is to show that the gteaux derivatives are continuous operators from the neighborhood @xmath52 into the banach space @xmath443 . from the identities and and since @xmath349 is an algebra the problem amounts to showing the continuity of the terms @xmath375 , @xmath444 , @xmath445 , @xmath446 , @xmath447 and @xmath448 . we shall present here the complete details for the term @xmath375 , with @xmath350 and the other terms can be dealt via straightforward variations . set @xmath449 with @xmath380 and @xmath450 . we shall prove the estimate @xmath451 in view of we may write @xmath452 with @xmath453 the estimate of the last term in follows immediately from , that is , @xmath454 to control the remaining term we introduce the functional @xmath455 owing to the inequalities and one has @xmath456 now let @xmath362 , then we have @xmath457 in view of the derivative of @xmath458 with respect to @xmath358 is given by @xmath459 where @xmath460 we shall transform this quantity into , @xmath461 with @xmath462 @xmath463 and @xmath464 for the first and the second terms one readily gets by sobolev embeddings @xmath465 to estimate the last term we shall use the inequality combined with the estimate , @xmath466 and consequently , @xmath467 putting together and we find , @xmath468 therefore @xmath469 inserting this inequality into the estimate we get @xmath470 putting together the last estimate with the estimate , we obtain @xmath471 this concludes the proof of the proposition . the main goal of this section is to perform a spectral study of the linearized operator of @xmath309 at the annular solution @xmath472 and denoted by the differential @xmath473 . in particular , we shall identify the values @xmath22 for which the kernel of @xmath474 is not trivial leading to what we call the dispersion relation . therefore the next step is to look among the `` nonlinear eigenvalues '' @xmath22 those corresponding to one - dimensional kernels which is an important assumption in crandall - rabinowitz s theorem . this task is very complicate compared to the previous cases discussed in @xcite . this is due to the multiple parameters @xmath475 and @xmath26 in this problem and especially to the nonlinear and implicit structure of the coefficients appearing in the dispersion relation . we will be able to validate only a sufficient , but still a satisfactory result compared to euler equation , with a restriction on the symmetry of the v - states . this will be deeply discussed in the subsection [ subsec12 ] devoted to the monotonicity of the eigenvalues . we propose to compute explicitly the differential @xmath474 and show that it acts as a fourier multiplier . since @xmath321 then for given @xmath476 we have @xmath477 d_{f_1 } g_2(\omega,0,0)h_1+d_{f_2 } g_2(\omega,0,0)h_2 \end{pmatrix}.\end{aligned}\ ] ] where we recall the function spaces @xmath478 and @xmath479 putting together the formulas , and with @xmath480 and @xmath481 , where we replace @xmath142 by @xmath155 and @xmath143 by @xmath156 we get @xmath482 @xmath483 with @xmath484 @xmath485}{\vert w-\tau\vert^{\alpha+2}}d\tau\bigg\},\end{aligned}\ ] ] @xmath486d\tau}{\vert w - b\tau\vert^{\alpha+2}}\bigg\},\end{aligned}\ ] ] and @xmath487}{\vert bw-\tau\vert^{\alpha+2}}d\tau\bigg\}.\end{aligned}\ ] ] we shall now compute the fourier series of the mapping @xmath488 with @xmath489 where @xmath490 and @xmath491 are real for all @xmath492 . this is summarized in the following lemma . [ lem0]let @xmath493 and @xmath40 . we set @xmath494 and @xmath46 where @xmath47 refers to the bessel function of the first kind . then , we have @xmath495 where the matrix @xmath79 is given for @xmath80 by @xmath496 the determinant of this matrix is given by @xmath497 first we shall compute @xmath498 . for this goal we start with calculating the term @xmath499 of the right - hand side of which is easy compared to the other terms . thus by straightforward computations we obtain @xmath500 the computation of the second term @xmath501 was done in the paper @xcite dealing with the simply connected domain . it is given by @xmath502 we shall later establish the identity which gives here @xmath503 to compute the term @xmath504 we first split it into two parts as follows , @xmath505 with @xmath506 and @xmath507 by using the fourier expansions of @xmath508 and @xmath509 we get , @xmath510 then by applying the formula with @xmath511 to the first term and the formula to the second term we find @xmath512\notag\\\triangleq & -\frac{\alpha}{2}b^2\sum_{n\geq 0}\big(\,a_n\gamma_n+&&c_n\delta_n\big)w^{n+1},\end{aligned}\ ] ] where we have used in the last equality the notation , @xmath513 and @xmath514 similarly , the second term @xmath515 may be written in the form , @xmath516 using the elementary fact @xmath517 combined with the formulas and we obtain @xmath518\notag\\ = & -\frac{\alpha } { 2}\sum_{n\geq 1}\overline{w}^{n+1}\bigg[&&a_nb^2\frac\alpha2f(\frac\alpha2 + 1,\frac\alpha2 + 1;2;b^2)\notag\\ & & & + c_n\frac{(1+\frac\alpha2)_{n-1}}{(n-1)!}b^{n}\bigg(f\big(\frac\alpha2,n+\frac\alpha2;n+1;b^2\big)-f\big(\frac\alpha2,n+\frac\alpha2;n;b^2\big)\bigg)\bigg].\end{aligned}\ ] ] thus owing to the formula applied with @xmath519 and @xmath520 one gets @xmath521 where @xmath522 and @xmath523 are defined by , @xmath524 and @xmath525 inserting the identities and into we find @xmath526.\end{aligned}\ ] ] to compute @xmath527 we shall use the formula which gives @xmath528 similarly we have @xmath529\end{aligned}\ ] ] and therefore using once again the identity we find @xmath530 consequently the fourier expansion of @xmath531 is described by the formula @xmath532\big(w^{n+1}-\overline{w}^{n+1}\big).\notag\end{aligned}\ ] ] by virtue of the identity we get @xmath533\big(w^{n+1}-\overline{w}^{n+1}\big).\end{aligned}\ ] ] finally inserting , and into we find @xmath534\\ & \times \big(w^{n+1}-\overline{w}^{n+1}\big ) .\end{aligned}\ ] ] next , we shall move to the computations of @xmath535 defined in . the first two terms are done in the preceding step and therefore it remains just to compute the term @xmath536 . it may be splitted into two terms , @xmath537 with @xmath538 and @xmath539 to compute the first term @xmath540 we write @xmath541 thus applying successively the formula to the first term with @xmath511 and the to the second one we get @xmath542\notag \\ = & -\frac{\alpha } { 2}\,\sum_{n\geq 1}\big(\,a_n&&\tilde{\gamma}_n+c_n\,\tilde{\delta}_n\,\big){w}^{n+1}\end{aligned}\ ] ] with @xmath543 and @xmath544 as to the term @xmath545 we write @xmath546 owing to and using the formulae and , one gets @xmath547\notag\\ = & - \frac{\alpha } { 2}\sum_{n\geq 1}\big(a_n\tilde{\alpha}_n+c_n\tilde{\beta}_n\big)\overline{w}^{n+1},\end{aligned}\ ] ] with @xmath548\ ] ] and @xmath549 now inserting the identities and into we find @xmath550\big(w^{n+1}-\overline{w}^{n+1}\big).\end{aligned}\ ] ] from the foregoing expressions for @xmath551 and @xmath552 one may write , @xmath553\\ & = \,\,\frac{(1+\frac\alpha2)_{n-1}}{n!}b^{n}&&\bigg[b^2\frac{\frac\alpha2\big(\frac\alpha2+n\big)\big(\frac\alpha2 + 1+n\big)}{(n+2)(n+1)}f\big(1+\frac\alpha2,n+2+\frac\alpha2;n+3;b^2\big)\\ & & & - nf\big(\frac\alpha2,n+\frac\alpha2;n+1;b^2\big)-\frac\alpha2f\big(1+\frac\alpha2,n+\frac\alpha2;n+1;b^2\big)\bigg].\end{aligned}\ ] ] hence using the formula with @xmath554 and @xmath555 yields @xmath556.\end{aligned}\ ] ] applying the formula with @xmath519 and @xmath555 we get @xmath557 this implies , @xmath558 using the expressions of @xmath559 and @xmath560 combined with the identity applied with @xmath561 and @xmath562 we find the compact formula @xmath563 putting together the preceding identities allows to write @xmath564.\end{aligned}\ ] ] according to the identity we get @xmath565 finaly , inserting the preceding identity and the expressions and into one can readily verify that @xmath566\\ & \times \big(w^{n+1}-\overline{w}^{n+1}\big).\end{aligned}\ ] ] this concludes the proof of the lemma [ lem0 ] . we shall collect some useful properties on the asymptotic behavior of the sequences @xmath567 and @xmath568 introduced in lemma [ lem0 ] . the study is done with respect to the parameters @xmath5 and @xmath181 . this is summarized in the next lemma . [ lem1 ] let @xmath493 and @xmath40 . then the following results hold true . 1 . for all @xmath569 , @xmath570 , @xmath571 . moreover , @xmath572 is strictly increasing , @xmath573 is strictly increasing and @xmath574 is strictly decreasing . 2 . let @xmath575 , then @xmath576 and @xmath577 3 . for @xmath181 sufficiently large , @xmath578 @xmath579 4 . the determinant of the matrix @xmath580 introduced in satisfies @xmath581 with @xmath582 and @xmath583 with @xmath185 denotes euler constant , @xmath584 is the sum of the series @xmath585 and @xmath586 is the riemann zeta function . the assertion @xmath587 from the last lemma shows that the spectrum is continuous with respect to @xmath5 . in other words , we have @xmath588 hence , by the change of variable @xmath589 we can see that @xmath590 is exactly the same matrix obtained in @xcite . however , for @xmath12 the dispersion relation established in @xcite involves the following matrix @xmath591 this discrepancy with the matrix @xmath592 is due to the parametrization used in @xcite for the interior curve . indeed , in that paper the perturbation of the interior curve is dilated by @xmath24 . thus with our parametrization we should multiply the second column of @xmath593 by @xmath24 and the matrix @xmath593 becomes @xmath594 as we can easily see , this matrix has exactly the same determinant of the matrix @xmath592 and therefore we find the same dispersion relations . let us now prove lemma [ lem1 ] . * ( 1 ) * to study the sign of @xmath595 we shall make use of sonine - schafheitlin s formula leading to the identity @xmath596 which is obviously positive for all @xmath597 . + let us now prove that the mapping @xmath598 is decreasing . for this end , we rewrite the hypergeometric series @xmath599 appearing in right - hand side of according to the identity , which yields @xmath600 from the relation we get @xmath601 therefore it is easily seen that @xmath602 is increasing and @xmath603 is decreasing . this implies in turn that @xmath604 is increasing and thus it should be positive . notice that these properties can be also proven from the series * ( 2 ) * passing to the limit in the formula defining @xmath605 when @xmath5 goes to one yields @xmath606 as to the second limit , we have @xmath607 since @xmath40 we can use the following identity , @xmath608 whose proof can be found for example in @xcite . consequently , @xmath609 now to compute the limits of @xmath610 when @xmath5 goes to the values @xmath199 and @xmath71 we shall rewrite @xmath610 by using the identity in the form @xmath611.\end{aligned}\ ] ] this gives in view of the formula and , @xmath612 this expression coincides with the `` eigenvalues '' in the simply connected case , see @xcite . it follows that , @xmath613 we note that these values coincide with the `` eigenvalues '' for euler equations in the simply connected case . to compute the second limit , we shall introduce for a fixed @xmath181 the function @xmath614 therefore we obtain according to , and the relation @xmath615 @xmath616 by applying the logarithm function to @xmath617 and differentiating with respect to @xmath5 one obtains the relation @xmath618 now using the fact that @xmath619 combined with the preceding identity and , we find @xmath620 which is the desired result . * ( 3)-(4 ) * the asymptotic behavior of @xmath595 may be easily obtained from the integral formula . the proof of was done in details in @xcite . finally , by combining , and the expression of @xmath621 given by one can deduce the identity . in this section we shall discuss some important properties concerning the monotonicity of the eigenvalues associated to the matrix @xmath622 already seen in lemma [ lem0 ] . this will be crucial in the study of the kernel of the linearized operator @xmath474 . recall that @xmath623 the determinant of this matrix given by is a second order polynomial on the variable @xmath22 and therefore it has two roots depending on all the parameters @xmath624 and @xmath625 for our deal it is important to formulate sufficient conditions to avoid the eigenvalues crossing in order to guarantee a one - dimensional kernel which is an essential assumption in crandall - rabionwitz s theorem . in what follows we shall use the variable @xmath626 instead of @xmath22 in the spirit of the work of @xcite . thus easy computations show that the determinant takes the form , @xmath627 with @xmath628 and @xmath629\theta_n-4b^2\big(\lambda_1 ^ 2(b)-\lambda_n^2(b)\big)\\ - & 2\big(1-b^2\big)\lambda_1(b)+1\notag . \end{aligned}\ ] ] note that the quantities @xmath605 and @xmath610 have been introduced in lemma [ lem0 ] . it is easy to check through straightforward computations that the reduced discriminant of the second order polynomial appearing in is given by @xmath630 our result reads as follows . [ lem2c2 ] there exists @xmath631 such that the following holds true . 1 . for all @xmath632 we get @xmath633 and the equation admits two different real solutions given by @xmath634 2 . the sequences @xmath635 and @xmath636 are strictly increasing and @xmath637 is strictly decreasing . 3 . for all @xmath638 we have @xmath639 [ rmq1 ] 1 . the number @xmath56 in @xmath587 of the previous proposition is the smallest integer satisfying , @xmath640 2 . in the known cases like the simply connected case with @xmath641 or the doubly connected case with @xmath11 the analysis is more easier because the dispersion relation is a kind of fractional polynomial but in the present case it is highly nonlinear with respect to the frequencies and the parameters @xmath5 and @xmath642 therefore the program is achieved with only a sufficient condition on the existence of the eigenvalue and which is given by . this condition coincides with that given in @xcite for @xmath11 . * ( 1 ) * we intend to discuss the conditions leading to the positivity of the discriminant defined in which ensures in turn that the polynomial has two real solutions . we can see that @xmath643 can be extended to a smooth function defined on @xmath644 as follows @xmath645 it is strictly positive if and only if @xmath646 or @xmath647 remark first that for @xmath648 and @xmath24 verifying the inequality we have @xmath649 . on other hand , from lemma [ lem1 ] one has : for all @xmath40 the mapping @xmath650 is strictly increasing , continuous and satisfying @xmath651 @xmath652 consequently the set @xmath653 is connected and takes the form @xmath6\beta,+\infty[$ ] , with @xmath654 e_b()=0 . @xmath655 hence the integer @xmath56 is chosen as @xmath656.\ ] ] * ( 2 ) * to prove that @xmath657 is decreasing we shall compute its derivative with respect to @xmath648 . pain computations give , @xmath658{\partial_x \theta_x}-8b^2\lambda_x(b){\partial_x \lambda_x(b)}.\end{aligned}\ ] ] since @xmath659 is decreasing , @xmath660 is increasing and @xmath661 ( see ( 1 ) from lemma [ lem1 ] ) then we deduce that @xmath662{\partial_x \theta_x}\\ & > 2(b^{-\alpha}+1)e_x(b){\partial_x \theta_x}.\end{aligned}\ ] ] hence , for all @xmath663 we have @xmath664 this shows that @xmath665 is strictly increasing . + now , recall that @xmath666 using the fact that @xmath660 is increasing ( according to lemma [ lem1 ] ) combined with the increasing property of @xmath665 we get the desired result . so it remains to establish that the mapping @xmath667 is strictly decreasing . for this aim we calculate its derivative with respect to @xmath648 , @xmath668{\partial_x \theta_x}+4b^2{\partial_x \lambda_x(b)}\dfrac{\lambda_x(b)}{\sqrt{\delta_x}}\\ & < & & b^{-\alpha}\big[1-\frac{(b^{-\alpha}+1)\theta_x-\big(1+b^{2}\big)\lambda_1(b)}{\sqrt{\delta_x}}\big]\partial_x \theta_x\\ & & & -\big[1+\frac{(b^{-\alpha}+1)\theta_x-\big(1+b^{2}\big)\lambda_1(b)}{\sqrt{\delta_x}}\big]\partial_x \theta_x.\end{aligned}\ ] ] where we have used in the last inequality the decreasing property of the mapping @xmath659 and the fact that @xmath661 . now from the inequality @xmath669 and the expression of @xmath670 we deduce that @xmath671 this gives the desired result . * ( 3 ) * this follows easily from ( 2 ) and the obvious fact @xmath672 in this section we shall prove theorem [ main ] which is deeply related to the spectral study developed in the preceding section combined with crandall - rabinowitz s theorem . to construct the function spaces where the bifurcation occurs we shall take into account the restriction to the high frequencies stated in proposition [ lem2c2 ] and include the @xmath26-fold symmetry of the v - states . to proceed , fix @xmath673 and @xmath674 , where @xmath56 is defined in and remark [ rmq1 ] . set , @xmath675 where @xmath676 is the space of the @xmath169periodic functions @xmath677 whose fourier series is given by @xmath678 this space is equipped with its usual norm . we define the ball of radius @xmath679 by @xmath680 and we introduce the neighborhood of zero , @xmath681 the set @xmath682 is endowed with the induced topology of the product spaces . + take @xmath683 then the expansions of the associated conformal mappings @xmath684 outside the unit disc @xmath685 are given successively by @xmath686 and @xmath687 this structure provides the @xmath688fold symmetry of the associated boundaries @xmath319 and @xmath320 , via the relation @xmath689 for functions @xmath690 and @xmath691 with small size the boundaries can be seen as a small perturbation of the boundaries of the annulus @xmath692 set @xmath693 and define the product space @xmath694 by @xmath695 from proposition [ lem2c2 ] recall the definition of the eigenvalues @xmath696 and the associated angular velocities are @xmath697 with @xmath698 and @xmath699 note that @xmath700 and @xmath701 were introduced in lemma [ lem0 ] . the v - states equations are described in and which we restate here : for @xmath67 , @xmath702 with @xmath703 now , to apply crandall - rabinowitz s theorem it suffices to show the following result . [ prozq ] let @xmath56 be as in the part @xmath378 of proposition @xmath704 and @xmath705 and take @xmath706 . then , the following assertions hold true . 1 . there exists @xmath707 such that @xmath708 is well - defined and of class @xmath176 . the kernel of @xmath474 is one - dimensional and generated by @xmath709 3 . the range of @xmath474 is closed and is of co - dimension one in @xmath694 . transversality assumption : if @xmath22 is a simple eigenvalue @xmath710 then @xmath711 * ( 1 ) * compared to proposition [ reg ] we need just to check that @xmath321 preserves the @xmath688fold symmetry and maps @xmath712 into @xmath694 . for this end , it is sufficient to check that for given @xmath713 , the coefficients of the fourier series of @xmath714 vanish at frequencies which are not integer multiple of @xmath26 . this amounts to proving that , @xmath715 this property is obvious for the first term @xmath716 . for the two last terms in the expression of @xmath717 it is enough to check the identity , @xmath718 with @xmath719 this follows easily by making the change of variables @xmath720 and from . indeed , @xmath721 this concludes the proof of the following statement , @xmath722 we shall describe the kernel of the linear operator @xmath723 and show that it is one - dimensional . let @xmath724 be two functions in @xmath725 such that @xmath726 recall from lemma [ lem0 ] the following expression , @xmath727 where the matrix @xmath79 is given for @xmath80 by @xmath728 now , if @xmath729 then @xmath730 thus , the kernel of @xmath474 is non trivial and it is one - dimensional if and only if @xmath731 this condition is ensured by the part @xmath378 of the proposition [ lem2c2 ] . then , @xmath732 is in the kernel of @xmath733 if and only the fourier coefficients in the identity vanish , namely , @xmath734 hence , a generator of ker@xmath735 can be chosen as the pair of functions @xmath736 * ( 3 ) * we are going to show that for any @xmath737 the range @xmath738 coincides with the space of the functions @xmath739 such that @xmath740 where @xmath741 for all @xmath569 and there exists @xmath742 such that @xmath743 for the sake of simple notation we remove in this part the parameter @xmath5 from @xmath744 the range of operator @xmath474 is obviously included in the space defined above which is closed and of co - dimension @xmath71 in @xmath694 . therefore it remains to check just the converse . let @xmath745 and @xmath746 be two functions in @xmath349 with fourier series expansions as in and . we shall prove that the equation @xmath747 admits a solution @xmath748 in the space @xmath749 where the fourier series expansions of these functions are given in . then according to , the preceding equation is equivalent to @xmath750 for @xmath511 , the existence follows from the condition and therefore we shall only focus on @xmath751 owing to the sequences @xmath752 and @xmath753 are uniquely determined by the formulae @xmath754 by computing the matrix @xmath755 we deduce that for all @xmath756 @xmath757 and @xmath758 therefore the proof of @xmath759 amounts to showing that @xmath760 we shall develop the computations only for the first component and the second one can be done in a similar way . we set @xmath761 and @xmath762 then in view of the function @xmath763 can be rewritten as follows @xmath764 with @xmath765 and @xmath766 two constants . the kernels are defined by @xmath767 and @xmath768 the convolution is understood in the usual one : for two continuous functions @xmath769 we define @xmath770 assume for a while that @xmath771 belongs to @xmath772 . then by virtue of the classical convolution law @xmath773 , it suffices to show that the kernels @xmath774 and @xmath775 belong to @xmath776 . the second and the third kernels are easy to analyze because the series converge absolutely , @xmath777 similarly , owing to one has @xmath778 and therefore @xmath779 . note that to bound the series we have used the fact that the sequence @xmath780 does not vanish and converges to a strictly positive number @xmath781 defined it remains to show that @xmath782 . for this end we shall use the following estimate : for any @xmath783 latexmath:[\[\label{kern1 } which is true for all @xmath48 and for a proof we can see @xcite . now to complete the reasoning it remains to prove the preceding claim asserting that the function @xmath785 belongs to the space @xmath772 . to prove this we write in view of , @xmath786 where @xmath39 and @xmath787 are two constants ( depending on @xmath26 ) and@xmath788 this allows to get @xmath789 then one may use the general decomposition : for @xmath790 @xmath791 which yields , @xmath792 since the sequence @xmath793 is bounded then by we obtain @xmath794 thus for @xmath795 large enough we get @xmath796 concerning the estimate of @xmath344 we shall restrict the analysis to @xmath797 and @xmath480 and the higher terms can be treated in a similar way . we write @xmath798 using the cauchy - schwarz inequality we get @xmath799 by the embedding @xmath800 we conclude that @xmath801 it remains to prove that @xmath802 . for this end , one needs first to check that we can differentiate the series term by term . fix @xmath803 and define @xmath804 from cauchy - schwarz inequality we find @xmath805 hence , @xmath806 differentiating @xmath807 term by term one gets @xmath808 put @xmath809 then using the continuity of szeg protection : @xmath810 on hlder spaces @xmath349 for @xmath811 we may conclude that @xmath812 belongs to @xmath349 , ( for more details see for example @xcite ) . by virtue of a classical result on fourier series one gets @xmath813 and consequently @xmath814 putting together and we deduce that @xmath815 is differentiable and @xmath816 this concludes that @xmath817 now , as before , we can easily get @xmath818 and we shall check that @xmath819 . arguing in a similar way to @xmath815 we can differentiate term by term the series defining @xmath344 leading to @xmath820 note that with the same kernels @xmath821 and @xmath822 as before one can write @xmath823 using the fact that @xmath812 belongs to @xmath349 and @xmath824 we obtain the desired result . + let @xmath825 be a simple eigenvalue associated to the frequencies @xmath26 and @xmath826 be the generator of the kernel @xmath474 defined in the part @xmath587 of proposition [ prozq ] . we shall prove that @xmath827 with @xmath828 differentiating and with respect to @xmath22 we get @xmath829 and @xmath830 hence , @xmath831 this pair of functions is in the range of @xmath474 if and only if the vector @xmath832 is a scalar multiple of one column of the matrix @xmath833 seen in , which happens if and only if @xmath834 combining this equation with @xmath835 we get @xmath836 this yields @xmath837=0\ ] ] which is equivalent to @xmath838 this first possibility is excluded by because @xmath839 and the second one is also impossible because it corresponds to a multiple eigenvalue which is not the case here . this concludes the proof of proposition [ prozq ] . even if there is a number of references on the numerical obtention of rotating @xmath52-states for the vortex patch problem ( see for instance @xcite and @xcite , and more recently @xcite ) , up to our knowledge nothing similar has been done for the quasi - geostrophic problem . therefore , for the sake of completeness , we will discuss in this section the numerical obtention of @xmath52-states for the quasi - geostrophic problem in both the simply - connected case and the doubly - connected case . since the procedure is very similar to that in the vortex patch problem , we will omit some details , which can be consulted in @xcite . we gather the main theoretical arguments from @xcite . given a simply - connected domain @xmath14 with boundary @xmath840 , where @xmath841 is the lagrangian parameter , and @xmath842 is counterclockwise parameterized , the contour dynamics equation for the quasi - geostrophic problem is @xmath843 we use a pseudo - spectral method to find @xmath26-fold @xmath52-states from . we discretize @xmath841 in @xmath56 equally spaced nodes @xmath847 , @xmath848 . observe that , although and are trivially equivalent , the addition of @xmath849 in the numerator cancels the singularity in the denominator ; indeed , @xmath850 where the mean radius is @xmath71 ; and we are imposing that @xmath854 , i.e. , we are looking for @xmath52-states symmetric with respect to the @xmath648-axis . for sampling purposes , @xmath56 has to be chosen such that @xmath855 ; additionally , it is convenient to take @xmath56 a multiple of @xmath26 , in order to be able to reduce the @xmath56-element discrete fourier transforms to @xmath856-element discrete fourier transforms . if we write @xmath857 , then @xmath858 . remark that , for any value of the parameters @xmath5 and @xmath22 , we have trivially @xmath863 , i.e. , the unit circumference is a solution of the problem . therefore , the obtention of a simply - connected @xmath52-state is reduced to finding numerically a nontrivial root @xmath864 of . to do so , we discretize the @xmath865-dimensional jacobian matrix @xmath866 of @xmath867 using first - order approximations . fixed @xmath868 ( we have chosen @xmath869 ) , we have @xmath870 where @xmath874^{-1}$ ] denotes the inverse of the jacobian matrix at @xmath871 . this iteration converges in a small number of steps to a nontrivial root for a large variety of initial data @xmath875 . in particular , it is usually enough to perturb the unit circumference by assigning a small value to @xmath876 , and leave the other coefficients equal to zero . our stopping criterion is @xmath877 where we have used a conformal mapping . however , a caveat should be make here . indeed , unlike in the vortex patch problem , given a @xmath52-state @xmath882 , and @xmath883 , @xmath884 is no longer a @xmath52-state , but , from , @xmath885 is . therefore , since we bifurcate from the unit circumference at a certain angular velocity @xmath886 , we always obtain , by uniqueness , the same @xmath52-states up to a scaling that implies also a modification on @xmath22 , irrespectively of the chosen numerical representation of @xmath842 . an equivalent observation can be done for the doubly - connected case , etc . given an @xmath22 slightly smaller than @xmath890 , it is straightforward to obtain the corresponding @xmath52-state with the technique described above . then , we can use that @xmath52-state as a new initial datum to obtain another @xmath52-state with smaller @xmath22 , and so on . however , it seems impossible to obtain numerically @xmath52-states for @xmath22 strictly larger than @xmath890 . this means that the bifurcation is pitchfork and this fact follows from a symmetry argument : if @xmath891 is a solution of then @xmath892 is a solution too . bearing in mind , we are able to obtain @xmath52-states for an arbitrary large number of symmetries @xmath26 . for instance , in figure [ f : vstate0holealpha0_5omegas ] , we have plotted simultaneously the @xmath52-states corresponding to @xmath893 , @xmath894 , for @xmath895 and @xmath896 . in all the numerical experiments in this section , we take @xmath897 nodes . since , according to , @xmath898 , the @xmath52-state corresponding to @xmath895 , in black , is practically a circumference , as expected . on the other hand , the @xmath52-state corresponding to @xmath899 is plotted in red . observe that we have been unable to obtain the @xmath52-state corresponding to , say , @xmath900 ; this makes us wonder whether the @xmath52-state in red might be close from developing some kind of singularity . it is an established fact that simply - connected limiting @xmath52-states do exist for @xmath901 , which corresponds to the vortex patch problem . these @xmath52-states are obtained after bifurcating from the circumference at @xmath902 , which corresponds to evaluated at @xmath901 , and decreasing @xmath22 as much as possible , until corner - shaped singularities appear . furthermore , it has been proved in @xcite that the angle at the corners is always @xmath903 , irrespectively of the number @xmath26 of symmetries . therefore , we are interested in understanding what happens when @xmath904 . in figure [ f : limitingvstates ] , we have plotted @xmath52-states corresponding to @xmath905 ; for the vortex patch problem , and for @xmath906 , @xmath893 , and @xmath907 ( i.e , a value of @xmath5 rather close to zero , an intermediate value , and a value rather close to one ) . we have used the smallest possible ( four - digit ) values of @xmath22 , which are offered in table [ t : limitingomegas ] , in such a way that the experiments become numerically instable for smaller values of @xmath22 . in order to facilitate the comparison between different @xmath26 , we have plotted @xmath908 ) , rather than @xmath840 . figure [ f : limitingvstates ] confirms graphically that the angles developed by the five limiting @xmath52-states in the vortex patch problem are identical . on the other hand , when @xmath906 , the @xmath52-states depicted are very similar to the limiting @xmath52-states in the vortex patch problem , whereas , for @xmath893 , and especially for @xmath907 , it is unclear whether any singularity has happened at all . in order to shed some light on this , we have plotted in figure [ f : bifurcation ] the respective bifurcation diagrams of @xmath909 in with respect to @xmath22 . in the vortex patch problem , we have a family of monotonic curves already shown in @xcite . when @xmath906 , the curves are very similar to those in the vortex patch problem , but slightly bigger and more spaced . then , as @xmath5 grows , the curves become bigger and bigger , and more and more spaced . furthermore , when @xmath907 , the curves are partially superposed ; for example , there are 2-fold and 3-fold @xmath52-states with the same @xmath22 . this phenomenon also happens in the last four curves , when @xmath893 . however , the most striking fact from figure [ f : bifurcation ] is that all the fifteen curves , corresponding to @xmath906 , @xmath893 and @xmath907 , lose monotonicity at their left ends . in fact , especially for @xmath907 , incipient hooks are clearly visible . this seems to suggest the presence of saddle - node bifurcation points ( see for instance @xcite ) at a certain @xmath910 , which is indeed the case , as we will show in the following lines . we work with the bifurcation curve corresponding to @xmath907 and @xmath911 in figure [ f : bifurcation ] , because it has the most pronounced hook , but everything that follows is applicable to the other bifurcation curves as well . we need to estimate the corresponding @xmath912 with enough accuracy . in our case , we have taken @xmath913 . then , given @xmath914 ( we have taken here @xmath915 ) , we calculate the @xmath52-states corresponding to @xmath916 , @xmath917 , @xmath918 , @xmath919 , and @xmath920 , whose coefficients in are respectively denoted as @xmath921 , @xmath922 , @xmath923 , @xmath924 , and @xmath925 . the main idea is to introduce a new parameter @xmath194 , instead of @xmath22 , in such a way that @xmath926 corresponds to @xmath927 , and so on . we set @xmath928 , @xmath929^{1/2}$ ] , @xmath930^{1/2}$ ] , @xmath931^{1/2}$ ] , @xmath932^{1/2}$ ] . therefore , our problem is reduced to finding the @xmath52-state corresponding to some @xmath194 slightly larger than @xmath933 ; and a fairly good initial guess for that @xmath52-state can be obtained by means of a four - degree lagrange interpolation polynomial . more precisely , let us define @xmath934 remark that a couple of trials may be needed until a good choice of @xmath940 , and of @xmath194 is found , i.e. , values that enable us to continue the bifurcation curve , and not to come back to some already known @xmath52-state . in our case , we have chosen @xmath194 equal to @xmath933 plus the mean of the four previous increments of @xmath194 , i.e. , @xmath941 after applying this technique just once , we have successfully obtained a @xmath52-state corresponding to @xmath942 , i.e. , a @xmath52-state beyond the critical point . it may be useful ( and sometimes even convenient ) to iterate several times the procedure , after updating @xmath943 , @xmath944,@xmath945 , @xmath946 , and @xmath947 . in fact , it can be even applied from the very beginning , to obtain all the bifurcation curves in [ f : bifurcation ] in their integrity , with an important spare of computational time . in figure [ f : bifurcation3updated ] , we plot on the left - hand side the completed bifurcation curve until @xmath948 ; the piece of curve beyond the saddle - node bifurcation point , absent in figure [ f : bifurcation ] , is shown in thicker stroke . it is possible to still continue the bifurcation curve , although the results are to be taken with prudence , because higher spatial resolution is needed . further numerical experiments , which include the use of alternative parameterizations of @xmath842 , would suggest the eventual formation of cusp - shaped singularities at the corners . they would also suggest the presence of additional saddle - node bifurcation points , in such a way that the bifurcation curve in [ f : bifurcation3updated ] would show spiral - like structures at its ends . nevertheless , since our results are still inconclusive , we postpone this challenging issue for the future . on the other hand , based on the previous pages and on additional numerical experiments that we have carried on , we conjecture the existence of saddle - node bifurcation points for all @xmath949 and for all @xmath887 . however , as @xmath5 decreases , smaller and smaller structures are expected at the ends of the bifurcation curves , until @xmath901 , when they disappear . indeed , in the vortex patch problem , as mentioned above , the bifurcation curves are always monotonic . we can not finish this section , without saying something about the case @xmath950 , which has a pretty different behavior and is interesting per se . in figure [ f : vstate0holen2alpha0_01omegas ] , we have plotted 2-fold @xmath52-states for @xmath951 and different values of @xmath22 , starting from @xmath952 , which is close to @xmath953 , so the corresponding @xmath52-state , in black , is practically a unit circumference . since @xmath5 is small , we might expect to have a similar behavior to that in the vortex patch problem , where the @xmath52-states tend to degenerate to a segment as @xmath22 decreases . however , although this is true for @xmath22 close enough to @xmath954 , there is an instant when convexity is lost , and the @xmath52-states get a more and more pronounced @xmath955-shape , as @xmath22 decreases . remark that @xmath22 can not be smaller than a certain value , which corresponds approximately to @xmath956 , and whose corresponding @xmath52-state is plotted in red . let us mention that the situation is very similar for greater @xmath5 , even for those close to 1 . in figure [ f : vstate0holen2alpha0_01omegas ] , the @xmath52-state in red seems to have developed no singularity . again , insight into what is happening is given by the bifurcation curve of @xmath909 in with respect to @xmath22 , for @xmath951 , which is plotted in figure . in that figure , we have also plotted the bifurcation curves for @xmath957 , being the four curves very similar to each other . as with @xmath949 , the bifurcation curves suggest the existence of saddle - node bifurcation points . to see whether this is indeed the case , we have used the previously described continuation method for @xmath951 , taking @xmath958 , and @xmath915 . figure [ f : bifurcation2updated ] confirms our suspicions . on the left - hand side , we plot the completed bifurcation curve until @xmath959 ; the piece of curve beyond the saddle - node bifurcation point , absent in figure [ f : bifurcation2 ] , is shown in thicker stroke . remark that , unlike figure [ f : bifurcation3updated ] , a zoom is necessary in order for the hook to be appreciated . on the right - hand side , we plot the new @xmath52-state corresponding to a slightly larger @xmath22 , i.e. , @xmath960 ( and such that @xmath961 is unstable ) , with twice as many nodes , i.e , @xmath962 . apparently , a self - intersection has happened , although a powerful zoom shows that the distance between the two inner pieces of curve is approximately @xmath963 ; moreover , there are apparently enough nodes in that region , so it seems that we could decrease that distance even further , by increasing the eight decimal of @xmath22 , and so on . even if we rather think that a self - intersection will eventually occur ( see @xcite and @xcite for similar phenomena in the vortex patch problem ) , further study is required here . given a doubly - connected domain @xmath14 with outer boundary @xmath964 and inner boundary @xmath965 , where @xmath841 is the lagrangian parameter , and @xmath966 and @xmath967 are counterclockwise parameterized , the contour dynamics equations for the quasi - geostrophic problem are @xmath968 the doubly - connected domain @xmath14 is a @xmath52-state if and only if its boundaries satisfy the following equations : @xmath969 & = 0 , \\ \label{e : qgcondition2 } \operatorname{re}\bigg[\bigg(\omega z_2(\theta ) - \frac{c_\alpha}{2\pi i}\int_0^{2\pi}\frac{z_{1,\phi}(\phi)d\phi}{|z_1(\phi ) - z_2(\theta)|^\alpha } + \frac{c_\alpha}{2\pi i}\int_0^{2\pi}\frac{z_{2,\phi}(\phi)d\phi}{|z_2(\phi ) - z_2(\theta)|^\alpha}\bigg)\overline{z_{2,\theta}(\theta)}\bigg ] & = 0.\end{aligned}\ ] ] however , as we did in , it is convenient to rewrite them in the following equivalent form : @xmath970 = 0 , \\ \label{e : qgcondition2a } \operatorname{re}\bigg[\bigg(\omega z_2(\theta ) & - \frac{c_\alpha}{2\pi i}\int_0^{2\pi}\frac{z_{1,\phi}(\phi)d\phi}{|z_1(\phi ) - z_2(\theta)|^\alpha } \cr & + \frac{c_\alpha}{2\pi i}\int_0^{2\pi}\frac{(z_{2,\phi}(\phi ) - z_{2,\theta}(\theta))d\phi}{|z_2(\phi ) - z_2(\theta)|^\alpha}\bigg)\overline{z_{2,\theta}(\theta)}\bigg ] = 0.\end{aligned}\ ] ] we use again a pseudo - spectral method to find @xmath52-states . we discretize @xmath841 in @xmath56 equally spaced nodes @xmath847 , @xmath848 . since @xmath966 and @xmath967 never intersect , the second integral in and the first integral in can be evaluated numerically with spectral accuracy at a node @xmath851 by means of the trapezoidal rule ; e.g. , @xmath971 in order to obtain doubly connected @xmath26-fold @xmath52-states , we approximate @xmath966 and @xmath967 as in : @xmath973 , \qquad z_2(\theta ) = e^{i\theta}\left[b + \sum_{k = 1}^m a_{2,k}\cos(m\,k\,\theta)\right],\ ] ] where @xmath841 , the mean outer and inner radii are respectively @xmath71 and @xmath24 ; and we are imposing that @xmath974 and @xmath975 , i.e. , we are looking for @xmath52-states symmetric with respect to the @xmath648-axis . again , if we choose @xmath56 of the form @xmath857 , then @xmath858 . we introduce into - , and approximate the error in those equations by an @xmath859-term sine expansion : @xmath976 = \sum_{k = 1}^m b_{1,k}\sin(m\,k\,\theta ) , \\ \operatorname{re}\bigg[\bigg(\omega z_2(\theta ) & - \frac{c_\alpha}{2\pi i}\int_0^{2\pi}\frac{z_{1,\phi}(\phi)d\phi}{|z_1(\phi ) - z_2(\theta)|^\alpha } \cr & + \frac{c_\alpha}{2\pi i}\int_0^{2\pi}\frac{(z_{2,\phi}(\phi ) - z_{2,\theta}(\theta))d\phi}{|z_2(\phi ) - z_2(\theta)|^\alpha}\bigg)\overline{z_{2,\theta}(\theta)}\bigg ] = \sum_{k = 1}^m b_{2,k}\sin(m\,k\,\theta ) . \end{split}\ ] ] remark that , for any value of the parameters @xmath24 , @xmath5 and @xmath22 , we have trivially @xmath979 , i.e. , any circular annulus is a solution of the problem . therefore , the obtention of a doubly - connected @xmath52-state is reduced to finding numerically @xmath980 and @xmath981 , such that @xmath982 is a nontrivial root of . to do so , we discretize the @xmath983-dimensional jacobian matrix @xmath866 of @xmath984 as in , taking @xmath869 : @xmath985 then , the sine expansion of gives us the first row of @xmath866 , and so on . hence , if the @xmath181-th iteration is denoted by @xmath986 , then the @xmath872-th iteration is given by @xmath987^{-1 } , \end{split}\ ] ] where @xmath874^{-1}$ ] denotes the inverse of the jacobian matrix at @xmath986 . to make this iteration converge , it is usually enough to perturb the annulus by assigning a small value to @xmath988 or @xmath989 , and leave the other coefficients equal to zero . our stopping criterion is @xmath990 where @xmath878 . as in the vortex patch problem , @xmath991 , so , for the sake of coherence , we change eventually the sign of all the coefficients @xmath980 and @xmath981 , in order that , without loss of generality , @xmath992 and @xmath993 . as we have seen , the procedure to find doubly - connected @xmath26-fold @xmath52-states is very similar in the vortex patch and in the quasi - geostrophic problems . however , as evidenced in the simply - connected case , the numerical study of the @xmath52-states for the quasi - geostrophic problem reveals itself as a much richer task . indeed , unlike in the vortex patch problem , where we had just two parameters @xmath24 and @xmath22 , we have now a third parameter @xmath5 . furthermore , since and are not homogeneous , choosing the mean outer radius not to be equal to one , as we are doing , would introduce a fourth parameter . therefore , we will limit ourselves here to exposing a few relevant facts . the hypergeometric function @xmath599 is commonly implemented in most scientific packages ; for instance , in matlab , @xmath997 can be evaluated by means of the command ` hypergeom([a , b ] , c , z ) ` , so we can find the only value @xmath994 satisfying efficiently and with the greatest possible accuracy , by means a simple bisection technique . in figure [ f : alphab0 ] , we have plotted @xmath994 against 200 different values of @xmath5 , i.e. , @xmath998 , and @xmath999 . observe that @xmath994 tends very quickly to @xmath71 ; for example , @xmath1000 . on the other hand , @xmath1001 , which is coherent with the vortex patch problem , where there is no lower bound for @xmath24 . however , we have discovered in our numerical experiments that is not sharp . indeed , we have been able to find @xmath52-states with @xmath24 much smaller than @xmath994 . in all the numerical experiments in this section , we take @xmath1002 nodes . in figure [ f : vstate1holealpha0_9b0_2omegas ] , we have plotted @xmath52-states corresponding to @xmath907 and @xmath1003 , where @xmath1004 . on the left - hand side , we have started to bifurcate from @xmath1005 . observe that @xmath967 is practically a circumference , for all @xmath22 . moreover , the @xmath52-state corresponding to @xmath1006 , in black , is practically a circular annulus , whereas the outer boundary of the @xmath52-state corresponding to @xmath1007 , in red , has a marked star shape . for @xmath22 slightly smaller than @xmath1007 , the numerical experiments become instable . on the right - hand side , we have started to bifurcate from @xmath1008 . the most remarkable fact is that @xmath22 is always negative , i.e , the @xmath52-states rotate clockwise , which is an important difference with respect to the vortex patch problem . on the other hand , @xmath966 is practically a circumference , for all @xmath22 . the @xmath52-state corresponding to @xmath1009 , in black , is practically a circular annulus , whereas the inner boundary of the @xmath52-state corresponding to @xmath1010 , in red , has a marked star shape . for @xmath22 slightly larger than @xmath1010 , the numerical experiments become instable . figure [ f : vstate1holealpha0_9b0_2omegas ] shows the obvious parallelism with @xcite in the vortex patch problem : as @xmath24 becomes smaller , bifurcating at @xmath1011 yields doubly - connected @xmath52-states closer and closer to simply - connected @xmath52-states ; whereas , bifurcating at @xmath1012 yields double - connected @xmath52-states closer and closer to the unit circumference . this explains why there are two bifurcation values of @xmath22 in the doubly - connected case , and just one single bifurcation value of @xmath22 in the simply - connected case . nonetheless , unlike what would have happened in the vortex patch problem , the @xmath52-states corresponding to @xmath1007 and to @xmath1010 seem to have developed no singularity . again , as in the simply - connected case , the explanation is given by the loss of monotonicity in the bifurcation curves of @xmath1013 and @xmath1014 in with respect to @xmath22 , plotted in figure [ f : bifurcation1hole ] , which predicts the existence of saddle - node bifurcation points . observe also that , when we bifurcate from @xmath1015 , @xmath1013 clearly dominates ; whereas , when we bifurcate from @xmath1016 , @xmath1017 dominates and @xmath1013 is of the order of @xmath1018 . this confirms our previous observations . following the procedure explained in the simply - connected case ( where we set @xmath928 , @xmath1019^{1/2}$ ] , and so on ) , we have continued the bifurcation curves at @xmath1020 , and @xmath1021 , until @xmath1022 and @xmath1023 , respectively , as is shown in figure [ f : bifurcation1holeupdated ] . it is still possible to go a bit further , but , in order not to lose accuracy , a larger number of nodes is convenient . the pieces of curve beyond the saddle - node bifurcation points are shown in thicker stroke . in figure [ f : vstate1holealpha0_9b_2 ] , we have plotted the @xmath52-states corresponding to @xmath1024 and @xmath1023 , but beyond the saddle - node bifurcation points . the differences with respect to the @xmath52-states in red from figure [ f : vstate1holealpha0_9b0_2omegas ] are evident . it would be interested to calculate which kind of limiting @xmath52-states develops . finally , whether the lower bound restriction for @xmath24 can be ignored completely or not is another relevant question . another relevant fact is the possibility to find examples where @xmath52-states exist for all @xmath1025 $ ] ; such situation was also found in the vortex patch problem . we have considered , for instance , @xmath1026 , @xmath893 , and @xmath1027 , with @xmath1028 and @xmath1029 . observe that @xmath1030 , so we are violating the restriction on @xmath994 again . we have computed successfully all the @xmath52-states with @xmath1031 . on the left - hand side of figure [ f : bifurcation1whole ] , we plot the @xmath52-states corresponding to @xmath1032 . the @xmath52-state corresponding to @xmath1033 ( in red ) , and to @xmath1034 ( in black ) , are practically circular annuli ( we use no thicker stroke here , because all the @xmath52-states are very close to each other ) . on the right - hand side of figure [ f : bifurcation1whole ] , we plot the bifurcation curves of @xmath1013 and @xmath1014 in , with respect to @xmath22 . as expected , the curves are closed ; however , the curve corresponding to @xmath1014 is much more symmetrical and reminds us of an ellipse . let us finish this section by mentioning the existence of stationary doubly - connected @xmath52-states when @xmath904 , i.e. , @xmath52-states with @xmath1035 . like the examples with @xmath1036 shown above , they have no particularity from a numerical point of view , yet they are a completely new phenomenon with respect to the vortex patch problem . to obtain them , it is necessary to choose @xmath26 , @xmath5 and @xmath24 , such that @xmath1037 , but @xmath1038 , since we bifurcate from the annulus at @xmath1039 . we have chosen @xmath1026 and @xmath893 , as in the last experiment , but with a @xmath24 even smaller , @xmath1040 , in such a way that @xmath1041 . figure [ f : stationary ] shows the corresponding stationary @xmath52-state . francisco de la hoz was supported by the spanish ministry of economy and competitiveness , with the project mtm2011 - 24054 , and by the basque government , with the project it641 - 13 . zineb hassainia and taoufik hmidi were partially supported by the anr project dyficolti anr-13-bs01 - 0003 - 01 . 99 m. h. p. ambaum , b. j. harvey , x. j. carton . _ instability of shielded surface temperature vortices_. j. atmos . 68 ( 2010 ) 964971 . m. h. p. ambaum and b. j. harvey , _ perturbed rankine vortices in surface quasi - geostrophic dynamics_. geophysical and astrophysical fluid dynamics , 105 ( 4 - 5 ) ( 2011 ) 377391 . s. a. chaplygin , _ on a pulsating cylindrical vortex . _ translated from the 1899 russian original by g. krichevets , edited by d. blackmore and with comments by v. v. meleshko . chaotic dyn . 12 ( 2007 ) , no . 1 , 101116 . chemin , _ fluides parfaits incompressibles _ , , astrisque 230 ( 1995 ) ; _ perfect incompressible fluids _ translated by i. gallagher and d. iftimie , oxford lecture series in mathematics and its applications , vol . 14 , clarendon press - oxford university press , new york ( 1998 ) . h.m . wu , e.a . overman ii and n.j . steady - state solutions of the euler equations in two dimensions : rotating and translating v - states with limiting cases i. algorithms ans results _ , j. comput . 53 ( 1984 ) , 4271 .

in this paper , we prove the existence of doubly connected v - states for the generalized sqg equations with @xmath00,1[.$ ] they can be described by countable branches bifurcating from the annulus at some explicit `` eigenvalues '' related to bessel functions of the first kind . contrary to euler equations @xcite , we find v - states rotating with positive and negative angular velocities . at the end of the paper we discuss some numerical experiments concerning the limiting v - states .

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Proceed to summarize the following text: the field of single - molecule electronics has been expanding rapidly during recent years , as techniques to electrically contact and control single molecules in a transport junction have improved @xcite . by studying the electric current through the molecule as function of the applied voltage - bias , spectroscopic information can be extracted @xcite . in setups with a gate - electrode , which can be used to control the electrostatic potential on the molecule , a detailed spectroscopy can be performed @xcite . by applying a temperature - bias and measuring the induced electric current or voltage , additional information can be extracted , such as the type of carriers ( holes / electrons ) dominating transport @xcite . this emerging field of molecular thermoelectrics @xcite is also interesting for applications . molecules have been predicted to be particularly efficient for conversion of heat into electric energy @xcite ( or analogously for cooling , using electric energy to pump heat ) , the reason being their very sharp electronic resonances when weakly coupled to electrodes @xcite . this is similar to the large thermoelectric efficiency of e.g. , semi - conducting nanowires with highly peaked densities of states @xcite . most theoretical works on meso- and nano - scale thermoelectrics have focused on the _ linear , equilibrium _ regime , where one operates close to the small voltage @xmath0 which exactly cancels the current induced by the small thermal bias @xmath1 . here the thermopower ( or seebeck coefficient ) @xmath2 is the decisive quantity , where ( @xmath3 ) @xmath4 is the ( thermal ) conductance . a large efficiency @xmath5 of the device operated as a heat to electric energy converter is then related to a large dimensionless thermoelectric figure of merit @xmath6 , where @xmath7 is the operating temperature and @xmath8 the thermal conductance . in bulk systems , @xmath9 is normally limited by the wiedemann - franz law , stating that @xmath10 is a system independent constant . however , the wiedemann - franz law is a result of fermi - liquid theory and breaks down in mesoscopic and nanoscopic systems , e.g. , due to large coulomb interaction , as has been demonstrated for quantum dots @xcite and metallic islands @xcite , allowing much larger values of @xmath9 to be reached . as @xmath11 , the efficiency approaches the ideal carnot value @xmath12 @xcite . however , in the linear regime , @xmath13 , the efficiency stays low even if @xmath9 can be made very large : @xmath14 . the _ non - linear _ thermoelectric properties of molecular junctions are therefore of great interest . recent experiments @xcite probing the thermopower of thiol end - capped organic molecules showed non - linearities in the measured @xmath15 already at @xmath16 . earlier measurements of thermopower in metallic island single - electron transistors even displayed a change of the sign of the thermopower for very large @xmath1 @xcite . in the interesting regime of sharp electronic resonances , the electron tunnel coupling @xmath17 is small and the main factor limiting the efficiency of molecular energy converters is expected to be the heat current from phonon exchange with rate @xmath18 @xcite . nonetheless , to our knowledge , its effect has this far not been systematically investigated . only by making the tunnel coupling larger , @xmath19 , the phonon contribution to the heat current becomes negligible . in this case , however , the efficiency becomes instead limited by the large electronic life - time broadening of the molecular resonances . the thermoelectric efficiency in this limit of coherent transport was studied very recently in the non - linear regime @xcite using both a many - body transport approach and a ( non - interacting ) approach based on hckel theory . except for the latter work and a few others @xcite , most theoretical studies of molecular thermoelectrics have focused on non - interacting models , using a landauer type approach . however , in the regime of weak tunnel coupling between molecule and electrodes , intra - molecular interactions typically constitute the largest energy scales of the problem . in this paper , we calculate the thermoelectric efficiency and converted electric power of a molecular device , including a single dominant molecular orbital , strong coulomb interaction and coupling to a discrete vibrational mode , as well as coupling to lead phonons and lead electrons . importantly , we include on equal footing the phonon and electron contributions to the heat current , both of which contribute in establishing the stationary occupation of the molecular vibrational mode . the coulomb repulsion and electron - vibration coupling on the molecule are treated non - perturbatively in the limit of weak electron and phonon exchange in which thermoelectric efficiency is high . a central finding is that optimal thermoelectric operation typically is achieved in the _ non - linear , non - equilibrium _ regime . here concepts of figure of merit and thermopower are no longer meaningful and the molecular occupancies , efficiency and output power must be explicitly calculated . the paper is organized as follows : sect . [ sec : model ] introduces the dissipative anderson - holstein model and the thermoelectric transport equations . in sect . [ sec : transport ] we present results for the efficiency and output power as function of the applied bias voltage and energy of the molecular orbital dominating transport . the heating of the molecule is analyzed in sect . [ sec : heat ] and the optimal choice of molecule and junction parameters is discussed in sect . [ sec : optimal ] . section [ sec : conclusions ] summarizes and provides an outlook . throughout the paper we set @xmath20 , where @xmath21 is planck s constant , @xmath22 the boltzmann constant and @xmath23 the electron charge . despite polarization and screening effects in molecular junctions @xcite , the electronic level - spacing in molecular devices is typically large compared to applied voltage- and temperature - bias . we therefore restrict our attention to a single molecular orbital dominating transport . in fact , the thermoelectric properties have also been predicted to be optimal in this case @xcite . however , the quantized vibrational modes , which couple to the charge localized on a molecular device , can not be neglected @xcite . the vibrations additionally couple to bulk phonon modes of the electrodes @xcite . the goal of this paper is to clarify the importance of these excitations , characteristic of a molecular device , for the thermoelectric properties . we consider a thermoelectric junction as sketched in fig . [ fig:1 ] . its basic physics is captured by the following _ dissipative _ anderson - holstein model hamiltonian @xmath24 , where @xmath25 the molecular hamiltonian , @xmath26 , describes a spin - degenerate orbital level ( operator @xmath27 for spin - projection @xmath28 ) with energy @xmath29 and coulomb repulsion @xmath30 . the electron number @xmath31 , with @xmath32 , is linearly coupled to the vibrational coordinate of the harmonic mode of frequency @xmath33 ( operator @xmath34 ) . the dimensionless electron - vibration coupling @xmath35 is the shift of the vibrational potential as the molecule is charged , measured in units of the vibrational zero - point amplitude . the reservoir hamiltonian , @xmath36 , describes the combined electron and phonon degrees of freedom in the two reservoirs , conveniently referred to as the hot ( @xmath37 ) and cold ( @xmath38 ) electrodes . non - interacting reservoir electrons with energy @xmath39 are created ( annihilated ) by @xmath40 ( @xmath41 ) ; @xmath42 ( @xmath43 ) are the corresponding phonon operators for an electrode phonon mode with frequency @xmath44 . in each electrode electrons and phonons are assumed to be in equilibrium with temperatures @xmath45 and @xmath46 respectively . the coupling between the reservoirs and the molecule is described by the hamiltonian @xmath47 , where the first term describes tunneling of electrons with amplitude @xmath48 and the second term couples the molecular and electrode vibrational coordinates with ( in general energy - dependent ) amplitude @xmath49 . in view of the thermoelectric efficiency we consider the case where both couplings are weak , i.e. , we want a small tunnel broadening and a small heat current carried by the phonons . therefore , @xmath47 can be treated perturbatively below . the electron - electron interaction and electron - vibration coupling on the molecule are however allowed to take arbitrary values , which is a crucial aspect for addressing the important regime @xmath50 and @xmath51 . ( color online ) . ( a ) : sketch of a thermoelectric junction with a single molecule , drawn by way of illustration as c@xmath52 . operated as a thermal- to electric power converter , a temperature - bias is applied across the device . for an energy level above the electrochemical potentials , @xmath53 , this can drive a net flow of _ electrons _ from the hot ( h ) to the cold ( c ) electrode by tunneling through the junctions . in addition , combined electron and phonon heat currents @xmath54 and @xmath55 are driven through the molecule . ( b ) : the thermoelectric circuit is loaded by a resistor @xmath56 , and as a result a voltage bias is applied to the cold electrode , partially opposing the thermally induced electron flow . the voltage thus ranges from @xmath57 ( corresponding to @xmath58 ) to the value where @xmath59 ( corresponding to @xmath60 ) . in the rest of the paper we will however consider a test device as drawn in ( a ) , where @xmath61 , rather than @xmath56 , is the free parameter . we investigate how to adjust @xmath61 and the other parameters to obtain a maximal efficiency @xmath5 and output power @xmath62 . ] we consider the thermoelectric junction in fig . [ fig:1](a ) operated as a heat to electric power converter . one electrode is heated ( referred to as hot ( h ) ) with the other electrode ( referred to as cold ( c ) ) kept at the ambient temperature . the hot electrode is grounded ( chemical potential @xmath63 measured relative to the electrode fermi levels at zero bias ) and a ( negative ) voltage @xmath64 is applied to the cold electrode ( @xmath65 ) . for simplicity we assume the capacitances associated with the tunnel junctions to both electrodes to be equal , resulting in a voltage - dependence of the molecular orbital , @xmath66 . we note that in an actual device which also makes use of the converted power , the voltage is not applied , but rather controlled by the temperature - bias and the resistance of the external circuit , see fig . [ fig:1](b ) . to formulate the transport equations , the linear coupling term in ( [ eq:3_h ] ) is first eliminated by a standard transformation @xcite , which leads to a renormalization of the onsite and charging energies : @xmath67 , @xmath68 . after this transformation the eigenstates of @xmath26 are easily found to be given by @xmath69 , where @xmath70 is the electronic state and @xmath71 denotes the vibrational excitation number . the corresponding eigenenergies are @xmath72 with @xmath73 , @xmath74 and @xmath75 . furthermore , the electron tunnel amplitude is renormalized to @xmath76 $ ] , thereby incorporating the franck - condon factors for electron tunneling . the resulting transport characteristics under an applied voltage - bias have been analyzed in many works , see e.g. , @xcite . we note that in principle the additional coupling to reservoir phonon modes requires a more involved transformation , leading to more complicated expressions for the renormalized parameters @xcite . these corrections can be neglected in the regime considered here , where the coupling between the reservoir phonons and the molecular vibrational mode is weak . as mentioned above , the maximum efficiency of energy conversion is expected in the limit of weak electron tunneling @xcite and weak coupling between molecular- and electrode vibrations : @xmath77 . here the rate for electron tunneling involving electrode @xmath78 is @xmath79 , where @xmath80 is the density of states , which is assumed energy independent ( wide band limit ) . the relevant rate for phonon exchange with electrode @xmath78 is @xmath81 , with @xmath82 defined by @xmath83 . here the phonon density of states , @xmath84 , and the coupling strength , @xmath49 , are in general energy - dependent . however , in the weak coupling limit only their value at @xmath83 enters into the problem due to the selection rule @xmath85 in lowest order perturbation theory in the coupling to electrode phonons ( see the expression ( [ eq : phonon_rate ] ) for the rate matrix below ) . in the regime of non - linear temperature- and/or voltage - bias addressed in this paper , the molecular density matrix is not known a priori and needs to be calculated . for the weak coupling considered here this can be done using a standard master equation approach . we can neglect contributions from non - diagonal elements of the density matrix , since the molecular states in our model are non - degenerate on the scale set by the rates ( @xmath86 ) and spin - degeneracy does not lead to off - diagonal contributions . we note that this holds only in the weak coupling limit where the transport rates are evaluated to lowest order perturbation theory in @xmath87 and @xmath88 @xcite . the transition rates for electron tunneling ( @xmath89 ) and phonon exchange ( @xmath90 ) can then be calculated from fermi s golden rule @xcite : @xmath91 \right\ } , \\ \label{eq : phonon_rate } w_{am , a'm'}^{(p ) } & = & \sum_{r } \gamma_r \delta_{a a ' } \left\ { \delta_{m , m'+1 } m b_r(\omega ) + \delta_{m , m'-1 } m ' \left [ b_r(\omega ) + 1 \right ] \right\}.\end{aligned}\ ] ] here @xmath92 is a franck - condon factor @xcite , @xmath93 the spin - projection onto the @xmath94-axis of state @xmath95 , which has electron number @xmath96 , and @xmath97 + 1\}$ ] and @xmath98 - 1\}$ ] are respectively the fermi- and bose distribution functions of lead @xmath78 with electro - chemical potential @xmath99 . the stationary state master equation to be solved for the occupations @xmath100 then reads @xmath101 where eq . ( [ eq : prob_norm ] ) expresses probability normalization . at this point , we note that the interplay of charge and phonon tunneling is still non - trivial , as they do `` interact '' via the vibrational occupation number . a finite electric current tends to highly excite the vibrational mode , leading to high effective molecular temperatures ( see also fig . [ fig:3](c ) ) and even clear deviations from an equilibrium ( boltzmann ) shape of the distribution . this effect is particularly pronounced when the electron - vibration coupling is not too large , @xmath102 . the phonon current , on the other hand , tends to thermalize the vibration towards a temperature , that depends only on the temperatures of the hot ( @xmath103 ) and cold ( @xmath104 ) lead and the relative size of the couplings @xmath105 and @xmath106 . however , through the excitations created by the electric current , the phonon current acquires an indirect dependence on both voltage and level position . therefore accurate calculation of the non - equilibrium molecular state accounting for both electron and phonon effects is crucial . the electric current , @xmath107 , and heat current , @xmath108 , going out of lead @xmath78 , are given by @xmath109 p_{a ' m ' } , \nonumber \\\end{aligned}\ ] ] where @xmath110 and @xmath111 . the electron current matrix @xmath112 is similar to ( [ eq : electronic_rate ] ) , but includes only processes involving reservoir @xmath78 and a plus / minus sign for processes adding electrons to the molecules ( first term in ( [ eq : electronic_rate ] ) ) / removing electrons from the molecules ( second term in ( [ eq : electronic_rate ] ) ) . analogously , the heat current matrices , @xmath113 and @xmath114 , are similar to ( [ eq : electronic_rate ] ) and ( [ eq : phonon_rate ] ) , respectively , but including only processes involving reservoir @xmath78 and with the rate multiplied by the energy of the tunneling electron ( measured relative to @xmath99 ) or phonon . we note that there is in general some ambiguity associated with the definition of the heat current , see @xcite , which however does not matter in the weak coupling limit discussed here . beyond the linear regime , thermopower and figure of merit are no longer suitable quantities and we instead directly calculate the efficiency of the energy converter as follows . driven by the thermal bias , electrons can gain potential energy by tunneling from the hot to the cold electrode via the molecule . the resulting electric output power is @xmath115 , where @xmath116 . the input heat power , required to maintain the temperature bias , is equal to the heat current , @xmath54 , flowing out of the hot electrode . the efficiency is thus given by @xmath117 note that there is no conservation of the stationary heat current , as there is for the electric current , @xmath118 . instead the first law of thermodynamics guarantees that @xmath119 . we start by studying the efficiency and output power at fixed thermal bias , here chosen to be @xmath120 , as function of applied voltage - bias @xmath61 and level position @xmath121 . the efficiency of a single level quantum dot ( spin - less electrons and no vibrational mode ) was studied in @xcite , where it was shown that the ideal carnot efficiency is reached in the equilibrium limit of vanishing current , requiring the fermi functions to be equal , @xmath122 , defining a line in the @xmath123 plane : @xmath124 . in fig . [ fig:2 ] this equilibrium line corresponds to the boundary of the white areas . however , along this line also the output power vanishes ( corresponding to reversible , infinitely slow operation without entropy loss ) . for vanishing couplings to the phonon mode , @xmath125 and @xmath126 , we recover this result in the non - interacting limit , @xmath127 , as well as for very strong interactions , @xmath128 . in the intermediate regime , the efficiency is slightly reduced . switching on the electron - vibration coupling , but keeping @xmath129 , the efficiency is decreased and never reaches the ideal value ( @xmath130 for @xmath120 ) , see fig . [ fig:2](a ) . in fact , @xmath5 vanishes close to the zero electric current line ( boundary of the white area ) , the reason being that , in contrast to the single - level discussed above , the heat current does not vanish completely when the charge current does . inside the white area the current has been reversed by a too large voltage - bias and flows from high- to low - biased electrode and therefore does not accomplish any useful electric work ( note that this regime can not be reached in the thermoelectric circuit of fig . [ fig:1](b ) ) . ( color online ) . ( a)(c ) : efficiency @xmath5 at thermal bias @xmath120 , as function of voltage - bias @xmath61 and level position @xmath121 for increasing coupling to substrate phonons , @xmath131 ( a ) , @xmath132 ( b ) , @xmath133 ( c ) . in all plots @xmath134 , @xmath135 , @xmath136 and the couplings are symmetric , @xmath137 , @xmath138 . ( d ) : output power @xmath62 as function of @xmath61 and @xmath121 for the parameters used in ( b ) ( the power depends only weakly on @xmath18 ) . ] the maximal efficiency is reached in the non - linear regime when the level is far above the fermi edges of both leads . in this case electron transport involves very few thermally excited states in the heated electrode ( tail of the fermi function ) and electron - induced vibrational excitations are exponentially suppressed , minimizing electronic heat loss . however , in this regime the current is highly suppressed , leading to a very small output power , see fig . [ fig:2](d ) . additionally , even a small coupling to the substrate phonons , @xmath132 in fig . [ fig:2](b ) , drastically decreases the efficiency in this low - current regime , while having a much smaller effect in the regime where the current is larger ( @xmath121 is smaller ) . thus , already a weak coupling to substrate phonon modes , @xmath139 , drastically changes the ideal operating conditions for maximum efficiency by introducing a heat loss which depends only weakly on @xmath121 and @xmath61 ( the dependence is indirect , through the vibrational occupations ) . when the coupling to the substrate phonons becomes comparable to the tunnel coupling , @xmath140 in fig . [ fig:2](c ) , the efficiency is significantly decreased also in the high current regime . the output power , shown in fig . [ fig:2](d ) for the parameters used in ( b ) , depends only weakly on @xmath18 and is maximal for @xmath141 and @xmath142 . next we fix the level position to a value with both large power and efficiency , @xmath143 , and vary instead @xmath61 and @xmath1 . the resulting efficiency and output power is shown in fig . [ fig:3](a ) and ( b ) respectively , for the same parameters as in fig . [ fig:2](b ) . ( color online ) . efficiency @xmath5 ( a ) , output power @xmath62 ( b ) , and molecular temperature @xmath144 ( c ) , as function of @xmath61 and @xmath1 , with @xmath143 and other parameters as in fig . [ fig:2](b ) . along the edge of the white areas in ( a ) and ( b ) @xmath59 , and this edge therefore defines the non - linear thermopower @xmath145 at @xmath59 . the standard linear response thermopower , @xmath146 , is thus given by the slope close to zero , indicated by the green dashed line in ( a ) . note that @xmath144 in ( c ) is plotted also in the regime where the current has been reversed , even though the device would normally not be operated under such conditions . ( d ) : @xmath5 ( thick lines ) and @xmath144 ( thin lines , normalized by @xmath147 ) as function of @xmath1 at @xmath148 for @xmath149 ( red solid lines ) , @xmath150 ( green dashed lines ) , and @xmath151 ( blue dotted lines ) . ] as above , a too large voltage - bias compared to the temperature - bias reverses the current and no useful electric work is accomplished ( white areas ) . the non - linear thermopower can be defined through @xmath152 at @xmath59 , i.e. , @xmath61 is the finite voltage needed to compensate the temperature - bias and give zero electric current . thus , @xmath153 is given by the slope of the line which passes through zero voltage at zero temperature - bias and hits the edge of the white areas at @xmath1 in fig . [ fig:3](a ) and ( b ) . for large temperature - bias there are clear deviations from the linear response thermopower , @xmath146 , given by the slope of the green dashed line in fig . [ fig:3](a ) . as expected , the efficiency and ( even more so ) the power is increased by an increased temperature - bias . figure [ fig:3](a)(b ) shows the dependence on the temperature - bias over a wide range , all the way up to @xmath154 . such large temperature - bias could be obtained if the device is operated at low temperatures . in applications , however , @xmath7 is most likely room temperature and junction stability limits operation to lower relative temperature - bias ( e.g. , @xmath155 would mean @xmath156 k , which is a realistic value ) . however , it is actually possible to keep the ( non - equilibrium ) molecular temperature , @xmath144 , much lower than the average electrode temperature , @xmath157 , allowing operation at higher temperature - bias . to calculate @xmath144 we use an idea suggested in @xcite and couple an additional phonon bath ( `` thermometer '' ) very weakly to the molecule the temperature of the thermometer bath is varied and @xmath144 is defined as the bath temperature where the heat current between this bath and the molecule vanishes . figure [ fig:3](c ) shows @xmath144 as function of @xmath61 and @xmath1 , where it is seen that for small voltages @xmath144 exceeds the average electrode temperature @xmath158 . a larger voltage - bias , however , reduces the non - equilibrium electron current and @xmath144 approaches @xmath159 . still , it is desirable to reduce @xmath144 further , thereby allowing operation at higher @xmath1 without breaking the molecule . in designing molecular thermoelectric junctions it is therefore important to choose the electrode material and molecular anchoring groups such that the molecular vibration couples more strongly to the substrate phonons of the colder electrode ( @xmath160 ) . this is shown in fig . [ fig:3](d ) , where @xmath161 ( red solid lines ) , @xmath162 ( green dashed lines ) and @xmath163 ( blue dotted lines ) . the asymmetric phonon coupling significantly reduces @xmath144 ( thin lines ) by preventing phonons from accumulating on the molecule : they enter slowly ( @xmath105 ) and exit quickly ( @xmath106 ) . however , the asymmetry has a rather small effect on the efficiency ( thick lines ) since heat is still prevented from `` leaking '' through the molecule via the phonons as long as @xmath105 stays small compared to @xmath17 . in contrast , with also @xmath105 large the efficiency goes down much more , and @xmath144 goes up ( the red solid and blue dotted lines for @xmath144 still show a very small difference : the electron tunneling and substrate phonon couplings drive the system toward slightly different molecular temperatures , causing @xmath144 to depend on the ratio @xmath164 ) . in realizing the mechanical coupling asymmetry @xmath160 , it is important keep the electronic tunnel coupling symmetric . introducing an asymmetry in the electron tunnel couplings , @xmath165 , while keeping @xmath166 fixed , reduces the efficiency since the current level is set by the smallest coupling , while the phonon leakage current depends only weakly on @xmath167 and @xmath168 ( only indirectly through the vibrational occupations ) . comparing fig . [ fig:3](a ) and ( b ) , we reach an important result for optimizing molecular thermoelectric junctions , namely that for a given temperature - bias , maximum efficiency and maximum output power is achieved at almost the same voltage - bias . this is also seen in fig . [ fig:4 ] , where ( a ) shows the maximum efficiency , @xmath169 , and ( b ) the efficiency @xmath5 at maximum power @xmath170 , both obtained by adjusting the voltage at given thermal bias @xmath1 . ( color online ) . ( a ) : maximum efficiency , @xmath169 , as function of @xmath1 , i.e. , the maximum obtained from vertical cuts in a plot such as fig . [ fig:3](a ) . parameters as in fig . [ fig:3](a)(c ) , but the strength of the coupling to substrate phonons is varied . here @xmath171 and @xmath172 . ( b ) : same as ( a ) , but instead showing efficiency at maximum output power @xmath173 , i.e. , @xmath5 taken at the maximum @xmath62 obtained from vertical cuts in a plot such as fig . [ fig:3](b ) . ] for non - zero coupling @xmath18 to substrate phonons the maximum efficiency and the efficiency at maximum output power are very close . the reason is seen from the relation @xmath174 , where @xmath175 ( @xmath176 ) is the electron ( phonon ) contribution to the heat current . since @xmath176 only has a weak ( indirect ) dependence on the voltage - bias , @xmath5 and @xmath62 can be simultaneously maximized by adjusting the bias when the phonon heat loss dominates ( @xmath177 ) . as fig . [ fig:3 ] and fig . [ fig:4 ] shows , this holds approximately also when @xmath178 . when the electron - vibration coupling becomes strong , @xmath179 , the tunnel amplitudes involving the vibrational ground state become suppressed ( franck - condon blockade @xcite ) . this reduces both the efficiency and output power since the current is decreased . additionally , heat dissipation is increased since transport through excited vibrational states is favored , the typical energy transferred to the vibrational mode by a tunneling electron being given by the classical displacement energy , @xmath180 . this is shown in fig . [ fig:5](a ) and ( b ) , where @xmath181 ( cf . , fig . [ fig:3](a ) and ( b ) ) . ( color online ) . dependence on the molecular vibration frequency @xmath33 and its coupling to the electron charge @xmath35 , all other parameters fixed to those of fig . [ fig:3](a)(c ) . ( a)(b ) : efficiency @xmath5 ( a ) and power @xmath62 ( b ) , as function of @xmath61 and @xmath1 for @xmath181 and @xmath182 . ( c)(d ) : efficiency @xmath5 as function of @xmath61 and @xmath1 for @xmath183 and @xmath182 ( c ) and @xmath134 and @xmath184 ( d ) . ] similarly , a smaller electron - vibration coupling enhances the efficiency , see fig . [ fig:5](c ) , where @xmath183 . in choosing @xmath135 ( @xmath185 mev assuming room temperature ) we have investigated the influence of a rather high energy vibrational mode . molecules often also have vibrational modes with much lower frequency , down to a few mev , especially when contacted to electrodes by linker wires @xcite . however , as is shown in fig . [ fig:5](d ) , where @xmath184 , a low - energy vibrational mode leads to a much smaller decrease of @xmath5 compared to the ideal case of no vibrational mode , as long as @xmath35 is not too large . the reason is simply that a low frequency mode essentially can be seen as a broadening of the electronic resonance of width @xmath186 , setting the scale for additional heat loss from electron tunneling compared to the case of no vibrational mode . almost all decrease in efficiency in this case comes from the coupling to substrate phonon modes . in contrast , a vibrational mode with a frequency much larger than the involved temperature- and voltage - bias , does not contribute at all to electron or heat transport ( other than through the trivial shift of the electronic parameters @xmath29 , @xmath30 and @xmath187 through the electron - vibration coupling ) . finally , we mention that in the simple model analyzed here , the strength of the coulomb interaction does not play a crucial role . in the presence of a coupling to substrate phonons , reducing @xmath188 leads to a somewhat larger efficiency ( and output power ) as the electric current is increased by the presence of another `` transport channel '' . we have analyzed the efficiency and output power of a non - linear molecular thermoelectric device operated as a power converter . accounting for the molecular vibration and its coupling to substrate phonons turned out to be crucial in comparison with results for quantum dot models without these , as it qualitatively changes the operating conditions for optimal efficiency away from the equilibrium regime . by investigating a generic model system we can now identify some basic criteria for efficient molecular energy converters : _ ( i ) _ the coupling between substrate phonon modes and molecular vibrations should be asymmetric and minimal . coupling more strongly to the colder lead reduces the molecular temperature and allows operation at higher temperature - bias , improving efficiency and output power . _ ( ii ) _ the electron tunnel couplings should be symmetric , @xmath189 . furthermore , they should be small as to minimize the life - time broadening , but still larger than the phonon coupling , @xmath190 . _ ( iii ) _ the local electron - vibration coupling energy should be small compared to the zero - point energy of the vibrational mode ( @xmath191 ) . this is most crucial for vibrational modes with frequencies around the operating temperatures and voltages . modes with much higher frequencies do not contribute at all , and those with much lower frequencies only contribute to the heat loss through the coupling to substrate phonons . _ ( iv ) _ ideal operating conditions for high efficiency and power is achieved when the conducting orbital energy is at a few @xmath192 from the fermi edges of the electrodes ( @xmath193 mev at room temperature ) . control of the thermopower by adding electron donating or withdrawing groups to benzenedithiol molecules , thereby shifting the position of the homo and lumo , was recently demonstrated @xcite . the temperature - bias should be chosen as high as is allowed by molecular stability and the heat source . the ideal voltage - bias depends on the other parameters , but is nearly the same when optimizing output power as when optimizing efficiency . additionally , the efficiency at maximum power is very close to the maximum efficiency . the general insights obtained in this exhaustive study of the most basic molecular thermoelectric model can serve as a guide for more complex molecular modeling , incorporating multiple vibrational modes , multiple electronic states , breakdown of the born - oppenheimer picture ( pseudo jahn - teller mixing @xcite ) , etc . in general , one expects deviations from a single - orbital model to give a less efficient energy - converter , as additional heat is lost by population of excited states . it might however be possible to find special circumstances under which excited states can instead be desirable , e.g. , by effectively cooling the vibrational mode . atomistic studies of specific configurations of molecules , anchoring groups and electrodes may identify suitable systems which satisfy the above criteria and thereby further assist in advancing the chemical engineering of molecular thermoelectric junctions . for device applications , engineering of molecular monolayer devices , rather than ones based on a single molecule , presents a challenge to supramolecular chemistry , nanodevice fabrication and surface science . we acknowledge financial support from the dfg under contract no . spp-1243 ( m. l. , m. w ) and the european union under the fp7 strep program single ( m. l. , k. f. ) . this work was carried out partly in the danish - chinese centre for molecular nano - electronics supported by the danish national research foundation .

we present a detailed study of the _ non - linear _ thermoelectric properties of a molecular junction , represented by a dissipative anderson - holstein model . a single orbital level with strong coulomb interaction is coupled to a localized vibrational mode and we account for both electron and phonon exchange with both electrodes , investigating how these contribute to the heat and charge transport . we calculate the efficiency and power output of the device operated as a heat to electric power converter and identify the optimal operating conditions , which are found to be qualitatively changed by the presence of the vibrational mode . based on this study of a generic model system , we discuss the desirable properties of molecular junctions for thermoelectric applications .

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Proceed to summarize the following text: in quantum information science it is a crucial problem to develop techniques for generating entanglement among stationary qubits . entanglement as unique feature of quantum mechanics can be used not only to test fundamental quantum - mechanical principles @xcite , but to play a central role in applications @xcite . especially , multipartite entanglement has been recognized as a powerful resource in quantum information processing and communication . there are two typical multipartite entangled states , greenberger - horne - zeilinger ( ghz ) and w states , which are usually referred to as maximal entanglement . numerous protocols for the preparation of such states have been proposed @xcite . most of them are scattering - based schemes which utilize two processes : the natural dynamic process of an always on system and the final project process carried out by a subsequent measurement . another feature of such kind of schemes is that there are two kinds of qubits involved in : target qubits and flying qubit . the target qubits are the main entities that will be entangled by the above two processes , which are usually stationary and can be realized by atoms , impurities , or other quantum devices . the flying qubit is amediator to establish the entanglement among the target qubits via the interaction between them , which is usually realized by photon or mobile electron . in this sense , the type of interaction between stationary qubits and the flying qubit as a mediator and the transfer of the flying qubit are crucial for the efficiency of the entanglement creation . in general , such two processes are mutually exclusive . the scattering between stationary and flying qubits can convert information between them , while it also reduces the fidelity of the flying qubit , which will affect the efficiency of the entanglement , especially for multi - particle system . it is still a challenge to create entanglement among massive , or stationary qubits . in this paper , we consider whether it is possible to use an arrangement of qubits , a spin network , to generate multipartite entanglement among stationary qubits via scattering process . we introduce a scheme that allow the generation of the ghz and w states of stationary qubits in spin networks . in the proposed scheme , the flying qubit is a gaussian type single - flip moving wave packet on the ferromagnetic background , which can propagate freely in @xmath4 chain . the stationary qubit is consisted of two spins coupled by ising type interaction with strength @xmath5 . a single spin flip can be confined inside such two spins by local magnetic field @xmath6 to form a double - dot ( dd ) qubit . the system of an @xmath4 spin chain with a dd qubit embedded in exhibits a novel feature under the resonance scattering condition @xmath7 , that a single - flip moving wave packet can completely pass over a dd qubit and switch it from state @xmath8 to @xmath9 simultaneously . we show that the scattering between a flying qubit and a dd qubit can induce the entanglement between them and the operation on the dd qubit can be performed by the measurement of the output flying qubit . it allows simple schemes for generation of multipartite entanglement , such as ghz and w states by simply - designed spin networks . we also investigate the influence of near - resonance effects on the success probabilities of the schemes . it is found that the success probabilities are @xmath10 @xmath11 and @xmath2 for the generation of ghz and w states , respectively . here @xmath3 is the transmission probability amplitude for a single dd qubit and @xmath0 is the number of the dd qubits . this paper is organized as follows . in sec . ii the dd qubit and spin network are presented . in sec . iii we investigate the resonance - scattering process between the flying and stationary qubits . iv and v are devoted to the application of the resonance scattering on schemes of creating ghz and w states . section vi is the summary and discussion . the spin network we consider in this paper is consisted of spins connected via the @xmath12 interaction . the hamiltonian is @xmath13 \\ & & + \sum_{i}h_{i}\sigma _ { i}^{z } , \notag\end{aligned}\]]where @xmath14 , and @xmath15 @xmath16 are the pauli spin matrices for the spin at site @xmath17 . the total @xmath18-component of spin , or the number of spin flips on the ferromagnetic background , is conserved as it commutes with the hamiltonian . for @xmath19 , it reduces to @xmath4 spin network , which has received a wide study for the purpose of quantum state transfer and creating entanglement between distant qubits by using the natural dynamics @xcite . for @xmath20 , the hamiltonian describes isotropic heisenberg model . in the antiferromagnetic regime ( @xmath21 ) , a ladder geometry spin network , a gapped system @xcite , has been employed as a data bus for the swapping operation and generation of entanglement between two distant stationary qubits . it has been shown that a moving wave packet can act as a flying qubit @xcite like photon in a fiber . on the other hand , the analogues of optical device , beam splitter can be fabricated in quantum networks of bosonic plenio , spin and ferimonic systems @xcite . in this paper , we consider a new type of qubit , double - dot qubit , which can be embedded in such spin networks . a dd qubit consists of two ordinary spins at sites @xmath22 and @xmath23 connected via ising type interaction in the form@xmath24when such kinds of two spins are embedded in the spin networks with @xmath25 and @xmath26 , a spin flip is confined within it and forms a dd qubit with the notations @xmath27 and @xmath28 . we will show that such a new type of qubit has a novel feature when it interacts with another spin flip in the spin networks . the main building block in the spin network of our scheme is the dd qubit . it acts as a massive or stationary qubit , like atoms or ions in cavity - qed - based schemes . to demonstrate the property of a dd qubit in a spin chain , we investigate a small system of @xmath29-site , a dd qubit connecting to two spins . in order to provide a clear exposition , we firstly assume a specific coupling configuration with @xmath7 , which leads to the following @xmath29-site hamiltonian@xmath30there is a quasi - invariant subspace with the diagonal energy being @xmath6 and under the condition @xmath31 , which is spanned by basis@xmath32the matrix of the hamiltonian in this subspace reads@xmath33\]]with eigenstates @xmath34 and eigen energies @xmath35 @xmath36 being @xmath37obviously , in the invariant subspace , such a @xmath29-site system acts as a normal @xmath38-site system . note that the dd qubit as the center of the @xmath38-site system can be in two different states @xmath8 or @xmath39 while another spin flip is at left or right site . thus a time evolution process can accomplish the transformation @xmath40 or @xmath41 with 100% success probability . such a feature is desirable for the quantum information processing . it is because that , on one hand , a spin flip passing over a dd qubit can operate the qubit state ; and on the other hand , the state of a dd qubit can indicate whether there is a spin flip passing over it . a similar transformation has been proposed through the cavity input - output process in adiabatic limit @xcite . a particular merit of the present scheme is that it is based on a natural dynamic process rather than an adiabatic process . now we consider the dynamic process of the interaction between a moving wave packet and a dd qubit . we embed a dd qubit into a chain as illustrated in fig . [ fig1](a ) . it has been shown that a single - spin - flip wave packet in the form @xcite@xmath42can propagate along a uniform spin chain without spreading approximately , where the vacuum state is fully ferromagnetic state @xmath43 . here @xmath44 is the normalization factor , @xmath45 is the center of the wave packet at @xmath46 and @xmath47 is the number of sites of the chain . at time @xmath3 , it will evolves to @xmath48 . let us firstly assume that initially the qubit is in the state @xmath49 , while a wave packet of type ( [ single p ] ) @xmath50 @xmath51 is coming from the left . similarly , we define @xmath52 @xmath53 , @xmath54 @xmath55 to denote a transmitted or reflected wave packet after scattering . in the strong local field regime , @xmath56 , the spin flip is firmly confined in the dd qubit . from the analysis of the above @xmath29-site system , which is called the resonant case with @xmath7 , the wave packet will pass freely through the dd qubit . comparing to the case without the embedded dd qubit , the output wave packet gets a forward shift with a lattice space , while switches the dd qubit from @xmath57 to @xmath58 , i.e. , @xmath59 in contrast , if the qubit is in state @xmath9 , the scattering process is @xmath60 i.e. , the incoming wave packet is totally reflected and maintains the qubit to be in state @xmath58 . interestingly , the states of wave packet and the dd qubit are both altered through this process , i.e. , being shifted with a lattice space . however , such a shift brings about totally different effects on the dd qubit and the wave packet , respectively : it switches the dd qubit from @xmath61 to @xmath62 , but does not alter the wave packet in the same manner as _ classical _ perfect elastic collision . it is worthy to point out that the dd qubit and the wave packet are not entangled in such resonant case . in the case of non - resonance @xmath63 , or initially the dd qubit and/or the incident wave packet are in a superposition states as @xmath64 where @xmath65 and @xmath66 are arbitrary coefficients satisfying @xmath67 . in practice , the difference between @xmath6 and @xmath5 will leads to the reflection of the incident wave packet . then the scattering process can be expressed as @xmath68 with @xmath69 . the transmission coefficient @xmath70 is a crucial factor in the following schemes for quantum information processing . on the other hand , the strength of the local field @xmath6 also results the spreading of the spin flip from state @xmath58 and reduces @xmath71 . we perform numerical simulation for the scattering process in order to investigate such phenomenon . numerical result for @xmath72 with @xmath73 is plotted in fig . it shows that the transmission coefficient is close to @xmath74 if @xmath75 , which is feasible in practice . it ensures that a spin network with an embedded dd qubit in a spin chain can perform the transformation ( [ tran1 ] , [ tran2 ] ) via a natural dynamic process rather than an adiabatic process . now we focus on the practical application of resonant scattering effect on the quantum information processing . as mentioned above , although the totally transmitted wave packet switches the state of the dd qubit from @xmath76 to @xmath58 , there is no entanglement between the dd and the wave packet arising from such a process . however , if the incoming wave packet is not polarized in @xmath18 direction , but in the form @xmath77 where @xmath65 and @xmath66 are restricted to be real for simplicity in the following context , the entanglement between the dd qubit and the scattered wave packet can be established . actually , the corresponding resonant scattering process can be expressed as @xmath78 the reduced density matrix of the final state in the basis @xmath79 , @xmath80 , @xmath81 , @xmath82 is@xmath83 , \]]which has concurrence @xmath84 . for @xmath85 , the concurrence between them reaches to the maximum @xmath86 . a flying qubit ( [ wp_in ] ) can be generated from wave packet ( [ single p ] ) via a @xmath87-beam or @xmath88-beam splitter @xcite . we consider the spin network with the geometry of two connected @xmath87-beam splitters . these two @xmath88-beam splitters are characterized by @xmath89 and @xmath90 respectively , which are schematically shown in fig . fig3(a ) . there is a dd qubit embedded in one of the two arms . now we consider the dynamic process with the initial state being @xmath91 , where @xmath92,denotes an incoming wave packet along the left chain . in the first step , through the beam splitter @xmath93 , @xmath94 is divided into two wave packets @xmath95 and @xmath96 along @xmath97 and @xmath98 arms , respectively . in the second step , sub - wave packet @xmath99 passes over dd qubit and becomes @xmath100 , while sub - wave packet @xmath101 propagates along @xmath98 arm and meets @xmath102 at the joint of beam splitter @xmath90 . in the third step , wave packets @xmath103 and @xmath101are reflected and divided by beam splitter @xmath104 , and contribute to the output wave packet @xmath105 . then the whole process can be expressed as @xmath106 from step 2 to step 3 we have used formulas ( [ alfa in ] , [ beta in ] ) derived in appendix a. when @xmath107 is measured in the output lead , the operation @xmath108is implemented . the success probability of this operation is @xmath109 . in the optimal case with @xmath110 @xmath111 , we can perform the operation @xmath112 by the process of resonant scattering and subsequent measurement with the success probability up to @xmath113 . the measurement of @xmath107 can be implemented by embedding another dd qubit to record the passing of the output wave packet . now we consider the case of multiple dd qubits embedded in @xmath97 arm , which is schematically illustrated in fig . [ fig3](b ) . all the @xmath0 stationary dd qubits are prepared initially in state @xmath114 . similarly , we have @xmath115with the notation @xmath116 . in the optimal case with @xmath110 @xmath111 , we can perform the operation @xmath117by the subsequent measurement with the success probability up to @xmath113 . then by using natural dynamics and subsequent measurement , multipartite entangled ghz state can be generated . this provides a simple way of entangling @xmath0 stationary qubits through scattering with a flying qubit . in the near - resonance scattering case , the transmission probability amplitude @xmath3 will effect the success probability to be @xmath118under the optimal conditions @xmath119 , @xmath120 . note that the success probability is reduced exponentially as the number of qubits increases . now we turn to the scheme of the generation of another type of multipartite entangled state , w state . the configuration of the spin network we utilized consists of two @xmath121-beam splitters with one dd qubit embedded in each parallel arm in the same way , as illustrated in fig . [ fig3](c ) . we start our analysis by considering @xmath122 case with two @xmath87-beam splitters being characterized by @xmath93 and @xmath123 respectively . denoting the qubit states of two dd qubits embedded in arms @xmath97 and @xmath98 as @xmath124 and @xmath125 respectively , the dynamic process can be written as @xmath126 in the optimal case with @xmath127 , we can perform the operation @xmath128by the subsequent measurement of the output spin flip with the success probability up to @xmath113 . now we extend the above conclusion to @xmath0-dd qubits case . for a @xmath121-beam splitter , we only consider the simplest case with identical hopping constant being @xmath129 between the input lead and each arm , which is schematically illustrated in fig . [ fig3](c ) . an incident wave packet will experience the following process . in the first step , @xmath130 is divided into @xmath0 wave packets through the beam splitter , with @xmath131 @xmath132 being the wave packet along the @xmath133th arm . in the second step , every sub - wave packet @xmath134 passes over the corresponding dd qubit embedded in the @xmath133th arm , and switches its state from @xmath135 to @xmath136 simultaneously . in the third step , all the sub - wave packets @xmath137 are reflected and divided at the node of the right beam splitter , and contribute to the output wave packet @xmath138 . the whole process can be expressed as @xmath139from step 2 to step 3 we have used the formula ( [ l_in ] ) derived in appendix b. when @xmath107 is measured in the output lead , the operation @xmath140is implemented with the success probability @xmath141 . then by using natural dynamics and subsequent measurement , multipartite entangled w state can be generated . in the near - resonance scattering case , the transmission probability amplitude @xmath3 reduces the success probability to @xmath142 as the comparison of the success probabilities of creating ghz and w states of @xmath0 qubits with the transmission coefficient @xmath143 , we plot eqs . ( p_ghz ) and ( [ p_w ] ) in fig . [ ghz_w ] for the cases with @xmath144 , @xmath145 , @xmath146 , and @xmath147 ; @xmath122 , @xmath38 , @xmath148 , @xmath149 . it shows that when @xmath143 is close to @xmath74 , the difference between @xmath150 and @xmath151 becomes large as @xmath152 increases . as @xmath143 decreases from @xmath74 , the difference between @xmath150 and @xmath151 becomes smaller for fixed @xmath0 . we have shown how a spin network can be used to generate multipartite entanglement among stationary qubits . the key of this scheme is the alternative of massive or stationary qubit in a spin network , dd qubit . the resonance scattering between a dd qubit and a sfwp , which acts as the alternative of a flying qubit in a spin network , allows a perfect transformation : an incident wave packet can totally pass through a dd qubit and switch it from state @xmath8 to state @xmath153 . the resonance scattering condition is investigated analytically and numerically . it shows that the resonance scattering is feasible in practice . this ensures that through the natural dynamical evolution of an incident single - spin - flip wave packet in a spin network and the subsequent measurement of the output single - spin - flip wave packet , multipartite entangled states among @xmath0 stationary qubits , ghz and w states can be generated . there are two merits in our scheme . firstly , the massive or stationary qubit , dd qubit , is constructed by the element , two neighbor spins of the spin network , which is applicable to all types of the scalable multi - qubit systems . secondly , it is based on a natural dynamic process rather than an adiabatic process . there is certainly significant potential for spin networks to find applications in solid state quantum processing and communication . we acknowledge the support of the cnsf ( grant no . 10874091 , 2006cb921205 ) . in this appendix , we present the exact results for the dynamics of wave packets in beam splitters . in the work of ref . @xcite the dynamics of a wave packet in the spin networks based on the @xmath4 model was studied . a @xmath87-beam splitter is consisted of three uniform spin chains with coupling constant @xmath154 . the connections between three uniform chains are @xmath155 and @xmath156 as in fig . it has been shown that under the condition @xmath157 , an input moving wave packet @xmath158 will be divided into two wave packets @xmath159 and @xmath160 without any reflection , which can be expressed as@xmath161similarly , the inverse process also holds , i.e.,@xmath162where states @xmath158 , @xmath163 , and @xmath164 ( @xmath107 , @xmath165 , and @xmath166 ) represent the wave packets coming in ( out ) of the node along the three chains respectively . contrarily , for two wave packets along the two arms , which interference destructively at the node , we have @xmath167combining the above eqs . ( [ a(1 ) ] and [ a(2 ) ] ) , we have@xmath168and@xmath169then for a given incident wave packet along any branch of a @xmath87 beam splitter , the probability amplitudes of all the output wave packets can be obtained exactly . now we consider a @xmath121 beam splitter consists of one chain of length @xmath170 and @xmath0 arms of length @xmath171 with identical connecting coupling strength @xmath129 for each arm , which is shown in fig . the hamiltonian of such quantum network reads where @xmath173 and @xmath174 are particle operators at site @xmath17 of chain @xmath175 and site @xmath176 of the arm @xmath133 . they can be boson or fermion operators . the conclusion for such model is available for the dynamics of a single flip in the analogue of spin network . similarly as shown in ref . @xcite , we can perform the following transformations@xmath177where @xmath178 and @xmath179 are also corresponding boson or fermion operators . under the transformations , hamiltonian ( [ quantum network ] ) can be written as which indicate that the original quantum network , @xmath121 beam splitter can be decomposed into @xmath0 independent chains , one of them is the length of @xmath182 and the rest are all the length of @xmath171 . based on the fact that a moving wave packet can propagate freely along the @xmath0 independent chains , we have the following processes through a straightforward algebra , we obtain a set of expressions @xmath184then for a given incident wave packet along any branch of a @xmath121 beam splitter , the probability amplitudes of all the output wave packets can be obtained exactly .

we propose a simple scheme to establish entanglement among stationary qubits based on the mechanism of resonance scattering between them and a single - spin - flip wave packet in designed spin network . it is found that through the natural dynamical evolution of an incident single - spin - flip wave packet in a spin network and the subsequent measurement of the output single - spin - flip wave packet , multipartite entangled states among @xmath0 stationary qubits , greenberger - horne - zeilinger ( ghz ) and w states can be generated with success probabilities @xmath1 and @xmath2 respectively , where @xmath3 is the transmission amplitude of the near - resonance scattering .

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Proceed to summarize the following text: the magnetocrystalline anisotropy energy ( mae ) is at the heart of magnetic properties of materials . it is of crucial importance from the fundamental or technological point of views since it provides an energy scale for the stability of magnetic domains where for example magnetic information is stored . when the mae is large and favors an out - of - plane orientation of the magnetic moments , perpendicular magnetic recording or magneto - optical recording is possible ( see e.g. refs@xcite ) . copt binary bulk alloy in the * l1@xmath0 * structure ( see fig.[cell_bulk ] ) is by now a classical example of a material exhibiting a large perpendicular mae , around 1 mev @xcite . there has been a tremendous number of studies related to the magnetic properties of this alloy in its bulk phase , as nanoparticles or in nanostructures combining co and pt ( see e.g. refs . @xcite ) . * alloy . the primitive cell is also sketched using dashed lines . ] a large amount of work has been devoted to unveil the origin of the large perpendicular mae in binary bulk alloys , see e.g. refs@xcite . the interplay of the tetragonality of the alloy , band filling , hybridization between the constituents affect certainly the magnitude of the mae . for instance , tetragonality leads to the lifting of the degeneracy of the electrons by the tetragonal crystal field and produces thereby an additional contribution to the mae . thus , and as expected from perturbation theory , the mae becomes proportional to @xmath1 instead of @xmath2 as found for cubic symmetry , where @xmath3 is the spin - orbit coupling constant . indeed , in cubic bulk systems , the high symmetry allows only for a fourth - order anisotropy constant , and thus they are characterized by a small mae . razee et al.@xcite argued however that the tetragonal distortion of copt , given by the axial ration c / a = 0.98 , contributes by only 15% of the mae . it was then concluded that the compositional order of the alloy is an important ingredient for a large mae . sakuma@xcite shows that by changing the axial ratio ( c / a ) defining the tetragonality of copt and fept alloys , the mae first smoothly decreases by increasing c / a till reaching a minimum at @xmath40.8 before a smooth increase in magnitude . interestingly , except a small window of axial ratios ( @xmath5 ) , the mae favors an out - of - plane orientation of the magnetic moments . the tetragonalization is then thought to provide an effect similar to the band filling@xcite . in the context of thin films , zhang et al.@xcite demonstrated with ab - initio simulations that for copt films terminated by co layers , a thickness of at least 9 monolayers exhibit a rather converged mae , with a bulk contribution of 1.36 mev favoring a perpendicular orientation of the magnetic moments and a counter - acting surface contribution of -0.76 mev favoring , interestingly , an in - plane orientation of the moments . their interest in copt was motivated by the experimental demonstration of coercivity manipulation of * l1@xmath0 * fept and fepd thin films@xcite by external electric field . their ab - initio simulations predicted a higher sensitivity of copt to electric field than that of fept films . pustogowa et al.@xcite investigated from first - principles several components made of co and pt deposited on pt(100 ) and pt(111 ) surfaces . they found that ordered superstructures of ( copt)@xmath6 deposited on both mentioned substrates are characterized by a perpendicular mae , which is heavily affected by chemical disorder in line with the analysis of razee et al.@xcite . the goal of this manuscript is to present a systematic ab - initio investigation on the effect of reduced dimensionality on the magnetic properties of copt(100 ) films with a focus on their mae and by addressing the impact of the termination type of the films . contrary to previous investigations@xcite , we consider not only co - terminated films but also pt - terminated films and several types of surface defects ( see fig.[cell_surf ] ) . for instance , we found that decreasing the thickness of the films leads to a sign change of the surface mae . pt covered thin films can boost the total perpendicular mae by a large amount stabilizing , thereby , more strongly the out - of - plane orientation of the moments . molecular dynamics simulations demonstrated the likeliness of having pt on the surface of copt alloy@xcite and thus the pertinence of our predictions . after a careful study of different defective terminations types ( stacking faults , anti - site defects ) , we provide the ingredient to increase the mae of the thin films . if we label the co and pt layers by respectively a and b , the perfect stacking along the [ 001 ] direction is given for example by ababab for 6 layers . possible stacking faults , which are planar defects , could be the sequence ababaa ( see figs.[cell_surf](c - d ) ) . anti - site defects on the surface means that instead of having at the surface a pure layer a , or layer b , we have an alloy , for example , made of a and b. in our work , we considered an alloy of the type a@xmath7b@xmath8 in the surface layer instead of the perfect b layer of our example ( see figs.[cell_surf](e - f ) ) . we simulate the thin films by adopting the slab approach with periodic boundary conditions in two directions while the periodic images in the third direction are separated by a sufficient amount of vacuum ( 15 ) to avoid interaction between neighboring supercells . we have chosen to use symmetrical calculation cells with an odd number of planes to avoid the pulay stress . some representative slabs are shown in fig.[cell_surf ] . here it can be observed that for equiatomic * l1@xmath0 * type of alloys two different surfaces exist when the slabs are stacked along the [ 001 ] direction . in the perfect cases , the surface termination can be made of either purely co atoms or pt atoms . the self - consistent calculations are carried out with the vienna ab initio simulation package ( vasp ) using a plane wave basis and the projector augmented wave ( paw ) approach @xcite . the exchange - correlation potential is used in the functional from of perdew , burke and ernzerhof ( pbe ) @xcite . the cut - off energies for the plane waves is 478 ev . the integration over the brillouin zone was based on finite temperature smearing ( methfessel - paxton method ) for the thin films while for the bulk case the tetrahedron method with blchl corrections has been used . the k - points grids are @xmath9 for the bulk calculation , and @xmath10 for the ( 001 ) surface calculations . the energy convergence criterion is set to @xmath11 ev while the geometrical atomic relaxations for the surfaces calculations were stopped when the forces were less than 0.01 ev / . the mae is extracted from the difference between the total energies of the two configurations : out - of - plane versus in - plane orientations of the magnetic moments . a positive value indicates a preference for the out - of - plane orientation of the magnetic moments . copt alloy as function of the axial ratio c / a under constant volume . two possible in - plane orientation of the magnetic moments are considered , [ 100 ] and [ 110 ] , but the obtained mae are very similar . the closed circle represents the experimental value@xcite , which is well reproduced by our simulations . other experimental values can be 50% larger , see e.g. ref . to start our investigations , we revisited the bulk alloy phase by finding the tetragonal lattice structure minimizing its energy . the optimal value of c / a ratio is equal to 0.984 with a lattice parameter value a of 3.80 , in good agreement with values available in the literature ( see e.g. @xcite ) . the calculated magnetic moments ( @xmath12 = 1.91 @xmath13 , @xmath14 = 0.40 @xmath13 ) are also in line with previous works @xcite . we also note the well - known emergence of an induced moment in pt , which is due to the hybridization of its 5d orbitals with the exchange splitted 3d orbitals of co. in order to calculate the mae , we considered two possible directions for the in - plane moments orientations , [ 110 ] and [ 001 ] , and we found a negligible difference in the obtained perpendicular mae . for the optimized structure , the mae reaches a value of 0.91 mev when the in - plane orientation of moments is along [ 100 ] and 0.97 mev for [ 110 ] as an in - plane orientation of the moments . both values are close to the experimental values that are given around 1 mev @xcite . in fig.[mae_bulk ] , we plot the bulk mae as function of the ratio c / a under the constant unit cell volume in a similar fashion then that of sakuma @xcite . the obtained curve agrees well with the one published in the latter article . for ratios between 0.6 and 1.2 , the mae experiences one minimum and two maxima . the largest in - plane mae is found for a ratio of 0.8 . as expected , if c / a = @xmath15 , i.e. c / a@xmath16 = 1 , the mae drops to zero since this corresponds to a primitive cell of the cubic b2 structure . as discussed by sakuma , upon tetragonalization , the electronic states of the atoms are shifted and the band filling changes , which affect the magnitude and sign of the mae . [ cols="^ " , ] another path for the analysis of the calculated maes is to use of the celebrated bruno s formula @xcite , which translates to the neglect of spin - flip contributions to the mae as given in eq.[mae_perturbation ] : @xmath17 } } - l^i_{\mathrm{[100]}}),\ ] ] where @xmath18 labels the different atoms , and @xmath19 being the orbital magnetic moment calculated when the spin magnetic moment points along the [ 100 ] or the [ 001 ] directions . the essence of bruno s formula is to relate the mae to the orbital moment anisotropy ( oma ) , i.e. @xmath20 } } - l_{\mathrm{[001]}}$ ] , and leads to the conclusion that the orientation of the magnetic moments is favored when the orbital magnetic moment is maximized . this formula is known to work reasonably well when the majority - spin states are occupied . thus , its validity is probably limited to some of the co atoms discussed in this manuscript but certainly not for pt . it is however instructive to analyze the results obtained with this well known formulation since it should correlate with the previous discussion . in fig.[anisotropy_orbital](a ) , the omas in the bulk of copt is plotted for co and pt as function of the c / a ratio in fashion similar to that used in fig.[mae_bulk ] . one notices that the pt contribution counteracts the one of co. while the anisotropy of the co orbital moment increases with c / a , favoring thereby an out - of - plane orientation of the magnetic moment , the anisotropy of the pt orbital moment has an opposite slope and favors an in - plane orientation of the magnetic moment . when summing up the two curves , considering the spin - orbit coupling constant , @xmath3 , to be the same for co and pt , which is of course is not true since @xmath21 is one order of magnitude larger than @xmath22 , one recovers the shape of the curve obtained in fig,[mae_bulk ] , i.e. having a minimum of the curve at c / a = 0.8 . similar to the bulk , the behavior of the co and pt oma in copt thin films counteract each other . in general , the co oma favors an out - of - plane orientation of the moment contrary to the pt oma . in fig.[anisotropy_orbital](b ) , we plot the surface mae of the thin films characterized by @xmath23 as function of the pt concentration on the layer deposited on the co - terminated thin film . thus , in the case of one perfect pt overlayer the pt concentration is 100% , while the investigated anti - site corresponds to a pt concentration of 50% . for the specific case of co stacking fault , the pt concentration is -100% . the surface mae seems to increase with the pt concentration but not in a regular manner . we plot on the same figure the average co oma per co atom and find that this quantity increases smoothly in magnitude with pt concentration . in addition the contribution of the average pt oma per pt atom is shown in fig.[anisotropy_orbital](b ) . the pt oma seems to correlate the irregular behavior of the surface mae . interestingly , we find that thin films with large pt oma per pt atom compared to the co oma leads to an in - plane surface mae . only the pt - terminated thin film , with a large perpendicular surface mae , is characterized by a large co oma . we investigated from ab - initio the magnetic behavior of copt thin films as function of thickness considering different types of terminations : perfect co or pt layers or different types of defects : anti - site or stacking faults . after this systematic study , we found that the mae is the largest when the thin films are terminated by a perfect pt overlayer . surprisingly in the latter case , the mae can be 1000% times larger than the one of co - terminated thin films . we also find that all types of investigated defects reduce dramatically the mae . the surface mae experiences a sign change when increasing the thickness of several investigated films . except for the pt - terminated films , the surface mae favors an in - plane orientation of the moments when the thickness @xmath24 is smaller then four . we proceeded to an analysis of the electronic structure of the thin films with a careful comparison to the copt bulk case and related the behavior of the mae to the location of the different virtual bound states utilizing second order perturbation theory . finally , the correlation between the mae and the oma is studied . we are grateful to claude demangeat , vasile caciuc , julen ibanez - azpiroz and manuel dos santos dias for helpful discussions . also we thank hongbin zhang for discussion and for providing us his data . this work was supported by c. n. e. p. r. u project ( d 00520090041 ) of the algerian government , the hgf yig program vh - 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the impact of reduced dimensionality on the magnetic properties of the tetragonal * l1@xmath0 * copt alloy is investigated from ab - initio considering several kinds of surface defects . by exploring the dependence of the magnetocrystalline anisotropy energy ( mae ) on the thickness of copt thin films , we demonstrate the crucial role of the chemical nature of the surface . for instance , pt - terminated thin films exhibit huge maes which can be 1000% larger than those of co - terminated films . besides the perfect thin films , we scrutinize the effect of defective surfaces such as stacking faults or anti - sites on the surface layers . both types of defects reduce considerably the mae with respect to the one obtained for pt - terminated thin films . a detailed analysis of the electronic structure of the thin films is provided with a careful comparison to the copt bulk case . the behavior of the maes is then related to the location of the different virtual bound states utilizing second order perturbation theory .

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Proceed to summarize the following text: recent evolution of mobile devices such as smart - phones and tablets has facilitated access to multi - media contents anytime and anywhere but such devices result in an explosive data traffic increase . the cisco expects by 2019 that these traffic demands will be grown up to 24.3 exabytes per month and the mobile video streaming traffic will occupy almost 72% of the entire data traffic @xcite . interestingly , numerous popular contents are asynchronously but repeatedly requested by many users and thus substantial amounts of data traffic have been redundantly generated over networks @xcite . motivated by this , caching or pre - fetching some popular video contents at the network edge such as mobile hand - held devices or small cells ( termed as _ local caching _ ) has been considered as a promising technique to alleviate the network traffic load . as the cache - enabled edge node plays a similar role as a local proxy server with a small cache memory size , the local wireless caching has the advantages of i ) reducing the burden of the backhaul by avoiding the repeated transmission of the same contents from the core network to end - users and ii ) reducing latency by shortening the communication distance . in recent years , there have been growing interests in wireless local caching . the related research has focused mainly on i ) femto - caching with cache - enabled small cells or access points ( called as caching helpers ) @xcite , ii ) device - to - device ( d2d ) caching with mobile terminals @xcite , and iii ) heterogeneous cache - enabled networks @xcite . for these local caching networks , varieties of content placements ( or caching placements ) were developed @xcite and for given fixed content placement , the performance of cache - enabled systems with different transmission or cache utilization techniques was investigated @xcite . specifically , content placement to minimize average downloading delay @xcite or average ber @xcite was proposed for fixed network topology . in a stochastic geometric framework , various content placements were also proposed either to minimize the average delay @xcite and average caching failure probability @xcite or to maximize total hit probability @xcite , offloading probability @xcite . however , these caching solutions were developed in limited environments ; they discarded wireless fading channels and interactions among multiple users , such as interference and loads at caching helpers . recently , the content placement on stochastic geometry modeling of caching was studied in @xcite . a tradeoff between content diversity and cooperative gain according to content placement was discovered well in @xcite but the caching probabilities were determined with numerical searches only . moreover , in @xcite , cache memory size is restricted to a single content size and loads at caching helpers are not addressed . the optimal geographical caching strategy to maximize the total hit probability was studied in cellular networks in @xcite . however , only hit probability whether the requested content is available or not among the covering base stations was investigated . none of the previous works successfully addressed the channel selection diversity and interactions among multiple users such as network interference and loads according to content placement . success of content delivery in wireless cache network depends mainly on two factors : i ) _ channel selection diversity gain _ and ii ) _ network interference_. for given realization of nodes in a network , these two factors dynamically vary according to what and how the nodes cache at their limited cache memory . specifically , if the more nodes store the same contents , they offer the shorter geometric communication distance as well as the better small - scale fading channel for the specific content request , which can be termed as channel selection diversity gain . on the contrary , if the nodes cache all contents uniformly , they can cope with all content requests but channel selection diversity gain can not help being small . moreover , according to content placement , the serving node for each content request dynamically changes , so the network interference from other nodes also dynamically varies . thus , it might be required to properly control the channel selection diversity gain and network interference for each content . recently , in @xcite , a tradeoff between content diversity and channel diversity was addressed in caching helper networks , where each caching helper is capable of storing only _ one content_. however , although pathloss and small - scale fading are inseparable in accurately modeling wireless channels , the channel diversity was characterized with only small - scale fading and the effects of pathloss and network interference depending on random network geometry were not well captured . in this context , we address the problem of content placement with a more generalized model considering pathloss , network interference according to random network topology based on stochastic geometry , small - scale channel fading , and arbitrary cache memory size . in this generalized framework , we develop an efficient content placement to desirably control cache - based channel selection diversity and network interference . the main contributions of this paper are summarized as follows . * we model the stochastic wireless caching helper networks , where randomly located caching helpers store contents independently and probabilistically in their finite cache memory and each user receives the content of interest from the caching helper with the largest instantaneous channel power gain . our framework generalizes the previous caching helper network models @xcite by simultaneously considering small - scale channel fading , pathloss , network interference , and arbitrary cache memory size . * with stochastic geometry , we characterize the channel selection diversity gain according to content placement of caching helpers by deriving the cumulative distribution function of the smallest reciprocal of the channel power gain in a noise - limited network . we derive the optimal caching probabilities for each file in closed form to maximize the average content delivery success probability for given finite cache memory size , and propose an efficient algorithm to find the optimal solution . * in interference - limited networks , we derive a lower bound of the average content delivery success probability in closed form . based on this lower bound with rayleigh fading , we derive near - optimal caching probabilities for each content in closed form to appropriately control the channel selection diversity and the network interference depending on content placement . * our numerical results demonstrate that the proposed content placement is superior to other content placement strategies because the proposed method efficiently balances channel selection diversity and network interference reduction for given content popularity and cache memory size . we also numerically investigate the effects of the various system parameters , such as the density of caching helpers , nakagami fading parameter , memory size , target bit rate , and user density , on the caching probability . the rest of this paper is organized as follows . in section ii , we describe the system model and performance metric considered in this paper . we analyze the average content delivery success probability and desirable content placement of caching helpers in a noise- and interference - limited network in sections iii and iv , respectively . numerical examples to validate the analytical results and to investigate the effects of the system parameters are provided in section v. finally , the conclusion of this paper is given in section vi . we consider a downlink wireless video service network , where the caching helpers are capable of caching some contents in their limited caching storage , as depicted in fig . [ fig : system_model ] . we assume that all contents have the same size normalized to one for analytic simplicity . the caching helpers are randomly located and modeled as @xmath0-d homogeneous poisson point process ( ppp ) with intensity @xmath1 . the caching helpers are equipped with a single antenna and their cache memory size is @xmath2 , so @xmath2 different contents can be cached at each helper since each content has unit size . the total number of contents is @xmath3 and the set ( library ) of content indices is denoted as @xmath4 . the contents have own popularity and their popularity distributions are assumed to follow the zipf distribution as in literature @xcite : @xmath5 where the parameter @xmath6 reflects the popularity distribution skewness . for example , if @xmath7 , the popularity of the contents is uniform . the lower indexed content has higher popularity , i.e. , @xmath8 if @xmath9 . note that our content popularity profile is not necessarily confined to the zipf distribution but can accommodate any discrete content popularity distribution . the users are also randomly located and modeled as @xmath0-d homogeneous poisson point process ( ppp ) with intensity @xmath10 . based on slivnyak s theorem @xcite that the statistics observed at a random point of a ppp @xmath11 is the same as those observed at the origin in the process @xmath12 , we can focus on a reference user located at the origin , called a _ typical user_. in this paper , we adopt _ random content placement _ where the caching helpers independently cache content @xmath13 with probability @xmath14 for all @xmath15 . according to the caching probabilities ( or policies ) @xmath16 , each caching helper randomly builds a list of up to @xmath2 contents to be cached by the probabilistic content caching method proposed in @xcite . 2 presents an example of the probabilistic caching method @xcite and illustrates how a caching helper randomly chooses @xmath2 contents to be cached among total @xmath17 contents according to the caching probability @xmath16 when the cache memory size is @xmath18 and total number of contents is @xmath19 . in this scheme , the cache memory of size @xmath2 is equally divided into @xmath20 ( @xmath21 ) blocks of unit size . then , starting from content 1 , each content sequentially fills the @xmath2 discontinuous memory blocks by the amount of @xmath22 from the first block . if a block is filled up in the filling process of content @xmath23 , the remaining portion of content @xmath23 continuously fills the next block . then , we select a random number within @xmath24 $ ] and the contents at the position specified by the random number in each block are selected . because one content is selected from each block by the selected random number , total @xmath20 ( @xmath21 ) contents can be selected in a probabilistic sense according to @xmath16 . in this way , in fig . [ rev : caching_explain ] , the contents @xmath25 are chosen to be cached . the contents selected in a probabilistic sense at each helper are cached in advance by either its request or overhearing . the caching helpers storing content @xmath13 can be modeled as independent ppp with intensity @xmath26 and the locations of the caching helpers storing content @xmath13 can be represented by @xmath27 where @xmath28 . the typical user requests one among @xmath17 contents according to the content popularity @xmath29 ; the content with a higher popularity is requested with higher likelihood . if the typical user requests content @xmath13 and selects a serving helper to maximize the instantaneous channel power gain among the helpers storing content @xmath30 , the received signal power becomes @xmath31 where @xmath32 is the transmit power of a caching helper , @xmath33 and @xmath34 denote the channel fading coefficient and the distance from the typical user to the caching helper located at @xmath35 , respectively , and @xmath36 is the path loss exponent . for each content @xmath13 , we denote a set of the reciprocals of the channel power gains from @xmath37 to the typical user in ascending order as @xmath38 , where @xmath39 . the notation @xmath40 and @xmath41 represent the distance and the channel fading coefficient from the typical user to the caching helper with the @xmath42-th smallest reciprocal channel power gain among the caching helpers storing content @xmath13 , respectively . note that the caching helper with the largest instantaneous channel power gain is equivalent to that with the smallest reciprocal of the channel power gain ( i.e. , @xmath43 ) . assuming gaussian signaling and time / frequency resource sharing among the users associated with the same caching helper , the mutual information between the typical user requesting content @xmath13 and its serving caching helper is @xmath44 where @xmath45 is the load of the serving caching helper , @xmath46 is the noise power variance , and @xmath47 is the interference received at the typical user , given by @xmath48 where @xmath49 is the set of caching helpers which do not cache content @xmath13 in their cache memory . the small - scale channel fading terms of the desired link and the interfering links follow the independent nakagami - m distributions with parameters @xmath50 and @xmath51 , respectively . similar to @xcite , we define the average content delivery success probability as a performance metric to properly account for the success events of content delivery as @xmath52,\label{def : ftsp}\end{aligned}\ ] ] where @xmath29 is the content requesting probability and @xmath53 is the target bit rate of content @xmath13 [ bits / s / hz ] to successfully support the real - time video streaming service of content @xmath13 without playback delay . in this section , in order to investigate how channel selection diversity affects the optimal caching solution , we first consider a noise - limited network ; when the number of active users is much smaller than the number of caching helpers , the impact of interference is negligible compared to the noise power and the typical user can be served without resource sharing with other users . in noise - limited networks , assuming gaussian signaling , the mutual information between the typical user requesting content @xmath13 and its serving helper is obtained as @xmath54 where @xmath55 is the signal - to - noise ratio ( snr ) . the power gain distribution of a nakagami-@xmath56 fading channel is given by @xmath57 where @xmath58 is the gamma function , @xmath59 , and @xmath60 is the fading parameter for link @xmath61 where @xmath62 represents either the desired link ( @xmath63 ) or the i.i.d . interfering links ( @xmath64 ) . if @xmath65 , the power gain distribution follows the exponential distribution corresponding to rayleigh fading . for @xmath66 , the channel is a deterministic channel . when the typical user receives content @xmath13 from the caching helper with the smallest reciprocal of the channel power gain ( i.e. , the largest channel power gain ) , the cumulative distribution function ( cdf ) of the smallest reciprocal of the channel power gain ( i.e. , @xmath43 ) is derived in lemma 1 . the cdf of the smallest reciprocal of the channel power gain , @xmath43 , in a nakagami-@xmath50 fading channel is given by @xmath67 where @xmath68 and @xmath69 . for @xmath30 , let @xmath70 be the path losses between the typical user and the caching helpers caching content @xmath13 . from the mapping theorem [ theorem 2.34 , @xcite ] , @xmath71 is a non - homogeneous ppp and its intensity function is given by @xmath72 where @xmath73 . note that @xmath74 are also mutually independent due to independence among @xmath75 . using the displacement theorem [ theorem 2.33 , @xcite ] , we can also derive the intensity function of @xmath76 for a general nakagami-@xmath50 fading channel as @xmath77 since the ppp of @xmath37 is transformed by the displacement and mapping theorems , @xmath78 is also a ppp @xcite . therefore , the cdf of @xmath43 is obtained as @xmath79 where @xmath80 denotes the number point of @xmath78 in a circle with a radius @xmath81 and @xmath68 . _ remark : _ as @xmath82 or @xmath83=\frac{\gamma(\delta+m_d)}{m_d^{\delta}\gamma(m_d)}$ ] increases , the cdf of @xmath43 grows faster to 1 because the intensity @xmath84 of ppp @xmath78 is proportional to them . in other words , as the number of caching helpers that are storing the content of interest and accessible by the typical user increases or the small - scale fading channel becomes more deterministic , the intensity of ppp @xmath78 representing the reciprocal channel power gains grows and thus the smallest reciprocal @xmath43 becomes smaller . especially , for given @xmath1 and @xmath50 , the largest channel power gain ( i.e. , @xmath85 ) grows as @xmath22 increases , which implies an increase of the _ channel selection diversity gain _ according to the content placement . [ fig : cdf ] validates the accuracy of lemma 1 for varying @xmath1 and @xmath50 . the cdf of @xmath43 increases faster to 1 as either @xmath1 or @xmath50 increases . however , the cdf of @xmath43 depends more on @xmath1 than on @xmath50 , so optimal caching probabilities are affected more by the density of caching helpers than channel fading . from lemma 1 , the average success probability for content delivery is derived in the following theorem . when the typical user receives content @xmath13 from the caching helper with the largest instantaneous channel power gain , the average success probability for content delivery @xmath86 in a nakagami-@xmath50 fading channel is obtained as @xmath87 where @xmath68 , @xmath69 , @xmath88 , and @xmath53 is the target bit rate of content @xmath13 . @xmath89 & = \mathbb{p}\left[\log_2\big(1 + \frac{\eta}{\xi_{i,1 } } \big)\geq \rho_i\right]\\ & = \mathbb{p}\left[\xi_{i,1 } \leq \frac{\eta}{2^{\rho_i } - 1}\right]\\ & = f_{\xi_{i,1}}\left(\frac{\eta}{2^{\rho_i } - 1}\right)\\ & = 1-e^{-\kappa p_i\left(\frac{\eta}{2^{\rho_i}-1}\right)^{\delta}},\label{eqn : success_prob}\end{aligned}\ ] ] where is obtained from lemma 1 . substituting into , we obtain . from lemma 1 , we know that the channel selection diversity gain for a specific content increases as the number of caching helpers storing the content increases , i.e. , @xmath22 increases . however , due to limited memory space @xmath2 , i.e. , the constraint @xmath90 , storing the same content at more caching helpers ( @xmath22 increases ) loses the chance of storing the other contents and the corresponding channel diversity gains . in this subsection , we derive the optimal solution of problem @xmath92 , the optimal caching probabilities , in closed form . for each @xmath13 , the function @xmath93 is convex with respect to @xmath22 since @xmath94 . since a weighted sum of convex functions is also convex function , problem @xmath92 is a constrained convex optimization problem and thus a unique optimal solution exists . the lagrangian function of problem @xmath92 is @xmath95 where @xmath96 is a constant , @xmath97 and @xmath98 are the nonnegative lagrangian multipliers for constraints and . after differentiating @xmath99 with respect to @xmath22 , we can obtain the necessary conditions for optimal caching probability , i.e. , _ karush - kuhn - tucker_(kkt ) condition as follows : @xmath100 from the constraint in , the optimal caching probabilities are given by @xmath101^{+}\\ & = \frac{1}{\kappa t_i}\left[\log\left(f_i\kappa t_i\right)-\log\left(\omega \!+\ ! \mu_i\right)\right]^{+ } , ~\forall i \!\in\ ! \mathcal{f},\label{eqn : opt}\end{aligned}\ ] ] where @xmath102^{+}=\max\{z,0\}$ ] . the caching probability of content @xmath13 grows as the content popularity @xmath29 becomes large , but is regulated by the term of @xmath103 . for the constraint in , @xmath97 is not necessarily zero because the optimal solution should always satisfy @xmath104 . based on the kkt conditions in - , lagrangian multipliers @xmath97 and @xmath98 range , according to @xmath22 , as , which is placed at the top of next page . @xmath105^{+ } & ~~\textrm{for}~~p_i=1,\\ f_i\kappa t_ie^{-\kappa t_i}<\omega < f_i\kappa t_i,&\mu_i=0~ & ~~\textrm{for}~~0<p_i<1,\\ \omega \geq f_i\kappa t_i,&\mu_i=0 ~ & ~~\textrm{for}~~p_i=0 . \label{multiplier_range } \end{array } \right.\end{aligned}\ ] ] reveals that the caching probability @xmath22 is determined according to lagrangian multiplier @xmath97 only since @xmath98 is a function of @xmath97 ; if @xmath106 where @xmath107 , then @xmath108 and thus @xmath109 . if @xmath110 where @xmath111 , then @xmath112 and thus @xmath113 . when @xmath114 , @xmath115 is bounded by @xmath116 since @xmath115 is decreasing with respect to @xmath97 . therefore , using the fact that @xmath117 for the optimal @xmath118 , one - dimensional bisection search can find the optimal @xmath118 and the corresponding @xmath119 given by @xmath120^{+},1\right),~\forall i\in\mathcal{f}.\label{opt_sol_noise}\end{aligned}\ ] ] the proposed algorithm to find the optimal caching probabilities @xmath119 is presented in algorithm 1 . consequently , the content delivery success probability maximized with @xmath119 becomes @xmath121.\end{aligned}\ ] ] [ cols="<",options="header " , ] in the previous section , the cache - based channel selection diversity gain for each content has been highlighted and the optimal caching probabilities to balance them were derived without consideration of interference . in this section , in the presence of network interference , we derive near - optimal content placement and analyze the effects of network interference on the content placement . we assume that the density of users is much higher than that of caching helpers , i.e. , @xmath122 , so the effect of noise is almost negligible relative to interference . when the typical user receives content @xmath13 from the caching helper with the smallest reciprocal of the instantaneous channel power among the caching helpers storing content @xmath13 , the other caching helpers interfere with the typical user because they are assumed to serve other users . then , the received signal - to - interference ratio ( sir ) at the typical user is represented as @xmath123 where @xmath47 is the interfering signal power and given by @xmath124 where @xmath49 is a set of the caching helpers which do not cache content @xmath13 and @xmath78 is a set of the reciprocals of the channel power gains from @xmath37 . note that the interfering signal power dynamically changes according to content placement of caching helpers since it is a function of @xmath43 and @xmath125 . therefore , optimal caching probabilities are expected to be obtained by optimally controlling channel selection diversity and network interference for given content popularity and cache memory size . in interference - limited networks , the average success probability of content delivery in is represented by @xmath126 , \label{eqn : avg_ps_inter}\end{aligned}\ ] ] where @xmath127 is a random load of the tagged caching helper when an arbitrary user receives content @xmath13 from the caching helper with the largest instantaneous channel power gain . to characterize , we require both the probability mass function ( pmf ) of the load at the tagged caching helper and the sir distribution when multiple contents are cached at each helper and the association is based on the instantaneous channel power gains . however , unfortunately , the exact statistics of the required information are unavailable because they are complicatedly determined by many interacting factors , such as multiple cached contents , locations of caching helpers and users , content request of users , instantaneous channel fading gains , etc . thus , the optimal caching probabilities to maximize have to be found by numerical searches of which complexity is prohibitively high for a huge number of contents . in this context , we propose near - optimal content placement to obtain some useful insights in interference - limited scenarios . to this end , we first approximate with the average load of the tagged caching helper @xcite as @xmath128,\label{rev:5_1}\end{aligned}\ ] ] where @xmath129 is the average load of the tagged caching helper when the user requests content @xmath13 to the caching helper with the largest instantaneous channel power gain . the validity of approximation is demonstrated in fig . [ fig : approx_check ] , where red star and blue circle represent the monte - carlo simulation and its approximation , respectively . this figure verifies that the approximation is quite tight to for arbitrary @xmath130 . moreover , a lowerbound of is obtained in the following theorem . when the typical user receives the requesting content from the caching helper with the smallest reciprocal of instantaneous channel power gain , the average success probability of content delivery is bounded below by @xmath131 where @xmath132 , @xmath133 is a constant independent of @xmath13 and makes the inequality hold for all ranges of @xmath16 , @xmath50 and @xmath51 are the nakagami fading parameters of the desired and interfering links , respectively , and @xmath134vdv\right.\nonumber\\ & \left.+~2\pi p_i\lambda\int_0^r \!\left[1 -\frac{m_i}{(spv^{-\alpha}+m_i)^{m_i}}\right]vdv\right),\\ f_{|x_i|}(r ) & = 2\pi p_i\lambda r\exp\left(-\pi p_i\lambda r^2\right).\end{aligned}\ ] ] see appendix [ appendixa ] . based on the lower - bounded average success probability of content delivery , we formulate an alternative optimization problem as @xmath135 although it is still non - trivial to obtain the solution of this alternative optimization problem , fortunately , when @xmath136 , i.e. , a rayleigh fading channel , the objective function ( i.e. , the lower bound of delivery success probability ) becomes more tractable and sheds light on intuitively understanding the impacts of network interference on content placement . therefore , in the following subsection , we focus on the case of @xmath136 ( i.e. , rayleigh fading ) . for rayleigh fading channels ( i.e. , @xmath136 ) , the lower - bound of delivery success probability in is simplified as @xmath137 where @xmath132 , @xmath138 , @xmath139 and @xmath140 is the gauss hypergeometric function . we omit the proof since it can be readily obtained by substituting @xmath136 in theorem 2 . with arbitrary cache memory size of @xmath2 at each helper , the alternative optimization problem * p2 * is rewritten as @xmath141 now we show that the objective function in * p3 * is concave and optimization problem * p3 * is also the constrained convex optimization problem . if we define @xmath142 as @xmath143 where @xmath144 and @xmath145 , its first derivative is @xmath146 ^ 2 } > 0 $ ] because @xmath147 always holds and @xmath148 for @xmath149 . note that @xmath150 for all @xmath13 because @xmath151 the second derivative of @xmath142 is @xmath152 ^ 3}\leq 0 $ ] and thus @xmath142 is a strictly increasing concave function . since a weighted sum of concave functions still satisfies concavity , optimization problem * p3 * is a constrained convex optimization problem . applying the same approach in section [ section : opt ] , we obtain the optimal caching probability of problem * p3 * as @xmath153^{+}\!\!\!\!,~\forall i\!\in\!\mathcal{f},\\ & = \frac{1}{1 \!-\ ! a_i}\left[-b_i+\sqrt{\frac{f_ib_i}{\omega^{\star}\!+\!\mu_i^{\star}}}~\right]^{+}\!\!\!,~\forall i\!\in\!\mathcal{f } , \label{eqn : opt2}\end{aligned}\ ] ] where lagrangian multipliers @xmath97 and @xmath98 range , according to @xmath22 , as , which is placed at the top of next page . replacing with and letting @xmath155 and @xmath156 in algorithm 1 , we can find the optimal @xmath118 and @xmath157 with one - dimensional bisection search and the corresponding near - optimal caching probabilities @xmath158 given by @xmath159^{+}\!\!,~1\right),~\forall i\!\in\!\mathcal{f}. \label{eqn : opt_sol_interlimited}\end{aligned}\ ] ] _ remark : _ unlike noise - limited networks , the solution of content placement obtained in is independent of the transmit power of caching helpers . the caching probability is a function of @xmath160 , @xmath29 and @xmath132 . in other words , the content placement is determined by the pathloss exponent , content popularity , and target bit rate . in this section , we evaluate the average success probability of content delivery to validate our analytical results in the previous sections . we also examine how various system parameters , such as @xmath161 , content popularity exponent ( @xmath162 ) , nakagami fading parameter ( @xmath50 and @xmath51 ) , pathloss exponent ( @xmath160 ) , density of caching helpers ( @xmath1 ) , user density ( @xmath10 ) , maximum target content bit rate ( @xmath163 ) , and cache memory size ( @xmath2 ) affect on caching probabilities . unless otherwise stated , the baseline setting of simulation environments is as follows : @xmath164 , @xmath165 , @xmath18 , @xmath136 , @xmath166 = 20 ( db ) , @xmath167 , @xmath168 ( units/@xmath169 ) , @xmath170 ( units/@xmath169 ) and @xmath171 ( bits / s / hz ) . the target bit rate for each content is uniformly generated as @xmath172 $ ] and all simulation results are averaged over 10,000 realizations . [ fig : caching_comparison ] compares the average success probabilities of content delivery in a noise - limited network for three different content placement strategies ; i ) caching the @xmath2 most popular contents ( mpc ) , ii ) caching the contents uniformly ( uc ) , and iii ) proposed content placement found by algorithm 1 ( proposed ) . this figure demonstrates that the proposed content placement in is superior to both uc and mpc in terms of average success probability of content delivery . mpc is closer to the proposed content placement than uc for high @xmath162 , and vice versa for low @xmath162 . for varying @xmath1 and @xmath50 , the optimal caching probability of each content @xmath13 in a noise - limited network is plotted in fig . [ fig : opt_sol_lambda_m ] , where the lower index indicates the higher popularity , i.e. , @xmath8 if @xmath173 . as @xmath1 or @xmath50 increases , the optimal caching probability becomes more uniform . it implies that it would be beneficial to increase hitting probability for all contents instead of focusing on channel selection diversity for a few specific contents . this is because channel power gains become higher as either the number of caching helpers increases or channels become more deterministic although channel selection diversity can be limited . this figure also exhibits that the optimal caching probability depends more on the geometric path loss than on small - scale fading , which matches the implication of fig . [ fig : cdf ] . fig . [ fig : opt_sol_target_bit_rate ] shows the optimal caching probability of each content @xmath13 in a noise - limited network for varying maximum target bit rate @xmath163 . as @xmath163 grows , the optimal caching probability becomes biased toward caching the most popular contents . if @xmath163 is large , increasing channel selection diversity gains of the most popular contents is more beneficial to improve success probability of content delivery . in fig . [ fig : opt_sol_m ] , the optimal caching probability of each content @xmath13 in a noise - limited network is plotted for varying cache memory size @xmath2 . the optimal caching probabilities scale with the cache memory size @xmath2 , but they become more uniform as @xmath2 increases . this is because less popular contents are accommodated in memory of larger size . [ fig : ps_with_opt_and_subopt ] compares the average success probabilities of content delivery with optimal @xmath174 obtained from by brute - force searches , with the proposed sub - optimal @xmath175 obtained from * p3 * , and the lower bound with the sub - optimal @xmath175 versus @xmath176 , when @xmath177 ( units/@xmath169 ) , @xmath164 , @xmath178 , and @xmath179 . for each @xmath180 and @xmath10 , the value of @xmath181 for a tighter lower bound is numerically found . since the content placement obtained from the lower bound is sub - optimal , the average content delivery success probability with the sub - optimal @xmath175 is bounded below that with optimal @xmath174 . although there is a large gap between the lower bound in and @xmath86 , the gap between the average content delivery success probabilities with the optimal @xmath174 and the proposed @xmath175 is small for an arbitrary target bit rate because and have quite similar shapes . consequently , the proposed sub - optimal caching probability is close to optimal caching probability although the sub - optimal caching probability is found from the lower bound in . [ fig : inter_comparision ] compares the average content delivery success probabilities among the proposed content placement schemes with numerically found @xmath181 yielding a tight lower bound and with @xmath182 , uc , and mpc versus the content popularity exponent @xmath162 . although the value of @xmath181 needs to be numerically found , any suboptimal solution even with the value @xmath181 which does not always satisfy the inequality in yields a lower average success probability of content delivery because of its suboptimality . from this fact , a suboptimal solution can be found by setting the value of @xmath181 to be the average load of a typical caching helper as @xmath183 for simplicity . [ fig : inter_comparision ] demonstrates that that both the proposed content placement schemes with numerically found @xmath181 and @xmath182 are superior to both uc and mpc in terms of average content delivery success probability for general @xmath162 . the average content delivery success probability with @xmath183 is quite similar to that with numerically found @xmath181 and outperforms uc and mpc . in an interference - limited network , for varying user density @xmath10 , the proposed caching probability of each content @xmath13 obtained by solving the convex optimization problem in * p3 * is plotted in fig . [ fig : inter_opt_sol_user ] , where the value of @xmath181 yielding a tight lower bound is numerically found . as the user density @xmath10 decreases , the optimal content placement tends to cache all contents with more uniform probabilities . we studied probabilistic content placement to desirably control cache - based channel selection diversity and network interference in a wireless caching helper network , with specific considerations of path loss , small - scale channel fading , network interference according to random network topology based on stochastic geometry , and arbitrary cache memory size . in a noise - limited case , we derived the optimal caching probabilities for each content in closed form in terms of the average success probability of content delivery and proposed a bisection based search algorithm to efficiently reach the optimal solution . in an interference - limited case , we derived a lower bound on the average success probability of content delivery . then , we found the near - optimal caching probabilities in closed form in rayleigh fading channels , which maximize the lower bound . our numerical results verified that the proposed content placement is superior to the conventional caching strategies because the proposed scheme efficiently controls the channel selection diversity gain and the interference reduction . we also numerically analyzed the effects of various system parameters , such as caching helper density , user density , nakagami @xmath184 fading parameter , memory size , target bit rate , and user density , on the content placement . since the pathloss dominates the small - scale fading effects according to lemma 1 , @xmath129 is approximated as the load of the tagged caching helper with the largest channel power gain averaged over fading ( i.e. , the load based on the association with long - term channel power gains ) , @xmath185 . moreover , the received sir with the association based on instantaneous channel power gains is larger than that with the association based on long - term channel power gains . accordingly , can be further bounded below as @xmath186,\label{rev:5_2}\end{aligned}\ ] ] where @xmath187 which is also validated in fig . [ fig : approx_check ] , where blue circle and green solid line represent and , respectively . in case of @xmath178 , a closed form expression of @xmath188 is available as @xmath189 @xcite , but with multiple contents ( @xmath190 ) analytic evaluation of is hard due to the complicated form of @xmath188 . to circumvent this difficulty , we again take a lower bound of as @xmath191 , \label{rev:5_3}\end{aligned}\ ] ] where @xmath133 is a constant independent of @xmath13 and makes the inequality hold for all ranges of @xmath16 , and @xmath132 . note that since is a decreasing function with respect to @xmath181 and bounded below by zero , there must exist a certain value of @xmath192 which makes the inequality hold . the value of @xmath181 yielding a tight lower bound can be numerically determined ; in general @xmath181 becomes larger as @xmath193 diminishes and @xmath162 grows . [ fig : approx_check ] validates , where green and black dotted lines represent and our lower bound in , respectively . it is verified that there exists a finite value of @xmath181 yielding a lower bound of regardless of @xmath16 . in our setting , the value of @xmath181 for a tighter lower bound is @xmath194 . although there exists a gap between and its lower bound , the shape of those two functions looks quite similar and thus the caching probabilities obtained from are close to the optimal caching probabilities . & \stackrel{(a)}{=}\sum_{i=1}^f f_i \int_0^{\infty}\!\mathbb{e}_{i_i}\!\left[\frac{\gamma(m_d , m_dp^{-1}\tau_i r^{\alpha}i_i)}{\gamma(m_d)}\right]\ ! f_{|x_i|}(r)dr , \label{lower_aftsp}\end{aligned}\ ] ] where @xmath132 , @xmath196 is the gamma function defined as @xmath197 , @xmath198 is the upper incomplete gamma function defined as @xmath199 , @xmath200 is the location of the nearest caching helper storing content @xmath13 , @xmath201 is the pdf of the distance to the nearest caching helper storing content @xmath13 , and @xmath202 the equality ( a ) is obtained from the nakagami-@xmath50 fading channel power gain . since @xmath203}{\gamma(m)}=e^{-my}\sum_{k=0}^{m-1}\frac{m^k}{k!}y^k$ ] , we have @xmath204\\ & = \sum_{k=0}^{m_d-1}\frac{1}{k!}\left(m_d p^{-1}\tau_i r^{\alpha } \right)^k\mathbb{e}_{i_i}\left[i_i^ke^{-m_dp^{-1}\tau_i r^{\alpha}i_i}\right]\\ & \stackrel{(b)}{=}\!\sum_{k=0}^{m_d-1}\!\frac{1}{k!}\left(-m_d p^{-1}\tau_i r^{\alpha}\right)^k \!\frac{d^k}{ds^k}\mathcal{l}_{i_i}(s)|_{s=\frac{m_d\tau_i r^{\alpha}}{p } } , \label{inner}\end{aligned}\ ] ] where ( b ) is from @xmath205 and @xmath206 is the laplace transform of @xmath207 given by @xmath208 = \mathbb{e}\left[e^{-s\sum_{y\in \phi\setminus x_i}p|h_y|^2|y|^{-\alpha}}\right]\\ & \stackrel{(c)}{=}\mathbb{e}\left[\prod_{y\in \phi\setminus x_i } \mathbb{e}_{|h_y|^2}\left[e^{-sp|h_y|^2|y|^{-\alpha}}\right]\right]\\ & \stackrel{(d)}{= } \exp\left(-2\pi p_i\lambda\int_{r}^{\infty}\left[1-\mathbb{e}_g\left[e^{-spgv^{-\alpha}}\right]\right]vdv\right)\nonumber\\ & ~~~\times\exp\!\left(\!-2\pi(1 \!-\ ! p_i ) \lambda\!\int_0^{\infty}\!\left[1\!-\!\mathbb{e}_g\!\left[e^{-spgv^{-\alpha}}\right]\right]vdv\!\right)\\ & \stackrel{(e)}{= } \exp\left(-2\pi p_i\lambda\int_{r}^{\infty}\frac{(spv^{-\alpha}+m_i)^{m_i}-m_i}{(spv^{-\alpha}+m_i)^{m_i}}vdv\right)\nonumber\\ & ~\times\exp\!\left(\!-2\pi(1\!-\!p_i ) \lambda\!\!\int_0^{\infty}\!\!\frac{(spv^{-\alpha}\!+\ ! m_i)^{m_i}}vdv\!\right)\\ & = \exp\left(-2\pi\lambda\int_0^{\infty}\frac{(spv^{-\alpha}+m_i)^{m_i}-m_i}{(spv^{-\alpha}+m_i)^{m_i}}vdv\right)\nonumber\\ & ~~~\times\exp\left(2\pi p_i\lambda\int_0^r\frac{(spv^{-\alpha}+m_i)^{m_i}-m_i}{(spv^{-\alpha}+m_i)^{m_i}}vdv\right ) , \label{laplace}\end{aligned}\ ] ] where ( c ) is due to independence of the channel ; ( d ) comes from the probability generating functional ( pgfl ) of ppp ; ( e ) is from the mogment generating function ( mgf ) of the nakagami-@xmath51 distribution . substituting into , we obtain @xmath209 where @xmath50 and @xmath51 are the nakagami fading parameters of the desired and interfering links , respectively , and @xmath210vdv\right.\nonumber\\ & \left.+2\pi p_i\lambda\!\int_0^r \left [ 1 - \frac{m_i}{(spv^{-\alpha}+m_i)^{m_i}}\right]vdv\right),\\ f_{|x_i|}(r ) & = 2\pi p_i\lambda r\exp\left(-\pi p_i\lambda r^2\right).\end{aligned}\ ] ] 1 cisco , `` cisco visual networking index : global mobile data traffic forecast update , 2014 - 2019 , '' available at http://www.cisco.com . n. golrezaei , a. f. molisch , a. g. dimakis , and g. caire , `` femtocaching and device - to - device collaboration : a new architecture for wireless video distribution , '' _ ieee commun . mag . 142 - 149 , apr . 2013 . k. shanmugam , n. golrezaei , a. g. dimakis , and a. f. molisch , and g. caire , `` femtocaching : wireless content delivery through distributed caching helpers , '' _ ieee trans . inform . theory _ 8402 - 8413 , dec . 2013 . j. song , h. song , and w. choi , `` optimal caching placement of caching system with helpers , '' in proc . _ ieee int . 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content delivery success in wireless caching helper networks depends mainly on cache - based channel selection diversity and network interference . for given channel fading and network geometry , both channel selection diversity and network interference dynamically vary according to what and how the caching helpers cache at their finite storage space . we study probabilistic content placement ( or caching placement ) to desirably control cache - based channel selection diversity and network interference in a stochastic wireless caching helper network , with sophisticated considerations of wireless fading channels , interactions among multiple users such as interference and loads at caching helpers , and arbitrary memory size . using stochastic geometry , we derive optimal caching probabilities in closed form to maximize the average success probability of content delivery and propose an efficient algorithm to find the solution in a noise - limited network . in an interference - limited network , based on a lower bound of the average success probability of content delivery , we find near - optimal caching probabilities in closed form to control the channel selection diversity and the network interference . we numerically verify that the proposed content placement is superior to other comparable content placement strategies . probabilistic content placement , caching probability , stochastic geometry , channel selection diversity

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Proceed to summarize the following text: the calar alto observatory is located at 2168 m height above the sea level , in the sierra de los filabres ( almeria - spain ) at 45 km from the mediterranean sea . it is the second largest european astronomical site in the northern hemisphere just behind the observatorio del roque de los muchachos ( located in the island of la palma ) , and the most important in the continental europe ( with excellent communications , making logistics easy , unexpensive and reliable ) . currently there are six telescopes located in the complex , three of them directly operated by the centro astronmico hispano alemn a.i.e . , a partnership between the spanish national research council ( csic ) and the german max - plank society ( mpg ) . these telescopes include the zeiss 3.5 m , the largest telescope in the continental western europe . the observatory is under operations since 1975 , when its 1.23 m zeiss reflector saw first light . the observatory operates a very large array of optical and near - infrared astronomical instrumentation , including imagers and spectrographs with different field - of - view and resolutions . there has been different attempts to characterize some of the main astronomical properties during its 35 years of operations : ( i ) leinert et al . ( 1995 ) determined the sky brightness corresponding to the year 1990 ; ( ii ) hopp & fernandez ( 2002 ) studied the extinction curve corresponding to the years 1986 - 2000 ; ( iii ) ziad et al . ( 2005 ) estimated the median seeing in the observatory from a single campaign in may 2002 ; and more recently ( iv ) snchez et al . ( 2007 ) where the optical night sky spectrum was presented , including an analysis of the light pollution , together with a more accurate estimation of the night - sky extinction , the typical seeing , the night - sky brightness and the fraction of useful time ; and ( v ) snchez et al . ( 2008 ) , where the night sky brightness in the near - infrared and the fraction of useful time was presented . several of these features are discussed below . the comprehensive database for the weather is public and it can also be obtained upon request . * left.- * extinction due to dust at calar alto . the values are comparatively smaller than representative values of observatories closer to the equator , both in the northern and the southern hemispheres . the behaviour is highly seasonal . * right.- * seeing distribution ( june 2001 sept . 2005 ) at the calar alto observatory . the median value is about 0.9 arcsec ( snchez et al . , title="fig:",width=257 ] * left.- * extinction due to dust at calar alto . the values are comparatively smaller than representative values of observatories closer to the equator , both in the northern and the southern hemispheres . the behaviour is highly seasonal . * right.- * seeing distribution ( june 2001 sept . 2005 ) at the calar alto observatory . the median value is about 0.9 arcsec ( snchez et al . , title="fig:",width=283 ] the optical spectrum at the calar alto observatory shows a strong contamination produced by different chemicals , in particular from mercury lines , which contribution to the sky - brightness in the different bands is of the order of 0.09 mag , 0.16 mag and 0.10 mag in b , v and r respectively . regarding the strength of the sodium pollution line in comparison with the air - glow emission , it is very strong , a problem which we expect to address in the near future in collaboration with the regioanl governament and the nearby towns . in any case , the observatory complies with the iau recommendations of a dark site . as a matter of fact , caha is classified as a class c site `` major observatory sites with operating telescopes having apertures @xmath02.5 m and zenith light pollution levels less than the natural variation in night - sky brightness associated with the 11-year solar cycle '' ] . a light - pollution regulation was been recently approved by the andalusian regional government , and the observatory is involved in its development , in order to reduce and revert the effect of human light pollution . the effect of such laws have been strong in other astronomical sites , as it was demonstrated for the kitt - peak observatory . we expect a similar influence in the next years . therefore , the darkness of the observatory should improve considerably in the near future , making it even more attractive for new instrumentation . * left.- * wind - gust at calar alto observatory during the last 13 years . note the logarithmic scale in the frequency . the absolute velocity maximum is below 47 m / s . * right.- * the pie chart shows the distribution of time spent at different velocities . , title="fig:",width=298 ] * left.- * wind - gust at calar alto observatory during the last 13 years . note the logarithmic scale in the frequency . the absolute velocity maximum is below 47 m / s . * right.- * the pie chart shows the distribution of time spent at different velocities . , title="fig:",width=257 ] the zenith - corrected values of the moonless night - sky surface brightness , for a typical dark night , are 22.39 , 22.86 , 22.01 , 21.36 , 19.25 , 15.95 , 13.99 , and 12.39 mag arcsec2 in u , b , v , r , i , j , h and k bands , which indicates that calar alto is a particularly dark site for optical and near - ir bands . these values are similar to those at other astronomical sites , including paranal , la silla , la palma and mauna kea . only the last one is clearly darker in the near - ir bands than any of the others . the typical extinction in the observatory in the v - band is 0.15 mag in the winter season , with little dispersion . in summer the extinction has a wider range of values , due to an increase of the aerosol extinction ( dust grains ) , although it does not reach the extreme peaks observed at other sites ( figure 1a ) . in particular , the extinction in the summer season is much lower than those reported for the observatories in the canary islands ( benn & ellison 1998 ) , where the influence of the sahara desert in that archipelago makes the extinction to rise above 0.25 mag in the v - band for a 20% of the time in this season . the presence of ozone is negligible at the calar alto observatory . the derived extinction curve shows that the typical extinction in the u - band is 0.4 - 0.5 mag , a remarkable good value for observatories at this height . the distribution of seeing was derived using the information gathered by the seeing monitor installed in the observatory for more than 4 years ( 300.000 individual measurements ) , and the science data from the alhambra ( moles et al . 2008 ) survey , from @xmath15000 individual frames . in both cases it was obtained that the median seeing was about 0.9 " , with a sub - arcsecond seeing in @xmath170% of the time ( figure 1b ) . the historical record in the observatory shows the maximum registered wind speed was 47 m / s in the last 28 years . however , values above 25 m / s are very rare , and these events are smooth and last for short periods of time . figure 2 displays the frequency of the gust - wind , and the duration for each type of event . as can be seen , there is no sudden variations . this fact makes the observatory a excellent site for large infrastructures . the fraction of astronomical useful time at the observatory was derived independently using the information obtained using the time when the extinction monitor is under operation , and the time when the telescopes were open for a long term project ( alhambra , 358 nights distributed along 6 years , moles et al . a total of 70% of total time was useful to perform astronomical observations under the following criteria : no obvious clouds or cirrus , extinction under 0.2 mag in the v - band , relative humidity under 95% . the fraction of complete clear nights ( more than six continuous hours of useful time ) , drops to a 50% . finally , a 30% of the nights were photometric , defined as nights for which the extinction was stable within the a 20% of the average value ( ie . , it was possible to perform direct photometric calibration with an accuracy of 0.2 mags ) . * left.- * clear nights at calar alto observatory during the period 1976 - 2009 . clear nights are defined as those having at least 6 observing hours of clear or mostly clear sky ( clouds cover not more than 25% ) . * right.- * total number of hours per year when astronomical observations were carried out , for the period 1976 - 2009 . , title="fig:",width=298 ] * left.- * clear nights at calar alto observatory during the period 1976 - 2009 . clear nights are defined as those having at least 6 observing hours of clear or mostly clear sky ( clouds cover not more than 25% ) . * right.- * total number of hours per year when astronomical observations were carried out , for the period 1976 - 2009 . , title="fig:",width=268 ] figure 3ab contains two histograms with the historical record for the observatory ( for the period 1976 - 2009 ) . the first one represents the number of clear nights ( defined as those having at least 6 observing hours of clear or mostly clear sky , with clouds cover below 25% ) . the data collected during these 30 years ( about three solar activity cycles ) clearly indicate that the number of clear nights are around 170 nights per year . see second histogram displays the number of useful hours per year , a more representative quantity since it is closely connected with the observations and the amount of data which can be produced . the histogram is highly variable , but during the last years an average of 2000 hours has been achieved . this fact might be related with the extensively use of service mode , much more effective than programs executed under a conventional visitor mode . assuming 9 useful hours per night , 2000 hours are equivalent to 222 full nights per year . * left.- * average for the number of clear nights per month during the period 2000 - 2009 . the bars provide an idea of the range of variability for each month . * right.- * monthly average for the number of useful hours for the period 2000 - 2009.,title="fig:",width=260 ] * left.- * average for the number of clear nights per month during the period 2000 - 2009 . the bars provide an idea of the range of variability for each month . * right.- * monthly average for the number of useful hours for the period 2000 - 2009.,title="fig:",width=264 ] as in the case of the extinction ( subsect . [ subsection : extinction ] ) , the effectiveness of the observatory , in terms of nights or hours per month , is highly seasonal . figure 4 illustrates this fact . in the first panel , we represent the number of clear nights per month . there is a conspicuous peak during the summer and a minimum in december , but january shows a very large dispersion , with years containing 20 clear nights and other with less than 10 clear nights . when expressed in terms of useful hours ( figure 4b ) the situation is more balanced . the summer peak is still present , but modulated by the night length . therefore , the minimum is located during early spring . surprisingly , january can be either very poor in useful hours or outstanding . .current and future instrumentation [ cols="^,^,^,^,^ " , ] + @xmath2 commissioning 2011 . + @xmath3 commissioning 2012 . + @xmath4 commissioning 2014 . + additional information at : + www.caha.es/telescopes-overview-and-instruments-manuals.html [ tab1 ] the regional government of andalusia ( junta de andaluca ) approved in 2007 an environmental law that includes a chapter on light pollution . this law becomes fully effective after the publication of its ancillary regulation ( august 2010 ) . calar alto has been advising junta de andaluca from the beginning of this legal project , with one representative in the advisory committee established to define the general terms of the law and its regulation . the law foresees creating an office for the protection of night sky against light pollution , and calar alto has secured its participation in this official institution . a protocol is under preparation and its implementation will allow caha to perform specific and individual negotiations with all the relevant cities and towns in the neighbourhood of the observatory . therefore , the quality as an outstanding astronomical site will be secured for many years to come . the observatory operates three telescopes at the present time , with primary mirrors of 3.5 m 2.2 and 1.23 meters . the instrumental suite is very diverse in the first two cases , and include optical and near - infrared imagers and spectrographs ( six instruments for the 3.5 m and four for the 2.2 m ) . in order to optimize the observatory resources ( man - power , financial and observing time ) , we are introducing a number of changes in our operational mode , including the implementation of a public archive and a simplified suite of instruments , following the recommendations of the european telescopes strategic review committee ( etsrc ) , which was appointed by the astronet board in coordination with the opticon executive committee . the current and possible future suite of instruments are summarized in table 1 . an example of a query in the caha public archive , developed under vo standards.,title="fig:",width=574 ] in collaboration with the center of astrobiology ( cab ) , which hosts the spanish node of the virtual observatory ( solano 2006 ) , we have developed a web - based tool in order to provide access to the data acquired at calar alto observatory . all data taken after july 1st 2010 will be made public after one year of proprietary rights . for data collected prior that date ( starting january 2008 ) , we have requested permit from the principal investigators for each individual program . the final goal is to provide raw and reduced data , as well as meta - data and links to other vo - compliant archives and tools ( as an example , see vosa , bayo et al . 2009 ) , in order to maximize the scientific exploitation of the data . figure 5 includes an example of a caha public archive query . we expect that the utility will be opened in early 2011 . the observatory has recently approved its first legacy program , the the califa survey ( calar alto legacy integral field area survey , snchez et al . 2010 , 2011 ; marino et al . this program will provide the largest and most comprehensive wide - field ifu survey of galaxies carried out to date , addressing several fundamental issues in galactic structure and evolution . we will observe a statistically well - defined sample of @xmath1600 galaxies in the local universe using 210 observing nights with the pmas / ppak integral field spectrophotometer , mounted on the calar alto 3.5 m telescope . the defining science drivers for the project are : a ) star formation and chemical history of galaxies , b ) the physical state of the interstellar medium , c ) stellar and gas kinematics in galaxies , and d ) the influence of the agns on galaxy evolution . califa will provide a valuable bridge between large single - aperture surveys such as sdss and more detailed studies of individual galaxies with ppak and other instruments . additional legacy program might be implemented in the future , specially at the 2.2 m telescope . * left.- * scheme of the optical design of cafe , including its main components . * right.- * first assembling of cafe of its optical bench , in september 2010 . the laser is used to align the different components . , title="fig:",width=283 ] * left.- * scheme of the optical design of cafe , including its main components . * right.- * first assembling of cafe of its optical bench , in september 2010 . the laser is used to align the different components . , title="fig:",width=238 ] at the present time , we are developing three new instruments , namely a high - spectral resolution optical spectrograph for general use at the 2.2 m ( cafe ) , a wide field - of view ( 30x30 arcmin ) near - infrared camera for the 2.2 m and the 3.5 m ( panic ) , and a high - spectral near - infrared spectrograph for exoplanetary searches ( carmenes ) . more information can be found in these proceedings or below for cafe . the calar alto fiber - fed echelle spectrograph ( cafe ) is an instrument under construction at caha to replace foces , the high - resolution echelle spectrograph at the 2.2 m telescope of the observatory . foces is a property of the observatory of the munich university , and it was recalled from calar alto by this institution in 2009 . the use of this instrument was very extensive , and represented a substantial fraction of the 2.2 m telescope time during its operational life - time . due to this fact , the observatory decided in 2008 to to build an improved replacement . cafe share its basic characteristics with those of foces . however , significant improvements have been introduced in the original design ( adding new calibration units ) , the quality of the materials , and the overall stability of the system . it is expected that the overall efficiency and the quality of the data will be significantly improved with respect to its predecessor . in particular , cafe is design and built to achieve resolutions of r@xmath170000 , which will be kept in the final acquired data , allowing it to compete with current operational extrasolar planets hunters . lcc parameter & foces & cafe + design & echelle spectrograph & echelle spectrograph + telescope & calar alto 2.2 m & calar alto 2.2 m + operational live - time & 1997 - 2009 & starting in 2011 + resolution & 46000/64000 & 70000 + ccd pixel scale & 24/15@xmath5 m & 13.5@xmath5 m + ccd cooling system & liquid nitrogen & peltier + wavelength range & 3800 - 7450@xmath6@xmath2 & 3800 - 7450@xmath6 + stability system & un - stabilized & vibrations and temperature + moving parts & slit , grating and prisms & no moving parts + calibration system & th - ar lamps & th - ar lamps and iodine - cell + quality of optical components & @xmath7/10 & @xmath7/20 + expected s / n@xmath3 & @xmath1100 & @xmath0150 + + + [ tab1 ] it is expected that the final assembling of the instrument take place before the end of 2010 . telescope commissioning is expected for spring 2011 . we expect that panic will be operational in 2012 and carmenes , which has received green light very recently , in 2014 . the calar alto observatory has two main properties : a very well - characterized and excellent astronomical site ( specially for spectroscopy ) and outstanding logistics ( excellent communications and a location in continental europe ) . the current and future suite of instruments for the optical and near - ir telescopes , and the support of two of the main research institutions in europe ( the spanish national research council and the german max - plank society , channeled through the instituto de astrofsica de andaluca and the max - plank institut fr astronomie ) , provide the necessary conditions for its continuation as relevant astronomical facility . we thank the _ `` viabilidad , diseo , acceso y mejora '' _ funding programs , icts-2008 - 24 and icts-2009 - 10 , of the spanish ministry of science , and the _ proyecto de excelencia _ funding program fqm-08 - 00360 of the _ junta de andaluca _ , for the support given to this project . the observatory will not be possible without the superb professionalism of its staff .

the calar alto observatory , located at 2168 m height above the sea level in continental europe , holds a significant number of astronomical telescopes and experiments , covering a large range of the electromagnetic domain , from gamma - ray to near - infrared . it is a very well characterized site , with excellent logistics . its main telescopes includes a large suite of instruments . at the present time , new instruments , namely cafe , panic and carmenes , are under development . we are also planning a new operational scheme in order to optimize the observatory resources .

You are an expert at summarizing long articles. Proceed to summarize the following text: the cataclysmic variable hu aqr currently consists of a 0.80 white dwarf that accretes from a 0.18 main - sequence companion star . the transfer of mass in the tight @xmath12 orbit is mediate by the emission of gravitational waves and the strong magnetic field of the accreting star . since its discovery , irregularities of the observed - calculated variations have led to a range of explanations , including the presence of circum - binary planets . detailed timing analysis has eventually led to the conclusion that the cv is orbited by two planets @xcite , a 5.7 planet in a @xmath13 orbit with an eccentricity of @xmath14 and a somewhat more massive ( 7.6 ) planet in a wider @xmath15 and eccentric @xmath16 orbit @xcite . although , the two - planet configuration turned out to be dynamically unstable on a 100010,000 year time scale ( * ? ? ? * see also [ sect : stability ] ) , a small fraction of the numerical simulations exhibit long term dynamical stability ( for model b2 in * ? ? ? * see tab.[tab : huaqr ] for the parameters ) . it is peculiar to find a planet orbiting a binary , in particular around a cv . while planets may be a natural consequence of the formation of binaries @xcite , planetary systems orbiting cvs could also be quite common . in particular because of recently timing residual in nn serpentis , dp leonis and qs virgo @xcite were also interpreted is being caused by circum - cv planets . although the verdict on the planets around hu aqr ( and the other cvs ) remains debated ( tom marsh private communication , and * ? ? ? * ) , we here demonstrate how a planet in orbit around a cv , and in particular two planets , can constrain the cv evolution and be used to reconstruct the history of the inner binary . we will use the planets to perform a precision reconstruction of the binary history , and for the remaining paper we assume the planets to be real . because of their catastrophic evolutionary history , cvs seem to be the last place to find planets . the original binary lost probably more than half its mass in the common - envelope phase , which causes the reduction of the binary separation by more than an order of magnitude . it is hard to imagine how a planet ( let alone two ) can survive such turbulent past , but it could be a rather natural consequence of the evolution of cvs , and its survival offers unique diagnostics to constrain the origin and the evolution of the system . after the birth of the binary , the primary star evolved until it overflowed it roche lobe , which initiated a common - envelope phase . the hydrogen envelope of the primary was ejected quite suddenly in this episode @xcite , and the white dwarf still bears the imprint of its progenitor : the mass and composition of the white dwarf limits the mass and evolutionary phase of its progenitor star at the moment of roche - lobe overflow ( rlof ) . for an isolated binary the degeneracy between the donor mass at the moment of rlof ( @xmath17 ) , its radius @xmath18 and the mass of its core @xmath19 can not be broken . the presence of the inner planet in orbit around hu aqr @xcite allows us to break this degeneracy and derive the rate of mass loss in the common - envelope phase . the outer planet allows us to validate this calculation and in addition to determine the conditions under which the cv was born . the requirement that the initial binary must have been dynamically stable further constrains the masses of the two stars and their orbital separation . during the cv phase little mass is lost from the binary system @xmath20constant ( but see * ? ? ? * ) , and the current total binary mass ( @xmath21 ) was not affected by the past ( and current ) cv evolution @xcite . the observed white dwarf mass then provides an upper limit to the mass of the core of the primary star at the moment of roche - lobe contact , and therefore also provides a minimum to the companion mass via @xmath22 . with the mass of the companion not being affected by the common envelope phase , we constrain the orbital parameters at the moment of rlof by calculating stellar evolution tracks to measure the core mass @xmath19 and the corresponding radius @xmath23 for stars with zero - age main - sequence mass @xmath24 . in fig.[fig : amcoreformzams3msun ] we present the evolution of the radius of a 3 star as a function of @xmath19 , which is a measure of time we adopted the henyey stellar evolution code mesa @xcite to calculate evolutionary track of stars from @xmath25 to 8 using amuse @xcite to run mesa and determine the mass of the stellar core . the latter is measured by searching for the mass - shell in the stellar evolution code for which the relative hydrogen fraction @xmath26 . at the moment of rlof the core mass is @xmath19 and the stellar radius @xmath27 . via the relation for the roche radius @xcite , we can now calculate the orbital separation at the moment of rlof @xmath28 as a function of @xmath17 . this separation is slightly larger than the initial ( zero - age ) binary separation @xmath29 due to the mass lost by the primary star since its birth @xmath30 . the long ( main - sequence ) time scale in which this mass is lost guarantees an adiabatic response to the orbital separation , i.e. @xmath31 constant . for each @xmath24 we now have a range of possible solutions for @xmath28 as a function of @xmath19 and @xmath32 . this reflects the assumption that the total mass ( @xmath33 ) in the observed binary with mass @xmath34 is conserved throughout the evolution of the cv . in fig.[fig : amcoreformzams3msun ] we present the corresponding stellar radius @xmath18 and @xmath35 as a function of @xmath19 for @xmath36 . this curve for @xmath28 is interrupted when rlof would already have been initiated earlier for that particular orbital separation . we calculate this curve by first measuring the size of the donor for core mass @xmath19 , and assuming that the primary fills its roche - lobe we calculate the orbital separation at which this happens . during the common envelope phase the primary s mantle is blown away beyond the orbit of the planets . the latter responds to this by migrating from the orbits in which they were born ( semi - major axis @xmath37 and eccentricity @xmath38 , the subscript @xmath39 indicates the inner planet , we adopt a @xmath40 to indicate the outer planet ) to the currently observed orbits . using first order analysis we recognize two regimes of mass loss : fast and slow . in the latter case the orbit expands adiabatically without affecting the eccentricity : the minimum possible expansion of the planet s orbit is achieved when the common envelope is lost adiabatically . fast mass loss leads to an increase in the eccentricity as well and may even cause the planet to escape @xcite . a planet born at the shortest possible orbital separation to be dynamically stable will have @xmath41 @xcite , which is slightly smaller than the distance at which circum binary planets tend to form @xcite . in fig.[fig : amcoreformzams3msun ] we present a minimum to the semi - major axis for a planet that was born at @xmath42 and migrated by the adiabatic loss of the hydrogen envelope from the primary star in the common - envelope phase . the planet can have migrated to a wider orbit , but not to an orbit smaller than the solid black curve ( indicated with @xmath43 ) in fig.[fig : amcoreformzams3msun ] . for the 3 star , presented in fig.[fig : amcoreformzams3msun ] , rlof can successfully result in the migration of the planet to the observed separation in hu aqr for @xmath44 , which occurs for @xmath45 . a core mass @xmath46 would , for a 3 primary star , result in an orbital separation that exceeds that of the inner planet in hu aqr ; in this case the core mass of the primary star must have been smaller than 0.521 . another constraint on the initial binary orbit is provided by the requirement that the mass transfer in the post common - envelope binary should be stable when the companion starts to overfill its roche lobe . to guarantee stable mass transfer we require that @xmath47 . the thick part of the red curve in fig.[fig : amcoreformzams3msun ] indicates the valid range for the initial orbital separation and core - mass for which the observed planet can be explained ; the thin parts indicate where these criteria fail . we repeat the calculation presented in fig.[fig : amcoreformzams3msun ] for a range of masses from @xmath48 to 8 with steps of 0.02 , the results are presented as the shaded region in fig.[fig : am0_distribution_hu ] . the response of the orbit of the planet to the mass loss depends on the total amount of mass lost in the common envelope and the rate at which it is lost . numerical common - envelope studies indicate that for an in - spiraling binary @xmath49 @xcite . at this rate the entire envelope @xmath505.8 is expelled well within one orbital period of the inner planet , which leads to an impulsive response and the possible loss ( for @xmath51 ) of the planet . the fact that the hu aqr is orbited by a planet indicates that at the distance of the planet @xmath52 . the eccentricity of the inner planet in hu aqr ( see tab.[tab : huaqr ] ) can be used to further constrain the rate at which the common - envelope was lost from the planetary orbit . the higher eccentricity of the outer planet indicates a more impulsive response , which is a natural consequence of its wider orbits with the same @xmath53 . this regime between adiabatic and impulsive mass loss is hard to study analytically @xcite . [ cols="<,<,<,<,<,<,<,<",options="header " , ] we calculate the effect of the mass loss on the orbital parameters by numerically integrating the planet orbit . the calculations are started by selecting initial conditions for the zero - age binary hu aqr @xmath24 , @xmath29 and consequently @xmath19 from the available parameter space ( shaded area ) in fig.[fig : am0_distribution_hu ] , and integrate the equations of motion of the inner planet with time . planets ware assumed to be born in a circular orbit ( @xmath54 ) in the binary plane with semi - major axis @xmath37 . the equations of motions are integrated using the high - order symplectic integrator huayno @xcite via the amuse framework . during the integration we adopt a constant mass - loss rate @xmath53 applied at every 1/100th of an orbit , and we continued the calculation until the entire envelope is lost ( see [ sect : ce ] and fig.[fig : am0_distribution_hu ] ) , at which time we measure the final semi - major axis and eccentricity of the planetary orbit . during the integration we allow the energy error to increase up to at most @xmath55 . by repeating this calculation while varying @xmath37 and @xmath56 we iterate ( by bisection ) until the result is within 1% of the observed @xmath57 and @xmath58 of the inner planet observed in hu aqr . the converged results of these simulations are presented in fig.[fig : am0_distribution_hu ] ( circles ) , and these represent the range of consistent values for the inner planet s orbital separation @xmath59752 as a function of @xmath608 and consistently reproduce the observed inner planet when adopting @xmath610.267/yr . the highest value for @xmath53 is reached for @xmath62 at an initial orbital separation of @xmath63 . the orbital solution for the inner planet is insensitive to the semi - major axis of the zero - age binary @xmath29 ( for a fixed @xmath24 ) , and each of these solutions were tested for dynamical stability , which turned out to be the case irrespective of the initial binary semi - major axis ( as discussed in [ sect : stability ] ) . we now adopt the in [ sect : innerplanet ] measured value of @xmath53 to integrate the orbit of the outer planet . the effect of the mass outflow on the planet is proportional to the square of the density in the wind at the location of the planet . we correct for this effect by reducing the mass loss rate in the common envelope that affects the outer planet by a factor @xmath64 . we use the same integrator and assumptions about the initial orbits as in [ sect : innerplanet ] , but we adopt the value of @xmath53 from our reconstruction of the inner planet ( see [ sect : innerplanet ] ) . to reconstruct the initial orbital separation of the outer planet @xmath65 , we vary this value ( by bisection ) until the final semi - major axis is within 1% of the observed orbit ( see tab.[tab : huaqr ] ) . the results are presented in fig.[fig : am0_distribution_hu ] ( triangles ) . the post common - envelope eccentricity of the outer planet then turn out to be @xmath66 . after having reconstructed the initial conditions of the binary system with its two planets we test its dynamical stability by integrating the entire system numerically for 1myr using the huayno integrator @xcite . to test the stability we check the semi - major axis and eccentricity of both planets every 100years . if any of these parameters change by a factor of two compared to the initial values or if the orbits cross we declare the system unstable , otherwise they are considered stable . the calculations are repeated with the 4th order hermite predictor - corrector integrator ph4 @xcite within amuse to verify that the results are robust , which turned out to be the case . we then repeated this calculation ten times with random inital orbital phases and again with a 1% gaussian variation in the initial planetary semi - major axes . in fig.[fig : am0_distribution_hu ] we present the resulting stable systems by coloring them red ( circled ) and blue ( triangles ) , the unstable systems are represented by open symbols . from the wide range of possible systems that can produce hu aqr only a small range around @xmath67 turns out to be dynamically stable . the eccentricity of the outer orbit of the stable systems ( which ware stable for initial conditions within 1% ) @xmath68 , which is somewhat smaller than the observed value for hu aqr ( * ? ? ? * @xmath69 ) . these values are obtained with @xmath70 . the small uncertainty in the derived value of @xmath53 is a direct consequence of its sensitivity to @xmath38 and the small error on @xmath24 from the requirement that the initial system is dynamically stable . we have adopted the suggestive results from the timing analysis of hu aqr , that the cv is orbited by two planets , to reconstruct the evolution of this complex system . a word of caution is well placed in that these observations are not confirmed , and currently under debate ( tom marsh private communication , and comments by the referee ) . however , the predictive power that such an observation would entail is interesting . the possibility to reconstruct the initial conditions of a cv by measuring the orbital parameters of two circum binary planets is a general result that can be applied to other binaries . for cvs in particular it enables us to constrain the value of fundamental parameters in the common - envelope evolution . this in itself makes it interesting to perform this theoretical exercise , irrespective of the uncertainty in the observations . on the other hand , the consistency between the observations and the theoretical analysis give some trust to the correctness of these observations . the presence of one planet in an eccentric orbit around a cv allow us to calculate the rate at which the common - envelope was lost from the inner binary . a single planet provides insufficient information to derive the initial mass of the primary star , but allows us to derive the initial binary separation and planetary orbital separation to within about factor of 5 , and the initial rate of mass loss from the common envelope to about a factor 2 . a second planet can be used to further constrain these parameters to a few per cent accuracy and allows us to make a precision reconstruction of the evolution of the cv . we have used the observed two planets in orbit around the cv hu aqr to reconstruct its evolution , to derived its initial conditions ( primary mass , secondary mass , orbital separation , and the orbital separations of both planets ) and to measure the rate of mass lost in the common - envelope parameters @xmath53 . by comparing the binary parameters at birth with those after the common - envelope phase we subsequently calculate the two parameters @xmath71 and @xmath72 . the measured rate of mass loss for hu aqr of @xmath73 from the inner planetary orbit , which from the binary system itself would entail a mass - loss rate of @xmath74 , when we adopt the initial binary to have a semi - major axis of @xmath75 , which is bracketed by our derived range of @xmath76160 . this is consistent with a mass - loss rate of @xmath77 from numerical common - envelope studies @xcite . by adopting that the binary survives its common envelope at a separation between @xmath78 ( at which separation the secondary star will just fill it s roche - lobe to the white dwarf ) and @xmath79 ( for gravitational wave radiation to drive the binary into roche - lobe overflow within 10gyr ) , we derive the value of @xmath802.0 ( for @xmath75 we arrive at @xmath81 ) . this value is a bit small compared to numerous earlier studies , which tend to suggest @xmath82 . the alternative @xmath72-formalism for common - envelope ejection gives a value of @xmath831.80 ( for @xmath75 we arrive at @xmath84 ) , which is consistent with the determination of @xmath72 in 30 other cvs @xcite . the inner planet in hu aqr formed at @xmath85@xmath86 , with a best value of @xmath87 , which is consistent with the planets found to orbit other binaries , like kepler 16 @xcite and for kepler 34 and 35 @xcite , although these systems have lower primary mass and secondary mass stars . it seems unlikely that more planets were formed inside the orbit of the inner most planet , even though currently there is sufficient parameter space for many more stable planets ; in the zero - age binary there has not been much room for forming additional planets further in . it is however possible that additional planets formed further out and those , we predict , will have even higher eccentricity than those already found . * acknowledgements * it is a pleasure to thank edward p.j . van den heuvel , tom marsh , inti pelupessy , nathan de vries , arjen van elteren and the anonymous referee for comments on the manuscript and discussions . this work was supported by the netherlands research council nwo ( grants # 612.071.305 [ lgm ] , # 639.073.803 [ vici ] and # 614.061.608 [ amuse ] ) and by the netherlands research school for astronomy ( nova ) .

cataclysmic variables ( cvs ) are binaries in which a compact white dwarf accretes material from a low - mass companion star . the discovery of two planets in orbit around the cv hu aquarii opens unusual opportunities for understanding the formation and evolution of this system . in particular the orbital parameters of the planets constrains the past and enables us to reconstruct the evolution of the system through the common - envelope phase . during this dramatic event the entire hydrogen envelope of the primary star is ejected , passing the two planets on the way . the observed eccentricities and orbital separations of the planets in hu aqr enable us to limit the common - envelope parameter @xmath0 or @xmath1 and measure the rate at which the common envelope is ejected , which turns out to be copious . the mass in the common envelope is ejected from the binary system at a rate of @xmath2 . the reconstruction of the initial conditions for hu aqr indicates that the primary star had a mass of @xmath3 and a @xmath4 companion in a @xmath5160 ( best value @xmath6 ) binary . the two planets were born with an orbital separation of @xmath7 and @xmath8 respectively . after the common envelope , the primary star turns into a @xmath9 helium white dwarf , which subsequently accreted @xmath10 from its roche - lobe filling companion star , grinding it down to its current observed mass of @xmath11 . methods : numerical planets and satellites : dynamical evolution and stability planet star interactions planets and satellites : formation stars : individual : hu aquarius stars : binaries : evolution

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Proceed to summarize the following text: the possibility of manipulating magnetization by spin - currents in a very efficient way is a key requirement for the design of novel spintronic devices @xcite , which promise to change the way digital information is processed and stored . in particular , the advantageous scaling of current - induced spin manipulation compared to the oersted field - induced switching allows for lower power operation at small design rules . the driving mechanism behind current - induced magnetization dynamics pioneered for use in metallic ferromagnets in the last twenty five years has been the spin - transfer torque @xcite . however , recently a novel and possibly more efficient approach to current - driven magnetization manipulation has been developed . in particular , very efficient current - induced spin dynamics has been observed in multilayer systems with an ultra - thin ferromagnetic layer sandwiched between two different non magnetic materials @xcite . these new current - induced torques originate from spin - orbit effects ( at least one of the two non - magnetic layers consists of an heavy metal ) and so they are referred to as spin - orbit torques ( sots ) @xcite . two different origins have been suggested for the sots . one is the spin accumulation induced at the [ heavy metal]@xmath0ferromagnet interface due to the bulk spin - hall effect ( she ) in the heavy metal layer @xcite . a second possible origin of the sots is the inverse spin - galvanic effect ( isge ) @xcite , which generates a non - equilibrium spin - density at both the top and the bottom interfaces of the ferromagnet . both effects are expected to induce an effective torque whose strength is a function of the ferromagnetic layer thickness . + in the same kind of materials stacks the presence of topologically non - trivial spin textures , such as homo - chiral domain walls has also been observed @xcite . chiral domain walls are reported to be very stable against annihilation @xcite and to exhibit interesting transport properties @xcite , thus being very promising for technological applications . the origin of these chiral spin structures is the interfacial dzyaloshinskii - moriya interaction ( dmi ) @xcite . the dmi is expected to primarily originate from the interface between the heavy metal and the ferromagnet , predicting a direct scaling of its effective strength with the inverse of the ferromagnetic layer thickness @xcite . accordingly , both the dmi and the sots are expected to depend strongly on the materials system as well as on the layers thickness . so , the ferromagnetic layer thickness dependence of dmi and sots is directly linked to their origin . a pure interface - like effect on one side and a more complicated mechanism which includes bulk - like processes on the other are expected to scale differently with the thickness of the ferromagnetic layer @xcite . in particular , there are predictions that the dmi and the isge can have a common origin , which should result in a similar dependence on the thickness @xcite . so , studying the sign and amplitude of dmi and sots as a function of the ferromagnetic layer thicknesses is a key necessity to reveal their origins . + materials stacks with a strong dmi as well as large sots are particularly promising , due to their rich physics as well as the possibility to be used in novel spintronic devices @xcite . one of the most promising of those materials stacks is the trilayer pt@xmath0co@xmath0alo@xmath1 @xcite . several manuscripts have reported the characterization of the dmi @xcite and of the sots @xcite in this system . however , reports from different groups for nominally identical samples have often shown contradicting results possibly originating from different thicknesses and growth conditions and no systematic study has been provided to date . in particular , varying the co thickness is a key challenge as it potentially allows to distinguish between effects from the interface with the pt and the alo@xmath1 and from the bulk of the materials . accordingly , a systematic and combined study of both the dmi and sots as a function of co thickness is the key step needed to understand and tailor the spin - orbit effects in this system . + here we report on the characterization of the dmi as well as the sots in identical samples of pt@xmath0co@xmath0alo@xmath1 . we extract the sign and the magnitude of dmi and sots as a function of the co layer thickness combining two key techniques , namely : current - induced dw motion ( cidwm ) and second harmonic hall measurements . a detailed study of cidwm in magnetic tracks is presented , which allows us to characterize the thickness dependence of the dmi and of the effective torque driving the dw motion . this is complemented by an in depth characterization of the sots employing second harmonic measurements in a hall bar geometry . comparing the thickness dependence of the dmi and the torques allows us to draw conclusions about their origin . 2 we describe the experimental techniques and the corresponding set - ups used in our study . in sec . 3 we report the characterization of current - induced dw motion in magnetic nanowires . sec . 4 reports the dmi values extracted from the analysis of this dw motion data . 5 presents the measured thickness dependence of sots . in sec . 6 we present a discussion of the main results , where we discuss the ferromagnetic thickness dependence of the dmi and the sots and compare to literature . the material system that is employed for the patterning of the magnetic devices is the multilayer : ta(4.0)@xmath0pt(4.0)@xmath0co(0.81.8)@xmath0alo@xmath1(2.0 ) ( all thicknesses in nm ) . the stack was deposited by magnetron sputtering technique on a si@xmath0sio@xmath2 substrate . the magnetic layer consists of a wedged co layer , with a nominal increase in thickness of 1 nm over a length of 2 cm in one of the two in - plane directions ( @xmath3-direction in the following ) . the wedged co layer is obtained by an _ in - situ _ moving shadow mask @xcite . + before patterning , the material stack is characterized by brillouin light scattering ( bls ) technique @xcite and magneto - optic kerr effect ( moke ) magnetometry . the stack exhibits perpendicular magnetic anisotropy ( pma ) , with a spontaneous magnetization pointing along the out - of - plane ( oop ) direction ( see fig . [ fig_characterization ] ) . the observed pma is induced by a strong interface anisotropy at the pt@xmath0co interface @xcite . the effective anisotropy energy density , @xmath4 , is found to decrease with increasing co thickness ( see fig . [ fig_k_eff ] ) , consistent with the presence of an interface anisotropy of @xmath5 mj / m@xmath6 and a saturation magnetization of @xmath7 a / m , both obtained by bls measurements @xcite . this is consistent with the results of magneto - optic kerr effect ( moke)-magnetometry measurements shown in fig . [ fig_moke_ptcoalox ] . hysteresis loops for an applied oop magnetic field , @xmath8 , are measured at different positions on the surface of the material stack with different @xmath9 . the coercive field is observed to decrease with increasing @xmath9 , in agreement with previous observations @xcite . the loss of pma with increasing @xmath9 explains the decreasing coercive field as a function of an increasing co thickness visible in fig . [ fig_moke_ptcoalox ] . after magnetic characterization , the wedge sample is patterned into several devices with different thicknesses of the co layer . the patterned devices consist of an array of several nanowires ( nws ) in parallel ( 1.5 - 2.0 @xmath10 m in width and 25 - 28 @xmath10 m in length , see fig . [ fig_cidwm_protocol ] ) used for current - induced domain wall motion ( cidwm ) experiments , and in hall - crosses ( 1 - 2 @xmath10 m in width and 50 @xmath10 m in length , see fig . [ fig_sh_setup ] ) used for the measurements of effective spin - orbit fields by the second harmonic ( @xmath11 ) technique @xcite . the devices are patterned by electron - beam lithography and ar - ion milling , at different positions on the sample surface corresponding to different @xmath9 . + in both experimental setups a 50 @xmath12-resistor is used to terminate the circuit to ground . an oscilloscope is used for measuring the pulse waveform , across its 50 @xmath12-internal resistance , @xmath13 . the total current flowing through the device is obtained by measuring the voltage across @xmath13 . for the evaluation of the current density , @xmath14 , the nominal thicknesses of the layers are used . in these type of thin film systems , the resistivity of the ta layer is known to be around 4 - 5 times larger than is for pt , while the co layer and the pt layer have a similar resistivity value @xcite . accordingly , the calculated current densities are obtained considering the 4 nm - thick ta bottom layer equivalent to a 1 nm - thick pt layer . the conventional current density @xmath15 is taken to be positive when it flows in the @xmath16-direction ( see fig . [ fig_setups ] ) , corresponding to an electron current density @xmath17 flowing in the @xmath18-direction . + concerning the cidwm experiment , the magnetic configuration of the wires is imaged by a wide field kerr microscope in the polar configuration @xcite . a magnetic coil is used for the generation of an external in - plane magnetic field . the experiments are carried out at t=300 k. the second harmonic measurements are carried out in hall - crosses ( see fig . [ fig_sh_setup ] ) . a small - amplitude sinusoidal ac current is applied with a frequency ( @xmath19 ) of 13.7 hz . this induces periodic oscillations of the magnetization about its equilibrium direction . these periodic oscillations can be attributed to the effective fields generated by the injected current . the periodic oscillation of the magnetization results in an oscillation of the hall resistance . the resulting hall voltage has a second harmonic component ( @xmath11 ) that relates directly to the current - induced fields @xcite . the measurement is performed in two schemes : longitudinal ( x - axis ) and transverse ( y - axis ) . in both schemes , the hall voltage is measured during a sweep of the in - plane magnetic field ( -400 mt to + 400 mt ) . in the longitudinal ( transverse ) scheme the direction of the magnetic field is applied along ( perpendicular to ) the direction of the injected current . it is important to note that the hall voltage also includes contributions from the planar hall effect @xcite , nernst effect and joule heating @xcite . these effects are taken into account to extract artifact - free current induced effective fields . the first type of experiment reported here is the study of current - induced dw motion ( cidwm ) . this allows us to establish if either the standard spin - transfer torque or the sots are the main driving mechanisms behind dw motion in this materials system and gauge the strength . furthermore , the dw motion study allows one to determine if chiral dws are present in the system , to extract their chirality and finally to obtain the sign and magnitude of the dmi for each investigated device . current - induced dw motion experiments are carried out in four different devices . the nominal thickness of the co layer in the different devices is : 0.93 nm , 1.31 nm , 1.37 nm and 1.43 nm . the measurement protocol is described in fig . [ fig_cidwm_protocol ] . first , the magnetic wires are saturated in the ( @xmath20 , @xmath21 ) or ( @xmath22 , @xmath23 ) magnetization state by an external out - of - plane magnetic field . second , the magnetization is reversed in all the nws by the switching process presented in previous papers @xcite , where each combination ( parallel / anti - parallel ) of field , @xmath24 , and current , @xmath14 , corresponds to a specific final state of the magnetization in the nws . as a result , a dw ready to be displaced is obtained in each nw . the differential kerr microscopy images at the bottom left ( right ) of fig . [ fig_cidwm_protocol ] show the cidwm experiment for @xmath25- ( @xmath26- ) dws . at @xmath27 dws are nucleated . at @xmath28 the same dws are moved by the injection of a train of current pulses . + the average dw velocity is extracted as the ratio between the total displacement , @xmath29 , visible in the differential kerr microscopy images and the total pulsing time , @xmath30 . the total pulsing time is given by the total number of pulses , @xmath31 , times the single pulse length , @xmath32 ( full width at half maximum , 10 - 15 ns ) : @xmath33 . two consecutive current pulses are separated by 1 ms , in order to fully magnetically relax the domain wall in between pulses to have reproducible conditions . the average dw velocity , @xmath34 , as a function of the current density , @xmath14 , is reported in fig . [ fig_vdw - ja ] , for the four different devices . the measurement is carried out for both @xmath25-dws ( red dots ) and @xmath26-dws ( blue dots ) , for positive and negative @xmath14 . + in all the four devices , @xmath25- and @xmath26-dws are observed to move in the same direction of the conventional current , @xmath15 , at approximately the same speed ( within the error bars ) . the critical current density for inducing significant dw motion in each device is observed to be in the range @xmath35 a / m@xmath6 , in line with what has been reported in previous works on the same materials system @xcite . from the result that both domain wall types ( @xmath25 and @xmath26 ) move against the electron flow , we can conclude that sots are the dominating torques responsible for the dw displacement and that the dws are all homo - chiral due to a finite dmi . next , we use cidwm to determine the dmi in these samples . to this end , the dw velocity is measured as a function of an applied magnetic field along the wire axis ( @xmath36 ) for a fixed current density . the measurement protocol is the following . first , one type of dw ( @xmath25 or @xmath26 ) is nucleated in each pre - saturated nw by current - induced magnetization switching @xcite . a typical switching pulse used in the experiment has a current density amplitude of @xmath37 a / m@xmath6 and a time duration of @xmath38 ns , assisted in the switching process by a fixed external longitudinal field of about 50 mt . once the dws are nucleated , they are displaced by the injection of a burst ( n=1 - 50 ) of current pulses with a measured duration of 15 ns . + the motion of @xmath25-dws in the presence of an applied longitudinal field is shown in fig . [ fig_vdw - hx_protocol ] . the total dw displacement is strongly affected by the presence of a finite @xmath24 . a positive field makes the @xmath25-dws move slower , while a negative field makes them move faster . . the velocity of the dws is observed to be reduced for @xmath39 and increased for @xmath40 , for both @xmath41 and @xmath42 . the gold dashed lines indicate the initial position of the dws . the gold arrows indicate the dw direction of motion . the injected current density is @xmath43 @xmath44 , while the longitudinal field amplitudes are : @xmath45 mt , @xmath46 mt , @xmath47 mt . the yellow numbers indicate how many pulses are used to generate the shown dw displacements . the different dw displacements in the two wires reflect the statistical spread of our observations . the images show current - induced motion of @xmath25-dws in the device with @xmath48 nm.,width=302 ] the measured average dw velocities as a function of the longitudinal field , @xmath49 , for all the devices are reported in fig . [ fig_vdw - hx ] ( symbols ) . red ( blue ) symbols refer to @xmath25- ( @xmath26- ) dws , while squares ( stars ) refer to @xmath41 ( @xmath42 ) . + as visible in fig . [ fig_vdw - hx_protocol ] and fig . [ fig_vdw - hx ] , while at zero - field the velocity of both types of dws is the same , in the presence of a finite longitudinal field the two types of dws move at different velocities . the change in the field amplitude affects differently the velocity of the two types of dws , making it possible to obtain @xmath25-dws and @xmath26-dws moving in opposite directions , when the field amplitude is large enough . a symmetric behavior is observed for the velocity of the two dw types with respect to @xmath24 , which can be described as : @xmath50 . + considering an @xmath25-dw , a sufficiently large positive @xmath24 slows them down , while a negative @xmath24 speeds the walls up . for very large positive @xmath24 the @xmath25-dws are also observed to change their direction of motion . however , differently from the case of the ta@xmath0cofeb@xmath0mgo system @xcite , it is not possible in this case to access the regime of fast dw motion in the reversed propagation direction . indeed , for in - plane fields larger than the ones shown in the graphs , local spontaneous magnetization reversal events @xcite start to occur , making the dw motion measurement not possible anymore . the amplitude @xmath24 at which domains start to nucleate is observed to decrease with an increasing @xmath9 . this results in a reduction of the range of in - plane fields which can be used for the investigation of dw motion , moving from the thinnest to the thickest device . + there is no single simple reason why the nucleation probability is observed to increase with increasing @xmath9 , but different effects can play a role . first , an increasing @xmath9 results , for ultra - thin layers like the ones under investigation , in an increasing conductivity of the ferromagnetic layer @xcite . accordingly , the observed increasing nucleation probability could be linked with the increase of the current flowing in the ferromagnetic layer , due to an increasing joule heating produced directly in the ferromagnetic material @xcite . secondly , as reported in fig . [ fig_characterization ] , the magnetic anisotropy of the magnetic layer decreases as @xmath9 increases . thus , the required amplitude of the in - plane field , at a fixed current density , for driving the magnetization reversal in the nws is expected to decrease with an increasing thickness @xcite . the nucleation process is observed to be particularly prevalent in the thickest investigated device , where already at applied fields of 10 - 20 mt reverse domains start to appear in the nws after the current pulse injection . this results in the impossibility of observing any reversal in the dw motion direction , as shown in fig . [ fig_vdw - hx_5 ] . + all that has been described above concerning @xmath25-dws is equally valid for @xmath26-dws for a symmetric reversal of the field @xmath24 . in this case , a positive in - plane field speeds up dw motion , while a negative field slows them down ( see fig . [ fig_vdw - hx ] ) . + these observations suggest strong spin - orbit torques acting in the materials stack , in combination with the presence of an interfacial dzyaloshinskii - moriya interaction @xcite . having established that both sots and dmi are present , the key task is to determine the sign and the strength of dmi and sots as a function of the co - layer thickness , as reported next . we use the cidwm results to determine the dmi present in the different devices . in order to do so , the so - called stopping fields for the @xmath25-dw and @xmath26-dw need to be extracted . the stopping field is the external longitudinal field which needs to be applied in order to make the dw stop moving @xcite . this can be extracted from the graphs shown in fig . [ fig_vdw - hx ] . + it is known that dmi stabilizes nel dws with the same chirality @xcite . this can be interpreted as due to the presence of an effective dmi field , @xmath51 , along the longitudinal direction . accordingly , the stopping field is the field needed to counteract the dmi field and turn the dw spin texture into the bloch configuration , making the spin - orbit torques acting on the dw zero . + the common approach used so far in order to extract the stopping fields from the cidwm graphs is based on a linear fitting of the high velocity data points ( for positive and negative @xmath24 ) , for each @xmath52 combination @xcite . the crossing point between the zero - velocity axis and the fitting curve would define the stopping field . however , it is not possible to use this approach in the present case . the reason is the lack of high velocity data points for one of the two field signs ( positive , for @xmath25-dws ; negative , for @xmath26-dws ) in the reported graphs , as discussed above . a simple linear fitting of the high velocity data points ( see fig . [ fig_vdw - hx_linfit ] ) results in the extraction of stopping field values that are outside the observed pinning ranges in fig . [ fig_vdw - hx ] . according to that , an alternative approach to extract the stopping fields is used . -dw with respect to @xmath24 , for the four studied devices . the reported data points refer to measurements with @xmath53 , while @xmath24 is parallel to the intrinsic nel - component of the dw internal magnetization . the lines are the linear fitting curves for the experimental data . the slope is found to increase with increasing @xmath9.,width=302 ] + the range of in - plane magnetic fields where the dws move at low velocities and become pinned is centered around the stopping field . indeed , the damping - like sot driving the dw motion is proportional to the nel component of the dw , which in turn results from the interplay of the external field and the dmi field . so , the stopping field where the dw does not feel a force is equal in magnitude and opposite in sign compared to the dmi field @xcite . as a consequence , the pinning range can be expressed as : @xmath54 $ ] ; where @xmath55 is the dmi effective field for @xmath56 dws , @xmath57 the pinning range of longitudinal magnetic fields . accordingly , the dmi effective field can be extracted as the center of the observed pinning ranges of @xmath24 . finally , the strength of the dmi can be obtained by the equation @xmath58 @xcite . + to determine the two stopping fields , @xmath59 and @xmath60 , we first ascertain the magnetic field value , @xmath61 , to which corresponds the highest measured @xmath34 in the reversed direction . this velocity is used as the threshold velocity , @xmath62 , which defines the boundary between the low velocity regime ( pinning regime ) and the high velocity regime . secondly , the data point with the same absolute value of the velocity , but with opposite sign is determined , together with the magnetic field at which it occurs , @xmath63 . these two values of the applied magnetic field are used as extremes of the pinning range , and the corresponding stopping field , @xmath64 , is calculated as the arithmetic mean of the two . two stopping fields are calculated for each type of dw ( when this is possible ) , corresponding to @xmath53 ( @xmath65 ) and @xmath66 ( @xmath67 ) , and the average stopping field for each dw type is obtained as @xmath68 . + the described protocol for the extraction of the effective dmi field is now applied to the experimental data reported in fig . [ fig_vdw - hx_1 ] , referring to the magnetic device with @xmath48 nm . the extracted stopping field for the @xmath25-dw is @xmath69 mt ( error defined as @xmath70 ) ; while for the @xmath26-dw @xmath71 mt . since @xmath72 , an effective dmi field of @xmath73 mt is obtained . + in order to extract the effective dmi coefficient @xmath74 , the dw width parameter needs to be calculated first . in the case of the thinnest device , @xmath75 nm ( where the used value of the exchange stiffness for the co layer @xmath76 j / m is chosen as reported in literature @xcite ) . accordingly , the effective dmi results to be @xmath77 mj / m@xmath6 . + the same process is repeated for @xmath78 nm and @xmath79 nm , extracting the respective effective dmi field and dmi strength , which are reported in fig . [ fig_dmi_1 ] and fig . [ fig_dmi_2 ] , respectively . for the device with thickness of @xmath80 nm it is not possible to extract a reliable dmi value . the lack of observations of dw motion in the pinning regime , does not allow to employ the described dmi extraction method for that specific device . the injection of an in - plane current through a nm@xmath0fm hetero - structure generates two different types of torques : a damping - like ( dl ) torque and a field - like ( fl ) torque , corresponding to two current - induced effective fields : @xmath81 and @xmath82 , respectively . these torques are responsible for the efficient current - induced domain wall motion and magnetization switching observed in those hetero - structures @xcite . here the thickness dependence of sots is obtained by two complementary approaches : on the one hand , the effective field moving the dws is extracted by analyzing the cidwm data by a collective - coordinate model ; on the other hand , both the dl - field and the fl - field are characterized by second harmonic hall measurements . to determine the spin - orbit torques acting on the dw , a collective - coordinate model ( ccm ) @xcite based on the extension of the one - dimensional model ( 1 dm ) is employed to reproduce the experimental observations reported in fig . [ fig_vdw - hx ] ( see appendix a for more details ) . in the framework of the ccm , the dw dynamics is described by three degrees of freedom : the position of the dw in the track , @xmath83 , the in - plane angle of the dw magnetic moment with respect to the @xmath3-axis , @xmath84 , and the angle defined by the normal to the dw surface with respect to the @xmath3-axis , @xmath85 , describing the tilt of the dw plane . + the action of the sots is equivalent to the presence of two effective magnetic fields : @xmath81 and @xmath82 @xcite . in the present work the fl - field is defined as @xmath86 , with @xmath87 being the proportionality factor between the two effective fields . however , in this section only the zero fl - field scenario ( @xmath88 ) is discussed , due to the fact that the dl - sot is the main source of the domain wall motion . in particular , we find that the same final dw velocity is predicted by the ccm calculations with or without the inclusion of a finite fl - field into the calculations ( more details can be found in appendix b ) . + the resulting ccm fitting curves are shown in fig . [ fig_vdw - hx ] ( solid lines ) . as it can be seen , they reproduce the experimental data well . the effective field ( @xmath81 ) is used as a free parameter in the calculations , while the dmi values used for the fitting procedure are the experimentally extracted ones . by using the best fitting curves , the amplitude and sign of the effective sot - field is extracted , for each device . the corresponding current - field efficiency , @xmath89 , is reported in fig . [ fig_1dm_sots ] as a function of the co thickness . the sign of the effective field is in agreement with a positive spin - hall angle , if the she is assumed as the main source of the observed sot . this is in agreement with what previously reported in literature for pt - based systems @xcite . + as shown in fig . [ fig_1dm_sots ] , the extracted effective field is observed to increase for a co thickness between 0.9 and 1.4 nm and then to level off around @xmath90 [ mt/(@xmath91a / m@xmath6 ) ] for larger thicknesses . this indicates that the effective sot - field generating the observed dw motion is initially scaling with the ferromagnetic layer thickness and then becomes independent of it . , as extracted by the 1 dm calculations . the values of @xmath89 shown here are the ones used for the generation of the fitting curves reported in fig . [ fig_vdw - hx].,width=302 ] next , we use the established @xmath11 technique @xcite to study the current - induced sots in pt@xmath0co@xmath0alo@xmath1 hall cross devices . in fig . [ fig_sots ] the measured current - induced effective fields along the longitudinal ( fig . [ fig_sl - sot ] ) and transverse ( fig . [ fig_fl - sot ] ) direction are shown with respect to the injected current , for one of the studied devices . both effective fields are found to scale linearly with the current amplitude . + in order to learn more about the microscopic origins of sots in the present materials system , their ferromagnetic thickness dependence is studied . accordingly , the current - induced sots are measured for different devices with a co thickness of : 0.93 nm , 0.99 nm , 1.31 nm 1.37 nm and 1.43 nm . + the measured effective fields as a function of the co thickness are reported in fig . [ fig_sot - vs - tco ] . first of all , the two effective fields are found to have the same qualitative dependence on the co thickness , suggesting a possible common origin for such a dependence . second , the two effective fields are observed to first increase and then decrease with the co thickness , clearly indicating a more complicated character than a simple interface - like one . indeed , in the pure interface - like case we would observe a simple 1/t dependence , which is clearly not the case here . third , the dl - field is found to be always larger than the fl - field , for all the investigated co thicknesses . in general , in [ heavy metal]@xmath0ferromagnet systems the primary origin of the dl - torque is usually attributed to the she due to the large soc characterizing the heavy metal @xcite . accordingly , the spin - hall effect is most probably an important source of both sots measured here , where the weaker fl - field is generated by the precession , around the exchange field , of the itinerant spins diffusing in the ferromagnetic layer @xcite . this interpretation is also in agreement with the observed non - monotonous thickness dependence . the she - induced spin - current diffuses in the ferromagnet and interacts with the local magnetization by generating the two sots . the length scale defining the thickness dependence of the corresponding effective fields is the transverse spin diffusion length in co @xcite , reported to be around 1.2 nm @xcite . indeed , after diffusing across the ferromagnet for a distance equal to the transverse spin - diffusion length , the spin - current is absorbed and no further effect on the magnetization beyond this thickness is produced . accordingly , in the first spin diffusion length the effective fields build up , then decay with a further increase of the co thickness . while this qualitatively describes the measurements , we can not rule out further origins that would be influenced by an identical effect , resulting in the similar thickness dependence reported here . , black squares ) and transverse ( @xmath82 , red dots ) effective fields as a function of the ferromagnetic thickness.,width=302 ] in multilayer systems like the one discussed here , the dmi is predicted to originate from the interface between the heavy metal ( pt ) and the ferromagnetic material ( co ) @xcite . accordingly , the dmi is expected to be an interface - like effect , where its effective strength scales with the inverse of the ferromagnetic thickness . in fig . [ fig_dmi_1 ] the measured dmi fields ( blue spheres ) are shown to be proportional to @xmath92 . furthermore , the extracted values of @xmath93 are in agreement with what has already been reported in literature for cidwm experiments @xcite . safeer et al . @xcite reported a 100 mt stopping field for dw motion parallel to the current flow in a pt(3 nm)@xmath0co(0.6 nm)@xmath0alo@xmath1(2 nm ) sample . ryu et al . @xcite reported a 140 mt stopping field in pt(1.5 nm)@xmath0[co(0.3 nm)@xmath0ni(0.7 nm)@xmath0co(0.15 nm ) ] nws . accordingly , the range of stopping fields reported here is in line with previously reported values for other material systems with a pt buffer layer , even though in those previous reports no systematic thickness dependence , as we provide here , is given . [ fig_dmi_1 ] we also show the calculated dw width parameter , @xmath94 ( red pentagons ) , as a function of @xmath92 . using the extracted @xmath93 and the calculated @xmath95 for calculating the dmi strength , results in a dmi that scales with the inverse ferromagnetic thickness ( see fig . [ fig_dmi_2 ] ) . this is in agreement with an interfacial dmi scenario , where the effective dmi strength is expected to be proportional to @xmath96 : @xmath97 , with @xmath98 being the pure interfacial dmi . however , a linear fitting of the data points ( not shown here ) does not generate a linear curve that crosses the origin of the axes . the crossing for @xmath99 would happen at a finite value of the ordinate axis . these observations are in agreement with what reported in ref . @xcite for a pt@xmath0co system , and in ref . @xcite for a pt@xmath0ni@xmath100fe@xmath101 system . cho and co - authors @xcite observed a linear dependence of the effective dmi strength on @xmath92 for @xmath102 nm@xmath103 ( similar to the present study ) , and a rapid drop of it for @xmath104 nm@xmath103 , going towards zero at @xmath99 . nembach and co - authors @xcite obtained similar results and attributed their observations to a non - trivial dependence of the interfacial dmi strength , @xmath98 , on @xmath105 . in the case reported here , devices with thicknesses larger than the studied ones exhibit a weak pma and so the formation of a multi - domain state . this prevents the study of cidwm and so the extraction of dmi , so that here we do not probe the expected faster decay of the effective dmi with respect to the inverse ferromagnetic thickness in thicker devices . + moving the analysis from a qualitative to a quantitative level , we compare now the dmi strengths extracted here with the values obtained by kim and co - authors @xcite , who carried out bls measurements on the very same material stack used for the patterning of the devices reported in this manuscript . in fig . [ fig_dmi_total ] the dmi values by kim et al . @xcite ( squares ) and our values ( dots ) are reported in the same graph . from the graph it is clearly visible that the values of @xmath106 ( solid dots ) obtained by cidwm are in quantitative disagreement with the ones obtained by bls . the two differently extracted values are off by about a factor 3 . indeed , if we multiply the extracted dmi values by a factor @xmath107 ( arbitrary choice , any value between 3 and 4 could have been chosen ) , the values of @xmath108 ( open dots ) are very close to the bls values . ( solid dots ) , is compared with the values obtained by bls by kim et al . @xcite ( open squares ) . the latter result to be about 3 times larger then the former . the open dots are the values of an arbitrarily defined @xmath108.,width=302 ] this could indicate that the procedure of comparing the values of dmi extracted by different techniques needs to be adjusted . if , on the one hand , the @xmath106 obtained here are in good agreement with the outcome of other cidwm - based measurements @xcite , on the other hand @xmath109 is in good agreement with the values obtained by bls measurements on the same sample @xcite . these results clearly motivate a theoretical analysis about the comparability of dmi measurements obtained with different experimental techniques . similar differences were also found by soucaille et al . @xcite , where the dmi values extracted by dw creep motion and bls studies are compared , for several materials stacks with low gilbert damping ( @xmath110 ) . + one possible interpretation proposed by soucaille et al . for the different values of dmi extracted by the two experimental techniques is the gilbert damping experienced by the dw during the creep motion . they observed a larger discrepancy between the two techniques results for materials stacks with a smaller damping factor , suggesting a better agreement for systems with large damping . however , despite the very large damping factor usually reported for pt@xmath0co@xmath0alo@xmath1 ( @xmath111 @xcite ) we do observe a quantitative discrepancy between the dmi values extracted by cidwm and the ones obtained by bls . this rules out the small damping as the key factor generating the discrepancy between the cidwm- and the bls - extracted dmi values . + a second possible interpretation of such a discrepancy relies on the different type of physical process probed by the two techniques . in the bls measurements spin waves with wavelengths of hundreds of nanometers are probed on an area of tens of microns ( defined by the laser spot ) @xcite . on the other hand , in cidwm experiments the object at the center of the study is the dw , whose width is usually @xmath112 nm in pma systems . accordingly , the dw motion is much more sensitive to the local variations in magnetic properties such as anisotropy , magnetization and dmi due to grain boundaries , interface roughness etc . , while the spin waves probed in the bls measurements are affected by the average values of those physical properties . indeed , it has been suggested @xcite that pinning sites could be characterized by a weaker dmi compared to the defect - free regions of the sample , which might help to explain the lower dmi probed by the dw compared to thermal magnons . + a final possible interpretation for the lower dmi extracted by the cidwm study relies on the presence of a non - negligible stt , even though for a related system no significant stt was claimed @xcite . it is known that the standard stt drives the dw motion along with the electron flow , unless exotic materials properties like a negative non - adiabaticity parameter characterize the system under study . however , to the best of our knowledge , no experimental observations of a negative non - adiabaticity parameter , in pt@xmath0co@xmath0alo@xmath1 and related systems , has ever been reported so far . accordingly , the presence of a finite stt would result in the observation of a stopping field that is smaller than the actual dmi field ( as previously reported by torrejon et al . nevertheless , if this was the main origin of the observed discrepancy , the strength of the stt needed to generate such effect would be @xmath113 times the actual sot s strength . this would correspond to a @xmath114 underestimation of the actual strength of the sot by the used 1d model calculations . in the scenario of a simple model based on the she - induced sot , where @xmath115 , this would result in the extraction of a max sot efficiency of @xmath116 . such a large sot efficiency has never been reported before in the study of cidwm , and thus , most likely , this can not explain our results . so the reported discrepancy between cidwm- and bls - extracted dmi values might be understood if a combination of all the three interpretations reported above is taken into account . however , it also shows that care needs to be taken when comparing values of the dmi determined by different techniques . furthermore , this should encourage future comparative studies of different systems using different approaches and in particular theoretically analyze the different probed properties to identify the origin of the discrepancies , which will be reserved for a future study . when comparing the effective sot - fields ( @xmath81 ) extracted by the ccm on the one side and the @xmath11 technique on the other side , two major observations can be made . first , the effective fields obtained by the two techniques are of comparable magnitude . second , while the effective fields extracted by the dw motion study are all contained in a range of values as large as the 25% of the largest extracted value ( see fig . [ fig_1dm_sots ] ) , for the effective fields obtained by hall measurements the largest value is 4 times as large as the smallest one ( see black squares in fig . [ fig_sot - vs - tco ] ) . furthermore , while the effective field increase with increasing thickness is seen by both techniques , the second harmonics technique detects also a stronger decrease for large thicknesses that is not directly visible in the dw motion results . + there may be several possible explanations for the different thickness dependence of the dl - field extracted by the two experiments , and here we discuss two of them . the first explanation is based on the symmetry of the sots . it is well known that sots depend on the polar angle ( angle between the magnetic moment and the z - axis ) @xcite , and the thickness dependence of the same sot ( dl - torque in this case ) does not need to be necessarily the same for different polar angles . furthermore , the magnetic moment in a dw is characterized by a polar angle of 90 deg ( contained in the x - y plane of the sample ) , while the magnetization in the hall measurements is almost collinear to the z - axis ( @xmath20 or @xmath22 state ) . accordingly , the observed different results obtained by cidwm and @xmath11 measurements can be interpreted as a polar angle - dependent scaling of the dl - field with the ferromagnetic thickness . a second possible interpretation is based on the different spin structures that are probed in the two experiments . in the dw motion study , the effective sot - field acts on a highly non - collinear spin texture , while in the hall measurements the probed magnetic state is a single domain . accordingly , the spin - transfer process in the two cases can be different , resulting in the extraction of different effective field amplitudes . furthermore , the dmi present in this system plays a key role in the dw dynamics @xcite , while it does not affect the current - induced dynamics of the saturated magnetic state . finally , the 1d - model used to analyze the torques exerted in the domain wall motion is limited by the virtue that it does not capture the evolution of the structure of the domain walls during their motion and this could lead to a limited understanding of the precise magnitude of the torques . in a future work , using independent determination of the damping and more refined modeling might lead to a more quantitative agreement . + in conclusion , we observe that the analysis of the results of @xmath11 and dw motion experiments provide us with effective sot - fields that are of similar magnitude . however a detailed correlation between the trend of the effective fields measured by the two methods calls for further investigation , which is beyond the scope of this work . a detailed characterization of the thickness dependence of dzyaloshinskii - moriya interaction ( dmi ) and spin - orbit torques ( sots ) in pt@xmath0co(t)@xmath0alo@xmath1 was obtained by mean of two different experimental techniques : current - induced domain wall motion and second - harmonic hall measurements . the sign and strength of the dmi are extracted by measuring the domain wall motion stopping fields . a negative dmi , corresponding to the presence of left - handed homo - chiral dws , is observed to decrease in strength with an increasing co thickness . this confirms that the dmi originates from the pt@xmath0co interface and its measured effective strength decreases for thicker ferromagnetic layers . the extracted dmi strengths are in agreement with values reported in previous current - induced dw motion studies on similar materials stacks , however they are quantitatively different from the values extracted for the very same sample by brillouin light scattering ( bls ) measurements . the dmi values presented here are about a factor 3 smaller than the values extracted by bls . this quantitative disagreement highlights that care has to be taken when comparing the results of the two techniques and the differences possibly originate from the different length scales and physical processes probed in the two experiments . + from the analysis of the dw motion using a collective - coordinates model we extracted the driving sot - field , while the symmetry and magnitude of both damping - like and field - like sot - fields acting on a magnetic single domain state were extracted by @xmath11 hall measurements . the effective damping - like sot - field driving the dw motion is observed to increase with an increasing ferromagnetic layer thickness up to @xmath117 nm . from the @xmath11 measurements both effective fields acting on the homogeneous spin structure are found to initially increase up to a co thickness of about 1.3 nm , and then to decrease with a further increase of the ferromagnetic thickness . the differences in the thickness dependence of the effective fields obtained by the two techniques can have different origins : the non - ability of the 1d - model to capture the real dynamics of the dw internal magnetization during its motion , or the different parts of the polar angle dependence of the torques probed in the two experiments . + the similar non - monotonic thickness dependence observed for the damping - like and field - like effective fields suggests a possible common origin for the two fields : the spin - hall effect in the pt bottom layer . in this scenario , the final effective fields are defined by the transverse spin diffusion length in co , which is expected to be around 1.2 nm . accordingly , for co thicknesses smaller than the transverse spin diffusion length the effective fields are observed to increase , while for co thicknesses larger than the spin diffusion length the effective fields are observed to decrease . however , while qualitatively the behavior fits a dominating she origin we can not rule out additional effects that affect the two torques similarly leading to a similar thickness dependence . finally , the qualitatively different thickness dependence of dmi and sots shows that both effects clearly do not have a common origin in our investigated system . we acknowledge support by the graduate school of excellence materials science in mainz ( mainz ) gsc 266 , staudinger weg 9 , 55128 , germany ; the dfg ( kl1811 , sfb trr 173 spin+x ) ; the eu ( ifox , nmp3-la-2012 246102 ; maspic , erc-2007-stg 208162 ; multirev erc-2014-poc 665672 ; wall , fp7-people-2013-itn 608031 ) and the research center of innovative and emerging materials at johannes gutenberg university ( cinema ) . e. m. acknowledges the support by project mat2011 - 28532-c03 - 01 from spanish government and project sa163a12 from junta de castilla y leon . c .- y . you acknowledges the support by nrf - dfg collaborative research program ( no . 2014k2a5a6064900 ) . kim acknowledges the support by the leading foreign research institute recruitment program ( no . 2012k1a4a3053565 ) through the national research foundation of korea . in the framework of the collective - coordinates model ( ccm ) the dw dynamics is described by three degrees of freedom : the position of the dw in the track , @xmath83 , the in - plane angle of the dw magnetic moment with respect to the @xmath3-axis , @xmath84 , and the angle defined by the normal to the dw surface with respect to the @xmath3-axis , @xmath85 , describing the tilt of the dw surface . the system of equations governing the dw dynamics reads : @xmath118 \\ & + \alpha\gamma_{0}[h_{pin}(q)+q\frac{\pi}{2}h_{dl}cos\phi ] , \end{split}\ ] ] @xmath119 \\ & + q\gamma_{0}[h_{pin}(q)+q\frac{\pi}{2}h_{dl}cos\phi ] , \end{split}\ ] ] @xmath120\frac{dq}{dt } & = -[\frac{2k_{eff}}{\mu_{0}m_{s}}+q\frac{\pi}{2}h_{d}cos(\phi-\chi)+h_{k}cos^{2}(\phi-\chi ) \\ & -\frac{\pi}{2}h_{x}cos\phi-\frac{\pi}{2}h_{fl}sin\phi]tan\chi - q\frac{\pi}{2}h_{dmi}sin(\phi-\chi ) \\ & -\frac{\pi}{2}h_{k}sin2(\phi-\chi ) , \end{split}\ ] ] where @xmath121 , with @xmath122 ( @xmath123 j / m @xcite ) being the dw parameter at rest , @xmath124 , @xmath125 and @xmath126 . @xmath127 mj / m@xmath6 is the pma anisotropy constant , @xmath128 the ferromagnetic layer thickness and @xmath129 a / m the saturation magnetization @xcite . @xmath130 is the dw shape anisotropy field with @xmath131 being the demagnetization field factor @xcite . a large gilbert damping parameter @xmath132 is used as taken from literature @xcite . @xmath133/@xmath134 for the @xmath25/@xmath26 configuration of dw . the dmi is modeled as an effective field along the @xmath3-axis with its amplitude given by @xmath135 , where @xmath74 is the dmi parameter @xcite . @xmath24 is the applied longitudinal field , @xmath136 is the pinning field . the spatially - dependent pinning field accounts for local imperfections ( such as edge and surface roughness or defects ) , and can be derived from an effective spacial - dependent pinning potential , @xmath137 , as @xmath138 ( @xmath139 is the width of the magnetic wire ) . a periodic potential is employed to describe the experimental results , @xmath140 , where @xmath141x@xmath142 j is the energy barrier of the pinning potential and @xmath143 is its periodicity . [ 1dm_1 ] , [ 1dm_2 ] , and [ 1dm_3 ] are numerically solved by means of a 4th runge - kutta algorithm with a time step of 0.1 ps over a temporal window of 100 ns . the analysis based on the 1 dm calculations allows us to gauge the role of the fl - torque on the dw motion . in fig . [ fig_1dm_fl - sot ] the experimental results ( open dots ) for the device with @xmath48 nm ( a ) and ( b ) , and @xmath79 nm ( c ) and ( d ) are reported together with several fitting curves obtained considering different @xmath82 amplitudes ( different @xmath87 values ) but always the same @xmath81 used to generate the fitting curves in fig . [ fig_vdw - hx ] . in fig . [ fig_1dm_posfl_d1 ] and [ fig_1dm_posfl_d4 ] the case @xmath88 ( solid squares ) is compared with the case @xmath144 ( open diamonds ) . while , in fig . 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we report the thickness dependence of dzyaloshinskii - moriya interaction ( dmi ) and spin - orbit torques ( sots ) in pt@xmath0co(t)@xmath0alo@xmath1 , studied by current - induced domain wall ( dw ) motion and second - harmonic experiments . from the dw motion study , a monotonous decay of the effective dmi strength with an increasing co thickness is observed , in agreement with a dmi originating at the pt@xmath0co interface . the study of the ferromagnetic thickness dependence of spin - orbit torques reveals a more complex behavior . the effective sot - field driving the dw motion is found to initially increase and then saturate with an increasing ferromagnetic thickness , while the effective sot - fields acting on a saturated magnetic state exhibit a non - monotonic behavior with increasing co - thickness . the observed thickness dependence suggests the spin - hall effect in pt as the main origin of the sots , with the measured sot amplitudes resulting from the interplay between the varying thickness and the transverse spin diffusion length of the co layer .

You are an expert at summarizing long articles. Proceed to summarize the following text: graphene , one monolayer of carbon atoms tightly packed into a two - dimensional honeycomb lattice , is actively being pursued as a material for next - generation electronics due to its promising electronic properties , such as high carrier mobility @xcite , long phase coherence lengths @xcite . on the other side , the unique two - dimensional atomic structure of graphene implies unique confinement on electron system and offers a perfect platform to explore the amazing physics phenomenons , such as quantum hall effect @xcite and massless dirac fermions @xcite . the first task for experimentalists to study graphene electronics is to fabricate high quality single layer graphene . until now , several different experimental methods have been proposed and realized to prepare single layer ( or few layers ) graphene , including mechanical exfoliation of highly oriented pyrolytic graphite @xcite , patterned epitaxially grown graphene on silicon carbide or transition metal ( e.g. ru , ni ) substrates @xcite , liquid - phase exfoliation of graphite @xcite , substrate - free gas - phase synthesis @xcite , and chemical vapor deposition @xcite . the success in fabricating single layer graphene has stimulated the extensive research efforts ( both theoretical and experimental ) in graphene related research area . the ultimate goal of the use of graphene in next - generation electronics is to realize all - graphene circuit with functional devices built from graphene layers or graphene nanoribbons ( gnrs ) @xcite . as the basic building blocks of such circuit , the concept of electronic devices based on graphene have been proposed theoretically and realized by experiments recently , such as field effect transistors @xcite , _ junctions @xcite , gas molecule sensor @xcite , and so on . in this article , we will focus our discussion on the basic electronic and transport properties of gnrs and their application to electronic devices . in particular , the theoretical investigations of gnrs physics and the technical aspects of gnr based electronic devices will be reviewed in detail . for other topics on the recent experimental and theoretical research efforts on graphene , please refer to the reviews by katsnelson @xcite , geim _ et al . _ @xcite , beenakker @xcite , and castro neto _ the realization of graphene electronics relies on the ability to modify the electronic properties of finite - size graphenes ( for example , from semiconducting to metallic ) by varying their size , shape , and edge orientation . such unique property compared to traditional semiconductor materials , such as silicon , would ultimately enable the design and miniaturization of future electronic circuit by patterned graphene . one of the most important issues in patterned graphene fabrication is the control of the nanoribbon width . in order to take advantage of quantum confinement effects in graphene , the ribbon width should go down to nanometer scale . to realize the patterning of graphene with nano - scale width , several different techniques have been proposed including standard e - beam lithography ( fig . 1a ) @xcite , microscope lithography ( fig . 1b ) @xcite , chemical method ( fig . 1c ) @xcite , metallic nanoparticle etching @xcite , and e - beam irradiation of ultrathin poly(methylmethacrylate ) ( pmma ) @xcite . as shown in fig . 1a , the scanning electron microscopy ( sem ) image reveals the graphene can be patterned by traditional e - beam lithography technique into nanoribbons with various widths ranging from 20 to 500 nm @xcite . figure 1b shows 10-nm - wide nanoribbon etched via scanning tunnelling microscope ( stm ) lithography . by setting the optimal lithographic parameters , it is possible to cut gnrs with suitably regular edges , which constitutes a great advance towards the reproducibility of gnr - based devices @xcite . figure 1c shows atomic force microscopy images of chemically derived gnrs with various widths ranging from 50 nm to sub-10 nm . these gnrs have atomic - scale ultrasmooth edges @xcite . [ tbp ] ( color online ) various gnrs got from different experimental methods : ( a ) the sem image of gnrs patterned by e - beam lithography . reprinted with permission from ref . @xcite , z. chen et al . , physica e * 40 * , 228 ( 2007 ) . 2007 , elsevier . ( b ) an 8-nm - wide 30@xmath0 gnr bent junction connecting an armchair and a zigzag nanoribbon etched by stm lithography . reprinted with permission from ref . @xcite , l. tapaszt et al . , nat . nanotechnol . * 3 * , 397 ( 2008 ) . 2008 , nature publishing group . ( c ) gnrs are got by using simple chemical methods . reprinted with permission from ref . @xcite , x. li et al . , science * 319 * , 1229 ( 2008 ) . 2008 , american association for the advancement of science.,title="fig:",scaledwidth=48.0% ] the electronic properties of gnrs exhibit a strong dependence on the orientation of their edges . as two typical types , armchair gnrs ( agnrs ) and zigzag gnrs ( zgnrs ) can be obtained by lithography technology along the specific orientation on graphene ( fig . 2b ) @xcite . actually , the detailed edge structures ( both armchair and zigzag ) have already been clearly observed in recent experiments @xcite . one of the most serious obstacle to graphene electronic application is the reliable control of the edge structure of gnrs . theoretical studies predict that edge states ( in a manner similar to the well - known concept of surface states of a 3d crystal ) in graphene are strongly dependent on the edge termination and affect the physical properties of gnrs @xcite . however , until now there is no reliable experimental method which is able to exactly control the edge structures and reduce their roughness . an interesting experimental observation is that the band gaps of gnrs show little orientation dependence @xcite and all fabricated gnrs show semiconducting behavior @xcite , which seems inconsistent with theoretical results @xcite . one of the reason for such inconsistency comes from the roughness of gnr edges and our explanation is also given in the following part of the article . another issue related to gnr edges is the edge passivation . since the dangling bonds from the edge carbon atoms have relatively high chemical activity , there is the possibility that other chemical elements present in the material fabrication process ( such as c , o , n , h and other chemical groups formed by these atoms ) would interact with the edge atoms and modify the electronic properties of gnrs . to the best of our knowledge , this issue has not been properly solved experimentally . [ tbp ] ( color online ) the structures of h - passivated 11-agnr ( a ) and 6-zgnr ( b ) , where big green balls and small blue balls represent carbon atoms and hydrogen atoms , respectively . integer @xmath1 is their width index.,title="fig:",scaledwidth=48.0% ] next we will review some basic electronic and transport properties of gnrs from the theoretical viewpoint . figs . 2a and 2b show two typical models of armchair and zigzag gnrs in first - principles or other atomic - level electronic structure calculations , noting as 11-agnr and 6-zgnr , respectively . here the numbers 11 and 6 are defined as the width index , @xmath1 . in order to remove the effect of dangling bonds , the edges of gnrs are saturated by hydrogen atoms . as geometrically terminated graphene , the electronic structure of gnrs can be modelled by imposing appropriate boundary conditions on schrdinger s equation with simple tight - binding ( tb ) approximations based on @xmath2-states of carbon @xcite . another way to get the band structure is to solve two - dimensional dirac s equation of massless free particles with an effective speed of light to model gnr system @xcite . within these models , it is predicted that gnrs with armchair - shaped edges can be either metallic or semiconducting depending on their widths , as shown in fig . 3a . on the other side , the gnrs with zigzag - shaped edges are metallic with peculiar edge states on both sides of ribbons regardless of their widths , as shown in fig . 4a . @xcite [ tbp ] ( color online ) the variation of band gaps of n@xmath3-agnrs as a function of width ( w@xmath3 ) obtained ( a ) from tb calculations and ( b ) from first - principles calculations ( symbols ) . ( c ) first - principles band structures of n@xmath3-agnrs with n@xmath3= 12 , 13 , and 14 , respectively . reprinted with permission from ref . @xcite , y. -w . son et al . lett . * 97 * , 216803 ( 2006 ) . 2006 , american physical society.,title="fig:",scaledwidth=48.0% ] further detailed _ ab initio _ and gw quasiparticle calculations show that all of the agnrs exhibit semiconducting behavior and the energy gaps decrease as a function of increasing ribbon widths . the variation in energy gaps can be separated into three distinct family behaviors @xcite , as shown in fig . 3b . as mentioned above , such dependence of band gap on the geometrical structure of gnr offers unique possibility to modify the electronic properties of gnrs simply by controlling the width and edge orientation in order to realize all - graphene functional devices . [ tbp ] ( color online ) electronic structures of graphene nanoribbons . in all figures , the fermi energy ( e@xmath4 ) is set to zero . a , the spin - unpolarized band structure of a 16-zgnr . b , the spatial distribution of the charge difference between @xmath5-spin and @xmath6-spin for the ground state when there is no external field . the magnetization per edge atom for each spin on each sublattice is 0.43@xmath7 with opposite orientation , where @xmath7 is the bohr magneton . the graph is the electron density integrated in the z direction , and the scale bar is in units of 10@xmath8e@xmath8 . c , from left to right , the spin - resolved band structures of a 16-zgnr with the external field of 0.0 , 0.05 and 0.1v , respectively . the red and blue lines denote bands of @xmath5-spin and @xmath6-spin states , respectively . reprinted with permission from ref . @xcite , y. -w . son et al . , nature * 444 * , 347 ( 2006 ) . 2006 , nature publishing group.,title="fig:",scaledwidth=48.0% ] upon inclusion of the spin degrees of freedom within density functional theory ( dft ) calculations , zgnrs are predicted to have a magnetic insulating ground state with ferromagnetic ordering at each zigzag edge and antiparallel spin orientation between the two edges @xcite , as shown in fig . the spin polarization originates from the edge states that introduce a high density of state ( dos ) at the fermi energy . it can be qualitatively understood in terms of the stoner magnetism of @xmath9 electrons ( in analogy to conventional @xmath10 electrons ) , which occupy a very narrow edge band and render instability of spin - band splitting @xcite . what is more interesting , the zigzag gnrs show half - metallic behavior when external transverse electric field is applied across the zgnrs along the lateral direction @xcite , as shown in fig . however , such spin related half - metallic phenomenon becomes weak with increasing ribbon width ( since the total energy difference per edge atom between spin - unpolarized and spin - polarized edge states is only about 20 mev in their simulation system and decreases with increasing width ) and is not energetically stable if the width of gnr is significantly larger than the decay length of the spin - polarized edge states @xcite . on the other hand , it is predicted that the half - metallicity can be also achieved in edge - modified or doped zgnrs @xcite another important issue regarding the basic electronic structures of gnrs relies on the edge states . due to the presence of the edge states , the @xmath2 and @xmath11 subbands of metallic zgnrs ( in the spin - unpolarized state ) do not cross with each other at the fermi level to span the whole energy range like metallic armchair carbon nanotubes ( cnts ) ( the left panel of fig . this leads to the fact that the transport property of zgnrs under a low bias voltage ( or a small potential step ) is only determined by the transmission between @xmath2 and @xmath11 subbands ( as shown in fig . 5a ) . with the presence of such unique band structures , zgnrs exhibit two distinct transport behaviors depending on the existence of @xmath12 mirror symmetry with respect to the midplane between two edges @xcite , although all the zgnrs have similar metallic energy band structures . since the @xmath2 and @xmath11 subbands of symmetric zgnrs ( i.e. , width index @xmath1 is an even number ) have opposite definite @xmath12 parities , the transmission between them is forbidden ( the left panel of fig . 5b ) . for asymmetric zgnrs ( i.e. , width index @xmath1 is an odd number ) , however , their @xmath2 and @xmath11 subbands do not have definite @xmath12 parities , so the coupling between them can contribute to about one conductance quantum ( the right panel of fig . this transport difference can be clearly reflected in the current - bias - voltage ( @xmath13-@xmath14 ) characteristics of zgnrs by using the first - principle transport simulation , as shown in fig . 5c . although metallic armchair carbon nanotubes also have @xmath2 and @xmath11 subbands with definite parities , such symmetry - depending ( or band - selective ) @xmath13-@xmath14 characteristics can not be observed in them because of the crossover of their subbands , i.e. , the absence of edge states . recently , theoretical work predicts a very large magnetoresistance in a graphene nanoribbon device due to the existence of edge states @xcite . [ tbp ] ( color online ) ( a ) schematic band structure around the fermi level of a zgnr under a positive ( @xmath15 ) and a negative ( @xmath16 ) potential step . ( b ) conductance of 8-zgnr and 7-zgnr under two potential steps shown in ( a ) . ( c ) @xmath13-@xmath14 curves of the two - probe system ( see the inset ) made of zgnrs with different widths @xmath1 . reprinted with permission from ref . @xcite , z. li et al . lett * 100 * , 206802 ( 2008 ) . 2008 , american physical society.,title="fig:",scaledwidth=47.0% ] besides the fabrication and theoretical study of monolayer graphene and gnrs , recent experimental @xcite and theoretical @xcite studies are also carried out on bilayer graphene and gnrs . theoretically , it is shown that the bilayer gnrs and monolayer gnrs have some similar electronic properties such as edge states localized at the zigzag edges and semiconducting behavior of armchair bilayer gnrs @xcite . experimentally , it is found that the bilayer graphene has unique features such as anomalous integer quantum hall effects @xcite , which is absent in single layer graphene . and the size of energy gap of such bilayer structures can be controlled by adjusting carrier concentration @xcite as well as by an external electric field @xcite . these unique properties open an opportunity to implement bilayer graphene or gnrs in various electronic applications . as mentioned above , current experimental techniques ( such as lithography ) are not able to realize exact control of the edge structures of gnrs and the edges are always very rough due to the limitation of the fabrication technology @xcite . there are theoretical evidences that such edge disorders can significantly change the electronic properties of gnrs @xcite , and lead to some unexpected physics effect , such as the anderson localization @xcite and coulomb blockade effect @xcite . these effects have already been observed in lithographically obtained graphene nanoribbons @xcite . the edge roughness is also crucial for the spin polarized properties of gnrs . as we know , the magnetic properties of gnrs depend on the highly degenerate edge states . in principle a perfect edge structure is necessary for stabilizing magnetic properties of gnrs as theoretically predicted . an important question is , how robust the spin - polarized state is in the presence of edge defects and impurities ? the answer to this question is not only scientifically interesting to better understand the physical mechanism of spin polarization in gnrs but also has important technological implications in the reliability of gnrs as a new class of spintronic materials . first - principles theoretical studies reveals the effect of edge vacancies and substitutionally doped boron atoms , as typical examples of structural edge defects and impurities , on the spin - polarization of zgnrs @xcite . the calculated energy difference between the magnetic state [ both antiferromagnetic ( af ) and ferromagnetic ( fm ) ] and the nonmagnetic state is found to rapidly decrease with increasing defect concentration and eventually decrease to zero ( nonmagnetic ) , as shown in fig . the critical defect ( impurity ) concentration is found to be @xmath17 0.10/ when the ribbon width is larger than 2 nm . [ tbp ] ( color online ) the energy difference per edge atom between the magnetic ( af or fm ) and paramagnetic ( pa ) state as a function of vacancy concentration in the edge . two different ribbon widths of @xmath1=6 and @xmath1=4 are shown . the inset shows the critical concentration as a function of the ribbon width up to 5 nm . reprinted with permission from ref . @xcite , b. huang et al . b * 77 * , 153411 ( 2008 ) . 2008 , american physical society.,title="fig:",scaledwidth=40.0% ] evidently , the magnetism in gnrs depends on a high density of state ( dos ) around the fermi energy coming from the highly degenerate edge states in a perfect ribbon edge ( @xmath18 ) that renders instability of spin polarization @xcite . the presence of edge vacancies and impurities would decrease the dos at @xmath18 since they do not contribute to the same edge state . from stoner model , such decrease of dos will suppress the spin polarization of gnr systems . therefore , the practical realization of the spin polarization in gnrs for spintronics applications could be rather challenging @xcite . recently , an interesting theoretical work systematically studied the spin current in rough gnrs and predicted that only gnrs with imperfect edges exhibit a nonzero spin conductance while there is no spin current in perfect gnrs @xcite . it confirms that the edge effect is of great importance to spin related properties of gnrs . furthermore , the problem of edge passivation has not yet clearly resolved by experiment until now . from the theoretical viewpoint , the edge passivation can be well modeled by the modifications of the hopping energies in the tight - binding approach @xcite or via additional phases in the boundary conditions @xcite . recent theoretical modeling and calculations have indicated that the edge passivation has a strong effect on the electronic and spin - polarized properties of gnrs @xcite . the possible passivation species include hydrogen , carbon , oxygen , nitrogen , and other chemical groups . further experimental works are needed to explore the realistic edge structures of gnrs at atomic scale and determine which types of edge passivation are favorable . the interesting and unique electronic properties of gnrs , such as orientation and width dependence of transport behavior , offer great possibilities for their electronic device applications . compared with other electronic materials , one of the most promising advantage of gnrs is that gnr - based devices and even integrated circuits can be fabricated by a single process of patterning a graphene sheet @xcite . figures 7a-7c illustrate three basic device building blocks : ( i ) metal - semiconductor junction , ( ii ) _ p - n _ junction , and ( iii ) hetero - junction , which can be , respectively , made by patterned gnrs ( i ) along different direction , ( ii ) with different edge doping , and ( iii ) with different widths . it was proposed that a variety of devices can be constructed from these building blocks . for example , a field effect transistor ( fet ) can be made simply by two metal - semiconductor junctions , as shown in fig . 7d . there are some potential key advantages in designing and constructing device architectures based on gnrs . the first advantage is the perfect atomic interface , a feature that is difficult to achieve for the interconnection between nanotubes of different diameter and chirality . second , it is generally difficult to find a robust method to make contact with the molecular device unit , because there exists usually a large contact resistance between the metal electrodes and molecules ( _ e.g. _ , single - walled cnt ) due to a very small contact area . this difficulty may be circumvented by using gnrs , because the gnr - based devices can be connected to the outside circuits exclusively via metallic gnrs ( or graphene ) , as illustrated in fig . 7d , which serve as extensions of metal electrodes to make contact with the semiconducting gnrs so that an atomically smooth metal - semiconductor interface is maintained with minimum contact resistance . last but not the least , the edges of gnrs may serve as effective sites for doping . in principle , by introducing different types of dopants at different sections of gnr edges , one can realize a _ p - n _ junction by selective doping , as shown in fig . 7b . [ tbp ] ( color online ) schematics of three device building blocks : ( a ) a metal - semiconductor junction between a zigzag and an armchair gnr , ( b ) a _ p - n _ junction between two armchair gnrs with different edge doping , and ( c ) a heterojunction between two armchair gnrs of different widths ( band gaps ) . ( d ) schematics of a gnr - fet , made from one semiconducting 10-agnr channel and two metallic 7-zgnr leads connected to two external metal electrodes . reprinted with permission from ref . @xcite , q. yan et al . , nano lett . * 7 * , 1469 ( 2007 ) . 2007 , american chemical society.,title="fig:",scaledwidth=38.0% ] one of the most important electronic applications based on gnrs is field effect transistors . recently , experimental studies @xcite have indicated the possibility of fabricating gnr - based transistors . the advantage of gnrs as an alternative material for transistors is that it could bypass the chirality challenge of cnts while retaining the excellent electronic properties of graphene sheets , such as the high @xmath19/@xmath20 ratio and excellent electron / hole mobilities . the performance of one sub-10-nm gnr - fet in the latest work from dai s group is shown in figs . 8a and 8b ( the transfer and output characteristics , respectively , for the gnr device with the width of @xmath17 2 @xmath21 0.5 nm and the channel length of @xmath17 236 nm)@xcite . this device delivered @xmath19 @xmath17 4 @xmath22a at @xmath23 = 1 v , @xmath19/@xmath20 ratio @xmath24 10@xmath25 at @xmath23 = 0.5 v , subthreshold slope @xmath26=d@xmath27/@xmath28 @xmath17 210 mv / decade and transconductance @xmath17 1.8 @xmath22s ( @xmath17900 @xmath22s/@xmath22 m ) . the device performance is comparable with the best cnt - based transistors . however , the dirac point was not observed around zero gate bias in this measurement , indicating _ p_-doping effects at the edges or by physisorbed species during the chemical treatment steps . [ tbp ] ( color online ) ( a ) and ( b ) transistor performance of gnr - fets with width of @xmath17 2 nm and channel length of @xmath17 236 nm [ ( c ) and ( d ) , width of @xmath17 60 nm , and channel length of @xmath17 190 nm ] . ( a ) transfer characteristics ( current vs gate voltage @xmath29-@xmath30 ) under various @xmath23 . @xmath19/@xmath20 ratio of @xmath24 10@xmath25 is achieved at room temperature . ( b ) output characteristics ( @xmath29-@xmath23 ) under various @xmath30 . on current density is @xmath17 2000 @xmath22a/@xmath22 m in this device . ( c ) transfer and ( d ) output characteristics of the 60 nm width gnr - fet device . reprinted with permission from ref . @xcite , x. wang et al . lett . * 100 * , 206803 ( 2008 ) . 2008 , american physical society.,title="fig:",scaledwidth=48.0% ] together with experimental progress on gnr - based transistors , theoretical studies using semiclassical and quantum transport models show that gnr - based fets could have a similar performance as cnt - based fets and might outperform traditional si - based fets @xcite . figure 9 shows a first - principles study on the performance of a typical gnr - based fet made with a 5.91 nm long intrinsic semiconducting 10-agnr channel connected to two metallic 7-zgnr leads ( source and drain ) @xcite . in fig . 9a , the near - symmetric @xmath31 curve shows an excellent ambipolar transistor with on / off ratio @xmath32/@xmath20 @xmath17 2000 and subthreshold swing of @xmath26 @xmath17 60 mv / decade , which are comparable to those of high performance cnt - fets . such the field effect can be clearly reflected in the change of @xmath33 characteristics under different gate voltages ( fig . figure 9c shows the @xmath31 curves of the gnr - fets made from the same 10-agnr channel with its length ranging from 1.69 to 6.76 nm , from which the values of @xmath26 are derived as a function of @xmath34 as shown in figure 9d . clearly , @xmath26 decreases with increasing @xmath34 , and gradually approaches @xmath1760 mv / decade when @xmath34 becomes longer than 6 nm . meanwhile , the on - current stays the same , independent of @xmath34 , but the off - state leakage current increases rapidly with decreasing @xmath34 , which gives rise to a large @xmath26 . the performance of the ambipolar gnr - fets made of intrinsic semiconductor channels can be understood in terms of metal - semiconductor tunneling junctionh within the semiclassical band - bending model . [ tbp ] ( color online ) ( a ) @xmath31 curve for a 5.91 nm long intrinsic 10-agnr channel ( @xmath14=20 mv ) . ( b ) @xmath33 curves under different gate voltage ( @xmath35 ) . ( c ) @xmath31 curves for different channel lengths ( @xmath14=20 mv ) . ( d ) the subthreshold swing ( @xmath26 ) as a function of channel length @xmath34 . ( e),(f ) @xmath31 curves for a 5.91 nm long 10-agnr channel with selective n and b doping , respectively ( @xmath14=20 mv ) . reprinted with permission from ref . @xcite , q. yan et al . , nano lett . * 7 * , 1469 ( 2007 ) . 2007 , american chemical society.,title="fig:",scaledwidth=48.0% ] compared with the basic ambipolar fets , it is well known that _ n_-type ( or _ p_-type ) fets serve as critical transistor devices for digital electronics applications @xcite . to realize such device design based on gnrs , a method was proposed using n ( or b ) atoms as selective dopants at the channel region of perfect gnr - fets ( the positions of b or n are indicated by arrows in fig figure 9e ( 9f ) shows the calculated @xmath31 curves under @xmath14 = 20 mv , exhibiting the typical behavior of a _ n_-type ( _ p_-type ) fet@xcite . it is suggested that all of the functional transistor devices that work in traditional si - based circuits could be realized by gnrs and gnr - based junctions in principle . [ tbp ] ( color online ) ( a ) the schematic structure of the field effect transistor ( fet ) made from a single 5-zgnr . the semiconducting channel is obtained by edge doping of n in a finite - length region ( the center region ) . ( b ) simulated @xmath13-@xmath27 curves of n - doped gnr - fets under @xmath14 = 0.01 v. the channel length is 8.54 nm and the linear doping concentration is 0.1365 @xmath36 . ( c ) the dependence of the subthreshold swing @xmath26 ( blue line ) and the on / off current ratio ( red line ) on the channel length @xmath34 . reprinted with permission from ref . @xcite , b. huang et al . . lett . * 91 * , 253122 ( 2007 ) . 2007 , american institute of physics.,title="fig:",scaledwidth=48.0% ] noting the current experimental difficulty to get an accurate z - shape junction ( i.e. , fet shown in fig . 7d ) due to the limitation of lithography technique , a new type of field effect transistor has also been proposed taking advantage of the metal - semiconductor transition in zgnrs induced by substitutional doping of nitrogen or boron atoms at their edges @xcite , as shown in fig . 10a . besides simplifying the fabrication process , such a linear configuration can also increase the device density in electronic circuits . figure 10b shows a typical @xmath37 curve for the n - doped gnr - fet ( with the channel length of 8.54 nm ) under the bias voltage @xmath14 = 0.01 v. clearly , the doped fet exhibits ambipolar characteristics , similar to the z - shape fets . the relationship between the device performance and the channel length is demonstrated by calculating @xmath31 curve of n - doped gnr - fets as a function of the doped channel length while keeping the bias voltage @xmath14 at 0.01 v. as shown in fig . 10c , the subthreshold swing @xmath26 of these doped gnr - fets decreases and the on / off current ratio increases exponentially . it can be seen that for good device performance with small @xmath26 value ( e.g. , below 100 mv / decade ) and high on / off current ratio ( e.g. , above 100 ) , the doped channel length should be longer than 5 nm . the minimum leakage current of those fets with the doped channels shorter than this critical length will be greatly enhanced by direct tunneling , which lowers the device performance . besides ideal case , some more practice issues concerning gnr - based fets are discussed in recent theoretical works . for example , the effects of the various contact types and shapes on the performance of schottky - barrier - type gnr - fets have been investigated theoretically @xcite , which indicates that the semi - infinite normal metal can potentially provide promising performance . in addition , the effect of edge roughness and carrier scattering on gnr - fets have been studied @xcite . the presence of edge disorder significantly reduces on - state currents and increases off - state currents ( the on / off ratio decreases ) , and introduces wide variability across devices . these effects become weaker for gnrs with larger width and smoother edges . however , the band gap decreases with increasing width , thereby increasing the band - to - band tunneling mediated subthreshold leakage current even with perfect gnrs . obviously , without atomically precise edge control during fabrication , it is hard to get reliable and stable performance of gnr - fets . due to their unusual basic properties , gnrs as well as graphene are promising for a large number of applications @xcite , from spin filters @xcite , valley filters @xcite , to chemical sensors @xcite . gnrs can be chemically and/or structurally modified in order to change its functionality and hence its potential applications . in summary , we review the basic electronic and transport properties of graphene nanoribbons , and discuss recent theoretical and experimental progress on gnr - based field effect transistors from the viewpoint of device application . due to the interesting electronic and magnetic properties , gnrs have been demonstrated as a promising candidate material for future post - silicon electronics such as transport materials , field effect transistors , and spin injection or filter . more experimental efforts will focus on fabricating high quality nanoribbon samples with accurate control of the edge structures . d. martoccia , p. r. willmott , t. brugger , m. bjorck , s. gunther , c. m. schleputz , a. cervellino , s. a. pauli , b. d. patterson , s. marchini , j. wintterlin , w. moritz , and t. greber , phys . * 101 * , 126102 ( 2008 ) . y. hernandez , v. nicolosi , m. lotya , f . m. blighe , z. sun , s. de , i. t. mcgovern , b. holland , m. byrne , y. k. gunko , j. j. boland , p. niraj , g. duesberg , s. krishnamurthy , r. goodhue , j. hutchison , v. scardaci , a. c. ferrari , and j. n. coleman , nat . nanotechnol . * 3 * , 563 ( 2008 ) .

the successful fabrication of single layer graphene has greatly stimulated the progress of the research on graphene . in this article , focusing on the basic electronic and transport properties of graphene nanoribbons ( gnrs ) , we review the recent progress of experimental fabrication of gnrs , and the theoretical and experimental investigations of physical properties and device applications of gnrs . we also briefly discuss the research efforts on the spin polarization of gnrs in relation to the edge states .

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Proceed to summarize the following text: low dimensional interacting spin systems have long been a laboratory for the discovery of novel quantum states of matter . enhanced quantum fluctuations due to reduced dimensionality enable the appearance of multiple quantum phases with unique characteristics driven by the interplay between strong interactions , external ( e.g. , pressure and magnetic field ) and internal ( e.g. , crystal field effects ) potentials and lattice geometry that are suppressed in higher dimensions . the relative simplicity of the microscopic models facilitates the development of exact analytical solutions in many cases and powerful field theoretic and computational approaches in others providing greater insight into the emergence of these complex phases and their physical properties . concurrent rapid advances in the synthesis and characterization of low - dimensional quantum magnets have kept the study of these systems one of the most active frontiers in condensed matter physics ( see landee and turnbull@xcite for a recent pedagogical review ) . one of the most remarkable results in the study of quantum spin models one that substantially enhanced our understanding of long range order in quantum many body systems is the pioneering work by haldane.@xcite by studying the non - linear sigma model in ( 1 + 1 ) dimensions , haldane conjectured that the ground state of the one - dimensional ( 1d ) heisenberg antiferromagnet ( hafm ) has gapless excitations for half - odd integer spins , whereas that for integer spins is separated from all excited states by a finite spin gap ( haldane gap ) . haldane s conjecture has inspired numerous theoretical studies of integer spins in low dimensions , primarily @xmath0 spins where the haldane phase is most robust . these include chain mean field theory ( cmft),@xcite exact diagonalization,@xcite density matrix renormalization group ( dmrg),@xcite and quantum monte carlo ( qmc ) simulations.@xcite the theoretical studies have been complemented by the discovery of several quasi - one - dimensional ( q1d ) spin @xmath0 quantum magnets , such as agvp@xmath1s@xmath2@xcite ndmap,@xcite nenb,@xcite nenp,@xcite nino,@xcite pbni@xmath1v@xmath1o@xmath3,@xcite srni@xmath1v@xmath1o@xmath3,@xcite tmnin,@xcite and y@xmath1banio@xmath4.@xcite in these materials , the magnetic ions are arranged in chains , with weak but finite inter - chain couplings , which affect the ground state phases . additionally , in most known @xmath0 magnets , the ubiquitous heisenberg exchange is complemented by a single - ion anisotropy . the expanded hilbert space of the @xmath0 spins , and the interplay between multiple competing interactions , external magnetic field and different lattice geometries result in a rich variety of ground state phases . in addition to the gapped haldane phase , examples of exotic quantum states realized in low dimensional interacting spin systems include experimentally realized bose einstein condensation ( bec ) of magnons,@xcite quantum paramagnet,@xcite and the recently proposed spin supersolid@xcite and ferronematic@xcite phases . recent advances in synthesis techniques have made it possible to engineer quasi - low - dimensional materials where the `` effective dimensionality '' ( that is , inter - chain or inter - layer couplings ) and hamiltonian parameters ( such as the ratio of exchange interaction and single - ion anisotropy ) can be controlled.@xcite this raises the possibility of preparing materials with desired predetermined properties . the search for such tailor - made materials has grown in recent years , as these are believed to drive the next generation of electronics . in addition to condensed matter systems , rapid advances in the field of ultracold atoms in optical lattices have opened up a new frontier in the study of interacting many - body systems in arbitrary dimensions . the unprecedented control over number of atoms , interactions , and lattice geometry makes it an ideal testbed for preparing and studying novel quantum states . the most remarkable property of the haldane phase is that even though there is no long - range magnetic order all spin - spin correlations decay exponentially there exists a hidden or nonlocal order measured by the string order parameter introduced by den nijs and rommelse@xcite . the introduction of a nonlocal order parameter was a marked departure from the conventional practice of characterizing quantum many body states in terms of local order parameters and ushered in the study of topological phases an area of intense current research . much insight into the nature of the haldane phase can be gained from the aklt ( affleck - kennedy - lieb - tasaki ) state the exact ground state of a 1d @xmath0 spin chain where the near - neighbor heisenberg interaction is supplemented by an additional interaction between neighboring spins.@xcite the aklt state may be understood by noting that the spin at each site can be thought of as a symmetric combination of two @xmath5 spins . pairs of these spins on neighboring sites form a singlet on each bond . for a periodic chain , this forms a unique valence bond solid ground state a singlet on each bond with a gap to lowest excitations . but for open chains , there remains an unpaired @xmath5 moment at each end which is doubly degenerate ( see fig . [ aklt ] for an illustration of this state ) . consequently , the ground state is gapped in the `` bulk '' and has degenerate gapless edge states . the state is protected by this @xmath6 symmetry . using the current terminology of topological states , this state is a symmetry protected topological ( spt ) phase . the haldane phase is adiabatically connected to the aklt state in other words , it has the same qualitative character . in fact , the haldane phase is widely recognized as the earliest and best understood spt phase in interacting many body systems.@xcite degrees of freedom composed of the symmetric combination of two spin @xmath5 spins ( small solid circles ) . in the aklt state , pairs of @xmath5 spins on neighboring sites form a singlet state ( blue lines ) . for open boundaries , this leaves an unpaired @xmath5 degree of freedom at each boundary . in the thermodynamic limit , these spins are completely decoupled from the rest of the lattice , leading to a fourfold degenerate ground state . ] what happens to the haldane phase in the presence of additional interactions , such as , inter - chain coupling and single - ion anisotropy ? the question is not simply of academic interest all real materials have nonzero interchain coupling and in many quantum magnets , the crystal electric field lifts the degeneracy of the local hilbert space in the form of a single - ion anisotropy . theoretical studies have shown that both these interactions destroy the haldane phase at sufficiently strong interaction strengths the single - ion anisotropy drives a transition to a quantum paramagnetic phase whereas interchain coupling favors long - range afm ordering but the ground state remains gapped up to finite values of the couplings . what is the nature of the gapped ground state away from the isotropic heisenberg point ? does it still retain its spt character ? the string order parameter the only definitive probe for the haldane phase is strictly defined only in one dimension and its extension to coupled chains can not be trusted without corroboration from additional measurements . fortunately , recent advancements in the study of spt phases have yielded new probes that provide better insight into the nature of the putative haldane phase away from the isotropic heisenberg point . in this brief review , we summarize some of our recent progress towards fully characterizing the haldane phase in quasi - one - dimensional geometries . we consider a model of low dimensional quantum magnets consisting of an array of weakly coupled spin-1 hafm chains . this model can be described by a spatially anisotropic hamiltonian of @xmath0 spin operators with nearest - neighbor spin exchange interactions and on - site single - ion anisotropy : @xmath7}{\vec s}_{i}\cdot{\vec s}_{j}+d\sum_{i}\left(s_{i}^{z}\right)^{2}.\ ] ] the first two sums are over all pairs of interacting spins @xmath8 , with @xmath9 and @xmath10 $ ] referring to nearest neighbor spin pairs along and perpendicular to the chain direction , respectively , while the third sum is over all lattice sites @xmath11 . here , we limit ourselves to nearest neighbor interactions on bipartite lattices to avoid the sign problem in qmc , and tune the interactions @xmath12 and @xmath13 to generate systems of weakly interacting spin chains . this is illustrated in fig . [ lattice ] for a spatially anisotropic square lattice composed of weakly interacting spin chains . in this paper , we consider several bipartite lattices in the quasi - one - dimensional limit . for each lattice , we define a chain length @xmath14 as the system size along the chain direction , and a perpendicular length @xmath15 such that for 2d lattices the total number of spins @xmath16 while for 3d lattices @xmath17 . the ratio of these two length scales forms the aspect ratio @xmath18 , which we set at @xmath19 unless otherwise stated . we henceforth set the intrachain interaction strength to unity ( @xmath20 ) , and study the ground state properties of the above model for various single - ion anisotropy @xmath21 and interchain coupling @xmath13 . ( blue lines ) and weak interchain coupling @xmath13 ( red lines ) . ] quantum monte carlo methods map quantum systems onto equivalent classical representations in @xmath22 dimensions , where the configuration space can be stochastically sampled by markov chain monte carlo methods.@xcite the numerical results presented here were obtained using the stochastic series expansion ( sse ) formulation of qmc.@xcite briefly , the density matrix can be expanded as a taylor series , @xmath23 . for a spin hamiltonian consisting of onsite fields and pair interactions , we can write @xmath24 , where the sum is over all pair interactions , or _ bonds_. the powers of @xmath25 appearing in the taylor expansion of @xmath26 can then be written as a sum over products of bond operators , @xmath27 , where @xmath28 represents a sequence of bond operators called the _ operator string_. note that the overall sign of a particular operator string depends on the number of bond operators with negative weight . to avoid the sign problem in qmc , all the bond operators must therefore be positive definite , which in practice limits the study of heisenberg antiferromagnets to bipartite lattices , where a suitable sublattice rotation is available . the configuration space consists of all possible rearrangements of the operator string , the length of which takes into account the expansion variable @xmath29 , while its sequence dictates the states @xmath30 appearing in the partition function , @xmath31 . loop algorithms exist to efficiently sample this configuration space.@xcite in addition to the sse method , we implement a projective qmc method that allows us to access both ground state expectation values as well as wave function overlaps between the ground state and a trivial product state . this method is quite similar to sse , and details are given elsewhere.@xcite here we present recent qmc results on the ground state properties of the spin @xmath0 hafm described by the hamiltonian in eq . ( [ model ] ) . first , we show how finite size scaling can be used to accurately determine the ground state phase boundaries between the haldane phase and neighboring magnetically ordered phases . next , we examine the effect of lattice geometry on the critical coupling of bipartite lattices . the phase diagram is then presented in the quasi - one - dimensional regime using a spatially anisotropic rectilinear lattice . this phase diagram is directly compared to the experimentally determined parameters of haldane gap materials . we also show the magnetization curves of the q1d model for the various zero - field ground state phases , and calculate the low - lying excitation spectrum of the haldane phase near the haldane gap minimum . finally , the haldane phase in q1d geometries is shown to belong to the new class of symmetry protected topological states . even with a simulation method such as sign - free quantum monte carlo that scales polynomially in system size , it is impossible to directly access results for macroscopic system sizes . to overcome this difficulty , we employ finite - size scaling ( fss ) methods to extrapolate our results to the ground state thermodynamic limit . since quantum systems map onto classical systems in @xmath32 dimensions , where @xmath33 is the dynamic critical exponent , similar procedures can be used to access the @xmath34 and @xmath35 limits . for ground states that break a continuous spin symmetry , the spin stiffness is a useful observable to determine the boundary of such a gapless phase ( as per the goldstone theorem ) with a gapped state of any order . this is because the spin stiffness must be finite in the gapless phase and zero in the gapped phase . the scaling function of the spin stiffness in the vicinity of a quantum critical point is known to be @xmath36 where @xmath37 is a hamiltonian parameter that can tune the ground state system to the quantum critical point at @xmath38 , @xmath39 is the critical exponent governing the growth of the ( spatial ) correlation length @xmath40 near @xmath38 , and @xmath33 relates @xmath40 to the correlation length in imaginary time by the relation @xmath41 . by taking the inverse temperature to scale as the system size @xmath14 ( with @xmath42 for the quantum phase transitions considered in this work ) , it is clear that @xmath43 is independent of system size at the critical point . this is demonstrated in the right panels of fig . [ scaling ] , where we plot @xmath44 for a @xmath45 dimensional quantum system and use @xmath46 to reach the ground state limit . the critical point is clearly visible at @xmath47 where the quantity @xmath44 is independent of system size . a similar scaling function is known for the square of the staggered magnetization along the @xmath33 axis @xmath48 that acts as an order parameter for the ising antiferromagnetic phase . it is @xmath49 where @xmath50 is the anomalous dimensionality . thus we can also see that @xmath51 will display a crossing behavior for varying @xmath14 at the phase boundaries of the ising afm phase . this is shown in the left panels of fig . [ scaling ] for the ising afm to haldane phase transition . , left panels ) and spin stiffness ( @xmath52 , right panels ) at the ising afm to haldane and haldane to @xmath53 afm phase transitions , respectively . interchain coupling is fixed at @xmath54 . the critical points @xmath55 and @xmath56 are determined by the crossing criterion ( upper panels ) assuming a dynamic critical exponent @xmath42 . good finite - size scaling collapse is achieved near the critical points assuming mean field exponents ( @xmath57 and @xmath58 ) for a continuous phase transition at the upper critical dimension @xmath59 ( lower panels ) . inverse temperature is scaled as @xmath46 to ensure convergence to the ground state limit . ] starting from a system of uncoupled chains in the haldane phase and slowly turning on the interchain coupling @xmath13 , the system will remain in the gapped haldane phase until a critical coupling @xmath60 is reached at which point the gap closes and long - range magnetic order develops . within the chain mean field approximation ( cmfa ) , the critical coupling @xmath60 depends only on the coordination number of chains @xmath61 . thus , the quantity @xmath62 might be expected to be universal for unfrustrated lattices . in fig . [ geometry ] we show results for chains arranged into square ( @xmath63 ) and honeycomb ( @xmath64 ) superlattices . while they give different results for the critical coupling ( @xmath65 and @xmath66 , respectively ) , the scaled values @xmath62 are almost in statistical agreement : @xmath67 vs. @xmath68 . these can be compared to the cmfa value @xmath69 that acts as a lower bound.@xcite ) at the haldane to nel phase transition for chain coordination * ( a ) * @xmath63 ( i.e. square superlattice of chains ) , and * ( b ) * @xmath64 ( i.e. honeycomb superlattice of chains ) . single - ion anisotropy is fixed at the isotropic point , @xmath70 . inverse temperature is scaled as @xmath71 to ensure convergence to the ground state limit . adapted from k. wierschem and p. sengupta , jps conf . * 3 * ( 2014 ) 012005 . ] the ground state phase diagram of @xmath25 is shown in fig . [ phase1 ] . for small @xmath72 and @xmath13 the system is in the haldane phase . for sufficiently strong interchain couplings , the haldane gap is quenched and three - dimensional long - range magnetic order sets in . this magnetic order is the nel antiferromagnetic state in the case of isotropic spins ( @xmath70 ) , while axial ( @xmath73 ) and planar ( @xmath74 ) anisotropy lead to ising afm and @xmath53 afm states , respectively . additionally , there is a quantum paramagnetic ( qpm ) phase for large @xmath75 . the haldane to @xmath53 afm phase boundary is determined by fss of the spin stiffness , while the haldane to ising afm phase boundary is determined by fss of the staggered magnetization . in the case of the haldane to nel phase transition at the isotropic point ( @xmath70 ) , fss of both the spin stiffness and staggered magnetization yield values in agreement up to the statistical uncertainty given . @xmath13 plane with phase boundaries as indicated by the dotted black lines . the borders of the haldane and qpm phases are obtained as polynomial fits to the present work and represent guides for the eye . data points for the present work are determined by qmc simulations at constant @xmath21 and @xmath13 ( solid red circles and blue squares , respectively ) . several haldane compounds are plotted as large crosshatched symbols using estimates for @xmath21 and @xmath13 from the indicated sources . adapted from k. wierschem and p. sengupta , phys . * 112 * , 247203 ( 2014 ) . ] one feature of interest in the phase diagram of fig . [ phase1 ] is the ability to change the effective hamiltonian parameters of a given material by the application of hydrostatic pressure . this has been demonstrated for nenp by zaliznyak _ et al._@xcite however , as can be seen in fig . [ phase1 ] , nenp under pressure is actually closer to an ideal heisenberg chain , i.e. it moves towards the origin and _ away _ from the boundaries of the haldane phase . thus , it is not a good candidate for the observation of pressure - induced quantum criticality , as has been found in the spin-1/2 dimer compound tlcucl@xmath76.@xcite by contrast , the isostructural compounds pbni@xmath1v@xmath1o@xmath3 and srni@xmath1v@xmath1o@xmath3 are found to already lie very near to a quantum critical point , making them prime candidates for the study of pressure - induced quantum criticality . in addition to pressure , chemical substitution represents another mechanism by which the effective hamiltonian parameters of magnetic compounds can be modified . this is already somewhat apparent in the isostructural compounds pbni@xmath1v@xmath1o@xmath3 and srni@xmath1v@xmath1o@xmath3 , where the differing influence of the pb@xmath77 and sr@xmath77 ions leads srni@xmath1v@xmath1o@xmath3 to be closer to the ising afm phase than pbni@xmath1v@xmath1o@xmath3 . indeed , srni@xmath1v@xmath1o@xmath3 was initially thought to magnetically order below @xmath78 based on experiments on powder samples,@xcite whereas recent results on polycrystalline@xcite and single crystals@xcite have established its low temperature magnetic behavior to be consistent with a non - magnetic haldane ground state . the prospect of chemical fine - tuning is even more exciting for molecule - based magnets , where the choice of bridging ligands has been shown to allow for the synthesis of a wide range of low dimensional structures . of particular interest is the recently synthesized compound [ ni(hf@xmath79)(3-clpy)@xmath80bf@xmath81 that is believed to lie near the 1d gaussian critical point.@xcite we note here that this critical point is well - described by the tomonaga - luttinger liquid theory with luttinger parameter @xmath82.@xcite it is important to take the crystal symmetry into consideration , as it determines which symmetry breaking and conserving phases can occur . for example , nenp is known to have a rhombohedral crystal electric field component @xmath83 that explicitly breaks the onsite @xmath84 spin symmetry of @xmath25 down to @xmath85 . additionally , application of a uniform longitudinal magnetic field is known to induce a staggered transverse field at the nickel sites due to a zigzag chain structure.@xcite also , note that the related compound nenc falls into the qpm phase since @xmath86,@xcite but because @xmath87 we do not include it here . we also have calculated the phase boundary between the @xmath53 afm and qpm phases from the q1d limit all the way to the spatially isotropic 3d lattice . this is shown in fig . [ phase2 ] where we also plot the location of some quantum magnets in the phase diagram . note that the phase boundary between the @xmath53 afm and qpm phases is a line of critical points belonging to the @xmath88 universality class in 3 + 1 dimensions . within the qpm phase , application of a magnetic field leads to a field induced quantum phase transition into the _ canted _ @xmath53 afm phase . this field induced transition has been studied by zhang _ et al._@xcite and shown to be a mean field transition because the effective dimension is @xmath89 ( the dynamical exponent is z=2 because these field induced transitions belong to the bec universality class@xcite ) . regime of the @xmath21@xmath13 plane . data points for the phase boundary are extracted by finite - size scaling analysis of qmc simulations performed at constant @xmath13 . dotted line is obtained as a polynomial fit to the qmc data and delineates the @xmath53 afm and qpm phase boundary . ] the theoretical effect of uniaxial hydrostatic pressure applied to dtn has been examined , with encouraging signs that dtn may be tuned to a quantum critical point ( qcp).@xcite this pressure - induced qcp would be of the @xmath88 universality class in 3 + 1 dimensions , as opposed to the field - induced qcp in dtn that has been shown to belong to the bec universality class in 3 + 2 dimensions.@xcite there also are several compounds much deeper in the qpm phase , such as sr@xmath76nipto@xmath90.@xcite the parameters for the compounds shown in figs . [ phase1 ] and [ phase2 ] are listed in table [ table1 ] . for completeness , we also mention the haldane chain compounds ndmaz,@xcite nenb,@xcite ninaz,@xcite and nino,@xcite which could also fit into the phase diagram in fig . [ phase1 ] near the related compounds ndmap and nenp but are not shown here in the interest of clarity . additionally , the hexagonal compounds csnicl@xmath76 and tmnin are outside the scope of the present model as they can not be well approximated by a cubic lattice . tmnin is a haldane gap material while csnicl@xmath76 exhibits haldane behavior above its magnetic ordering temperature @xmath91.@xcite finally , the low temperature phase of tl@xmath79ru@xmath79o@xmath92 has been interpreted as a haldane chain system that spontaneously arises from a fundamentally three - dimensional crystal structure due to the orbital ordering.@xcite c c r r r r + compound & bravais lattice & @xmath93 & @xmath94 & @xmath12 & ref . + + agvp@xmath1s@xmath95 & monoclinic & 0.006 & @xmath96 & 780k & [ ] + ndmap & orthorhombic & 0.25 & @xmath97 & 33.1k & [ ] + nenp & orthorhombic & 0.16 & @xmath98 & 46k & [ ] + nenp@2.5gpa & orthorhombic & 0.09 & @xmath99 & 48k & [ ] + [ ni(hf@xmath79)(3-clpy)@xmath80bf@xmath81 & monoclinic & 0.88 & & 4.86k & [ ] + pbni@xmath1v@xmath1o@xmath3 & tetragonal & -0.05 & @xmath100 & 104k & [ ] + srni@xmath1v@xmath1o@xmath3 & tetragonal & -0.04 & @xmath101 & 100k & [ ] + y@xmath1banio@xmath4 & tetragonal & -0.039 & @xmath102 & 240k & [ ] + + dtn & tetragonal & 4.05 & @xmath103 & 2.2k & [ ] + [ ni(hf@xmath79)(pyz)@xmath79]sbf@xmath90 & tetragonal & 1.22 & @xmath105 & 9k & [ ] + sr@xmath76nipto@xmath90 & rhombohedral & 8.8 & & 11k & [ ] + [ table1 ] the ground state magnetization curve can be a good way to characterize quantum magnets at low temperatures . the direct application of an external magnetic field is an efficient probe of the underlying magnetic phases of a magnetic compound . for longitudinal magnetic fields , the hamiltonian in eq.([model ] ) comes with a conservation law such that the ground state at any field must occur within a single magnetization sector defined by the quantum number @xmath106 . this can lead to several interesting features . first , note that the fully polarized state @xmath107 is an exact eigenstate of the hamiltonian , since the off - diagonal terms come in pairs @xmath108 such that @xmath107 is always annihilated , leaving only the action of the diagonal terms to set its energy . additionally , the saturation field @xmath109 can often be obtained by setting the energy difference between @xmath107 and the lowest energy state in the @xmath110 sector to zero . for the model considered here on an anisotropic cubic lattice , it can be shown that @xmath111 as long as @xmath21 is not strongly easy - axis . it has previously been demonstrated that for @xmath73 in the @xmath112 limit it is the @xmath113 sector that instead determines the saturation field , while for intermediate @xmath21 a discontinuity in @xmath114 develops at a first order transition between the fully polarized and canted @xmath53 afm phases.@xcite gapped and gapless phases respond differently to an applied magnetic field . for the longitudinal magnetic field @xmath115 the ising afm , haldane , and qpm phases are all gapped and therefore stay in the @xmath116 magnetization sector until the field reaches some critical value @xmath117 . in the case of the ising afm phase , there is a first order phase transition the so - called _ spin flop _ transition that occurs , whereby one sublattice of spins that was previously aligned against the field `` flops '' over and becomes partially aligned with the field instead , while at the same time the other sublattice of spins reduces its relative polarization . these spins retain some antiferromagnetism by developing a staggered magnetization in the @xmath53 plane the so - called _ canted _ @xmath53 afm phase . in the case of the haldane and qpm phases , there is instead a continuous phase transition at @xmath117 , with field - induced critical points belonging to the bec universality class . in fig . [ magnetization ] we plot the magnetization curves for @xmath118 for zero - field ground states in the haldane , @xmath53 afm and qpm phases . it is easy to see that the gapless @xmath53 afm phase immediately responds to the applied field , while the magnetization curves for the haldane and qpm phases show signs of a small gap , with @xmath119 . all three curves appear similar in their approach to the fully polarized state , with the increase in slope due to a combined effect of low spin number ( @xmath0 , where we remind the reader that for the most quantum spin @xmath5 the slope diverges at @xmath109 as is known from the bethe ansatz ) and reduced dimensionality . these features are most easily seen in the corresponding differential susceptibility , @xmath120 . another feature to notice is the pronounced hump in the differential susceptibility around @xmath121 when @xmath70 , followed by a local minimum . this feature has previously been pointed out by kashurnikov _ _ in the magnetization curve of the haldane phase of a 1d chain,@xcite and is characteristic of the haldane phase . , the interchain coupling is set to @xmath118 with inverse temperature @xmath46 sufficient to reach the ground state regime for @xmath122 . ] one of the most characteristic features of the haldane phase is the haldane gap . this arises directly in the haldane conjecture due to the difference between integer and half - odd - integer spins . it was also one of the first features to be confirmed , with early exact diagonalization results showing a finite gap for the spin-1 heisenberg model.@xcite the accurate determination of this gap was also one of the first applications of the powerful dmrg method.@xcite the low lying states can also be estimated with an upper bound estimator @xmath123 that can be derived from well known sum rules.@xcite when the single - mode approximation holds reasonably well , this estimator can be quite accurate . the spectrum of the haldane phase in 1d is known from dmrg studies,@xcite and in particular at @xmath124 the spectrum is sharply peaked at a single - magnon mode ( corresponding to the haldane gap ) , with a further gap to the higher multi - magnon modes . thus , the estimator should be very accurate for @xmath124 . we use this to calculate the low lying spectrum of coupled chains with @xmath124 along the chain direction . our results are shown in fig . [ dispersion ] , where the tendency for the gap to close as the interchain coupling is increased is quite clear , as is the trend for the modes about @xmath125 to become linear , indicative of the fact that the system is approaching the gapless nel afm phase . . dashed line represents the haldane gap of uncoupled chains ( @xmath126 ) . data shown for length @xmath122 and inverse temperature @xmath127 . adapted from k. wierschem and p. sengupta , phys . lett . * 112 * , 247203 ( 2014 ) . ] a nonlocal string order captures the hidden symmetry breaking in the haldane phase of the spin-1 hafm in 1d . for two sites @xmath11 and @xmath128 in a chain , the string correlation function is given by @xmath129}s_j^z\right>. \label{eq : string}\ ] ] in the limit that the distance between sites @xmath11 and @xmath128 goes to infinity , @xmath130 becomes exactly @xmath131 in the aklt state,@xcite while in the haldane state it has been estimated to be @xmath132.@xcite although it is not possible to generalize the string order parameter to higher dimensional correlations , it is possible to calculate the string correlation function in systems of coupled spin chains for spins residing within the same chain . in fig . [ strings ] we plot @xmath133 , which as @xmath35 should scale exponentially to zero in the absence of string order , or to a nonzero value when string order is present . thus , @xmath133 acts as a string order parameter . for comparison , we also show the scaled spin stiffness @xmath44 , which shows that below @xmath134 the system is in the gapped haldane phase , while above this value @xmath135 is finite and we are in the gapless @xmath53 afm phase . it is interesting to note that the string order parameter shows a qualitative change in behavior at @xmath136 . below @xmath136 , @xmath133 shows very weak decay with system size , and may even saturate to a finite value as @xmath35 . by contrast , in the @xmath53 afm phase , @xmath133 exhibits rapid decay with system size , and will scale to zero as @xmath35 . while this would seem to indicate that there is long - range string order in the haldane phase in q1d geometries , a note of caution must be sounded . this is because the string correlation function has been argued to always scale exponentially to zero with distance for any nonzero value of the interchain coupling.@xcite thus , it remains a possibility that the correlation length of string order grows rapidly at @xmath136 , yet remains finite in the q1d haldane phase . and * ( b ) * the scaled spin stiffness @xmath137 across the haldane to @xmath53 afm phase boundary with aspect ratio @xmath138 , interchain coupling @xmath118 , and inverse temperature @xmath46 . adapted from k. wierschem and p. sengupta , phys . 112 * , 247203 ( 2014 ) . ] due to the shortcomings of the string order parameter in more than one spatial dimension , it would be nice to find alternative ways to characterize the haldane phase in q1d geometries . since the haldane state in 1d is a nontrivial spt state , we might expect it to remain such in q1d systems . still , direct evidence for this has been lacking . this may have to do with the difficulty of characterizing spt states in general . one common method is to look at the low lying entanglement spectrum,@xcite where an overall degeneracy signals a nontrivial spt state.@xcite while the entanglement spectrum can be easily accessed using powerful dmrg techniques in 1d , systems in higher dimension pose a tougher challenge . additionally , there is the possibility that the `` cut '' used to form a bipartition of the system during the construction of the reduced density matrix may break an off - site symmetry , and so misdiagnose spt phases protected by such symmetries . fortunately , in 1d and 2d there exists an alternative way to distinguish between trivial and nontrivial spt states . this is the so - called strange correlator of you _ et al._,@xcite which uses a mapping between the space - time correlations at the spatial boundary of a nontrivial spt state with a trivial spt state ( or vacuum ) and ( through a laplace transform ) the spatial correlations at the temporal boundary between the time - evolved quantum field theories of the respective trivial and nontrivial spt states . since the space - time correlations at the aforementioned spatial boundary are known to possess long - range ( or , possibly quasi - long - range ) order in one and two dimensions,@xcite a long - range or quasi - long - range strange correlator is then a direct measure of spt order in 1d and 2d , while short - range behavior is a sign of a trivial product state . in 3d , the possibility of a topologically ordered edge state allows for the construction of nontrivial spt phases with short - range strange correlators ; however , long - range or quasi - long - range behavior remains indicative of spt order . the strange correlator is calculated as a `` mixed '' correlation function between the nontrivial spt state @xmath139 and the trivial spt state @xmath140 , @xmath141 in practice , @xmath140 is taken as a symmetric product state , which is by definition a trivial spt state . the state @xmath139 can be our putative spt state in question , and thus a ( quasi- ) long - range strange correlator becomes a direct probe of the nontrivial spt character of the state @xmath139 . it should also be mentioned that this only applies to states that are gapped and symmetric , i.e. they are either trivial or nontrivial spt states to begin with . for the case considered here , there is a host of evidence that the q1d haldane phase is just such a state . the strange correlator has recently been implemented using qmc methods.@xcite in fig . [ strange ] we show the finite - size scaling of a finite - size order parameter @xmath142 constructed from the strange correlator . for the q1d haldane phase , we see that @xmath143 remains finite , signaling a nontrivial spt state . this is true for both 2d and 3d lattices . by contrast , in the qpm phase we see evidence for exponential decay to zero , as expected for a nontrivial spt state . thus , we have provided direct evidence that the q1d haldane phase is indeed a nontrivial spt state . , black circles ) and the quantum paramagnetic phase ( @xmath144 , red squares ) . results in the ground state limit are obtained by scaling the operator string length as @xmath145 . solid lines and filled symbols are results in 3d with @xmath118 , while dashed lines and symbols are in 2d with @xmath146 . adapted from k. wierschem and p. sengupta , phys . lett . * 112 * , 247203 ( 2014 ) . ] the identification of the quasi - one - dimensional haldane phase as a nontrivial spt state in its own right naturally leads us to consider what sort of topological edge modes might be present in this system . although these edge states are yet to be directly determined using numerical methods , there exists a solid theoretical framework from which we can speculate on their nature . first , consider that the haldane phase in 1d supports degenerate spin-1/2 edge states . next , if we couple haldane chains into an @xmath147-leg ladder , the edge states will form an overall singlet for @xmath147 even , but retain a degeneracy for @xmath147 odd due to kramers theorem . this explains why even leg ladders form trivial spt states , while odd leg ladders form non - trivial spt states . what happens to the edge states as @xmath148 ? for infinitesimal interchain couplings , they must be none other than the spin-1/2 heisenberg antiferromagnetic chain , with a gapless linear dispersion near the antiferromagnetic ordering wave vector . similarly , the edge states of a three - dimensional q1d haldane phase correspond to the ground state of the spin-1/2 heisenberg square lattice antiferromagnet , which is likewise gapless . it is natural to assume the edge state spectrum will not significantly alter as @xmath13 is increased while remaining in the quasi - one - dimensional haldane phase . further , as long as the protecting symmetry of space inversion about chain bonds remains intact , these edge states are robust to any symmetry conserving perturbations . the question then becomes : how can we measure these directly , both in simulation as well as in experiment ? in this context , we mention that a method has recently been proposed for the experimental probe of fractional edge states in @xmath0 heisenberg chains.@xcite string order has been predicted to decay exponentially , with correlation length @xmath149 , for any finite coupling @xmath13 between chains.@xcite if this is the case , we might expect essentially 1d behavior when the length of individual chains @xmath150 . since spin-1/2 edge states have been observed in doped haldane chain compounds using non - magnetic dopant ions ( which break the spin chains into finite segments ) , we may surmise that the finite segments @xmath151 in such systems obey the relation @xmath152 , where we use the expectation value to reflect the distribution of @xmath151 values due to random placing of the dopant ions ( for a doping concentration @xmath153 , we expect @xmath154 ) . in a system such as y@xmath1banio@xmath4 where the interchain coupling is very small ( @xmath155),@xcite the resulting correlation length is so large ( @xmath156 ) that almost any finite doping @xmath153 is sufficient to guarantee that @xmath152 . indeed , spin-1/2 edge states are observed in the system y@xmath1bani@xmath157mg@xmath158o@xmath4 with @xmath159 and @xmath160.@xcite however , it is interesting to speculate what might occur in haldane materials with relatively strong @xmath13 , such as pbni@xmath1v@xmath1o@xmath3 and srni@xmath1v@xmath1o@xmath3 : might there exists a critical doping @xmath161 , below which @xmath162 ? if so , are spin-1/2 edge states still detectable ? one exciting possible use of the symmetry protected edge states in haldane chains is for measurement - based quantum computation ( mbqc).@xcite in this quantum computing scheme , an entangled state is initially prepared , after which single qubit measurements are performed . these measurements destroy the resource state , so this method is also referred to as _ one - way _ quantum computing . the aklt state , and haldane ground states in general , have been shown to be a possible resource state for mbqc.@xcite however , in order to achieve universal quantum computation , a two dimensional resource state is required . in this fashion , the spin-3/2 aklt state on the honeycomb lattice has been proposed as a universal resource for mbqc.@xcite it is interesting to speculate that any vbs state with protected edge modes in two dimensions might prove to be a universal resource state for mbqc , in which case the haldane phase of spin-1 heisenberg chains weakly coupled into a two dimensional array ( as in fig . [ lattice ] ) might be a way to achieve such a state . this would be very useful since aklt states of higher spins require additional hamiltonian terms beyond the bilinear and biquadratic ones needed for spin-1 . spin-1 chains also have the distinction that the aklt state is adiabatically connected to the haldane phase at the heisenberg point ( i.e. as the biquadratic term is tuned to zero , the vbs state remains intact ) . in higher dimensions , the heisenberg ground state tends towards nel order , which is the case on honeycomb and square lattices . the remarkable properties of the haldane phase in spin-1 heisenberg antiferromagnets have spurred continued interest in haldane gap materials and their low - energy effective spin models . yet surprisingly , no accurate determination of the phase diagram for these models in experimentally relevant geometries has been completed until recently . here , we have presented some of our recent work on these systems , including an accurate phase diagram for a square superlattice of weakly interacting heisenberg antiferromagnetic spin chains . further , we have shown the magnetization and differential susceptibility curves for longitudinal magnetic fields , and calculated the low - lying dispersion along the @xmath163 plane in reciprocal space . we have also presented direct evidence for the symmetry protected topological character of the haldane phase in quasi - one - dimensional geometries . this includes extended string correlations along the weakly coupled chains , as well as a direct characterization through the long - range behavior of the strange correlator . these calculations lead to a picture of the quasi - one - dimensional haldane phase as a symmetry protected topological phase with an effective gapless spin-1/2 heisenberg antiferromagnetic state at its surface . in this brief review , we have endeavored to introduce the haldane phase within the current classification of short - range entangled symmetric states of matter namely , the symmetry protected topological states . this exciting development is stimulating renewed interest in this field . at the same time , recent developments in the growth and chemical synthesis of molecule - based magnets allow for greater control over the microscopic properties of magnetic materials . the convergence of these two lines of inquiry will be a promising direction for future research . in particular , the experimental identification of edge states at the surface of a quantum antiferromagnet would potentially realize the first known example of a bosonic spt state . such a state could have potential uses in measurement - 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we review the basic properties of the haldane phase in spin-1 heisenberg antiferromagnetic chains , including its persistence in quasi - one - dimensional geometries . using large - scale numerical simulations , we map out the phase diagram for a realistic model applicable to experimental haldane compounds . we also investigate the effect of different chain coupling geometries and confirm a general mean field universality of the critical coupling times the coordination number of the lattice . inspired by the recent development of characterization of symmetry protected topological states , of which the haldane phase of spin-1 heisenberg antiferromagnetic chain is a preeminent example , we provide direct evidence that the quasi - one - dimensional haldane phase is indeed a non - trivial symmetry protected topological state .

You are an expert at summarizing long articles. Proceed to summarize the following text: * b72 hereafter ) show that one need an extra @xmath0 relative velocity boost after the plunge in order for the inbound debris to reach the ergosphere negative - energy orbits , to ultimately extract energy from the negative - energy orbits to be carried away by the outbound debris . note that , while for a kerr black hole the apparent " radii of the event horizon , the photon orbit and and isco ( @xmath1 ) on a prograde orbit are the same in the boyer - lindquist coordinates , they are separate in proper coordinates and their energies are distinctly different . for example , for a kerr black hole , the orbital velocity at isco is @xmath2 ( not @xmath3 ) and the minimum energy of a plunge orbit that results from decay of a bound stable orbit is @xmath4 ( where @xmath5 is the mass of the orbiting particle ) . thus , in order to get to the negative - energy ergosphere orbits , the orbiting particle at isco need to cross over this velocity gap of @xmath6 . this condition was deemed to be astrophysically unlikely ( b72 ) . we suggest here that , when two black holes merge , if one or both have orbital material at close to their respective isco , then some of the orbital material may be able to jump " over this velocity gap of @xmath6 to land in the ergosphere negative - energy orbits , while portion of the orbiting material that escapes may be able to extract energy from the black hole via the @xcite process at the expense of the rotation of the black hole @xcite . our subsequent argument , in a large part , hinges on the assumption that at least some smbh are kerr black holes , because only their isco orbits are energetic enough to possess orbiting matter with a velocity dispersion equal to @xmath3 . this is supported by at least two observational lines of evidence : the observational inference of high radiative efficiency of luminous quasars of @xmath7 ( e.g. , * ? ? ? * ) and x - ray observations of iron k line profiles ( e.g. , * ? ? ? theoretically , kerr ( or near kerr ) smbh are fairly easily reached by gas accretion ( e.g. , * ? ? ? the existence of accretion gas around smbh is beyond any reasonable doubt , given the fact that we see quasars shine and there is no reasonable alternative to smbh accretion . we shall discuss briefly the possible mix of the matter that may be orbiting smbh in the inner region ; in other words , if things other than gas , such as stars or dense stellar debris , also exist there that are sufficiently long - lived . we shall only consider solar - type stars subsequently . compact objects such as neutron stars and white dwarfs are always swallowed whole by the smbh of mass of interest here and we assume that they do not produce astrophysically tangible signals ( even if they were launched from the center with very high velocities ) . the tidal disruption radius for a solar - type star of mass @xmath8 and solar radius @xmath9 is @xmath10 where @xmath11 and @xmath12 is the schwarzschild radius . equation ( [ eq : rt ] ) merely states the well known fact that @xmath13 smbh swallow solar - type stars whole , but smaller smbh tidally disrupt solar - type stars and are predicted to produce a distinct class of optical - uv flashes that have now been observationally confirmed ( e.g. , * ? ? ? it seems likely that , for smbh with @xmath14 sufficiently smaller than one , the stellar debris from tidal disruption that becomes bound to the smbh is gasified through a variety of processes ( distortion , compression , precession , possibly nuclear flash and ultimately shocks ) and accreted by the smbh in a relatively short period of time ( e.g. , * ? ? ? thus , it may be that there is not a significant amount of stars or stellar debris orbiting @xmath15 smbh at @xmath16 . we will now turn our attention to smbh with @xmath17 , in light of the strong observational evidence of their existence at the center of every giant elliptical galaxy ( e.g. , * ? ? ? for smbh of @xmath13 three primary routes may be able to capture a star without having to severely damaging it . a star on a low angular momentum parabolic orbit may lose its orbital energy through repeated pericenter passages to be eventually captured . in order to be captured to an orbit of radius @xmath18 , it needs to lose an amount of orbital energy equal to @xmath19 . under the assumption that the star is not destroyed ( e.g. , not puffed up to become a giant ) , the maximum rate of energy loss may not exceed its luminosity @xmath20 , which translates to a tidal capture timescale @xmath21 being @xmath22 where @xmath23 is the lifetime of the star and @xmath24 is assumed . this seems too long to be interesting , given that the star also has to have a pericenter distance not too much larger than @xmath25 to experience significant tidal effect for @xmath13 smbh ( see equation [ eq : rt ] ) and our assumed dissipation rate is likely on the generous side already . of course , if the star gets overheated thus expands , the envelope region that is puffed up would be tidally stripped off to become gas " . interaction between the smbh with a stellar binary may allow one member of the binary to be captured to a bound orbit around the smbh . the tidal capture radius through this three - body interaction is @xmath26 ( e.g. , * ? ? ? * ) , which is larger than the tidal radius of a single star ( equation [ eq : rt ] ) by a factor of @xmath27 with @xmath28 being the semi - major axis . this factor , possibly substantially greater than unity , allows @xmath13 smbh to be able to capture stars , although the exact rate depends on too many factors to be certain . during the merger of two massive galaxies each hosting a smbh , it is possible that a significant number of stars ( some fraction of them are binaries ) may be driven into low energy orbits to reach the inner regions of each smbh , given the significant amount of torque exerted by one galaxy on the other . repeated interaction between stars and an accretion disc around the smbh may also bring the star to a bound orbit . in fact , @xcite show that for @xmath13 smbh , star - disc drag may become the dominant capture mechanism . given these considerations , it seems that smbh of @xmath13 with an accretion disc ( i.e. , luminous quasars ) could possibly have both gas and stars at @xmath29 , while @xmath15 smbh may carry a disc that is mostly composed of gas . what might happen to stars embedded in an accretion disc is a subject beyond this study . we shall only note two points . first , for a star of mass @xmath5 on a circular orbit at radius @xmath18 around a smbh of mass @xmath30 , the orbital decay time scale due to its own gravitational radiation is @xmath31 @xcite , where @xmath32 is the gravitational constant , @xmath3 is speed of light and @xmath33 . thus , if the supply rate of solar - type stars to regions of @xmath34 , from either decay of outer orbits and/or directly captured by some processes , is larger than @xmath35yr around an @xmath13 smbh , one should expect to see some accumulation of solar - type stars near the innermost stable circular orbit @xmath36 . other drag process experienced by stars or dense debris , such as by the accretion disc , is likely to be less strong than the gravitational radiation drag at @xmath16 . for our purpose , it suffices that it is astrophysically plausible that smbhs of @xmath13 prior to their merger possibly carry stars or stellar debris as well as gas in their inner orbits . second , for @xmath37 , the orbiting bound stars may be significantly deformed or disrupted approaching @xmath1 , thus producing stellar debris orbiting at @xmath16 . if that happens , the velocity dispersion of the debris will be of order @xmath38 . therefore , the debris would be confined to a relatively narrow range in radius following the disruption . consider the inspiral of the two kerr smbh , @xmath39 and @xmath40 ( @xmath41 ) , via gravitational radiation , with only @xmath40 ( for simplicity ) carrying a disc of gas and/or stellar debris orbiting at @xmath42 . we assume that all angular momentum vectors are parallel and all orbits are on the same plane . when @xmath40 arrives at @xmath1 of @xmath39 , one finds that at any instant some material is moving at the same direction as the orbital velocity of @xmath40 around @xmath39 with a relative velocity of @xmath2 , while some other material is moving at the opposite direction to the orbital velocity of @xmath40 around @xmath39 with a relative velocity of @xmath2 . still , there is some other material that is in - between . assuming at least some debris are still on some elliptical - like orbits ( may or may not be closed ) , given that the relative velocity of the debris around @xmath40 spans the whole range from @xmath43 to @xmath2 relative to the orbital velocity of @xmath40 about @xmath39 , it seems plausible that at least some of this material is able to fill in the required velocity gap of @xmath6 relative to the orbital velocity at @xmath36 of @xmath40 to reach the ergosphere negative - energy orbits of @xmath39 . assuming that the part of the orbiting material that reaches the ergosphere negative - energy orbits is able to communicate with the remainder of the material that eventually escapes , the originally proposed @xcite process to extract energy from the smbh may become possible . however , caution should be taken . it may be that , when the two smbh get that close , the assumptions on which our simple argument is based break down . the primary assumption is that the orbiting material at @xmath29 around @xmath40 , at least some of it , manages not to plunge into @xmath40 during the process in the presence of a nearby @xmath39 , but instead plunges to a minimum - energy orbit of @xmath39 . the other assumption is that the orbiting material around @xmath40 has not lost the velocity dispersion of @xmath3 that it had at isco , when plunging into @xmath39 . the second assumption at least seems feasible , because it is rather unlikely that during the short time of the plunge one is able to remove " or dissipate the velocity dispersion of order the speed of light for the orbiting material around @xmath40 . on the contrary , it would seem plausible that , during the plunge , en route to the event horizon , the velocity dispersion of the orbiting material may be further increased . detailed black hole merger simulations ( e.g. , * ? ? ? * ) , carrying some extra amount of orbiting material , will be necessary to provide some insight . if what is outlined above really happens , what the escaping material that is leaving with a total energy higher than its initial total energy ( but with reduced rest mass ) will look like is just a guess at this time , in the absence of detailed simulations . but it is conceivable that the debris may form a relativistic jet or , strictly speaking , a mixed flow of gas and stellar debris moving at ( nearly ) the speed of light , in a fashion perhaps not unlike what was suggested earlier @xcite . if the escaping amount of material is substantial and especially a substantial fraction of that is stars or dense stellar debris , the flow may be quite bulletstic " hence would remain cruising in a straight line , like a jet , for a while . we have argued that it is astrophysically plausible that kerr smbh carry a significant amount of material ( gas and stars or stellar debris ) in the inner region close to the innermost stable circular orbit , after their host galaxies merge . the orbiting matter at isco has a velocity dispersion equal to the speed of light about the orbital velocity of their host smbh . as a result , we suggest that , when the two smbh merge , some of the debris may be able to jump over the velocity gap between the isco orbit and negative - energy ergosphere orbits . if that indeed happens , we would have found an astrophysically plausible way to tap into the vast rotational energy reservoir of smbh via the penrose process . , d. , ajhar , e. a. , bender , r. , bower , g. , dressler , a. , faber , s. m. , filippenko , a. v. , gebhardt , k. , green , r. , ho , l. c. , kormendy , j. , lauer , t. r. , magorrian , j. , & tremaine , s. 1998 , , 395 , a14 +

if a supermassive black hole has some material orbiting around it at close to its innermost stable circular orbit ( isco ) , then , when it plunges into a second supermassive black hole , the orbiting material has a velocity dispersion of order of speed of light about the orbital velocity of its host black hole . it becomes plausible that some of the orbiting material will be catapulted " to the negative - energy ergosphere orbits of the second black hole at the plunge . this may provide an astrophysically plausible way to extract energy from the black hole , originally suggested by penrose . = 1

You are an expert at summarizing long articles. Proceed to summarize the following text: in this data driven area , the amount and complexity of the available data grows at an almost incredible speed . therefore , there is a high need to develop novel tools to cope with such complex data structures . whereas the first statistical techniques were designed only to manage either quantitative or qualitative data , we can now find statistical procedures to handle functional data ( see for instance arribas - gil and mller @xcite ; febrero - bande and gonzlez - manteiga @xcite ; jacques and preda @xcite ) , fuzzy - valued data ( see , for instance , ferraro and giordani @xcite ; gonzlez - rodrguez _ et al . _ @xcite ; coppi _ et al . _ @xcite ) ; incomplete / missing data ( see , for instance , bianco _ et al . _ @xcite ; ferraty _ et al . _ @xcite ; lin @xcite ; zhao _ et al . _ @xcite ) , and several other types of data . interval - valued data are a type of complex data that requires specific statistical techniques to analyze them . interval - valued data may arise for different reasons . in some cases the underlying random variable is intrinsically interval - valued , e.g. the daily fluctuation of the systolic blood pressure . in other cases , there is an underlying real - valued but to preserve a level of confidentiality respondents are only asked to indicate the interval containing their value , e.g. their salary . it may also happen that the real - valued measurement is only partially known due to certain limitations , such as is the case for interval censored data . finally , aggregation of a typically large dataset may lead to e.g. interval - valued symbolic data which include interval variation and structure . the @xmath0-median considered here does not make any assumption about the source of the interval - valued data . in particular , it does not matter whether the random experiment that generated the data involves an underlying observable real - valued random variable or not . an important remark is that the space of intervals is only a semilinear space , but not a linear space due to the lack of the opposite of an interval . therefore , although intervals can be identified with two - dimensional vectors ( with first component the mid - point / centre and second component the nonnegative spread / radius ) , it is not advisable to treat them as regular bivariate data . indeed , common assumptions for multivariate techniques do not hold in this case . statistical procedures for random interval - valued data have already been proposed in the literature for different purposes , such as regression analysis ( e.g gil _ et al . _ @xcite ; gonzlez - rodrguez _ et al . _ @xcite ; blanco - fernndez _ et al . _ @xcite ; lima neto _ et al . _ @xcite ; fagundes _ et al . _ @xcite ; giordani @xcite ) ; testing hypotheses ( e.g. montenegro _ et al . _ @xcite ; nakama _ et al . _ @xcite ; gonzlez - rodrguez _ et al . _ @xcite ) , clustering ( e.g. de carvalho _ et al . _ @xcite ; durso _ et al . _ @xcite ; giusti and grassini @xcite ; da costa _ et al . _ @xcite , etc . ) , principal component analysis ( e.g. billard and diday @xcite ; durso and giordani @xcite ; makosso - kallyth and diday @xcite , etc . ) , modelling distributions ( see brito and duarte silva @xcite ; sun and ralescu @xcite ) . one of the most commonly used location measures is the aumann - type mean ( see aumann @xcite ) . it is indeed supported by numerous valuable properties , including laws of large numbers , and is also coherent with the interval arithmetic . the main disadvantage is that it is strongly influenced by outliers and data changes , which makes this measure not always suitable as a summary measure of the distribution of a random interval . this drawback is in fact inherited from the standard real / vectorial - valued case . in the real case , the most popular alternative is the median . in the real case , the most popular robust alternative to the mean is the median . for multivariate data the spatial median ( also called the @xmath1-median , as introduced by weber @xcite ) is a popular robust alternative to estimate the center of the multivariate data . the spatial median is defined as the point in multivariate space with minimal average euclidean distance to the observations . for more details and extensions , see for instance gower ( @xcite ) , brown ( @xcite ) , milasevic and ducharme ( @xcite ) , ( cadre @xcite ) , roelant and van aelst ( @xcite ) , debruyne _ et al . _ ( @xcite ) , fritz _ et al . _ ( @xcite ) , zuo ( @xcite ) . sinova and van aelst ( 2014 ) adapted the spatial median to interval - valued data by using a suitable @xmath2 metric on this space ( see also sinova et al . they used the versatile generalized metric introduced by bertoluzza _ ( @xcite , see also gil _ et al . _ @xcite ; trutschnig _ et al . _ @xcite ) the resulting @xmath0-median estimator has been shown to be robust with high breakdown point and good finite - sample properties . in this paper we show another important property of the estimator , which is its strong consistency . the rest of this paper is organized as follows : in section 2 the basic concepts related to the interval - valued space , interval arithmetic and metric for intervals will be introduced , as well as the usual location measure . in section 3 , the @xmath0-median for random intervals and its main properties are recalled . the strong consistency of the @xmath0-median is proven in section 4 . finally , some concluding remarks are presented in section 5 . let @xmath3 denote the class of nonempty compact intervals . any interval @xmath4 in the space @xmath5 can be characterized in terms of either its infimum and supremum , @xmath6 $ ] , or its mid - point and spread or radius , @xmath7 $ ] , where @xmath8 the usual interval arithmetic provides the addition , i.e. @xmath9 $ ] with @xmath10 and the product by a scalar , i.e. @xmath11 $ ] with @xmath12 and @xmath13 . with these two operations the space @xmath3 is semilinear , but not linear due to the lack of a difference of intervals . therefore , statistical techniques for interval - valued data are based on distances . to measure the distance between two interval - valued observations , we consider the _ @xmath0 metric _ introduced by bertoluzza _ et al . _ ( @xcite ) , which can be defined as ( see gil _ et al . _ @xcite ) : @xmath14 where @xmath15 and @xmath16 . following the general random set approach , a _ random interval _ can usually defined as a borel measurable mapping @xmath17 , where @xmath18 is a probability space with respect to @xmath19 and on @xmath5 the borel @xmath20-field generated by the topology induced by the @xmath0 metric . as a consequence from the borel measurability , crucial concepts in probabilistic and inferential developments , such as the ( induced ) distribution of a random interval or the stochastic independence of random intervals , are well - defined . one of the most used location measures is the _ aumann - type mean value_. it is defined , if it exists , as the interval @xmath21=[e(\inf x),e(\sup x)]$ ] or @xmath21=[e({\hbox { \rm mid}}\ , x)-e({\hbox { \rm spr}}\ , x),e({\hbox { \rm mid}}\ , x)+e({\hbox { \rm spr}}\ , x)]$ ] ( both expressions are equivalent ) . moreover , it is the frchet expectation with respect to the @xmath0 metric , i.e. , it is the unique interval that minimizes , over @xmath12 , the expression @xmath22 $ ] . as a robust alternative to the aumann - type mean , sinova and van aelst ( 2014 ) proposed the @xmath0-median as measure of location , which is defined as follows . @xmath23-median(s ) _ of a random interval @xmath24 is(are ) the interval(s ) @xmath25\in \mathcal k_c(\mathbb r)$ ] such that @xmath26 ) ) = \min_{k\in\mathcal k_c(\mathbb r)}e(d_{\theta}(x , k)),\ ] ] whenever the involved expectations exist . analogously , the sample @xmath0-median statistic is defined as follows . let @xmath27 be a simple random sample from a random interval @xmath24 with realizations @xmath28 . the _ sample @xmath23-median _ ( or medians ) @xmath29}_n$ ] is ( are ) the random interval that takes , for @xmath30 , the interval value(s ) @xmath31}$ ] that is ( are ) the solution(s ) of the following optimization problem : @xmath32 where @xmath4 , @xmath33 and @xmath34 depend on @xmath30 ( which has been omitted from the notation for the sake of simplicity ) and the fixed value @xmath35 . sinova and van aelst ( 2014 ) showed the existence of the sample @xmath0-median estimator and its uniqueness whenever not all the two - dimensional sample points @xmath36 are collinear . moreover , the robustness was shown by its finite sample breakdown point ( donoho and huber @xcite ) which is given by @xmath37}_n,\mathbf{x}_n , d_\theta\big)=\frac{1}{n}\cdot\lfloor\frac{n+1}{2}\rfloor,\vspace{-0.25cm}\ ] ] where @xmath38 denotes the floor function . in this section we investigate the strong consistency of the sample @xmath0-median under general conditions . [ consistency ] let @xmath39 be a random interval associated with a probability space @xmath40 such that the @xmath0-median exists and is unique . then , the sample @xmath0-median is a strongly consistent estimator of the @xmath0-median , that is , @xmath41_n},m_\theta[x])=0 \quad \text{a.s . } [ p].\ ] ] _ proof . _ sufficient conditions for the strong consistency of an estimator are given in huber ( @xcite ) . we will check that these conditions , detailed below , are satisfied in our case : * the parameter set ( @xmath42 in our case , with the topology induced by the @xmath0-metric ) is a locally compact space with a countable base and @xmath18 is a probability space . let @xmath43 be the following real - valued function on @xmath44 : @xmath45)}. \end{array}\ ] ] * assuming that @xmath46 are independent @xmath47-valued random elements with + common probability distribution @xmath48 , the sequence of functions @xmath49 , defined as @xmath50}_n$ ] , satisfies that almost surely ( obviously because of the definition of the sample @xmath0-median ) . _ assumption ( a-1 ) _ for each fixed @xmath51 , the function @xmath52)}\\[0.8ex ] & & & & \displaystyle{=\sqrt{({\rm mid}\ , x(\omega)-y_0)^2+\theta \cdot ( { \rm spr}\ , x(\omega)-z_0)^2}}\vspace{-0.1 cm } \end{array}\ ] ] is @xmath19-measurable and separable in doob s sense : there is a p - null set @xmath53 and a countable subset @xmath54 such that for every open set @xmath55 and every closed interval @xmath56 , the sets @xmath57 @xmath58 differ by at most a subset of @xmath53 . _ assumption ( a-2 ) _ the function @xmath59 is a.s . lower semicontinuous in @xmath60 , that is , @xmath61 as the neighborhood @xmath62 of @xmath60 shrinks to @xmath63 . _ assumption ( a-3 ) _ there is a measurable function @xmath64 such that @xmath65 ^ -<\infty \quad \text { for all } ( y , z)\in \mathbb r \times [ 0,\infty),\vspace{-0.2cm}\ ] ] @xmath65^+<\infty \quad \text { for some } ( y , z)\in \mathbb r \times [ 0,\infty).\ ] ] thus , @xmath66 $ ] is well - defined for all @xmath67 . _ assumption ( a-4 ) _ there is a @xmath68 such that @xmath69 @xmath70 for all @xmath71 _ assumption ( a-5 ) _ there is a continuous function @xmath72 such that * for some integrable @xmath73 , @xmath74 * the following condition is satisfied : @xmath75 * it is also fulfilled that : @xmath76\geq 1.\ ] ] we now verify these conditions of huber : _ ( a-1 ) _ for each fixed @xmath51 , the function @xmath77 is @xmath19-measurable ( because @xmath78 and @xmath79 are measurable functions since @xmath39 is a random interval ) and separable in doob s sense : choosing @xmath80 as countable subset , for every open set @xmath81 and every closed interval a , it will be seen that the sets @xmath82 coincide . obviously , @xmath83 . by _ reductio ad absurdum _ , it is now supposed that @xmath84 . let @xmath85 : * since @xmath86 , @xmath87 for all @xmath88 ; * since @xmath89 , there exists @xmath90 such that @xmath91 . @xmath92 is an open set , so there exists a ball of radius @xmath93 such that @xmath94 notice now that , for a fixed @xmath95 , the function @xmath96 is continuous . therefore , @xmath97 is an open set of @xmath42 and @xmath98 too . @xmath99 is a dense set of @xmath42 , so @xmath100 let @xmath101 . then , @xmath102 , so @xmath103 . but also , @xmath104 this is a contradiction , so the conclusion is that @xmath105 . _ ( a-2 ) _ indeed , it will be proved for all @xmath95 . let @xmath106 be any element of @xmath47 and let @xmath60 be any ( fixed ) point of @xmath42 . first , notice that it is fulfilled for a sequence of neighborhoods @xmath107 of @xmath60 when @xmath108 for all @xmath109 that @xmath110)\right\}_{n\in \mathbb n}\ ] ] is a monotonically increasing sequence . furthermore , this sequence is bounded since @xmath111 ) \leq d_\theta(x(\omega),[y_0-z_0,y_0+z_0])\ ] ] for all @xmath112 because @xmath113 . therefore , the sequence converges to its supremum , which will be @xmath114)$ ] . by _ reductio ad absurdum _ , suppose that there is a smaller upper bound @xmath115)-\varepsilon,\ ] ] for an arbitrary @xmath116 . let s denote by @xmath117 a neighborhood of @xmath60 satisfying that @xmath118 . then , it can be seen that @xmath119),\ ] ] so @xmath120 can not be the supremum . thus , using the triangular inequality , @xmath121)\geq \underset{(y , z)\in b((y_0,z_0),\frac{\varepsilon}{2})}{\inf}d_\theta(x(\omega),[y - z , y+z])\ ] ] @xmath122)-d_\theta([y - z , y+z],[y_0-z_0,y_0+z_0])\right]\ ] ] @xmath123)-\underset{(y , z)\in b((y_0,z_0),\frac{\varepsilon}{2})}{\sup}d_\theta([y - z , y+z],[y_0-z_0,y_0+z_0])\ ] ] @xmath124)-\varepsilon = c.\ ] ] now this result will be extended to general sequences @xmath125 . consider the suprema and the infima radii reached in every neighborhood , namely , @xmath126,[y - z , y+z]),\ ] ] @xmath127,[y - z , y+z]).\ ] ] it is known that @xmath128 , since @xmath125 shrinks to @xmath63 . moreover , @xmath129 as @xmath130 for all @xmath112 . let @xmath131 be any nonnegative number . as @xmath132 , there exists @xmath133 such that for all @xmath134 , @xmath135 . then , @xmath136 and @xmath111)\geq \underset{(y , z)\in b((y_0,z_0),r_n)}{\inf}d_\theta(x(\omega),[y - z , y+z])\ ] ] @xmath137)-\underset{(y , z)\in b((y_0,z_0),r_n)}{\sup}d_\theta([y_0-z_0,y_0+z_0],[y - z , y+z])\ ] ] @xmath138)-\varepsilon.\ ] ] analogously , as @xmath139 , there exists @xmath140 such that for all @xmath141 , @xmath142 . therefore , @xmath143 and @xmath111 ) \leq \underset{(y , z)\in b((y_0,z_0),s_n)}{\inf}d_\theta(x(\omega),[y - z , y+z])\ ] ] @xmath144)+\underset{(y , z)\in b((y_0,z_0),s_n)}{\inf}d_\theta([y - z , y+z],[y_0-z_0,y_0+z_0])\ ] ] @xmath145)+\varepsilon.\ ] ] so for any @xmath116 , there exists @xmath146 , such that for all @xmath147 , @xmath148)-\varepsilon < \underset{(y , z)\in u_n}{\inf}d_\theta(x(\omega),[y - z , y+z])\ ] ] @xmath145)+\varepsilon,\ ] ] that is to say , @xmath149)-d_\theta(x(\omega),[y_0-z_0,y_0+z_0])\right|<\varepsilon,\ ] ] so the sequence @xmath150)\right\}_{n\in \mathbb n}$ ] converges to @xmath151).$ ] _ ( a-3 ) _ let @xmath152 be the measurable function ( see ( a-1 ) ) : @xmath153)=\sqrt{({\rm mid}\ , x(\omega))^2+\theta \cdot ( { \rm spr}\ , x(\omega))^2.}}\vspace{-0.1 cm } \end{array}\ ] ] fixed any arbitrary @xmath154 , @xmath65 ^ -\]]@xmath155)-d_\theta(x(\omega),[0,0]),0\}\ , dp(\omega)\ ] ] @xmath156)\\>d_\theta(x(\omega),[y - z , y+z])\}\end{array}}}\big[d_\theta(x(\omega),[0,0])-d_\theta(x(\omega),[y - z , y+z])\big ] dp(\omega).\ ] ] by the triangular inequality , @xmath157)\\>d_\theta(x(\omega),[y - z , y+z])\}\end{array}}}\big[d_\theta(x(\omega),[y - z , y+z])+d_\theta([y - z , y+z],[0,0])\]]@xmath158)\big ] dp(\omega)\ ] ] @xmath159,[0,0])\cdot p\big(\omega : d_\theta(x(\omega),[0,0])>d_\theta(x(\omega),[y - z , y+z])\big ) < \infty.\ ] ] analogously , @xmath65^+\]]@xmath160)-d_\theta(x(\omega),[0,0]),0\}\ , dp(\omega)\ ] ] @xmath156)\\\leq d_\theta(x(\omega),[y - z , y+z])\}\end{array}}}\big[d_\theta(x(\omega),[y - z , y+z])-d_\theta(x(\omega),[0,0])\big ] \ , dp(\omega).\ ] ] by the triangular inequality , @xmath157)\\ \leq d_\theta(x(\omega),[y - z , y+z])\}\end{array}}}\big[d_\theta(x(\omega),[0,0])+d_\theta([0,0],[y - z , y+z])\]]@xmath161)\big ] dp(\omega)\ ] ] @xmath162,[y - z , y+z])\cdot p\big(\omega : d_\theta(x(\omega),[0,0])\leq d_\theta(x(\omega),[y - z , y+z])\big ) < \infty.\ ] ] so the second inequality also holds for all @xmath163 in this case . _ ( a-4 ) _ the @xmath0-median exists and is unique , so that @xmath164 , \hbox { \rm spr}\ , m_\theta[x])=\arg \underset{(y , z)\in \mathbb r\times [ 0,\infty)}{\min}e\left[d_\theta(x(\omega),[y - z , y+z])\right]\ ] ] @xmath165)\right]-e\left[d_\theta(x(\omega),[0,0])\right]\]]@xmath165)-d_\theta(x(\omega),[0,0])\right]\]]@xmath166 and @xmath167 , { \hbox { \rm spr}}\ , m_\theta[x])$ ] fulfills this assumption . there is a continuous function @xmath72 @xmath168,[0,0])+1}\vspace{-0.1 cm } \end{array}\ ] ] such that * for the integrable function @xmath169 , @xmath170)-d_\theta(x(\omega),[0,0])}{d_\theta([y - z , y+z],[0,0])+1}\geq -1\ ] ] because using the triangular inequality , @xmath170)-d_\theta(x(\omega),[0,0])}{d_\theta([y - z , y+z],[0,0])+1}\ ] ] @xmath171)-d_\theta([y - z , y+z],[0,0])-d_\theta(x(\omega),[0,0])}{d_\theta([y - z , y+z],[0,0])+1}\ ] ] @xmath172,[0,0])}{d_\theta([y - z , y+z],[0,0])+1}\geq -1.\ ] ] * the following condition is satisfied : @xmath75 let @xmath173 be any sequence with @xmath174 ( i.e. , @xmath175,[0,0])\underset{n}{\longrightarrow}\infty$ ] ) and @xmath176)-d_\theta(x(\omega),[0,0])\right]=\gamma((y_0,z_0))\in \mathbb r,\ ] ] where @xmath60 represents the minimum found in ( a-4 ) . then , there exists @xmath177 such that for all @xmath178 , @xmath179,[0,0])>m.\ ] ] so , for all @xmath178 , @xmath180,[0,0])+1\right)\geq m+1.\ ] ] finally , @xmath181 * it is also fulfilled that : @xmath182)-d_\theta(x(\omega),[0,0])}{b((y , z))}\right]\geq 1.\ ] ] let s see that @xmath183)-d_\theta(x(\omega),[0,0])}{d_\theta([y - z , y+z],[0,0])+1}\geq 1,\ ] ] so the result follows . @xmath183)-d_\theta(x(\omega),[0,0])}{d_\theta([y - z , y+z],[0,0])+1}\ ] ] @xmath184)-d_\theta(x(\omega),[0,0])}{d_\theta([y_k - z_k , y_k+z_k],[0,0])+1}\right)\ ] ] for any fixed @xmath185 . the sequence @xmath186)-d_\theta(x(\omega),[0,0])}{d_\theta([y_k - z_k , y_k+z_k],[0,0])+1}\right\}_{n\in\mathbb n}\ ] ] is monotonically increasing and is upper bounded by @xmath187 : for all @xmath188 , using the triangular inequality , @xmath189)-d_\theta(x(\omega),[0,0])}{d_\theta([y_k - z_k , y_k+z_k],[0,0])+1}\ ] ] @xmath190,[0,0])}{d_\theta([y_k - z_k , y_k+z_k],[0,0])+1}\leq 1.\ ] ] + so it converges to its supremum : @xmath191)-d_\theta(x(\omega),[0,0])}{d_\theta([y_k - z_k , y_k+z_k],[0,0])+1}\right)\ ] ] @xmath192)-d_\theta(x(\omega),[0,0])}{d_\theta([y_k - z_k , y_k+z_k],[0,0])+1}\right)\ ] ] let s finally see that this supremum is at least equal to @xmath187 . by _ reductio ad absurdum _ , let s suppose that @xmath193)-d_\theta(x(\omega),[0,0])}{d_\theta([y_k - z_k , y_k+z_k],[0,0])+1}\right)=1-\varepsilon,\ ] ] for some @xmath116 . one gets then a contradiction because one finds an @xmath194 such that @xmath195)-d_\theta(x(\omega),[0,0])}{d_\theta([y_k - z_k , y_k+z_k],[0,0])+1}>1-\varepsilon\ ] ] since for all @xmath196 , @xmath189)-d_\theta(x(\omega),[0,0])}{d_\theta([y_k - z_k , y_k+z_k],[0,0])+1}\geq 1-\frac{\varepsilon}{2}>1-\varepsilon\ ] ] as we will show now . recall that @xmath174 , so for all @xmath197 , there exists @xmath194 such that for all @xmath198 , @xmath175,[0,0])>m$ ] . therefore , @xmath179,x(\omega))\geq d_\theta([y_n - z_n , y_n+z_n],[0,0])-d_\theta(x(\omega),[0,0])\]]@xmath199).\ ] ] taking @xmath200)\in \mathbb r$ ] ( for the fixed arbitrary @xmath185 ) , we can easily check that @xmath201 is a lower bound of the sequence @xmath202)-d_\theta(x(\omega),[0,0])}{d_\theta([y_k - z_k , y_k+z_k],[0,0])+1}\right\}_{k\geq n^*}.\ ] ] + for any @xmath196 , @xmath203)-d_\theta(x(\omega),[0,0])\ ] ] @xmath204)+\frac{\varepsilon}{2}d_\theta(x(\omega),[y_k - z_k , y_k+z_k])\ ] ] @xmath161)\ ] ] @xmath205,[0,0])-\left(1-\frac{\varepsilon}{2}\right)d_\theta(x(\omega),[0,0])\ ] ] @xmath206)-d_\theta(x(\omega),[0,0])\ ] ] @xmath207,[0,0])+\frac{\varepsilon}{2}d_\theta(x(\omega),[y_k - z_k , y_k+z_k])\ ] ] @xmath208)\ ] ] @xmath209,[0,0])+\frac{\varepsilon}{2}\left(\frac{2}{\varepsilon}-1+\big(\frac{4}{\varepsilon}-1\big)d_\theta(x(\omega),[0,0])\right)\ ] ] @xmath208)=\left(1-\frac{\varepsilon}{2}\right)d_\theta([y_k - z_k , y_k+z_k],[0,0])+1-\frac{\varepsilon}{2}\ ] ] @xmath210,[0,0])+1\big).\hspace{2.4cm}\square\ ] ] this paper complements the study of the properties of the @xmath23-median as a robust estimator of the center of a random interval by showing its strong consistency which is one of the most important basic properties of an estimator . we obtained this result by showing that all the sufficient conditions of huber ( @xcite ) are fulfilled . these results open the door to further develop robust statistical inference for random intervals based on the @xmath23-median such as the development of hypotheses testing procedures . authors are grateful to mara ngeles gil for her helpful suggestions to improve this paper . the research by beatriz sinova was partially supported by / benefited from the spanish ministry of science and innovation grant mtm2009 - 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the sample @xmath0-median is a robust estimator of the central tendency or location of an interval - valued random variable . while the interval - valued sample mean can be highly influenced by outliers , this spatial - type interval - valued median remains much more reliable . in this paper , we show that under general conditions the sample @xmath0-median is a strongly consistent estimator of the @xmath0-median of an interval - valued random variable .

You are an expert at summarizing long articles. Proceed to summarize the following text: a small number of close binary stars are thought to dominate the dynamic evolution of many globular star clusters ( hut et al . 1992 , bailyn 1995 ) , yet classes of such objects which are relatively easily found in the field have proven frustratingly difficult to discover in clusters . a prime example is cataclysmic variables ( cvs ) , which call attention to themselves via large amplitude light outbursts , and peculiar , ultraviolet - excess colors in quiescence . with quiescent absolute magnitudes @xmath2 , modern ground - based photometric techniques should easily uncover such objects in clusters with typical distances of @xmath3 , even with modest telescopes , unless all such objects are lost to the crowded cores . yet prior to the launch of _ hubble space telescope _ ( _ hst _ ) , we are aware of only two candidate identifications of cvs in globular clusters , m5 v101 ( margon et al . 1981 ) , whose classification and cluster membership seems secure ( naylor et al . 1989 , shara et al . 1990 ) , and m30 v4 ( margon & downes 1983 ) , whose membership is unclear ( shara et al . 1990 , machin et al . 1991 ) . observations from _ hst _ have certainly improved the situation , although perhaps not as much as many would have expected . a few clusters are now known to have a handful of spectroscopically - confirmed cvs ; a recent review is given by grindlay ( 1999 ) . however despite intensive photometric and color - selected searches deep into the cores of a number of clusters , few outbursting objects are found , and most authors believe there is a serious discrepancy with theoretical predictions ( shara & drissen 1995 ; livio 1996 ; shara et al . 1996 , hereafter s96 ) . whether the problem lies with formation / destruction rates , or some unique property of cluster cvs , remains to be clarified , and will surely require a larger sample of objects . here we discuss _ hst _ observations of the cluster ngc6624 using the _ space telescope imaging spectrograph _ ( stis ) . this cluster contains near its center a highly luminous bursting x - ray source with an 11-minute period , the shortest - known binary star ; the system is thought to be a double - degenerate ( stella et al . 1987 , king et al . 1993 , anderson et al . another object in the cluster has also been suggested by s96 as a candidate cv , although we discuss its nature further in 3 . as part of a program to study the central bright x - ray source , we obtained deep stis spectroscopic exposures of this object , and these results will be discussed elsewhere ( deutsch et al . 2000 ) . here we report the completely serendipitous discovery of @xmath1 emission line object , which fell by good fortune in the stis slit , and has the properties of a classical cv . this object deepens the mystery of the cv content of clusters . s96 report a multi - epoch sensitive photometric search of the core of ngc6624 which identifies only one candidate cv , and suggest these objects are very rare , at least in this cluster . yet we have found another such an object completely by accident , in the first long - slit ultraviolet spectral exposure made of the cluster . on 1998 march 14 we obtained 12150 s of integration in the center of ngc6624 over 5 orbits with the _ hst _ stis fuv - mama , using a @xmath4 wide , @xmath5 long slit . the spectra are reprocessed and extracted using calstis 2.0 and the latest calibration files available as of 1999 april 20 , where nearly half of the calibration files are updated from the original pipeline processing . in fig . 1 we show a @xmath6 long region of the cluster spectrum , which has been background subtracted for display purposes . the primary target , the x - ray source hereafter denoted star k , is clearly visible at the top . two other objects , labeled stars 1 and 2 , are visible . no other objects are discernible in the entire long - slit spectrum , including the region not shown in fig the extracted spectrum of star 1 , binned @xmath7 to 1.17 per bin , is displayed in fig . strong , broad emission lines of n v @xmath81238,1242 , si iv @xmath81394,1403 and/or o iv ] @xmath91401 , c iv @xmath81548,1550 , and he ii @xmath91640 are clearly detected . the o i @xmath91304 feature is most likely imperfectly subtracted geocoronal emission . the location of a few other common emission lines which may contribute to this spectrum are also marked . the continuum is very weakly detected at @xmath10 erg @xmath11 s@xmath12 @xmath12 . this spectrum is that of a classical cataclysmic variable in a quiescent state ( e.g. , wu et al . 1992 ) . we show in 3 that both the optical and uv magnitudes of the object agree with those expected for a cv at the distance of the cluster , and the probability of a chance superposition of a non - member within @xmath13 of the cluster core is small . we conclude that we have serendipitously discovered a cv in the cluster . we have attempted to independently verify cluster membership via radial velocity measurements of the spectrum , but the results are inconclusive . we were forced to use an undesirably large slit width due to potential problems in target acquisition in this complex field , and the resulting configuration is poorly suited for absolute velocity determinations . if we accept the nominal calstis wavelength calibration , the inferred radial velocity , obtained from several strong emission lines , is @xmath14 km s@xmath12 . the uncertainty in this value is however dominated by the zero point error introduced by the unknown location of the object within the slit , which could be as large as @xmath15 km s@xmath12 , so the constraint is not meaningful . further , we expect the radial velocity of the star to vary , with unknown period and probably large amplitude , and so can not make a cogent estimate of the mean velocity after one observation . we have also searched for radial velocity variations from orbit to orbit , again with inconclusive results due to large uncertainties in the subsets of the data . the spectrum of star 2 is a flat continuum of @xmath16 erg @xmath11 s@xmath12 @xmath12 , and featureless except for possible detections of the common interstellar absorption lines si ii @xmath17 , c ii @xmath81334,1335 , and c iv @xmath81548,1551 . when dereddened , the spectrum and photometry are reasonably well described by a kurucz model ( kurucz 1992 ) of t@xmath18,000 k and @xmath19 . this is consistent with the properties of the non - flickering `` nf '' objects discussed by cool et al . ( 1998 ) and edmonds et al . ( 1999 ) . we attempt to determine the precise location of the objects seen in our spectroscopic observations by examining archival _ we find f140w and f430w ( pre - costar ) foc images taken on 1992 august 13 which cover this field , as well as wfpc2 images taken on 1994 april 17 and 1994 october 15 , obtained with a variety of filters . there exist two additional epochs of wfpc2 data , but as this field falls on the lower resolution wide field camera ccds in those observations , they provide no additional useful information . given the crowding in the cluster core , determination of which objects are responsible for the observed spectra is not trivial . based on information in the stis header , the distance of stars 1 and 2 from the bright star k is @xmath20 and @xmath21 , respectively , at a position angle of @xmath22 . in fig . 3 we show 5@xmath235@xmath24 regions of the various foc and wfpc2 images . overlaid are two arcs indicating the calculated radii of these objects from star k ( not present in this field ) . the nearly horizontal lines indicate the edges of a @xmath4 slit . therefore , we expect to find the objects responsible for the spectra near the intersections of the arcs and the slit . residual uncertainty in the slit angle is less than 1@xmath25 and thus not important at this scale . finally , the figure assumes that star k is centered on the slit . if star k is in fact shifted slightly in the slit , then the location of the slit we have drawn may be displaced up or down by up to @xmath26 ; the target acquisition scenario employed should place star k very near the center of the slit , however . in the 1400 foc image , we find a faint object at the expected location of star 2 , and label it star b ( recognizing that the association is hardly guaranteed . ) we find no evidence for any detection at the expected location of star 1 in this f140w image . examining successively longer wavelength images , we find that star b exhibits uv - excess between the f336w and f439w images , but is hopelessly contaminated by light from neighboring stars at longer wavelengths . at the expected location of star 1 , we find a faint star which we label a at f439w and longer wavelengths , although there is no evidence of it at f336w and f140w . star a is positively detected at multiple epochs as well as multiple wavelengths , although it falls just outside the foc f430w image . there is also another faint object @xmath27 nw of star a , which is also a plausible candidate . no small adjustments in slit angle or slit shift yield a better alignment . in the astrometric frame of _ hst _ image u2as0101 t , the coordinates for star a are @xmath28@xmath29 , @xmath30@xmath31 . although internally precise , this position has a probable uncertainty of @xmath32 with respect to external frames . based on our estimate of the cluster center in the same image , we find that star a is @xmath33 , or 1.8 core radii , from the center of ngc6624 ( assuming @xmath34 from harris 1996 ) . a detailed discussion of a precise measurement of the cluster center , not relevant to our analysis , is given by king et al . two radio pulsars are located within @xmath35 of star a ( biggs et al . 1994 ) , but the positions are definitely disjoint , given the quoted uncertainties . in a cluster of @xmath34 , it is perhaps not surprising that multiple interesting but unrelated objects are in such proximity . in table 1 we present photometry for selected objects in the field . we use a combination of profile - fitting and aperture photometry to derive these magnitudes . for wfpc2 images , aperture corrections are taken from table 2(a ) in holtzman et al . ( 1995b ) . the photometric measurements have not been corrected for geometric distortions , nor is any correction for charge transfer efficiency losses ( holtzman et al . 1995b ) applied ; for most of the images , these effects should contribute errors of only a few percent . we use the photometric zero points for the stmag system from table 9 ( @xmath36 ) in holtzman et al . ( 1995a ) . approximate @xmath37 measurement uncertainties are also provided in the table . systematic errors for all magnitudes due to uncertainties in detector performance and absolute calibration are @xmath385% . magnitude measurements presented here are denoted @xmath39 , where @xmath9 is the filter designation , approximately indicating the central wavelength of the filter in nanometers . in the stmag system , zero points are set such that a flat spectrum ( @xmath40 ) source will have identical magnitudes at all wavelengths . for the foc measurements , we use the calibration provided in the header of the images . a stsdas _ synphot _ calculation yields a calibration value which differs by @xmath41% . we use a tiny tim ( krist 1993 ) synthetic profile to determine an aperture correction . since the f140w images were obtained in an unusual mode and far uv calibration is often difficult in any case , we suspect that the absolute calibration may not be accurate to better than a factor of two . in order to further check that the associations between stars 1 and a , and stars 2 and b , are plausible , we estimate @xmath42 for stars 1 and 2 at the time of the stis observations by convolving the spectra with the f140w bandpass and integrating the observed flux . we estimate @xmath43 for star 1 , and @xmath44 for star 2 , and estimate uncertainties of 0.2 mag principally because the spectra do not fully cover the f140w bandpass . given the various uncertainties in this estimate and foc calibration , the agreement between the stis and foc ( table 1 ) magnitude estimates is excellent . aside from the observed level of excitation in the spectrum , the lack of any brighter and/or spatially extended image near the correct location for star 1 provides confirmation that the observed emission spectrum can not be due to a cluster planetary nebula or the background superposition of a low redshift agn . we conclude that star a is the likely source of the emission line spectrum . we point out in passing yet another unusual object in the same field . star c ( table 1 , fig . 3 ) is extremely bright in these f140w images , but a rather faint object in longer wavelength passbands . when dereddened , the @xmath42 , @xmath45 , and @xmath46 measurements are well fit by a kurucz model of t@xmath47,000 k. this temperature is most sensitive to the f140w measurement for which the calibration is poorest . if the foc calibration is adjusted by 0.4 mag such that the foc observed and stis convolved magnitudes are equal for star b , the implied temperature for star c is t@xmath48,000 k. this extraordinary uv excess is noteworthy but otherwise not relevant to the present discussion , except as further evidence of a multitude of exotic objects near the center of this cluster . our photometry in 2.2 together with the known distance and reddening of the cluster permit a comparison of the luminosity of this object with that of the far better - studied field cvs . adopting @xmath49 and @xmath50 ( harris 1996 ) , our measured @xmath51 implies @xmath19 . there are now four trigonometric parallaxes for classical field cvs ( harrison et al . 1999 , mcarthur et al . 1999 ) , which collectively imply @xmath52 , similar to the @xmath53 often quoted from much larger samples ( warner 1995 ) . given the uncertainties in our photometry for this faint object in a very crowded field , and our single - epoch measurement of an undoubtedly variable star , the agreement of the inferred luminosity of the new ngc6624 cv with those in the field is gratifying . if the system is strongly magnetic , a possibility we consider below , our observed magnitude is perhaps somewhat brighter than expected from polars in the field , but , again given the uncertainties , not alarmingly so . we are aware of few if any ultraviolet spectra of globular cluster cvs . edmonds et al . ( 1999 ) display a low signal - to - noise spectrum of an object in ngc 6397 , where heii @xmath91640 is termed marginally detected " in emission by those authors . through absolutely no credit to the current authors , our spectrum is far better exposed . the prominent heii @xmath91640 emission in our spectrum is deserving of comment . this line is normally not strong in classical cvs , but is seen in polars . our spectrum is indeed quite similar to that of am her in this wavelength range ( greeley et al . unfortunately the simplest defining spectroscopic characteristic of am her stars , very strong heii @xmath94686 emission , is not accessible to us . although little can be inferred from one object , we note that grindlay et al . ( 1995 ) have suggested that magnetic white dwarfs are preferentially produced in globular clusters ( see also grindlay 1999 and references therein ) . these authors also discuss the issue of possible confusion of the spectra of cluster cvs with those of quiescent low - mass x - ray binaries ( lmxbs ) , and those considerations also apply here . our spectrum does not unambiguously distinguish between the two cases ; for example , the quiescent lmxb cen x-4 ( v822 cen ) shows quite weak heii @xmath94686 ( mcclintock & remillard 1980 ) and little or no @xmath91640 emission ( blair et al . 1984 ) , but aql x-1 ( v1333 aql ) displays strong heii @xmath94686 in quiescence ( garcia et al . 1999 ) . as more cvs than quiescent lmxbs are known in clusters , it seems most conservative , but still uncertain , to continue the discussion of cvs . as this object lies only a few arcsec from the brightest x - ray source in any globular cluster , its x - ray properties are as yet unknown , but may be accessible to observations from _ chandra_. although our detection of the ultraviolet continuum is weak , it is of interest to ask if the observed ultraviolet flux agrees with that expected from the field objects , as we have determined above is the case for the visible band . if we assume @xmath54 ( cardelli et al . 1989 ) , our observed @xmath55 erg @xmath11 s@xmath12 @xmath12 corresponds to an extinction - corrected monochromatic luminosity of @xmath56 erg s@xmath12 @xmath12 . for the quiescent u gem , for example , @xmath57 erg @xmath11 s@xmath12 @xmath12 ( wu et al . 1992 , long et al . 1994 , long & gilliland 1999 ) ; this ratio of 30,000 in observed fluxes speaks well to advances in ultraviolet spectroscopic capabilities from the _ international ultraviolet explorer _ ( _ iue _ ) to the _ hst _ era . adopting @xmath58 pc ( harrison et al . 1999 ) and @xmath59 ( panek and holm 1984 , long & gilliland 1999 ) for u gem , we infer @xmath60 erg s@xmath12 @xmath12 for that object . while this excellent level of agreement is almost surely fortuitous , it reaffirms that the cv in ngc6624 , with the exception of its unusual environment , seems a totally normal cataclysmic . as various abnormal mass transfer scenarios have been at times invoked to explain the lack of cluster cv outbursts , this agreement is interesting . there is one previous candidate for a cv in ngc6624 , as discussed by s96 . the identification was based on the presence of the object on two consecutive _ hst _ images , but its absence in all other observations . we have recovered the candidate in the archival _ hst _ images , and derive its position as @xmath28@xmath61 , @xmath30@xmath62 using the astrometric header information in _ hst _ image u2kl0406 t . this position is @xmath63 from star a , and thus this object can not be responsible for our spectrum . moreover , upon close examination of these data , we find that the radial profile of this object is not compatible with a stellar one . we suggest it is an image artifact of uncertain origin . in fig . 4 , we show a surface plot of the pixels around this object and a nearby , typical star of similar total counts . the s96 object has a profile much too sharp as compared with this and other stars . it is indeed detected in two frames , and thus is not a charged particle hit , but rather probably a group of hot pixels or calibration file defect . we stress however that the removal of the s96 object from the list of cluster cvs merely _ strengthens _ the primary scientific conclusion of that paper , namely that erupting cvs are remarkably rare in clusters . we have serendipitously discovered a mass - transfer close binary , presumably with a degenerate companion , close to the core of ngc 6624 . it is likely that the system is a classical cataclysmic variable , although a quiescent lmxb can not be ruled out . the situation is a curious inversion of the normal problem in globular cluster observations , where one finds an interesting star but has difficulty obtaining the spectrum due to severe crowding . we have easily obtained the spectrum , but are uncertain of which object is responsible for the emission , although we advance as a reasonable candidate star a. the uncertainty in the exact coordinates of the system does not however change the surprising result that an object supposedly so rare has been found accidentally . the object was found in a single stis long - slit exposure encompassing @xmath64 arcsec@xmath65 at an arbitrary position angle passing quite close to the cluster center . a hypothetical observing program that included all possible position angles with this slit , thus completely mapping the cluster center to @xmath66 , would cover @xmath67 more area . tempered by the usual uncertainties of _ a posteriori _ statistics , it seems quite likely that the center of ngc6624 contains several , and possibly many , more cvs of the type we have discovered here . if we now accept that these systems most certainly are present as has indeed been theoretically expected for some time then one or more mechanisms which suppress the expected outbursts and/or odd colors , and thus hide globular cluster cvs , must certainly be operative .

despite indications that classical cataclysmic variable ( cv ) stars are rare in globular clusters in general , and in the cluster ngc6624 in particular , we have serendipitously discovered such a star @xmath0 from the cluster center . a _ hubble space telescope _ spectrum of the @xmath1 object shows strong , broad emission lines typical of numerous field cvs , and the inferred optical and uv luminosity are also similar . our accidental observation also provides the first high - quality ultraviolet spectrum of a globular cluster cv . that we have detected such an object in an observation that includes just a few percent of the central area of the cluster may indicate that cluster cvs are more common than previously thought , at least near the core . -0.6 in 9.6 in 0.210 in accepted for publication in the astronomical journal + to appear in the 1999 december issue , volume 118 + _ received 1999 july 28 ; accepted 1999 august 31 _

You are an expert at summarizing long articles. Proceed to summarize the following text: the benchmarks for optimal performance of heat engines and refrigerators , under reversible conditions , are the carnot efficiency @xmath4 , and the carnot coefficient of performance @xmath5 respectively , where @xmath6 is the ratio of cold to hot temperatures of the reservoirs . for finite - time models such as in the endoreversible approximation @xcite and the symmetric low - dissipation carnot engines @xcite , the maximum power output is obtained at the so called curzon - ahlborn ( ca ) efficiency , @xmath7 @xcite . however , ca - value is not as universal as @xmath8 . for small temperature differences , its lower order terms are obtained within the framework of linear irreversible thermodynamics @xcite . thus models with tight - coupling fluxes yield @xmath9 as the efficiency at maximum power . further , if we have a left - right symmetry , then the second - order term @xmath10 is also universal @xcite . on the other hand , the problem of finding universal benchmarks for finite - time refrigerators is non - trivial . for instance , the rate of refrigeration ( @xmath11 ) , which seems a natural choice for optimization , can not be optimized under the assumption of a newtonian heat flow ( @xmath12 ) between a reservoir and the working medium @xcite . in that case , the maximum rate of refrigeration is obtained as the coefficient of performance ( cop ) @xmath13 vanishes . so instead , a useful target function @xmath14 has been used @xcite , where @xmath11 is the heat absorbed per unit time by the working substance from the cold bath , or the rate of refrigeration . the corresponding cop is found to be @xmath15 , for both the endoreversible and the symmetric low - dissipation models . so this value is usually regarded as the analog of ca - value , applicable to the case of refrigerators . in any case , the usual benchmarks for optimal performance of thermal machines are decided by recourse to optimization of a chosen target function . the method also presumes a complete knowledge of the intrinsic energy scales , so that , in principle , these scales can be tuned to achieve the optimal performance . in this letter , we present a different perspective on this problem . we consider a situation where we have a limited or partial information about the internal energy scales , so that we have to perform an inference analysis @xcite in order to estimate the performance of the machine . inference implies arriving at plausible conclusions assuming the truth of the given premises . thus the objective of inference is not to predict the `` true '' behavior of a physical model but to arrive a rational guess based on incomplete information . in this context , the role of prior information becomes central . in the spirit of bayesian probability theory , we treat all uncertainty probabilistically and assign a prior probability distribution to the uncertain parameters @xcite . we define an average or expected measure of the performance , using the assigned prior distribution . the approach was proposed by one of the authors @xcite and has been then applied to different models of heat engines @xcite . these works show that ca - efficiency can be reproduced as a limiting value when the prior - averaged work or power in a heat cycle is optimized . in particular , for the problem of maximum work extraction from finite source and sink , the behavior of efficiency at maximum estimate of work shows universal features near equilibrium @xcite , e.g. @xmath16 $ ] . similarly , other expressions for efficiency at maximum power , such as in irreversible models of stochastic engines @xcite , which obey a different universality near equilibrium , can also be reproduced from the inference based approach @xcite . however , so far the approach has not been applied to other kinds of thermal machines such as refrigerators . it is not obvious , beforehand , that the probabilistic approach can be useful in case of refrigerators also . the purpose of this paper is to extend the prior probability approach by taking the paradigmatic feynman s ratchet and pawl model @xcite . we show that the prior information infers not only the ca - efficiency @xmath17 in the engine mode , but also the @xmath18 value in the refrigerator mode of the model . further , we point out that the expected heat flows in the averaged model behave as newtonian flows . the present paper is organized as follows . in section 2 , we describe the model of feynman s ratchet as heat engine and discuss its optimal configuration . in section 2.1 , the approach based on prior information is applied to the case when the efficiency of the engine is fixed , but the internal energy scales are uncertain . the approach is extended to the refrigerator mode , in section 3 . in section 4 , we discuss alternate models where also the use of jeffreys prior leads to emergence of ca efficiency . finally , section 5 is devoted to discussion of results and conclusions . the model of feynman s ratchet as a heat engine consists of two heat baths with temperatures @xmath19 and @xmath20 . a vane , immersed in the hot bath , is connected through an axle with a ratchet in contact with the cold bath , see fig.[fig - ratchet ] . the rotation of the ratchet is restricted in one direction due to a pawl which in turn is connected to a spring . the axle passes through the center of a wheel from which hangs a weight . so the directed motion of the ratchet rotates the wheel , thereby lifting the weight . to raise the pawl , the system needs @xmath21 amount of energy to overcome the elastic energy of the spring . suppose that in each step , the wheel rotate an angle @xmath22 and the torque induced by the weight be @xmath23 . then the system requires a minimum of @xmath24 energy to lift the weight . hence the rate of forward jumps for lifting the weight is given as @xmath25 where @xmath26 is a rate constant and we have set boltzmann s constant @xmath27 . the statistical fluctuations can produce a directed motion at a finite rate , only if the ratchet - pawl system is mesoscopic . hence the pawl can undergo a brownian motion by bouncing up and down as it is immersed in a finite temperature bath . this turns the wheel in backward direction and lowers the position of the weight . this is the reason that the system can not work as an engine if @xmath28 @xcite . the rate of the backward jumps is @xmath29 thus one can regard @xmath30 and @xmath31 as the work done by and on the system , respectively . in an infinitesimally small time interval @xmath32 , the work done by the system is given as @xmath33 thus the power output of the engine is defined as @xmath34 . similarly , the rate of heat absorbed from the hot reservoir , is given as @xmath35 or the amount of heat absorbed in the small time interval is @xmath36 . then the efficiency of the engine is given by @xmath37 the rate at which waste heat is rejected to the cold reservoir is @xmath38 , which follows from the conservation of energy flux . the power output , optimized with respect to energy scales @xmath39 and @xmath21 @xcite , is given by @xmath40 the corresponding efficiency at maximum power is @xmath41 further , it was discussed in ref . @xcite that the above expression for efficiency shares some universal properties of efficiency at optimal power found in other finite - time models @xcite . now we consider a situation where the efficiency of the engine has some pre - specified value @xmath42 , but the energy scales ( @xmath43 ) are not given to us in a priori information . since @xmath42 is known , the problem is reduced to a single uncertain parameter , due to eq . ( [ efficiency - ratchet ] ) . one can cast the problem either in terms of @xmath39 or @xmath21 . in terms of the latter , we can write power as @xmath44 analogous to quantification of prior information in bayesian statistics , we assign a prior probability distribution for @xmath21 in some arbitrary , but a finite range of positive values : @xmath45 $ ] . later we consider an asymptotic range in which the analysis becomes simplified and we observe universal features . now consider two observers @xmath46 and @xmath47 who respectively assign a prior for @xmath39 and @xmath21 . taking the simplifying assumption that each observer is in an equivalent state of knowledge , we can write @xcite @xmath48 where @xmath49 is the prior distribution function , taken to be of the same form for each observer . at a fixed known value of efficiency , it implies that @xmath50 , where the normalization constant , @xmath51^{-1}$ ] . this is also known as jeffreys prior for a one - dimensional scale parameter @xcite . now the expected value of power , over this prior , is defined to be @xmath52 where @xmath53^{-1}. \label{defc}\ ] ] upon performing the integration , we get @xmath54 now this expected power depends on the extreme values defining the range of the prior . we chose a finite range in order to define a normalized prior distribution . otherwise , information on the finite values of these scales is not available . on the other hand , as the range is made arbitrarily large , the average power becomes increasingly small . thus a comparison between the absolute magnitudes of optimal power ( eq . ( [ maxpower ] ) ) and the prior - averaged power does not seem fruitful . however , the expected power is seen to become optimal at a certain value of the given efficiency . further , universal features are shown by this efficiency in the asymptotic limit . it also provides a good estimate of the actual values of efficiency at maximum power . hence , on maximizing @xmath55 with respect to @xmath42 , we get @xmath56 for given values of the limits , we obtained numerical solution for @xmath42 . as shown in fig . [ finite - limits - engine ] , the efficiency at maximum expected power versus @xmath57 is plotted , for a given value of the upper limit @xmath58 . alternately , setting the lower limit @xmath57 as relatively small in magnitude , one can visualise the behaviour of the efficiency with @xmath58 . interestingly , these solutions show convergence to the ca - value , @xmath2 . ( scaled by @xmath19 ) , while @xmath59 . the upper and lower curves correspond to @xmath60 and @xmath61 , respectively . the dashed lines represent corresponding ca values . the efficiency is also plotted versus @xmath58 ( inset ) , assuming @xmath62 . for larger values of @xmath58 , the efficiency approaches ca value . , width=283 ] the convergence to the ca value as observed in fig . 1 , can be argued as follows . let us assume that the temperature gradient is not very large , i.e. @xmath0 is not close to zero . or in other words , @xmath8 is small compared to unity . this implies that @xmath42 is also small since it is bounded from above by @xmath8 . now let us consider the limits which satisfy , @xmath63 and @xmath64 @xcite , referred to as _ asymptotic range _ in the following . then the condition ( [ deta ] ) simplifies to the form @xmath65 this implies that the efficiency at optimal @xmath66 , approaches the ca value . _ uniform prior _ : on the other hand , maximal ignorance about the likely values of a parameter may be represented by a uniform prior density , @xmath67 . then the expected power , is given as @xmath68 where @xmath69 . integrating the above equation , we get @xmath70 . \end{aligned}\ ] ] here , we are interested in the efficiency at maximum expected power ( @xmath71 ) in the asymptotic range . therefore , by putting @xmath72 and then considering the asymptotic limit , we get @xmath73 whose real solution is given by @xmath74 where @xmath75 . these efficiencies are compared in fig . [ fig - ratchet - engine ] . in particular , we note that in the asymptotic range , the efficiency depends only on the ratio of the reservoir temperatures . further , the use of jeffreys prior gives a closer approximation to the actual behavior of efficiency at optimal performance of the engine . ) ) when a uniform prior is used , also in the asymptotic range . the dashed curve represents the efficiency at optimal power @xmath76 , eq . ( [ efopt]).,width=302 ] to compare these efficiencies near equilibrium i.e. @xmath8 close to zero , we expand these expressions as taylor series for small values of @xmath8 , @xmath77 \;\ ; ( { \rm eq . ( \ref{efopt } ) } ; \mbox { at optimal power } ) \label{seropt}\\ \eta^*&=&\frac{\eta_c}{2}+\frac{\eta_c^2}{8}+\frac{6 \eta_c^3}{96}+o[\eta_c^4]\ ; \ ; { \rm ( ca\ ; value\ ; from\ ; 1/\epsilon_2 \ ; prior ) } \label{serp } \\ \eta_u&=&\frac{\eta_c}{2}+\frac{\eta_c^2}{16}+\frac{\eta_c^3}{64}+o[\eta_c^4].\;\ ; { \rm ( with\ ; uniform\ ; prior)}\end{aligned}\ ] ] the series in eqs . ( [ seropt ] ) and ( [ serp ] ) were obtained in ref . we note that @xmath9 term in the optimal performance can be faithfully reproduced by the expected power irrespective of the chosen prior . however , the second order term follows from the use of jeffreys prior . in this section , we consider the function of feynman s ratchet as a refrigerator @xcite . it is analogous to bttiker - landauer model @xcite , as discussed in @xcite . by optimizing the target function @xmath78 for feynman s ratchet , the cop at optimal performance @xmath79 satisfies a transcendental equation @xcite . the solution can be approximated by an interpolation formula @xmath80 similar to the case of heat engine , we now show using the prior based approach , that cop at optimal performance can be obtained for feynman s ratchet as refrigerator . the cop for certain values of @xmath39 and @xmath21 is given by @xmath81 . also the rate of refrigeration is given by @xmath82 in terms of @xmath13 and one of the scales say , @xmath21 , the @xmath1-criterion is given by @xmath83 now we suppose that the cop is fixed at some value @xmath13 , and @xmath21 is uncertain , within the range @xmath45 $ ] . then jeffreys prior for @xmath21 can be argued , similar to eq . ( [ choice - of - prior ] ) . now we define the expected value of @xmath1 as @xmath84 where @xmath85 is given by eq . ( [ defc ] ) . upon integrating the above equation , we get @xmath86 as with power output for the engine , the average @xmath87 becomes increasingly small in the asymptotic limit . in the following , we focus on cop at maximal @xmath87 , in the asymptotic limit . so the maximum of @xmath87 with respect to @xmath13 , is evaluated as @xmath88 the numerical solution for @xmath13 versus one of the limits is shown in fig . [ finite - limits - refri ] . -criterion is plotted versus @xmath58 ( scaled by @xmath19 ) , while @xmath62 . the upper and lower curves correspond to @xmath61 and @xmath60 , respectively . the dashed lines represent corresponding @xmath18 values . for larger values of @xmath58 , cop approaches the corresponding @xmath18 . in inset , cop is plotted versus @xmath57 , when @xmath59 . the cop approaches @xmath18 as @xmath57 takes smaller values . , width=302 ] finally , in the asymptotic range , the above expression reduces to @xmath89 so the permissible solution ( @xmath90 ) of the above quadratic equation , which maximizes @xmath87 , is given as @xmath91 _ uniform prior _ : on the other hand , with uniform prior , the expected @xmath1-criterion is given as @xmath92 upon integrating the above equation , we get @xmath93.\end{aligned}\ ] ] now , we want to estimate @xmath94 , the cop at maximum expected @xmath1-criterion in asymptotic range . hence , by putting @xmath95 and imposing the asymptotic range , we obtain the following equation @xmath96 whose acceptable solution can be finally written in the following form @xmath97 - 1 . \label{unicop}\ ] ] again , we see that in the asymptotic range , the cop is given only in terms of the ratio of the reservoir temperatures . we show in fig . [ fig - fridge ] , a comparison amongst the different expressions for cop at optimized performance versus this ratio . ) is plotted versus @xmath0 . the solid curve shows the cop at optimal expected performance ( @xmath87 ) when jeffreys prior is assigned and the asymptotic range is applied . the dashed line represents the interpolation formula for cop corresponding to the optimum @xmath1 value @xcite . the top , dotted line is the result of uniform prior , again in the asymptotic range . the inset shows the same three quantities for close - to - equilibrium values of @xmath0.,width=302 ] in near - equilibrium regime , the carnot cop @xmath98 , as well as @xmath18 become large in magnitude . one can then write the series expansion for @xmath18 relative to @xmath98 as follows : @xmath99.\ ] ] in this case , @xmath94 relative to @xmath98 behaves as follows : @xmath100.\ ] ] according to refs . @xcite close to equilibrium and upto the leading order , @xmath101 behaves as @xmath102 . the optimal behavior is thus reproduced by the use of jeffreys prior , but uniform prior is not able to generate this dependence . similarly , for large temperature differences , @xmath103 , we get the limiting behavior as @xmath104 while @xmath105 . the interpolation formula at optimal performance , gives @xmath106 @xcite . before closing this section , we point out that performing the same analysis in terms of @xmath39 as the uncertain scale , we obtain a similar behavior in the asymptotic range of values , and the same figures of merit , @xmath17 and @xmath18 , are obtained with the choice of jeffreys prior . so far , we have focused on the performance of feynman s ratchet . in the following , we wish to point out that the above inference analysis can also be performed on other classes of heat engines / refrigerators @xcite . the model which we discuss below is a four - step heat cycle performed by a few - level quantum system ( working medium ) . further , the cycle is accomplished using infinitely slow processes . the particular cycle is the quantum otto cycle @xcite . consider a quantum system with hamiltonian @xmath107 , with eigenvalue spectrum of the form @xmath108 . here @xmath109 is characterised by the energy quantum number and other parameters / constants which remain fixed during the cycle . we assume there are @xmath110 non - degenerate levels . the parameter @xmath111 represents an external control , equivalent to applied magnetic field for a spin system . initially , the system is in thermal state @xmath112 at temperature @xmath19 , where @xmath113 , @xmath114 , and the partition function @xmath115 . the quantum otto cycle involves the following steps @xcite : \(i ) the system is detached from the hot bath and made to undergo a quantum adiabatic process , in which the external control is slowly changed from the value @xmath111 to @xmath116 . thus the hamiltonian changes from @xmath117 to @xmath118 with eigenvalues @xmath119 . following quantum adiabatic theorem , the system remains in the instantaneous eigenstate of the hamiltonian and so the occupation probabilities of the levels remain unchanged . for @xmath120 , this process is the analogue of an adiabatic expansion . the work done _ by _ the system in this stage is equal to the change in mean energy @xmath121)$ ] . the change in energy spectrum is such that the ratio of energy gaps between any two levels before and after the quantum adiabatic process is the same . this makes it possible to assign temperature to the system along the adiabatic process . thus after step ( i ) , this temperature is given by @xmath122 . \(ii ) the system with changed spectrum @xmath123 is brought to thermal state @xmath124 by contact with cold bath at inverse temperature @xmath125 , where @xmath126 and @xmath127 . on average , the heat rejected to the bath in this step , is defined as @xmath128 h_2)$ ] . \(iii ) the system is now detached from the cold bath and made to undergo a second quantum adiabatic process ( compression ) during which the control is reset to value @xmath111 . work done _ on _ the system in this step is @xmath129)$ ] . \(iv ) finally , the system is put in contact with the hot bath again . heat is absorbed by the system in this step , whence it recovers its initial state @xmath130 . on average , the total work done in one cycle , is calculated to be @xmath131 similarly , heat exchanged with hot bath in step ( iv ) is given by @xmath132 heat exchanged by the system with the cold bath is @xmath133 . the efficiency of the engine @xmath134 , is given by @xmath135 clearly , this cycle has two internal energy scales and the efficiency is also similar to that of feynman s ratchet , eq . ( [ efficiency - ratchet ] ) . one can seek an optimal engine configuration , by optimising work output per cycle over the parameters @xmath111 and @xmath116 . however , unlike the case of feynman s ratchet as engine , a closed - form expression for the efficiency at optimal work seems difficult to obtain here @xcite . we can formulate a problem of estimation here , for performance of the engine , assuming that the absolute magnitudes of internal scales are not known . further , we simplify by assuming that the ratio of energy scales , or in other words , the efficiency is specified . in the following , we briefly outline the emergence of ca efficiency in this problem . the following treatment generalizes the analysis of ref . @xcite . it is convenient to express @xmath136 , using eq . ( [ eta ] ) . due to analogy with the ratchet problem , we may take the prior for the uncertain parameter @xmath111 to be jeffreys prior : @xmath137 , where @xmath138^{-1}$ ] . the expected work per cycle for a given @xmath42 , is then given by @xmath139 to perform the integration , we write @xmath140 and integrate by parts . the result can be written as : @xmath141 thus the average work is evaluated to be @xmath142 , \label{wev}\ ] ] or , which is written briefly as : @xmath143,\ ] ] where @xmath144 and @xmath145 can be easily identified from eq . ( [ wev ] ) . now we wish to find the efficiency at optimal average work , and so we apply the condition @xmath146 the resulting equation is , in general , a function of @xmath147 and @xmath148 . however , we are interested in the asymptotic limit of large @xmath147 and vanishing @xmath148 . in this limit , the dominant term in the sum @xmath149 is given by @xmath150 , where @xmath151 is the ground - state energy . therefore , @xmath152 . similarly , in the said limit @xmath153 finally , using the above limiting forms in eq . ( [ wzero ] ) , we obtain : @xmath154 = 0,\ ] ] which implies that the expected work becomes optimal at @xmath155 , or at ca - efficiency . we observed in feynman s ratchet that for small temperature differences , the figures of merit at optimal values of @xmath66 and @xmath87 , agree with the corresponding expressions at the optimal values of @xmath156 and @xmath1 . the important conditions which hold in this comparison are , jeffreys prior as the underlying prior and an asymptotic range of values over which the prior is defined . in contrast , the uniform prior is not able to generate the optimal behavior in the near equilibrium regime . further we note that for endoreversible models with a newtonian heat flow between a reservoir and the working medium , the efficiency at optimal power is exactly @xmath17 @xcite . correspondingly , the cop at optimal @xmath1-criterion is given by @xmath18 @xcite . in this paper , these values are obtained with an inference based approach assuming incomplete information in a mesoscopic model of heat engine . we have also shown that our analysis applies to a broader class of idealized models of heat engines / refrigerators , driven by quasi - static processes . here also , ca efficiency emerges from the use of jeffreys prior , under the given conditions of the model . we conclude with an argument to support as to why our approach yields the familiar results of finite - time thermodynamics . to exemplify , in the case of feynman s ratchet , the asymptotic range has been considered _ after _ we optimized the expected power output ( in case of engine ) over the efficiency . one may consider these two steps in the opposite order , i.e. take the asymptotic range first and then perform the optimization . for that we rewrite eq . ( [ q1dot ] ) as follows : @xmath157 and define the expected heat flux as @xmath158 then in the asymptotic range , we obtain the approximate expression as @xmath159,\ ] ] where @xmath85 is as in eq . ( [ defc ] ) . here we can draw a parallel with newtonian heat flow : @xmath160 $ ] where @xmath161 is an effective temperature . similarly , the prior - averaged rate of heat rejected to the cold reservoir can be written as @xmath162.\ ] ] here also , we may identify another newtonian heat flow @xmath163 $ ] , with the same effective heat conductance @xmath85 , between an effective temperature @xmath164 and temperature @xmath165 of the cold reservoir . then it is easily seen that the maximum of expected power @xmath166 , is obtained at ca value . similarly , one can argue for the emergence of @xmath18 in the case of refrigerator mode , in terms of effective heat flows which are newtonian in nature . interestingly , the above expressions seem to suggest an analogy between the expected mesoscopic model with limited information , and a finite - time thermodynamic model with newtonian heat flows . if we compare with the endoreversible models @xcite , then we observe that the assumption of a newtonian heat flow goes together with obtaining ca efficiency at maximum power , and cop @xmath18 at optimum @xmath1-criterion . we however note that the analogy does not hold in entirety . the effective temperatures defined above do not have physical counterpart in the ratchet model , although in the endoreversible picture , these denote the temperatures of the working medium while in contact with hot or cold reservoirs . secondly , the heat conductances need not be equal for the endoreversible model with newtonian heat flows . further , the intermediate temperatures @xmath167 and @xmath168 as above , are equal in magnitude at the maximum expected power @xmath66 . however , for the endoreversible model , these temperatures are not equal at maximum power @xcite . still , the form of expressions for the rates of heat transfer do provide a certain insight into the emergence of the familiar expressions for figures of merit at optimal expected performance within the prior - averaged approach . finally , we close with a few observations on future lines of enquiry . it was seen in fig . 1 , that for a specified finite range for the prior , the estimates of efficiency at maximum power are either above , or below the estimates in the asymptotic range . in particular , the estimates are function of the values @xmath57 and @xmath58 . we obtain universal results , dependent on the ratio of reservoir temperatures , only in the asymptotic range . further , the smaller values of the upper limit , overestimate the efficiency ( inset in fig . 1 ) whereas the larger values of the lower limit , underestimate the efficiency . an opposite behavior is seen for the refrigerator mode ( fig . moreover , this trend for a chosen mode ( engine / refrigerator ) is specific to the choice of the uncertain variable . thus the trend is reversed , if instead of choosing @xmath21 , we perform the analysis with @xmath39 as the uncertain variable . this behavior is seen in both the engine as well as the refrigerator mode . investigation into the relation between inferences derived from the two choices for the uncertain variable , may yield further insight into the behavior of estimated performance and the approach in general . the point may be appreciated by noting that by specifying a finite - range for the prior we add new information to the probabilistic model . in order that inference may provide a useful and practical guess on the actual performance of the device , this additional prior information has to be related to some objective features of the model . these considerations are relevant for further exploring the intriguing relation between the subjective and the objective descriptions of thermodynamic models @xcite . the authors acknowledge financial support from the department of science and technology , india under the research project no . sr / s2/cmp-0047/2010(g ) , titled : `` quantum heat engines : work , entropy and information at the nanoscale '' . y. wang , m. li , z. c. tu , a. c. hernndez , j. m. m. roco , coefficient of performance at maximum figure of merit and its bounds for low - dissipation carnot - like refrigerators , phys . e 86 ( 2012 ) 011127 . y. hu , f. wu , y. ma , j. he , j. wang , a. c. hernndez , j. m. m. roco , coefficient of performance for a low - dissipation carnot - like refrigerator with nonadiabatic dissipation , phys . e 88 ( 2013 ) 062115 .

we estimate the performance of feynman s ratchet at given values of the ratio of cold to hot reservoir temperatures ( @xmath0 ) and the figure of merit ( efficiency in the case of engine and coefficienct of performance in the case of refrigerator ) . the latter implies that only the ratio of two intrinsic energy scales is known to the observer , but their exact values are completely uncertain . the prior probability distribution for the uncertain energy parameters is argued to be jeffreys prior . we define an average measure for performance of the model by averaging , over the prior distribution , the power output ( heat engine ) or the @xmath1-criterion ( refrigerator ) which is the product of rate of heat absorbed from the cold reservoir and the coefficient of performance . we observe that the figure of merit , at optimal performance close to equilibrium , is reproduced by the prior - averaging procedure . further , we obtain the well - known expressions of finite - time thermodynamics for the efficiency at optimal power and the coefficient of performance at optimal @xmath1-criterion , given by @xmath2 and @xmath3 respectively . this analogy is explored further and we point out that the expected heat flow from and to the reservoirs , behaves as an effective newtonian flow . we also show , in a class of quasi - static models of quantum heat engines , how ca efficiency emerges in asymptotic limit with the use of jeffreys prior .

You are an expert at summarizing long articles. Proceed to summarize the following text: in cold dark matter cosmology , the initially smooth distribution of matter in the universe is expected to collapse into a complex network of filaments and voids , structures which have been termed the `` cosmic web '' . the filamentary distribution of galaxies in the nearby universe has been revealed in detail by recent large galaxy redshift surveys such as the 2dfgrs ( colless et al . 2001 , baugh et al . 2004 ) , the sloan digital sky survey ( sdss , stoughton et al . 2002 , doroshkevich et al . 2004 ) and the 2@xmath19 all sky survey ( 2mass , maller et al . numerical simulations successfully reproduce this network ( jenkins et al . 1998 ; colberg et al . 2004 ) and indicate that galaxies are only the tip of the iceberg in this cosmic web ( katz et al . 1996 ; miralda - escud et al . hydrodynamic simulations suggest that at the present epoch , in addition to dark matter and galaxies , the filaments are also composed of a mixture of cool , photoionised gas ( the low@xmath0 remnants of the forest ) and a shock heated , low - density gaseous phase at temperatures between @xmath20 k and @xmath21 k that contains most of the baryonic mass , the `` warm - hot '' intergalactic medium ( whim , cen & ostriker 1999 ; dav et al . 1999 ) . observational constraints on the physical conditions , distribution , a nd metal enrichment of gas in the low - redshift cosmic web are currently quite limited . the existence of the whim appears to be a robust prediction of cosmological simulations ( dav et al . thus , observational efforts are increasingly being invested in the search for whim gas and , more generally , the gaseous filamentary structures predicted by the models . large - scale gaseous filaments have been detected in x - ray emission ( wang et al . 1997 ; scharf et al . 2000 ; tittley & henriksen 2001 ; rines et al 2001 ) . however , x - ray emission studies with current facilities predominantly reveal gas which is hotter and denser than the whim ; this x - ray emitting gas is not expected to contain a substantial portion of the present - epoch baryons ( dav et al . the most promising method for observing the whim in the near term is to search for uv ( o@xmath4roman6 , ne@xmath4roman8 ) and x - ray ( o@xmath4roman7 , o@xmath4roman8 , ne@xmath4roman9 ) absorption lines due to whim gas in the spectra of background qsos / agns ( tripp et al . 2000 , 2001 ; savage et al . 2002,2005 ; nicastro et al . 2002 ; bergeron et al . 2002 ; richter et al . 2004 ; sembach et al . 2004 ; prochaska et al . 2004 ; danforth & shull 2005 ) . while absorption lines provide a sensitive and powerful probe of the whim , the pencil - beam nature of the measurement along a sight line provides little information on the context of the absorption , e.g. , whether the lines arise in an individual galaxy disk / halo , a galaxy group , or lower - density regions of a large - scale filament or void . thus , to understand the nature of highly ionised absorbers at low redshifts , several groups are pursuing deep galaxy redshift surveys and observations of qsos behind well - defined galaxy groups or clusters . for example , to study gas located in large - scale filaments , bregman et al . ( 2004 ) have searched for absorption lines indicative of the whim in regions between galaxy clusters / superclusters and have identified some candidates . in this paper , we carry out a similar search as part of a broader program that combines a large _ hst _ survey of low@xmath0 o@xmath4roman6 absorption systems observed on sight lines to low@xmath0 quasars ( tripp et al . 2004 ) and a ground based survey to measure the redshifts and properties of the galaxies foreground to the background qsos . the ground based survey is done in two steps : first , multi - band ( u , b , v , r and i ) imagery is obtained to identify the galaxies and to estimate their photometric redshifts . then , spectroscopic redshifts are obtained for the galaxies that are potentially ( according to the photometric redhshifts ) at lower redshift that the background object . as part of the large _ hst _ survey , we have observed the quasar hs0624 + 6907 ( @xmath22 = 0.3700 ) with the e140 m echelle mode of the space telescope imaging spectrograph ( stis ) on board the _ hubble space telescope_. we have also obtained multiband images and spectroscopic redshifts of galaxies in the 0624 field . the sight line to 0624 passes by several foreground abell clusters ( [ sec : abell_clusters ] ) and provides an opportunity to search for gas in large - scale filaments . we shall show that gas ( absorption systems ) and galaxies are detected at the redshifts of the structures delineated by the abell clusters in this direction . while the absorbing gas is intergalactic , and it is likely that we are probing gas in cosmic web filaments , the properties of these absorbers are surprising . instead of low - metallicity whim gas , we predominantly find cool , photoionised , and high - metallicity gas in these large - scale structures . this paper is organized as follows . the observations and data reduction procedures are described in 2 , including _ hst_/stis and _ far ultraviolet spectroscopic explorer _ ( ) observations as well as ground - based imaging and galaxy redshift measurements . in 3 , we present information on the foreground environments probed by the 0624 sight line , derived from the literature on abell clusters and from our new galaxy redshift survey . the absorption - line measurement methods are described in 4 , and we investigate the physical state and metallicity of the absorbers in 5 . section 6 reviews the properties of the full sample of ly@xmath16 lines derived from the stis spectrum with emphasis on the search for broad ly@xmath16 lines . section 7 discusses the implications of this study , and we summarize our conclusions in 8 . throughout this paper , we use the following cosmological parameters : @xmath23 , @xmath24 and @xmath25 . 0624 was observed with stis on 2 jan . 2002 and 23 - 24 feb . 2002 as part of a cycle 10 _ hst _ observing program ( id=9184 ) . the echelle spectrograph was used with the e140 m grating which provides a resolution of 7 fwhm and covers the 1150@xmath261730 range with only a few small gaps between orders at wavelengths greater than 1630 . the @xmath27 entrance aperture was used to minimize the effect of the wings of the line spread function . the total exposure time was 61.95 ksec . the data were reduced as described in tripp et al . ( 2001 ) using the stis team version of calstis at the goddard space flight center . the final signal - to - noise ( s / n ) per resolution element is 3 at 1150 , increases linearly to 14 at 1340 and then decreases to 7 at 1730 . for further information on the design and performance of stis , see woodgate et al . ( 1998 ) and kimble et al . ( 1998 ) . hs0624 + 6907 was also observed by the pi team on several occasions between 1999 november and 2002 february ( program ids p1071001 , p1071002 , s6011201 , and s6011202 ) . records spectra with four independent spectrographs ( `` channels '' ) , two with sic coatings for coverage of the 905@xmath261105 wavelength range , and two with lif coatings optimized to cover 1100@xmath261187 ( see moos et al . 2000,2002 for details about design and performance ) . the spectrograph resolutions range from 20@xmath2630 km s@xmath28 ( fwhm ) . for hs0624 + 6907 , the total integration time in the lif1 channel was 110 ksec ; the other channels had somewhat lower integration times due to channel coalignment problems during some of the observations . we have retrieved the spectra from the archive and have reduced the data using calfuse version 2.4.0 as described in tripp et al . ( 2005 ) . because the spectra in the individual channels have modest s / n ratios , we have aligned and combined all available lif channels to form the final spectra that we used for our measurements ( we find that combining all available lif data does not degrade the spectral resolution ) . for the spectral range uniquely covered by the sic channels , we used only the sic2a data . finally , we compared absorption lines of comparable strength ( e.g. , fe@xmath4roman2 @xmath291144.94 vs. fe@xmath4roman2 @xmath291608.45 ) observed by _ fuse _ and stis in order to align the _ fuse _ spectrum with the stis spectrum and thereby correct the wavelength zero point of the data . one of the primary goals of our low@xmath0 qso absorption line program is to study the connections between galaxies and absorption systems . these studies require good imaging ( for galaxy target selection and information on individual galaxies of interest ) followed by optical spectroscopy for accurate redshift measurements . to initiate the galaxy - absorber study toward 0624 , we first obtained a @xmath30 mosaic of images centered on the qso with spicam on the apache point observatory ( apo ) 3.5 m telescope on 2002 october 5 . subsequently , we obtained images of a larger field in better seeing with the noao 8k@xmath318k ccd mosaic camera ( mosa , muller et al . 1998 ) , on the kitt peak national observatory ( kpno ) 4 m telescope . the spicam images were used to select targets for the first spectroscopic observing run , but thereafter we only used the better - quality mosa images . [ cols="<,^,^,^,^,^ " , ] 0624 was observed with mosa on the 4 m on 2003 january 29 - 30 . the field of view is @xmath32 with a scale of @xmath33/pixel . as summarized in table [ tab : obslog ] , images were recorded in @xmath34 and @xmath35 with a standard dithering pattern for filling in gaps between the ccds and for rejection of cosmic rays . photometric standard stars from landolt ( 1992 ) were also observed at regular intervals . during these observations , the seeing ranged from @xmath360 to 1@xmath373 . the data were reduced with the iraf software package mscred following standard procedures . the final r - band mosa image of 0624 is shown in figures [ fig : field ] and [ fig : fieldzoom ] . galaxy targets for follow - up spectroscopy were selected from the images using the sextractor software package ( bertin & arnouts 1996 ) . redshifts of 29 galaxies were obtained using the double imaging spectrograph ( dis ) on the apo 3.5 m telescope on the following dates : 2002 november 12 , 2003 january 29@xmath2631 , 2003 april 03 , 2003 april 21 , and 2003 december 25 . spectra were recorded using a single 1.5 arcsec wide slit with total exposure times ranging from 360 to 1800 s per object . the data were processed in the conventional manner , and were wavelength calibrated using helium - neon - argon arc - lamp exposures . small zero - point offsets in wavelength were applied as needed , after comparing observed skyline wavelengths with their rest values . the spectra were typically recorded at resolutions of @xmath38 fwhm . llccccccccc & & & & & & & & & + & & & & & & & & & & & qso & 06:30:02.50 & 69:05:03.99 & 0.3700 & 0.0 & 0.000 & 13.8 & 14.2 & 0.1 & -27.7 + 01 & se12 & 06:30:41.70 & 68:58:32.71 & 0.0327 & 7.4 & 0.290 & 16.6 & 17.8 & 1.0 & -19.2 + 02 & se3 & 06:30:55.32 & 69:02:41.99 & 0.0424 & 5.3 & 0.265 & 19.6 & 20.5 & 0.9 & -16.7 + 03 & ne1 & 06:30:56.14 & 69:08:00.90 & 0.0547 & 5.6 & 0.358 & 15.9 & 17.0 & 0.9 & -21.0 + 04 & nw2 & 06:29:46.66 & 69:08:03.59 & 0.0560 & 3.3 & 0.216 & 18.6 & 19.5 & 0.9 & -18.4 + 05 & se4 & 06:30:33.00 & 68:53:02.00 & 0.0622 & 12.3 & 0.887 & 16.8 & 17.9 & 0.9 & -20.5 + 06 & se5 & 06:32:55.20 & 68:56:59.99 & 0.0637 & 17.4 & 1.282 & 16.8 & 18.3 & 1.3 & -20.5 + 07 & se8 & 06:31:01.79 & 68:57:35.89 & 0.0638 & 9.2 & 0.675 & 16.0 & 17.1 & 0.9 & -21.2 + 08 & se1 & 06:30:11.22 & 69:02:09.61 & 0.0640 & 3.0 & 0.222 & 18.8 & 19.4 & 0.7 & -18.5 + 09 & sw3 & 06:29:07.80 & 69:03:32.01 & 0.0650 & 5.1 & 0.384 & 17.4 & 18.3 & 0.9 & -20.0 + 10 & se13 & 06:30:58.30 & 69:04:34.11 & 0.0650 & 5.0 & 0.375 & 16.9 & 18.5 & 1.3 & -20.4 + 11 & se6 & 06:32:50.70 & 68:56:03.00 & 0.0652 & 17.6 & 1.318 & 15.9 & 17.3 & 1.2 & -21.5 + 12 & ne3 & 06:30:21.40 & 69:05:39.70 & 0.0655 & 1.8 & 0.135 & 16.6 & 18.1 & 1.3 & -20.8 + 13 & nw11 & 06:29:23.48 & 69:22:43.29 & 0.0660 & 18.0 & 1.367 & 16.3 & 17.9 & 1.3 & -21.1 + 14 & se7 & 06:32:49.20 & 68:56:00.39 & 0.0664 & 17.5 & 1.334 & 17.0 & 18.0 & 0.7 & -20.4 + 15 & ne2 & 06:32:25.55 & 69:20:05.81 & 0.0733 & 19.7 & 1.646 & 16.5 & 18.0 & 1.3 & -21.1 + 16 & nw1 & 06:29:43.65 & 69:09:35.33 & 0.0760 & 4.8 & 0.417 & 15.9 & 17.3 & 1.1 & -21.8 + 17 & sw2 & 06:28:33.03 & 68:59:26.30 & 0.0763 & 9.8 & 0.849 & 16.6 & 18.3 & 1.3 & -21.1 + 18 & se9 & 06:30:14.81 & 68:49:44.79 & 0.0764 & 15.4 & 1.334 & 16.4 & 18.0 & 1.3 & -21.3 + 19 & nw12 & 06:29:11.59 & 69:07:07.89 & 0.0764 & 5.0 & 0.433 & 16.4 & 18.0 & 1.4 & -21.3 + 20 & se10 & 06:32:52.40 & 68:57:59.01 & 0.0764 & 16.8 & 1.457 & 16.5 & 18.1 & 1.3 & -21.2 + 21 & nw3 & 06:29:53.77 & 69:08:20.51 & 0.0766 & 3.4 & 0.293 & 18.3 & 19.6 & 1.1 & -19.4 + 22 & sw1 & 06:29:33.24 & 69:05:01.00 & 0.0903 & 2.6 & 0.264 & 17.1 & 18.2 & 0.9 & -21.0 + 23 & se11 & 06:30:06.84 & 68:52:22.20 & 0.1001 & 12.7 & 1.407 & 16.5 & 18.1 & 1.2 & -21.9 + 24 & nw7 & 06:26:43.70 & 69:14:06.91 & 0.1009 & 19.9 & 2.215 & 16.7 & 18.2 & 1.4 & -21.6 + 25 & nw9 & 06:29:03.89 & 69:17:33.40 & 0.1108 & 13.5 & 1.639 & 16.4 & 18.2 & 1.3 & -22.2 + 26 & nw4a & 06:29:35.43 & 69:07:25.80 & 0.1125 & 3.4 & 0.415 & 18.5 & 20.2 & 1.2 & -20.1 + 27 & nw8 & 06:28:29.39 & 69:17:28.89 & 0.1126 & 14.9 & 1.832 & 16.7 & 18.5 & 1.3 & -21.9 + 28 & nw10 & 06:29:05.25 & 69:17:46.69 & 0.1129 & 13.7 & 1.685 & 18.1 & 19.9 & 1.3 & -20.5 + 29 & nw6 & 06:29:35.30 & 69:09:44.99 & 0.1429 & 5.3 & 0.794 & 19.4 & 20.4 & 0.8 & -19.8 + 30 & se2 & 06:30:44.35 & 69:01:08.09 & 0.1664 & 5.4 & 0.927 & 19.3 & 20.8 & 1.0 & -20.2 + 31 & nw5 & 06:29:43.52 & 69:09:19.01 & 0.2061 & 4.6 & 0.927 & 18.4 & 20.3 & 1.4 & -21.7 + 32 & nw4b & 06:29:35.62 & 69:07:37.91 & 0.3008 & 3.5 & 0.940 & 21.0 & 22.2 & 0.4 & -19.9 the redshift measurements were made following the procedure described by jenkins et al . we used the iraf routine fxcor to cross - correlate the galaxy spectra with that of the radial velocity standard hd 182572 . in general we only used the blue channel dis data for the cross - correlation , where the 4000 break and stellar absorption lines were most apparent . red channel data were usually used to identify and measure the wavelengths of redshifted emission lines ( [ o iii ] , h@xmath39 , h@xmath16 , etc . ) when present . the galaxy redshifts obtained in this way are summarized in table [ tab : spec_red ] and are accurate to between 70 and 170 ( which corresponds to a sight line distance displacement uncertainty of 1.0 to 2.4 mpc for an unperturbed hubble flow ) . we also observed three galaxies with the echellette spectrometer and imager ( esi ; sheinis et al . 2001 ) on the 10 m keck ii telescope on the nights of 2004 september 10 and 11 during morning twilight . we observed galaxy ne3 ( see table [ tab : spec_red ] ) in echellette mode with the 0.5@xmath40 slit which provides @xmath41 spectral resolution ( fwhm ) . the fainter se13 and sw3 galaxies were observed in low dispersion mode using a 1@xmath40 slit which affords @xmath42 at @xmath43 . the exposures were flat fielded and wavelength calibrated with the esiredux package ( prochaska et al . 2003xavier / esi / index.html ] ) . the ne3 redshift was derived from the centroids of the high - resolution na@xmath4roman1 and h@xmath16 absorption lines , and the redshift uncertainty is @xmath4430 km s@xmath28 . for se13 and sw3 , redshifts were measured by fitting na@xmath4roman1 , h@xmath39 , and ca@xmath4roman2 h and k , and the uncertainties are @xmath44150 km s@xmath28 . the completeness of our galaxy redshift survey ( i.e. , the percentage of targets brighter than a given magnitude in the sextractor galaxy catalog with good spectroscopic redshifts ) is graphically summarized in figure [ fig : completeness ] as a function of limiting @xmath45 magnitude and angular separation from the sight line ( @xmath46 ) . in the @xmath47 region centered on 0624 , we have measured spectroscopic redshifts for all galaxies brighter than @xmath45 = 19.0 , and the survey is @xmath48 per cent complete for @xmath49 . as we shall see , there is a prominent cluster of absorption lines in the 0624 spectrum at @xmath50 0.0635 ; at this redshift , 5@xmath51 corresponds to a projected distance of 367 @xmath7 kpc , and @xmath45 = 19.0 corresponds to @xmath52 or @xmath53 ( taking @xmath54 from lin et al . 1996 ) . for comparison , the large magellanic cloud has a magnitude equal to @xmath55 or @xmath56 . at this redshift , we have good completeness even for low luminosity galaxies . at larger radii , a substantial number of bright galaxies are found , and our redshift survey is shallower . nevertheless , within a 10 radius circle , our survey is still 60 per cent complete for galaxies brighter than @xmath57 . using information gleaned from the literature in combination with our galaxy redshift survey , we can identify several large - scale structures that are pierced by the 0624 sight line . in this section we comment on these structures including nearby abell clusters ( [ sec : abell_clusters ] ) as well as smaller ( and closer ) galaxy groups ( [ sec : close_groups ] ) . clusters are clustered and often reveal even larger cosmic structures , i.e. , superclusters ( einasto et al . 2001 and references therein ) . in cosmological simulations , clusters are found at the nodes where large - scale filaments connect . to test the fidelity of cosmological simulations , which are now being used in a wide variety of astrophysical analyses , it is important to search for observational evidence of the expected _ gaseous _ filaments feeding into clusters and to measure the properties of the filaments . the sight line to 0624 passes through a region of relatively high abell cluster density and is well - suited for investigation of this topic . shows the positions of abell clusters around the sight line to 0624 , including the cluster richness class and spectroscopic redshift ( when available from the literature ) . the density of abell clusters in this region is relatively high compared to the vicinity of the other clusters in the abell catalog : the number of abell clusters within 2@xmath58 ( 3@xmath59 ) of a557 ( the cluster closest to the 0624 sight line ) is 2 ( 1.6 ) times larger than the average number within 2@xmath60 ( 3@xmath59 ) of all abell clusters . einasto et al . ( 2001 ) have identified two superclusters in the direction of 0624 . their supercluster scl71 ( at @xmath61 ) includes a554,a557 , a561 , a562 , and a565 while scl72 ( at @xmath62 ) includes a559 and a564 . however , a557 and a561 do not have spectroscopic redshifts , and based on our spectroscopic redshifts in the field of hs0624 + 6907/abell557 ( see table [ tab : spec_red ] ) , it appears likely that the visually identified a557 is a false cluster due to the superposition of galaxy groups at several different redshifts . to be conservative , we only use clusters with spectroscopic redshifts to identify large - scale structures . the clusters at ( a564 and a559 ) are separated by 4.7 from each other , and the clusters at ( a565 , a562 and a554 ) are separated by 3.9 and 8.7 . according to colberg et al . ( 2004 ) , in cosmological simulations , more than 85 per cent of the clusters with a separation lower than 10 are connected with a filament . we will show in subsequent sections that both absorption lines in the spectrum of 0624 and galaxies close to the sight line are detected at the redshifts of both of these abell cluster structures , which indicates that gaseous filaments connect into the clusters . in this section we offer some brief comments about specific galaxies and galaxy groups close to the sight line of 0624 , as revealed by our optical spectroscopy . we place these observations in the context of the abell clusters described in the previous section . we plot in the redshift distribution of the galaxies from table [ tab : spec_red ] . from this figure we can identify three galaxy groups : two galaxy groups appear to be present at redshifts similar to those of the abell clusters , i.e , at @xmath63 ( 7 galaxies ) and @xmath64 ( 6 galaxies ) . this indicates that the filament of galaxies connecting a559 to a564 must extend more then 3 degrees ( 15 ) west from a559 and structure linking a562 and a554 likely extends by at least 3 degrees ( 22 ) in order to cross the qso sight line . however , the most prominent group in includes 10 galaxies at , which does not match up with any abell cluster with a known spectroscopic redshift . to show the spatial distribution of galaxies in the three prominent groups in the mosa field , provides a colour - coded map of projected spatial coordinates of the galaxies . we see that the group at @xmath65 is mostly southeast of the sight line while the galaxies associated with abell 554/562/565 at are predominantly northwest of 0624 . the galaxies associated with the abell 559/564 supercluster appear to extend from the southwest across the sight line to the northeast . we note that spectroscopic redshifts are not available for several of the clusters shown in fig . [ fig : abell_map ] , including the one that is closest to the sight line , a557 . however , abell clusters are visually identified without the aid of spectroscopy , and it can be seen from figures [ fig : field ] and [ fig : gal_colour ] that several discrete groups are found at the location of a557 . it is likely that a557 is not a true cluster but rather is the superposition of several groups in projection . is gas also present in these large - scale cosmic filaments ? to address this question , we searched the spectrum of 0624 for any absorption counterparts at the redshifts of galaxies and galaxy structures near the qso sight line . the redshifts of the h@xmath4roman1 systems that we have identified and measured ( see [ sec : absline ] ) in the spectrum of 0624 are plotted at the top of ; the length of the line is proportional to the rest equivalent width of the line . this search has revealed three interesting results : first , when a galaxy is located at an impact parameter @xmath66 from the sight line , absorption is almost always found within a few hundred of the galaxy redshift ( compare table [ tab : spec_red ] to table [ tab : lyalist ] ) , consistent with the findings of previous studies ( e.g. , lanzetta et al . 1995 ; tripp et al . 1998 ; impey et al . 1999 ; chen et al . 2001 ; bowen et al . 2002 ; penton et al . 2002 ) . second , strong absorption is clearly detected at the redshift of the @xmath67 abell 564/559 supercluster . this absorption system is detected in , , and the c@xmath4roman4 @xmath68 1548.20 , 1550.78 doublet ( [ ss : z076 ] ) , and the absorption redshift is very similar to that of a559 . weak absorption is also detected at @xmath13 = 0.10822 , which is within 500 of the abell 554/562/565 filament . evidently , and not surprisingly , gas is also found in the filaments that feed into the clusters near 0624 . third , figure [ fig : zgaldist ] qualitatively indicates that the strongest lines are situated in the regions of highest galaxy density , which is similar to the conclusions of bowen et al . ( 2002 ) and c^ ot et al . ( 2005 ) . could these absorbers simply arise in the halos of individual galaxies ? as we show in the next section , the strong h@xmath4roman1 system at is comprised of a large number of components spread over 1000 . such kinematics are unprecedented in single galaxy halos . in addition , we find no obvious pattern that shows a connection between individual lines and individual galaxies in this complex . in the case of this strong h@xmath4roman1 system at , the closest observed galaxy to the sight line around this redshift has @xmath69 ( ne3 in table [ tab : spec_red ] ) . however , a closer and fainter galaxy could have been missed by the spectroscopic survey . using photometric redshifts ( measured as described in chen et al . 2003 ) to cull the distant background galaxies with photometric redshifts @xmath70 , we find that there are only four galaxies closer to the sight line than ne3 that could be near . these four objects are only candidates since photometric redshifts have substantial uncertainties . however , if we assume the redshift of these candidates to be @xmath71 , the closest one to 0624 has still a large impact parameter @xmath72 kpc . in the case of at @xmath13 = 0.07573 , the closest galaxy in projection is nw3 at @xmath73 kpc . it is conceivable that this absorption originates in the large halo of this particular galaxy , but we note that three galaxies are found at @xmath74 kpc at this @xmath75 , and many other origins are possible ( e.g. , intragroup gas or tidally stripped debris ) . finally , the absorption at @xmath76 is at a substantial distance ( 415 ) from the nearest known galaxy ( nw4a ) and is unlikely to be halo gas associated with that object . we now turn to the absorption - line measurements . as discussed in the previous section , figure [ fig : zgaldist ] compares the distributions of galaxies and ly@xmath16 lines in the direction of 0624 . we have measured the column densities and doppler parameters of the ly@xmath16 lines in the spectrum of 0624 using the voigt profile decomposition software vpfit ( see webb 1987rfc / vpfit.html ] ) . table [ tab : lyalist ] summarizes the h@xmath4roman1 equivalent widths , column densities , and @xmath77values measured in this fashion ( some of the lines are strongly saturated and consequently voigt profile fitting does not provide reliable measurements ; we discuss our treatment of these cases below ) . a particularly dramatic cluster of ly@xmath16 lines is evident at @xmath78 0.0635 in figure [ fig : zgaldist ] . the portion of the stis spectrum of hs0624 + 6907 covering this ly@xmath16 cluster is shown in figure [ fig : hi00635 ] . to avoid confusion with galaxy clusters , we will hereafter refer to this group of lines as a `` complex '' . this complex contains at least 13 h@xmath4roman1 components spread over a velocity range of 1000 km s@xmath28 . we will refer to the strongest ly@xmath16 absorption in figure [ fig : hi00635 ] at @xmath13 = 0.06352 as `` component a '' . component a is detected in absorption in the h@xmath4roman1 ly@xmath16 , ly@xmath39 , and ly@xmath79 transitions as well as the si@xmath4roman3 @xmath291206.50 , si@xmath4roman4 @xmath681393.76 , 1402.77 , and c@xmath4roman4 @xmath681548.20 , 1550.78 lines . low ionisation stages such as o@xmath4roman1 , c@xmath4roman2 , and si@xmath4roman2 are not detected at the redshift of component a or at the redshifts of any of the other components evident in figure [ fig : hi00635 ] . the o@xmath4roman6 doublet at the redshifts of the ly@xmath16 cluster in falls in a region that is partially blocked by galactic h@xmath80 and fe@xmath4roman2 lines . nevertheless , much of the region is free from blending , and we find no evidence for o@xmath4roman6 absorption . we also do not see the n@xmath4roman5 doublet . lccl|lccl & @xmath81 & @xmath82 & @xmath83 & & @xmath81 & @xmath82 & @xmath83 + @xmath75 & ( m ) & ( ) & ( ) & @xmath75 & ( m ) & ( ) & ( ) 0.017553@xmath84 1.0e-5 & 45@xmath8410 & 12.96@xmath840.05 & 29@xmath84 4.3 & 0.207540@xmath84 0.5e-5 & 150@xmath84 9 & 13.48@xmath840.02 & 27@xmath84 1.5 + 0.030651@xmath84 0.4e-5 & 99@xmath84 9 & 13.36@xmath840.03 & 22@xmath84 1.7 & 0.213232@xmath84 1.6e-5 & 98@xmath8414 & 13.22@xmath840.05 & 45@xmath84 5.6 + 0.041156@xmath84 0.8e-5 & 104@xmath8411 & 13.33@xmath840.03 & 41@xmath84 3.0 & 0.219900@xmath84 2.3e-5 & 143@xmath8415 & 13.39@xmath840.05 & 60@xmath84 8.6 + 0.053942@xmath84 0.6e-5 & 85@xmath84 7 & 13.26@xmath840.04 & 24@xmath84 2.3 & 0.223290@xmath84 0.3e-5 & 256@xmath8412 & 13.86@xmath840.02 & 25@xmath84 0.9 + 0.054367@xmath84 4.1e-5@xmath85 & 65@xmath8413 & 13.09@xmath840.11 & 60@xmath8419.2@xmath86 & 0.232305@xmath84 2.8e-5@xmath87 & 125@xmath8413 & 13.33@xmath840.08 & 44@xmath84 7.7@xmath86 + 0.054829@xmath88 & 458@xmath8410 & @xmath4414.5@xmath89 & @xmath4435@xmath89 & 0.232547@xmath84 2.3e-5@xmath87 & 44@xmath8410 & 12.86@xmath840.21 & 24@xmath84 7.3 + 0.055153@xmath84 7.8e-5@xmath87 & 237@xmath8414 & 13.68@xmath840.17 & 84@xmath8430.7@xmath86 & 0.240599@xmath84 0.6e-5 & 110@xmath8410 & 13.33@xmath840.04 & 20@xmath84 2.0 + 0.061879@xmath84 0.4e-5 & 184@xmath84 6 & 13.77@xmath840.03 & 21@xmath84 1.4 & 0.252251@xmath84 1.2e-5 & 55@xmath8411 & 12.96@xmath840.06 & 24@xmath84 4.2 + 0.062014@xmath84 1.0e-5 & 21@xmath84 4 & 12.63@xmath840.17 & 8@xmath84 4.7 & 0.268559@xmath84 2.1e-5 & 68@xmath8414 & 13.03@xmath840.05 & 51@xmath84 7.2 + 0.062150@xmath84 1.5e-5@xmath87 & 13@xmath84 4 & 12.41@xmath840.22 & 10@xmath84 7.9 & 0.272240@xmath84 0.6e-5 & 37@xmath84 8 & 12.80@xmath840.06 & 12@xmath84 2.2 + 0.062343@xmath84 0.8e-5@xmath87 & 128@xmath84 7 & 13.45@xmath840.05 & 30@xmath84 4.0 & 0.279771@xmath84 1.7e-5@xmath87 & 174@xmath8413 & 13.50@xmath840.06 & 34@xmath84 4.9 + 0.062647@xmath84 2.5e-5@xmath87 & 101@xmath84 7 & 13.31@xmath840.14 & 35@xmath8412.3 & 0.280171@xmath84 0.7e-5@xmath87 & 576@xmath8415 & 14.32@xmath840.02 & 43@xmath84 1.9@xmath86 + 0.062762@xmath84 0.7e-5@xmath87 & 39@xmath84 3 & 12.95@xmath840.28 & 8@xmath84 3.7 & 0.295307@xmath84 0.7e-5 & 309@xmath8415 & 13.80@xmath840.02 & 42@xmath84 2.0 + 0.062850@xmath84 1.2e-5@xmath87 & 110@xmath84 5 & 13.42@xmath840.14 & 20@xmath84 7.0 & 0.296607@xmath84 0.9e-5 & 203@xmath8418 & 13.54@xmath840.02 & 52@xmath84 2.9 + 0.063037@xmath84 1.4e-5@xmath87 & 101@xmath84 6 & 13.33@xmath840.13 & 27@xmath84 8.8 & 0.308991@xmath84 0.6e-5 & 167@xmath8412 & 13.49@xmath840.03 & 28@xmath84 1.8 + 0.063456@xmath84 1.6e-5@xmath87 & 569@xmath84 9 & 14.46@xmath840.30 & 48@xmath84 8.4@xmath86 & 0.309909@xmath84 5.5e-5@xmath87 & 246@xmath8418 & 13.61@xmath840.10 & 66@xmath8412.3@xmath86 + 0.063481@xmath84 1.6e-5@xmath87 & 443@xmath84 6 & 15.27@xmath840.13 & 24@xmath84 5.5 & 0.310454@xmath84 8.0e-5@xmath87 & 170@xmath8417 & 13.43@xmath840.33 & 62@xmath8440.3@xmath86 + 0.063620@xmath84 2.7e-5@xmath87 & 153@xmath84 4 & 14.29@xmath840.38 & 10@xmath84 5.6 & 0.310881@xmath8414.4e-5 & 88@xmath8416 & 13.13@xmath840.43 & 51@xmath8427.7 + 0.064753@xmath84 0.9e-5@xmath87 & 257@xmath84 9 & 13.87@xmath840.04 & 33@xmath84 3.0 & 0.312802@xmath84 4.4e-5 & 257@xmath8418 & 13.65@xmath840.10 & 54@xmath84 9.3 + 0.065016@xmath84 0.8e-5@xmath87 & 282@xmath84 8 & 13.97@xmath840.04 & 31@xmath84 2.7 & 0.313028@xmath84 1.6e-5 & 72@xmath8410 & 13.09@xmath840.24 & 17@xmath84 6.8 + 0.075731@xmath84 0.2e-5 & 292@xmath84 8 & 14.18@xmath840.03 & 24@xmath84 0.8 & 0.313261@xmath84 4.7e-5 & 244@xmath8419 & 13.62@xmath840.10 & 55@xmath8410.9 + 0.090228@xmath84 4.2e-5 & 106@xmath8412 & 13.29@xmath840.08 & 76@xmath8413.7 & 0.317901@xmath84 1.2e-5 & 139@xmath8418 & 13.37@xmath840.04 & 34@xmath84 3.6 + 0.130757@xmath84 1.0e-5 & 114@xmath84 9 & 13.34@xmath840.04 & 34@xmath84 3.6 & 0.320889@xmath84 0.4e-5 & 349@xmath8414 & 13.97@xmath840.02 & 31@xmath84 1.2 + 0.135966@xmath84 3.9e-5 & 119@xmath8411 & 13.33@xmath840.10 & 57@xmath8410.7 & 0.327245@xmath84 5.0e-5@xmath87 & 316@xmath8421 & 13.73@xmath840.32 & 69@xmath8415.6@xmath86 + 0.160541@xmath84 5.0e-5@xmath87 & 69@xmath84 7 & 13.08@xmath840.21 & 34@xmath8410.3 & 0.327721@xmath8438.7e-5@xmath87 & 264@xmath8426 & 13.61@xmath840.43 & 115@xmath8462.1@xmath86 + 0.160744@xmath84 1.0e-5@xmath87 & 200@xmath84 7 & 13.66@xmath840.05 & 30@xmath84 2.4 & 0.332674@xmath84 1.1e-5 & 202@xmath8418 & 13.55@xmath840.04 & 38@xmath84 3.4 + 0.199750@xmath84 0.6e-5 & 87@xmath84 9 & 13.24@xmath840.05 & 17@xmath84 2.0 & 0.339759@xmath84 0.3e-5 & 647@xmath8413 & 14.45@xmath840.03 & 42@xmath84 1.3 + 0.199946@xmath84 1.2e-5@xmath87 & 83@xmath8411 & 13.17@xmath840.06 & 26@xmath84 4.6 & 0.346824@xmath84 0.6e-5 & 221@xmath8416 & 13.59@xmath840.02 & 39@xmath84 1.9 + 0.204831@xmath84 0.3e-5 & 208@xmath84 9 & 13.72@xmath840.02 & 24@xmath84 1.0 & 0.348645@xmath84 0.9e-5 & 40@xmath8410 & 12.78@xmath840.06 & 18@xmath84 3.0 + 0.205326@xmath84 0.2e-5 & 322@xmath84 8 & 14.12@xmath840.03 & 25@xmath84 0.8 figure [ fig : syst0063 ] compares the absorption profiles of the ly@xmath16 , ly@xmath39 , ly@xmath79 , si@xmath4roman3 , si@xmath4roman4 , and c@xmath4roman4 lines at @xmath13 = 0.06352 , and table [ tab : cldnlist0064 ] lists the equivalent widths , column densities , and @xmath77values of the metals detected at this redshift as well as upper limits on undetected species of interest . both voigt - profile fitting and direct integration of the `` apparent column density '' profile ( see savage & sembach 1991 ; jenkins 1996 ) were used to estimate the metal column densities . these independent methods are generally in good agreement for the metal lines , which do not appear to be strongly affected by unresolved saturation . the si@xmath4roman3 @xmath90 absorption at @xmath91 0.06352 is slightly blended with a strong h@xmath4roman1 line at @xmath91 0.05486 ( see figure [ fig : syst0063 ] ) . however , the metal lines at this redshift have a distinctive two - component profile ( see figures [ fig : syst0063 ] and [ fig : civ064 ] ) , and the si@xmath4roman3 profile shape is in good agreement with those of the c@xmath4roman4 and si@xmath4roman4 lines . this indicates that ( unrelated ) blended from @xmath13 = 0.05486 contributes little optical depth to the wavelength range where the si@xmath4roman3 absorption occurs . we fitted the and h@xmath4roman1 at @xmath920.05486 simultaneously , assuming all of the components are centered shortward of the si@xmath4roman3 line . the resulting joint fit is shown in figure [ fig : syst0063 ] . the profile parameters derived for most of the lines in the @xmath93 cluster are reasonably well - constrained . some of the components are strongly blended and are consequently more uncertain than the formal profile - fitting error bars indicate ; these are marked in table [ tab : lyalist ] . component a was also difficult to measure for a different reason : the three usable lyman series lines ( ly@xmath16 , ly@xmath39 and ly@xmath79 ) are all saturated , and consequently voigt profile fitting did not provide good constraints for the determination of the component a h@xmath4roman1 column densities . as shown in figure [ fig : cog ] , using a single - component curve of growth and the observed equivalent widths of ly@xmath94 , and @xmath79 , we find that the component a h@xmath4roman1 absorption lines can be reproduced by two distinct sets of values for the h@xmath4roman1 column density and the doppler parameter . one of the two sets implies n(h@xmath4roman1)@xmath95 , which should produce a strong absorption edge characteristic of a lyman limit system ( lls ) at an observed wavelength of 970 . this wavelength region is covered by the spectrum in the sic2a channel and is shown in figure [ fig : lls_rutr ] . the s / n of the sic2a channel is low but is adequate to constrain @xmath96(h@xmath4roman1 ) . the optical depth @xmath97 of the lyman limit absorption and the h@xmath4roman1 column density are approximately related by @xmath98 where @xmath99 ) is the absorption cross section and @xmath29 is the rest - frame wavelength . the sic2a spectrum does not show any compelling evidence of a lyman limit edge at the expected wavelength , but the continuum placement is somewhat uncertain and because of this , a small lyman limit decrement could be present . based on the small depth of the flux decrement at @xmath100 = 970 , we derive a @xmath101 upper limit of @xmath96(h@xmath4roman1 ) @xmath102 @xmath103 ( upper black solid curve in figure [ fig : lls_rutr ] ) . we also show in figure [ fig : lls_rutr ] the lyman limit absorption expected for @xmath96(h@xmath4roman1 ) = @xmath104 @xmath103 ( lower black curve ) , which is too strong with our adopted continuum placement . the absence of a strong lyman limit edge rules out the higher h@xmath4roman1 column density of 10@xmath105 @xmath103 predicted by the curve of growth shown in figure [ fig : cog ] . the lower @xmath96(h@xmath4roman1 ) derived from the curve of growth ( 10@xmath106 @xmath103 ) is consistent with the lack of a lyman limit edge . to be conservative , we present below the metallicities derived both from the upper limit [ @xmath96(h@xmath4roman1 ) @xmath107 and from the somewhat lower best value from the curve of growth shown in figure [ fig : cog ] . we next examine the physical conditions and metal enrichment of the absorption systems implied by the column densities and the doppler parameters obtained from voigt profile fitting . we concentrate on the absorbers at @xmath13=0.06352 and 0.0757 because these systems show metal absorption and can be associated with nearby galaxies / structures as discussed in [ sec : environment ] . to derive abundances from the detected metals in these systems ( si@xmath4roman3 , si@xmath4roman4 , and c@xmath4roman4 ) , we must apply ionisation corrections , which depend on the ionisation mechanism and physical conditions of the gas . we will show that the gas is predominantly photoionised , and that the implied metallicities are relatively high . to investigate the absorber ionisation corrections and metallicities , we employ cloudy photoionisation models ( v96 , ferland et al . 1998 ) as described in tripp et al . ( 2003 ) . in these models , the absorbers are approximated as constant density , plane - parallel gas slabs with a thickness that reproduces the observed h@xmath4roman1 column density . the gas in the cloud is photoionised by the uv background from quasars at @xmath108 . we used the uv background spectrum shape computed by haardt & madau ( 1996 ) with the intensity normalized to @xmath109 at 1 rydberg . this value is consistent with theoretical and observational constraints ( shull et al . 1999 ; weymann et al . 2001 ; dav & tripp 2001 , and references therein ) . with the models required to match the observed @xmath96(h@xmath4roman1 ) , the predicted metal column densities depend mainly on the ionisation parameter @xmath110 ( @xmath111 ionising photon density / total hydrogen number density ) , the overall metallicity , = log ( @xmath96(x)/@xmath96(y ) ) - log ( x / y)@xmath112 . ] and the assumed relative abundances of the metals . we assume solar relative abundances , and we adopt recent revisions reported by allende prieto et al . ( 2001,2002 ) and holweger ( 2001 ) for oxygen , carbon , and silicon , respectively . the high - ion column densities predicted by the photoionisation models depend on the assumed uv background shape . in this paper , we primarily use the uv background shape computed by haardt & madau ( 1996 ) , but we note that other assessments of the uv background ( e.g. , madau , haardt , & rees 1999 ; shull et al . 1999 ) adopt a somewhat steeper euv spectral index for quasars , which changes the cloudy ionisation patterns for a given metallicity and ionisation parameter . we investigate how these uv background uncertainties affect our results by modeling our systems with both uv background shapes ( see below ) . @l@ c@ c@ ccc@ & @xmath114 & & @xmath81 & & + species & ( ) & @xmath115 & ( m ) & @xmath116 & @xmath117h i & 1215.670 & 2.704 & 603@xmath84 9 & 15.37@xmath118 & @xmath119 + & 1025.722 & 1.909 & 349@xmath8421 & & @xmath119 + & 972.537 & 1.450 & 302@xmath8422 & & @xmath119 + o i & 1302.168 & 1.804 & 6@xmath84 9 & & @xmath120 + o vi & 1037.617 & 1.836 & 28@xmath8417 & & @xmath121 + c ii & 1334.532 & 2.232 & 104@xmath84 9 & & @xmath122 + c iv & 1548.204 & 2.470 & 143@xmath84 9 & 13.67@xmath123 & 13.61@xmath840.03 + & 1550.781 & 2.169 & 90@xmath8410 & & 13.63@xmath840.05 + si ii & 1260.422 & 3.104 & 23@xmath8412 & & @xmath124 + si iii & 1206.500 & 3.304 & 151@xmath8410 & 13.02@xmath125 & 13.08@xmath840.03 + si iv & 1393.760 & 2.855 & 76@xmath8410 & 13.05@xmath126 & 12.95@xmath840.06 + & 1402.773 & 2.554 & 49@xmath8411 & & 13.05@xmath840.10 + n v & 1242.804 & 1.988 & 23@xmath8410 & & @xmath127 as discussed above , the h@xmath4roman1 absorption lines detected at @xmath128 are spread over a wide velocity range of @xmath129 ( see figure [ fig : hi00635 ] ) . from the velocity centroids of the 13 fitted voigt profiles and by using the biweight statistic as described in beers et al . ( 1990 ) , we estimate that the line - of - sight velocity dispersion of this h@xmath4roman1 absorption complex is @xmath130 . the velocity dispersion of galaxies near this redshift is comparable to this value though substantially more uncertain ( due to the larger uncertainties in the galaxy redshifts ) . this velocity dispersion is comparable to those observed in elliptical - rich galaxy groups ( e.g. zabludoff & mulchaey 1998 ) , which is interesting because elliptical - rich groups often show diffuse x - ray emission ( mulchaey & zabludoff 1998 ) indicative of hot gas in the intragroup medium . however , we will show in [ sec : discussion ] that the available information suggests that this group is predominantly composed of late - type spiral and s0 galaxies . spiral - rich groups are much fainter in x - rays but could still contain hot intragroup gas if the gas is somewhat cooler ( @xmath131 k ) or has a much lower density than that found in elliptical - rich groups ( mulchaey 2000 ) . however , we argue that most of the gas in the complex at @xmath132 is unlikely to be hot , collisionally ionised gas for several reasons : first , the h@xmath4roman1 lines in the complex are generally too narrow . if the line broadening is dominated by thermal motions , then the doppler parameter is directly related to the gas temperature , @xmath133 , where @xmath134 is the mass , @xmath135 is the atomic mass number , and the numerical coefficient is for @xmath83 in km s@xmath28 and @xmath136 in k. since other factors such as turbulence and multiple components can contribute to the line broadening , @xmath77values provide only upper limits on the temperature . applying this equation to the line @xmath77values from table [ tab : lyalist ] , we find that most of the h@xmath4roman1 lines in the @xmath137 complex indicate that @xmath8 k , which is colder than expected for the diffuse intragroup medium based on observed group velocity dispersions , even in spiral - rich groups ( e.g. , mulchaey et al . 1996 ) . in a complex cluster of lines , it is easy to hide a broad component indicative of hot gas ( see , e.g. , figure 6 in tripp & savage 2000 ) , so the narrow lines do not preclude the presence of hot gas , but they do indicate that many cool clouds are present in the intragroup medium . second , the metal line profiles in component a favor cool , photoionised gas . if component a metal lines were to arise in gas in collisional ionisation equilibrium ( cie ) , the @xmath96(c@xmath4roman4)/@xmath96(si@xmath4roman4 ) and @xmath96(si@xmath4roman4)/@xmath96(si@xmath4roman3 ) column density ratios ( integrated across both components seen in these species , see table [ tab : cldnlist0064 ] ) would require a gas temperature @xmath138 k ( sutherland & dopita 1993 ) . however the c@xmath4roman4 component at @xmath139 km s@xmath28 is rather narrow . to show this , figure [ fig : civ064 ] plots an expanded view of the c@xmath4roman4 doublet . we see that the @xmath139 km s@xmath28 is marginally resolved at the stis e140 m resolution of @xmath140 km s@xmath28 . voigt profile fitting for this component formally yields @xmath141 , which is significantly lower than the @xmath77value implied by the cie temperature , i.e. , @xmath142 km s@xmath28 . the stronger component at @xmath143 km s@xmath28 is broader ( see figure [ fig : civ064 ] ) , but the @xmath96(c@xmath4roman4)/@xmath96(si@xmath4roman4 ) and @xmath96(si@xmath4roman4)/@xmath96(si@xmath4roman3 ) ratios are similar in the components at @xmath144 and 0 km s@xmath28 , and we expect the ionisation mechanism and physical conditions to be similar in both components . third , cloudy models photoionised by the uv background from qsos are fully consistent with the measured si@xmath4roman3 , si@xmath4roman4 , and c@xmath4roman4 column densities ( and upper limits on undetected species ) at @xmath13 = 0.06352 . figure [ fig : cloudym ] shows the relevant metal column densities predicted by cloudy models ( small symbols connected with solid lines ) with log @xmath96(h@xmath4roman1 ) = 15.37 and [ m / h ] = @xmath145 compared to the observed column densities ( large symbols ) . we can see that the metal column densities are in agreement ( within the 1@xmath146 observational uncertainties ) with this model at log @xmath147 ( log @xmath148 ) . the narrow h@xmath4roman1 and c@xmath4roman4 components at this redshift could still arise in shock - heated material if they originate in gas that is not in ionisation equilibrium . many papers have considered the properties of gas that is initially shock - heated to some high temperature and then cools more rapidly than it can recombine ( e.g. , shapiro & moore 1976 ; edgar & chevalier 1986 ) . however , if component a metal lines were to arise in gas in such a state , according to both computations from shapiro & moore ( 1996 ) and schmutzler & tscharnuter ( 1992 ) , the @xmath96(si@xmath4roman3)/@xmath96(si@xmath4roman4 ) column density ratio would require a gas temperature similar ( @xmath138 k ) to the one found for the collisional ionisation equilibrium hypothesis . moreover , assuming solar abundances , the predicted o@xmath4roman6 column density at this temperature is always higher than our observed upper limit ( @xmath445 times higher in the schmutzler model ) . finally , the @xmath96(c@xmath4roman4)/@xmath96(si@xmath4roman4 ) column density ratio implies an even higher temperature than 10@xmath149 k ( 2.5 times higherr in the shapiro model ) . because of these points , this non - equilibrium cooling gas scenario seems unlikely to apply to the @xmath113 absorber toward 0624 . the cloudy modeling has some other interesting implications in addition to the basic conclusion that the gas is photoionised . for example , the photoionisation model indicates that the absorber has a relatively high metallicity of @xmath150 even though we have found no luminous galaxies within @xmath151 . a similarly high metallicity ( [ o / h]@xmath152 ) was recently reported by jenkins et al . ( 2005 ) for a lls in the spectrum of phl1811 , but that system is much closer in projection to a luminous galaxy . if we adopt the more conservative upper limit on @xmath96(h@xmath4roman1 ) from the absence of a lyman limit edge ( @xmath96(h@xmath4roman1)@xmath153 , see figure [ fig : lls_rutr ] ) instead of the curve - of - growth h@xmath4roman1 column , we obtain [ m / h ] @xmath154 . this lower limit is still substantially higher than metallicities typically observed in analogous absorbers at higher redshifts ( e.g. , schaye et al . 2003 ) and is comparable to abundances seen in high - velocity clouds near the milky way ( e.g. , sembach et al . 2001,2004 ; collins , shull , & giroux 2003 ; tripp et al . 2003 ; ganguly et al . 2005 ; fox et al . 2005 ) . to derive confidence limits on parameters extracted from our cloudy models , for combinations of metallicity @xmath155 and ionisation parameter @xmath110 we calculated the @xmath156 statistic , @xmath157 where @xmath96 indicates column density and the sum is over the three ions si@xmath4roman3 , si@xmath4roman4 , and c@xmath4roman4 . with the minimum @xmath156 obtained at [ m / h ] = @xmath145 and log @xmath158 , we evaluated confidence limits by finding parameters that increased @xmath159 by the amount appropriate for a given confidence level ( see lampton , margon , & bowyer 1976 ; press et al . 1992 ) . in this way , we find [ m / h ] = @xmath160 at the 2@xmath146 confidence level . of course , these confidence limits do not fully reflect potential sources of systematic error such as uncertainties in the shape of the ionising flux field or accuracy of the atomic data incorporated into cloudy . when we used the steeper uv background shape ( e.g. madau , haardt , & rees 1999 ; shull et al . 1999 ) , the observations are still consistent with the cloudy model for a lower metallicity of [ m / h ] = @xmath161 and a larger ionisation parameter log @xmath162 . we can also place constraints on physical quantities such as the absorber size ( the length of the path through the absorbing region ) and the thermal gas pressure ( but see the caveats discussed in 5 of tripp et al . figure [ fig : absorbersize ] shows confidence interval contours for the absorber size @xmath163 and thermal pressure @xmath164 implied by the photoionisation model . the best fit implies that @xmath165 3.5 kpc . if spherical , the baryonic mass of this cloud would be @xmath166 . however , we can see from figure [ fig : absorbersize ] that the model allows a large range for @xmath163 at the 2@xmath146 level . the low thermal pressure implied is also notable . when the steeper uv background shape is used for the cloudy model , the predicted pressure is lower by a factor 1.5 and the absorber size increases to @xmath1678 kpc.the range of pressures within the contours in figure [ fig : absorbersize ] is several orders of magnitude lower than the gas pressure measured in the disk of the milky way ( see jenkins & tripp 2001 ) and is even lower than pressures measured in hvcs in the milky way halo ( e.g. wakker , oosterloo , & putman 2002 ; fox et al . . however , sembach et al . ( 1995 , 1999 ) found similar pressure for civ hvcs surrounding the milky - way with somewhat lower metallicity . moreover , pressures this low are predicted in some theoretical models of galactic halos ( wolfire et al . finally , the derived pressure depends on the intensity used to normalize the ionising flux field ( see tripp et al . 2005 ) and both the particle density and the pressure could be higher if the radiation field is brighter than we assumed . as discussed in [ sec : environment ] , the absorption lines detected at @xmath13 = 0.07573 occur in a large - scale structure connected to a559 and a564 . only h@xmath4roman1 ly@xmath16 , ly@xmath39 and the c@xmath4roman4 @xmath681548.20,1550.78 doublet are clearly detected at this redshift ; the profiles of these absorption lines are shown in figure [ fig : syst0076 ] . the weakness of ly@xmath39 and the absence of ly@xmath79 suggest that the h@xmath4roman1 lines are not badly affected by unresolved saturation , and profile fitting measurements are robust . likewise , comparison of the c@xmath4roman4 apparent column density profiles shows no evidence of unresolved saturation . we have fitted these lines with only one component . the results of the fit are listed in table [ tab : cldnlist076 ] . the width of the h@xmath4roman1 line ( @xmath169 ) indicates a temperature for the gas lower than @xmath170 k , which again favors a photoionisation process . no o@xmath4roman6 is evident at this redshift , but several strong unrelated lines of various elements are found close to the expected wavelength of the o@xmath4roman6 doublet , and these lines might mask weak o@xmath4roman6 absorption . despite the fact that c@xmath4roman4 is the only metal detected in this system , we can nevertheless place an interesting lower limit on the absorber metallicity . the carbon abundance can be expressed as [ c / h]= log[/ ] + log[/ ] - log ( c / h)@xmath171 , where and are the ion fractions of h@xmath4roman1 and c@xmath4roman4 , respectively combination . the maximum is not evident in figure [ fig : cloudym ] because it occurs at a higher value of @xmath110 than the range shown . ] . with the h@xmath4roman1 column from the ly@xmath16+ly@xmath39 fit ( table [ tab : cldnlist076 ] ) , and again assuming that the gas is photoionised by the uv background from qsos ( haardt & madau 1996 ) , we find that log[/]@xmath172 , and therefore @xmath173>-0.6 $ ] in the absorber . once again , this metallicity lower limit is relatively high despite the fact that no luminous galaxies have been found near the sight line ( the closest galaxy is nw3 at @xmath174 kpc , see table [ tab : spec_red ] ) . @l@c@c@c@c@c@ & & & & & + & & & & & h i & 1215.670 & 309@xmath8410 & 14.18@xmath840.04 & 24.6@xmath841.0 & 14.06@xmath840.03 + & 1025.722 & 108@xmath8421 & & & 14.28@xmath840.08 + o vi & 1037.617 & 0@xmath8419 & & & @xmath175 + c iv & 1548.204 & 51@xmath84 7 & 13.18@xmath840.04 & 13.1@xmath841.5 & 13.24@xmath840.04 + & 1550.781 & 27@xmath84 8 & & & 12.09@xmath840.22 + si iii & 1206.500 & 0@xmath8410 & & & @xmath176 + si iv & 1393.760 & 17@xmath8414 & & & @xmath124 + n v & 1238.821 & 25@xmath84 9 & & & @xmath177 in [ sec : environment ] we noted that the clusters a554 , a562 , and a565 indicate that a large - scale filament / supercluster is foreground to 0624 . our galaxy redshift survey has also revealed four galaxies close to 0624 at the redshift of the abell 554/562/565 structure ( see figure [ fig : field ] and table [ tab : spec_red ] ) , which suggests that the large - scale filament extends across the 0624 field . however , we only find a couple of weak ly@xmath16 lines near the redshift of this structure at redshifts substantially offset from those of the galaxies and abell clusters . it is possible that the ly@xmath16 lines are weak / absent because the gas in the filament is so hot that the h@xmath4roman1 ion fraction makes the ly@xmath16 line undetectable , but we also note that the nearest galaxies are farther from the sight line in this case ( @xmath179 mpc ) than in the structures at @xmath180 and discussed above , so it is also possible that the sight line does not penetrate the part of the dark matter filament where the potential is deep enough to accumulate gas and galaxies ( see discussion in bowen et al . how do the properties of the ly@xmath16 absorbers at , , and compare to the other ly@xmath16 lines in the 0624 spectrum ( and in other sight lines ) ? we have found that the systems at and arise in photoionised cool gas ; is this true of the majority of the ly@xmath16 lines in the spectrum ? in particular , do we find ly@xmath16 lines that arise in hot gas ? richter et al . ( 2004 , 2005 ) and sembach et al . ( 2004 ) have recently identified a population of broad ly@xmath16 lines ( blas ) with @xmath181 40 km s@xmath28 in the spectra of several low@xmath0 qsos ( pg0953 + 415 , pg1116 + 215 , pg1259 + 593 , and h1821 + 643 ) . williger et al . ( 2005 ) similarly find a substantial number of blas in the spectrum of pks0405 - 123 . bowen et al . ( 2002 ) have also identified some bla candidates using somewhat lower resolution data . if these lines are mainly broadened by thermal motions , then they trace the warm - hot igm , and moreover , they in this case would contain a substantial portion of the baryons in the universe at the present epoch ( see richter et al . 2004 , 2005 ; sembach et al . 2004 ) . based on simulations , richter et al . ( 2005 ) and williger et al . ( 2005 ) find that some of the blas are not predominantly thermally broadened but instead are due to line blends that are difficult to recognize at the s / n afforded by typical stis echelle spectra . however , richter et al . ( 2005 ) conclude that approximately 50 per cent of the blas are mainly thermally broadened , and some high s / n examples in the above papers are remarkably smooth and broad and appear to entirely consistent with a single broad gaussian ( see , e.g. , figures 4 and 5 in richter et al . 2005 ) . in this paper , in addition to using a different sight line , we have employed methods that are independent from ( e.g. , using different software ) the techniques used in the papers above for continuum normalization , line detection , and profile fitting . consequently , we have an opportunity to independently check the bla findings reported in these papers . for the 0624 sight line , we find that the mean @xmath77value for all ly@xmath16 lines is @xmath182 = 37 km s@xmath28 , and the median @xmath183 km s@xmath28 . however , as noted in table [ tab : lyalist ] , some of the ly@xmath16 lines are significantly blended , and the line parameters are accordingly uncertain . if we exclude these uncertain blended cases , we find @xmath184 km s@xmath28 and @xmath185 km s@xmath28 . these ensemble @xmath77values are in reasonable agreement with the previous high - resolution ly@xmath16 studies . in figure [ fig : larg ] we compare our measurements of the h@xmath4roman1 @xmath77values and column densities from table [ tab : lyalist ] to the measurements reported by richter et al . ( 2004 ) and sembach et al . the solid line indicates @xmath83 vs. @xmath96(h@xmath4roman1 ) for a gaussian line with central optical depth = 0.1 . this is effectively a detection threshold ; lines that have ( @xmath186 ) combinations to the left of this line are not likely to be detected . the dotted line in figure [ fig : larg ] shows the minimum @xmath77value as a function of @xmath96(h@xmath4roman1 ) predicted by dav & tripp ( 2001 , see their equation 5 ) from the hydrodynamic cosmological simulations of dav et al . this predicted lower envelope appears to be in reasonable agreement with the observed lower envelope for the three sight lines shown in the figure . from figure [ fig : larg ] , we see that the ( @xmath186 ) distribution that we have obtained from the 0624 sight line appears to be generally similar to those obtained by sembach et al . ( 2004 ) and richter et al . sembach et al . and richter et al . did find more extremely broad ly@xmath16 lines ( @xmath181 80 km s@xmath28 ) than we have been able to positively identify in the 0624 spectrum . this may be partly due to signal - to - noise differences the data employed by richter et al . and sembach et al . have higher s / n because such broad and shallow lines are difficult to detect in the 0624 data . for @xmath187 km s@xmath28 , the different sight lines appear to be in broad agreement . excluding lines within 5000 km s@xmath28 of the qso redshift , of the qso redshift can arise in intrinsic gas ejected by the qso , and these lines can be rather broad ( see , e.g. , yuan et al . 2002 ) , even if not part of a full - blown broad absorption line outflow . ] the 0624 spectrum can be used to search for blas between @xmath188 and @xmath189 . accounting for regions in which broad lines could have been masked by igm or ism lines , we obtain a blocking - corrected total redshift path @xmath190 . with 21 ly@xmath16 lines in the sample with @xmath181 40 km s@xmath28 , we thus obtain @xmath191(bla ) = 64@xmath8416 . this is somewhat larger than the values reported by sembach et al . ( 2004 ) and richter et al . ( 2004,2005 ) . however , richter et al . have excluded blas that are located in complex blends on the grounds that these cases are more likely to be affected by non - thermal broadening . if we follow the same procedure , we must reject 10 blas ( see table [ tab : lyalist ] ) ; the remaining 11 blas would then imply @xmath191(bla ) = 33@xmath8410 . using equations 1 , 5 , and 6 from sembach et al . ( 2004 ) , but adjusted for the somewhat different cosmological parameters assumed in this paper , we find that our full sample implies that the bla baryonic content is @xmath192(bla ) = 0.017 @xmath7 ( in the usual notation , i.e. , @xmath193 ) . this high value probably substantially overestimates the bla baryonic content , largely because of false blas that arise from blends . if we exclude blas that are located in complex blends , this drops to @xmath192(bla ) = 0.0036 @xmath7 , which is similar to values obtained by richter et al . and sembach et al . the uncertainties in @xmath192(bla ) due to , e.g. , lines that are broad due to blends or other non - thermal broadening mechanisms , are large and currently difficult to assess ( see discussion in richter et al . however , these initial calculations suggest that blas may harbor an important quantity of baryons . with future uv spectrographs , it would be valuable to obtain high - resolution spectra with substantially better s / n in order to accurately assess the baryonic content of blas as part of the general census of ordinary matter in the nearby universe . we have acquired detailed information about the abundances , physical conditions , and galaxy proximity of absorption systems in the direction of 0624 . what are the implications of these measurements for broader questions of galaxy evolution and cosmology ? the processes that add gas to galaxies ( e.g. accretion ) and remove gas from galaxies ( e.g. , winds , dynamical stripping ) can have profound effects on galaxy evolution , and the `` feedback '' of matter and energy from galaxies into the igm is now believed to play an important role in shaping structures that subsequently grow out of the igm ( voit g. m. , 2005 ) . the quantity and implications of the @xmath194 k whim gas is a topic of particular interest currently . the galaxies and absorption systems in the direction of 0624 , particularly the galaxy group and ly@xmath16 complex at @xmath75 = , have some interesting , and perhaps surprising , implications regarding these questions , which we now discuss . rosat observations of diffuse x - ray emission have established that galaxy groups that are dominated by early - type galaxies often contain diffuse , hot intragroup gas ( mulchaey 2000 , and references therein ) . based on the observed relation between intragroup gas temperature and velocity dispersion @xmath146 in x - ray bright groups ( @xmath195 ) and the fact that spiral - rich groups have lower velocity dispersions than elliptical - rich groups , mulchaey et al . ( 1996 ) have hypothesized that spiral - rich groups might have somewhat cooler intragroup media that could give rise to qso absorption lines at whim temperatures ( e.g. , o@xmath4roman6 ) . however , the galaxy group at @xmath75 = appears to have properties that are not consistent with the elliptical - rich groups detected with rosat nor with the idea that spiral - rich groups contain warm - hot intragroup gas . it is unclear if the galaxy group at @xmath75 = is a spiral - rich group . figure [ fig : galpics ] shows @xmath45 images from the mosa data of the 10 galaxies that we have found in this group . most of the galaxies in the group show evidence of disks and bulges ( both in the direct images and in radial brightness profiles ) . we find from visual inspection that 4 - 5 of the 10 galaxies have indications of spiral structure ( sw3 , se1 , se8 , se6 , and possibly se4 ) . however , the remaining galaxies could be early - type s0 galaxies , and therefore the early - type fraction might be comparable to groups that show diffuse x - ray emission ( see , e.g , figure 7 in zabludoff & mulchaey 1998 ) . the ne3 , se13 , and se5 galaxies , which morphologically appear to be early - type galaxies , have colours and magnitudes consistent with the `` red sequence '' colour - magnitude relation observed in clusters ( e.g. , bower , lucey , & ellis 1992 ; mcintosh et al . 2005 ) ; these galaxies are likely s0s ( the other 7 galaxies have blue colours characteristic of late types ) . the velocity dispersion of the group at @xmath75 = , albeit uncertain , is more comparable to those of elliptical - dominated groups than spiral - rich groups ( mulchaey et al . 1996 ; zabludoff & mulchaey 1998 ) . regardless of whether the group is elliptical- or spiral - rich , it is surprising that we find a large number of cool , photoionised clouds in the intragroup medium ( [ ss : z064 ] ) . in the hot intragroup medium of an elliptical - rich group , h@xmath4roman1 lines should be extremely broad and weak , but instead we find strong , narrow lines ( see figure [ fig : hi00635 ] ) . even in the cooler gas predicted to be found in late - type dominated groups , the h@xmath4roman1 lines should be broader . we could entertain models of cooling intragroup gas , but in such models o@xmath4roman6 is expected to be stronger . likewise , if the intragroup gas is a multiphase medium with cooler clouds ( which cause the h@xmath4roman1 absorption lines ) embedded in a hotter phase , then we might expect to detect o@xmath4roman6 from the interface between the phases ( fox et al . 2005 ) , unless conduction is somehow suppressed . the lack of evidence of hot gas leads us to question whether this group is a bound , virialized system . an alternative possibility is that our sight line passes along the long axis of a large - scale filamentary structure in the cosmic web . in this case , the projection of the galaxies and ly@xmath16 clouds along the sight line could give a false impression of a group in which hot gas would be expected . however , in cosmological simulations of large - scale filaments , whim gas is expected to be widespread at the present epoch , even in modest - overdensity regions ( see , e.g. , figure 4 in cen & ostriker 1999b ) , so it is interesting that we find a substantial number of cool clouds at @xmath12 , somewhat contrary to theoretical expectations . as noted above , our data do not preclude the presence of whim gas at @xmath12 , but we find no clear evidence of it . other sight lines show similar clusters of ly@xmath16 lines , e.g. , the ly@xmath16 complex at @xmath196 0.057 toward pks2155 - 304 ( shull et al . 1998 ; shull , tumlinson , & giroux 2003 ) or the ly@xmath16 lines at @xmath196 0.121 toward h1821 + 643 ( tripp et al . however , unlike the 0624 ly@xmath16 complex , the pks2155 - 304 and h1821 + 643 examples both show evidence of warm - hot intragroup gas . to test whether the observations and simulations are in accord , it would be useful to assess the frequency and physical properties of these ly@xmath16 complexes in cosmological simulations for comparison with the observations . it is also interesting that the two systems for which we have obtained abundance constraints ( at @xmath75 = and ) both indicate relatively high metallicities , but both of these systems are at least 100 kpc away ( in projection ) from the nearest known galaxy . this naturally raises a question : how did gas that is so far from a galaxy attain such a high metallicity ? the gas could have been driven out of a galaxy by a galactic wind ; some wind models predict that the outflowing material will have a high metallicity ( mac low & ferrara 1999 ) , the difficulty with this interpretation is that winds from nearby galaxies are usually observed to contain substantial amounts of hot gas ( e.g. , strickland et al . 2004 ) , which seems to be inconsistent with the absorption line properties as we have discussed . a more likely explanation is that the high - metallicity gas we have detected in absorption has been tidally stripped out of one of the nearby galaxies . there are indications that tidal stripping could be a more gentle process for removing gas from galaxies , and a tidally stripped origin can therefore more easily accommodate the observed low - ionisation state of the gas . for example , in the direction of ngc3783 , the galactic high - velocity cloud ( hvc ) at @xmath197 247 is now recognized to be tidally stripped debris from the smc . while this tidally stripped material shows a wide array of low - ionisation absorption lines , it has little or no associated high - ion absorption ( lu et al . 1994 ; sembach et al . moreover , the tidally - stripped hvc contains molecular hydrogen , which sembach et al . argue formed in the smc and survived the rigors of tidal stripping ( as opposed to forming in situ in the stream ) . both the absence of high ions and the survival of h@xmath80 suggest that the stripping process did not substantially ionise and heat this hvc . several galaxies are close enough to the 0624 sight line to be plausible sources of the gas in a tidal stripping scenario . one of the nearby galaxies , se1 , has a distorted spiral morphology . this galaxy is a plausible source of tidally stripped matter . we have presented a study of absorption - line systems in the direction of 0624 using a combination of high - resolution uv spectra obtained with _ hst_/stis and plus ground - based imaging and spectroscopy of galaxies within @xmath4430 of the sight line . in addition to presented the basic measurements and ancillary information , we have reported the following findings : \1 . there are several abell galaxy clusters in the foreground of 0624 , including two clusters at @xmath12 0.077 ( a559 and a564 ) and three at @xmath12 0.110 ( a554 , a562 , and a565 ) . these clusters trace large - scale dark matter structures , i.e. , superclusters or filaments of the `` cosmic web '' . our galaxy redshift survey has revealed galaxies at these supercluster redshifts in the immediate vicinity of the 0624 sight line , and therefore our qso spectra provide an opportunity to study the gas in large - scale intergalactic filaments . the most prominent group of galaxies found in our galaxy redshift survey is at @xmath12 and is not associated with an abell cluster with a spectroscopic redshift from the literature . however , a557 , for which no spectroscopic redshift has been reported , is at least partly due to the galaxy group at @xmath12 . the two strongest ly@xmath16 absorption systems at @xmath198 arise in galaxy groups at @xmath199 and . the ly@xmath16 absorption at is particularly dramatic : at this redshift , we find 13 ly@xmath16 lines spread over a velocity range of 1000 km s@xmath28 with a line - of - sight velocity dispersion of 265 km s@xmath28 . the second - strongest system at is associated with the abell 559/564 large - scale structure , and this indicates that a filament containing gas and galaxies feeds into the abell 559/564 supercluster . analysis of the ly@xmath16 absorption - line complex at @xmath200 provides strong evidence that the gas is photoionised and relatively cool ; we find no compelling evidence of warm - hot gas in this large - scale filament . we detect si@xmath4roman3 , si@xmath4roman4 , and c@xmath4roman4 in the strongest component in this ly@xmath16 complex , and photoionisation models indicate that the gas metallicity is high , [ m / h ] = @xmath201 . this is somewhat surprising because we do not find any luminous galaxies close to the sight line ; the closest galaxy is at a projected distance @xmath202 kpc . the ly@xmath16 system at is only detected in the c@xmath4roman4 doublet , but nevertheless we find a similar result : the lower limit on the metallicity is relatively high ( [ c / h ] @xmath14 ) while the nearest galaxy is at @xmath203 kpc . we have compared the distribution of ly@xmath16 doppler parameters and h@xmath4roman1 column densities to high - resolution measurements obtained from other sight lines , and we find good agreement . we find that the number of broad ly@xmath16 absorbers with @xmath181 40 km s@xmath28 per unit redshift is in agreement with results recently reported by richter et al . ( 2004,2005 ) and sembach et al . the baryonic content of these broad h@xmath4roman1 lines is still highly uncertain and requires confirmation with higher s / n data , but it is probable that some of these broad line arise in warm - hot gas and contain an important portion of the baryons in the nearby universe . we also find that the lower bound on @xmath83 vs. @xmath96(h@xmath4roman1 ) is in agreement with predictions from cosmological simulations . the absence of warm - hot gas in the galaxy group / ly@xmath16 complex at @xmath200 is difficult to reconcile with x - ray observations of bound galaxy groups . it seems more likely that this in this case we are viewing a large - scale cosmic web filament along its long axis . the filament contains a mix of early- and late - type galaxies and many cool , photoionised clouds . the high - metallicity , cool cloud at @xmath13 = 0.06352 is probably tidally stripped material . this origin can explain the high metallicity and the lack of hot gas . one of the nearby galaxies has a disturbed morphology consistent with this hypothesis . similar clusters of ly@xmath16 lines have been observed in other sight lines , and additional examples are likely to be found as we continue to analyse stis data . we look forward to comparisons of these observations to predictions from cosmological simulations and other theoretical work in order to better understand the processes that affect the evolution of galaxies and the intergalactic medium . we thank dan mcintosh and neal katz for useful discussions . the stis observations of hs0624 + 6907 were obtained for _ hst _ program 9184 with financial support through nasa grant hst go-9184.08-a from the space telescope science institute . this research was also supported in part by nasa through long - 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we present high - resolution ultraviolet spectra of absorption - line systems toward the low@xmath0 qso 0624 ( @xmath1 ) . coupled with ground - based imaging and spectroscopic galaxy redshifts , we find evidence that many of these absorbers do not arise in galaxy halos but rather are truly integalactic gas clouds distributed within large - scale structures , and moreover , the gas is cool ( @xmath2 k ) and has relatively high metallicity ( @xmath3 ) . _ hst _ space telescope imaging spectrograph ( stis ) data reveal a dramatic cluster of 13 h@xmath4roman1 lines within a 1000 interval at @xmath5 . we find 10 galaxies at this redshift with impact parameters ranging from @xmath6 kpc to 1.37 @xmath7 mpc . the velocities and velocity spread of the lines in this complex are unlikely to arise in the individual halos of the nearby galaxies ; instead , we attribute the absorption to intragroup medium gas , possibly from a large - scale filament viewed along its long axis . contrary to theoretical expectations , this gas is not the shock - heated warm - hot intergalactic medium ( whim ) ; the width of the lines all indicate a gas temperature @xmath8 k , and metal lines detected in the complex also favor photoionised , cool gas . no o@xmath4roman6 absorption lines are evident , which is consistent with photoionisation models . remarkably , the metallicity is near - solar , [ m / h ] @xmath9 ( @xmath10 uncertainty ) , yet the nearest galaxy which might pollute the igm is at least 135 @xmath7 kpc away . tidal stripping from nearby galaxies appears to be the most likely origin of this highly enriched , cool gas . more than six abell galaxy clusters are found within @xmath11 of the sight line suggesting that the qso line of sight passes near a node in the cosmic web . at @xmath12 0.077 , we find absorption systems as well as galaxies at the redshift of the nearby clusters abell 564 and abell 559 . we conclude that the sight line pierces a filament of gas and galaxies feeding into these clusters . the absorber at @xmath13 = 0.07573 associated with abell 564/559 also has a high metallicity with [ c / h ] @xmath14 , but again the closest galaxy is relatively far from the sight line ( @xmath15 kpc ) . the doppler parameters and h@xmath4roman1 column densities of the ly@xmath16 lines observed along the entire sight line are consistent with those measured toward other low@xmath0 qsos , including a number of broad ( @xmath17 ) lines . intergalactic medium galaxies : abundances large - scale structure of the universe quasars : individual ( hs0624@xmath186907 )

You are an expert at summarizing long articles. Proceed to summarize the following text: consider a process which is repeated @xmath1 times and each repetition has @xmath0 possible outcomes . for concreteness we may think of assigning @xmath1 balls to @xmath0 labelled boxes , where each box can hold any number of balls . the first ball can go into any box , the second ball can go into any box , ... , and the @xmath1th ball can go into any box . each assignment or allocation is thus a sequence of @xmath1 box labels and results in some number @xmath3 of balls in box 1 , @xmath4 in box 2 , etc . , where the @xmath5 are @xmath6 and sum to @xmath1 . there are @xmath7 possible assignments in all , and many of them can lead to the same count vector @xmath8 . we refer to these assignments as the _ realizations _ of the count vector . the arrangement of @xmath1 balls into @xmath0 boxes can represent the construction of any object consisting of @xmath0 distinguishable parts out of @xmath1 identical units . so if the balls represent pixels of an image , the attributes of color and ( suitably discretized ) intensity are ascribed to the boxes to which the pixels are assigned . then the count vector is thought of as a 2-dimensional matrix with rows labelled by intensity and columns by color . other examples are people categorized by age , height , and weight , vehicles classified by weight , size , and fuel economy , packets in a communications network with attributes of origin , destination , size , and timestamp , etc . the object can even be a ( discrete ) probability distribution . when the process simply represents the classification of @xmath1 units by @xmath0 discrete or discretized attibutes , it is known as a ( multi - dimensional ) contingency table . now consider imposing constraints @xmath9 on the allowable assignments , expressed as a set of _ linear _ relations on the elements of the _ frequency _ vector @xmath10 corresponding to the counts @xmath11 e.g. @xmath12 , @xmath13 , etc . as @xmath1 grows , the frequency vectors of more and more of the assignments that satisfy the constrtaints will have _ entropy _ closer and closer to that of a particular @xmath0-vector @xmath14 , the vector of _ maximum entropy _ @xmath15 subject to the constraints @xmath9 . ( we denote this vector by @xmath14 , as opposed to @xmath16 , to emphasize that its entries are , in general , not rational . ) this result is known , more or less , in many forms : the original is e.t . jaynes s `` entropy concentration theorem '' @xcite , @xcite , in the information theory literature it is the `` conditional limit theorem '' @xcite , and in computer science there is `` strong entropy concentration '' @xcite , @xcite . all these results involve limits or asymptotics in one way or another , i.e. in the statement [ cols= " < , < " , ] to interpret the first line of table [ tab : q ] , suppose we choose @xmath17 . we then find @xmath18by corollary [ cor : pd ] , this rational approximation to the maxent p.d . ( [ eq : phisq ] ) has at least @xmath19 times more realizations than the entire set of p.d.s which satisfy the constraints to accuracy @xmath20 but differ from the p.d . ( @xmath21 ) by more than 0.01 in @xmath22 norm . the phenomenon of entropy concentration appears when a large number of units is allocated to containers subject to constraints that are linear functions of the numbers of units in each container : most allocations will result in frequency ( normalized count ) vectors with entropy close to that of the vector of maximum entropy that satisfies the constraints . asymptotic proofs of this phenomenon are known , beginning with the work of e. t. jaynes , but here we presented a formulation entirely devoid of probabilities and provided explicit bounds on how large the number of units must be for concentration to any desired degree to occur . our formulation also deals with the fact that constraints can not be satisfied exactly by rational frequencies , but only to some prescribed tolerances , and also eliminates ( as opposed to `` resolves '' ) the well - known issue of expectations vs. measurements in constraints . in addition , we established a perhaps more useful version of the concentration result , in terms of deviation from the maximum entropy vector , instead of the usual maximum entropy value , as well as results that pertain to the maximum entropy vector itself and not to a whole set of vectors around it . because of its conceptual simplicity and minimality of assumptions , entropy concentration is a powerful justification of the widely - used discrete maxent method ( the other being axiomatic formulations ) , and we believe that the explicit , non - asymptotic bounds strengthen it considerably . all of our results were illustrated with detailed numerical examples . thanks to david applegate for discussing with me what can be done , howard karloff for telling me what ca nt be done , steve korotky for our many discussions on entropy and networks , and neil sloane for many informative discussions , answering many questions , in particular about latttice points , and for his careful reading of the paper . rounding ensures that @xmath23 . from the explanation after definition [ def : fstar ] , the adjustment of @xmath24to @xmath25 ensures @xmath26 , which establishes the @xmath27 claim . in more detail , the 0 elements of @xmath25 coincide with those of @xmath14 , and this adjustment causes at most @xmath28 of the non - zero elements of @xmath25 to differ from the corresponding elements of @xmath29 by @xmath30 , so @xmath31 . hence @xmath32 , which establishes the claim for the @xmath22 norm . beginning with the equality constraints ( [ eq : constr2e ] ) , note that @xmath33 . set @xmath34 . then we have @xmath35 , with @xmath36 . now @xmath37 , where @xmath38 denotes the matrix infinity norm ( also known as the `` maximum row sum '' norm ) . the inequality holds because the vector norm @xmath39 is _ compatible _ with the ( rectangular ) matrix norm @xmath40 ( see @xcite , 5.7 ) . so if we make @xmath41 , we will have @xmath42 , as required by ( [ eq : constr2e ] ) . the inequality constraints ( [ eq : constr2i ] ) are handled in exacty the same way . theorem 16.3.2 of @xcite is a similar result , but in terms of the @xmath22 norm . the function @xmath45 , @xmath46 $ ] , is concave and has a maximum at @xmath47 . let @xmath48 be @xmath49 , and consider the difference of the values of @xmath50 at two points that are @xmath51 apart : @xmath52 . since @xmath53 always , the maximum of @xmath54 occurs at @xmath55 and equals @xmath56 . so if @xmath57 , @xmath58 ( this is tighter than what we would get by simply applying the defining inequality of concavity to @xmath59 . ) we have now shown that if @xmath60 , then @xmath61 . the result of the proposition follows . using proposition [ prop : norment0 ] , we want to find a @xmath62 s.t . @xmath63 for all @xmath64 . setting @xmath65 and @xmath66 , we want to find a @xmath67 s.t . for all @xmath68 , @xmath69 . we claim that this inequality , where @xmath70 and @xmath71 is expected to be @xmath72 , is satisfied by @xmath73 , for any @xmath74 . indeed , @xmath75 which is possible for any @xmath74 if @xmath76 is large enough . with @xmath77 , this condition is @xmath78 . but this holds for @xmath79 , a very mild requirement . finally , the function @xmath80 is increasing for @xmath81 , so the l.h.s . of ( [ eq : haty ] ) will hold for all @xmath68 as desired . we have now shown that @xmath82 will hold if @xmath59 is s.t . @xmath83 where the `` @xmath84 '' can be tightened to @xmath85 . begin with @xmath86 and use the fact that @xmath87 which is defined for all @xmath88 by @xmath89 . then we find @xmath90 finally , for the upper bound in prop . [ prop : s ] , the sum of the last two terms is maximized when @xmath91 . for the lower bound , it is minimized when @xmath92 . we begin by observing that the sum over @xmath93 is bounded above by the sum over all of @xmath94 and then use the bound of proposition [ prop : s ] on @xmath95 to find @xmath96 to evaluate the last sum , let @xmath97 be the subset of @xmath94 consisting of vectors with @xmath98 non - zero elements . since the @xmath97 form a partition of @xmath94 , @xmath99 where the @xmath100 comes from the fact that as pointed out in proposition [ prop : s ] , @xmath95 depends only on the non - zero elements and not on their positions . thus @xmath101 we now need an auxiliary result on the inner sum in ( [ eq : ss ] ) : proposition [ prop : sumint ] is proved separately later . using this result in ( [ eq : ss ] ) , @xmath104 for the first inequality we used @xmath105 and for the second we assumed that @xmath106 and @xmath107 . combining the above with ( [ eq : nrb1 ] ) , and again assuming @xmath106 we obtain the result of the lemma . ignoring the rational requirement for the moment and denoting @xmath59 by @xmath108 , only @xmath109 are independent , so our set is the subset of @xmath110 belonging to @xmath111 we will construct inside this set an @xmath112-dimensional rectangular parallelepiped @xmath113 whose intersection with @xmath94 is easy to count . to construct @xmath113 we will determine its two extreme points , the one with the largest coordinates , @xmath114 , and the one with the smallest , @xmath115 , where @xmath116 . if @xmath114 satisfies ( [ eq : krs ] ) , then @xmath117 the 4th inequality is true , and the 2nd implies the first . the 2nd and 3d inequalities are satisfied if @xmath118 , and @xmath119 will be maximized if @xmath120 since @xmath14 has @xmath121 non - zero elements , we can assume w.l.o.g . that @xmath122 . similarly , for the other extreme point @xmath115 we must have @xmath123 and @xmath124 . if some @xmath125 are 0 the corresponding @xmath126 are 0 , and w.l.o.g . we can take the non - zero @xmath126 to be @xmath127 . then the @xmath126 that satisfy the inequalities and maximize the product @xmath128 are @xmath129 but this needs @xmath130 for all the non - zero @xmath125 , which we have assumed . thus from ( [ eq : py ] ) and ( [ eq : pz ] ) @xmath131 sides of @xmath113 have length @xmath132 , and the other @xmath133 sides have length @xmath134 . again . we can take @xmath135 to be @xmath136 , the largest element of @xmath14 . so if we assume that @xmath137 , @xmath113 has @xmath138 sides of length @xmath139 and @xmath133 sides of length @xmath140 . now a @xmath141-dimensional parallelepiped with sides of lengths @xmath142 , irrespective of its location in @xmath143 , contains at least @xmath144 lattice points , i.e. points in @xmath145 . ( this can be established by induction on @xmath141 . for @xmath146 it says that a segment of length @xmath147 on the real axis contains at least @xmath148 integers . ) applying this to @xmath113 with all its @xmath149 dimensions scaled up by @xmath1 , the scaled @xmath113 must contain at least @xmath150 points whose coordinates are rational numbers with denominator @xmath1 , i.e. vectors in @xmath94 . the first factor and its attendant condition @xmath151 are absent if @xmath152 . let @xmath153 be some constant whose purpose is expained later , in the proof of theorem [ th:1 ] . we begin by deriving a lower bound on the size of @xmath154 , a subset of @xmath155 . consider the set @xmath156 . since @xmath157 , proposition [ prop : theta_inf ] implies that any@xmath59 in this set also belongs to @xmath158 . further , by the middle expression in the definition of @xmath159 , proposition [ prop : norment ] implies that any such @xmath59 also has entropy at least @xmath160 . thus all @xmath59 in the set @xmath161 belong to @xmath162 . finally @xmath159 satisfies the conditions of proposition [ prop : anlb ] , hence the size of @xmath161 is bounded from below by @xmath163 . but @xmath164 , so we established the first claim of the lemma . now suppose that all @xmath59 in @xmath154 have at least @xmath165 non - zero elements ; for the purposes of this proof we may take these to be the first @xmath98 elements . then by proposition [ prop : s ] , if @xmath166 is an arbitrary element of @xmath154 , @xmath167 let @xmath168 ; this vector has integral entries , all positive , and summing to @xmath1 . the maximum of @xmath169 equals @xmath170 , occurring when @xmath171 and @xmath172 . thus the exponential in ( [ eq : a ] ) is at least @xmath173 . further , the maximum of @xmath174 subject to @xmath175 occurs at @xmath176 , so the last factor in ( [ eq : a ] ) is at least @xmath177 . finally @xmath178 , and so ( [ eq : a ] ) implies the second result of the lemma , but with the number @xmath98 still undetermined . by requiring @xmath179 to be less than the smallest non - zero element of @xmath14 , we can ensure that there is no element of @xmath59 which is 0 while the corresponding element of @xmath14 is positive ; this is accomplished by the last term on the r.h.s . of ( [ eq : theta0 ] ) . thus we can take @xmath98 equal to @xmath180 , the number of non - zero elements of @xmath14 . the upper bound on @xmath181 and the lower bound on @xmath182 are given by lemmas [ le : nrbn ] and [ le : nran ] . both these bounds _ increase _ when the entropy tolerance @xmath183 decreases towards 0 , as makes sense . to simplify the proof we assume that @xmath184 . then combining the two bounds and unifying some numerical constants @xmath185 when everything else is fixed , this lower bound on @xmath186 ( eventually ) increases as @xmath187 , as we want it to . in general , this behavior would have been impossible if @xmath188 were 1 . this is why we introduced @xmath188 and required it to be @xmath189 : it serves to strictly separate our bounds on @xmath182 and @xmath181 . there is freedom in choosing the value of @xmath188 , which we exploit below . to establish ( [ eq : ratio ] ) we need the l.h.s . of ( [ eq : nranrb ] ) to be @xmath190 . this reduces to requiring @xmath191 where the constants @xmath192 are @xmath193 we will now show that ( [ eq : n1 ] ) is satisfied by @xmath194 first , assume @xmath195 . setting @xmath196 in ( [ eq : n1 ] ) with @xmath146 we reduce to @xmath197 the r.h.s . of this condition is a convex combination of @xmath198 and 1 , so the first condition will hold if @xmath199 , which is true when @xmath200 . now let @xmath201 . putting @xmath202 in ( [ eq : n1 ] ) , we reduce to establishing @xmath203 , which will hold if @xmath204 , always true . the r.h.s . of ( [ eq : n11 ] ) depends on @xmath205 , which has up to this point been left unspecified . we finally need @xmath206 and @xmath207 , and we observe that as @xmath208 , the first of these bounds increases while the second decreases . further , the first bound is finite at 0 and infinite at 1 , whereas the second is infinite at 0 and finite at 1 . thus there is an optimal @xmath188 which makes the two bounds equal , the @xmath209 which solves @xmath210 . consider the simplest case @xmath212 first . we can bound the sum as follows : @xmath213 to see this , note that @xmath214 because the sum is a lower riemann sum for the integral . since the summand is symmetric about @xmath215 , doubling this produces the desired result . now consider the case of even @xmath98 , i.e. @xmath216 . divide the @xmath5 into @xmath217 pairs , each of which sums to some number @xmath218 and these numbers in turn sum to @xmath1 : @xmath219 here the inequality follows by applying ( [ eq : mu=2 ] ) , which does not depend on @xmath1 , to each of the inner sums . further , @xmath220 where in the first equality we assume w.l.o.g . that @xmath221 , and the 2nd equality follows from the fact that the number of compositions of @xmath222 into @xmath223 parts ( i.e. the solutions of @xmath224 , @xmath225 ) , is @xmath226 . finally we bound the binomial coefficient by @xmath227 , to arrive at @xmath228 now we turn to the case of odd @xmath98 , i.e. @xmath229 . similarly to what we did above , @xmath230 by ( [ eq : mueven ] ) , the r.h.s . does not exceed @xmath231 and this last sum can be bounded by the integral @xmath232 we have thus shown that for @xmath229 , @xmath233 eqs . ( [ eq : mueven ] ) and ( [ eq : muodd ] ) establish the proposition for all @xmath102 . first we show that if @xmath234 then @xmath235 . by proposition [ prop : n1 ] , @xmath236 , so by proposition [ prop : theta_inf ] @xmath237 . further , @xmath238 means that @xmath239 by proposition [ prop : norment ] . therefore @xmath240 implies that @xmath16 belongs to the set @xmath241 as claimed . next we put a lower bound on @xmath242 . applying proposition [ prop : s ] we see that @xmath95 is @xmath243 the r.h.s . of ( [ eq : a ] ) in the proof of lemma [ le : nran ] with @xmath244 , so @xmath95 is @xmath243 the bound of lemma [ le : nran ] on @xmath245 with @xmath246 . then from the proof of theorem [ th:1 ] , we see that @xmath247 is @xmath243 the r.h.s . of ( [ eq : nranrb ] ) , but with the condition on @xmath1 being @xmath248 . the rest of the proof of theorem [ th:1 ] then applies , to the point where @xmath1 has to satisfy @xmath206 and @xmath249 . @xmath209 equalizes these bounds , and the completion of the proof of theorem [ th:1 ] then establishes that if @xmath250 , @xmath251 . the function @xmath252 , @xmath107 , is increasing for @xmath253 $ ] . the first implication in the proposition then follows immediately from theorem 16.3.2 of @xcite , the @xmath22 norm bound on entropy , which states that if two @xmath0-vectors @xmath254 are s.t . @xmath255 , then @xmath256 . to prove the second implication we use pinsker s inequality and the `` triangle inequality '' for cross- or relative entropy , or divergence @xmath257 . applied to @xmath59 and @xmath14 , pinsker s inequality states that @xmath258 ( see @xcite , lemma 12.6.1 ) . then the triangle inequality , using the uniform distribution as the prior or reference distribution ( @xcite , theorem 12.6.1 ) , implies that @xmath259 . what we want to prove follows from the above two inequalities . pinsker s inequality can be tightened in two ways : @xcite show that the @xmath260 can be replaced by a factor @xmath261 , and @xcite give right - hand sides that are polynomials involving powers of the norm beyond the square . entirely analogous to that of lemma [ le : nrbn ] , except that the set @xmath262 is defined by ( [ eq : newanbn ] ) instead of ( [ eq : bn ] ) , and the factor @xmath263 in ( [ eq : nrb1 ] ) , coming from the upper bound on @xmath95 of proposition [ prop : s ] , is replaced by the factor @xmath264 of proposition [ prop : norment2 ] . the proof follows that of lemma [ le : nran ] : first we lower - bound the size of @xmath265 and then the entropy of the @xmath59 in it . the basic difference is that here we have @xmath22 norms . if @xmath266 , so is @xmath267 , and then proposition [ prop : anlb ] says that the size of @xmath268 is at least @xmath269 . second , concerning the entropy of @xmath270 , by proposition [ prop : norment2 ] @xmath271 implies that @xmath272 is at least @xmath273 . the proof then follows that of lemma [ le : nran ] , except that the term @xmath274 in ( [ eq : a ] ) is replaced by @xmath275 . the proof uses lemmas [ le : nrbn1 ] and [ le : nran1 ] and is completely analogous to that of theorem [ th:1 ] . the main feature is that @xmath15 falls out of the new ( [ eq : nranrb ] ) , the exponential is @xmath281 with @xmath282 , and the condition on @xmath1 is now @xmath283 . @xmath192 are the same as in theorem [ th:1 ] , except for the denominators . finally , @xmath284 is finite at @xmath285 and increases to @xmath286 at @xmath287 , whereas @xmath288 is infinite at @xmath285 and decreases to a finite value at @xmath289 . thus the equation @xmath290 has a root @xmath209 between 0 and @xmath291 , which equates the two sides and is therefore the optimal @xmath188 . the proof is analogous to that of lemma [ cor:1 ] . first , by proposition [ prop : n1 ] , @xmath292 implies that @xmath293 . second , if @xmath294 then @xmath295 by the 2nd claim of proposition [ prop : n1 ] . hence if @xmath296 , @xmath16 belongs to the set @xmath265 . next , by the argument in the proof of lemma [ cor:1 ] , @xmath242 can be lower - bounded by the bound of lemma [ le : nran1 ] with @xmath246 . so @xmath297 is lower - bounded by the new ( [ eq : nranrb ] ) as in the proof of theorem [ th:2 ] but the condition on @xmath1 is now @xmath298 . the rest follows as in the proof of theorem [ th:2 ] . in this case there are no constraints , so by proposition [ prop : theta_inf ] @xmath299 . also , @xmath300 and @xmath301 further , if @xmath302 , then @xmath303 , so the condition of corollary [ cor : pd ] on @xmath1 is @xmath304 . the conditions @xmath302 and @xmath278 are satisfied if @xmath305 . finally , @xmath306 . m. benoit and s. verdoolage . polynomial approximations in the polytope model : bringing the power of quasi - polynomials to the masses . in _ proceedings of 6th workshop on optimizations for dsp and embedded systems ( odes-6 ) _ , pages 4554 . boston , massachusetts , april 2008 . s. verdoolaege , k. woods , m. bruynooghe , and r. cools . computation and manipulation of enumerators of integer projections of parametric polytopes . technical report , katholieke universiteit leuven , department of computer science , celestijnenlaan 200a , b-3001 heverlee ( belgium ) , march 2005 .

consider the construction of an object composed of @xmath0 parts by distributing @xmath1 units to those parts . for example , say we are assigning @xmath1 balls to @xmath0 boxes . each assignment results in a certain count vector , specifying the number of balls allocated to each box . if only assignments satisfying a set of constraints that are linear in these counts are allowable , and @xmath0 is fixed while @xmath1 increases , most assignments that satisfy the constraints result in frequency vectors ( normalized counts ) whose entropy approaches that of the maximum entropy vector satisfying the constraints . this phenomenon of `` entropy concentration '' is known in various forms , and is one of the justifications of the maximum entropy method , one of the most powerful tools for solving problems with incomplete information . the appeal of entropy concentration comes from the simplicity of the argument : it is based purely on counting and does not need probabilities . existing proofs of the concentration phenomenon are based on limits or asymptotics . here we present non - asymptotic , explicit lower bounds on @xmath1 for a number of variants of the concentration result to hold to any prescribed accuracies , taking into account the fact that allocations of discrete units can satisfy constraints only approximately . the results are illustrated with examples on die tossing , vehicle or network traffic , and the probability distribution of the length of a @xmath2 queue .

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Proceed to summarize the following text: a great deal of effort has gone into observing and analyzing disintegration of sunspots ( and starspots ) . the sunspot decay is usually characterized by the rate of decrease of the sunspot area @xmath3 , and numerous observations appear to be consistent with a parabolic decay law , with @xmath4 a decreasing quadratic function of time @xmath5 . early theories invoked turbulent diffusion of the magnetic field within the spot to model the observed rate of decay , yet such models predicted a linear decay law , corresponding to a constant area decay rate @xmath6 @xcite . in order to explain a parabolic decay , @xcite developed a model of sunspot disintegration by turbulent `` erosion '' of penumbral boundaries , which occurs when bits of magnetic field are sliced away from the edge of a sunspot and swept to the supergranular cell boundaries by supergranular flows @xcite . a key feature of the erosion model is that the turbulent diffusivity , associated with the flows , is suppressed within the spot . the assumption is justified by the theoretical prediction that the diffusivity @xmath0 should rapidly decrease if the magnetic field @xmath2 exceeds an energy equipartition value @xcite , which is why the diffusivity in the turbulent erosion model may be assumed to be a decreasing function of the magnetic field strength . @xcite presented observational evidence in support of the parabolic decay law and its theoretical explanation by turbulent erosion . @xcite recently revisited the theory of sunspot decay by turbulent erosion , considered as a moving boundary problem . while some of the earlier results were confirmed for moderate sunspot magnetic field strengths , the new analytical and numerical solutions yielded a significantly improved theoretical description of sunspot disintegration . in particular , the dependence of the spot area was shown to be a nonlinear function of time , which in a certain parameter regime can be approximated by a parabola . more accurate expressions for the spot lifetime in terms of an initial magnetic field were derived analytically and verified numerically . following @xcite , @xcite assumed in their study that the turbulent diffusivity @xmath7 within a decaying sunspot is much less than that outside the spot . a more realistic model should incorporate a more realistic dependence of the turbulent diffusivity on the field strength within the spot . our aim in this paper is further to develop the theory of turbulent erosion by exploring the effect of a non - vanishing diffusivity within a sunspot on the rate of its disintegration . theoretical mechanisms and observable features of the diffusive transport of the photospheric magnetic field have been a subject of intense research activity . our detailed analysis of a simple nonlinear model , reinforced by numerical solutions , can complement more detailed magnetohydrodynamic simulations ( e.g. * ? ? ? * ; * ? ? ? * ) and guide empirical models @xcite in studies of sunspot and starspot evolution . the determination of lifetimes of spots is a topic of general and current astrophysical interest . quantitative models for starspot evolution may help estimate the magnetic diffusion timescale and thus yield important constraints in stellar dynamo models , which provides further motivation for our exploration of the turbulent erosion model . following @xcite and @xcite , we model a decaying sunspot as a cylindrically symmetric flux tube . the evolution of the magnetic field @xmath8 is governed by the following nonlinear diffusion equation : @xmath9 where @xmath5 is time , and @xmath10 is the distance from the @xmath11-axis . the turbulent diffusivity @xmath0 is suppressed in a magnetic field exceeding an energy equipartition value @xmath12 @xcite , where @xmath13 g for typical parameters of the solar photosphere @xcite . it follows that @xmath0 is a decreasing function of @xmath2 , although the exact functional dependence remains uncertain . below we investigate several choices for @xmath7 , which yield analytical solutions . as previously @xcite , we choose dimensionless units so that @xmath14 , where @xmath15 and @xmath16 is the initial radius of the fluxtube . consequently the time is measured in units of @xmath17 . the initial value problem is specified by the dimensionless magnetic field profile @xmath18 at @xmath19 , where @xmath20 is the maximum field within the spot . another important parameter is the total magnetic flux of the spot @xmath21 an initially present sunspot corresponds to @xmath22 ( or @xmath23 in dimensional units ) . in the following we define the dimensionless fluxtube radius @xmath24 by the condition that @xmath25 at its edge . our choice of the length scale @xmath16 to be the initial radius of the spot means that @xmath26 and it follows that @xmath27 finally , we define the spot decay time @xmath28 by the condition @xmath29 which is equivalent to @xmath30 similarity solutions to partial differential equations have a large number of applications ( e.g. * ? ? ? in particular self - similar solutions to nonlinear diffusion equations have been considered for several forms of the diffusion coefficient ( e.g. * ? ? ? these similarity reductions not only lead to exact solutions of specific initial - value problems but also serve as intermediate asymptotics that approximate solutions of a much larger class of problems . here we consider a similarity reduction of the nonlinear two - dimensional diffusion equation ( [ eq - diffusion ] ) , assuming that the dimensionless diffusion coefficient has a power - law form : @xmath31 where we take @xmath32 in order to model the supression of magnetic diffusivity in the strong magnetic field of a sunspot . the self - similar solution to equation ( [ eq - diffusion ] ) , which satisfies the flux conservation condition @xmath33 , is known to take the form @xmath34 with @xmath35 @xcite . on substituting this form into equation ( [ eq - diffusion ] ) , solving for @xmath36 , and using equation ( [ eq - phi0-def ] ) to specify an integration constant , we get the following expression for an evolving field profile : @xmath37^{-1/\nu } . \label{eq - essim - solution}\ ] ] it follows that the maximum field ( at @xmath38 ) is given by @xmath39 and so @xmath40 now equation ( [ eq - b-1 - 0 ] ) yields the magnetic flux @xmath41 to find the decay time @xmath28 for a given initial magnetic field @xmath42 , we express the parameter @xmath43 in terms of @xmath42 and substitute the resulting expression into equation ( [ eq - b-0-t - selfsim ] ) . on solving equation ( [ eq - b-0-t ] ) for @xmath28 and using equation ( [ eq - phi0-selfsim ] ) to eliminate @xmath44 , we obtain the sunspot decay time in terms of the parameters @xmath42 and @xmath45 : @xmath46 in the limit @xmath47 , equation ( [ eq - t - selfsim ] ) reduces to the expression for the case of linear diffusion : @xmath48 note that the self - similar solution has a curious feature : equation ( [ eq - t - selfsim ] ) predicts that @xmath49 as @xmath50 ( for any value of @xmath45 ) , whereas it is physically obvious that @xmath51 when @xmath52 . this singular limit behavior is related to the fact that the magnetic flux @xmath53 as @xmath50 . in practice this does not cause any problems since we always assume the initial field @xmath22 in order to model a sunspot . the self - similar solution above is applicable only for @xmath54 since the flux integral in equation ( [ eq - phi0-def ] ) diverges for larger values of @xmath45 , which physically corresponds to an instanteneous flux transfer to infinity @xcite . more generally , @xmath55 is required to avoid the divergence of the flux integral for diffusion in @xmath56 dimensions . mathematical issues of existence and uniqueness of solutions were analyzed by @xcite . while solutions for @xmath57 formally violate the total flux conservation , they are mathematically correct and may provide a useful local description of nonlinear diffusion . as an illustration , consider the case @xmath58 . it is straightforward to derive a separable solution to equation ( [ eq - diffusion ] ) . by assuming @xmath59 , we reduce the problem to a second - order ordinary differential equation for the spatial part @xmath60 . the solution is as follows : @xmath61 where @xmath62 is an integration constant , and the other integration constant is determined by the requirement that @xmath63 be finite at @xmath38 . in a different context , equation ( [ eq - fnu1 ] ) in a particular case @xmath64 was given by @xcite . on using equations ( [ eq - b-0 - 0 ] ) and ( [ eq - phi0-def ] ) to express @xmath28 and @xmath62 in terms of @xmath42 and @xmath44 , we obtain @xmath65 here @xmath66 is the initial magnetic flux of the spot ( at @xmath19 ) , which decreases with time according to @xmath67 the localized magnetic field profile is seen to shrink with time until the spot vanishes at @xmath68 the termination of the process in a finite time was referred to by @xcite as `` superfast '' diffusion . this unusual feature of the solution is related to the singular behavior of @xmath69 as @xmath70 : in sharp contrast to the solution of a linear problem , the continuity flux @xmath71 is independent of time and approaches a constant value , @xmath72 , as @xmath73 , which results in `` flux suction at infinity '' @xcite . physically , because @xmath70 as @xmath74 , the diffusivity @xmath75 . consequently the diffusion time scale @xmath76 , and so a diffusive description breaks down as @xmath74 . we now consider a different model for the diffusivity suppression ( quenching ) within a spot , which complements the analysis by @xcite . we assume that the evolution of the magnetic field is described by the nonlinear two - dimensional diffusion equation ( [ eq - diffusion ] ) with a step dependence in the dimensionless diffusion coefficient : @xmath77 and @xmath78 @xcite considered the case @xmath79 , which corresponds to a very strong suppression of turbulent diffusivity within the spot . a more realistic model would correspond to a less severe turbulent suppression of the diffusivity within the spot . therefore we generalize our nonlinear diffusion model to incorporate the effect of a non - vanishing diffusivity within a sunspot on the rate of its disintegration . to obtain an analytical solution for the case of a moderate diffusivity suppression within the spot , we use the heat - balance integral method @xcite , which proved to yield accurate approximations in problems of nonlinear diffusion @xcite . the basic idea is to require that an approximate solution satisfy an integral of a nonlinear equation rather than the equation itself . we apply the method to describe the turbulent erosion of a sunspot , modeled as a cylindrically symmetric fluxtube . we assume that @xmath80 and seek an approximate solution of the form @xmath81 . \label{eq - b - form}\ ] ] the fluxtube radius @xmath24 is defined by equation ( [ eq - edge ] ) . consequently we have @xmath82 integration of equation ( [ eq - diffusion ] ) over @xmath10 from @xmath83 to @xmath84 and substitution of the self - similar form ( [ eq - b - form ] ) into the resulting equation yields @xmath85 = 4 \epsilon g r_e^2 , \label{eq - re - ode}\ ] ] where equation ( [ eq - edge ] ) was used to simplify the right - hand side , and equation ( [ eq - fg - rel ] ) was used to eliminate @xmath86 . in the linear case @xmath87 , the solution of the initial value problem with a gaussian profile at @xmath19 is given by @xmath88 here the parameters of an evolving gaussian profile are chosen to satify the initial conditions , given by equations ( [ eq - b-0 - 0 ] ) and ( [ eq - b-1 - 0 ] ) . the decay time , defined by equations ( [ eq - t - def ] ) and ( [ eq - b-0-t ] ) , is easily shown to be @xmath89 where , as previously , we assume @xmath22 in order to exclude solutions with an infinite magnetic flux . note for clarity that , if @xmath90 , we have @xmath91 at @xmath19 in the linear solution , and so diffusion causes the fluxtube radius to increase until it reaches a maximum at @xmath92 and then to decrease . motivated by the form of the linear solution , we substitute @xmath93 into the approximate equation ( [ eq - b - form ] ) describing a weakly nonlinear case @xmath94 . thus equation ( [ eq - re - ode ] ) becomes an ordinary differential equation for the fluxtube radius @xmath24 , which should be solved subject to the initial condition given by equation ( [ eq - r=1 ] ) . equation ( [ eq - t - def ] ) then yields the spot decay time @xmath28 . keeping in mind that @xmath95 is assumed to be small , we solve equation ( [ eq - re - ode ] ) by iteration . on substituting a simple linear function @xmath96 into the right - hand side of equation ( [ eq - re - ode ] ) and integrating from @xmath83 to @xmath5 , we get an approximate analytical expression for @xmath97 : @xmath98 \right\ } . \label{eq - re - approx}\ ] ] on setting @xmath99 , we obtain an algebraic equation for the decay time @xmath28 : @xmath100 = 1 . \label{eq - t - algebr}\ ] ] alternatively , integration of equation ( [ eq - re - ode ] ) over @xmath5 from @xmath83 to @xmath28 yields an expression for @xmath28 , which makes clear that @xmath101 leads to a slower decay : @xmath102 now substitution of equations ( [ eq - g - fun ] ) and ( [ eq - re2-lin ] ) into the right - hand side of equation ( [ eq - t - integral ] ) yields equation ( [ eq - t - algebr ] ) . to solve equation ( [ eq - t - algebr ] ) in the case of a small @xmath95 , we replace @xmath28 by @xmath103 from equation ( [ eq - t - lin ] ) in all terms containing @xmath95 . the result is as follows : @xmath104 . \label{eq - t - approx}\ ] ] for a fixed @xmath105 , @xmath106 is an increasing function of @xmath42 . we note that the assumed form for the magnetic diffusivity @xmath0 , given by equations ( [ eq - dc - step1 ] ) and ( [ eq - dc - step2 ] ) , is finite in the limiting case @xmath107 , whilst the real magnetic diffusivity is expected to vanish in this limit . however , since @xmath108 remains finite in our solution for any @xmath109 , the behavior of @xmath0 as @xmath107 is irrelevant . our assumption of a suppressed but nonzero diffusivity inside the spot is physically meaningful as long as the magnetic field @xmath2 remains finite , as it does in our solution . we have previously considered a solution for @xmath110 valid when the diffusivity vanishes within the spot @xcite , which is accurate for a very strong magnetic field @xmath42 in the spot . the new solution presented here quantifies the effects of a nonvanishing @xmath0 . to quantify the accuracy of the analytical results , equation ( [ eq - diffusion ] ) is solved numerically . we use an explicit scheme which maps the region @xmath111 $ ] in radius @xmath10 to the region @xmath112 $ ] in a transformed independent variable @xmath113 , as explained in the appendix . the approach allows an exact boundary condition to be imposed at @xmath114 , which provides more accurate solutions than an approximate boundary condition at a finite radius . we present solutions with a grid spacing @xmath115 in @xmath113 and with a time step one quarter of the stability limit identified in the appendix . to test the numerical method , and to illustrate the properties of the analytic solutions presented in section 3 , we consider solution with the power - law form for the diffusion coefficient , equation ( [ eq - plaw - dcoeff ] ) , which admits self - similar solutions . first we consider a flux - conserving case ( @xmath116 ) , with @xmath117 . the solid curves in the upper panel of figure 1 show the analytic magnetic field profile @xmath110 given by equation ( [ eq - essim - solution ] ) at the three times during the spot evolution @xmath19 , @xmath118 , and @xmath119 , where @xmath28 is the decay time defined by equation ( [ eq - t - selfsim ] ) . the numerical solutions are also shown in this panel by dashed curves , but they coincide with the solid curves and are not visible . the lower panel shows the absolute error in the numerical solution at the times @xmath118 ( circles ) and @xmath119 ( plus signs ) . the maximum error is @xmath120 . the numerical solution conserves flux throughout the time evolution ( @xmath121 time steps ) to within @xmath122 . second we consider the non - flux conserving case @xmath123 , which exhibits superfast diffusion , i.e. vanishing of the spot at the time @xmath28 defined by equation ( [ eq - t - sfast ] ) . the upper panel of figure 2 shows the analytic and numerical solutions at the three times @xmath19 , @xmath118 , and @xmath119 , for the case @xmath124 , with a log - linear display used to illustrate the behaviour for large @xmath10 . the analytic solutions are the solid curves , and the numerical solutions at times @xmath118 and @xmath119 are shown by the circles and by the plus signs , respectively . the analytic solution at time @xmath119 is identically zero . the numerical solution is accurate initially , but becomes inaccurate as @xmath125 . the reason for the error is that the boundary condition at infinity , equation ( [ eq - bc2 ] ) , does not reproduce the behaviour of the continuity flux at large @xmath10 , as shown in the lower panel . in particular the numerical solution does not maintain a large positive value of the continuity flux as @xmath126 , which produces the `` flux suction at infinity '' . this example demonstrates the need to accurately represent the boundary conditions at large @xmath10 in the solutions . the numerical solution for the case with a diffusion coefficient defined by equations ( [ eq - dc - step1 ] ) and ( [ eq - dc - step2 ] ) provides a test of the heat - balance integral method results presented in section 4 , and in particular of the expression ( [ eq - t - approx ] ) for the lifetime . in this case we do not have an exact solution to ensure accuracy , but the numerical solutions presented here conserve flux during the time evolution to within @xmath128 , in the worst case . first we consider the field profile @xmath110 for the heat - balance solution , specified by equations ( [ eq - b - form ] ) , ( [ eq - fg - rel ] ) , ( [ eq - g - fun ] ) , and ( [ eq - re - approx ] ) . figure 3 compares the heat - balance method profiles ( solid curves ) with the numerical solutions ( dashed curves ) at the three times @xmath19 , @xmath118 , and @xmath119 . the solutions assume an initial magnetic field strength @xmath117 . the left panel shows the case @xmath129 , and the right panel shows the case @xmath130 . for the smaller value of @xmath95 the heat - balance integral method provides a good approximation to the field profile throughout the evolution . the method is somewhat less accurate for the larger value of @xmath95 . figure 4 repeats the display in figure 1 , but shows the case @xmath131 . the accuracy of the field profile obtained with the heat - balance integral method is not strongly dependent on the choice of @xmath42 . second , we consider the accuracy of the heat - balance integral estimate for the spot lifetime , equation ( [ eq - t - approx ] ) . the estimate has the form @xmath132 where @xmath133 is the lifetime for the linear case ( @xmath134 ) , given by equation ( [ eq - t - lin ] ) . figure 5 compares equation ( [ eq - t - approx - taylor ] ) ( solid curves ) with the numerical solution ( circles ) as a function of @xmath42 , for the range @xmath135 . the upper panel shows @xmath28 for the three cases @xmath87 , @xmath136 , and @xmath130 ( bottom to top ) and the lower panel shows @xmath137 for the nonlinear cases @xmath129 and @xmath130 . figure 5 demonstrates that the heat - balance method provides a good approximation to the lifetime of the spot for the choice @xmath129 , over the range of initial field strengths considered . the approximation is worse for the larger value of @xmath95 , as expected . it is interesting to consider replacing equation ( [ eq - t - approx - taylor ] ) by the @xmath138 pad approximant : @xmath139 the dotted curves in figure 5 show the lifetimes given by equation ( [ eq - t - approx - pade ] ) , and the results show that the pad expression provides a better approximation for the lifetime . this might be expected based on the final steps in the derivation of equation ( [ eq - t - approx ] ) . the physical mechanism of sunspot erosion was proposed by @xcite . @xcite introduced a nonlinear diffusion equation of a magnetic flux tube as a model of turbulent erosion . @xcite identified the maximum magnetic field @xmath42 in a sunspot as the key parameter that determines the lifetime @xmath28 of the spot and derived a sunspot decay law due to turbulent erosion . as @xcite demonstrated , however , the accuracy of the predictions was limited : for instance , @xcite have shown that the sunspot lifetime in the model is about a half of that originally predicted . we have presented in this paper a further development of the quantitative theory of sunspot disintegration by turbulent erosion . our analysis makes it clear that a variety of scalings @xmath140 are theoretically possible , depending on initial conditions and the dependence of the turbulent diffusivity on the magnetic field strength within the spot . the results may have implications for the use of the model for estimating the magnetic diffusion timescale in starspots , which is an important parameter in stellar dynamo models . experimental studies typically attempt to infer the anomalous , turbulent - driven magnetic diffusivity by equating the lifetime of a spot to a theoretical diffusion time scale @xcite . we expect that the more detailed analytical models we have developed may help to improve the accuracy of the procedure . more generally , we expect that the new models may allow insight into differences and similarities between sunspots and starspots . the original formulation of the turbulent erosion model predicted a parabolic dependence of the sunspot area on time @xcite . yet the deviations from the parabolic decay are large for any one spot @xcite , motivating the use of continuous linear piecewise functions in modeling of the starspot growth and decay @xcite . such an approach is consistent with the more general decay laws predicted in our analysis , ultimately controlled by the dependence of the magnetic diffusivity on the magnetic field strength within a spot . as pointed out by the referee , however , apparent deviations from the parabolic decay law for individual spots can be caused by the difficulty of identifying the individual spots within a decaying sunspot group or by the effect of a varying external plage field outside the spots @xcite . the turbulent erosion model can be further improved . recall that the solution for @xmath79 in @xcite describes turbulent erosion of a sunspot as a moving boundary problem in which the rate of sunspot decay is controlled by the inward speed of a current sheet around the spot . in other words , the decay rate is determined by the local diffusion rate of magnetic field within the sheet , modeled as a tangential discontinuity at @xmath141 . by contrast , the new solution given by equations ( [ eq - re - approx ] ) and ( [ eq - t - approx ] ) describes the sunspot decay determined by global magnetic field diffusion , and the field discontinuity is ignored in the smooth profile of the evolving magnetic field , which should be a reasonable assumption if @xmath94 . the diffusive evolution of exact self - similar solutions is also a global process . yet physically the sunspot disintegration rate is likely to be influenced by both mechanisms , and so a more accurate solution for intermediate values of @xmath95 should incorporate both local and global diffusion processes by considering a more general initial field profile in the spot and more realistic diffusivity dependence on the magnetic field strength . more detailed models of spot decay should also quantify the effects of flux cancellation , caused by photospheric magnetic reconnection ( e.g. * ? ? ? application of the model to the data on sunspot and starspot decay may shed light on the physics of turbulent diffusion in magnetized astrophysical plasmas . an anonymous referee s comments and suggestions are gratefully acknowledged . @xcite numerically solved equation ( [ eq - diffusion ] ) using a crank nicolson scheme . the method imposed a boundary condition at a finite outer boundary @xmath142 which allowed flux transport across the boundary , using one - sided spatial derivatives . this approach was an improvement over the assumption of zero flux at an outer radius used by @xcite , but it was still a source of error for the long time integrations necessary to determine the lifetimes for the model spots . here we use a simpler explicit scheme , but with a transformation of the infinite @xmath10-domain to a finite domain , to allow an exact outer boundary condition . we consider solution of equation ( [ eq - diff - xr ] ) on a uniformly spaced grid in @xmath113 defined by @xmath147 , with @xmath148 , where @xmath149 is the grid spacing . similarly we consider discrete times @xmath150 , with @xmath151 , where @xmath152 is a constant time step . a suitable forward - time , centred - space ( ftcs ) discretisation of equation ( [ eq - diff - xr ] ) is @xcite : @xmath153 where @xmath154 and @xmath155 , and where @xmath156 . the diffusion coefficients at intermediate grid points may be approximated by @xmath157 equations ( [ eq - diff - xr - ftcs ] ) and ( [ eq - dcoeff - approx ] ) give the numerical scheme @xmath158 . \label{eq - numerical - scheme}\ ] ] equation ( [ eq - numerical - scheme ] ) is an explicit prescription for time evolution at grid locations @xmath159 . for @xmath160 ( @xmath38 ) the physical boundary condition is @xmath161 for all times . equation ( [ eq - bc1 - 1 ] ) may be enforced using a one - sided ( forward ) approximation to the derivative with respect to @xmath113 at @xmath162 : @xmath163 giving the update for the grid point @xmath160 : @xmath164 this approach requires that the initial conditions satisfy equation ( [ eq - bc1 - 2 ] ) , i.e. @xmath165 . under the transformation ( [ eq - xtor ] ) the total magnetic flux , defined by equation ( [ eq - phi0-def ] ) , becomes : @xmath169b[r(x),t)\,dx . \label{eq - phi0-def - x1}\ ] ] equation ( [ eq - phi0-def - x1 ] ) may be evaluated in our discrete version of the problem using the trapezoidal rule @xcite : @xmath170 where @xmath171 for @xmath172 , and @xmath173 . equation ( [ eq - phi0-def - x2 ] ) is used to check that flux is approximately conserved by the numerical solution . a simple estimate of the stability condition for the method may be made as follows . we expect that an ftcs discretisation of equation ( [ eq - diffusion ] ) is stable at a given time step subject to @xcite : @xmath174 where @xmath175 is the grid spacing . ( strictly this requires a uniform grid in @xmath10 . ) from equation ( [ eq - xtor ] ) we have @xmath176 so equation ( [ eq - ftcs - stability1 ] ) becomes @xmath177 in the case of the model with a step dependence of @xmath0 on @xmath2 ( section 4 ) we can take @xmath178 numerical experimentation suggests that equation ( [ eq - ftcs - stability - hbal ] ) provides a good estimate for the actual stability constraint .

quantitative models of sunspot and starspot decay predict the timescale of magnetic diffusion and may yield important constraints in stellar dynamo models . motivated by recent measurements of starspot lifetimes , we investigate the disintegration of a magnetic flux tube by nonlinear diffusion . previous theoretical studies are extended by considering two physically motivated functional forms for the nonlinear diffusion coefficient @xmath0 : an inverse power - law dependence @xmath1 and a step - function dependence of @xmath0 on the magnetic field magnitude @xmath2 . analytical self - similar solutions are presented for the power - law case , including solutions exhibiting `` superfast '' diffusion . for the step - function case , the heat - balance integral method yields approximate solutions , valid for moderately suppressed diffusion in the spot . the accuracy of the resulting solutions is confirmed numerically , using a method which provides an accurate description of long - time evolution by imposing boundary conditions at infinite distance from the spot . the new models may allow insight into differences and similarities between sunspots and starspots .

You are an expert at summarizing long articles. Proceed to summarize the following text: wormholes are hypothetical objects which connect two distant parts of the same spacetime or two different spacetimes by a throat - like object which has the minimum radius of the spacetime . although a wormhole solution first entered the physics literature in 1916 @xcite , the concept was first considered seriously in 1935 by einstein and rosen @xcite which was later called einstein - rosen bridge but the word wormhole was first time coined by wheeler @xcite in 1957 . a more interesting analysis of wormholes was performed by morris and thorne in 1988 and they presented a new kind of wormhole ( traversable wormhole ) for the first time @xcite . it was known from before that matter we need to support such a geometry violates the weak and strong energy conditions near the throat @xcite . morris and thorne reconsidered these conditions for a traversable wormhole @xcite . since the matter that supports this geometry does nt satisfy the common energy conditions they called it exotic. an example of exotic matter is matter with negative energy density @xcite . another property of these wormholes is the possibility of transforming them into time machines for backward time traveling @xcite and thereby , perhaps for causality violation by closed timelike curves . teo @xcite found that the null energy condition ( @xmath1 ) is violated by stationary , axially symmetric , traversable wormholes but there can be classes of geodesics which do not cross energy condition violating regions . another case of evolving wormholes may alter this situation@xcite . it is known that we can have violations of weak energy condition ( @xmath2 ) due to some quantum mechanical effects ( such as casimir effect ) @xcite . if we search for this effect in the history of evolving cosmos , we can find such a situation at the quantum cosmological era when quantum gravity is dominant . following the inflation theory by a. guth @xcite it has been supposed that non - trivial topological objects such as microscopic wormholes may have been formed during that era and then enlarged to macroscopic objects with expansion of the universe @xcite . although most studies have focused on four dimensions , with the advent of string theory that demands higher dimensional spacetimes , it is natural to examine the possibility of wormholes beyond the ordinary four dimensions . euclidean wormholes in string theory have recently been studied @xcite as solutions of supergravities . with this motivation , we are going to choose the generalized friedmann - robertson - walker ( @xmath3 ) metric in @xmath0-dimensions and investigate exact lorentzian wormhole geometries with spherically symmetry . this paper is organized in the following manner : in sec . [ field ] we present the ansatz metric and the resulting solutions in 4-dimensions . in section [ ndim ] , we extend the solutions to ( n+1)-dimensions . in section [ energy ] , we investigate the corresponding energy - momentum tensor and determine the exoticity parameter . the last section is devoted to conclusions and closing remarks . it is very common today that we start studying cosmology with the so - called cosmological principle. it states that the universe at large scale is homogeneous and isotropic . with this assumption , we find out that the metric we need to demonstrate such a spacetime in 4-dimensions is as follows @xmath4 , \label{fe1}\ ] ] which is known as the robertson - walker ( rw ) metric @xcite . the coordinate system @xmath5 used here is the so - called co - moving coordinate system. @xmath6 is the scale factor and the only dynamical parameter to determine . @xmath7 correspond to spatially flat , closed and open spacetimes respectively . this metric contains a high degree of symmetry which is demonstrated by its six killing vectors . for our aim , we need to generalize this metric to @xmath0-dimensions and reduce its symmetry . we break its homogeneity by replacing @xmath8 with @xmath9 . this metric is still isotropic about @xmath10 but not necessarily homogeneous . we therefore write our ansatz metric as @xmath11 , \label{fe2}\ ] ] where @xmath12 is an unknown function . it is clear that the robertson - walker metric is a special case of this metric . with this ansatz metric , we look at our equations . we start with @xmath13 and then extend the solutions to arbitrary @xmath14 . in the first step we write our ansatz metric for 5-dimensions ( n=4 ) @xmath15 . \label{fe201}\ ] ] the non - vanishing components of the einstein tensor for our ansatz metric read @xmath16 @xmath17 @xmath18 @xmath19 in which dot is derivative with respect to @xmath20 , while prime is derivative with respect to @xmath21 . such a geometry is supported by an anisotropic , diagonal energy - momentum tensor : @xmath22 , \label{fe7}\ ] ] @xmath23 , \label{fe8}\ ] ] @xmath24.\label{fe9}\ ] ] in these relations @xmath25 and @xmath26 are the transverse and radial pressures , respectively . here , we assume the following equation of state @xmath27 where @xmath28 depends on @xmath14 the dimension of the space . this equation reduces to the vacuum equation of state @xmath29 when @xmath30 . using equations ( [ fe7]-[fe9 ] ) and relation ( [ fe10 ] ) in 5-dimensions ( n=4 ) , we obtain : @xmath31 + \\\nonumber & & \frac{1}{1+\gamma}\left [ \left[-3(\frac{r\ddot r+\dot r^2}{r^2})-3\frac{a}{r^2r^2(1+a)}\right ] + \gamma \left[-3(\frac{r\ddot r+\dot r^2}{r^2})-\frac{ra^{\prime}+a(1+a)}{r^2r^2(1+a)^2}\right]\right]=0.\label{fe11}\end{aligned}\ ] ] fortunately , this equation can be separated into radial and temporal equations . @xmath32 this equation can be easily solved and has the following solutions : @xmath33 @xmath34 the energy - momentum tensor needed to support this geometry has the following components : @xmath35 @xmath36 and @xmath37 the solutions we obtained in 5-dimensional spacetime can be easily extended to @xmath0-dimensions . the resulting solutions in ( n+1)-dimensions read @xmath38 @xmath39 following these solutions , the energy - momentum tensor which supports this structure in ( n+1)-dimensions will be : @xmath40 @xmath41 @xmath42 as a check for the correctness of these solutions one can see that they reduce to the solutions of @xcite for n=3 with proper definition for the integration constants . now let us have a look at radial behavior of the solutions . in the paper by morris and thorne @xcite , the metric of the wormhole is written in the form : @xmath43 in which @xmath44 is the shape function and the throat radius satisfies @xmath45 . if the equation @xmath46 has any root @xmath47 and simultaneously @xmath48 for @xmath49 then we will have a wormhole and @xmath47 gives the throat radius of the wormhole . in our solutions , with choosing @xmath50 the condition for the existence of wormholes will be @xmath51 and @xmath52 where @xmath53 and @xmath54 are constants related to the integration constants : @xmath55 solving equation ( [ fend7 ] ) analytically is not possible except for special values of n , as presented in @xcite for n=3 . then for investigating the properties of the solutions , we plot @xmath56 against @xmath21 in figure ( [ i1]-[i2 ] ) . we classify the solutions in the following manner : in the case ( a ) , @xmath57 and @xmath58 , we see from fig.[i1 ] that we have a lower limit on @xmath21 which corresponds to the throat radius of the wormhole and we see also that there is no upper limit on @xmath21 which reminds us that we have an open spacetime . for the choice ( b ) , @xmath59 and @xmath58 , we have lower and upper limits on @xmath21 . the lower limit corresponds to the throat radius of the wormhole and the upper limit signifies a closed spacetime . in the cases ( a ) and ( b ) , the kretschman scalar blows up at @xmath10 but , this point is not included in our physical spacetime with proper signature . case ( c ) , @xmath60 and @xmath61 represents a naked singularity in a closed cosmological background , because the kretschman scalar blows up at the origin ( @xmath10 ) . case ( d ) , @xmath62 and @xmath63 leads to @xmath64 , which corresponds to the @xmath65 @xmath3 metric . case ( e ) , @xmath66 and @xmath67 , @xmath9 becomes negative and the metric s signature is not of cosmological interest . and finally , ( f ) , @xmath68 and @xmath69 , @xmath9 is regular everywhere and represent a naked singularity , again but in an open universe . we can also have a discussion on the cases which include @xmath70 . for case @xmath71 , from ( [ fend2 ] ) we obtain @xmath72 this equation with @xmath73 and @xmath58 , leads to a wormhole centered , open universe which is illustrated in fig.[i2 ] part ( g ) . it is worth having a look at the ricci scalar ( @xmath74 ) . in ( n+1)-dimensional spacetime @xmath74 is : @xmath75 we can see that choosing @xmath76 or @xmath50 , constant curvature spacetime is retrieved . in the case @xmath76 , we have a maximally symmetric de sitter spacetime . we expect this result because , we see that the constant part of the ricci scalar reminds us a maximally symmetric spacetime curvature . the wormholes discussed in this paper are traversable . we present two reasons here . the first reasoning is based on the redshift of a signal emitted at the comoving coordinate @xmath77 and received by a distant observer . using the metric ( [ fe201 ] ) and for a radial beam , we obtain @xmath78 using this relation for two signals separated by @xmath79 in time when emitted ( and @xmath80 when detected ) , we obtain @xmath81 in which @xmath82 is the scale factor at the time of observation , and @xmath83 is the scale factor at the time of emission . this leads to the exactly same relation as the cosmological redshift relation which shows that the wormhole does not introduce extra ( local ) redshift . light signals , therefore can travel to the both sides of the throat and there is no horizon . the second is based on the geodesic equation , which -for the metric ( [ fe201])- leads to @xmath84 and @xmath85 in which @xmath86 is an affine parameter along the geodesic . the first equation has the following first integral @xmath87 since the proper distant element is @xmath88 , we see that there is no radial turning point and any particle can move in either radial directions at any point near to the wormhole , which clearly shows that the wormhole is traversable . let us have a brief discussion on the observational consequences of these solutions . we can have a variety of geodesics in a wormhole spacetime @xcite . geodesics can pass through the throat right to the other part of the spacetime ( the parallel universe ) , or be deflected back to the same universe . as pointed out by cramer et . @xcite , it should be possible -in principle- to detect wormholes via the gravitational lensing they cause . this , however , depends on the wormhole being stable , which is not addressed in the present paper . the @xmath0-dimensional version of diagonal elements of the energy- momentum tensor are were given in equations ( [ fend3]-[fend5 ] ) . some points are interesting to mention about the energy - momentum tensor . since we are looking for spherical structures in a cosmological background , @xmath25 , @xmath26 and @xmath89 should become almost @xmath21-independent at large @xmath21 . in particular , for suitable values of @xmath28 , the solutions approach @xmath90 which correspond to dark energy ( cosmological constant ) . the second point is the conservation equation of the energy - momentum tensor : @xmath91 which leads to @xmath92 our solutions satisfy this equation , as expected . let us now investigate the exoticity parameter . for an isotropic fluid , the energy - momentum tensor is in the form @xmath93 and the exoticity parameter is defined according to @xmath94 where @xmath95 corresponds to the exotic matter and @xmath96 to the non - exotic matter . since we have an anisotropic medium , the energy - momentum tensor has the form @xmath97 . we therefore take the average pressure @xmath98 taking this into account , we adopt the following , more general definition for @xmath99 : @xmath100 and modify ( [ e7 ] ) according to @xmath101 from ( [ fe10 ] ) we see that the modified exoticity parameter is everywhere zero and for all cases , which shows that we are on the border line between exotic and non - exotic matter according to the criterion . the weak energy condition @xmath102 requires @xmath103 for every nonspacelike @xmath104 which leads to @xcite @xmath105 these equations and relations ( [ fend3]-[fend5 ] ) lead to @xmath106 @xmath107 @xmath108 these relations show that , with the choice @xmath109 , we ca nt have any wormhole without violating @xmath2 , or at least the border line at which we have the equality relation in ( [ e13 ] ) and ( [ e14 ] ) . in the case @xmath110 the relations ( [ e13 ] ) and ( [ e14 ] ) are both simultaneously satisfied and we have to investigate the relation ( [ e12 ] ) in order to see wether @xmath2 is satisfied . it is interesting to note that for particular ranges of constants we have wormhole solutions which satisfy @xmath2 throughout the spacetime . as an example of these solutions one can see case ( g ) in fig.[i2 ] . we introduced time - dependent wormholes in an @xmath0-dimensional expanding cosmological background . we presented an ansatz metric and a linear relation between the diagonal elements of the energy - momentum tensor . with these assumptions , we separated and solved the field equations . solutions were classified into different categories with distinct geometries for the central object . we distinguished two classes of solutions which represented lorentzian wormholes with expanding throat in closed and open universes . we also found two classes of solutions that contain an intrinsic singularity in open and closed universes . the other interesting solution was the familiar maximally symmetric de sitter spacetimes . the ricci scalar was calculated and the constant curvature class of solutions were discussed . the issue of the traversability of the wormhole solutions was considerd . we investigated the energy - momentum tensor required to support these solutions . our solutions led to an energy - momentum tensor which approaches the cosmological constant case far from the wormhole for @xmath111 . finally , we investigated the exoticity parameter . the exoticity parameter for the linear equation of state is zero everywhere which is border line between exotic and non - exotic matter . we also checked the weak energy condition and found that the weak energy condition in the choice @xmath112 is violated except for the border line case . for the case @xmath70 , @xmath2 is satisfied for suitable values of constants .

we discuss @xmath0-dimensional dynamical wormholes in an evolving cosmological background with a throat expanding with time . these solutions are examined in the general relativity framework . a linear relation between diagonal elements of an anisotropic energy - momentum tensor is used to obtain the solutions . the energy - momentum tensor elements approach the vacuum case when we are far from the central object for one class of solutions . finally , we discuss the energy - momentum tensor which supports this geometry , taking into account the energy conditions .

You are an expert at summarizing long articles. Proceed to summarize the following text: compactness is a fundamental and important property in both theory and applications @xcite . compactness also plays an important role in the applications of fuzzy sets @xcite . the compact criteria attracts much attention . it s well - known that arzel ascoli theorem(s ) provide compact criteria in classic analysis and topology . there also exist many important and interesting works including @xcite which characterized compactness in fuzzy sets spaces equipped with different topologies . since diamond and kloeden @xcite introduced @xmath4 metric which is a @xmath0-type metric , it has become one of the most often used convergence structure on fuzzy sets . naturally , people have started to consider the characterizations of compactness in fuzzy sets spaces with @xmath4 metric . diamond and kloeden @xcite gave compact criteria of fuzzy number space @xmath5 with @xmath4 metric . ma @xcite modified the characterization given by diamond and kloeden . wu and zhao @xcite pointed out that characterization proposed by ma is still wrong , and they gave a right characterization in @xmath6 . convexity is a very useful property . star - shapedness is a natural extension of convexity . of cause , research on fuzzy counterparts of star - shaped sets has aroused the interest of people @xcite . diamond @xcite introduced the set of fuzzy star - shaped numbers which is denoted by @xmath7 . he characterized the compact sets in @xmath8 , where @xmath9 denotes the set of all the fuzzy star - shaped numbers with respect to the origin . @xmath5 and @xmath9 do not include each other . they both are subsets of @xmath7 . wu and zhao @xcite pointed out that this characterization is wrong and gave a characterization of compactness in @xmath8 . based on the results in @xcite , zhao and wu @xcite further proposed a characterization of compactness in @xmath10 . in these discussions , it is found that the concept `` @xmath11-mean equi - left - continuous '' proposed by diamond and kloeden @xcite plays an important role in establishing and illustrating characterizations of compactness in fuzzy sets spaces with @xmath4 metric . compare the characterizations in @xcite to arzel ascoli theorem , we find that the latter provides the compact criteria by characterizing the totally bounded sets while the former does not seem to characterize the totally bounded sets . since , in metric space , totally boundedness is a key feature of compactness , it is a natural and important problem to consider how to characterize totally bounded sets in spaces of fuzzy sets with @xmath4 metric ? qiu et al.@xcite introduced @xmath12 , the set of all general fuzzy star - shaped numbers , which take fuzzy star - shaped numbers as a special case . it is a more natural fuzzy extension of star - shaped sets in some sense . so this has caused an important problem : how to characterize compact sets in @xmath13 ? as diamond @xcite pointed out , the four spaces @xmath6 , @xmath8 , @xmath10 and @xmath13 are not complete . krtschmer @xcite presented the completion of @xmath6 which is described by the support functions of fuzzy numbers . it raises a natural and basic problem : what are the completions of all the rest spaces ? in this paper , we want to answer the above three questions , they are relevant to each other . an interesting fact is that our discussion can be put in a more general framework which does not have any assumptions of convexity or star - shapedness . so we consider the set of all normal , upper semi - continuous , compact - support fuzzy sets on @xmath3 , which is denoted by @xmath14 . further we introduce @xmath15 , the @xmath0-type extension of @xmath14 . all the fuzzy sets spaces mentioned in this paper are subspaces of @xmath16 . the remainder part of this paper is organized as follows . since @xmath4 metric are based on the well - known hausdorff metric , section 2 introduces and discusses some properties of hausdorff metric . in section 3 , we recall and introduce some concepts and results of fuzzy sets related to our paper . then , in section 4 , we present characterizations of relatively compact sets , totally bounded sets and compact sets in @xmath17 . section 5 shows that @xmath15 is in fact the completion of @xmath18 according to @xmath4 metric . then it gives the characterizations of relatively compact sets , totally bounded sets and compact sets in @xmath19 . these are key results of this paper . based on the conclusions in sections 4 and 5 and discussions on convexity and star - shapedness of fuzzy sets , in section 6 , it shows that @xmath20 , @xmath21 , @xmath22 and @xmath23 are just the completions of @xmath24 , @xmath6 , @xmath10 and @xmath13 , respectively . we clarify the relation among the ten fuzzy sets spaces discussed in this paper . as consequences of preceding results , it obtains characterizations of totally bounded sets , relatively compact sets and compact sets in these spaces . let @xmath25 be the set of all natural numbers , @xmath26 be the set of all rational numbers , @xmath27 be @xmath28-dimensional euclidean space , @xmath29 be the set of all the nonempty compact and convex sets in @xmath27 , @xmath30 be the set of all nonempty compact set in @xmath27 , and @xmath31 be the set of all nonempty closed set in @xmath27 . the well - known hausdorff metric @xmath32 on @xmath33 is defined by : @xmath34 for arbitrary @xmath35 , where @xmath36 @xcite [ kcs ] @xmath37 is a complete metric space in which @xmath38 and @xmath39 are closed subsets . hence , @xmath38 and @xmath29 are also complete metric spaces . @xcite [ sca ] a nonempty subset @xmath40 of @xmath41 is compact if and only if it is closed and bounded in @xmath41 . @xcite [ mce ] let @xmath42 satisfy @xmath43 then @xmath44 and @xmath45 . on the other hand , if @xmath46 and @xmath47 , then @xmath45 . a set @xmath48 is said to be star - shaped relative to a point @xmath49 if for each @xmath50 , the line @xmath51 joining @xmath52 to @xmath53 is contained in @xmath54 . the kernel ker@xmath54 of @xmath54 is the set of all points @xmath49 such that @xmath55 for each @xmath50 . the symbol @xmath56 is used to denote all the star - shaped sets in @xmath27 . obviously , @xmath57 . it can be checked that @xmath58 for all @xmath59 . we say that a sequence of sets @xmath60 converges to @xmath61 , in the sense of kuratowski , if @xmath62 where @xmath63 in this case , we ll write simply @xmath64 . the following two known propositions discuss the relation of the convergence induced by hausdorff metric and the convergence in the sense of kuratowski . the readers can see @xcite for details . [ hms ] suppose that @xmath65 , @xmath66 , @xmath67 , are nonempty compact sets in @xmath27 . then @xmath68 as @xmath69 implies that @xmath70 . [ klhe ] suppose that @xmath65 , @xmath66 , @xmath67 , are nonempty compact sets in @xmath27 and that @xmath66 , @xmath67 , are connected sets . if @xmath71 , then @xmath68 as @xmath69 . [ ksc ] @xmath56 is a closed set in @xmath72 . * proof * suppose that @xmath73 , @xmath74 and @xmath68 as @xmath69 . in the following , we will prove that @xmath75 . choose @xmath76 , @xmath67 , then there exists an @xmath77 such that @xmath78 for all @xmath79 , where @xmath80 . note that @xmath40 is a compact set , we know that there is a subsequence @xmath81 of @xmath82 such that @xmath83 . so @xmath84 , it then follows from proposition [ hms ] that @xmath85 . now , we show that @xmath65 is star - shaped and @xmath86 . it suffices to show that @xmath87 for all @xmath88 and @xmath89 $ ] . in fact , given @xmath88 , since @xmath90 , there is a sequence @xmath91 such that @xmath92 . hence , for each @xmath89 $ ] , @xmath93 and thus , by proposition [ hms ] , @xmath94 . @xmath95 [ kef ] let @xmath65 , @xmath66 be star - shaped sets , @xmath67 . if @xmath68 , then @xmath96 . * proof * from the proof of theorem [ ksc ] , we get the desired results . we do not know whether theorem [ ksc ] and corollary [ kef ] are known conclusions , so we give our proofs here . in this section , we recall and introduce various spaces of fuzzy sets including fuzzy numbers space , fuzzy star - shaped numbers space and general fuzzy star - shaped numbers space . it will be shown in this paper that these spaces form pairs of original spaces and their completions with @xmath0-metric . some basic properties of these spaces are discussed . we use @xmath97 to represent all fuzzy subsets on @xmath27 , i.e. functions from @xmath27 to @xmath98 $ ] . for details , we refer the readers to references @xcite . @xmath99 can be embedded in @xmath100 , as any @xmath101 can be seen as its characterization function , i.e. the fuzzy set @xmath102 for @xmath103 , let @xmath104_{{\alpha}}$ ] denote the @xmath105-cut of @xmath65 , i.e. @xmath106_{{\alpha}}=\begin{cases } \{x\in \mathbb{r}^m : u(x)\geq { \alpha}\ } , & \ { \alpha}\in(0,1 ] , \\ { \rm supp}\ , u=\overline{\{x \in \mathbb{r}^m : u(x)>0\ } } , & \ { \alpha}=0 . \end{cases}\ ] ] for @xmath103 , we suppose that + ( 1 ) @xmath65 is normal : there exists at least one @xmath107 with @xmath108 ; + ( 2 ) @xmath65 is upper semi - continuous ; + ( 3 - 1 ) @xmath65 is fuzzy convex : @xmath109 for @xmath110 and @xmath111;$ ] + ( 3 - 2 ) @xmath65 is fuzzy star - shaped , i.e. , there exists @xmath112 such that @xmath65 is fuzzy star - shaped with respect to @xmath52 , namely , @xmath113 for all @xmath114 ; + ( 3 - 3 ) given @xmath115 $ ] , then there exists @xmath116_\lambda$ ] such that @xmath117_\lambda$ ] for all @xmath118_\lambda$ ] ; + ( 4 - 1 ) @xmath104_0 $ ] is a bounded set in @xmath27 ; + ( 4 - 2 ) @xmath119_{\alpha } , \{0\ } ) ^p \ ; d\alpha \right)^{1/p } < + \infty $ ] , where @xmath120 and @xmath121 denotes the origin of @xmath27 ; + ( 4 - 3 ) @xmath104_{\alpha}$ ] is a bounded set in @xmath27 when @xmath122 . * if @xmath65 satisfies ( 1 ) , ( 2 ) , ( 3 - 1 ) and ( 4 - 1 ) , then @xmath65 is a fuzzy number . the set of all fuzzy numbers is denoted by @xmath5 . * if @xmath65 satisfies ( 1 ) , ( 2 ) , ( 3 - 2 ) and ( 4 - 1 ) , then we say @xmath65 is a fuzzy star - shaped number . the set of all fuzzy star - shaped numbers is denoted by @xmath7 . * if @xmath65 satisfies ( 1 ) , ( 2 ) , ( 3 - 3 ) and ( 4 - 1 ) , then we say @xmath65 is a general fuzzy star - shaped number . the set of all general fuzzy star - shaped numbers is denoted by @xmath12 . @xmath123 can be embedded in @xmath5 , as any @xmath124 can be viewed as the fuzzy number @xmath125 we can see that if @xmath126 , then @xmath127}\mbox{ker}\;[u]_\lambda \not= \emptyset$ ] , however this inequality may not hold when @xmath128 . so @xmath129 . the definition of fuzzy star - shaped numbers was given by wu and zhao @xcite . the concept of general fuzzy star - shaped numbers was given by qiu et al . @xcite . diamond and kloeden @xcite introduced the @xmath4 distance ( @xmath130 ) on @xmath12 which are defined by @xmath131_{\alpha } , [ v]_{\alpha } ) ^p \ ; d\alpha \right)^{1/p}\ ] ] for all @xmath132 . therein , they pointed out that @xmath4 is a metric on @xmath12 and that the three spaces @xmath6 , @xmath10 and @xmath133 are not complete . krtschmer @xcite has given the completion of @xmath6 which is described by the support functions of fuzzy numbers . to discuss properties of the above three spaces more clearly , we need to consider @xmath0type noncompact fuzzy sets . by relaxing assumption ( 4 - 1 ) a little to get ( 4 - 2 ) , we obtain the following @xmath0type noncompact fuzzy sets . suppose that @xmath134 . * if @xmath65 satisfies ( 1 ) , ( 2 ) , ( 3 - 1 ) and ( 4 - 2 ) , then we say @xmath65 is a @xmath0-type non - compact fuzzy number . the set of all such fuzzy numbers is denoted by @xmath135 . * if @xmath65 satisfies ( 1 ) , ( 2 ) , ( 3 - 2 ) and ( 4 - 2 ) , then we say @xmath65 is a @xmath0-type non - compact fuzzy star - shaped number . the set of all such fuzzy star - shaped numbers is denoted by @xmath136 . * if @xmath65 satisfies ( 1 ) , ( 2 ) , ( 3 - 3 ) and ( 4 - 2 ) , then we say @xmath65 is a @xmath0-type non - compact general fuzzy star - shaped number . the set of all such general fuzzy star - shaped numbers is denoted by @xmath137 . clearly , @xmath138 , @xmath139 , @xmath140 and @xmath141 . the discussions in this paper can be put in a more general framework without any assumptions of convexity or star - shapedness . so we consider the following types of fuzzy sets . suppose that @xmath134 . * if @xmath65 satisfies ( 1 ) , ( 2 ) and ( 4 - 1 ) , then @xmath65 is a normal upper semi - continuous compact - support fuzzy set on @xmath3 . the set of all such fuzzy sets is denoted by @xmath14 . * if @xmath65 satisfies ( 1 ) , ( 2 ) and ( 4 - 2 ) , then @xmath65 is a normal upper semi - continuous @xmath0-type non - compact - support fuzzy set on @xmath3 . the set of all such fuzzy sets is denoted by @xmath15 . * if @xmath65 satisfies ( 1 ) , ( 2 ) and ( 4 - 3 ) , then @xmath65 is a normal upper semi - continuous non - compact - support fuzzy set on @xmath3 . the set of all such fuzzy sets is denoted by @xmath142 . note that @xmath143 satisfies assumption ( 2 ) is equivalent to @xmath104_{\alpha}\in c ( \mathbb{r}^m)$ ] for all @xmath144 $ ] . so @xmath145 . it is easy to see that @xmath146 and @xmath147 are subsets of @xmath14 and @xmath148 , respectively . it can be checked that the @xmath4 distance , @xmath120 , ( see eq . ) is also a metric on @xmath15 . but @xmath4 distance is not a metric on @xmath149 because @xmath150 may equal @xmath151 for some @xmath152 . all the fuzzy sets spaces mentioned in this paper are subspaces of @xmath16 . in this paper , we show that @xmath153 is the completion of @xmath154 . then we find that @xmath155 , @xmath156 and @xmath23 are just the completion of @xmath6 , @xmath157 and @xmath158 respectively . the following lemma shows that for each @xmath159 , the only possible unbounded cut - set is the 0-cut , @xmath104_0 $ ] . [ ulce ] given @xmath159 , then for each @xmath160 $ ] , @xmath104_{\alpha}$ ] is a compact set . * proof * since @xmath104_{\alpha}$ ] is a closed set , it only need to show that @xmath104_{\alpha}$ ] is a bounded set , i.e. @xmath161_{\alpha } , \{0\ } ) < + \infty$ ] . we proceed by contradiction . if @xmath161_{\alpha } , \{0\})=+\infty$ ] . note that @xmath161_\beta , \{0\ } ) \geq h([u]_{\alpha } , \{0\ } ) $ ] when @xmath162 . then @xmath163_{\alpha } , \{0\ } ) ^p \ ; d\alpha \right)^{1/p } \geq \left ( \int_0^\alpha h([u]_{\alpha } , \{0\ } ) ^p \ ; d\alpha \right)^{1/p } = + \infty , $ ] which contradicts the fact that @xmath159 . @xmath95 [ act ] by lemma [ ulce ] , we know that @xmath104_{\alpha}\in k_c ( \mathbb{r}^m)$ ] for all @xmath164 and @xmath160 $ ] , and that @xmath104_{\alpha}\in k_s ( \mathbb{r}^m)$ ] for all @xmath165 and @xmath160 $ ] . denote @xmath166 } \mbox{ker}\ ; [ u]_{\alpha}$ ] for @xmath165 ( also see @xcite ) . it is easy to check that , given @xmath165 , then @xmath167 if and only if @xmath168 . the following representation theorem is used widely in the theory of fuzzy numbers . @xcite [ nr]given @xmath169 then + ( 1 ) @xmath104_{\lambda}\in k_c(\mathbb{r}^m)$ ] for all @xmath111 $ ] ; + ( 2 ) @xmath104_{\lambda}=\bigcap_{\gamma<\lambda}[u]_\gamma$ ] for all @xmath170 $ ] ; + ( 3 ) @xmath104_0=\overline{\bigcup_{\gamma>0}[u]_\gamma}$ ] . moreover , if the family of sets @xmath171\}$ ] satisfy conditions @xmath172 through @xmath173 then there exists a unique @xmath174 such that @xmath104_{{\lambda}}=v_\lambda$ ] for each @xmath111.$ ] similarly , we can obtain representation theorems for @xmath7 , @xmath12 , @xmath18 , @xmath135 , @xmath136 , @xmath137 and @xmath175 which will be used in the sequel . [ rslp ] given @xmath176 then + ( 1 ) @xmath104_{\lambda}\in k_s(\mathbb{r}^m)$ ] for all @xmath111 $ ] , and @xmath177 } \mbox{ker}\ ; [ u]_\alpha \not= \emptyset$ ] ; + ( 2 ) @xmath104_{\lambda}=\bigcap_{\gamma<\lambda}[u]_\gamma$ ] for all @xmath170 $ ] ; + ( 3 ) @xmath104_0=\overline{\bigcup_{\gamma>0}[u]_\gamma}$ ] . moreover , if the family of sets @xmath171\}$ ] satisfy conditions @xmath172 through @xmath173 , then there exists a unique @xmath178 such that @xmath104_{{\lambda}}=v_\lambda$ ] for each @xmath111 $ ] . [ rgslp ] given @xmath179 then + ( 1 ) @xmath104_{\lambda}\in k_s(\mathbb{r}^m)$ ] for all @xmath111 $ ] ; + ( 2 ) @xmath104_{\lambda}=\bigcap_{\gamma<\lambda}[u]_\gamma$ ] for all @xmath170 $ ] ; + ( 3 ) @xmath104_0=\overline{\bigcup_{\gamma>0}[u]_\gamma}$ ] . moreover , if the family of sets @xmath171\}$ ] satisfy conditions @xmath172 through @xmath173 , then there exists a unique @xmath180 such that @xmath104_{{\lambda}}=v_\lambda$ ] for each @xmath111 $ ] . [ rfbe ] given @xmath181 then + ( 1 ) @xmath104_{\lambda}\in k(\mathbb{r}^m)$ ] for all @xmath111 $ ] ; + ( 2 ) @xmath104_{\lambda}=\bigcap_{\gamma<\lambda}[u]_\gamma$ ] for all @xmath170 $ ] ; + ( 3 ) @xmath104_0=\overline{\bigcup_{\gamma>0}[u]_\gamma}$ ] . moreover , if the family of sets @xmath171\}$ ] satisfy conditions @xmath172 through @xmath173 , then there exists a unique @xmath182 such that @xmath104_{{\lambda}}=v_\lambda$ ] for each @xmath111 $ ] . [ rep ] given @xmath183 then + ( 1 ) @xmath104_{\lambda}\in k_c(\mathbb{r}^m)$ ] for all @xmath170 $ ] ; + ( 2 ) @xmath104_{\lambda}=\bigcap_{\gamma<\lambda}[u]_\gamma$ ] for all @xmath170 $ ] ; + ( 3 ) @xmath104_0=\overline{\bigcup_{\gamma>0}[u]_\gamma}$ ] ; + ( 4 ) @xmath119_{\alpha } , \{0\ } ) ^p \ ; d\alpha \right)^{1/p } < + \infty $ ] . moreover , if the family of sets @xmath171\}$ ] satisfy conditions @xmath172 through @xmath184 , then there exists a unique @xmath164 such that @xmath104_{{\lambda}}=v_\lambda$ ] for each @xmath111 $ ] . [ rs0p ] given @xmath185 then + ( 1 ) @xmath104_{\lambda}\in k_s(\mathbb{r}^m)$ ] for all @xmath170 $ ] and @xmath177 } \mbox{ker}\ ; [ u]_\alpha \not= \emptyset$ ] ; + ( 2 ) @xmath104_{\lambda}=\bigcap_{\gamma<\lambda}[u]_\gamma$ ] for all @xmath170 $ ] ; + ( 3 ) @xmath104_0=\overline{\bigcup_{\gamma>0}[u]_\gamma}$ ] ; + ( 4 ) @xmath119_{\alpha } , \{0\ } ) ^p \ ; d\alpha \right)^{1/p } < + \infty $ ] . moreover , if the family of sets @xmath171\}$ ] satisfy conditions @xmath172 through @xmath184 , then there exists a unique @xmath167 such that @xmath104_{{\lambda}}=v_\lambda$ ] for each @xmath111 $ ] . [ rsp ] given @xmath186 then + ( 1 ) @xmath104_{\lambda}\in k_s(\mathbb{r}^m)$ ] for all @xmath170 $ ] ; + ( 2 ) @xmath104_{\lambda}=\bigcap_{\gamma<\lambda}[u]_\gamma$ ] for all @xmath170 $ ] ; + ( 3 ) @xmath104_0=\overline{\bigcup_{\gamma>0}[u]_\gamma}$ ] ; + ( 4 ) @xmath119_{\alpha } , \{0\ } ) ^p \ ; d\alpha \right)^{1/p } < + \infty $ ] . moreover , if the family of sets @xmath171\}$ ] satisfy conditions @xmath172 through @xmath184 , then there exists a unique @xmath187 such that @xmath104_{{\lambda}}=v_\lambda$ ] for each @xmath111 $ ] . [ rfbp ] given @xmath188 then + ( 1 ) @xmath104_{\lambda}\in k(\mathbb{r}^m)$ ] for all @xmath170 $ ] ; + ( 2 ) @xmath104_{\lambda}=\bigcap_{\gamma<\lambda}[u]_\gamma$ ] for all @xmath170 $ ] ; + ( 3 ) @xmath104_0=\overline{\bigcup_{\gamma>0}[u]_\gamma}$ ] ; + ( 4 ) @xmath119_{\alpha } , \{0\ } ) ^p \ ; d\alpha \right)^{1/p } < + \infty $ ] . moreover , if the family of sets @xmath171\}$ ] satisfy conditions @xmath172 through @xmath184 , then there exists a unique @xmath189 such that @xmath104_{{\lambda}}=v_\lambda$ ] for each @xmath111 $ ] . in this section , we present a characterization of relatively compact sets in fuzzy sets space @xmath190 . based on this , we then give characterizations of totally bounded sets and compact sets in @xmath190 . the following concepts are important to establish and illustrate the characterizations . @xcite a set @xmath191 is said to be uniformly @xmath11-mean bounded if there is a constant @xmath192 such that @xmath193 for all @xmath194 . we can see that @xmath40 is uniformly @xmath11-mean bounded is equivalent to @xmath40 is a bounded set in @xmath195 . @xcite let @xmath196 . if for given @xmath197 , there is a @xmath198 such that for all @xmath199 @xmath200_\alpha , [ u]_{\alpha - h})^p \;d\alpha \right)^{1/p } < \varepsilon , \ ] ] where @xmath201 , then we say @xmath65 is @xmath11-mean left - continuous . suppose that @xmath40 is a nonempty set in @xmath175 . if the above inequality holds uniformly for all @xmath194 , then we say @xmath40 is @xmath11-mean equi - left - continuous . ma @xcite use @xmath202 to denote the fuzzy set @xmath202 induced by @xmath103 which is defined as follows : @xmath203 diamond @xcite characterized the compact sets in @xmath6 and @xmath8 , where @xmath9 denotes the set of all fuzzy star - shaped number @xmath65 with respect to @xmath121 , i.e. @xmath204 . obviously , @xmath205 and @xmath206 . @xcite [ dempc ] a closed set @xmath40 of @xmath6 is compact if and only if : + ( 1 ) @xmath207_0 : u\in u\}$ ] is bounded in @xmath208 ; + ( 2 ) @xmath40 is @xmath11-mean equi - left - continuous . @xcite [ dsm0pc ] a closed set @xmath40 of @xmath8 is compact if and only if : + ( 1 ) @xmath207_0 : u\in u\}$ ] is bounded in @xmath208 ; + ( 2 ) @xmath40 is @xmath11-mean equi - left - continuous . ma @xcite modified proposition [ dempc ] as follows . @xcite [ mgep ] a closed set @xmath40 of @xmath6 is compact if and only if : + ( 1 ) @xmath40 is uniformly @xmath11-mean bounded ; + ( 2 ) @xmath40 is @xmath11-mean equi - left - continuous ; + ( 3 ) for @xmath209 , if @xmath210 converges to @xmath211 in @xmath4 metric for any @xmath212 , then there exists a @xmath213 such that @xmath214 . wu and zhao @xcite pointed out the above three characterizations of compactness are wrong , and gave compactness criteria in spaces @xmath6 and @xmath8 . @xcite [ gep ] a closed set @xmath40 of @xmath6 is compact if and only if : + ( 1 ) @xmath40 is uniformly @xmath11-mean bounded ; + ( 2 ) @xmath40 is @xmath11-mean equi - left - continuous ; + ( 3 ) let @xmath215 be a decreasing sequence in @xmath216 $ ] converging to zero . for @xmath209 , if @xmath217 converges to @xmath218 in @xmath4 metric , then there exists a @xmath213 such that @xmath219_{\alpha}=[u(r_i)]_{\alpha}$ ] when @xmath220 . @xcite [ gs0p ] a closed set @xmath40 of @xmath8 is compact if and only if : + ( 1 ) @xmath40 is uniformly @xmath11-mean bounded ; + ( 2 ) @xmath40 is @xmath11-mean equi - left - continuous ; + ( 3 ) let @xmath215 be a decreasing sequence in @xmath216 $ ] converging to zero . for @xmath209 , if @xmath217 converges to @xmath221 in @xmath4 metric , then there exists a @xmath222 such that @xmath219_{\alpha}=[u(r_i)]_{\alpha}$ ] when @xmath220 . based on proposition [ gs0p ] , zhao and wu @xcite further presented compactness criteria of @xmath10 . @xcite[gsp ] a closed set @xmath40 of @xmath10 is compact if and only if : + ( 1 ) @xmath40 is uniformly @xmath11-mean bounded ; + ( 2 ) @xmath40 is @xmath11-mean equi - left - continuous ; + ( 3 ) let @xmath215 be a decreasing sequence in @xmath216 $ ] converging to zero . for @xmath209 , if @xmath217 converges to @xmath223 in @xmath4 metric , then there exists a @xmath224 such that @xmath219_{\alpha}=[u(r_i)]_{\alpha}$ ] when @xmath220 . compare propositions [ gep ] , [ gs0p ] , [ gsp ] with arzel ascoli theorem , we notice that the latter provides the compact criteria by characterizing the totally bounded set while the former does not seem to do so . since totally boundedness is a key feature of compactness , it is natural and important to consider characterizations of totally bounded sets . in this section , we consider characterizations of totally boundedness , relatively compactness and compactness in @xmath190 . some fundamental conclusions and concepts in classic analysis and topology are listed below , which are useful in this paper . the readers can see @xcite for details . * * lebesgue s dominated convergence theorem*. let @xmath225 be a sequence of integrable functions that converges almost everywhere to a function @xmath226 , and suppose that @xmath225 is dominated by an integrable function @xmath227 . then @xmath226 is integrable , and @xmath228 . * * fatou s lemma*. let @xmath225 be a sequence of nonnegative integrable functions that converges almost everywhere to a function @xmath226 , and if the sequence @xmath229 is bounded above , then @xmath226 is integrable and @xmath230 * * absolute continuity of lebesgue integral*. suppose that @xmath226 is lebesgue integrable on @xmath231 , then for arbitrary @xmath197 , there is a @xmath232 such that @xmath233 whenever @xmath234 and @xmath235 . * * minkowski s inequality*. let @xmath236 , and let @xmath237 be measurable functions on @xmath238 such that @xmath239 and @xmath240 are integrable . then @xmath241 is integrable , and minkowski s inequality @xmath242 holds . * a relatively compact subset @xmath243 of a topological space x is a subset whose closure is compact . * let @xmath244 be a metric space . a set @xmath40 in @xmath245 is totally bounded if and only if for each @xmath197 , it contains a finite @xmath246-approximation , where an @xmath246-approximation to @xmath40 is a subset @xmath247 of @xmath40 such that @xmath248 for each @xmath249 . * let @xmath244 be a metric space . then a set @xmath40 in @xmath245 is relatively compact implies that it is totally bounded . for subsets of a complete metric space these two meanings coincide . thus @xmath244 is a compact space iff @xmath245 is totally bounded and complete . firstly , we need some lemmas . [ unes ] suppose that @xmath40 is a bounded set in @xmath190 , then @xmath207_{\alpha } : u\in u\}$ ] is a bounded set in @xmath208 for each @xmath250 . * proof * if there exists an @xmath251 such that @xmath207_{{\alpha}_0 } : u\in u\}$ ] is not a bounded set in @xmath208 . then there is a @xmath194 such that @xmath104_{{\alpha}_0 } \notin k(\mathbb{r}^m)$ ] or @xmath207_{{\alpha}_0 } : u\in u\}$ ] is a unbounded set in @xmath252 . for both cases , there exist @xmath253 such that @xmath254_{{\alpha}_0 } , \{0\ } ) > n\cdot ( \frac{1}{{\alpha}_0})^{1/p}$ ] when @xmath67 , and hence @xmath255_{{\alpha } } , \{0\ } ) ^p \ ; d\alpha \right)^{1/p } > n,$ ] which contradicts the boundness of @xmath40 . @xmath95 lemma [ ulce ] can be seen as a corollary of lemma [ unes ] . [ lcf ] suppose that @xmath196 and @xmath144 $ ] , then @xmath161_{\alpha } , [ u]_{\beta } ) \to 0 $ ] as @xmath256 . * proof * the desired result follows immediately from theorem [ rfbp ] and proposition [ mce ] . @xmath95 [ uplc ] if @xmath196 , then @xmath65 is @xmath257mean left - continuous . * proof * given @xmath197 . note that @xmath196 , from the absolute continuity of lebesgue integral , we know there exists an @xmath258 such that @xmath259_\alpha , \{0\})^p \;d\alpha \right)^{1/p } \leq \varepsilon/3.\ ] ] by lemma [ ulce ] , @xmath104_{\frac{h_1}{2 } } \in k(\mathbb{r}^m)$ ] , then there exists an @xmath192 such that @xmath161_{\frac{h_1}{2 } } , \{0\})<m$ ] . this yields that @xmath260_\alpha , [ u]_{\alpha - h } ) \leq h([u]_\alpha , \{0\ } ) + h ( [ u]_{\alpha - h } , \{0\ } ) \leq 2m\ ] ] for all @xmath261 and @xmath262 . by lemma [ lcf],we know that @xmath161_\alpha , [ u]_{\alpha - h } ) \rightarrow 0 $ ] when @xmath263 , it then follows from the lebesgue s dominated convergence theorem and that @xmath264_\alpha , [ u]_{\alpha - h } ) ^p \;d\alpha \right)^{1/p } \rightarrow 0\ ] ] when @xmath265 . thus there exists an @xmath266 such that @xmath267_\alpha , [ u]_{\alpha - h } ) ^p \;d\alpha \right)^{1/p } < \varepsilon/3\ ] ] for all @xmath268 . now combined and , we know that , for all @xmath269 , @xmath270_\alpha , [ u]_{\alpha - h } ) ^p \;d\alpha \right)^{1/p } \nonumber \\ & \leq \left(\int_h^{h_1 } h([u]_\alpha , [ u]_{\alpha - h } ) ^p \;d\alpha \right)^{1/p } + \left(\int_{h_1}^{1 } h([u]_\alpha , [ u]_{\alpha - h } ) ^p \;d\alpha \right)^{1/p } \nonumber \\ & \leq \left(\int_h^{h_1 } h([u]_\alpha , \{0\ } ) ^p \;d\alpha \right)^{1/p } + \left(\int_h^{h_1 } h ( [ u]_{\alpha - h } , \{0\ } ) ^p \;d\alpha \right)^{1/p } + \varepsilon/3 \nonumber \\ & \leq \left(\int_0^{h_1 } h([u]_\alpha , \{0\ } ) ^p \;d\alpha \right)^{1/p } + \left(\int_0^{h_1 } h ( [ u]_{\alpha } , \{0\ } ) ^p \;d\alpha \right)^{1/p } + \varepsilon/3 \nonumber \\ & \leq \varepsilon/3 + \varepsilon/3 + \varepsilon/3 = \varepsilon.\end{aligned}\ ] ] from the arbitrariness of @xmath246 , we know that @xmath65 is @xmath257mean left - continuous . @xmath95 now , we arrive at the main results of this section . [ pcn ] @xmath40 is a relatively compact set in @xmath271 if and only if + ( 1 ) @xmath40 is uniformly @xmath11-mean bounded ; + ( 2 ) @xmath40 is @xmath11-mean equi - left - continuous . * proof * _ * necessity . * _ if @xmath40 is a relatively compact set in @xmath271 . since @xmath271 is a metric space , it follows immediately that @xmath40 is a bounded set in @xmath271 , i.e. @xmath40 is uniformly @xmath11-mean bounded . now we prove that @xmath40 is @xmath257mean equi - left - continuous . given @xmath197 . since @xmath40 is a relatively compact set , there exists an @xmath272-net @xmath273 of @xmath40 . from lemma [ uplc ] , we know that @xmath274 is @xmath11-mean equi - left - continuous . hence there exists @xmath232 such that @xmath275_\alpha , [ u_k]_{\alpha - h } ) ^p \;d\alpha \right)^{1/p } \leq \varepsilon/3.\ ] ] for all @xmath276 and @xmath277 . given @xmath194 , there is an @xmath278 such that @xmath279 , and thus , by , we know that for all @xmath276 , @xmath280_\alpha , [ u]_{\alpha - h } ) ^p \;d\alpha \right)^{1/p } \nonumber \\ & \leq \left(\int_h^1 h([u]_\alpha , [ u_k]_{\alpha } ) ^p \;d\alpha \right)^{1/p } + \left(\int_h^1 h([u_k]_\alpha , [ u_k]_{\alpha - h } ) ^p \;d\alpha \right)^{1/p } + \left(\int_h^1 h([u_k]_{\alpha - h } , [ u]_{\alpha - h } ) ^p \;d\alpha \right)^{1/p } \nonumber \\ & \leq \varepsilon/3 + \varepsilon/3 + \varepsilon/3 = \varepsilon.\end{aligned}\ ] ] from the arbitrariness of @xmath246 and @xmath194 , we obtain that @xmath40 is @xmath11-mean equi - left - continuous . _ * sufficiency . * _ if @xmath40 satisfies ( 1 ) and ( 2 ) . to show @xmath40 is a relatively compact set , it suffices to find a convergent subsequence of an arbitrarily given sequence in @xmath40 . let @xmath281 be a sequence in @xmath40 . to find a subsequence @xmath282 of @xmath281 which converges to @xmath283 according to @xmath4 metric , we split the proof into three steps . * step 1 . * find a subsequence @xmath282 of @xmath281 and @xmath284 such that @xmath285_\alpha , [ v]_\alpha ) { \stackrel } { \mbox{a.e . } } { \to } 0 \ ( [ 0,1 ] ) .\ ] ] since @xmath40 is uniformly @xmath257mean bounded , by lemma [ unes ] , @xmath207_\alpha : u\in u\}$ ] is a bounded set in @xmath38 for each @xmath144 $ ] . thus , by proposition [ sca ] , for each @xmath250 , @xmath207_\alpha : u\in u\}$ ] is a relatively compact set in @xmath208 . arrange all rational numbers in @xmath216 $ ] into a sequence @xmath286 . then @xmath281 has a subsequence @xmath287 such that @xmath288_{q_1}\}$ ] converges to @xmath289 , i.e. @xmath290_{q_1 } , u_{q_1 } ) \to 0 $ ] . if @xmath291 have been chosen , we can choose a subsequence @xmath292 of @xmath293 such that @xmath294 _ { q_{k+1 } } \}$ ] converges to @xmath295 . thus we obtain nonempty compact sets @xmath296 . with @xmath297 whenever @xmath298 . put @xmath299 for @xmath67 . then @xmath282 is a subsequence of @xmath281 and @xmath300_{q_k } , u_{q_k } ) \to 0 \ \ \ \mbox{as } \ \ \ n\to \infty\ ] ] for @xmath301 . define @xmath302 \ } $ ] as follows : @xmath303\hbox { ; } \\ \overline { \bigcup_{\alpha \in ( 0,1 ] } v_\alpha } , & \ \alpha=0 \hbox{. } \end{array } \right.\ ] ] then @xmath304 , @xmath305 $ ] , have the following properties : 1 . @xmath306 for all @xmath170 $ ] ; 2 . @xmath307 for all @xmath170 $ ] ; 3 . @xmath308 . in fact , by proposition [ mce ] , we obtain that @xmath309 for all @xmath310 $ ] . thus property ( 1 ) is proved . properties ( 2 ) and ( 3 ) follow immediately from the definition of @xmath304 . define a function @xmath311 $ ] by @xmath312 then @xmath313 is a fuzzy set on @xmath27 . from properties ( 1 ) , ( 2 ) and ( 3 ) of @xmath304 , we know that @xmath314_{\alpha}=v_{\alpha}.\ ] ] so @xmath315 . clearly if the following statements ( ) and ( ) are true , then we obtain , i.e. @xmath316_\alpha , [ v]_\alpha ) { \stackrel } { \mbox{a.e . } } { \to } 0 \ ( [ 0,1 ] ) $ ] . 1 . @xmath317 is at most countable , where @xmath318_{\alpha}\}$ ] , where @xmath319_{\beta } } $ ] . 2 . if @xmath320 , then @xmath321_\alpha , [ v]_\alpha ) \to 0 \ \hbox{as } \ n\to \infty.\ ] ] firstly , we show assertion ( ) . let @xmath322_\alpha \nsubseteq \overline { \{v > \alpha\ } } \ ; \}$ ] . notice that @xmath323 ( in fact , it can be checked that @xmath324 ) . by the conclusion in appendix of @xcite , @xmath325 is at most countable . so @xmath317 is at most countable . secondly , we show assertion ( ) . suppose that @xmath320 , then from proposition [ mce ] , @xmath326_\beta , [ v]_{\alpha } ) \to 0 $ ] as @xmath327 . thus , given @xmath197 , we can find a @xmath232 such that @xmath328 for all @xmath329 with @xmath330 . so @xmath331_\alpha , v_{\alpha } ) \leq h^*([v_n]_{q_1 } , v_{\alpha } ) \leq h^*([v_n]_{q_1 } , u_{q_1 } ) + \varepsilon\ ] ] for @xmath332 . hence , by and the arbitrariness of @xmath246 , we obtain @xmath333_\alpha , v_{\alpha } ) \to 0 \ ( n\to \infty).\ ] ] on the other hand , @xmath334_{\alpha } ) \leq h^*(v_{\alpha } , [ v_n]_{q_2 } ) \leq h^ * ( u_{q_2 } , [ v_n]_{q_2 } ) + \varepsilon\ ] ] for @xmath335 . hence , by and the arbitrariness of @xmath246 , we obtain @xmath336_\alpha ) \to 0 \ ( n\to \infty).\ ] ] combined with and , we thus obtain . * step 2 . * prove that @xmath337_\alpha , [ v]_\alpha ) ^p \;d\alpha \right)^{1/p } \to 0.\ ] ] given @xmath197 . it can be deduced that , for all @xmath338 , @xmath339_\alpha , [ v]_\alpha ) ^p \;d\alpha \right)^{1/p } \nonumber \\ \leq & \left ( \int_0^h h ( [ v_n]_\alpha , [ v_n]_{\alpha+h } ) ^p \;d\alpha \right)^{1/p } + \left ( \int_0^h h ( [ v_n]_{\alpha+h } , [ v]_{\alpha+h } ) ^p \;d\alpha \right)^{1/p } + \left ( \int_0^h h ( [ v]_{\alpha+h } , [ v]_\alpha ) ^p \;d\alpha \right)^{1/p } \nonumber \\ = & \left ( \int_h^{2h } h ( [ v_n]_{\beta - h } , [ v_n]_{\beta } ) ^p \;d\beta \right)^{1/p } + \left ( \int_h^{2h } h ( [ v_n]_{\beta } , [ v]_{\beta } ) ^p \;d\beta \right)^{1/p } + \left ( \int_h^{2h } h ( [ v]_{\beta } , [ v]_{\beta - h } ) ^p \;d\beta \right)^{1/p } \nonumber \\ \leq & \left ( \int_h^{1 } h ( [ v_n]_{\beta - h } , [ v_n]_{\beta } ) ^p \;d\beta \right)^{1/p } + \left ( \int_h^{1 } h ( [ v_n]_{\beta } , [ v]_{\beta } ) ^p \;d\beta \right)^{1/p } + \left ( \int_h^{1 } h ( [ v]_{\beta } , [ v]_{\beta - h } ) ^p \;d\beta \right)^{1/p}.\label{ean}\end{aligned}\ ] ] since @xmath40 is @xmath11-mean equi - left - continuous , there exists an @xmath340 such that @xmath341_{\beta - h } , [ v_n]_{\beta } ) ^p \;d\beta \right)^{1/p } < \varepsilon/4\ ] ] for all @xmath67 . from , we know if @xmath69 then @xmath342_{\beta - h } , [ v_n]_{\beta } ) \to h ( [ v]_{\beta - h } , [ v]_{\beta } ) $ ] a.e . on @xmath343 so , by fatou lemma , we have @xmath344_{\beta - h } , [ v]_{\beta } ) ^p \;d\beta \right)^{1/p } \leq \liminf_n \left ( \int_h^{1 } h ( [ v_n]_{\beta - h } , [ v_n]_{\beta } ) ^p \;d\beta \right)^{1/p } \leq \varepsilon/4,\ ] ] note that @xmath345_h$ ] and @xmath346_h$ ] are contained in @xmath207_\alpha : u\in u\}$ ] which is compact , it thus follows from the lebesgue s dominated convergence theorem and that @xmath347_\alpha , [ v]_\alpha ) ^p \;d\alpha \right)^{1/p } \to 0\ ] ] as @xmath69 . hence there is an @xmath348 such that @xmath349_\alpha , [ v]_\alpha ) ^p \;d\alpha \right)^{1/p } \leq \varepsilon/4\ ] ] for all @xmath79 . combined with , , , and , it yields that @xmath350_\alpha , [ v]_\alpha ) ^p \;d\alpha \right)^{1/p } \\ & \leq \left ( \int_0^h h ( [ v_n]_\alpha , [ v]_\alpha ) ^p \;d\alpha \right)^{1/p } + \left ( \int_h^1 h ( [ v_n]_\alpha , [ v]_\alpha ) ^p \;d\alpha \right)^{1/p } \\ & \leq \varepsilon/4 + \left ( \int_h^1 h ( [ v_n]_\alpha , [ v]_\alpha ) ^p \;d\alpha \right)^{1/p } + \varepsilon/4 + \left ( \int_h^1 h ( [ v_n]_\alpha , [ v]_\alpha ) ^p \;d\alpha \right)^{1/p } \\ & \leq \varepsilon\end{aligned}\ ] ] for all @xmath79 . thus we obtain from the arbitrariness of @xmath246 . * step 3 . * show that @xmath351 . by , we know that there is an @xmath77 such that @xmath352_\alpha , [ v_n]_\alpha ) ^p \;d\alpha \right)^{1/p } < 1,\ ] ] and then @xmath353_\alpha , \{0\ } ) ^p \;d\alpha \right)^{1/p } \\ & \leq \left ( \int_0 ^ 1 h ( [ v]_\alpha , [ v_n]_\alpha ) ^p \;d\alpha \right)^{1/p } + \left ( \int_0 ^ 1 h ( [ v_n]_\alpha , \{0\ } ) ^p \;d\alpha \right)^{1/p } \\ & \leq 1 + \left ( \int_0 ^ 1 h ( [ v_n]_\alpha , \{0\ } ) ^p \;d\alpha \right)^{1/p } < + \infty.\end{aligned}\ ] ] by properties ( 1),(2 ) and ( 3 ) of @xmath304 and theorem [ rfbp ] , this yields that @xmath351 . from steps 1 , 2 and 3 , we know that for arbitrary sequence @xmath281 of @xmath40 , there exists a subsequence @xmath282 of @xmath281 which converges to @xmath351 . this means that @xmath40 is a relatively compact set in @xmath190 . @xmath95 [ crm ] from theorem [ pcn ] , we can obtain the following conclusion . suppose that @xmath354 and that @xmath355 . let @xmath356 be a decreasing sequence in @xmath216 $ ] converging to zero . if @xmath357 satisfies conditions ( 1 ) and ( 2 ) in theorem [ pcn ] . then the following statements are equivalent . + ( 1 ) @xmath358 for all @xmath215 , @xmath359 . + ( 2 ) @xmath360_\alpha , [ u]_\alpha ) { \stackrel } { \mbox{a.e . } } { \to } 0 \ ( [ 0,1 ] ) $ ] . + ( 3 ) @xmath361 and @xmath196 . the readers can see @xcite for study on this subject , which consider the relations among @xmath4 convergence and other types of convergence on fuzzy sets spaces . [ tcn ] @xmath40 is a totally bounded set in @xmath271 if and only if + ( 1 ) @xmath40 is uniformly @xmath11-mean bounded ; + ( 2 ) @xmath40 is @xmath11-mean equi - left - continuous . * proof * notice that it only use the totally boundedness of @xmath40 to show the necessity part of the proof of theorem [ pcn ] . so the desired conclusion follows immediately from theorem [ pcn ] . @xmath95 [ gscn ] let @xmath40 be a closed subset of @xmath362 , then @xmath40 is compact in @xmath362 if and only if + ( 1 ) @xmath40 is uniformly @xmath11-mean bounded ; + ( 2 ) @xmath40 is @xmath11-mean equi - left - continuous . * proof * the desired result follows immediately from theorem [ pcn ] . in this section , we show that @xmath190 is the completion of @xmath363 , and present characterizations of totally bounded sets , relatively compact sets and compact sets in @xmath363 . diamond and kloeden @xcite pointed out that @xmath6 is not a complete space . ma @xcite gave the following example to show this fact . let @xmath364 ma pointed out that @xmath365 is a cauchy sequence in @xmath366 , but has no limit in @xmath367 . put @xmath368 then it can be checked that @xmath369 and @xmath66 converges to @xmath65 in @xmath370 . notice that @xmath371 and @xmath372 , it yields that both @xmath373 and @xmath374 are not complete . along this line , it can be shown that @xmath6 , @xmath10 , @xmath13 and @xmath363 are not complete . [ scp ] @xmath190 is a complete space . * proof * it suffices to prove that each cauchy sequence has a limit in @xmath190 . let @xmath375 be a cauchy sequence in @xmath190 , we assert that @xmath375 is a relatively compact set in @xmath190 . to show this assertion , by theorem [ pcn ] , it only need to prove that @xmath375 is a bounded set in @xmath190 and that @xmath375 is @xmath11-mean equi - left - continuous . the former follows immediately from the fact that @xmath375 is a cauchy sequence . now we prove the latter . given @xmath197 , since @xmath376 is a cauchy sequence , there exists an @xmath377 satisfies that @xmath378 for all @xmath379 . by lemma [ uplc ] , @xmath380 is @xmath11-mean equi - left - continuous , hence we can find an @xmath212 such that @xmath381_{{\alpha}-h } , [ u_k]_{{\alpha } } ) ^p \;d{\alpha}\right)^{1/p } \leq \varepsilon/3 \label{nevc}\ ] ] for all @xmath382 . if @xmath383 , then @xmath384_{{\alpha}-h } , [ u_k]_{{\alpha } } ) ^p \;d{\alpha}\right)^{1/p } \nonumber \\ & \hspace{-6mm}\leq \left ( \int_h^{1 } h ( [ u_k]_{{\alpha}-h } , [ u_n]_{{\alpha}-h } ) ^p \;d{\alpha}\right)^{1/p } + \left ( \int_h^{1 } h ( [ u_n]_{{\alpha}-h } , [ u_n]_{{\alpha } } ) ^p \;d{\alpha}\right)^{1/p } + \left ( \int_h^{1 } h ( [ u_n]_{{\alpha } } , [ u_k]_{{\alpha } } ) ^p \;d{\alpha}\right)^{1/p } \nonumber \\ & \hspace{-6 mm } \leq \varepsilon/3 + \varepsilon/3 + \varepsilon/3 = \varepsilon . \label{knvc}\end{aligned}\ ] ] from the arbitrariness of @xmath246 and ineqs . and , we know @xmath385 is @xmath11-mean equi - left - continuous . now , from the relatively compactness of @xmath385 in @xmath190 , there exists a subsequence @xmath386 of @xmath385 such that @xmath387 . note that @xmath385 is a cauchy sequence , we thus know that @xmath66 , @xmath67 , also converges to @xmath65 in @xmath388 . the proof is completed . @xmath95 by theorems [ pcn ] and [ tcn ] , a set @xmath40 in @xmath190 is totally bounded if and only if it is relatively compact . we can also deduce that @xmath190 is complete from this fact . [ sln ] @xmath389 is a dense set in @xmath190 . * proof * given @xmath196 . put @xmath390 , @xmath67 . then @xmath391_{\alpha}=\left\ { \begin{array}{ll } [ u]_{\alpha } , & \hbox{if } \ { \alpha}\geq 1/n , \\ \mbox{}[u]_{1/n } , & \hbox{if } \ \alpha \leq 1/n , \end{array } \right.\ ] ] for all @xmath392 $ ] . by lemma [ ulce ] , we know @xmath393_0 = [ u]_{1/n}\in k(\mathbb{r}^m)$ ] . it thus follows from theorems [ rfbe ] and [ rfbp ] that @xmath394 for @xmath67 . since @xmath196 , we know @xmath395_{{\alpha } } , \{0 \ } ) ^p \;d{\alpha}\right)^{1/p } < + \infty$ ] , thus , by the absolute continuity of the lebesgue s integral , it holds that , for each @xmath197 , there is a @xmath396 such that @xmath397_{{\alpha } } , \{0 \ } ) ^p \;d{\alpha}\right)^{1/p } < \varepsilon.\ ] ] note that @xmath398_{1/n } , [ u]_{{\alpha } } ) ^p \;d{\alpha}\right)^{1/p } \\ & \leq \left ( \int_0^{1/n } h ( [ u]_{{\alpha } } , \{0\ } ) ^p \;d{\alpha}\right)^{1/p } + \left ( \int_0^{1/n } h ( [ u]_{1/n } , \{0\ } ) ^p \;d{\alpha}\right)^{1/p } \\ & \leq 2 \left ( \int_0^{1/n } h ( [ u]_{{\alpha } } , \{0\ } ) ^p \;d{\alpha}\right)^{1/p},\end{aligned}\ ] ] it then follows from ineq . that @xmath399 as @xmath69 . so , for each @xmath196 , we can find a sequence @xmath400 such that @xmath401 converges to @xmath402 . this means that @xmath389 is dense in @xmath175 . @xmath95 [ undg ] from the proof of theorem [ sln ] , we know the following fact . + given @xmath196 , then @xmath403 for each @xmath404 , and @xmath405 as @xmath69 . from theorems [ scp ] and [ sln ] , we get the following theorem . @xmath406 is the completion of @xmath407 . let @xmath408 , then @xmath40 is totally bounded in @xmath409 if and only if + ( 1 ) @xmath40 is uniformly @xmath11-mean bounded ; + ( 2 ) @xmath40 is @xmath11-mean equi - left - continuous . * proof * note that @xmath410 , we thus know @xmath411 is a totally bounded set in @xmath412 if and only if @xmath40 is a totally bounded set in @xmath406 . so the desired conclusion follows immediately from theorem [ tcn ] . @xmath95 [ fbchra ] let @xmath408 , then @xmath40 is compact in @xmath409 if and only if + ( 1 ) @xmath40 is uniformly @xmath11-mean bounded ; + ( 2 ) @xmath40 is @xmath11-mean equi - left - continuous ; + ( 3 ) @xmath413 , where @xmath414 is the closure of @xmath40 in @xmath190 . * proof * the desired result follows immediately from theorem [ gscn ] . @xmath95 condition ( 3 ) in theorem [ fbchra ] involves the closure of @xmath40 in the completion space @xmath190 . next , we look for an characterization of compactness that only depends on @xmath40 itself . the following concept is needed . let @xmath415 and @xmath416 be the characteristic function of @xmath417 , where @xmath418 is a positive real number . given @xmath419 , then @xmath420 . define @xmath421_{\alpha } , [ \widehat{b_r } ] _ { \alpha } ) ^p \ , d{\alpha}\right ) ^ { 1/p}.\ ] ] it can be checked that , for @xmath419 , @xmath422 if and only if @xmath104_0 \subseteq b_r $ ] . note that @xmath423_{\alpha } , [ v \vee \widehat{b_r } ] _ { \alpha } ) \leq h ( [ u]_{\alpha } , [ v]_{\alpha } ) , \ ] ] it thus holds that @xmath424 [ fbsrchra ] let @xmath408 , then @xmath40 is relatively compact in @xmath409 if and only if @xmath40 satisfies conditions ( 1 ) , ( 2 ) in theorem [ pcn ] and the following condition ( 3@xmath425 ) . ( 3@xmath425 ) : : given @xmath426 , there exists a @xmath427 and a subsequence @xmath282 of @xmath281 such that @xmath428 . * proof * suppose that @xmath40 is a relatively compact set but does not satisfy condition ( 3@xmath425 ) . take @xmath429 , then there exists @xmath430 and a subsequence @xmath431 of @xmath432 such that @xmath433 for all @xmath67 . if @xmath291 and positive numbers @xmath434 have been chosen , we can find a subsequence @xmath292 of @xmath293 and @xmath435 such that @xmath436 for all @xmath67 . put @xmath299 for @xmath67 . then @xmath282 is a subsequence of @xmath281 and @xmath437 for @xmath301 . let @xmath438 be a accumulation point of @xmath282 . it then follows from and that @xmath439 for all @xmath301 . so we know @xmath440 . this contradicts the fact that @xmath40 is a relatively compact set in @xmath441 . suppose that @xmath442 satisfies condition ( 3@xmath425 ) . given a sequence @xmath281 in @xmath40 with @xmath443 . then , from , there exists a @xmath427 such that @xmath444 . hence @xmath104_0 \subseteq b_r $ ] , i.e. @xmath445 . so , by theorem [ pcn ] , we know that if @xmath40 meets conditions ( 1 ) , ( 2 ) and ( 3@xmath425 ) , then @xmath40 is a relatively compact set in @xmath446 . @xmath95 [ fbschra ] let @xmath40 be a closed set in @xmath447 , then @xmath40 is compact in @xmath409 if and only if @xmath40 satisfies conditions ( 1 ) , ( 2 ) and ( 3@xmath425 ) in theorem [ fbsrchra ] . * proof * the desired result follows immediately from theorem [ fbsrchra ] . in this section , we consider relationship among all fuzzy sets spaces mentioned in this paper . they are subspaces of @xmath409 or @xmath362 . the conclusions are summarized in fig.[sbpren ] . by using these facts and the results in sections [ crfpat ] and [ relation ] , we gives characterizations of totally bounded sets , relatively compact sets and compact sets in these subspaces . [ gsmc ] @xmath137 is a closed set in @xmath362 * proof * it only need to show that each accumulation point of @xmath137 belongs to itself . given a sequence @xmath281 in @xmath137 with @xmath448 , then clearly @xmath254_\alpha , [ u]_\alpha ) { \stackrel } { \mbox{a.e . } } { \to } 0 \ ( [ 0,1 ] ) $ ] . suppose that @xmath144 $ ] . if @xmath254_\alpha , [ u]_\alpha ) \to 0 $ ] , then by theorem [ ksc ] , @xmath104_{\alpha}\in k_s(\mathbb{r}^m)$ ] . if @xmath254_\alpha , [ u]_\alpha ) \not\to 0 $ ] , then there exists a sequence @xmath449 such that @xmath450_{\beta_n } \in k_s(\mathbb{r}^m)$ ] . note that @xmath104_{\alpha}= \bigcap_{n } [ u]_{\beta_n } $ ] , this implies that @xmath450_\alpha \in k_s(\mathbb{r}^m)$ ] . so we know @xmath451 . @xmath95 [ slne ] @xmath12 is a dense set in @xmath23 . * proof * given @xmath165 . put @xmath390 , @xmath67 . then @xmath452 . from remark [ undg ] , we know that @xmath399 as @xmath69 . so @xmath12 is dense in @xmath23 . @xmath95 [ sgmcde ] @xmath136 is a closed subset of @xmath23 . * proof * to show that @xmath136 is a closed set in @xmath453 , let @xmath281 be a sequence in @xmath136 which converges to @xmath451 , we only need to prove that @xmath167 . since @xmath454_{\alpha } , [ u]_{\alpha})^p \ ; d\alpha \right)^{1/p } \to 0 $ ] , it holds that @xmath455_{\alpha } , [ u]_{\alpha } ) \to 0 \ \hbox{a.e . on } \ [ 0,1].\ ] ] hence @xmath456 is a bounded set in @xmath208 , and therefore @xmath457 we assert that @xmath458_{\alpha}\ \hbox{for all } \ { \alpha}\in ( 0,1].\ ] ] so , from and , we know @xmath459 } \mbox{ker}\ ; [ u]_{\alpha}= \mbox{ker}\ ; u.\ ] ] it thus follows from theorem [ rs0p ] that @xmath167 . now we prove . the proof is divided into two cases . + * case 1*. @xmath160 $ ] satisfies that @xmath254_{\alpha } , [ u]_{\alpha } ) \to 0 $ ] . + in this case , by corollary [ kef ] , we have that @xmath460_{\alpha}\subset \mbox{ker}\ ; [ u]_{\alpha}.\ ] ] * case 2*. @xmath160 $ ] satisfies that @xmath254_{\alpha } , [ u]_{\alpha } ) \not\to 0 $ ] . + by , we know that there is a sequence @xmath461 $ ] , @xmath67 , such that @xmath462 and @xmath254_{{\alpha}_n } , [ u]_{{\alpha}_n } ) \to 0 $ ] . from case 1 , we obtain that @xmath463_{{\alpha}_n}.\ ] ] note that @xmath464_{\alpha } , [ u]_{{\alpha}_n } ) \to 0 $ ] , so , by corollary [ kef ] , @xmath465_{{\alpha}_n } \subset \mbox{ker}\ ; [ u]_{{\alpha}},\ ] ] combined and , we get that @xmath466_{{\alpha}}. \quad \box\ ] ] [ ukc ] suppose that @xmath281 is a sequence in @xmath22 and that @xmath165 . if @xmath399 , then @xmath167 and @xmath467 . * proof * the desired result follows immediately from the proof of theorem [ sgmcde ] . @xmath95 @xmath10 is a closed subspace of @xmath13 . * proof * the desired result follows immediately from corollary [ ukc ] . @xmath95 [ sdln ] @xmath7 is a dense set in @xmath22 . * proof * the proof is similar to the proof of theorem [ sln ] . @xmath95 [ vmce ] @xmath21 is a closed subspace of @xmath22 . * proof * by proposition [ kcs ] , we know that @xmath468 is a closed set in @xmath469 . in a way similar to the proof of theorem [ sgmcde ] , we can obtain the desired result by using this fact . @xmath95 @xmath6 is a closed subspace of @xmath10 . * proof * the desired results follows immediately from theorem [ vmce ] . @xmath95 [ edln ] @xmath5 is a dense set in @xmath21 . * proof * the proof is similar to the proof of theorem [ sdln ] . @xmath95 [ gsc ] @xmath470 is the completion of @xmath471 . * proof * the desired result follows from theorems [ scp ] , [ gsmc ] , [ slne ] . @xmath95 [ smc ] @xmath22 is the completion of @xmath10 . * proof * the desired result follows from theorems [ sgmcde ] , [ sdln ] , [ gsc ] . @xmath95 [ evmc ] @xmath21 is the completion of @xmath6 . * proof * the desired result follows from theorems [ vmce ] , [ edln ] , [ smc ] . @xmath95 we use @xmath472 to denote the closure of @xmath473 in @xmath22 . from corollary [ ukc ] and remark [ undg ] , we know that 1 . @xmath474 . 2 . @xmath9 is a closed subset of @xmath10 . 3 . @xmath475 is a closed subset of @xmath22 . . @xmath476 denotes that @xmath477 is a closed subspace of @xmath478 and @xmath479 means that @xmath478 is the completion of @xmath477 . , title="fig : " ] + we summarize above facts in figure [ sbpren ] . based on these facts , we will discuss characterizations of totally bounded sets , relatively compact sets and compact sets in various subspaces of @xmath480 in the sequel . [ spcn ] let @xmath481 ( @xmath482 , @xmath483 , @xmath484 ) , then @xmath40 is totally bounded if and only if it is relatively compact in @xmath485 ( @xmath21 , @xmath486 , @xmath487 ) , which is equivalent to + ( 1 ) @xmath40 is uniformly @xmath11-mean bounded , and + ( 2 ) @xmath40 is @xmath11-mean equi - left - continuous . * proof * note that in a complete space , a set is totally bounded is equivalent to it is relatively compact . so the desired results follow from theorems [ pcn ] and the completeness of @xmath485 , @xmath21 , @xmath486 and @xmath487 . @xmath95 [ smpcn ] let @xmath40 be a closed set in @xmath485 ( @xmath21 , @xmath486 , @xmath487 ) , then @xmath40 is compact in @xmath485 ( @xmath21 , @xmath486 , @xmath487 ) if and only if + ( 1 ) @xmath40 is uniformly @xmath11-mean bounded ; + ( 2 ) @xmath40 is @xmath11-mean equi - left - continuous . * proof * the desired results follow immediately from theorem [ spcn ] . @xmath95 let @xmath488 ( @xmath489 , @xmath490 , @xmath491 ) , then @xmath40 is totally bounded in @xmath492 ( @xmath493 , @xmath494 , @xmath495 ) if and only if + ( 1 ) @xmath40 is uniformly @xmath11-mean bounded ; + ( 2 ) @xmath40 is @xmath11-mean equi - left - continuous . * proof * the desired conclusion follows immediately from theorem [ tcn ] . @xmath95 [ gsmchra ] let @xmath488 ( @xmath489 , @xmath490 , @xmath491 ) , then @xmath40 is compact in @xmath492 ( @xmath496 , @xmath497 , @xmath498 ) if and only if + ( 1 ) @xmath40 is uniformly @xmath11-mean bounded ; + ( 2 ) @xmath40 is @xmath11-mean equi - left - continuous ; + ( 3 ) @xmath413 , where @xmath414 is the closure of @xmath40 in @xmath485 ( @xmath21 , @xmath486 , @xmath487 ) . * proof * the desired results follow from theorem [ gscn ] and the completeness of @xmath485 , @xmath21 , @xmath486 and @xmath487 . @xmath95 [ fbsrchrae ] let @xmath488 ( @xmath489 , @xmath490 , @xmath491 ) , then @xmath40 is relatively compact in @xmath158 ( @xmath499 , @xmath500 , @xmath10 ) if and only if + ( 1 ) @xmath40 is uniformly @xmath11-mean bounded ; + ( 2 ) @xmath40 is @xmath11-mean equi - left - continuous ; + ( 3@xmath425 ) given @xmath426 , there exists a @xmath427 and a subsequence @xmath282 of @xmath281 such that @xmath428 . * proof * note that @xmath501 , @xmath502 , @xmath503 , @xmath504 , so we can obtain the desired results by applying theorems [ fbsrchra ] and [ spcn ] . @xmath95 [ fbschrae ] let @xmath40 be a closed set in @xmath505 ( @xmath506 , @xmath507 , @xmath7 ) , then @xmath40 is compact in @xmath158 ( @xmath499 , @xmath500 , @xmath10 ) if and only if @xmath40 satisfies conditions ( 1 ) , ( 2 ) and ( 3@xmath425 ) in theorem [ fbsrchrae ] . * proof * the desired result follows immediately from theorem [ fbsrchrae ] . @xmath95 compare theorem [ smpcn ] with proposition [ gsp ] . we can see that , in contrast to the former , the latter has an additional condition ( 3 ) . the reason is that @xmath10 is not complete . the function of `` conditions ( 3 ) '' in proposition [ gsp ] is to guarantee completeness of the closed subspace @xmath508 of @xmath10 . in fact , under the assumptions of proposition [ gsp ] , by remark [ crm ] , we know that if @xmath217 converges to @xmath223 in @xmath4 metric , then there exists a @xmath509 such that @xmath510 and @xmath511_{\alpha}= [ u_0^{(r_i)}]_{\alpha}= [ u(r_i)]_{\alpha}$ ] when @xmath220 . so proposition [ gsp ] can also be written as : * proposition [ gsp]@xmath425 * a closed set @xmath40 in @xmath512 is compact if and only if : + ( 1 ) @xmath40 is uniformly @xmath11-mean bounded ; + ( 2 ) @xmath40 is @xmath11-mean equi - left - continuous ; + ( 3 ) @xmath513 , where @xmath514 is the closure of @xmath40 in @xmath22 . similarly , propositions [ gs0p ] and [ gep ] can also be written as : * proposition [ gs0p]@xmath425 * a closed set @xmath40 in @xmath8 is compact if and only if : + ( 1 ) @xmath40 is uniformly @xmath11-mean bounded ; + ( 2 ) @xmath40 is @xmath11-mean equi - left - continuous ; + ( 3 ) @xmath515 , where @xmath514 is the closure of @xmath40 in @xmath516 . * proposition [ gep]@xmath425 * a closed set @xmath517 is compact if and only if : + ( 1 ) @xmath40 is uniformly @xmath11-mean bounded ; + ( 2 ) @xmath40 is @xmath11-mean equi - left - continuous ; + ( 3 ) @xmath518 , where @xmath514 is the closure of @xmath40 in @xmath21 . the authors wish to thank the anonymous reviewers for their invaluable comments and suggestions .

diamond gave compact criteria in fuzzy numbers space endowed with @xmath0 metric and compact criteria in the space of fuzzy star - shaped numbers with respect to the origin endowed with @xmath0 metric . however , these compact criteria are wrong . wu and zhao proposed right characterizations in these two spaces . based on this result , zhao and wu further gave compact criteria in the space of fuzzy star - shaped numbers with @xmath0 metric . however , compare the existing compactness characterizations of fuzzy sets spaces endowed with @xmath0 metric with arzel ascoli theorem , it finds that the latter gives the compact criteria by characterizing the totally bounded sets while the former does not seem to characterize the totally bounded sets . since , in metric spaces , totally boundedness is a key feature of compactness . we present characterizations of totally bounded sets , relatively compact sets and compact sets in the fuzzy sets spaces @xmath1 and @xmath2 equipped with @xmath0 metric , where @xmath1 and @xmath2 are two kinds of fuzzy sets on @xmath3 which do not have any assumptions of convexity or star - shapedness . all fuzzy sets spaces considered in this paper are subspaces of @xmath2 endowed with @xmath0 metric . based on these characterizations and the discussions on convexity and star - shapedness of fuzzy sets , we construct the completions of every fuzzy sets space mentioned in this paper . then we clarify relation among all the ten fuzzy sets spaces discussed in this paper including the general fuzzy star - shaped numbers space introduced by qiu et al . at last , it gives characterizations of totally bounded sets , relatively compact sets and compact sets in all the fuzzy sets spaces mentioned in this paper . [ section ] [ section ] [ section ] [ section ] [ section ] [ section ] [ section ] fuzzy sets ; compact sets ; totally bounded sets ; @xmath0 metric ; star - shaped sets

You are an expert at summarizing long articles. Proceed to summarize the following text: relativistic jets in agn and microquasars carry energy from very small to very large scales . these jets form in the surroundings of compact objects , such as neutron stars or stellar black - holes in the case of microquasar jets , and supermassive black - holes ( smbh ) in the case of agn jets . the forming scales are of the order of a few times to about 30 times the radius of the central object , depending on the forming mechanism.@xcite from these regions , jets propagate up to nine powers of ten in distance . this implies formidable stability and collimation taking into account that different instabilities can affect the evolution of jets independently of their nature . moreover , their propagation is possibly changing the properties and evolution of the host galaxies and their environments through heating by shocks and/or mixing , and removing of gas by shocks and transfer of momentum . thus , jet stability has been a matter of study during the last four decades.@xcite depending on the nature of jets , i.e. , whether they are magnetically or particle dominated , different types of instabilities can arise . in the case of magnetically dominated jets , both current - driven ( cdi ) and kelvin - helmholtz ( khi ) instabilities can grow , whereas in the case of particle dominated jets , only the latter may be present.@xcite different stabilising mechanisms for both kinds of instabilities have been proposed , and some of them could explain the remarkable collimation and long distances covered by extragalactic jets , as opposed to the view that the reason for the stability of jets remains unknown.@xcite in this text i review the state - of - the - art of relativistic jet stability . section [ cd ] is focused on the cdi . section [ kh ] is devoted to the khi . in section [ nl ] i overview several non - linear processes that can also dramatically affect the evolution of jets and their collimation . finally , in section [ pv ] i discuss the possible reasons why relativistic jets manage to carry large amounts of energy from the central regions of galaxies or binary stars to very large distances . relativistic jets are thought to be formed by magnetohydrodynamical processes in the surroundings of compact objects.@xcite thus , it is reasonable to expect that jets are magnetically dominated close to these regions.@xcite in this regime , cdi is triggered by differences in the magnetic forces when a toroidal field is present . in the case that a pinch is produced , the larger force produced by the compressed lines allows the pinch to grow . in the case that a kink is produced , the magnetic lines get closer in the inner part of the kink , increasing the magnetic force and enlarging the curvature of the jet . both modes are stabilised by the presence of a poloidal field component , as the magnetic tension of these lines acts against the compression or distortion of the jet . cdi in magnetized flows has been studied from an analytical perspective in the non - relativistic regime by a number of authors.@xcite results show that this instability becomes dominant at large values of helicity ( ratio between the toroidal and poloidal field components ) and it is stabilised by a strong poloidal magnetic field . under the physical conditions typical of jets close to the forming regions , cdi is expected to be dominant , though with small growth rates.@xcite in the relativistic regime , numerical simulations of static columns have shown that growth rates decrease with decreasing alfvn speed and , regarding the non - linear regime , an increasing helicity of the magnetic field with increasing radius in the jet , stabilises the flow with respect to kink cdi.@xcite the introduction of a sub - alfvnic velocity shear generates important differences in the growth of this instability : in the case that the shear is inside the characteristic radius of the static column , the plasma flows through a temporally growing kink , whereas if the shear is outside that radius , the kink is advected with the flow and grows in distance.@xcite it has been suggested that the conversion from poynting to kinetic flux should occur in the first hundreds of gravitational radii , on the basis of observations and modelling of the jet in m87.@xcite magnetic acceleration has been proposed@xcite as an efficient mechanism . the growth of cdi to non - linear amplitudes can result in mass - load and acceleration of the entrained particles , so this process could also be related to a change from magnetically to particle dominated jets.@xcite however , there is no direct evidence for any of both . the role of cdi could be significantly reduced by jet expansion and rotation , which can have a stabilising effect . further simulations with realistic conditions are needed to solve this question . nevertheless , extragalactic jets embed stars and are necessarily entrained by gas clouds and stars rotating around the agn , so the mass - load and conversion of jet main energy channel from poynting flux to particles is difficult to avoid . it has been shown that sub - alfvnic flows are khi stable,@xcite but once the jets are super - alfvnic and particle dominated , khi may take over cdi and start to play an important role in the long - term stability of jets . the khi develops at the boundary between two fluids with relative velocity . it couples to any perturbation with a certain periodicity at which the system presents unstable modes . the sound - wave generated by the perturbation propagates through the body of the jet and grows in amplitude due to over - reflection at the boundaries.@xcite in the non - linear regime , the jet can be disrupted and decelerated by the entrainment of external matter . assuming cylindrical symmetry in jets , solutions to the linearised equations of relativistic hydrodynamics have the form @xmath0,@xcite @xmath1 being the wave - number , @xmath2 the azimuthal wave - number ( an integer giving the number of oscillations around the jet s circular cross - section ) , and @xmath3 the frequency . these solutions may have complex values of wave - number and/or frequency , the imaginary part giving the exponential growth of the amplitude . unstable modes are separated into surface modes and body modes . their distinctive property is the number of zeros that the the radial component of the wave - number ( @xmath4 ) of the unstable wave ( e.g. , the pressure wave ) has between the jet axis and its boundary . the surface mode shows no zeros between the axis and the boundary , whereas the body modes show as many zeros as their order indicates ( e.g. , one zero in the case of the first body mode ) . figure 1 shows some extracted modes from the solution to the linear problem of a sheared jet . the upper lines represent the real part of the solution ( frequency ) versus the wave - number , whereas the lower lines represent the imaginary part ( growth rate ) versus wave - number . [ f1 ] [ [ the - linear - regime ] ] the linear regime + + + + + + + + + + + + + + + + + the growth of khi has been shown to depend on the jet velocity ( unstable modes grow faster for slower jets ) , temperature ( faster growth for hotter jets ) and density ( faster growth for more dilute jets).@xcite there is a general correlation between this dependence and the long - term stability properties of jets , which is more remarkable in colder , faster and denser ones.@xcite the analytical , and numerical in some cases , knowledge acquired about the effect that the growth of khi modes could have on jets@xcite has been widely used in analysing the structures observed in different extragalactic jets and has allowed to derive estimates of the physical parameters that govern them ( e.g. , 3c 345 , 3c 120 , 3c 273 , m87 , s5 0836 + 710 ) . @xcite@xcite in particular , it was shown that expansion and acceleration can provide the jet in 3c 120 with long - term stability@xcite or that the jet in m87 could be decelerating and heating , leading to destabilization in the kpc scales.@xcite numerical plus analytical work has also led to the conclusion that the observed structures in the jet in 3c 273 could be coupling to khi modes , which may show up at different observing frequencies , depending on the section of the jet where they develop.@xcite this result also has the implicit conclusion that different observing frequencies may be showing different regions of a transversally structured jet . @xcite in this respect , several works have also shown that shear - layers are easily generated in jets by different mechanisms , including the growth of khi modes.@xcite in the classical regime , there is no limit to the growth of khi , other than disruption of the jet or the saturation of short wavelengths ( the latter will be discussed in the section about the non - linear phase ) , both involving an increase of the width of the velocity shear with a decrease of the radial gradient of velocity.@xcite for the case of relativistic jets , there is an analytical prediction of the saturation of the linear growth when the amplitude of the velocity perturbation reached the speed of light in the reference frame of the jet.@xcite figure 2 shows the saturation of the perturbation in velocity close to the speed of light . numerical simulations have confirmed this result and shown that the limiting amplitude is smaller for faster jets , thus making them more stable.@xcite overall , this shows that relativistic jets are intrinsically more stable than classical ones with respect to the growth of any kind of instability . on top of this , semi - analytic plus numerical studies of sheared jets have also shown that the growth of small - wavelength , fast - growing modes , as those shown on the right of fig . 1,@xcite can lead to long - term collimation of fast jets with moderately thick shear - layers ( @xmath5 , with @xmath6 the jet radius ) . different analytical and numerical works have certified that the presence of winds surrounding the jet@xcite or thick shear - layers@xcite reduce the growth rates of the khi modes . it has also been shown analytically that in the presence of a poloidal magnetic field , sub - alfvnic and even super - alfvnic jets can be stabilised by the presence of a magnetised surrounding sheath.@xcite [ f2 ] [ [ the - non - linear - regime ] ] the non - linear regime + + + + + + + + + + + + + + + + + + + + + starts typically after saturation.@xcite if the dominating mode is a low - order one with a relatively long wavelength , the jet shows strong deformations and can be disrupted by the generation of shocks at the boundary with the ambient medium.@xcite in this context , the jet develops a wide mixing layer and undergoes significant deceleration to mildly relativistic speeds . otherwise , if small - wavelength modes like resonant modes ( see fig . 1 ) dominate the growth of the instability , the jet develops a hot shear - layer and keeps collimation.@xcite figure 3 shows the schlieren plots for the last snapshot of two simulations of the temporal growth of khi in jets with different lorentz factors ( @xmath7 for the left image and @xmath8 for the right one ) . otherwise , both simulations are equal . the left panel , corresponding to a slower jet developing long - wavelength modes , shows turbulent mixing in a wide region , whereas the right panel , corresponding to the faster jet , which develops resonant modes , shows a significant degree of collimation . this result has been recently confirmed by 3d simulations and its implications extensively discussed@xcite ( see figure 4 ) . a similar , although not so efficient , mechanism for long - term jet stabilisation was also found for classical jets when the shortest modes dominate the growth of the instability up to the non - linear regime,@xcite which is also true for relativistic jets.@xcite the distribution of a number of simulated jets in a relativistic mach - number versus lorentz factor plane@xcite shows that there is a trend of larger stability in the non - linear regime in the case of faster and colder jets , which fall in the upper - right corner of the plane . [ f3 ] [ f4 ] in the previous section , the growth of instabilities has been discussed as starting in the linear regime , i.e. , from small perturbations . however , jets are destabilised by perturbations with large ( non - linear ) amplitudes and may undergo reconfinement shocks or meet irregularities in the ambient media . these processes are non - linear and can not be accounted for by linear theory of jet stability . they are of interest because generally imply strong shocks and thus represent candidate locations for very - high energy emission . [ [ extragalactic - jets ] ] extragalactic jets + + + + + + + + + + + + + + + + + + most probably jets are not in pressure equilibrium with the ambient medium because the pressure in the surrounding cocoon changes with time , as it expands . over - pressured jets expand and recollimate after becoming under - pressured with respect to the environment,@xcite or meet high density regions in an inhomogeneous environment ( e.g. , supernova remnants ( snr ) or massive stars with powerful winds ) . depending on the initial pressure difference , these shocks can trigger small amplitude pinching or helical motion that may couple to khi,@xcite or they can generate a large amplitude pinch and destroy the jet@xcite ( see figure 5 ) . only in the case that jets are close to pressure equilibrium with respect to the ambient medium , the generation and influence of reconfinement shocks can be neglected . when crossing the galaxy , the jet flow may encounter stars and clouds of gas that can be embedded in the jet or entrain it due to their proper motions.@xcite in the case of stars , the stellar wind has been claimed to efficiently mix with the jet flow and decrease the mean jet velocity,@xcite mainly in the case of massive stellar winds and very light jets . also , irregularities in the intergalactic medium have been claimed to play a role in the hybrid morphology of a number of sources that show both fri and frii structure in their two jets.@xcite large amplitude initial perturbations , such as those triggered by changes in the injection angle of the flow could generate non - linear distortions of jets and lead to their disruption.@xcite the injection of overdense plasma in jets has been invoked to explain the observation of components travelling through parsec - scale jets and the generation of trailing components.@xcite these have been shown to generate significant pinching that could couple to khi modes.@xcite in the kiloparsec scales , the arcs observed in some fri jets * * ? ? ? * * have also been suggested to be due to changes in the injection rates from the source.@xcite [ f5 ] [ [ microquasar - jets ] ] microquasar jets + + + + + + + + + + + + + + + + inhomogeneities in the ambient medium can also have a strong effect on microquasar jets in high - mass x - ray binaries ( hmxb ) with typical powers ( @xmath9).@xcite such inhomogeneities are encountered by the jet in the transition between the unshocked and shocked stellar wind from the massive companion . another change of ambient medium occurs for a young hmxb still embedded in the snr , at the transition between the shocked wind and the shocked snr , and finally from the latter to the interstellar medium ( ism ) through the shocked ism . for the case of older hmxb , the system has moved away from the snr or this has become very dilute and the region of interaction between the shocked stellar wind and the shocked ism also implies a change in the conditions encountered by jets . in these density jumps jets are decelerated and need several thousand years to drill the shocked ism before being able to propagate into the ism . proper motion of the hmxb provides a further destabilising mechanism , as the jet is impacted by the otherwise quasi - steady shocked stellar wind , propagating in the direction opposite to that of the proper motion . within the binary system , the stellar wind impacting the jet can also generate jet entrainment , deceleration and loss of collimation for jets with powers @xmath10 , mainly if a strong reconfinement shocks that decelerate the flow occur in this region , as has been shown to be the possible.@xcite see also figure 6 . in the previous sections , the main properties of instabilities that can affect the large - scale morphology of relativistic jets , namely cdi and khi , have been summarised . these have been proven to be disruptive enough to trigger efficient entrainment and decelerate jets . however , several mechanisms that may damp their growth and thus make relativistic jets stable have been given : * cdi : poloidal magnetic field , mildly relativistic sheaths and jet expansion . * khi : thick shear - layers or surrounding winds , the decrease of the cocoon density with time and jet expansion . most importantly , it has been shown that relativistic jets are less sensitive to the growth of instabilities than classical jets , due to the saturation of their growth at the speed of light . in addition , resonant modes that can show up in fast jets surrounded by thin shear - layers , or simply the development of short wavelength modes as compared to the jet radius can result in little mixing , restricted to the boundaries of jets , and thus little loss of jet thrust and collimation . if one or several of these mechanisms play a role , jets can remain collimated along large distances , as it has been shown by simulations . other mechanisms such as jet rotation or different configurations of the magnetic field are still to be explored in the relativistic regime . the properties of jets and the media outside the leading bow - shock change with distance to the source . the cocoons surrounding jets show a homogeneous pressure in long - term simulations of jets due to their sound - speeds being larger than the expansion velocities . as the cocoon expands , its density and pressure drop and , as long as the properties of the jet do not change , it will become denser with respect to its environment , gaining extra stability . the only caveat is that when the cocoon is very under - pressured , strong reconfinement shocks may form . nevertheless , even when jets are entrained and decollimated , they seem to preserve a large velocity that can be sub - relativistic , but could still be enough to keep the propagation of the jet to large scales . in fri jets , as compared to frii s , it is more likely that non - linear processes like mass - loading from stars@xcite or strong reconfinement shocks@xcite are the cause of their disrupted morphologies . these jets have apparently similar velocities in the parsec - scales as the friis,@xcite so other intrinsic properties such as density , temperature or composition should play a role to explain their lower powers . they are observed to have an opening angle in the inner regions , but there is not any hint of growth of instabilities before the jet deceleration and loss of collimation . in hybrid sources in which the jet and counterjet appear to have different morphologies,@xcite it is possible that irregularities in the ambient medium are the cause of the difference , for jets with powers between those typical of fris and friis . only in one case an frii classified jet with an apparently irregular structure in the largest scales,@xcite has been reported to develop instabilities that could cause the jet disruption.@xcite this implies that , under regular conditions , i.e. , without strong non - linear perturbations , the stabilising mechanisms proposed by different authors in the last years and summarised here play a significant role . it has also been suggested that frii jets can turn into fri jets in the long term@xcite or after the end of an active phase , as numerical simulations show.@xcite however , this possibility should be matched to the relativistic velocities measured in fri jets at the parsec scales . the long - term stability of relativistic jets allows them to carry energy from very small to very large scales . the bow - shocks generated by extragalactic jets need @xmath11 years to reach distances of the order of hundreds of kpc . in the most powerful radio - sources , strong shocks are triggered in the ism and icm , which can change the evolution of the galaxies themselves by removal of ism . this process can suppress star formation significantly and heating the surrounding gas , as shown by recent long - term simulations of relativistic jets.@xcite thus , the importance of the study of this interaction and the evolution of the jet itself . the research on jet stability in the next years should be addressed to investigate the effect of magnetic fields on the long - term evolution of jets . this should help to study , with input from observations and modelling , to which distance and extent the magnetic field is dynamically important and the role of instabilities in the eventual transfer of poynting flux into kinetic energy of particles . in addition , more realistic situations should be explored , such as jet rotation and expansion . these works will certify or falsify the role of the stabilising mechanisms summarised in this contribution and could even add further ones , suggesting that there may be more than one answer to the problem of jet stability . finally , it would be important to study , via detailed numerical simulations , the process of mixing and entrainment that jets undergo as a consequence not only of the growth of instabilities , but also of the interaction with stellar winds and clouds of cold gas . i acknowledge j.m . mart , m. hanasz and p.e . hardee for sharing their understanding of jet physics with me during the last years . financial support by the spanish `` ministerio de ciencia e innovacin '' ( micinn ) grants aya2010 - 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relativistic jets carry energy and particles from compact to very large scales compared with their initial radius . this is possible due to their remarkable collimation despite their intrinsic unstable nature . in this contribution , i review the state - of - the - art of our knowledge on instabilities growing in those jets and several stabilising mechanisms that may give an answer to the question of the stability of jets . in particular , during the last years we have learned that the limit imposed by the speed of light sets a maximum amplitude to the instabilities , contrary to the case of classical jets . on top of this stabilising mechanism , the fast growth of unstable modes with small wavelengths prevents the total disruption and entrainment of jets . i also review several non - linear processes that can have an effect on the collimation of extragalactic and microquasar jets . within those , i remark possible causes for the decollimation and deceleration of fri jets , as opposed to the collimated frii s . finally , i give a summary of the main reasons why jets can propagate through such long distances .

You are an expert at summarizing long articles. Proceed to summarize the following text: since the discovery of superconductivity in alkali - metal doped @xmath8 , extensive research on @xmath8 and other fullerenes has been carried out worldwide , aiming at understanding the mechanism for superconductivity and other related issues in fullerenes.@xcite most of the theoretical models assumed that electron - phonon interaction is important for superconductivity.@xcite based on the analysis of the linewidths in vibronic spectra excited either by light ( raman scattering ) or by neutrons , the electron - phonon coupling constant @xmath5 for @xmath10 ( a = alkali metal ) has been estimated . recently , winter and kuzmany observed that the low frequency @xmath11 and @xmath12 modes lose all degeneracy and split into five components , each of which couples differently to the @xmath13 electrons for single crystal of @xmath6 at 80 k.@xcite these results revealed that in the superconducting state , the pairing is mediated by phonons with weak or intermediate coupling . @xcite the lowest two unoccupied molecular orbitals of @xmath8 are both triply degenerated , having @xmath13 and @xmath14 symmetry . filling of @xmath13 and @xmath14 bands with electrons is achieved by intercalation of alkali metals and alkaline earth metals to @xmath8 solids , respectively . nevertheless , understanding of the `` @xmath14 superconductors '' is extremely poor in comparison with the well known @xmath13 superconductors . comparison of physical property in between the @xmath13 and @xmath14 superconductors is of particular interest from the view point of mechanism of superconductivity . from the @xmath13 symmetry of the electrons in the conduction band a coupling is only possible to the total symmetric @xmath15 modes and to the five - fold degenerate @xmath1 modes . while the coupling to the @xmath15 mode is expected to be weak due to an efficient screening effect , the @xmath1 modes may have a significantly strong coupling constant since they allow a jahn - teller mechanism . a similar coupling should take place in the case of the electrons with @xmath14 symmetry . superconductivity of ba - doped @xmath8 was first discovered by kortan et al , @xcite who claimed that the superconducting phase is bcc @xmath3 . recently , baenitz et al.,@xcite on the other hand , reported that the superconducting phase is not @xmath3 but @xmath2 . very recently , we succeeded to synthesize single phase @xmath2 , and unambiguously confirmed that the @xmath2 is the superconducting phase . in this work , we present results of a raman scattering study of single phase @xmath0 ( x=3 , 4 and 6 ) with @xmath14 states . the results indicate that the electron - phonon interaction is also important for the @xmath14 superconductor , particularly in superconducting @xmath2 . in addition , some amazing results were observed , particularly for the low frequency @xmath1 modes . ( 1 ) raman shift of the tangential @xmath15 mode for @xmath3 is much larger than the simple extrapolation relationship between raman shift and charge transfer in alkali metal doped @xmath8 ; while the radial @xmath15 mode nearly remains unchanged with increasing charge transfer . ( 2 ) the raman scattering behavior is quite different among the three phases of @xmath16 , @xmath2 and @xmath3 , especially for the low frequency @xmath1 modes . the low frequency @xmath1 modes lose all degeneracy and split into five ( or four ) peaks at room temperature for the @xmath2 and @xmath3 samples , each of which couples differently to electrons with @xmath14 symmetry . the splitting of low frequency @xmath1 modes into five components even at room temperature is similar to that observed in single crystal of @xmath6 at low temperature of 80 k. @xcite this is significant to understand the splitting and to evaluate the electron - phonon coupling constants for all directly coupling mode , estimating tc in ba - doped @xmath8 . samples of @xmath0 ( x=3 , 4 and 6 ) were synthesized by reacting stoichiometric amount of powers of ba and @xmath8 . a quartz tube with mixed powder inside was sealed under high vacuum of about @xmath17 torr . the samples of @xmath16 and @xmath3 were calcined at 600 @xmath18 for 216 hours with intermediate grindings of two times . in order to obtain high quality @xmath2 sample , thermal annealing was carried out at 600 @xmath18 for 1080 hours with five intermediate grindings . x - ray diffraction showed that all samples were single phase , which is also confirmed by the single peak feature of the pentagonal pinch @xmath7 mode in the raman spectra . raman scattering experiments were carried out using the 632.8 nm line of a he - ne laser in the brewster angle backscattering geometry . the scattering light was detected with a dilor xy multichannel spectrometer using a spectral resolution of 3 @xmath19 . decomposition of the spectra into individual lines was made with a peak - fitting routine after a careful subtraction of the background originating from the laser . in order to obtain good raman spectra , the samples were ground and pressed into pellets with pressure of about 20 @xmath20 , which were sealed in pyrex tubes under a high vacuum of @xmath21 torr . figure 1 shows room temperature raman spectra for the polycrystalline samples of @xmath16 , @xmath2 , and @xmath3 . for the three samples , only one peak of the pentagonal pinch @xmath7 mode is observed , providing an evidence that each sample is in a single phase . these agree fairly well with the x - ray diffraction patterns . interestingly , the three spectra have different strongest lines ; they are @xmath12 , @xmath22 , and @xmath7 modes for @xmath16 , @xmath2 , and @xmath3 , respectively . another thing to be noted is that the half - width of all corresponding peaks of @xmath2 is largest among @xmath0 ( x=3 , 4 and 6 ) samples except for the @xmath22 mode . this result is indicative of an importance of electron - phonon coupling in raman spectrum of @xmath2 . detailed discussion is given in the following . also , it is to be pointed out that the raman spectrum of @xmath16 sample is amazingly similar to that of @xmath23,@xcite suggesting that the electronic states of @xmath16 is similar to that of @xmath23 . this is in a fair agreement with a simple expectation that @xmath8 in both compounds is hexavalent . the frequency of the pentagonal pinch mode @xmath24 decreases with increasing ba concentration , similarly to the case of alkali - metal doped @xmath8.@xcite the raman shift of the @xmath7 mode is discussed in the following . by contrast , the frequency of the radial @xmath22 mode remains almost unchanged with ba concentration , being different from the case of @xmath25 , where a slight up - shift of the radial @xmath22 mode was observed.@xcite the low frequency @xmath1 modes show dramatic changes depending on the ba concentration . in particular , clear splittings are observed for the lowest frequency @xmath1 modes of @xmath2 and @xmath3 . the positions ( @xmath26 ) and halfwidths ( @xmath27 ) of the raman modes observed are listed in table i. for comparison , the lines for pure @xmath8 are included in table i. in the following , we show detailed analysis of @xmath1 modes first , and then , discuss on the @xmath15 modes . in fig.2 we show the results of a line - shape analysis of the raman spectra of the @xmath11 modes for @xmath16 , @xmath2 , and @xmath3 samples . all modes were fit to a lorentzian line shape . for @xmath16 and @xmath3 , a doublet with lorentzian components is observed , which has been observed in @xmath23.@xcite however , the @xmath11 mode has to be fit with four components for @xmath2 . this splitting may be attributed to the symmetry lowering due to the orthorhombic structure of this material . a similar behavior has been observed in single crystal @xmath6 at 80 k,@xcite in which the @xmath11 mode is split into five components . position of the @xmath11 components for @xmath2 sample is nearly the same as that observed in @xmath6 . figure 3 shows the higher resolution raman spectra in the vicinity of 400 @xmath19 for @xmath16 , @xmath2 , and @xmath3 . while the cubic @xmath16 shows a single peak at 432 @xmath19 . @xmath12 mode is apparently split into five components in @xmath3 . this splitting of @xmath12 mode in @xmath28 is unexpected since the group theoretical consideration predicts a splitting into two in the space group @xmath29 ( @xmath30 ) . the splitting of the @xmath12 mode might suggest a symmetry lowering which is not detected in the x - ray diffraction . this type of disagreement between microscopic spectroscopy and structural analysis was observed in @xmath31 , and still remains an open question . @xcite a characteristic feature of the @xmath12 mode of @xmath3 is that the widths @xmath27 of the components are almost the same except for the 428 @xmath19 component . by contrast , the @xmath12 mode of @xmath2 shows a strong peak at the high frequency edge associated with a long tailing structure towards lower frequencies . linewidth and lineshift for the components are clearly related . a theoretical calculation shows the electron - phonon coupling constants are very sensitive to the change in the normal coordinates , the different components of the mode correspond to the different coupling constants.@xcite it suggests that the fivefold degeneracy of the mode is lifted and each component couples with a different strength to the @xmath14 carriers in @xmath2 . results of a line - shape analysis of the raman spectra of the @xmath32 modes are shown in fig.4 . a doublet of @xmath32 is observed for @xmath16 , which is ascribed to symmetry - lowering relative to @xmath8 molecules . @xcite the @xmath32 mode also displays a splitting into four both in @xmath2 and @xmath3 . the splitting of the @xmath32 mode in @xmath3 also contradicts with the group theoretical consideration . it is to be pointed out that this anomalous splitting of @xmath3 @xmath1 modes is observed only in @xmath12 and @xmath32 modes . the other @xmath1 modes are singlet or doublet , being consistent with the group theoretical consideration . in reference 9 , winter and kuzmany gave several possible explanations for the splitting of the low frequency @xmath1 modes . ( 1 ) the splitting of the modes is understood from the merohedral disorder for the alkali derived metallic fullerides.@xcite this disorder is of low enough symmetry to allow only one dimensional representations for all modes . ( 2 ) the splitting originates from a jahn - teller type interaction . this interaction can give rise to a new vibrational system with rather large number of components , even more than five.@xcite also , a contribution to the splitting from an internal strain between the doped part of the crystal and the undoped part of the crystal . in our experiments , the low frequency @xmath1 modes almost lose all degeneracy for @xmath2 and @xmath3 , and is different from that of @xmath16 which is similar to that of @xmath23 at room temperature . in the case of @xmath2 , the splitting can be understood since the crystal structure is orthorhombic . however , the splitting of @xmath3 is not explained from the crystal structure . particularly , when one considers that @xmath3 is isostructural to @xmath23 , the splitting of @xmath12 and @xmath32 modes are considerably anomalous . this result might suggest that there exists a symmetry lowing which can not be detected by x - ray diffraction . similar symmetry lowing is observed in the nmr spectra of @xmath31.@xcite the next thing to be pointed out is that the splitting is observed even in polycrystalline samples and at room temperature , in contrast to the case of @xmath6 . in alkaline - earth - metal doped @xmath8 , the local - density approximation calculations show a strong hybridization between the alkaline - earth - atom states and the @xmath8 @xmath33 states.@xcite this hybridization , which is absent in the alkali - metal doped @xmath8 , may play an essential role for the splitting of low frequency @xmath1 modes at room temperature . for the components of low frequency @xmath1 modes , a clear relation between line shift and line broadening is observed in @xmath2 , which is similar to that of single crystal @xmath6 . winter and kuzmany have pointed out that the electron - phonon interaction plays an important role in the broadening and the shift of the lines , and they deduced electron - phonon coupling constants.@xcite the phonon linewidth broadening @xmath34 due to the electron - phonon interaction in a metal can be related to a dimensionless electron phonon coupling constant @xmath35 given by @xcite @xmath36 where n(0 ) the density of states at the fermi level per spin and molecule , and @xmath37 and @xmath38 the mode degeneracy and the frequency before any coupling to the electrons , respectively . the allen s formula given above will be used to derive the coupling constants for the eight @xmath1 modes . frequencies of pure @xmath8 were used as the bare phonon frequencies . in the framework of allen s theory there should be a linear relation of the form@xcite @xmath39 between @xmath27 the linewidth and @xmath40 the difference between the bare phonon frequency and the observed frequency . according to the experimental values of the three lowest frequency @xmath1 modes in table i , the relations between linewidth and frequency shift is plotted in fig.5 for @xmath2 and @xmath3 . the @xmath27 and @xmath40 relation for @xmath2 is linear and consistent with that expected from eq.(2 ) . n(0 ) can be deduced from the slope . the density of states obtained from the three @xmath1 modes are 7 ev@xmath41 , 4 ev@xmath41 and 3.2 ev@xmath41 , respectively . the discrepancy may arise from the fact that we could not use the real bare phonon frequencies for the evaluation . geometry effects may also contribute to the shift and may be different for the modes . for @xmath3 , there exists no relation between the linewidth and lineshift in fig.5b . n(0 ) is much less than 1 ev@xmath41 if it were deduced basing on the relation between linewidth and lineshift in fig.5b . it suggests @xmath3 could not follow electron - phonon coupling theory . it further supports that @xmath2 is superconducting phase , rather than @xmath42 . for the evaluation of the coupling constants as discussed below a value of 7 @xmath4 is used for n(0 ) . to our knowledge , no n(0 ) for @xmath2 is available . the calculated n(0 ) is 4.3 states per ev@xcite and an experimental value of 5.6 @xmath4 was reported for @xmath3 . @xcite the averaged linewidths and the overall coupling constants for each mode and for all @xmath1 modes for @xmath2 are listed in table ii , together with the frequencies for the pure @xmath8 . the averaged linewidths are directly evaluated from the linewiths listed in table i. the values for @xmath35 are evaluated using eq.(1 ) . the individual contributions to the coupling constant from each @xmath1 mode are listed in table ii . the three lowest frequency @xmath1 modes dominate the contribution to @xmath5 , yielding over 70% of the total value . large coupling constants of the low @xmath1 modes were also observed in @xmath6.@xcite within the bcs framework , the superconducting transition temperature @xmath43 can been evaluated basing on the experimental values for @xmath5 by the mcmillan equation @xmath44\ ] ] where @xmath45 is the logarithmic averaged phonon frequency , @xmath46 is the boltzmann constant , and @xmath47 is coulomb repulsion between conduction electrons . according to the observed frequencies and the evaluated coupling constants , the @xmath45 was determined as 490 @xmath19 . with this value and @xmath5 , the superconducting transition temperature of 7 k can be evaluated , assuming the @xmath47 value as 0.3 , however , which is anomalously large . the value for @xmath47 is much larger than 0.18 in @xmath6 in the same way for evaluation of @xmath43 . it might suggest a difference between @xmath13 and @xmath14 superconductors . to evaluate @xmath43 , on the other hand , the logarithmic averaged phonon frequency of 150 @xmath19 is obtained if the @xmath47 is set as a reasonable value 0.2 . in this case , the phonon frequency is significantly smaller than the intramolecular vibration range . interestingly , the small phonon energy associated with superconductivity is also suggested by analysis of another @xmath14 superconductor @xmath48.@xcite let us switch to the arguments on the totally symmetric @xmath15 modes . figure 6 shows the raman shift of the @xmath7 pentagonal pinch mode as a function of nominal charge transfer simply derived from the chemical formula for @xmath0 . in this figure , we plotted the present results of @xmath0 , as well as that of @xmath25 reported by duclos et al . @xcite and the theoretical results of jishi and dresselhaus @xcite for comparison . since the plots of @xmath0 approximately fall on an extrapolation of @xmath25 or theoretical line , the charge transfer value from ba to @xmath8 is almost complete . the molecular valences of @xmath0 ( x=3 , 4 , 6 ) are regarded as -6 , -8 , and -12 , respectively . however , the situation is more complicated than the case of alkali doped materials . several band calculations and experiments @xcite suggest a strong effect of hybridization of ba and @xmath8 orbitals . if this is the case , the net charge transfer to @xmath8 is expected to be incomplete . in the present result , however , the charge transfer is approximately complete . moreover , the slope of @xmath0 is steeper than that of @xmath25 or theory . these results indicate that the phonon mode should be reconsidered in the presence of metal - fullerene hybridization . especially , there is a difference of 10 @xmath19 between the experimental and theoretical values for @xmath3 . the theory of jishi and dresselhaus focuses on the mode softening of the tangential vibrational @xmath15 mode for the alkali - metal derived fullerides , the hydridization between intercalants and @xmath8 was not considered . it can be seen from table i that the frequency of the radial @xmath15 mode for @xmath49 is 506 @xmath19 . the upshift is as high as 13 @xmath19 relative to pure @xmath8 . but , upon further doping with barium , the frequency nearly remains unchanged , being different from alkali - metal doped @xmath8 , which shows a continuous hardening of @xmath22 mode as a function of alkali metal concentration.@xcite the mode - stiffening effect is due to electrostatic interactions which produces sufficient stiffening to encounter the softening of the mode expected on the basis of charge - transfer effects.@xcite in the case of ba derived fullerides , there exists a strong hybridization between the ba atoms and the @xmath9-type functions of the @xmath8 network . this may lead to a decrease in the electrostatic interactions , so that the frequency of the radial @xmath50 mode nearly remains unchanged with increasing ba concentration . raman scattering studies of single phase @xmath16 , @xmath2 , and @xmath3 have been carried out . the lowest frequency @xmath1 modes split in to five components for @xmath2 and @xmath3 . a characteristic relation between lineshift and linewidth is observed in @xmath2 , this is consistent with that expected by electron - phonon interaction.while @xmath3 does not exhibit such behavior . the characteristic relation is used to evaluate the n(0 ) , the electron - phonon coupling constants are evaluated basing on the raman results in the framework of allen s theory . the radial @xmath15 mode shows a different behavior from alkali derived fullerides , the frequency remains unchanged with increasing ba concentration ; the effect of charge transfer on the softening of the tangential @xmath15 mode is larger in the alkaline - earth metal doped @xmath8 than in alkali derived @xmath8 . these discrepancies may arise from the hybridization between intercalants and @xmath8 in alkaline - earth metal doped @xmath8 . x. h. chen would like to thank the inoue foundation for science for financial support . this work is partly supported by grant from the japan society for promotion of science ( rftf 96p00104 , mpcr-363/96 - 03262 ) and from the ministry of education , science , sports , and culture . . positions , averaged linewidths and electron - phonon coupling constants normalized to the density of states at the fermi energy for eight fivefold degenerate hg modes for @xmath2 sample . [ cols="^,^,^,^,^ " , ] raman spectra of the @xmath11 mode for @xmath16 , @xmath2 , and @xmath3 . the dash lines are computer fits for the individual components , which add up to the full line on the top of the experimental results . + figure 3 : plot of linwidth @xmath27 versus observed frequency shift @xmath51 for the individual components of the @xmath52 , @xmath12 , and @xmath32 modes , circles for the @xmath11 mode ; triangles for the @xmath12 mode ; squares for the @xmath32 mode . ( a ) for the sample @xmath2 ; ( b ) for the sample @xmath3 . + figure 6 : charge transfer - raman shift relation for the @xmath7 pinch mode . squares represent the experimental results of @xmath0 , circles are from the results of @xmath25 reported by duclos et al . ( ref.15 ) , and triangles refer to calculations from theory of jishi and dresselhaus ( ref.16 ) .

raman spectra are reported for ba doped fullerides , @xmath0 ( x=3 , 4 , and 6 ) . the lowest frequency @xmath1 modes split into five components for @xmath2 and @xmath3 even at room temperature , allowing us a quantitative analysis based on the electron - phonon coupling theory . for the superconducting @xmath2 , the density of states at the fermi energy was derived as 7 @xmath4 , while the total value of electron - phonon coupling @xmath5 was found to be 1.0 , which is comparable to that of @xmath6 . the tangential @xmath7 mode , which is known as a sensitive probe for the degree of charge transfer on @xmath8 molecule , shows a remarkable shift depending on the ba concentration , being roughly consistent with the full charge transfer from ba to @xmath8 . an effect of hybridization between ba and @xmath8 @xmath9 orbitals is also discussed . = 10000 * pacs numbers : 78.30.-j , 72.80.rj , 74.70.-b *

You are an expert at summarizing long articles. Proceed to summarize the following text: in recent years , there has been a growing interest in testing gravitational decoherence or possible gravitational effects on quantum mechanics ( qm ) in condensed matter and quantum - optical systems @xcite . decoherence can be studied in the framework of quantum mechanics and it does not require any additional assumptions . the dynamics of a system which is coupled with the environment follows from the schrdinger equation . an observer who has only access to system degrees of freedom observes a nonunitary dynamics which can be obtained by tracing out the environmental degrees of freedom from the total density matrix . this averaging generically reduces the coherence of the reduced density matrix describing the system . thus , a part of the phase information is distributed over the environmental variables . whether the phase information can be restored or not depends crucially on the form of the coupling and the properties of the environment . we will address decoherence via emission of gravitational waves in section [ decoherence ] . interestingly , it has also been argued that gravity might lead to a loss of coherence which can not be discussed in terms of standard quantum mechanics @xcite . a prominent example of this `` intrinsic decoherence '' was discussed in @xcite . there , it was argued that superpositions of static configurations have finite life - times and decay on a time - scale @xmath0 , where @xmath1 denotes the gravitational self - energy of the difference of the mass - distributions which are involved in the superposition . the main motivation for such an effect is the lack of a canonical time - like killing vector when superpositions of space - times are considered . in section [ noise ] , we want to address the question whether it is possible to derive such a decoherence rate from a tensor noise model . we point out several problems which come along when one introduces a tensor noise `` by hand '' . every physical system is coupled to gravity via its energy - momentum tensor . in the weak - field limit , the metric @xmath2 can be expanded around the minkowski background @xmath3 according to @xmath4 . the resulting action is quadratic in the metric perturbations @xmath5 . using the path integral formalism , the influence functional can be evaluated exactly . when the metric perturbations have a time - dependent quadrupole moment , the environmental modes can carry away phase - information about the system which will lead to a real loss decoherence @xcite . in the following , we discuss the loss of coherence of a particle which does not move on a geodesic and radiates gravitational waves . the analogue effect in electrodynamics due to emission of photons ( bremsstrahlung ) has been studied in @xcite . in this section , we follow notations used in @xcite . the total action of a mass distribution which is centered around @xmath6 and coupled to gravity can be written as @xmath7 the action @xmath8 contains in general also gravitational self - interaction terms which are of higher order and will not be considered here . we choose the transverse - traceless gauge , i.e. , @xmath9 . then the graviton action adopts the simple quadratic form @xmath10 the interaction between the energy - momentum tensor @xmath11 of the system and the external graviton field is bilinear , i.e. , @xmath12 here only the spatial components of the energy - momentum tensor contribute due to the gauge choice above . the graviton field can be expanded into plane waves according to @xmath13 where we introduced the transverse - traceless polarization tensors @xmath14 . the time - evolution of the system s density matrix is determined by the action @xmath15 and the influence functional @xmath16 . then the propagator of the reduced density matrix is given by @xmath17 where the subscripts @xmath18 indicate the initial and final values respectively . the influence functional adopts the explicit form @xmath19\nonumber\\\end{aligned}\ ] ] with the correlation functions @xmath20 we assumed the environmental modes to be in a thermodynamical state with the inverse temperature @xmath21 . the sum over polarizations gives the polarization tensor @xmath22 where the projection operators are @xmath23 . the real and imaginary parts of the phase @xmath24 in the influence functional ( [ influence ] ) lead to dissipation and decoherence , respectively . in this section , we calculate @xmath25 from which the dependence of the decoherence rate on the various parameters of the model follows . as particular example , we discuss the evaluation of ( [ phase ] ) using an interference device setup ( figure [ setup ] ) . a matter distribution is in superposition of either following a right trajectory @xmath6 ( blue ) or a left one @xmath26 ( red ) . for a small angle @xmath27 , figure [ setup ] can be interpreted as the double - slit experiment , whereas @xmath28 corresponds to an interferometer . on both trajectories , the system changes its direction at time @xmath29 due to a slit or mirror . the energy - momentum tensor of a particle in flat space - time takes the form @xmath30 with the velocity vector @xmath31 . we assume that a particle is smeared out over a space - region determined by @xmath32 , i.e. , @xmath33\ , . $ ] it is well - known that the divergence equation of the energy - momentum tensor gives the geodesic equation with respect to the background space - time . however , the two possible trajectories of a particle in figure [ setup ] do not correspond to geodesics in minkowski space - time , hence @xmath34 . generally , one would need to model a mirror , the interaction between a particle and a mirror and the interaction of gravity with the particle - mirror system . here , we follow a simpler approach by adding momentum densities which kick a particle at @xmath29 . thus , our system can be described now as `` particle + momentum density such that the energy - momentum tensor is divergence - free along the trajectories '' . in particular , we choose @xmath35 the time - time component @xmath36 can be chosen such that @xmath37 the additional momentum densities ensure that the divergence of the energy - momentum tensor @xmath38 vanishes along the trajectory . in order to obtain an interpretation of the momentum densities introduced , let us consider @xmath39 with @xmath40 and @xmath41 . then we have @xmath42 we interpret this expression as follows : when @xmath43 , a particle moves straight in positive @xmath44- and @xmath45-directions . at time @xmath46 , it gets a momentum transfer which changes its velocity in @xmath44-direction from @xmath47 to @xmath48 . this corresponds to the right trajectory @xmath6 in figure [ setup ] . a left trajectory @xmath49 which has a kick at @xmath46 towards the right can be constructed in the same way by defining @xmath50 . path.eps ( 50,45)@xmath51 ( 50,110)@xmath51 ( 117,110)@xmath51 ( 133,124)@xmath52 ( 117,45)@xmath51 ( 85,40)@xmath27 ( 75,3)@xmath53 ( 45,10)@xmath44 ( 8,48)@xmath45 ( -15,76)@xmath54 ( 147,76)@xmath54 ( 75,143)@xmath55 ( 130,26)@xmath52 the dependence of decoherence rate on velocity @xmath51 of an object with temperature of gravitational waves @xmath56k and @xmath57k . we chose the mass of an object @xmath58 kg . ] by substituting @xmath38 into ( [ phase ] ) and evaluating the thermal part of the expression ( [ phase ] ) along the trajectories , we find for velocities @xmath59 @xmath60 \sin^2\left(\frac{\theta}{2}\right)\nonumber\\ & \times\left(4\ln\left[\frac{\sinh\left(\pi t/(2\beta\right))}{\pi t/(2\beta)}\right ] -\ln\left[\frac{\sinh\left(\pi t/\beta\right)}{\pi t\beta}\right]\right)+\mathcal{o}(v^8)\end{aligned}\ ] ] where @xmath61 is a time when two trajectories meet again . with an angle @xmath62 , the effect is approximately proportional to @xmath63 . the authors of @xcite found the same velocity - dependence in the non - relativistic single - mode limit from a master equation approach . when @xmath64 and @xmath65 , we have @xmath66 the decoherence factor @xmath67 vanishes when two trajectories are parallel @xmath68 , that is , the trajectories coincide with each other . with @xmath69 , @xmath67 is proportional to @xmath70 , @xmath71 apart from thermal gravitational waves , vacuum fluctuations also contribute to the decoherence rate . for realistic parameters , this contribution is very small that it can be neglected . its expression diverges logarithmically with the inverse of the spread of the mass distribution , @xmath72 . in figure 2 , we plotted the decoherence rate with the temperature of the gravitational waves being 1k and @xmath73k . since gravitational waves decoupled earlier from photons and have therefore a lower temperature than the cosmic microwave background however , the actual temperature of the thermal gravitational waves background can be much lower and it depends on the details of reheating after inflation @xcite . here , we assume a thermal contribution with 1k as the upper bound and vacuum contribution would be the lower bound on the decoherence rate . we claim that the decoherence effect comes along with the emission of gravitational waves . the emitted energy of a particle along one of the trajectories depicted in figure [ setup ] can be calculated classically . the emission occurs when the direction of a particle is changed by a mirror . as long as the emitted radiation can be distinguished for trajectories , the environment measures which path a particle is moving to some extent , and this leads to environmental gravitational decoherence . for a trace - reversed perturbation , @xmath74 , the solution to the linearized einstein equations gives the expression @xmath75 when the particle is slowly moving , the far field is dominated by quadrupole moment tensor @xmath76 . in terms of the trace - free quadrupole moment , @xmath77 , the emitted power is @xmath78 here we will assume a smooth trajectory of a particle which moves along one path @xmath79\hat{\mathbf{x}}+ v\cos(\theta/2)[f(t,0)-f(t , t)]\hat{\mathbf{y}}\,\end{aligned}\ ] ] with @xmath80 . the time @xmath81 determines how fast the direction of a particle is changed at a mirror , how fast a particle is accelerated initially from a source and decelerated at a detector finally . in the limit @xmath82 , the radiated energy is crucially determined by @xmath81 , @xmath83 for vacuum fluctuations , the minimal possible time @xmath81 is determined by the extension of the particle , that is @xmath84 . for finite temperatures , the effective cutoff of the gravitational modes is determined by @xmath85 , so one should expect @xmath86 . it has been argued that the incompatibility between general relativity and quantum mechanics may lead to a form of intrinsic decoherence @xcite . such an effect , if it exists , is not related to any form of environmental degrees of freedom which carry away phase information from the system . the conflict between the principle of general covariance and the superposition principle of quantum mechanics is fundamental : when gravity is dynamical , the geometry of space - time must depend on the quantum state of the matter which it contains . thus , different matter states live in different space - times which can not be identified pointwise with each other . one might argue that the difference of three - forces in the superposed space - times give an estimate for the typical life - time of a superposition . this life - time @xmath87 is given by @xmath88 with @xmath89 [ \rho(\mathbf{x}')-\rho'(\mathbf{x}')]}{|\mathbf{x}-\mathbf{x'}|}.\end{aligned}\ ] ] although the relation ( [ gamma_penrose ] ) has not been derived from fundamental physics and although it is not clear yet , whether an intrinsic decoherence exists in nature at all @xcite , we want to address the question , whether this decoherence rate can be derived in some limit from a noise model . scalar noise models have been studied in the literature , see , for example @xcite . however , we want to couple the energy - momentum tensor to the noise which requires a tensor random current . furthermore , we assume that the life - time ( [ gamma_penrose ] ) is the newtonian limit of a more general expression . a possible generalization would be @xmath90 where @xmath91 is the green s function for linearized gravitational waves . since @xmath91 is not positive definite , we are immediately confronted with a problem : the expression ( [ gamma_penrose_mod ] ) can also adopt negative values , which invalidates the interpretation of a decoherence rate . in the context of quantum mechanics , this would violate the positivity of the density matrix since the off - diagonal elements would be unbounded , @xmath92 \langle i |\hat{\rho}(0)|j\rangle , \qquad i\neq j\ , . \end{aligned}\ ] ] however , it might be possible that terms such as ( [ gamma_penrose ] ) or ( [ gamma_penrose_mod ] ) are only a part of a more general _ positive _ definite quantity . the simplest covariant terms which can be added to the fierz - pauli action take the form @xmath93 with arbitrary constants @xmath52 and @xmath94 . here we introduced the tensor noise @xmath95 which couples to the linearized gravitational field and to the energy - momentum tensor . the first term would correspond to a stochastic contribution of the energy - momentum tensor whereas the second part resembles stochastic gravitational fluctuations . the gauge - invariance of the graviton field requires @xmath96 . furthermore we assume the gaussian distribution @xmath97 of the tensor noise to be @xmath98=\frac{1}{n}\exp \left(-\frac{1}{2}\int d^4x j_{\mu\nu } j^{\mu\nu}\right)\,.\end{aligned}\ ] ] contrary to section [ decoherence ] , we will use the de - donder gauge . this includes also the instantaneous interaction between stationary matter distributions in addition to the radiation part of the gravitational perturbations . after performing the integrals over @xmath95 and @xmath5 , additional contributions to the influence functional arise due to the presence of the noise . in the non - relativistic limit , we find for large times the expression @xmath99\,,\end{aligned}\ ] ] where @xmath100 denotes the influence functional due to gravitational waves . assuming an energy - momentum tensor of the form @xmath101 , the decoherence rates @xmath102 have a simple interpretation . @xmath103 measures the difference of the newtonian gravitational potentials in the respective spacetimes , @xmath104 @xmath105 is equal to the expression ( [ gamma_penrose ] ) up to a numerical factor and accounts for the difference of the newtonian forces , @xmath106 dimensional arguments might suggest that @xmath107 . finally , @xmath108 incorporates the difference of the mass densities , @xmath109 it is not yet clear whether this result is a gauge - independent statement . the time - evolution of the density matrix requires a definite choice of the time - parameter @xmath110 . ( we are not interested in scattering matrix elements where the interaction decreases adiabatically to zero for @xmath111 and lorentz - invariant quantities can be defined . ) a change in the time - parameter , corresponding to an transformation of the metric components , might alter the expression for the decoherence rate . we studied the decrease of coherence due to the emission of gravitational waves and gave an estimate for the emitted gravitational energy . the dominant contribution of the decoherence rate is given by thermal gravitational waves . due to the smallness of the gravitational interaction , this effect is negligible for elementary particles . if it would be possible to send an object with mass @xmath112 kg and velocity @xmath113 m/s through an interferometer , the decoherence rate would be roughly @xmath114s@xmath115 with temperature of gravitational waves being 1k . furthermore , we discussed a tensor noise model which partly resembles , in the newtonian limit , a form of intrinsic decoherence which was discussed in the literature before . the coherence decrease due to this noise model depends on the difference of the mass density , the gravitational potential and the newtonian 3-force of the superposition . the authors would like to thank g. w. semenoff , d. carney , d. scott , b. l. hu , c. anastopoulos and w. g. unruh for helpful discussions , and express our gratitude to p. c. e. stamp for support . f. s. is partially supported by a ubc international tuition award , ubc faculty of science graduate award , the research foundation for opto - science and technology , nserc discovery grant and cfi funds for crucs . the part of the work was supported by the templeton foundation ( grant number jtf 36838 ) . 9 marshall w , simon c , penrose r and bouwmeester d 2003 _ phys . lett . _ * 91 * 130401 kleckner d et al . 2008 _ new journal of physics . _ * 10 * 095020 kaltenbeck r et al . 2012 _ exp astron . _ * 34 * 123 penrose r 1996 _ general relativity and gravitation . _ * 28 * 581 - 599 penrose r 1986 , in _ quantum gravity 2 : a second oxford symposium _ , oxford university press penrose r 1986 , in _ quantum concepts in space and time _ , oxford university press disi l 1987 _ phys . lett . _ * 120 * a 377 - 381 krolyhzy f 1966 _ nuovo cim . _ * 52 * 390 krolyhzy f 1974 , magyar fizikai polyoirat * 12 * 24 komar a b 1969 , int . j. theor . * 2 * 157 omns r 1992 , rev . 64 * 339 unruh w g 2000 _ in relativistic quantum measurement and decoherence _ eds breuer h p and petruccione f ( springer ) pp 125 - 140 breuer h and petruccione f 2001 _ phys . rev . _ a * 63 * 032102 - 1 breuer h and petruccione f 2002 _ the theory of open quantum systems . _ ( oxford ) anastopoulos c 1996 _ phys . _ d * 54 * 1600 anastopoulos c and hu b l 2013 _ class . quantum grav . _ * 30 * 165007 hu b l 2014 _ j. phys . ser . _ * 504 * 012021 anastopoulos c and hu b l 2008 _ class . quantum grav . _ * 25 * 154003 cloutier j and semenoff g w 1991 _ phys . _ d * 44 * 3218 - 3229

we discuss decrease of coherence in a massive system due to the emission of gravitational waves . in particular we investigate environmental gravitational decoherence in the context of an interference experiment . the time - evolution of the reduced density matrix is solved analytically using the path - integral formalism . furthermore , we study the impact of a tensor noise onto the coherence properties of massive systems . we discuss that a particular choice of tensor noise shows similarities to a mechanism proposed by disi and penrose .

You are an expert at summarizing long articles. Proceed to summarize the following text: after years of study to understand how the nucleon spin originates from the constituent partons , an exhaustive answer is still missing . moreover , while a lot of information has been gathered concerning the longitudinal structure of a fast moving nucleon ( with respect to its direction of motion ) , very little is known about the transverse structure . in recent years these aspects have raised a lot of interest and after important theoretical developments and experimental findings , transverse spin and transverse momentum @xmath0 of the quarks are by now considered as fundamental ingredients in the description of the hadron structure . spin-@xmath0 correlations give rise to various observables in hard hadronic processes such as the azimuthal asymmetries seen both in unpolarized or transversely polarized semi - inclusive deep - inelastic scattering ( sidis ) , and led to the introduction of transverse momentum dependent ( tmd ) parton distribution functions ( pdf ) and fragmentation functions ( ff ) . among these functions of special interest are the sivers functions @xmath3 @xcite , which describe an azimuthal asymmetry in the parton distributions inside a transversely polarized nucleon , and by their chirally - odd partner @xmath4 , the boer - mulders functions @xcite , describing the transverse parton polarization inside an unpolarized hadron , and generating azimuthal asymmetries in unpolarized sidis . compass results on the sivers asymmetries are given elsewhere @xcite while here the unpolarized azimuthal asymmetries are presented . the cross - section for hadron production in lepton - nucleon dis @xmath5 for unpolarized targets and an unpolarized or longitudinally polarized beam is the following @xcite : @xmath6 & & \left . + \varepsilon_2 \cos(2\phi_h ) f^{\cos\ ; 2\phi_h}_{uu } + \lambda_\mu \varepsilon_3 \sin \phi_h f^{\sin \phi_h}_{lu } \right ] \end{array}\ ] ] where @xmath7 is the fine structure constant , @xmath8 , @xmath9 and @xmath10 are the inclusive dis variables , @xmath11 is the fraction of the virtual photon energy carried by the detected hadron , @xmath12 is the azimuthal angle of the outgoing hadron in the @xmath13-nucleon system . @xmath14 , @xmath15 , @xmath16 , @xmath17 and @xmath18 are structure functions , with the first and second subscripts which indicate the beam and target polarization respectively , and the last subscript which , if present , indicates the polarization of the virtual photon . finally @xmath19 is the beam longitudinal polarization and : @xmath20 \varepsilon_2 & = & \dfrac{2(1-y)}{1+(1-y)^2 } \\[2ex ] \varepsilon_3 & = & \dfrac{2 y \sqrt{1-y}}{1+(1-y)^2 } \end{array}\ ] ] are depolarization factors . the boer - mulders pdfs contribute to both the @xmath21 and the @xmath22 structure functions , together with the so called cahn effect @xcite which arises from the fact that the kinematics is non collinear when the @xmath0 is taken into account ( i.e. a kinematical higher twist ) , and with the perturbative gluon radiation , resulting in order @xmath23 qcd processes . pqcd effects are becoming important for high transverse momenta @xmath24 of the produced hadrons , while are small for @xmath24 up to 1 gev@xmath25 . the @xmath26 modulation , which arises from the natural polarization of the muon beam , does not have a clear interpretation in the parton model . in the past , azimuthal asymmetries have been measured by the emc collaboration @xcite , with a liquid hydrogen target and a muon beam at a slightly higher energy , but without separating hadrons of different charge . these data have been used @xcite to extract the average @xmath27 . azimuthal asymmetries have been also measured by e665 @xcite and at higher energies by zeus @xcite . more recent are the compass results first presented at @xcite , and the measurements done by hermes , first shown at @xcite . the compass experiment has been set up at the cern sps m2 beam line . it combines high rate beams with a modern two stage magnetic spectrometer@xcite . compass has collected data with a 160 gev positive muon beam impinging on a polarized solid target . the beam is naturally polarized by the @xmath28decay mechanism , and the beam polarization is estimated to be @xmath29 with a @xmath30 relative error . the beam intensity is @xmath31 muons per spill . up to 2006 the experiment has used @xmath1lid as deuteron target because its favorable dilution factor of @xmath320.4 , particularly important for the measurement of . in 2007 an ammonia nh@xmath33 target has been used as proton target . polarizations of 50% and 90% have been reached , respectively for the two target materials . the event selection requires standard dis cuts , i.e. @xmath34 , mass of the final hadronic state @xmath35 , @xmath36 , and the detection of at least one hadron in the final state . moreover only events with a vertex in the forward target cell are used in order to minimize nuclear interactions and for a better data / monte carlo agreement . finally , for the detected hadrons it is also required that : r0.50 * the fraction of the virtual photon energy carried by the hadron be @xmath37 to select hadrons from the current fragmentation region ; * @xmath38 , where @xmath24 is the hadron transverse momentum with respect to the virtual photon direction , for a better determination of the azimuthal angle @xmath12 . data taken both with a longitudinally polarized and a transversely polarized target have been used , mixing positive or negative orientations in order to cancel effects depending on the polarization of the nucleons . this statistics corresponds to almost 1 month of the 2004 data taking and after all the cuts consists of @xmath39 positive hadrons and @xmath40 negative hadrons entering the asymmetry calculations . in the measurement of unpolarized asymmetries all the methods used to cancel the experimental acceptance can not be adopted and the correction for this effect is mandatory before fitting the azimuthal modulation . this is done by using a full monte carlo chain , which starts from the sidis event generation performed by lepto@xcite , simulate the experimental setup and the particle interactions in the passive and active material of the detectors ( also including the detector response done by comgeant ) , and ends with the reconstruction of the generated events by the same program ( coral ) used to analyze the real data . the quality of this chain is evaluated by comparing distributions of real data and of generated events both for the dis variables and for the hadronic variables . the experimental acceptance as a function of the azimuthal angle @xmath41 is then calculated as the ratio of reconstructed over generated events for each bin of @xmath8 , @xmath11 and @xmath24 on which the asymmetries are measured . the overall @xmath9 , @xmath11 and @xmath24 acceptances are quite constant over the range used in the analysis , so that the effect coming by the integration over the unlooked variables when the asymmetries in one of the variables are extracted is well within the systematic error . the measured distribution , corrected for acceptance is fitted with the following functional form : @xmath42 an example of a measured azimuthal distribution , acceptance corrections and corrected azimuthal distribution is shown in fig . [ azidist ] , together with the resulting fit . the contribution of the acceptance corrections to the systematic error has been studied with care . as the asymmetries were extracted from data taken both with longitudinal and transverse target configurations , comparing the two results gives the effect of the acceptance changes due to the different configuration ( solenoid vs. dipole ) of the target magnet and to the different direction of the incoming beam ( for the transverse setup the beam is bent in order to leave the target with the same direction as in the longitudinal case ) . in order to check the effect of the simulation parameters the acceptances have been calculated using two different sets of lepto parameters . all the resulting asymmetries were compared in order to quantify the systematic error in each kinematical bin . further systematic tests , like splitting the data sample according to the event topology and to the time of the measurement , gave no significant contributions . r0.65 the @xmath43 asymmetries , not shown here , measured by compass are compatible with zero , at the present level of statistical and systematic errors , over the full range of @xmath8 , @xmath11 and @xmath24 covered by the data . the @xmath21 asymmetries extracted from compass deuteron data are shown in fig . [ f : cosphi ] for positive ( upper row ) and negative ( lower row ) hadrons , as a function of @xmath8 , @xmath11 and @xmath24 . the bands indicate the size of the systematic error . the asymmetries show the same trend for positive and negative hadrons with a slightly larger values for the positive one . values as large as 30@xmath4440% are reached in the last point of the @xmath11 range . the theoretical predictions @xcite in fig . [ f : cosphi ] takes into account the cahn effect only , which does not depend on the hadron charge . the boer - mulders pdfs are not taken into account in this case . the @xmath45 asymmetries are shown in fig . [ f : cos2phi ] together with the theoretical predictions of @xcite , which take into account the kinematical contribution given by the cahn effect , first order pqcd ( which , as expected , is negligible in the low @xmath24 region ) , and the boer - mulders pdfs ( coupled to the collins ff ) , which give a different contribution to positive and negative hadrons . in @xcite the boer - mulders pdfs are assumed to be proportional to the sivers function as extracted from the preliminary hermes data . the compass data show different amplitude for positive and negative hadrons , a trend which confirms the theoretical predictions . there is a satisfactory agreement between the data points and the model calculations , which hints to a non zero boer - mulders pdfs . 99 slides : + ` http://indico.cern.ch/contributiondisplay.py?contribid=308&sessionid=4&confid=53294 ` d.w . sivers , phys . * d41 * ( 1991 ) 83 . d. boer and p.j . mulders , phys . rev . * d57 * ( 1998 ) 5780 . a. bressan , `` compass results on collins and sivers asymmetries '' , these proceedings . bacchetta _ et al . _ , jhep * 0702 * ( 2007 ) 93 . cahn , phys . * b78 * ( 1978 ) 269 . the emc collaboration , m. arneodo _ et al . _ , z. phys . * c34 * ( 1987 ) 277 . the emc collaboration , j. ashman _ et al . _ , z. phys . * c52 * ( 1991 ) 361 . m. anselmino _ et al . _ , phys . rev . * d71 * ( 2005 ) 74006 the e665 collaboration , m.r . et al . _ , phys . rev . * d48 * ( 1993 ) 5057 . the zeus collaboration , j. breitweg _ et al . _ , b481 * ( 2000 ) 199 . w. kfer , `` measurements of unpolarized azimuthal asymmetries at compass '' , in the proceedings of ` transversity 2008 ' and arxiv:0808.0140 . f. giordano , `` measurements of azimuthal asymmetries of the unpolarized cross - section at hermes '' , in the proceedings of spin2008 and arxiv:0901.2438 . the compass collaboration , p. abbon _ et al . * a577 * ( 2007 ) 455 . g. ingelman , a. edin and j. rathsman , comp . commun . 101 ( 1997 ) 108 . m. anselmino _ et al . _ , eur . j. * a31 * ( 2007 ) 373 . v. barone , a. prokudin and b.q . ma , phys . rev . * d78 * ( 2008 ) 45022 .

azimuthal asymmetries measured in unpolarized semi - inclusive deep inelastic scattering bring important information on the inner structure of the nucleons , and can be used both to estimate the average quark transverse momentum @xmath0 and to access the so - far unmeasured boer - mulders functions . compass results using part of the 2004 data collected with a @xmath1lid target and a 160 gev @xmath2 beam are presented separately for positive and negative hadrons .

You are an expert at summarizing long articles. Proceed to summarize the following text: organic molecular crystals , namely crystals composed of organic molecules held together by weak van der waals forces , are emerging as excellent candidates for fabricating nanoscale devices . these have potential application in electronics and optoelectronics in particular in areas such as solar energy harvesting , surface photochemistry , organic electronics and spintronics @xcite . a feature common to such class of devices is that they are composed from both an organic and inorganic component , where the first forms the active part of the device and the second provides the necessary electrical contact to the external circuitry . clearly the electronic structure of the interface between these two parts plays a crucial role in determining the final device performance and needs to be understood carefully . in particular it is important to determine how charge transfers between the organic and the inorganic component and the energies at which the transfer takes place . this is a challenging task , especially in the single - molecule limit . upon adsorption on a substrate , the electron addition and removal energies of a molecule change value from that of their gas phase counterparts . this is expected since , when the molecule is physisorbed on a polarisable substrate , the removal ( addition ) of an electron from ( to ) the molecule gives rise to a polarisation of the substrate . the image charge accumulated on the substrate in the vicinity of the molecule alters the addition or removal energy of charge carriers from the molecule . a common way to calculate the addition and removal energies is to use a quasiparticle ( qp ) description . within the qp picture , one ignores the effects of relaxation of molecular orbitals due to addition or removal of electrons and consequently takes the relative alignment of the metal fermi level , @xmath1 , with either the lowest unoccupied molecular orbital ( lumo ) and highest occupied molecular orbital ( homo ) of a molecule as removal energy . this effectively corresponds to associate the electron affinity and the ionization potential respectively to the lumo and homo of the molecule . the adequacy of the qp description then depends on the level of theory used to calculate the energy levels of the homo and lumo . if the theory of choice is density functional theory ( dft ) @xcite , then a number of observations should be made . firstly , it is important to note that except for the energy of the homo , which can be rigorously interpreted as the negative of the ionization potential @xcite , in general the kohn - sham orbitals can not be associated to qp energies . this is , however , commonly done in practice and often the kohn - sham qp levels provide a good approximation to the true removal energies , in particular in the case of metals . for molecules unfortunately the situation is less encouraging with the local and semi - local approximations of the exchange and correlation functional , namely the local density approximation ( lda ) and the generalized gradient approximation ( gga ) , performing rather poorly even for the homo level . such situation is partially corrected by hybrid functionals @xcite or by functional explicitly including self - interaction corrections @xcite , and extremely encouraging results have been recently demonstrated for range separated functionals @xcite . the calculation of the energy levels alignment of a molecule in the proximity of a metal , however , presents additional problems . in fact , the formation of the image charge , although it is essentially a classical electrostatic phenomenon , has a completely non - local nature . this means that unless a given functional is explicitly non - local it will in general fail in capturing such effect . the most evident feature of such failure is that the position of the homo and lumo changes very little when a molecule approaches a metallic surface @xcite . such failure is typical of the lda and gga , and both hybrid and self - interaction corrected functionals do not improve much the situation . a possible solution to the problem is that of using an explicit many - body approach to calculate the qp spectrum . this is for instance the case of the gw approximation @xcite , which indeed is capable of capturing the energy levels renormalization due to the image charge effect @xcite . the gw scheme , however , is highly computationally demanding and can be applied only to rather small systems . this is not the case for molecules on surfaces , where the typical simulation cells have to include several atomic layers of the metal and they should be laterally large enough to contain the image charge in full . this , in addition to the gw necessity to compute a significant fraction of the empty states manifold , make the calculations demanding and it is often not simple even to establish whether convergence has been achieved . in this paper we approach the problem of evaluating the charge transfer energies of an organic molecule physisorbed on an inorganic substrate with the help of a much more resource - efficient alternative , namely constrained density functional theory ( cdft ) @xcite . in cdft , one transfers one electron from the molecule to the substrate ( and vice versa ) and calculates the difference in energy with respect to the locally charge neutral configuration ( no excess of charge either on the molecule or the substrate ) @xcite . as such cdft avoids the calculation of a qp spectrum , which is instead replaced by a series of total energy calculations for different charge distributions this approach is free of any interpretative issues and benefits from the fact that even at the lda level the total energy is usually an accurate quantity . finally , it is important to remark that , for any given functional , cdft is computationally no more demanding than a standard dft calculation , so that both the lda and the gga allow one to treat large systems and to monitor systematically the approach to convergence . here we use the cdft approach to study the adsorption of molecules on a 2-dimensional ( 2d ) metal in various configurations . it must be noted that in contrast to a regular 3d metal , in a 2d one the image charge induced on the substrate is constrained within a one - atom thick sheet . this means that electron screening is expected to be less efficient than in a standard 3d metal and the features of the image charge formation in general more complex . in particular we consider here the case of graphene , whose technological relevance is largely established @xcite . most importantly for our work , recently graphene has been used as template layer for the growth of organic crystals @xcite . it is then quite important to understand how such template layer affects the level alignment of the molecules with the metal . as a model system we consider a simple benzene molecule adsorbed on a sheet of graphene . this has been studied in the past @xcite , so that a good description of the equilibrium distance and the corresponding binding energy of the molecule in various configurations with respect to the graphene sheet are available . furthermore , a @xmath2 study for some configurations exists @xcite , so that our calculated qp gap can be benchmarked . our calculations show that the addition and removal energies decrease in absolute value as the molecule is brought closer to the graphene sheet . such decrease can be described with a classical electrostatic model taking into account the true graphene dielectric constant . as it will be discussed , a careful choice of the substrate unit cell is necessary to ensure the inclusion of the image charge , whose extension strongly depends on the molecule - substrate distance . we also reveal that the presence of defects in the graphene sheet , such as a stone - wales one , does not significantly alter the charge transfer energies . in realistic situations , _ e.g. _ at the interface between a molecular crystal and an electrode , a molecule is surrounded by many others , which might alter the level alignment . we thus show calculations , where neighboring molecules are included above , below and in the same plane of the one under investigation . interestingly , our results suggest that the charge transfer states are weakly affected by the presence of other molecules . in order to find the ground state energy of a system , kohn - sham dft minimises a universal energy functional @xmath3=\sum_{\sigma}^{\alpha,\beta}\sum_{i}^{n_\sigma}\langle \phi_{i \sigma}|-\frac{1}{2}\nabla^{2}|\phi_{i \sigma}\rangle+\int d\mathbf{r}v_n(\mathbf{r})\rho(\mathbf{r})+j[\rho]\\ + e_\mathrm{xc}[\rho^{\alpha},\rho^{\beta}]\ : , \end{split}\ ] ] where @xmath4 , @xmath5 and @xmath6 denote respectively the hartee , exchange - correlation ( xc ) and external potential energies . the kohn - sham orbitals , @xmath7 , for an electron with spin @xmath8 define the non - interacting kinetic energy @xmath9 , while @xmath10 is the total number of electrons with spin @xmath8 . the electron density , is then given by @xmath11 . in contrast to regular dft , in cdft one wants to find the ground state energy of the system subject to an additional constraint of the form @xmath12 where @xmath13 is a weighting function that describes the spatial extension of the constraining region and @xmath14 is the number of electrons that one wants to confine in that region . in our case @xmath15 is set to 1 inside a specified region and zero elsewhere . in order to minimise @xmath16 $ ] subject to the constraint , we introduce a lagrange multiplier @xmath17 and define the constrained functional @xcite @xmath18=e[\rho]+v_\mathrm{c}\left(\sum_{\sigma}\int w_\mathrm{c}^{\sigma}(\mathbf{r})\rho^{\sigma}(\mathbf{r})d\mathbf{r}-n_\mathrm{c}\right)\ ] ] now the task is that of finding the stationary point of @xmath19 $ ] under the normalization condition for the kohn - sham orbitals . this leads to a new set of kohn - sham equations @xmath20\phi_{i\sigma}\\ = \epsilon_{i\sigma}\phi_{i\sigma}\ : , \end{split}\ ] ] where @xmath21 is the exchange and correlation potential . equation ( [ equ4 ] ) does not compute @xmath17 , which remains a parameter . however , for each value of @xmath17 it produces a unique set of orbitals corresponding to the minimum - energy density . in this sense we can treat @xmath19 $ ] as a functional of @xmath17 only . it can be proved that @xmath19 $ ] has only one stationary point with respect to @xmath17 , where it is maximized @xcite . most importantly the stationary point satisfies the constraint . one can then design the following procedure to find the stationary point of @xmath19 $ ] : ( i ) start with an initial guess for @xmath22 and @xmath17 and solve eq . ( [ equ4 ] ) ; ( ii ) update @xmath17 until the constraint eq . ( [ equ2 ] ) is satisfied ; ( iii ) start over with the new @xmath17 and a new set of @xmath23s . here we use cdft to calculate the charge transfer energy between a benzene molecule and a graphene sheet . for any given molecule - to - substrate distance , @xmath24 , we need to perform three different calculations : 1 . a regular dft calculation in order to determine the ground state total energy @xmath25 and the amount of charge on each subsystem ( _ i.e. _ on the molecule and on the graphene sheet ) 2 . a cdft calculation with the constraint that the graphene sheet contains one extra electron and the molecule contains one hole . this gives the energy @xmath26 . 3 . a cdft calculation with the constraint that the graphene sheet contains one extra hole and the molecule one extra electron . this gives the energy @xmath27 . the charge transfer energy for removing an electron from the molecule and placing it on the graphene sheet is then @xmath28 . similarly , that for the transfer of an electron from the graphene sheet to the molecule is @xmath29 . since in each run the cell remains charge neutral , there is no need here to apply any additional corrections . however , we have to keep in mind that this method is best used when the two subsystems are well separated so that the amount of charge localized on each subsystem is a well defined quantity . in our calculations we use the cdft implementation @xcite for the popular dft package siesta@xcite , which adopts a basis set formed by a linear combination of atomic orbitals ( lcao ) . the constrain is introduced in the form of a projection over a specified set of basis orbitals and in particular use the lowdin projection scheme . throughout this work we adopt double - zeta polarized basis set with an energy cutoff of 0.02 ry . the calculations are done with norm - conserving pseudopotential and the lda is the exchange - correlation functional of choice . a mesh cutoff of 300 ry has been used for the real - space grid . we impose periodic boundary conditions with different cell - sizes and the @xmath30-space grid is varied in accordance with the size of the unit cell . for instance , an in plane 5@xmath315 @xmath30-grid has been used for a 13@xmath3113 graphene supercell . we begin this section with a discussion on the equilibrium distance for a benzene molecule adsorbed on graphene . this is obtained by simply minimizing the total energy difference @xmath32 , where @xmath33 is the total energy for the cell containing benzene on graphene , while @xmath34 ( @xmath35 ) is the total energy of the same cell when only the benzene ( graphene ) is present . this minimization is performed for two different orientations of the benzene molecule with respect to the graphene sheet : the _ hollow _ ( h ) configuration , in which all the carbon atoms of the benzene ring are placed exactly above the carbon atoms of graphene , and the _ stack _ ( s ) configuration , in which alternate carbon atoms of the benzene molecule are placed directly above carbon atoms of the graphene sheet [ see fig . [ fig : figure1](a , b ) ] . for the h configuration we find an equilibrium distance of 3.4 , while for the s one this becomes 3.25 . these results are in fair agreement with another lda theoretical study @xcite ( predicting 3.4 and 3.17 respectively for for the h and s orientations ) . note that a more precise evaluation of such distances requires the use of van der waals corrected functionals . this exercise , however , is outside the scope of our work and here we just wish to establish that the equilibrium distance is large enough for our constrain to remain well defined . it can also be noted that the equilibrium distance of 3.6 obtained with a vdw - df study @xcite is not very different from our lda result . , for different unit cell sizes of graphene sheet . the results are presented for two different molecule - to - graphene distances : 3.4 and 6.8 .,scaledwidth=45.0% ] we then study the dependence of the charge transfer energies on the size of the graphene unit cell used . this is achieved by looking at the charge transfer gap , @xmath36 , as a function of the unit cell size at various molecule - to - graphene distances ( see fig . [ fig : figure2 ] ) . when the molecule is very close to the graphene sheet , after transferring an electron , the image charge is strongly attracted by the oppositely charged molecule and thereby remains highly localized . however , as the molecule moves away from the substrate , the attraction reduces since the coulomb potential decays with distance , resulting in a delocalization of the image charge . this will eventually spread uniformly all over the graphene sheet in the limit of an infinite distance . if the unit cell is too small , the image charge will be artificially over - confined , resulting in an overestimation of @xmath37 and @xmath38 and , as a consequence , of the charge transfer energies . this effect can be clearly seen in fig . [ fig : figure2 ] , where we display the variation of the charge transfer energies as a function of the cell size . clearly , for the shorter distance ( 3.4 corresponding to the average equilibrium distance ) , the energy gap converges for supercells of about 10@xmath3110 ( 10@xmath3110 graphene primitive cells ) . at the larger distance of 6.8 the same convergence is achieved for a 13@xmath3113 supercell . next we compute the charge transfer energies as a function of the distance between the sheet and the molecule . in order to compare our results with the gap expected in the limit of an infinite distance , we need to evaluate first the ionization potential , @xmath39 , and the electron affinity , @xmath40 , of the isolated benzene molecule . this is also obtained in terms of total energy differences between the neutral and the positively and negatively charged molecule , namely with the @xmath41scf method . this returns a quasiparticle energy gap , @xmath42 , of 11.02 ev , in good agreement ( within 4.5% ) with the experimental value @xcite . likewise we also determine the fermi level ( @xmath43 ) of graphene , which is found to be 4.45 ev . in fig . [ fig : figure3](a ) we show the change in the charge transfer energy gap with the distance of the benzene from the graphene sheet for the h configuration . as expected , when the molecule is close to the surface , there is a considerably large attraction between the image charge and the opposite charge excess on the molecule , resulting in an additional stabilization of the system and a reduction in magnitude of @xmath44 and @xmath37 . hence , in such case the charge transfer energies have a reduced magnitude and the charge transfer gap is smaller than that in the gas phase . then , as the molecule moves away from the graphene sheet , the charge transfer energies increase and so does the charge transfer energy gap until it eventually reaches the value corresponding to the homo - lumo gap of the isolated molecule in the limit of an infinite distance . in figs . [ fig : figure3](b ) , ( c ) , ( d ) and ( e ) we show the excess charge - density , @xmath45 , in different parts of the system after transferring one electron for two different molecule - to - graphene distances . the excess charge - density @xmath45 is defined as @xmath46 , where @xmath47 and @xmath48 are respectively the charge densities of the system before and after the charge transfer . thus the portion of @xmath45 localized on the graphene sheet effectively corresponds to the image charge profile . clearly , due to the stronger coulomb attraction , the image charge is more localized for @xmath49 than for @xmath50 . at equilibrium for the s configuration , @xmath51 , the charge transfer energy gap is calculated to be 8.91 ev , which is in good agreement ( within 4% ) with the gap obtained by @xmath52 @xcite . in table [ table : configuration ] , for the purpose of comparison , we have listed the charge transfer energies and charge transfer gaps for two different heights , 3.4 and 6.8 , and in different configurations . the most notable feature is that for the case of a pristine graphene substrate the specific absorption site plays little role in determining the charge transfer levels alignment . in general actual graphene samples always display lattice imperfections @xcite . in order to determine the effect of such structural defects on the ct energies , we consider a reference system where a stone - wales ( sw ) defect ( in which a single c - c bond is rotated by 90@xmath53 ) is present in the graphene sheet . we have then calculated @xmath54 for two different positions of the molecule with respect to the defect on the sheet , namely the @xmath55 position , in which the molecule is placed right above the defect and the @xmath56 position , in which it is placed above the sheet far from the defect ( see fig . [ fig : figure1 ] ) . our findings are listed in tab . [ table : configuration ] , where we report the charge transfer energies for both the configurations , assuming the molecule is kept at the same distance from the graphene sheet . from the table it is evident that the structural change in graphene due to presence of such defect does not alter the charge transfer energies of the molecule . this is because the image charge distribution on graphene is little affected by presence of the sw defect . in addition , the density of states ( dos ) of graphene remains almost completely unchanged near its fermi energy after introducing such defect as can be seen in fig . [ figure4 ] , which shows that the partial density of states ( pdos ) of the atoms forming the sw defect has no significant presence near the fermi level . thus , after the charge transfer , the electron added to ( or removed from ) the graphene sheet has the same energy that it would have in the absence of the defect , i.e. it is subtracted ( added ) from a region of the dos where there is no contribution from the sw defect . in this context , it is noteworthy that a @xmath52 study @xcite has concluded that altering the structure of pristine graphene by introducing dopant ( which raises the fermi level of graphene by 1 ev ) also has minor effect on the qp gap of benzene , reducing it by less than 3% . .@xmath57 , @xmath58 and @xmath59 for various configurations of a benzene molecule on pristine and defective graphene . h and s denote adsorption of benzene on graphene in the _ hollow _ and _ stack _ configuration , respectively . @xmath55 and @xmath56 correspond to adsorption on graphene with sw defect , with the former corresponding to adsorption exactly on top of the defect and the latter corresponding to adsorption away from the site of the defect . the configurations @xmath60 and @xmath61 both correspond to adsorption of two benzene molecules in _ hollow _ configuration- one at height 3.4 and another at a height 6.8 . while in @xmath60 , the ct is calculated for the lower molecule , in @xmath61 , the ct is calculated for the upper one . finally @xmath62 represents the case in which we have a layer of non - overlapping benzenes adsorbed on graphene and one is interested in calculating the ct energy for one of them , which is placed in the _ hollow _ configuration . [ cols="^,^,^,^,^",options="header " , ] in real interfaces between organic molecules and a substrate , molecules usually are not found isolated but in proximity to others . it is then interesting to investigate the effects that the presence of other benzene molecules produce of the charge transfer energies of a given one . to this end we select three representative configurations . in the first one , @xmath60 , the graphene sheet is decorated with two benzene molecules , one at 3.4 while the other is placed above the first at 6.8 from the graphene plane . we then calculate the charge transfer energies of the middle benzene ( the one at 3.4 from the sheet ) . the excess charge on different parts of the system ( image charge ) , after transferring one electron to the sheet , is displayed in fig . [ fig : he](a ) and fig . [ fig : he](b ) . the second configuration , @xmath61 , is identical to the first one but now we calculate the charge transfer energies of the molecule , which is farther away from the graphene sheet , namely at a distance of 6.8 . for this configuration , the excess charge after a similar charge transfer is shown in fig . [ fig : he](c ) and fig . [ fig : he](d ) . in the third configuration , @xmath62 , we arrange multiple benzene molecules in the same plane . the molecules are in close proximity with each other although their atomic orbitals do not overlap . charge transfer energies are then calculated with respect to one benzene molecule keeping the others neutral and an isovalue plot for similar charge transfer is shown in fig . [ fig : he](e ) and fig . [ fig : he](f ) . the charge transfer energies calculated for these three configurations are shown in tab . [ table : configuration ] . if one compares configurations where the molecule is kept at the same distance from the graphene plane , such as the case of h(@xmath49 ) , @xmath60 and @xmath62 or of h(@xmath50 ) and @xmath61 , it appears clear that the presence of other molecules has some effect on the charge transfer energies . in particular we observe than when other molecules are present both @xmath57 and @xmath58 get more shallow , i.e. their absolute values is reduced . interestingly the relative reduction of @xmath57 and @xmath58 depends on the details of the positions of the other molecules ( e.g. it is different for @xmath60 and @xmath62 ) but the resulting renormalization of the homo - lumo gap is essentially identical [ about 25 mev when going from h(@xmath49 ) to either @xmath60 or @xmath62 ] . this behaviour can be explained in terms of a simple classical effect . consider the case of @xmath60 for example . when one transfers an electron from the middle benzene to the graphene sheet the second benzene molecule , placed above the first , remains neutral but develops an induced charge dipole . the moment of such dipole points away from the charged benzene and lowers the associated electrostatic potential . importantly , also the potential of graphene will be lowered . however , since the potential generated by an electrical dipole is inversely proportional to the square of distance , the effect remains more pronounced at the site of the middle benzene than at that of the graphene sheet . a similar effect can be observed for an electron transfer from the graphene sheet to the middle benzene and for the @xmath62 configuration . in the case of @xmath61 , the system comprising the topmost benzene ( from which we transfer charge ) and the graphene plane can be thought of as a parallel - plate capacitor . the work , @xmath64 , done to transfer a charge @xmath65 from one plate to the other is @xmath66 where @xmath67 is the capacitance , which in turn is proportional to the dielectric constant of the medium enclosed between the plates . hence , at variance with the case of h(d=6.8 ) , the space in between the molecule and the graphene sheet is occupied by a molecule with finite dielectric constant and not by vacuum . this results in a reduction of @xmath64 , so that the charge transfer energies for @xmath61 are smaller than those for h(d=6.8 ) . finally , we show that our calculated energy levels alignment can be obtained from a classical electrostatic model . if one approximates the transferred electron as a point charge and the substrate where the image charge forms as an infinite sheet of relative permittivity @xmath68 then , for a completely planar distribution of the bound surface charge , the work done by the induced charge to take an electron from the position of the molecule ( at a distance @xmath24 ) to infinity is @xmath69 hence , this electrostatic approximation predicts that the presence of the substrate lowers the @xmath70 of the molecule by @xmath71 with respect to the corresponding gas - phase value . however , the actual image charge is not strictly confined to a 2d plane but instead spills out over the graphene surface . we can account for such non - planar image charge distribution by introducing a small modification to the above expression @xcite and write the lumo at a height @xmath24 as @xmath72 where @xmath73 is the distance between the centre of mass of the image charge and the substrate plane and @xmath74 is the gas - phase lumo ( the electron affinity ) . a similar argument for the homo level shows an elevation of same magnitude due to the presence of the substrate . in fig . [ fig : classical_plot ] we plot the charge transfer energies and show that they compare quite well with the curves predicted by the classical model by using an effective dielectric constant of 2.4 for graphene @xcite . when drawing the classical curves we have used an approximate value , @xmath75 , which provides an excellent estimate for smaller distances , @xmath24 . it is worth noting that for larger distances , though the actual value of @xmath73 should be much less , the overall effect of @xmath73 is very small and almost negligible . in the same graph , we have also plotted the classical curves corresponding to benzene on a perfectly metallic ( @xmath76 ) surface . this shows that the level renormalization of benzene for physisorption on graphene is significantly different from that on a perfect metal , owing to the different screening properties of graphene . ( circles ) and @xmath77 ( squares ) calculated for different molecule - to - substrate distances . the cdft results are seen to agree well with the classically calculated curve given in red . the horizontal lines mark the same quantities for isolated an molecule ( gas - phase quantities ) . the continuous black line shows the position of the classically calculated level curve for adsorption on a perfect metal @xmath76.,scaledwidth=44.0% ] we have used cdft as implemented in the siesta code to calculate the energy levels alignment of a benzene molecule adsorbed on a graphene sheet . in general the charge transfer energies depend on the distance between the molecule and the graphene sheet , and this is a consequence of the image charge formation . such an effect can not be described by standard kohn - sham dft , but it is well captured by cdft , which translates a quasi - particle problem into an energy differences one . with cdft we have simulated the energy level renormalization as a function of the molecule - to - graphene distance . these agree well with experimental data for an infinite separation , where the charge transfer energies coincide with the ionization potential and the electron affinity . furthermore , an excellent agreement is also obtained with @xmath0 calculations at typical bonding distances . since cdft is computationally inexpensive we have been able to study the effects arising from bonding the molecule to a graphene structural defect and from the presence of other benzene molecules . we have found that a stone - wales defect does not affect the energy level alignment since its electronic density of state has little amplitude at the graphene fermi level . in contrast the charge transfer energies change when more then a molecule is present . all our results can be easily rationalized by a simple classical electrostatic model describing the interaction of a point - like charge and a uniform planar charge distribution . this , at variance to the case of a perfect metal , takes into account the finite dielectric constant of graphene . this work is supported by the european research council , quest project . computational resources have been provided by the supercomputer facilities at the trinity center for high performance computing ( tchpc ) and at the irish center for high end computing ( ichec ) . additionally , the authors would like to thank dr . ivan rungger and dr . a. m. souza for helpful discussions . 29ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1002/cphc.200700177 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrev.136.b864 [ * * , ( ) ] link:\doibase 10.1103/physrevb.18.7165 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevb.88.165112 [ * * , ( ) ] @noop * * , ( ) @noop ( ) \doibase http://dx.doi.org/10.1016/j.apsusc.2010.07.069 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.96.146107 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevb.88.235437 [ * * , ( ) ] @noop * * , ( ) \doibase http://dx.doi.org/10.1016/0022-1902(81)80486-1 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( )

constrained density functional theory ( cdft ) is used to evaluate the energy level alignment of a benzene molecule as it approaches a graphene sheet . within cdft the problem is conveniently mapped onto evaluating total energy differences between different charge - separated states , and it does not consist in determining a quasi - particle spectrum . we demonstrate that the simple local density approximation provides a good description of the level aligmnent along the entire binding curve , with excellent agreement to experiments at an infinite separation and to @xmath0 calculations close to the bonding distance . the method also allows us to explore the effects due to the presence of graphene structural defects and of multiple molecules . in general all our results can be reproduced by a classical image charge model taking into account the finite dielectric constant of graphene .

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Proceed to summarize the following text: carrier propagation in mott or charge - transfer insulators is a challenging problem particularly motivated by strongly correlated superconducting cuprates @xcite . while holes move incoherently in one - dimensional systems featuring charge and spin separation @xcite as well as in systems with antiferromagnetic ( af ) ising interactions @xcite , the hole motion becomes coherent and quasiparticles ( qps ) arise at low energy in the quantum af @xmath0-@xmath1 model @xcite . these qps are indeed observed in angle - resolved photoemission ( arpes ) experiments in cuprates @xcite . in general , low - energy qps coexist with incoherent processes at high energy , as in the af phase on the square @xcite or honeycomb lattice @xcite . this is however not always the case as shown by arpes experiments for the spin - orbit mott insulator na@xmath4iro@xmath5 @xcite , without clear evidence for qps at low energy @xcite . in a recent study hole - doped li@xmath4ir@xmath6ru@xmath7o@xmath5 with honeycomb structure was found insulating at all doping levels @xcite . electronic systems with honeycomb lattice include graphene @xcite , optical lattices @xcite , topological insulators @xcite , and frustrated magnets @xcite . interest in the latter was triggered by theoretical predictions @xcite that na@xmath4iro@xmath5 may host kitaev model physics and quantum spin hall effect . the ground state of this model is a kitaev spin liquid ( ksl ) characterized by finite spin correlations only for nearest neighbor ( nn ) spins @xcite . the ksl belongs to the spin disordered phases which are in the center of interest in quantum magnetism @xcite . in mott insulators where the strong spin - orbit interaction generates a kramers doublet from partly filled @xmath8 orbitals @xcite , as in na@xmath4iro@xmath5 , effective @xmath9 pseudospins stand for local , spin - orbital entangled @xmath8 states which form orbital moments @xcite . both kitaev and heisenberg interactions emerge from the spin and orbital coupling on the honeycomb lattice of iridium ions and form the kitaev - heisenberg ( kh ) model @xcite . experimental observations revealed zigzag ( zz ) magnetic order in na@xmath4iro@xmath5 @xcite it may be explained within the kh model with next - nearest neighbor ( nnn ) @xmath10 and third nn ( 3nn ) @xmath11 af interactions @xcite . the phase diagram of the frustrated heisenberg @xmath12-@xmath10-@xmath11 af model @xcite includes the nel af , zz , and also stripy ( st ) phase . these phases survive when more general spin interactions with symmetric off - diagonal exchange are considered @xcite . doping the ksl is particularly exciting as the ground state is spin disordered and it is unclear whether qps would form @xcite . moreover in the ksl interactions may emerge that lead to unexpected forms of superconductivity . indeed , slave - boson studies found here @xmath13-wave superconductivity at intermediate doping @xcite , whereas fermi liquid was claimed at light doping @xcite . an important first step to explore hole doping is however the study of single hole motion and the existence of qps . so far , spin - charge separation was shown for the kagome lattice , being a prototype of a spin liquid , while small qp peaks were found at some momenta in a frustrated checkerboard lattice @xcite . the @xmath0-@xmath1-like kh model provides here a unique opportunity to investigate hole propagation in quite different magnetic phases emerging from frustrated interactions . the purpose of this paper is : ( i ) to investigate the evolution of the spectral properties in the kh model under increasing frustration of magnetic interactions , ( ii ) to recognize the qp behavior in various magnetic phases of the frustrated kh model , and ( iii ) to establish whether the disordered ksl indeed realizes a paradigm of a fermi liquid at light doping as suggested in ref . @xcite . in our study we employ exact diagonalization ( ed ) of finite periodic clusters which has the important advantage that no approximations have to be made , which is particularly important as the analytical theory of hole - motion in the kh model is notoriously difficult and largely unexplored . the possible problems with the ed approach are finite size effects and the difficulty to extrapolate to the thermodynamic limit ( tl ) . as important results , we report the spectral functions of the different ordered phases and their respective quasiparticle features . moreover we report a totally incoherent spectral weight distribution for a hole moving in the kitaev spin - liquid phase in the strong coupling regime , i.e. , relevant for the iridate systems . the absence of qps in this case for finite systems clearly suggests that also in the tl there are no qps at strong coupling . for this reason we conclude that then a dilute gas of holes ( that are individually not qps ) will not turn into a fermi liquid . the paper is organized as follows : in the sect . ii we outline the kh model and discuss the calculation of various correlation functions that are used to determine the phase diagram of the kh - model . section iii deals with the motion of holes in the ordered phases of the model and discusses the different propagators and spectral functions used . in sect . iv we address the hole motion in the ksl and analyse the spectral weight distribution . the latter is discussed in the intermediate and strong coupling limit . moreover the dynamical spin structure factor is analysed for the ksl both for the kitaev and the kh model in order to explore the different scattering channels for holes . results are summarized in sect . v. we consider the following @xmath0-@xmath1-like kh model ( @xmath14 ) , on the honeycomb lattice [ fig . [ fig : defs](a ) ] : @xmath15 it consists of the kinetic energy term @xmath16 of projected fermions with flavor @xmath17 @xcite which move in the restricted space without double occupancies as a result of large on - site coulomb repulsion @xmath18 . the spins @xmath9 are defined in terms of fermionic creation ( annihilation ) operators @xmath19 ( @xmath20 ) with flavor @xmath17 at site @xmath21 : @xmath22 where @xmath23 are pauli matrices , @xmath24 . we emphasize that already the kitaev terms @xmath25 with different ising spin interactions that depend on bond direction introduce strong spin frustration in eq . . in the following we shall assume ferromagnetic ( fm ) kitaev ( @xmath26 ) and af heisenberg ( @xmath27 ) exchange , @xmath28 here @xmath1 is the energy unit and @xmath29 $ ] is a parameter that interpolates between the heisenberg and kitaev exchange couplings for nn spins @xmath9 . the model eq . ( [ model ] ) includes nnn @xmath30 and 3nn @xmath31 af terms as well . we use exact diagonalization ( ed ) within the lanczos algorithm for a periodic cluster of @xmath32 sites which accomodates all point group symmetries of the infinite lattice . the momenta corresponding to allowed symmetry representations are presented in the first brillouin zone ( bz ) in fig . [ fig : defs](b ) . we introduce @xmath33 , @xmath34 , and @xmath35 , where @xmath36 and @xmath37 . note that in absence of symmetry - breaking field , @xmath38 and @xmath39 are identical and only two distinct representations exist , @xmath40 . sites and the elementary translations @xmath41 that connect atoms @xmath42 within one sublattice , i.e. , connect unit cells consisting of one atom @xmath42 and one @xmath43 each . heisenberg couplings between nn ( @xmath12 ) , nnn ( @xmath10 ) and 3nn ( @xmath11 ) in @xmath44 ( [ model ] ) are indicated by solid and dashed lines ; kitaev couplings @xmath45 involve a single spin component for each bond direction , @xmath46 . ( b ) first bz with high symmetry @xmath47 and @xmath48 points ( in absence of symmetry breaking these points are equivalent to one another).,title="fig:",width=143 ] .0 cm sites and the elementary translations @xmath41 that connect atoms @xmath42 within one sublattice , i.e. , connect unit cells consisting of one atom @xmath42 and one @xmath43 each . heisenberg couplings between nn ( @xmath12 ) , nnn ( @xmath10 ) and 3nn ( @xmath11 ) in @xmath44 ( [ model ] ) are indicated by solid and dashed lines ; kitaev couplings @xmath45 involve a single spin component for each bond direction , @xmath46 . ( b ) first bz with high symmetry @xmath47 and @xmath48 points ( in absence of symmetry breaking these points are equivalent to one another).,title="fig:",width=151 ] + in the present ed approach of finite systems there is no spontaneous symmetry breaking , and the spin components @xmath49 , @xmath50 , and @xmath51 are equivalent . the intrinsic spin order parameter @xmath52 in the ground state can be determined by identifying and calculating the respective correlation functions that reflect the emerging long - range order @xcite , @xmath53 for each phase @xmath54 , where @xmath55 is the average over the ground state @xmath56 . in this definition we select @xmath57 sign for the spin components @xmath58 for the nel af phase and @xmath59 sign for the staggered and zz phase ; @xmath60 for the nel ( @xmath61 ) , @xmath62 for the st ( @xmath63 ) , and either @xmath64 or @xmath65 for the zz ( @xmath66 ) phase ( these points are equivalent in this case ) . here @xmath21 and @xmath67 label unit cells [ fig . [ fig : defs](a ) ] , and @xmath68 . the order parameter @xmath52 ( [ ssf ] ) is large when spin correlations are close to the ones expected for a magnetic phase @xmath54 ; in all other phases it is negligible . two examples are shown in fig . [ fig : defsb](a ) : ( i ) for @xmath69 one finds large @xmath70 , and ( ii ) for @xmath71 this order parameter drops to @xmath72 but @xmath73 increases to @xmath74 . indeed , a transition between the af and zz phase is found here , see dotted line in fig . [ fig : defsb](b ) , while the st phase is unstable here and @xmath75 is small . ( [ ssf ] ) representing the nel ( @xmath54=af ) , the zigzag ( @xmath54=zz ) and the kitaev invariant @xmath76 ( [ w ] ) obtained for @xmath77 and @xmath78 . ( b ) phase diagram of the kh model eq . in @xmath79 plane ( points ) for @xmath77 , with af , st , zz and ksl phases the insets show types of magnetic order ( arrows ) or disorder ( circles ) . , title="fig:",width=234 ] ( [ ssf ] ) representing the nel ( @xmath54=af ) , the zigzag ( @xmath54=zz ) and the kitaev invariant @xmath76 ( [ w ] ) obtained for @xmath77 and @xmath78 . ( b ) phase diagram of the kh model eq . in @xmath79 plane ( points ) for @xmath77 , with af , st , zz and ksl phases the insets show types of magnetic order ( arrows ) or disorder ( circles ) . , title="fig:",width=264 ] + in the ksl phase spin correlations @xmath80 vanish beyond nn spins at @xmath81 @xcite , and further neighbor correlations remain small in the kitaev liquid regime at @xmath82 @xcite . this results in @xmath83 for all conventional spin order parameters @xcite . to identify the ksl we introduce here an average of the kitaev invariant @xcite on a single hexagon @xmath84 , @xmath85 where @xmath86 labels the spin component @xmath87 interacting with a spin at site @xmath67 along the outgoing bond @xmath88 via @xmath89 . one finds that @xmath90 when the ground state of the kitaev model is approached at @xmath91 , see fig . [ fig : defsb](a ) . the phase diagram in the @xmath11 versus @xmath92 plane is displayed in fig . [ fig : defsb](b ) . we recognize the af phase at small @xmath92 , the intermediate st phase , and at large @xmath93 the kitaev liquid phase . at large further neighbor exchange interaction @xmath11 the zigzag phase emerges in the intermediate range of @xmath92 . theses phases were identified by the order parameters discussed above , while the phase boundaries were determined by a different powerful tool , namely the study of the fidelity susceptibility @xcite , i.e. , the changing rate of the overlap between ground states at adjacent points . note that the af@xmath94zz transition at @xmath95 follows from symmetry arguments and as such is independent of the cluster size . in the following we shall analyze the spectral properties of a hole inserted into the ordered ground state @xmath56 , being the quantum nel af , st , or zz phase . the ksl phase will be explored in detail in the subsequent section . we use here the standard numerical lanczos algorithm which spans efficiently the relevant krylov space and yields spectral functions in form of a continued fraction @xcite . the calculation begins with the determination of the ground state @xmath56 and the subsequent addition of a hole , that is the annihilation of an electron as in a photoemission experiment . therefore we consider in the following the hole creation operator , @xmath96 , in form of a plane wave that includes all sites of the honeycomb lattice equally , and alternatively a hole creation operator , @xmath97 , where holes are created only on one sublattice @xcite , namely sublattice @xmath42 , @xmath98 when a hole is created in the ground state @xmath56 , the spectral functions , @xmath99 correspond to the physical green s function @xmath100 that is measured in arpes experiments ( [ ac ] ) , or to the sublattice green s function @xmath101 ( [ ad ] ) . in the definition of the spectral functions ( [ ac ] ) and ( [ ad ] ) excitation energies are measured relative to the ground state energy @xmath102 of a mott insulator with @xmath103 electrons . ( [ ac ] ) for one hole excitations @xcite at strong coupling , @xmath104 , obtained by ed at four distinct momenta @xmath105 ( solid , dashed , dashed - dotted and dotted lines ) for : ( a ) the nel phase at @xmath106 , ( b , c ) st and zz phases at @xmath107 , and ( d ) the ksl phase at @xmath108 . parameters : @xmath77 , @xmath109 , and @xmath110 , except in ( c ) where @xmath78.,title="fig:",width=291 ] + in all phases the spectra @xmath111 and @xmath112 shown in figs . [ fig : acw ] and [ fig : adw ] , respectively , have the total width @xmath113 as for free hole motion on the honeycomb lattice . for small @xmath114 the model is weakly frustrated and @xmath56 is the quantum af nel state , see fig . [ fig : defsb](b ) . taking @xmath115 , a value representative for strong coupling ( @xmath116 ) @xcite , i.e. , when the kinetic energy of a hole is larger than the energy of a magnetic bond , one finds that the spectral function @xmath111 has a qp at low energy and its spectral weight is large at the @xmath117 and much weaker at the @xmath3 point , see fig . [ fig : acw](a ) . in this phase no qualitative differences between @xmath111 and @xmath112 functions are observed , see figs . [ fig : acw](a ) and [ fig : adw](a ) , except near @xmath118 for the @xmath3 point . the sublattice spectral function , @xmath112 , where holes are injected / removed on the same sublattice , reveals large spectral weight at low energy . for the st and zz phases the low energy features in @xmath112 can indeed be identified as qps accompanied by incoherent spectral weight they are more pronounced in the zz phase , cf . figs . [ fig : adw](b ) and [ fig : adw](c ) . yet , the qp features are suppressed at the @xmath3 and @xmath119 points in these phases in the physical spectral function @xmath111 , cf . figs . [ fig : acw](b ) and [ fig : acw](c ) . as shown before @xcite , this is consistent with the observed absence of qps in arpes for na@xmath4iro@xmath5 @xcite . ( [ ad ] ) for a hole excitation created on one sublattice @xcite . the method , lines and parameters are the same as in fig . [ fig : acw ] . , title="fig:",width=291 ] + the momentum @xmath105 dependence of the low energy qps shown in fig . [ fig : disp ] reveals the strong dependence of hole dispersion on the magnetic order . as in cuprates @xcite , the qp dispersion in the nel af phase is narrowed from the free ( unconstrained ) fermionic band width @xmath120 by strong correlations and is determined by the magnetic exchange @xmath121 . the dispersion has a minimum ( maximum ) at the @xmath122 ( @xmath3 ) point and is further reduced when frustration of magnetic exchange increases from @xmath123 to @xmath106 [ fig . [ fig : disp](a ) ] . at finite @xmath11 the hole energy decreases at the @xmath119 point , but otherwise the dispersions at @xmath106 are quite similar , cf . figs . [ fig : disp](a ) and [ fig : disp](b ) . ) by ed for @xmath115 , @xmath77 , and increasing @xmath92 : ( a ) @xmath110 , and ( b ) @xmath124 . hole energies are respective to the ground state energy @xmath102.,title="fig:",width=302,height=226 ] ) by ed for @xmath115 , @xmath77 , and increasing @xmath92 : ( a ) @xmath110 , and ( b ) @xmath124 . hole energies are respective to the ground state energy @xmath102.,title="fig:",width=302,height=226 ] + in contrast , the dispersion is absent in the st phase , see fig . [ fig : disp](a ) , as coherent hole propagation is hindered here due to the alternating af and fm bonds . instead , for fm chains in the zz phase the dispersion appears reversed with respect to that found for the nel phase now a minimum ( maximum ) is at the @xmath3 ( @xmath122 ) point . while this dispersion decreases from weak ( @xmath125 ) to strong ( @xmath126 ) coupling , its shape remains the same [ dashed line in fig . [ fig : disp](b ) ] . at first glance one might conclude from the spectral functions for the ksl , displayed in figs . [ fig : acw](d ) and [ fig : adw](d ) , that hole propagation in the ksl is similar to that in the zz phase [ figs . [ fig : disp](a ) and [ fig : disp](b ) ] . in this case the lowest excitation energy is found at the @xmath3 point . again , one finds a distinct peak in @xmath127 which is absent in @xmath128 , suggesting also here a hidden qp . as shall see below that the fine structure of the low energy peaks in the spectral function shown in fig . [ fig : adw](d ) does not represent well defined qps in the case of the ksl . next we show that the ksl phase is manifestly different from all the ordered phases discussed so far . we address the nature of low energy states by analyzing first the intermediate coupling regime of @xmath129 . again one finds distinct low energy peaks in @xmath127 @xcite , see fig . [ fig : ksl](b ) , missing in @xmath130 [ fig . [ fig : ksl](a ) ] . in contrast , both spectral functions are rigorously identical at the @xmath122 point , cf . @xmath131 and @xmath132 in figs . [ fig : ksl](a ) and [ fig : ksl](b ) . we note that for the honeycomb lattice @xmath122 corresponds to the dirac point which is the degeneracy point of noninteracting electrons @xcite . a surprise comes when the fine structure of the sublattice spectral function is analyzed in absence of spectral broadening [ at @xmath133 in eq . ( [ ad ] ) ] , @xmath134 which may be rewritten using spectral weights : @xmath135 in the following we shall see the advantage of studying the single - particle propagation directly in terms of the _ spectral weight distribution _ , @xmath136 @xmath137 and @xmath138 is an excited state in the space with one extra hole and total momentum @xmath139 that contributes with a finite spectral weight . we stress , that @xmath140 is in mathematical terms a distribution and not a function , although in the tl it may contain parts that can be represented by continuous curves , in some cases . when looking at the spectral weight distribution in fig . [ fig : ksl](c ) we recognize a robust qp , that is , a well separated bound state at lowest energy , only at the @xmath3 point with large spectral weight @xmath141 . at _ all other _ @xmath142 points no bound states exist but instead rather continuous distributions of weights @xmath140 is found in the range of the lowest eigenvalues @xmath143 ( [ omega ] ) . from the spectral weight distribution displayed in fig . [ fig : ksl](c ) it is clear that for example at the @xmath119 point the spectra are represented by a superposition of several many - body states in the @xmath144-particle sector with slightly different energies and there is no dominant pole that could be considered as a qp state . on the other hand , simply by looking at the spectral function @xmath145 in fig . [ fig : ksl](b ) one may easily overlook the breakdown of the qp picture . we stress that in all cases there is substantial incoherent spectral weight at higher energies that extends to the upper edge of the spectrum at @xmath146 . to understand this striking difference of the character of the spectra at low energy at @xmath3 and @xmath147 , respectively , we need to investigate the spin excitations in the kitaev liquid regime . finally it is the scattering of carriers from spin excitations that determines whether they can propagate as qps or whether they are completely overdamped . at four nonequivalent momenta @xmath142 ( solid , dashed , dashed - dotted and dotted lines ) with @xmath109 : ( a ) full spectral function @xmath111 , and ( b ) sublattice function @xmath112 . panel ( c ) shows the spectral weights @xmath148 ( [ alpha ] ) ( symbols ) larger than @xmath149 at low energy ( shaded ) for different @xmath142 values ; dashed lines are guides to the eye . parameters : @xmath108 , @xmath150 . , title="fig:",width=302 ] + our aim here is to explore the spin excitations of the kh model in the ksl regime . however , our discussion will be more transparent when we first focus on the pure kitaev model at @xmath81 and subsequently analyze by numerical simulation the changes of the spin structure factor for the kh case in the ksl regime ( @xmath151 ) . we shall employ here the usual representation of the kitaev model in terms of @xmath152 spin operators @xmath153 where @xmath154 and @xmath81 , according to eq . . spin excitations are most easily understood by transforming the kitaev model into the majorana representation @xcite . kitaev introduced a representation for the spin algebra @xmath155=i \epsilon_{abc}\sigma_i^c\delta_{ij}$ ] in terms of four majorana operators @xmath156 , @xmath157 , per lattice site which obey the anticommutation relations , @xmath158 . here each spin operator component is expressed by a product of two majorana fermions , @xmath159 where @xmath160 is associated with the bond direction and @xmath161 , with the respective vertex ( at site @xmath21 ) . we suppress the upper index in @xmath162 further on . using this representation one can write the hamiltonian as follows , @xmath163 where @xmath164 are the bond operators , @xmath165 to take care of the fermionic property @xmath166 we adopt a notation where @xmath167 is on the @xmath168 sublattice , respectively . the important point is now , that the so - defined bond operators commute with the hamiltonian , @xmath169=0.\ ] ] hence they are conserved quantities and due to their definition as products of two majoranas , see eq . , these operators can only take the values @xmath170 . in the ground state sector all @xmath164 have the same sign on all bonds , either @xmath171 or @xmath172 . a change of a bond variable leads to a _ vortex pair excitation _ with a gap @xmath173 . in addition there is a second class of spin excitations which result from the motion of the @xmath174 majorana particles described by the hamiltonian in eq . . here we concentrate on the symmetric case , with equal exchange constants along three directions in the honeycomb lattice , @xmath175 ; in this case the _ majorana excitations are gapless _ @xcite . for the undoped ksl in the case of the kitaev - heisenberg model eq . at four distinct momenta , for : ( a ) @xmath108 , and ( b ) @xmath176 . the curves are obtained by exact diagonalization of a 24-site cluster with periodic boundary conditions and are smoothened using a lorentzian broadening @xmath177.,title="fig:",width=302 ] for the undoped ksl in the case of the kitaev - heisenberg model eq . at four distinct momenta , for : ( a ) @xmath108 , and ( b ) @xmath176 . the curves are obtained by exact diagonalization of a 24-site cluster with periodic boundary conditions and are smoothened using a lorentzian broadening @xmath177.,title="fig:",width=313 ] next we shall investigate the dynamic spin - structure factor in the kitaev limit @xmath178 by exact diagonalization and explore the changes in the kitaev - liquid regime of the kh model for @xmath82 , which can not be solved analytically . this is demonstrated by a calculation of the dynamic spin - structure factor @xmath179 defined as follows @xcite , @xmath180 , and @xmath181 denotes spin raising ( lowering ) operators , respectively . here @xmath56 and @xmath182 are the ground and excited state , with energies @xmath183 and @xmath184 , respectively . the spin quantization axis is chosen parallel to the @xmath51-th spin axis of the kitaev term for convenience . figure [ fig : skw](a ) shows @xmath185 for the kh model eq . ( 1 ) in the ksl regime at @xmath108 for four different @xmath139 points : @xmath3 , @xmath122 , @xmath186 and @xmath119 . the numerical calculations were performed for a 24-site cluster with periodic boundary conditions . the main weight of the spin structure factor is concentrated between the vortex type spin gap at @xmath187 and @xmath188 . we observe a moderate momentum dependence with the vortex spin gap at @xmath3 , @xmath189 slightly larger than at @xmath122 , @xmath186 , and @xmath119 , where we find @xmath190 . moreover , we can conclude that the dispersion of the gap and of the peak structures is due to the heisenberg terms in the hamiltonian and disappears when one approaches the kitaev limit @xcite , as seen in fig . [ fig : skw](b ) . thus we find here that the main contributions to the spin - response as measured by dynamical spin - structure factor come from the gapped vortex excitations . furthermore the classification of spin excitations obtained for the pure kitaev model as well as the spin gap due to vortex excitations appears still relevant for the ksl phase of the kh model eq . ( 1 ) at @xmath108 . our central argument why qps at finite momentum are destroyed at _ intermediate coupling _ , as shown in fig . [ fig : ksl](c ) , will be outlined next . our explanation rests on the fact that there are two distinct types of elementary spin excitations from which the holes can scatter in the ksl . these excitations can be classified according to the exact solution given by kitaev @xcite as : ( i ) gapped vortex spin excitations with minimal gap @xmath191 , and ( ii ) gapless majorana excitations . we consider as intermediate coupling regime the range @xmath192 , where the kinetic energy of holes is comparable with the magnetic energy on the bonds , and not much larger as in the strong coupling regime , @xmath193 . an important aspect of the spectral functions at intermediate coupling , as in the case of fig . [ fig : ksl](c ) which was calculated for @xmath194 , is the large size of the spin gap , @xmath195 in units of the @xmath0-scale . as the excitation energies @xmath143 of the low energy states of holes at @xmath119 , @xmath186 and @xmath122 relative to the bottom of the band at the @xmath3 point @xmath196 , are much smaller than the vortex spin gap @xmath197 at @xmath194 , no decay via vortex excitations @xcite can occur . therefore , we conclude that the destruction of qps is due to the scattering of _ gapless _ majorana fermion excitations , and that fig . [ fig : ksl](c ) has to be seen as a fingerprint of fractionalization of electrons into holons and gapless majorana fermions of the kitaev model @xcite . because of the dominant role of scattering from gapless spin - excitations we expect that in larger clusters with a denser @xmath139-mesh , hole pockets near the @xmath3 point are not protected against strong scattering , and in consequence the ksl does not turn into a fermi liquid at low doping . , as obtained for : ( a ) the zz phase and ( b ) the ksl phase , both at intermediate coupling @xmath198 , and ( c ) the ksl phase at strong coupling @xmath126 . parameters : ( a ) @xmath199 and @xmath200 ; ( b ) and ( c ) @xmath108 and @xmath110 in eq . width=302 ] before moving to the strong coupling regime @xmath201 we shall highlight the striking difference of the spectral weight distribution @xmath202 at low energy in the zz and the ksl phase , respectively , at @xmath198 . in fig . [ fig : qp](a ) the spectral weight distribution @xmath202 for the zz phase shows well defined qp bound states . although the spectral weights of these qps are significantly reduced , the bound states are well separated from the continuum ( of incoherent states ) at higher energy , except for @xmath203 where this binding energy is much weaker . in the ksl phase fig . [ fig : qp](b ) , on the other hand , a well defined bound state is seen only at the @xmath3 point . because of the size of the vortex gap @xmath204 the absence of qps in fig . [ fig : qp](b ) at @xmath119 , @xmath186 and @xmath122 is a smoking gun for the important role played by scattering by gapless majorana excitations . this is consistent with the spectral shape of @xmath205 which is reminiscent of a continuum , that may result from a convolution of holons and gapless majorana fermions . we turn now to the _ strong coupling _ regime . in the strong coupling case of @xmath115 , i.e. , relevant for the iridates @xcite and displayed in fig . [ fig : qp](c ) , the vortex spin gap becomes small ( in units of @xmath0 ) , i.e. , @xmath206 t . thus the strong coupling result in fig . [ fig : qp](c ) highlights the effect of the additional vortex excitations which form a new decay channel and damp the excitations at the @xmath122 , @xmath186 , and @xmath119 points even further . compared to the result at intermediate coupling at @xmath207 [ fig . [ fig : qp](b ) ] where the vortex gap is @xmath208 , at strong coupling one finds a spin gap to the vortex excitations which is 20 times smaller , that is @xmath209 . furthermore , it may be instructive to go back to the spectral function @xmath210 in fig . [ fig : adw](d ) which basically contains the same information for the strong coupling case ( @xmath126 ) of the ksl as the spectral weight distribution shown in fig . [ fig : qp](c ) . the fine structure seen in @xmath202 can only be resolved in @xmath210 when the resolution parameter @xmath174 is taken small enough . for arpes experiments , where actually @xmath211 is measured , this implies that a sufficient momentum and energy resolution is required . we conclude , that the peak at @xmath3 , which appeared as a separate bound state at intermediate coupling in fig . [ fig : qp](b ) , appears now at strong coupling , see fig . [ fig : qp](c ) , rather as the edge of the continuum than as an isolated bound state . moreover , when a single hole does not propagate as a qp , then a dilute gas of holes will not form a fermi liquid . therefore we stress , that the absence of a well defined separate bound state is a _ new _ qualitative feature which speaks against qps and fermi liquid behavior at low doping , and this time the argument emerges from a diagnosis at the @xmath3 point itself ! we have shown that hole propagation is modified in a remarkable way as increasing kitaev interactions drive the system from the nel order via other ordered antiferromagnetic phases towards the kitaev spin liquid . quasiparticles are found in the nel phase , whereas coherent hole propagation is hindered in stripe and zigzag phases , where hidden quasiparticles with weak dispersion result from coexisting ferromagnetic and antiferromagnetic bonds . as the most unexpected result , in the kitaev liquid phase we have found unprecedented spectral weight distribution at low energy that signals the absence of quasiparticles , both at intermediate and strong coupling . thus , it appears clearly that carrier motion in the lightly doped kitaev liquid is non - fermi liquid like . the above conclusion follows from the short - range nature of spin correlations in the kitaev spin liquid @xcite . unlike in a quantum antiferromagnet on a square @xcite or honeycomb @xcite lattice where spin - flip processes couple to a moving hole and generate new energy scale for coherent hole propagation , the kitaev spin - liquid phase is characterized by ising - like nearest neighbor spin correlations of one spin component . such correlations are insufficient to generate coherent hole propagation . they are well captured by the present cluster size of @xmath32 sites , and qualitative changes of the described scenario are therefore unexpected when the cluster size is increased . summarizing , we have given clear arguments that shed serious doubts on the claim from a slave - boson approximation , namely that the low doped kitaev spin - liquid phase is a fermi liquid @xcite . the first of these arguments rests on the exact solution for the spin excitations in the kitaev model in the intermediate coupling regime , and on our finding that _ gapless _ majorana excitations are responsible for the absence of qps away from the @xmath3 point . from the gaplessness we conclude that also states in the closer vicinity of @xmath3 are not protected . our second argument emerges in the strong coupling case from the result for the spectral distribution @xmath212 at the @xmath3 point itself , which in this case has no similarity to a qp but appears rather as the edge of a continuum . we consider this as evidence that at strong coupling there are no qps in the single hole case near the @xmath3 point , and from this finding we can safely conclude that fermi liquid behavior is absent in the low doping regime . we thank bruce normand for valuable advice , as well as mona berciu , george jackeli and roser valent for insightful discussions . a.m.o . acknowledges support by the polish national science center ( ncn ) under project no . 2012/04/a / st3/00331 . acknowledges support by the natural science foundation of jiangsu province of china under grant no . bk20141190 . s. a. trugman , phys . b * 37 * , 1597 ( 1988 ) ; 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we explore with exact diagonalization the propagation of a single hole in four magnetic phases of the @xmath0-@xmath1-like kitaev - heisenberg model on a honeycomb lattice : the nel antiferromagnetic , stripe , zigzag and kitaev spin - liquid phase . we find coherent propagation of spin - polaron quasiparticles in the antiferromagnetic phase by a similar mechanism as in the @xmath0-@xmath1 model for high-@xmath2 cuprates . in the stripe and zigzag phases clear quasiparticles features appear in spectral functions of those propagators where holes are created and annihilated on one sublattice , while they remain largely _ hidden _ in those spectral functions that correspond to photoemission experiments . as the most surprising result , we find a totally incoherent spectral weight distribution for the spectral function of a hole moving in the kitaev spin - liquid phase in the strong coupling regime relevant for iridates . at intermediate coupling the finite systems calculation reveals a well defined quasiparticle at the @xmath3 point , however , we find that the gapless spin excitations wipe out quasiparticles at finite momenta . also for this more subtle case we conclude that in the thermodynamic limit the lightly doped kitaev liquid phase does not support quasiparticle states in the neighborhood of @xmath3 , and therefore yields a _ non - fermi liquid _ , contrary to earlier suggestions based on slave - boson studies . these observations are supported by the presented study of the dynamic spin - structure factor for the kitaev spin liquid regime .

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Proceed to summarize the following text: we stress that the main purpose of this paper is to extend and complete the study of the notion of generalized riesz product associated to the rank one flows on @xmath0 formulated in the same manner as peyrire in @xcite . the authors in @xcite mentioned that peyrire extended the notion of riesz product to the real line using a class of kernel functions . furthermore , it is noted by peyrire in his pioneer paper @xcite that an alternative extension of the classical riesz products can be done using the bohr compactification of @xmath0 . indeed , it is usual that the extension of some notions from the periodic setting to the almost periodic deals with the bohr compactification @xmath2 of @xmath0 ( or more generally , the bohr compactification of local abelian groups ) . the bohr compactification plays in the almost periodic case the same role played by the torus @xmath3 in the periodic case as the domain of the fast scale variables . as opposed to the torus , the bohr compactification is often a non - separable compact topological space and this lack of separability is a source of difficulties in trying to adapt the arguments from the periodic context to the almost periodic one . peyrire @xcite mentioned this difficulty and introduced the riesz products on @xmath0 associated to some class of kernels . + notice that here we use the notion of separability used in hirotada - kakutani paper @xcite , that is , the topological group ( or space ) is called separable if it satisfies the second countability axiom of hausdorff which means that it has a countable basis . in that paper , hirotada and kakutani established that the bohr compactification of a given locally compact abelian group has a countable basis if and only if the union of the spectrum of all almost periodic functions is countable . + our analysis here is also motivated by the recent growing interest in the problem of the flat polynomials suggested by a.a . prikhodko @xcite in the context of @xmath0 . it turns out that the main idea developed in @xcite does not seem well adapted to the context of our extension of generalized riesz products to the bohr compactification of @xmath0 . this is due to the fact that the sequence of trigonometric polynomials constructed by a.a . prikhodko is only locally @xmath4-flat and not @xmath4-flat in the usual sense ( see , for instance , remark [ rem - pri ] ) . + the paper is organized as follows . in section 2 we review some standard facts on the almost periodic functions including the ergodicity of the action of @xmath0 by translations on its bohr compactification . in section 3 we define the notion of the generalized riesz products on the bohr compactification of @xmath0 . in section 4 we summarize and extend the relevant material on the kakutani criterion and the bourgain criterion on the singularity of the generalized riesz products introduced in section 3 . in section 5 we state and prove the central limit theorem due to m. kac . in section 6 we apply the central limit theorem of kac to prove our main result concerning the singularity of a large class of generalized riesz products on @xmath2 . finally , in the appendix , we consider the problem of the flat polynomials on the bohr compactification of @xmath0 and we add a short note based on hirodata - kakutani paper @xcite on the bohr compactification compared with the stone - ech compactification . the bohr compactification of @xmath0 is based on the theory of almost periodic functions initiated by h. bohr @xcite in connection with the celebrated @xmath5-function of riemann . in this section we are going to recall the basic ingredients of this theory . for the basic facts about almost periodic functions and generalizations of this concept the reader is referred to the classical presentation of bohr @xcite and besicovitch @xcite . we point out that the theory of almost periodic functions can be extended to more general setting with applications in many context including the non - linear differential equations @xcite . let @xmath6 be a bounded continuous function and @xmath7 ; we say that @xmath8 is an @xmath9-almost period for @xmath10 if latexmath:[\[\displaystyle \sup_{x \in \r}|f(x+\tau)-f(x)|\stackrel{\rm { def}}{= } @xmath10 is said to be almost periodic if for any @xmath12 the set of @xmath9-almost periods of @xmath10 is relatively dense , i.e. , there is @xmath13 such that any interval with length @xmath14 contains at least one @xmath9-almost period . the space of all almost periodic functions is denoted by @xmath15 . from the above definition we easily deduce that @xmath15 is a subspace of the space of bounded continuous functions on @xmath0 . an important characterization of almost periodic functions is due to bochner and it can be stated as follows [ bochner ] a bounded function @xmath10 is almost periodic function if , and only if , the family of translates @xmath16 is relatively compact in the space bounded continuous functions on @xmath0 endowed with the sup - norm topology . the proof of theorem [ bochner ] can be found in @xcite , @xcite or @xcite . furthermore , we have the following fundamental theorem ( see for instance @xcite or @xcite ) . a bounded continuous function @xmath10 is almost periodic function if , and only if , @xmath10 is uniformly approximated by finite linear combinations of functions in the set @xmath17 . the space of all continuous functions on @xmath2 is denoted by @xmath18 . @xmath18 is a commutative @xmath19-algebra under pointwise multiplication and addition . below , we give an important topological characterization of the bohr compactification of @xmath0 due to gelfand , raikov and chilov . they obtain this characterization as an application of their theory of commutative banach algebras . [ gelfand]the group @xmath0 , equipped with the usual addition operation , may be embedded as a dense subgroup of a compact abelian group @xmath2 in such way as to make @xmath15 the family of all restrictions functions @xmath20 to @xmath0 of functions @xmath21 . the operator @xmath22 is an isometric @xmath23-isomorphism of @xmath18 onto @xmath15 . moreover , the addition operation @xmath24 extends uniquely to the continuous group operation of @xmath2 , @xmath25 the group @xmath2 is called the bohr compactification of @xmath2 . for simplicity of notation , for any @xmath10 in @xmath15 , we use the same letter @xmath10 for its canonical extension to @xmath2 . as a consequence of theorem [ gelfand ] combined with the riesz representation theorem we have the dual of the space @xmath15 is isometrically isomorphic to the space @xmath26 of all radon measures on the bohr compactification of @xmath0 . the isomorphism @xmath27 is given by the formula @xmath28 we recall in the following the definition of the characters . the characters play a important role in the abelian group and , by koopmann observation , in the spectral analysis of dynamical systems . let @xmath29 be an abelian group and @xmath30 its identity element , then a character of @xmath29 is a complex valued function @xmath31 defined on @xmath29 such that @xmath32 and @xmath33 for all @xmath34 let us recall the following basic fact on compact abelian groups due to peter and weyl . let @xmath29 be a compact abelian group , with @xmath35 its borel field and @xmath36 its haar measure . then the set of continuous characters is fundamental both in @xmath37 and in @xmath38 the proof of peter - weyl theorem can be found in @xcite and for the thorough treatment we refer the reader to @xcite . it is obvious that the continuous characters of @xmath2 are the functions @xmath39 . in addition the orthogonality of two distinct characters can be checked directly . indeed , @xmath40 for @xmath41 we denote by @xmath42 the asymptotic mean value of @xmath10 , given by @xmath43 as a consequence of the averaging properties of almost periodic functions we have the following [ mean ] for any @xmath41 we have @xmath44 where @xmath45 is the haar measure in @xmath2 , normalized to be a probability measure , and @xmath46 is the usual lebesgue measure in @xmath0 . moreover we have @xmath47 where @xmath48 is any bounded subset of @xmath0 with @xmath49 , @xmath50 is the lebesgue measure of @xmath48 . following @xcite , for any @xmath51 , we introduce the notation @xmath52 that is , @xmath53 are the fourier coefficients of @xmath10 relative to orthonormal family @xmath54 ; the inner product is defined by @xmath55 + since the functions @xmath41 correspond to restrictions , @xmath56 , of continuous functions @xmath10 on @xmath2 , a natural question is whether it is possible to define a class of functions @xmath10 which correspond to restrictions " @xmath56 , of functions @xmath57 . this motivates the following definition . given @xmath58 the space @xmath59 , of besicovitch s generalized almost periodic functions on @xmath0 , consists of those functions @xmath60 for which there exists a sequence @xmath61 satisfying @xmath62 we denote @xmath59 simply by @xmath63 the space of generalized almost periodic functions @xmath64 was introduced by besicovitch , who also gave them a structural characterization . we refer to @xcite for more details about functions in @xmath64 . we immediately have @xmath65 for any @xmath66 and it is easy to see that any @xmath67 has the mean value property , that is , for any bounded measurable @xmath68 , with @xmath69 , we have @xmath70 where @xmath71 . the space corresponds to @xmath72 in a way similar to the one in which the space @xmath15 corresponds to @xmath18 . indeed , notice first that the definition of @xmath64 immediately gives that the asymptotic mean value @xmath73 of a function in @xmath64 is well defined ; moreover , any approximating sequence @xmath61 satisfying can be viewed as a cauchy sequence in @xmath72 and , hence , there exists @xmath74 such that @xmath75 converge to @xmath10 in @xmath72 . since @xmath76 is easily seen to be independent of the approximating sequence , in this way we may associate with each @xmath77 a well determined function @xmath74 which we may view as an _ `` extension '' _ of @xmath10 to @xmath2 . notice that the map @xmath78 is a linear map and that the approximation procedure together with lemma [ mean ] show that @xmath79 as a consequence , the kernel of the map @xmath80 is made by the functions @xmath10 such that the asymptotic mean value of @xmath81 is @xmath82 . the corresponding quotient space is denoted by @xmath83 . for @xmath77 we set @xmath84 so that @xmath85 is a semi - norm on @xmath64 . it is well known that @xmath64 is complete with respect to the semi - norm @xmath85 ( see for instance @xcite ) . the space @xmath86 is endowed with the scalar product @xmath87 the second equality follows by with @xmath88 , implying that the scalar product is preserved under the map @xmath78 . finally , we define @xmath89 again , @xmath90 is a semi - norm on @xmath91 and the corresponding quotient space is denoted by @xmath92 . we finish this paragraph by stating the following important fact on the properties of the map @xmath93 . the proof of it is left to the reader . the map @xmath93 is an isometric isomorphism between the banach spaces @xmath83 and @xmath72 for any @xmath94 $ ] furthermore , we point out that a suitable extension of lebesgue and fatou s convergence results is obtained in @xcite . + gelfand - raikov - chilov theorem [ gelfand ] allows us to define the bohr compactification of @xmath0 . but , in the harmonic analysis , it is well known that the pontrygain theorem gives an alternative definition . we briefly recall it here and we refer the reader to the large literature on the subject @xcite and for a deeper discussion on the relation between the almost periodic functions and the bohr compactification of @xmath0 to @xcite , @xcite . + let @xmath29 be locally compact abelian group and let @xmath95 be its dual , i.e. , @xmath95 is the set of the characters endowed with the topology inherited from @xmath29 . let @xmath96 be @xmath95 with the discrete topology . then @xmath97 is the bohr compactification of @xmath29 . @xmath98 is a compact group such that @xmath29 is a dense subset of @xmath98 . we end this section by stating and proving the classical result on the ergodicity of the action of @xmath0 by translation on @xmath2 . we recall that the action of @xmath0 by translation is defined by @xmath99 where the extended addition is given by theorem [ gelfand ] . clearly , the family @xmath100 is a flow acting on @xmath2 since @xmath101 and the haar measure is invariant under translation . the action of @xmath0 on @xmath2 is ergodic , that is , for any borel set @xmath102 which is invariant under the translation action we have @xmath103 , where @xmath104 denotes the normalized haar measure of @xmath105 . moreover @xmath106 let @xmath102 be an invariant borel set . we have @xmath107 now , translations are strongly continuous on @xmath108 . indeed , this is a standard consequence of the density of @xmath18 in @xmath109 , which follows from theorem [ gelfand ] , and the invariance of the haar measure . therefore , the right - hand side is a continuous function of @xmath110 , and so the identity still holds with @xmath111 . hence we get , using fubini theorem and the invariance of the haar measure , @xmath112 from which it follows that @xmath113 , as asserted . it remains to show that @xmath114 . first we observe that @xmath0 is a borel subset of @xmath2 , since it is the union of a countable family of compact sets , e.g. , the images of the intervals @xmath115,~~ k \in \n$ ] . since @xmath0 is invariant under the translation action we have @xmath116 . but , for any @xmath117 , @xmath118 is also an invariant borel set and @xmath119 . by the invariance of the haar measure @xmath120 . hence , we conclude that @xmath114 and the proof is achieved . riesz products were discovered in 1918 by f. riesz @xcite to answer affirmatively a special question in the theory of fourier series , namely , whether there exists a continuous measure whose fourier coefficients do not vanish at infinity . roughly speaking , the riesz products are a kind of measures on the circle constructed inductively . the pioneer riesz product construction gives a concrete example . since then , the riesz construction proved to be the source of powerful ideas that can be used to produce concrete counterexample of measures with a number of desired properties ( controllability of the convergence of the fourier coefficients being the goal of the original construction ) . later , a. zygmund extended riesz construction and introduced what it is nowadays called classical riesz products @xcite . in 1975 , that riesz products appear as a spectral type of some dynamical systems was shown by f. ledrappier @xcite . ten years later , m. queffelec @xcite , inspired by the work of coquet - kamae and mandes - france @xcite , showed that the specific generalized riesz products are the right tool to describe the spectrum of the class of dynamical systems arising from the substitution ( see @xcite and the references therein ) . in 1991 , b. host , j .- f . mla , f. parreau in @xcite realized a large class of riesz products as the maximal spectral type of the unitary operator associated with a non - singular dynamical system and a cocycle . finally , in the more general setting , j. bourgain established the connections between some class of generalized riesz products on the circle and the maximal spectral type of a class of maps called rank one maps @xcite . one year later , an alternative proof is given by choksi - nadkarni using the host - mla - parreau argument @xcite,@xcite and at the same time a simple proof is obtained by klemes - reinhold @xcite using the standard fourier analysis argument . + in @xcite , el abdalaoui - lemaczyk - lesigne and ulcigrai proved that the generalized riesz products analogous to peyrire - riesz products can be realized as a spectral type of some class of rank one flows . + here our aim is to extend the notion of generalized riesz products to the bohr compactification of @xmath0 . it turns out that such generalization can be done directly . more precisely , in the study of the spectrum of some class of rank one flows , the following trigonometric polynomials on @xmath0 appears @xmath121 with @xmath122 is a sequence of positive integers greater than @xmath123 and @xmath124 is a sequence of positive real numbers with @xmath125 for any @xmath126 ; the sequence @xmath127 is defined inductively by @xmath128 for simplicity , for any @xmath129 we introduce the following sequence of real numbers @xmath130 and for any real number @xmath131 , we put @xmath132 let @xmath133 be a family of trigonometric polynomials given by . then the weak limit of the sequence of probability measures on @xmath2 @xmath134 exists and is denoted by @xmath135 let @xmath136 and @xmath137 . by the definition of @xmath138 we have @xmath139 and it is obvious that @xmath140 therefore , @xmath141 is a probability measure on @xmath2 . in addition , for any @xmath142 , we have @xmath143 hence @xmath144 consequently the limit @xmath145 of the sequence @xmath146 exists . now , since @xmath2 is compact and @xmath147 is a sequence of probability measure on @xmath2 we can extract a subsequence @xmath148 which converge weakly to some probability measure on @xmath2 . we deduce that the limit of @xmath147 exists in the weak topology and this finishes the proof . the proof above is largely inspired by lemma 2.1 in @xcite ; it gives more , namely , the polynomials @xmath138 can be chosen with positive coefficients and satisfying @xmath149 we mention also that we have @xmath150 for any given sequence of positive integers @xmath151 . + we can now formulate our main result whose proof occupies all section 6 . [ main - here ] let @xmath152 be a sequence of positive integers greater than @xmath123 and @xmath153 be a sequence of positive real numbers . assume that there exists a sequence of positive integers @xmath154 such that for any positive integer @xmath155 , the numbers @xmath156 are rationally independent . then the generalized riesz product @xmath157 where @xmath158 is singular with respect to the haar measure on @xmath2 . the famous dichotomy theorem of kakutani has a rather long history . in his 1948 celebrated paper @xcite , kakutani established a purity law for infinite product measures . precisely , if @xmath159 and @xmath160 are a infinite product measures , where @xmath161 are probability measures such that @xmath162 is absolutely continuous with respect to @xmath163 , for each positive integer @xmath164 , then @xmath165 or @xmath166 according as @xmath167 converges or diverges . there are a several proofs of kakutani criterion in literature ( see @xcite and the references given there ) . for a proof based on the hellinger integral we refer the reader to @xcite . kakutani s theorem was specialized to the gaussian measures on hilbert space with identical correlation operators in @xcite and it was extended to gaussian measures with non - identical correlation operators by segal @xcite , hajek @xcite , feldman @xcite and rozanov @xcite . later , in 1979 , ritter in @xcite , @xcite generalized kakutani s theorem to a certain non - products measures with application to the classical riesz products . here , applying the bourgain methods , we obtain a new extension of kakutani s theorem to the class of generalized riesz products on the bohr compactification of @xmath0 . indeed , we show that the independence along subsequence suffices to prove the singularity . nevertheless , our strategy is similar to the strategy of @xcite and it is based on the extension of bourgain methods to the generalized riesz products on the bohr compactification of @xmath0 combined with the central limit tools introduced in @xcite . moreover , having in mind applications beyond the context of this paper , we shall state and prove a guenais sufficient condition on the @xmath4 flatness of the polynomials which implies the existence of generalized riesz products on @xmath2 with haar component . we recall that the generalized riesz products @xmath168 is given by @xmath169 where @xmath170 [ bourg - cri]the following are equivalent 1 . @xmath168 is singular with respect to haar measure . 2 . @xmath171 _ the proof of @xmath2 version of bourgain criterion is based on the following lemma . [ cs - b ] the following are equivalent 1 . @xmath172 2 . _ @xmath173 _ the proof is a simple application of cauchy - schwarz inequality . consider @xmath174 and @xmath175 . denote @xmath176 and @xmath177 its complement in @xmath178 . let @xmath179 be two real numbers and define a probability measure on @xmath0 by @xmath180}(t)}{b - a}dt , { \textrm{~where~ } } dt { \textrm{~is~the~lebesgue~measure}}.\ ] ] then we have @xmath181 by letting @xmath182 goes to infinity , we get @xmath183 the last equality follows from . [ proof of theorem [ bourg - cri ] ] assume that ( i ) holds . to prove that @xmath168 is singular , it suffices to show that for any @xmath184 , there is a borel set @xmath185 with @xmath186 and @xmath187 . let @xmath188 . fix @xmath189 such that for any @xmath190 , we have @xmath191 . the set @xmath192 satisfies : @xmath193 and since @xmath194 is open set , it follows from the portmanteau theorem that we have @xmath195 for the converse , given @xmath188 , there exists a continuous function @xmath196 on @xmath2 such that : @xmath197 let @xmath198 . by cauchy - schwarz inequality , we have @xmath199 since @xmath168 is the weak limit of @xmath200 , we have @xmath201 thus , @xmath202 . since @xmath203 is arbitrary , we get @xmath204 and this completes the proof . from now on , let @xmath205 be a sequence of positive integers for which @xmath206 are linearly independent over the rationals and let us fix some subsequence @xmath207 of @xmath205 , @xmath208 and @xmath209 with @xmath210 . put @xmath211 we define the degree of any trigonometric polynomial @xmath10 by @xmath212 and we denote @xmath213 . from equations and , we have @xmath214 since @xmath215 , telescoping we get @xmath216 in the same spirit as above it is easy to see the following lemma . the proof of it in the case of the torus is given in @xcite and @xcite . [ inegalite1 ] with the above notations we have @xmath217 the following proposition is a simple extension of proposition 2.4 in @xcite . [ haar ] we have @xmath218 the sequence of probability measures @xmath219 converges weakly to the haar measure . from the proposition [ haar ] and lemma [ inegalite1 ] we deduce the following [ inegalite2]with the above notations we have @xmath220 now , in the following lemma , we state a sufficient condition for the existence of an absolutely continuous component with respect to the haar measure for the given generalized riesz product . in the case of @xmath221 action , the lemma is due to mlanie guenais @xcite , and the proof is similar . if @xmath222 , then @xmath168 admits an absolutely continuous component . we denote by @xmath223 the norm in @xmath224 . for all functions @xmath225 and @xmath226 in @xmath227 , by cauchy - schwarz inequality we have @xmath228 and by assumption , @xmath229 hence @xmath230 and the infinite product @xmath231 is convergent : @xmath232 let @xmath233 be a positive integers and take @xmath234 and @xmath235 , + then @xmath236 and @xmath237 ; hence by we have @xmath238 using the fact that @xmath239 , we obtain by induction @xmath240 from combined with we deduce that for large enough @xmath241 @xmath242 hence the sequence @xmath243 does not go to zero in @xmath4-norm . it follows from bourgain criterion that the generalized riesz product @xmath244 is not purely singular . as @xmath245 has only countably many zeros , we conclude that @xmath168 admits also an absolutely continuous component with respect to the haar measure . the kac central limit theorem in the context of the bohr compactification of @xmath0 is stated and proved in @xcite . for sake of completeness we prove it here using the standard probability arguments . the real numbers @xmath246 are called rationally independent if they are linearly independent over @xmath221 , i.e. for all @xmath247 @xmath248 [ kac - clt ] let @xmath249 be rationally independent . then , the functions @xmath250 are stochastically independent with respect to the haar measure of the bohr compactification of @xmath0 . it is sufficient to show that for any positive integer @xmath251 and for a given positive integers @xmath252 , we have @xmath253 write @xmath254 and recall that @xmath255 hence @xmath256 whence @xmath257 because of linear independence , @xmath258 can be zero only if @xmath259 , for any @xmath260 and thus it follows that @xmath261 we conclude that @xmath262 and this finish the proof of the theorem . in the following we recall the classical well known multidimensional central limit theorem in probability theory @xcite stated in the following forms [ mclt ] let @xmath263 be a triangle array of random variables vectors in @xmath264 and put @xmath265 suppose that 1 . the random variables @xmath266 are square integrable . 2 . for each @xmath267 , the @xmath268 , are independents . 3 . for any @xmath269 , @xmath270 4 . @xmath271 5 . for each @xmath272 , 1 . @xmath273 2 . @xmath274 then @xmath275 is a hermitian non - negative definite matrix and the sequence of random vectors @xmath276 converges in distribution to the complex gaussian measure @xmath277 on @xmath264 . we also need the following important and classical fact from probability theory connected to the notion of the uniform integrability . the sequence @xmath278 of random variables is said to be uniformly integrable if and only if @xmath279 it is well - known that if @xmath280 for some @xmath9 positive , then @xmath281 are uniformly integrable . + let us mention that the convergence in distribution or probability does not in general imply that the moments converge ( even if they exist ) . the useful condition to ensure the convergence of the moments is the uniform integrability . indeed , we have [ chung]if the sequence of random variables @xmath281 converges in distribution to some random variable @xmath282 and for some @xmath283 , @xmath284 , then for each @xmath285 , @xmath286 for the proof of theorem [ chung ] we refer the reader to @xcite or @xcite . + now let us state and prove the kac central limit theorem . let @xmath287 be a sequence of rationally independent real numbers . then the functions @xmath288 , are stochastically independent under the haar measure of the bohr compactification of @xmath0 and converge in distribution to the complex gaussian measure @xmath289 on @xmath290 . by theorem [ kac - clt ] the functions @xmath288 , are stochastically independent under the haar measure of @xmath2 and it is straightforward to verify that the hypotheses of central limit theorem [ mclt ] are satisfied . we conclude that the sequence @xmath291 converges in distribution to the complex gaussian measure @xmath289 on @xmath290 . using the analogous lemma of fjer s lemma @xcite combined with the clt methods introduced in @xcite , we shall give a direct proof of the singularity of a large class of generalized riesz products on @xmath2 . therefore , our strategy is slightly different from the strategy of the proofs given by many authors in the case of the torus @xcite , @xcite , @xcite , @xcite . indeed , the crucial argument in their proofs is to estimate the following quantity @xmath292 precisely , they showed that the weak limit point of the sequence @xmath293 is bounded below by a positive constant and it is well - known that this implies the singularity of the generalized riesz products ( see for instance @xcite or @xcite ) . + let us start our proof by proving the following lemma analogous to fjer s lemma @xcite [ fejer ] with the above notations we have @xmath294 by our assumption the sequence @xmath295 is rationally independent . hence , by kac theorem [ kac - clt ] , for @xmath296 , the function @xmath226 and @xmath297 are stochastically independent . this allows us to write @xmath298 which proves the lemma . applying lemma [ fejer ] , we proceed to construct inductively the sequence @xmath299 such that , for any @xmath300 , we have @xmath301 indeed , by our assumption combined with kac clt [ kac - clt ] , it follows that @xmath302 converges in distribution to the complex gaussian measure @xmath289 on @xmath290 . but , according to , @xmath302 is uniformly integrable . hence , from theorem [ chung ] , we get @xmath303 we remind that the density of the standard complex normal distribution @xmath289 is given by @xmath304 now assume that we have already construct @xmath305 and apply with @xmath306 combined with lemma [ fejer ] to get @xmath307 such that @xmath308 put @xmath309 . therefore the inequality holds and by letting @xmath310 we conclude that @xmath311 which yields by bourgain criterion that @xmath168 is singular with respect to haar measure and completes the proof . the fundamental argument in the proof above is based on lemma [ fejer ] and therefore strongly depended on the assumption that along subsequence the positive real numbers @xmath312 are linearly independent over the rationals . we argue that in the general case , one may use the methods of bourgain @xcite , klemes - reinhold @xcite , klemes @xcite and el abdalaoui @xcite to establish the singularity of a large class of generalized riesz products on @xmath2 . in particular , the case when @xmath313 is bounded . in the forthcoming paper , we will show how to extend a classical results from the torus and real line setting to the generalized riesz products on the bohr compactification of @xmath0 . we are concerned here with the flat polynomials issue in @xmath2 . first , we recall briefly the relevant fact on the flatness problem in the torus @xmath315 . + the problem of flatness go back to littlewood in his 1968 famous paper @xcite . in that paper , littlewood introduce two class of complex polynomials @xmath316 and @xmath317 where @xmath318 is a positive integer . the class @xmath316 is a class of those polynomials @xmath319 that are _ unimodular _ , that is , @xmath320 , for @xmath321 . @xmath317 is the subclass of @xmath316 with real coefficients , i.e. , @xmath322 , for @xmath321 . the polynomials @xmath225 in @xmath317 are nowadays called littlewood polynomials . by parseval s formula @xmath323 therefore , for all @xmath324 , @xmath325 in @xcite littlewood raised the problem of the existence of a sequence @xmath326 of unimodular polynomials such that @xmath327 such sequence of unimodular polynomials are called ultraflat . precisely , the usual definition of ultraflatness is given as follows let @xmath328 be a sequence of positive integers and @xmath329 a sequence of positive real numbers tending to @xmath82 , we say that a sequence @xmath330 of unimodular polynomials is @xmath329-ultraflat if @xmath331 the problem of existence of the ultraflat polynomials was solved affirmatively by kahane in his 1980 paper @xcite . precisely , kahane proved that there exists a sequence of @xmath332-ultraflat with @xmath333 but as noted by queffelec and saffari @xcite kahane proof is some kind of miracle . this is due to the fact that kahane work is inspired by krner paper @xcite and krner paper is based on byrnes paper @xcite . but , in 1996 j. benedetto and his student hui - chuan wu discovered that theorem 2 in byrnes paper @xcite was erroneous and as a consequence invalidated krner main result . fortunately kahane proof was independent of theorem 2 in byrnes paper . + the problem of the existence of the ultraflat littlewood polynomials is unsettled to this date and as pointed by erdlyi @xcite it is a common belief that there is no ultraflat sequence of littlewood polynomials . as a consequence no long barker sequences exist ( see @xcite and the references given there ) . + one more important class of polynomials is the class of polynomials with coefficients @xmath334 called a class of idempotent polynomials for obvious reasons of being convolution idempotents and denoted by @xmath335~~:~~ p(x)=\sum_{0}^{n-1}a_k x^k , a_k \in \{0,1\}\},~~n \in \n^*.\ ] ] a subclass of @xmath336 with constant term @xmath123 is called a class of newman polynomials and denoted by @xmath337 . another extremal open problem in the class of littewood polynomials or in the class of newman polynomials is the problem of @xmath4-flatness . in the same way as the ultraflatness , the @xmath4-flatness is defined as follows let @xmath328 be a sequence of positive integers and @xmath329 a sequence of positive real numbers tending to @xmath82 , we say that a sequence @xmath330 of polynomials is @xmath338-flat if @xmath339 the problem of the existence of @xmath4-flat sequence of unimodular polynomials was solved by newman in @xcite . later , m. guenais inspired by newman work constructed @xcite a sequence of @xmath4-flat littlewood polynomials on @xmath340 where @xmath341 is an increasing sequence of prime numbers . as a consequence she established a existence of countable group action with simple spectrum and haar component subsequently , el abdalaoui and lemaczyk @xcite proved that the sequence of littlewood polynomials constructed by guenais is ultraflat . before , j. bourgain in his 1993 paper @xcite conjectured that the supremum of the @xmath4 norm by @xmath342 norm over all idempotent polynomials on the circle must be strictly less than one . precisely , he make the following conjecture @xcite @xmath343 using the bourgain ideas , m. guenais @xcite connected the problem of the existence of @xmath4-flat sequence of littlewood polynomials or newman polynomials and the banach problem on whether there exist a dynamical system with simple lebesgue spectrum . ulam in his book @xcite stated the banach problem in the following form does there exist a square integrable function @xmath344 and a measure preserving transformation @xmath345 , @xmath346 , such that the sequence of functions @xmath347 forms a complete orthogonal set in hilbert space ? let us formulate the bourgain conjecture in context of bohr compactification of @xmath0 . for that , let @xmath348 be the subspace of trigonometric polynomials on @xmath0 and @xmath349 be an increasing sequence of real numbers . put @xmath350 then , we can state the bourgain conjecture in context of bohr compactification of @xmath0 in the following form for any increasing sequence @xmath349 of real numbers , we have @xmath351 where @xmath352 for the sequence @xmath349 of rationally independent numbers , we are able to prove the following proposition . let @xmath353 be a sequence of real numbers such that for any @xmath354 , the real numbers @xmath355 are rationally independent . then , we have @xmath356 since , for any @xmath354 , the real numbers @xmath355 are rationally independent , then we can apply the kac clt theorem [ kac - clt ] . but , the functions @xmath357 are in @xmath109 we deduce that they are uniformly integrable . hence , by theorem [ chung ] , we conclude that @xmath358 this finish the proof of the proposition . the discussion above allows us to formulate the following questions does there exist a sequence of @xmath4-flat littlewood polynomials or of the @xmath4-flat newman polynomials on @xmath2 ? subsequently does there exist a the ultraflat littlewood or newman polynomials on @xmath2 ? [ rem - pri]we remind here that in @xcite , a. a. prikhodko constructed a sequence of trigonometric polynomials @xmath133 which is locally @xmath4-flat , that is , for any @xmath359 and @xmath360 there exists a positive integer @xmath361 such that for any @xmath362 , we have @xmath363 the polynomials @xmath364 are given by @xmath365 where , @xmath366 @xmath367 is a sequence of positive integers with @xmath368 and @xmath369 , @xmath370 is a sequence of rational numbers which goes to @xmath82 as @xmath318 goes to @xmath371 . + but , this sequence is not @xmath4-flat on the bohr compactification of @xmath0 . indeed , one may show that we have @xmath372 in addition , it is proved in @xcite that this sequence is not @xmath4-flat on @xmath0 . let @xmath29 be a locally abelian compact group and denoted by @xmath373 its topology . let @xmath95 be a group of the characters on @xmath29 , that is , the continuous homomorphism from @xmath29 to the torus @xmath374 put @xmath375 the space of all functions from @xmath95 to @xmath315 equipped with the product topology ( i.e. ; the pointwise convergence topology ) . therefore , there is a canonical injective homomorphism from @xmath29 to @xmath376 given by @xmath377 @xmath30 is called a dual homomorphism . by abuse of notation we denote by the same letter @xmath29 the image of @xmath29 under @xmath30 . therefore , @xmath29 is equipped with the bohr topology denoted by @xmath378 and the topology inherited from @xmath29 . hence @xmath379 . by taking the closure of @xmath29 with respect to the bohr topology we get the bohr compactification of @xmath29 and we denoted it by @xmath380 . thus , by construction , @xmath380 is compact . + let us state the following useful lemma . by the definition of the product topology on @xmath376 , a sequence of functions @xmath385 converge to some function @xmath10 if and only if , for any @xmath383 , @xmath386 by taking @xmath387 and @xmath388 we get @xmath389 which means that for any @xmath383 , @xmath390 and the proof of the lemma is complete . we deduce from the lemma the following crucial fact about the separability of the bohr compactification of @xmath29 . we recall that the topological space is separable if it is contains a countable , dense subset . in the case of @xmath0 , the lemma [ cvg ] say that the sequence of real numbers @xmath391 converge in the sense of the bohr topology to @xmath110 if and only if , for any @xmath142 , @xmath392 but since the characters on @xmath0 are lipschitz we deduce easy that the bohr topology is contained in the usual topology and @xmath226 is dense in the usual topology and bohr topology . + nevertheless , by the hirodata - kakutani theorem , @xmath0 equipped with the bohr topology does nt have a countable basis and it is often that the bohr topology does nt have a countable basis . precisely , hirodata - kakutani theorem asserts we recall that the fourier coefficient of @xmath10 on @xmath393 is given by @xmath394 where @xmath36 is the haar measure on @xmath29 and @xmath393 is in the spectrum of @xmath10 if the fourier coefficients of @xmath10 on @xmath31 is not zero the author wishes to express his gratitude to m. lemaczyk , j - p . thouvenot , e. lesigne , a. bouziad , j - m . strelcyn and a. a. prikhodko for fruitful discussions on the subject . it is a pleasure for him to acknowledge the warm hospitality of the poncelet french - russian mathematical laboratory in moscow where a part of this work has been done . p. billingsley , _ convergence of probability measures , _ second edition . wiley series in probability and statistics : probability and statistics . a wiley - interscience publication . john wiley & sons , inc . , new york , 1999 . borwein and m. j. mossinghoff , _ barker sequences and flat polynomials , _ ( english summary ) number theory and polynomials , 71 - 88 , london math . soc . lecture note ser . , 352 , cambridge univ . press , cambridge , 2008 . g. brown and w. moran , _ products of random variables and kakutani s criterion for orthogonality of product measures , _ j. london math . ( 2 ) 10 ( 1975 ) , part 4 , 401 - 405 . j. bourgain , _ on the spectral type of ornstein class one transformations , _ isr . j. math . , 84 ( 1993 ) , 53 - 63 . j. s. byrnes,_on polynomials with coefficients of modulus one , _ bull . london math . , 9 ( 1977 ) , no . 2 , 171 - 176 . j. r. choksi and m. g. nadkarni , _ the maximal spectral type of rank one transformation , _ can . math . bull . , 37 ( 1 ) ( 1994 ) , 29 - 36 . k. l. chung , _ a course in probability theory , _ third edition . academic press , inc . , san diego , ca , 2001 . j. coquet , t. kamae and m. mends france , _ sur la mesure spectrale de certaines suites arithmtiques , _ ( french ) bull . france 105 ( 1977 ) , no . 4 , 369 - 384 . c. corduneanu , _ almost periodic functions , _ with the collaboration of n. gheorghiu and v. barbu . translated from the romanian by gitta bernstein and eugene tomer . interscience tracts in pure and applied mathematics , no . [ john wiley & sons ] , new york - london - sydney , 1968 . d. dacunha - castelle and m. duflo , _ probabilits et statistiques . tome 2 , _ ( french ) [ probability and statistics . vol . 2 ] , problmes temps mobile [ movable - time problems ] , collection mathmatiques appliques pour la matrise [ collection of applied mathematics for the master s degree ] , masson , paris , 1983 . n. dunford and j. t. schwartz , _ linear operators . i and ii , _ interscience publishers , inc . , new york , 1958 , 1963 . t. erdlyi , _ how far is an ultraflat sequence of unimodular polynomials from being conjugate - reciprocal ? , _ michigan math . j. , 49 ( 2001 ) , no . 2 , 259 - 264 . s. a. grigoryan and v. t. tonev , _ shift - invariant uniform algebras on groups , _ instytut matematyczny polskiej akademii nauk . monografie matematyczne ( new series ) [ mathematics institute of the polish academy of sciences . mathematical monographs ( new series ) ] , 68 . birkhuser verlag , basel , 2006 . m. kac , _ statistical independence in probability , analysis and number theory . _ the carus mathematical monographs , no . 12 published by the mathematical association of america . distributed by john wiley and sons , inc . , new york , ( 1959 ) . v. p. khavin and n. k. nikolki , _ commutative harmonic analysis . i. general survey . classical aspects , _ ( russian ) translation by d. khavinson and s. v. kislyakov , encyclopaedia of mathematical sciences , 15 . springer - verlag , berlin , 1991 . i. e. segal , _ distributions in hilbert space and canonical systems of operators , _ trans . 88 1958 12 - 41 . s. m. ulam , _ problems in modern mathematics , _ science editions john wiley & sons , inc . , new york 1964 . a. zygmund , _ trigonometric series vol . i _ , second ed . , cambridge univ . press , cambridge , 1959 .

we study a class of generalized riesz products connected to the spectral type of some class of rank one flows on @xmath0 . applying a central limit theorem of kac , we exhibit a large class of singular generalized riesz products on the bohr compactification of @xmath0 . moreover , we discuss the problem of the flat polynomials in this setting . + @xmath1dedicated to professors jean - paul thouvenot and bernard host . + ( 2010 ) primary : 42a05 , 42a55 ; secondary : 11l03 , 42a61 . + generalized riesz products , almost periodic functions , bohr compactification , mean value , besicovitch space , kac central limit theorem , kakutani criterion . +

You are an expert at summarizing long articles. Proceed to summarize the following text: [ sec : intro ] a _ reflection framework _ is a planar structure made of _ fixed - length bars _ connected by _ universal joints _ with full rotational freedom . additionally , the bars and joints are symmetric with respect to a reflection through a fixed axis . the allowed motions preserve the _ length _ and _ connectivity _ of the bars and _ symmetry _ with respect to some reflection . this model is very similar to that of _ cone frameworks _ that we introduced in @xcite ; the difference is that the symmetry group @xmath0 acts on the plane by reflection instead of rotation through angle @xmath1 . when all the allowed motions are euclidean isometries , a reflection framework is _ rigid _ and otherwise it is _ flexible_. in this paper , we give a _ combinatorial _ characterization of minimally rigid , generic reflection frameworks . formally a reflection framework is given by a triple @xmath2 , where @xmath3 is a finite graph , @xmath4 is a @xmath0-action on @xmath3 that is free on the vertices and edges , and @xmath5 is a vector of non - negative _ edge lengths _ assigned to the edges of @xmath3 . a _ realization _ @xmath6 is an assignment of points @xmath7 and a representation of @xmath0 by a reflection @xmath8 such that : @xmath9 the set of all realizations is defined to be the _ realization space _ @xmath10 and its quotient by the euclidean isometries @xmath11 to be the configuration space . a realization is _ rigid _ if it is isolated in the configuration space and otherwise _ flexible_. as the combinatorial model for reflection frameworks it will be more convenient to use colored graphs . a _ colored graph _ @xmath12 is a finite , directed , the orientation of the edges do not play a role , but we give the standard definition for consistency . ] graph @xmath13 , with an assignment @xmath14 of an element of a group @xmath15 to each edge . in this paper @xmath15 is always @xmath0 . there is a standard dictionary ( * ? ? ? * section 9 ) associating @xmath16 with a colored graph @xmath12 : @xmath13 is the quotient of @xmath3 by @xmath15 , and the colors encode the covering map via a natural map @xmath17 . in this setting , the choice of base vertex does not matter , and indeed , we may define @xmath18 and obtain the same theory . we can now state the main result of this paper . [ theo : reflection - laman ] a generic reflection framework is minimally rigid if and only if its associated colored graph is reflection - laman . the _ reflection - laman graphs _ appearing in the statement are defined in section [ sec : matroid ] . genericity has its standard meaning from algebraic geometry : the set of non - generic reflection frameworks is a measure - zero algebraic set , and a small _ geometric _ perturbation of a non - generic reflection framework yields a generic one . as in all known proofs of `` maxwell - laman - type '' theorems such as theorem [ theo : reflection - laman ] , we give a combinatorial characterization of a linearization of the problem known as _ infinitesimal rigidity_. to do this , we use a _ direction network _ method ( cf . @xcite ) . a _ reflection direction network _ @xmath19 is a symmetric graph , along with an assignment of a _ direction _ @xmath20 to each edge . the _ realization space _ of a direction network is the set of solutions @xmath21 to the system of equations : @xmath22 where the @xmath0-action @xmath23 on the plane is by reflection through the @xmath24-axis . a reflection direction network is determined by assigning a direction to each edge of the colored quotient graph @xmath12 of @xmath16 ( cf . * lemma 17.2 ) ) . since all the direction networks in this paper are reflection direction networks , we will refer to them simply as `` direction networks '' to keep the terminology manageable . a realization of a direction network is _ faithful _ if none of the edges of its graph have coincident endpoints and _ collapsed _ if all the endpoints are coincident . a basic fact in the theory of finite planar frameworks @xcite is that , if a direction network has faithful realizations , the dimension of the realization space is equal to that of the space of infinitesimal motions of a generic framework with the same underlying graph . in @xcite , we adapted this idea to the symmetric case when all the symmetries act by rotations and translations . as discussed in ( * ? ? ? * section 1.8 ) , this so - called `` parallel redrawing trick '' described above does _ not _ apply verbatim to reflection frameworks . thus , we rely on the somewhat technical ( cf . * theorem b ) , ( * ? ? ? * theorem 2 ) ) theorem [ theo : direction - network ] , which we state after giving an important definition . let @xmath19 be a direction network and define @xmath25 to be the direction network with @xmath26 . these two direction networks form a _ special pair _ if : * @xmath19 has a faithful realization . * @xmath27 has only collapsed realizations . [ theo : direction - network ] let @xmath12 be a colored graph with @xmath28 vertices , @xmath29 edges , and lift @xmath16 . then there are directions @xmath30 such that the direction networks @xmath19 and @xmath27 are a special pair if and only if @xmath12 is reflection - laman . briefly , we will use theorem [ theo : direction - network ] as follows : the faithful realization of @xmath19 gives a symmetric immersion of the graph @xmath3 that can be interpreted as a framework , and the fact that @xmath27 has only collapsed realizations will imply that the only symmetric infinitesimal motions of this framework correspond to translation parallel to the reflection axis . in this paper , all graphs @xmath31 may be multi - graphs . typically , the number of vertices , edges , and connected components are denoted by @xmath28 , @xmath32 , and @xmath33 , respectively . the notation for a colored graph is @xmath12 , and a symmetric graph with a free @xmath0-action is denoted by @xmath16 . if @xmath16 is the lift of @xmath12 , we denote the fiber over a vertex @xmath34 by @xmath35 , with @xmath36 , and the fiber over a directed edge @xmath37 with color @xmath38 by @xmath39 . we also use _ @xmath40-sparse graphs _ @xcite and their generalizations . for a graph @xmath13 , a _ @xmath40-basis _ is a maximal @xmath40-sparse subgraph ; a _ @xmath40-circuit _ is an edge - wise minimal subgraph that is not @xmath40-sparse ; and a _ @xmath40-component _ is a maximal subgraph that has a spanning @xmath40-graph . points in @xmath41 are denoted by @xmath42 , indexed sets of points by @xmath43 , and direction vectors by @xmath30 and @xmath44 . realizations of a reflection direction network @xmath19 are written as @xmath21 , as are realizations of abstract reflection frameworks . context will always make clear the type of realization under consideration . lt is supported by the european research council under the european union s seventh framework programme ( fp7/2007 - 2013 ) / erc grant agreement no 247029-sdmodels . jm is supported by nsf cdi - i grant dmr 0835586 . [ sec : matroid ] in this short section we introduce the combinatorial families of sparse colored graphs we use . let @xmath12 be a @xmath0-colored graph . since all the colored graphs in this paper have @xmath0 colors , from now on we make this assumption and write simply `` colored graph '' . we recall two key definitions from @xcite . the map @xmath18 is defined on cycles by adding up the colors on the edges . ( the directions of the edges do nt matter for @xmath0 colors . similarly , neither does the traversal order . ) as the notation suggests , @xmath45 extends to a homomorphism from @xmath46 to @xmath0 , and it is well - defined even if @xmath13 is not connected . let @xmath12 be a colored graph with @xmath28 vertices and @xmath32 edges . we define @xmath12 to be a _ reflection - laman graph _ if : the number of edges @xmath47 , and for all subgraphs @xmath48 , spanning @xmath49 vertices , @xmath50 edges , @xmath51 connected components with non - trivial @xmath45-image and @xmath52 connected components with trivial @xmath45-image @xmath53 this definition is equivalent to that of _ cone - laman graphs _ in ( * ? ? ? * section 15.4 ) . the underlying graph @xmath13 of a reflection - laman graph is a @xmath54-graph . another family we need is that of _ ross graphs _ ( see @xcite for an explanation of the terminology ) . these are colored graphs with @xmath28 vertices , @xmath55 edges , satisfying the sparsity counts @xmath56 using the same notations as in ( [ eq : cone - laman ] ) . in particular , ross graphs @xmath12 have as their underlying graph , a @xmath57-graph @xmath13 , and are thus connected @xcite . a _ _ ross - circuit__. ] is a colored graph that becomes a ross graph after removing _ any _ edge . the underlying graph @xmath13 of a ross - circuit @xmath12 is a @xmath57-circuit , and these are also known to be connected @xcite , so , in particular , a ross - circuit has @xmath58 , and thus satisfies ( [ eq : cone - laman ] ) on the whole graph . since ( [ eq : cone - laman ] ) is always at least ( [ eq : ross ] ) , we see that every ross - circuit is reflection - laman . because reflection - laman graphs are @xmath54-graphs and subgraphs that are @xmath57-sparse are , in addition , ross - sparse , we get the following structural result . [ prop : ross - circuit - decomp ] let @xmath12 be a reflection - laman graph . then each @xmath57-component of @xmath13 contains at most one ross - circuit , and in particular , the ross - circuits in @xmath12 are vertex disjoint . [ sec : reflection-22 ] the next family of graphs we work with is new . a colored graph @xmath12 is defined to be a _ graph , if it has @xmath28 vertices , @xmath47 edges , and satisfies the sparsity counts @xmath59 using the same notations as in ( [ eq : cone - laman ] ) . the relationship between ross graphs and reflection-@xmath57 graphs we will need is : [ prop : ross - adding ] let @xmath12 be a ross - graph . then for either * an edge @xmath37 with any color where @xmath60 * or a self - loop @xmath61 at any vertex @xmath62 colored by @xmath63 the graph @xmath64 or @xmath65 is reflection-@xmath57 . adding @xmath37 with any color to a ross @xmath12 creates either a ross - circuit , for which @xmath58 or a laman - circuit with trivial @xmath45-image . both of these types of graph meet this count , and so the whole of @xmath64 does as well . it is easy to see that every reflection - laman graph is a reflection-@xmath57 graph . the converse is not true . [ prop : reflection - laman - vs - reflection-22 ] a colored graph @xmath12 is a reflection - laman graph if and only if it is a reflection-@xmath57 graph and no subgraph with trivial @xmath45-image is a @xmath57-block.@xmath66 let @xmath12 be a reflection - laman graph , and let @xmath67 be the ross - circuits in @xmath12 . define the _ reduced graph _ @xmath68 of @xmath12 to be the colored graph obtained by contracting each @xmath69 , which is not already a single vertex with a self - loop ( this is necessarily colored @xmath63 ) , into a new vertex @xmath70 , removing any self - loops created in the process , and then adding a new self - loop with color @xmath63 to each of the @xmath70 . by proposition [ prop : ross - circuit - decomp ] the reduced graph is well - defined . [ prop : reduced - graph ] let @xmath12 be a reflection - laman graph . then its reduced graph is a reflection-@xmath57 graph . let @xmath12 be a reflection - laman graph with @xmath71 ross - circuits with vertex sets @xmath72 . by proposition [ prop : ross - circuit - decomp ] , the @xmath73 are all disjoint . now select a ross - basis @xmath74 of @xmath12 . the graph @xmath48 is also a @xmath57-basis of @xmath13 , with @xmath75 edges , and each of the @xmath73 spans a @xmath57-block in @xmath48 . the @xmath40-sparse graph structure theorem ( * ? ? ? * theorem 5 ) implies that contracting each of the @xmath73 into a new vertex @xmath70 and discarding any self - loops created , yields a @xmath57-sparse graph @xmath76 on @xmath77 vertices and @xmath78 edges . it is then easy to check that adding a self - loop colored @xmath63 at each of the @xmath70 produces a colored graph satisfying the reflection-@xmath57 counts ( [ eq : ref22a ] ) with exactly @xmath79 edges . since this is the reduced graph , we are done . a _ map - graph _ is a graph with exactly one cycle per connected component . a _ reflection-@xmath80 _ graph is defined to be a colored graph @xmath12 where @xmath13 , taken as an undirected graph , is a map - graph and the @xmath45-image of each connected component is non - trivial . [ lemma : reflection-22-decomp ] let @xmath12 be a colored graph . then @xmath12 is a reflection-@xmath57 graph if and only if it is the union of a spanning tree and a reflection-@xmath80 graph . by ( * ? ? ? * lemma 15.1 ) , reflection-@xmath80 graphs are equivalent to graphs satisfying @xmath81 for every subgraph @xmath48 . thus , ( [ eq : ref22a ] ) is @xmath82 the second term in ( [ eq : ref22redux ] ) is well - known to be the rank function of the graphic matroid , and the lemma follows from the edmonds - rota construction @xcite and the matroid union theorem . in the next section , it will be convenient to use this slight refinement of lemma [ lemma : reflection-22-decomp ] . [ prop : reflection-22-nice - decomp ] let @xmath12 be a reflection-@xmath57 graph . then there is a coloring @xmath83 of the edges of @xmath13 such that : * the @xmath45-image of every subgraph in @xmath84 is the same as in @xmath12 . * there is a decomposition of @xmath84 as in lemma [ lemma : reflection-22-decomp ] in which the spanning tree has all edges colored by the identity . it is shown in ( * ? ? ? * lemma 2.2 ) that @xmath45 is determined by its image on a homology basis of @xmath13 . thus , we may start with an arbitrary decomposition of @xmath12 into a spanning tree @xmath85 and a reflection-@xmath80 graph @xmath86 , as provided by lemma [ lemma : reflection-22-decomp ] , and define @xmath83 by coloring the edges of @xmath85 with the identity and the edges of @xmath86 with the @xmath45-image of their fundamental cycle in @xmath85 in @xmath12 . proposition [ prop : reflection-22-nice - decomp ] has the following re - interpretation in terms of the symmetric lift @xmath16 : [ prop : reflection - laman - decomp - lift ] let @xmath12 be a reflection-@xmath57 graph . then for a decomposition , as provided by proposition [ prop : reflection-22-nice - decomp ] , into a spanning tree @xmath85 and a reflection-@xmath80 graph @xmath86 : * every edge @xmath87 lifts to the two edges @xmath88 and @xmath89 . ( in other words , the vertex representatives in the lift all lie in a single connected component of the lift of @xmath85 . ) * each connected component of @xmath86 lifts to a connected graph . [ sec : direction - network ] we recall , from the introduction , that for reflection direction networks , @xmath0 acts on the plane by reflection through the @xmath24-axis , and in the rest of this section @xmath90 refers to this action . the system of equations ( [ eq : dn - realization1])([eq : dn - realization2 ] ) defining the realization space of a reflection direction network @xmath19 is linear , and as such has a well - defined dimension . let @xmath12 be the colored quotient graph of @xmath16 . to be realizable at all , the directions on the edges in the fiber over @xmath91 need to be reflections of each other . thus , we see that the realization system is canonically identified with the solutions to the system : @xmath92 from now on , we will implicitly switch between the two formalisms when it is convenient . let @xmath12 be a colored graph with @xmath32 edges . a statement about direction networks @xmath19 is _ generic _ if it holds on the complement of a proper algebraic subset of the possible direction assignments , which is canonically identified with @xmath93 . some facts about generic statements that we use frequently are : * almost all direction assignments are generic . * if a set of directions is generic , then so are all sufficiently small perturbations of it . * if two properties are generic , then their intersection is as well . * the maximum rank of ( [ eq : colored - system ] ) is a generic property . we first characterize the colored graphs for which generic direction networks have strongly faithful realizations . a realization is _ strongly faithful _ if no two vertices lie on top of each other . this is a stronger condition than simply being faithful which only requires that edges not be collapsed . [ prop : ross - realizations ] a generic direction network @xmath19 has a unique , up to translation and scaling , strongly faithful realization if and only if its associated colored graph is a ross graph . to prove proposition [ prop : ross - realizations ] we expand upon the method from ( * ? ? ? * section 20.2 ) , and use the following proposition . [ prop : reflection-22-collapse ] let @xmath12 be a reflection-@xmath57 graph . then a generic direction network on the symmetric lift @xmath16 of @xmath12 has only collapsed realizations . since the proof of proposition [ prop : reflection-22-collapse ] requires a detailed construction , we first show how it implies proposition [ prop : ross - realizations ] . let @xmath12 be a ross graph , and assign directions @xmath30 to the edges of @xmath13 such that , for any extension @xmath64 of @xmath12 to a reflection-@xmath57 graph as in proposition [ prop : ross - adding ] , @xmath30 can be extended to a set of directions that is generic in the sense of proposition [ prop : reflection-22-collapse ] . this is possible because there are a finite number of such extensions . for this choice of @xmath30 , the realization space of the direction network @xmath19 is @xmath94-dimensional . since solutions to ( [ eq : colored - system ] ) may be scaled or translated in the vertical direction , all solutions to @xmath19 are related by scaling and translation . it then follows that a pair of vertices in the fibers over @xmath62 and @xmath95 are either distinct from each other in all non - zero solutions to ( [ eq : colored - system ] ) or always coincide . in the latter case , adding the edge @xmath37 with any direction does not change the dimension of the solution space , no matter what direction we assign to it . it then follows that the solution spaces of generic direction networks on @xmath19 and @xmath96 have the same dimension , which is a contradiction by proposition [ prop : reflection-22-collapse ] . @xmath66 it is sufficient to construct a specific set of directions with this property . the rest of the proof gives such a construction and verifies that all the solutions are collapsed . let @xmath12 be a reflection-@xmath57 graph . [ [ combinatorial - decomposition ] ] combinatorial decomposition + + + + + + + + + + + + + + + + + + + + + + + + + + + we apply proposition [ prop : reflection-22-nice - decomp ] to decompose @xmath12 into a spanning tree @xmath85 with all colors the identity and a reflection-@xmath80 graph @xmath86 . for now , we further assume that @xmath86 is connected . [ [ assigning - directions ] ] assigning directions + + + + + + + + + + + + + + + + + + + + let @xmath44 be a direction vector that is not horizontal or vertical . for each edge @xmath87 , set @xmath97 . assign all the edges of @xmath86 the vertical direction . denote by @xmath30 this assignment of directions . : the @xmath24-axis is shown as a dashed line . the directions on the edges of the lift of the tree @xmath85 force all the vertices to be on one of the two lines meeting at the @xmath24-axis , and the directions on the reflection-@xmath80 graph @xmath86 force all the vertices to be on the @xmath24-axis . ] [ [ all - realizations - are - collapsed ] ] all realizations are collapsed + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + we now show that the only realizations of @xmath19 have all vertices on top of each other . by proposition [ prop : reflection - laman - decomp - lift ] @xmath85 lifts to two copies of itself , in @xmath3 . it then follows from the connectivity of @xmath85 and the construction of @xmath30 that , in any realization , there is a line @xmath98 with direction @xmath44 such that every vertex of @xmath3 must lie on @xmath98 or its reflection . since the vertical direction is preserved by reflection , the connectivity of the lift of @xmath86 , again from proposition [ prop : reflection - laman - decomp - lift ] , implies that every vertex of @xmath3 lies on a single vertical line , which must be the @xmath24-axis by reflective symmetry . thus , in any realization of @xmath19 all the vertices lie at the intersection of @xmath98 , the reflection of @xmath98 through the @xmath24-axis and the @xmath24-axis itself . this is a single point , as desired . figure [ fig : ref-22-collapse ] shows a schematic of this argument . [ [ x - does - not - need - to - be - connected ] ] @xmath86 does not need to be connected + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + finally , we can remove the assumption that @xmath86 was connected by repeating the argument for each connected component of @xmath86 separately . @xmath66 the full theorem [ theo : direction - network ] will reduce to the case of a ross - circuit . [ prop : ross - circuit - pairs ] let @xmath12 be a ross circuit with lift @xmath16 . then there is an edge @xmath99 such that , for a generic direction network @xmath100 with colored graph @xmath101 : * the solution space of @xmath100 induces a well - defined direction @xmath20 between @xmath62 and @xmath95 , yielding an assignment of directions @xmath30 to the edges of @xmath13 . * the direction networks @xmath19 and @xmath102 are a special pair . before giving the proof , we describe the idea . we are after sets of directions that lead to faithful realizations of ross - circuits . by proposition [ prop : reflection-22-collapse ] , these directions must be non - generic . a natural way to obtain such a set of directions is to discard an edge @xmath37 from the colored quotient graph , apply proposition [ prop : ross - realizations ] to obtain a generic set of directions @xmath103 with a strongly faithful realization @xmath104 , and then simply set the directions on the edges in the fiber over @xmath37 to be the difference vectors between the points . proposition [ prop : ross - realizations ] tells us that this procedure induces a well - defined direction for the edge @xmath37 , allowing us to extend @xmath103 to @xmath30 in a controlled way . however , it does _ not _ tell us that rank of @xmath19 will rise when the directions are turned by angle @xmath105 , and this seems hard to do directly . instead , we construct a set of directions @xmath30 so that @xmath19 is rank deficient and has faithful realizations , and @xmath27 is generic . then we make a perturbation argument to show the existence of a special pair . the construction we use is , essentially , the one used in the proof of proposition [ prop : reflection-22-collapse ] but turned through angle @xmath105 . the key geometric insight is that horizontal edge directions are preserved by the reflection , so the `` gadget '' of a line and its reflection crossing on the @xmath24-axis , as in figure [ fig : ref-22-collapse ] , degenerates to just a single line . let @xmath12 be a ross - circuit ; recall that this implies that @xmath12 is a reflection - laman graph . [ [ combinatorial - decomposition-1 ] ] combinatorial decomposition + + + + + + + + + + + + + + + + + + + + + + + + + + + we decompose @xmath12 into a spanning tree @xmath85 and a reflection-@xmath80 graph @xmath86 as in proposition [ prop : reflection - laman - decomp - lift ] . in particular , we again have all edges in @xmath85 colored by the identity . for now , we _ assume that @xmath86 is connected _ , and we fix @xmath99 to be an edge that is on the cycle in @xmath86 with @xmath106 ; such an edge must exist by the hypothesis that @xmath86 is reflection-@xmath80 . let @xmath107 . furthermore , let @xmath108 and @xmath109 be the two connected components of the lift of @xmath85 . for a vertex @xmath110 , we denote the lift in @xmath108 by @xmath111 and the lift in @xmath109 by @xmath112 . we similarly denote the lifts of @xmath113 and @xmath114 by @xmath115 and @xmath116 . [ [ assigning - directions-1 ] ] assigning directions + + + + + + + + + + + + + + + + + + + + the assignment of directions is as follows : to the edges of @xmath85 , we assign a direction @xmath44 that is neither vertical nor horizontal . to the edges of @xmath86 we assign the horizontal direction . define the resulting direction network to be @xmath19 , and the direction network induced on the lift of @xmath48 to be @xmath117 . [ [ the - realization - space - of - tildegvarphimathbfd ] ] the realization space of @xmath19 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + figure [ fig : ross - circuit - special - pair ] contains a schematic picture of the arguments that follow . [ lemma : rc - proof-1 ] the realization space of @xmath19 is @xmath94-dimensional and parameterized by exactly one representative in the fiber over the vertex @xmath62 selected above . in a manner similar to the proof of proposition [ prop : reflection-22-collapse ] , the directions on the edges of @xmath85 force every vertex to lie either on a line @xmath98 in the direction @xmath44 or its reflection . since the lift of @xmath86 is connected , we further conclude that all the vertices lie on a single horizontal line . thus , all the points @xmath118 are at the intersection of the same horizontal line and @xmath98 or its reflection . these determine the locations of the @xmath119 , so the realization space is parameterized by the location of @xmath120 . inspecting the argument more closely , we find that : in any realization @xmath21 of @xmath19 , all the @xmath118 are equal and all the @xmath119 are equal . because the colors on the edges of @xmath85 are all zero , it lifts to two copies of itself , one of which spans the vertex set @xmath121 and one which spans @xmath122 . it follows that in a realization , we have all the @xmath118 on @xmath98 and the @xmath119 on the reflection of @xmath98 . in particular , because the color @xmath123 on the edge @xmath99 is @xmath63 , we obtain the following . [ lemma : rc - proof-5 ] the realization space of @xmath19 contains points where the fiber over the edge @xmath99 is not collapsed . : the @xmath24-axis is shown as a dashed line . the directions on the edges of the lift of the tree @xmath85 force all the vertices to be on one of the two lines meeting at the @xmath24-axis . the horizontal directions on the connected reflection-@xmath80 graph @xmath86 force the point @xmath118 to be at the intersection marked by the black dot and @xmath119 to be at the intersection marked by the gray one . ] [ [ the - realization - space - of - tildegvarphimathbfd-1 ] ] the realization space of @xmath124 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the conclusion of lemma [ lemma : rc - proof-1 ] implies that the realization system for @xmath19 is rank deficient by one . next we show that removing the edge @xmath99 results in a direction network that has full rank on the colored graph @xmath74 . [ lemma : rc - proof-2 ] the realization space of @xmath19 is canonically identified with that of @xmath124 . in the proof of lemma [ lemma : rc - proof-1 ] , that @xmath86 lifts to a connected subgraph of @xmath3 was not essential . because a horizontal line is preserved by the reflection , realizations will take on the same structure provided that @xmath86 lifts to a subgraph with two connected components . removing @xmath99 from @xmath86 leaves a graph @xmath125 with this property since @xmath125 is a tree . it follows that the equation corresponding to the edge @xmath99 in ( [ eq : colored - system ] ) was dependent . [ [ the - realization - space - of - tildegvarphimathbfdperp ] ] the realization space of @xmath27 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + next , we consider what happens when we turn all the directions by @xmath105 . [ lemma : rc - proof-3 ] the realization space of @xmath27 has only collapsed solutions . this is exactly the construction used to prove proposition [ prop : reflection-22-collapse ] . [ [ perturbing - tildegvarphimathbfd ] ] perturbing @xmath19 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + to summarize what we have shown so far : * @xmath19 has a @xmath94-dimensional realization space parameterized by @xmath120 and identified with that of a full - rank direction network on the ross graph @xmath74 . * there are points @xmath21 in this realization space where @xmath126 . * @xmath19 has a @xmath63-dimensional realization space containing only collapsed solutions . what we have not shown is that the realization space of @xmath19 has _ faithful _ realizations , since the ones we constructed all have many coincident vertices . proposition [ prop : ross - realizations ] will imply the rest of the theorem , provided that the above properties hold for any small perturbation of @xmath30 , since some small perturbation of _ any _ assignment of directions to the edges of @xmath74 has only faithful realizations . [ lemma : rc - proof-4 ] let @xmath127 be a perturbation of the directions @xmath103 on the edges of @xmath48 . if @xmath127 is sufficiently close to @xmath103 , then there are realizations of the direction network @xmath128 such that @xmath126 . the realization space is parameterized by @xmath120 , and so @xmath129 varies continuously with the directions on the edges and @xmath120 . since there are realizations of @xmath130 with @xmath131 , the lemma follows . lemma [ lemma : rc - proof-4 ] implies that any sufficiently small perturbation of the directions assigned to the edges of @xmath48 gives a direction network that induces a well - defined direction on the edge @xmath99 which is itself a small perturbation of @xmath132 . since the ranks of @xmath100 and @xmath27 are stable under small perturbations , this implies that we can perturb @xmath103 to a @xmath133 that is generic in the sense of proposition [ prop : ross - realizations ] , while preserving faithful realizability of @xmath134 and full rank of the realization system for @xmath135 . the proposition is proved for when @xmath86 is connected . [ [ x - need - not - be - connected ] ] @xmath86 need not be connected + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the proof is then complete once we remove the additional assumption that @xmath86 was connected . let @xmath86 have connected components @xmath136 . for each of the @xmath137 , we can identify an edge @xmath138 with the same properties as @xmath99 above . assign directions to the tree @xmath85 as above . for @xmath139 , we assign directions exactly as above . for each of the @xmath140 with @xmath141 , we assign the edges of @xmath142 the horizontal direction and @xmath138 a direction that is a small perturbation of horizontal . with this assignment @xmath30 we see that for any realization of @xmath19 , each of the @xmath140 , for @xmath141 is realized as completely collapsed to a single point at the intersection of the line @xmath98 and the @xmath24-axis . moreover , in the direction network on @xmath143 , the directions on these @xmath137 are a small perturbation of the ones used on @xmath86 in the proof of proposition [ prop : reflection-22-collapse ] . from this is follows that , in any realization @xmath27 , is completely collapsed and hence full rank . we now see that this new set of directions has properties ( a ) , ( b ) , and ( c ) above required for the perturbation argument . since that argument makes no reference to the decomposition , it applies verbatim to the case where @xmath86 is disconnected . @xmath66 the easier direction to check is necessity . [ [ the - maxwell - direction ] ] the maxwell - direction + + + + + + + + + + + + + + + + + + + + + if @xmath12 is not reflection - laman , then it contains either a laman - circuit with trivial @xmath45-image , or a violation of @xmath54-sparsity . if there is a laman - circuit with trivial @xmath45-image , the parallel redrawing theorem ( * ? ? ? * theorem 4.1.4 ) in the form ( * ? ? ? * theorem 3 ) implies that this subgraph has no faithful realizations for @xmath144 only if it does in @xmath145 if rank - deficient . a violation of @xmath54-sparsity implies that the realization system ( [ eq : colored - system ] ) of @xmath27 has a dependency , since the realization space is always at least @xmath63-dimensional . [ [ the - laman - direction ] ] the laman direction + + + + + + + + + + + + + + + + + + + now let @xmath12 be a reflection - laman graph and let @xmath74 be a ross - basis of @xmath12 . for any edge @xmath146 , adding it to @xmath48 induces a ross - circuit which contains some edge @xmath99 having the property specified in proposition [ prop : ross - circuit - pairs ] . note that @xmath147 is again a ross - basis . we therefore can assume ( after edge - swapping in this manner ) for all @xmath146 that @xmath37 has the property from proposition [ prop : ross - circuit - pairs ] in the ross - circuit it induces . we assign directions @xmath103 to the edges of @xmath48 such that : * the directions on each of the intersections of the ross - circuits with @xmath48 are generic in the sense of proposition [ prop : ross - circuit - pairs ] . * the directions on the edges of @xmath48 that remain in the reduced graph @xmath68 are perpendicular to an assignment of directions on @xmath148 that is generic in the sense of proposition [ prop : reflection-22-collapse ] . * the directions on the edges of @xmath48 are generic in the sense of proposition [ prop : ross - realizations ] . this is possible because the set of disallowed directions is the union of a finite number of proper algebraic subsets in the space of direction assignments . extend to directions @xmath30 on @xmath13 by assigning directions to the remaining edges as specified by proposition [ prop : ross - circuit - pairs ] . by construction , we know that : [ lemma : laman-1 ] the direction network @xmath19 has faithful realizations . the realization space is identified with that of @xmath100 , and @xmath103 is chosen so that proposition [ prop : ross - realizations ] applies . [ lemma : laman-2 ] in any realization of @xmath25 , the ross - circuits are realized with all their vertices coincident and on the @xmath24-axis . this follows from how we chose @xmath30 and proposition [ prop : ross - circuit - pairs ] . as a consequence of lemma [ lemma : laman-2 ] , and the fact that we picked @xmath30 so that @xmath143 extends to a generic assignment of directions @xmath149 on the reduced graph @xmath68 we have : the realization space of @xmath27 is identified with that of @xmath150 which , furthermore , contains only collapsed solutions . observe that a direction network for a single self - loop ( colored @xmath63 ) with a generic direction only has solutions where vertices are collapsed and on the @xmath24-axis . consequently , replacing a ross - circuit with a single vertex and a self - loop yields isomorphic realization spaces . since the reduced graph is reflection-@xmath57 by proposition [ prop : reduced - graph ] and the directions assigned to its edges were chosen generically for proposition [ prop : reflection-22-collapse ] , that @xmath27 has only collapsed solutions follows . thus , we have exhibited a special pair , completing the proof . @xmath66 [ [ remark ] ] remark + + + + + + it can be seen that the realization space of a direction network as supplied by theorem [ theo : direction - network ] has at least one degree of freedom for each edge that is not in a ross basis . thus , the statement can not be improved to , e.g. , a unique realization up to translation and scale . [ sec : reflection - laman - proof ] let @xmath151 be a reflection framework and let @xmath12 be the quotient graph . the configuration space , which is the set of solutions to the quadratic system ( [ eq : lengths-1])([eq : lengths-2 ] ) is canonically identified with the solutions to : @xmath152 where @xmath23 acts on the plane by reflection through the @xmath24-axis . ( that `` pinning down '' @xmath23 does not affect the theory is straightforward from the definition of the configuration space : it simply removes rotation and translation in the @xmath153-direction from the set of trivial motions . ) computing the formal differential of ( [ eq : colored - lengths ] ) , we obtain the system @xmath154 where the unknowns are the _ velocity vectors _ @xmath155 . a standard kind of result ( cf . @xcite ) is the following . thus , we define a realization to be _ infinitesimally rigid _ if the system ( [ eq : colored - inf ] ) has maximal rank , and _ minimally infinitesimally rigid _ if it is infinitesimally rigid but ceases to be so after removing any edge from the colored quotient graph . [ prop : rigidity - vs - directions ] let @xmath6 be a realization of a reflection framework with @xmath23 acting by reflection through the @xmath24-axis . define the direction @xmath20 to be @xmath157 . then the rank of ( [ eq : colored - inf ] ) is equal to that of ( [ eq : colored - system ] ) for the direction network @xmath158 . the , more difficult , `` laman direction '' of the main theorem follows immediately from theorem [ theo : direction - network ] and proposition [ prop : rigidity - vs - directions ] : given a reflection - laman graph theorem [ theo : direction - network ] produces a realization with no coincident endpoints and a certificate that ( [ eq : colored - inf ] ) has corank one . @xmath66 the statement of proposition [ prop : rigidity - vs - directions ] is _ exactly the same _ as the analogous statement for orientation - preserving cases of this theory . what is different is that , for reflection frameworks , the rank of @xmath158 is _ not _ , the same as that of @xmath159 . by proposition [ prop : reflection-22-collapse ] , the set of directions arising as the difference vectors from point sets are _ always non - generic _ on reflection - laman graphs , so we are forced to introduce the notion of a special pair as in section [ sec : direction - network ] . matthew berardi , brent heeringa , justin malestein , and louis theran . rigid components in fixed - latice and cone frameworks . in _ proceedings of the @xmath160 annual canadian conference on computational geometry ( cccg ) _ , 2011 . url http://arxiv.org/abs/1105.3234 . mike develin , jeremy l. martin , and victor reiner . rigidity theory for matroids . _ comment . _ , 820 ( 1):0 197233 , 2007 . issn 0010 - 2571 . doi : 10.4171/cmh/89 . url http://dx.doi.org/10.4171/cmh/89 . audrey lee and ileana streinu . pebble game algorithms and sparse graphs . _ discrete math . _ , 3080 ( 8):0 14251437 , 2008 . issn 0012 - 365x . doi : 10.1016/j.disc.2007.07.104 . url http://dx.doi.org/10.1016/j.disc.2007.07.104 . ileana streinu and louis theran . slider - pinning rigidity : a maxwell - laman - type theorem . _ discrete & computational geometry _ , 440 ( 4):0 812837 , 2010 . issn 0179 - 5376 . doi : 10.1007/s00454 - 010 - 9283-y . url http://dx.doi.org/10.1007/s00454-010-9283-y . walter whiteley . some matroids from discrete applied geometry . in j. bonin , james g. oxley , and b. servatius , editors , _ matroid theory _ , volume 197 of _ contemporary mathematics _ , pages 171311 . american mathematical society , 1996 .

we give a combinatorial characterization of generic minimally rigid reflection frameworks . the main new idea is to study a pair of direction networks on the same graph such that one admits faithful realizations and the other has only collapsed realizations . in terms of infinitesimal rigidity , realizations of the former produce a framework and the latter certifies that this framework is infinitesimally rigid . = 1

You are an expert at summarizing long articles. Proceed to summarize the following text: the flavor evolution of supernova ( sn ) neutrinos , is strongly impacted by the self - induced effects , associated with instabilities induced by the the neutrino - neutrino interactions in the deepest stellar regions @xcite ( see also @xcite for a recent review ) . in this context a key ingredient in the characterization of these effects is related to the current - current nature of low - energy weak interactions , which implies a `` multi - angle term '' @xcite @xmath4 , where @xmath5 is the neutrino velocity @xcite . till recently , all studies have assumed the axial symmetry for this term , that would then depend only on the zenith - angle . it has been shown that the `` multi - zenith - angle '' ( mza ) term in some cases can hinder the maintenance of the coherent oscillation behavior for different neutrino modes @xcite . a valuable tool to diagnose the possible instabilities of a dense neutrino gas is given by the linearized stability analysis of the neutrino equations of motion . the linearized equations including generic azimuthal and zenith angle distributions for neutrinos were first worked out in @xcite . then , the formalism for the azimuthal symmetric case was further developed in @xcite . the stability method would allow one to determine the possible onset of the flavor conversions , seeking for an exponentially growing solution of the eigenvalue problem , associated with the linearized equations of motion for the neutrino ensemble . recently , this method has been applied in @xcite removing the assumption of axial symmetry in the @xmath6 propagation . as a result , a multi - azimuthal - angle ( maa ) instability was found in addition to the bimodal @xcite and mza ones @xcite . in particular , it was considered a neutrino ensemble with a strong excess of @xmath7 over @xmath8 , as expected during the sn accretion phase ( at post - bounce times @xmath9 s ) . in this situation , the instability has been found in normal mass hierarchy ( nh , @xmath10 ) , where the system would have been stable imposing the perfect axial symmetry . subsequently , the role of this instability has been clarified in @xcite with simple toy models . the discovery of the new maa effects , has also motivated first numerical studies of the non - linear neutrino propagation equations in sn , introducing the azimuthal angle as angular variable in the multi - angle kernel , in addition to the usual zenith angle @xcite . remarkably it was considered the @xmath6 propagation only along a radial direction , i.e. a local solution along a specific line of sight , under the assumption that the transverse variations of the global solution are small . in this approximation , for the unstable case discussed above , maa effects lead in nh to spectral swaps and splits analogous to what produced in inverted hierarchy ( ih , @xmath11 ) by the known bimodal instability @xcite also in a completely isotropic neutrino gas @xcite . all these recent works assume that the maa effects can develop without any matter hindrance . remarkably , a crucial ingredient to be considered to determine the impact of self - induced flavor instabilities is the ordinary matter term , associated with the net electron densities @xmath12 in sne . as pointed out in @xcite for the axial - symmetric case , when @xmath12 is not negligible with respect to the neutrino density @xmath13 , the large phase dispersion induced by the matter for @xmath6 s traveling in different directions , would partially or totally suppress the self - induced oscillations through peculiar mza effects . recent studies of this case performed with realistic sn models , indicates that this situation is realized during the supernova accretion phase ( at post - bounce times @xmath9 s ) . as a consequence , the self - induced flavor conversions found in ih in the axial - symmetric models are strongly inhibited @xcite . from a preliminary schematic study done in @xcite it results that the matter density required to suppress the mma instability in nh is larger than the one necessary to suppress the self - induced conversions in ih for the axial symmetric case . motivated by these previous results , we find it is mandatory to understand what is the role of the dense matter on the maa effects during the accretion phase . we will use as benchmark for the neutrino and matter density profiles the sn models from recent long term simulations of core - collapse explosions , performed by the basel - darmstadt model . these were already considered by some of us in @xcite . the plan of our work is as follows . in sec . 2 we present the neutrino equations of flavor evolution without imposing axial asymmetry . then , we describe the setup to perform the stability analysis of the linearized equations of motion . in sec . 3 we present the results of the stability analysis and we compare them with the numerical solution of the equations . we focus on the accretion phase for two different sn progenitor models . finally , in sec . 4 we comment on our results and we conclude . in the axial symmetric case , the sn neutrino flavor evolution is described by ordinary differential equations @xcite , characterizing the flavor changes along a radial direction . when axial symmetry is broken by the maa effects , in order to get the global solution of the problem in general one would consider also variations along the transverse direction to the neutrino propagation . this would imply passing from ordinary to partial differential equations , with a big layer of complication in the numerical solution . however , in our study we are mostly interested in answering the question of the stability of the dense neutrino gas under mma effects , in the presence of a large matter term . therefore , _ before _ the maa instability emerges , we can still consider the ordinary differential equations in the only radial direction . these are enough to determine which cases are completely stable under maa effects . in the other cases , in which the maa instability develops , these equations would be useful to determine the onset radius of the flavor conversions . nevertheless , the subsequent flavor evolution can be taken just as indicative , since it is based on the assumption that variations in the transverse direction always remain small . under this approximation , the flavor evolution depends only on @xmath14 , @xmath15 and @xmath5 . then , following @xcite we write the equations of motion for the flux matrices @xmath16 as function of the radial coordinate . we use negative energy @xmath15 for anti - neutrinos . following the usual prescription , we label the zenith angular mode in terms of the variable @xmath17 , where @xmath18 is the emission angle relative to the radial direction of the neutrinosphere radius @xmath19 @xcite . we call @xmath20 the azimuth angle of the neutrino velocity @xmath21 . we normalize the flux matrices to the @xmath22 number flux at the neutrinosphere . the diagonal @xmath16 elements are the ordinary number fluxes integrated over a sphere of radius @xmath14 . the off - diagonal elements , which are initially zero , carry a phase information due to flavor mixing . then , the equations of motion read @xcite @xmath23 \,\ , \label{eq : eom1}\ ] ] with the hamiltonian @xcite @xmath24 the matrix @xmath25 of neutrino mass - squares causes vacuum flavor oscillations . we work in a two - flavor scenario , associated with the atmospheric mass - square difference @xmath26 ev@xmath27 and a small ( matter suppressed ) in - medium mixing @xmath28 . we will always consider nh , where maa effect could emerge for the flux ordering we are considering . the matrix @xmath29 in flavor basis , contains the net electron density and is responsible for the mikheyev - smirnov - wolfenstein ( msw ) matter effect @xcite with the ordinary background . finally , the term at second line represents the @xmath6-@xmath6 refractive term , where @xmath30 . in the multi - angle term of eq . ( [ eq : eom2 ] ) , the radial velocity of a mode with angular label @xmath31 is @xmath32 @xcite and the transverse velocity is @xmath33 @xcite . the term @xmath34 is the responsible for the breaking of the axial symmetry . to solve numerically eq . ( [ eq : eom1 ] ) we use an integration routine for stiff ordinary differential equations taken from the nag libraries @xcite and based on an adaptive method . we have used @xmath35 modes for @xmath36 $ ] , @xmath37 for @xmath38 $ ] . concerning neutrino emission model , in order to simplify the complexity of our numerical simulations of the flavor evolution , we assume all @xmath6 s to be represented by a single energy , that we fix at @xmath39 mev . this approximation is reasonable since our main purpose is to determine only if the dense matter effects block the development of the self - induced transformations . in order to perform the stability analysis , we linearize the equations of motion [ eq . ( [ eq : eom2 ] ) ] , following the approach of @xcite . we write the flux matrices in the form @xmath40 where we switch to the frequency variable @xmath41 , and we introduce the neutrino flux difference distributions @xmath42 that represent the flavor fluxes @xmath43 at the neutrinosphere , normalized to the @xmath22 flux . in the following we will always assume axial symmetry of the neutrino emission . therefore @xmath44 . the @xmath45 is conserved and then irrelevant for the flavor conversions . the @xmath7 survival probability is @xmath46 , given in terms of the swap factor @xmath47 of the matrix in the second term on the right - hand side . the off - diagonal components @xmath48 are complex and @xmath49 . the initial conditions are @xmath50 and @xmath51 . self - induced flavor transitions start when the off - diagonal term @xmath48 grows exponentially . in the small - amplitude limit @xmath52 , and at far distances from the neutrinosphere @xmath53 , the linearized evolution equations for @xmath48 assume the form @xcite @xmath54 s \nonumber \\ & - & \mu \int d \gamma^\prime [ u + u^\prime - 2\sqrt{u u^\prime } \cos ( \varphi-\varphi^\prime)]g^\prime s^\prime \label{eq : lin } \,\ .\end{aligned}\ ] ] in this equation @xmath55 , quantifies the `` asymmetry '' of the neutrino spectrum , normalized to the @xmath22 number flux . in the sn models we are using , typically @xmath56 during the accretion phase ( see fig . 3 in @xcite ) . the @xmath6-@xmath6 interaction strength is given by @xmath57}{4 \pi r^2}\frac{r_{\nu}^2}{2 r^2 } \nonumber \\ & = & \frac{3.5 \times 10^{9}}{r^4 } \left(\frac{l_{{\overline\nu}_e}}{\langle e_{{\overline\nu}_e } \rangle } - \frac{l_{{\overline\nu}_x}}{\langle e_{{\overline\nu}_x } \rangle } \right ) \nonumber \\ & \times & \left(\frac{15 \,\ \textrm{mev } } { 10^{52 } \,\ \textrm{erg / s}}\right ) \left(\frac{r_{\nu}}{10 \,\ \textrm{km } } \right)^2 , \label{eq : mu}\end{aligned}\ ] ] while ordinary matter background term is given by @xmath58 where @xmath59 is the net electron fraction , and @xmath60 is the matter density . the radial distance @xmath14 is expressed in km , while the numerical values of @xmath61 and @xmath62 in the two previous equations are quoted in km@xmath63 , as appropriate for the sn case . one can write the solution of the linear differential equation [ eq . ( [ eq : lin ] ) ] in the form @xmath64 with complex frequency @xmath65 and eigenvector @xmath66 . a solution with @xmath67 would indicate an exponential increase in @xmath48 , i.e. an instability . the solution of eq . ( [ eq : lin ] ) can then be recast in the form of an eigenvalue equation for @xmath66 . one gets as consistency condition @xcite @xmath68 a flavor instability is present whenever eq . ( [ consit ] ) admits a solution @xmath69 . we consider the core - collapse supernova simulations of massive stars with 8.8 and 10.8 @xmath2 progenitor from ref . @xcite , taken as benchmark for our numerical study in @xcite . the first type of sn belongs to the class of o - ne - mg - core progenitor . the second one is an iron - core progenitor . under the single - energy approximation we are using , we characterize the neutrino energy spectra as @xmath70 \,\ , \label{singleen } \end{aligned}\ ] ] where @xmath71 are the total number fluxes of the species @xmath72 at the neutrinosphere . in order to fix the neutrinosphere radius @xmath73 , consistently with our choice in @xcite we take the radius at which the @xmath7 s angular distribution has no longer significant backward flux , i.e. a few % of the total one . this typically is in the range @xmath74 km ( see fig . 4 in @xcite ) . for our choice of neutrino representative energy ( @xmath39 mev ) , the corresponding frequency is @xmath75 the @xmath76 represent the zenith - angle distributions . in our study we assume two different models . at first , we consider the `` half - isotropic '' case , where in analogy with a black - body emission , it is assumed @xmath77 for all the species . we will then compare the results obtained in this widely used prescription , with the one obtained taking the @xmath76 directly from the output of the sn simulations . in this case , the zenith - angle distributions would be flavor - dependent and forward enhanced ( i.e. peaked at small @xmath31 ) with respect to the half - isotropic emission model ( see fig . 1 and the discussion in @xcite ) . we will see that the presence of forward peaked distributions will enhance the matter suppression of the maa instability . finally , we comment that our results are based on axial symmetric neutrino angular distributions . in this regard , it is interesting to mention that in @xcite it has been shown that if perfect cylindrical symmetry in the initial neutrino distributions were given up , super - fast flavor turnovers could be produced . we leave the investigation of this interesting issue for a future work . progenitor mass . radial evolution of the ratio @xmath78 between the matter @xmath62 and neutrino @xmath61 potentials at different post - bounce times.,scaledwidth=50.0% ] we start our investigation with the case of the @xmath1 @xmath2 iron - core supernova . for this model , the net electron density @xmath12 and the neutrino densities @xmath13 for different post - bounce times were shown in fig . 5 of ref . @xcite . in order to quantify the relative strength of the matter potential @xmath62 [ eq . ( [ eq : lambda ] ) ] with respect to the neutrino potential @xmath61 [ eq . ( [ eq : mu ] ) ] we plot in fig . 1 the ratio @xmath79 as a function of the radial coordinate @xmath14 at different post - bounce times @xmath80 in the range [ 0.1,0.6 ] s. we realize that @xmath81 before the abrupt discontinuity associated with the shock front position . as we will see with the stability analysis , this strong matter dominance would prevent the flavor conversions before the shock front . however , for some time snapshots ( i.e. @xmath82 s ) the ratio can become @xmath83 after the shock front , leaving the possibility of flavor conversions in this region . for the same post - bounce times of fig . 1 , we show in fig . 2 the radial evolution of the eigenvalue @xmath84 determined from the solution of eq . ( [ consit ] ) . we consider the following cases : _ ( a ) _ @xmath85 and a half - isotropic neutrino emission ( dashed curves ) , _ ( b ) _ dense matter effects and a half - isotropic neutrino emission ( continuous curves ) and , _ ( c ) _ dense matter effects and non - trivial neutrino angular distributions ( dotted curves ) . in fig . 3 we show the survival probability @xmath86 for electron antineutrinos @xmath87 for the same cases of fig . 2 , obtained solving the non - linear propagation equations [ eq . ( [ eq : eom2 ] ) ] . we start discussing our results for the case of @xmath85 [ case _ ( a ) _ ] . we realize that when the neutrino system enters the unstable regime ( @xmath67 ) , the @xmath84 function rapidly grows from zero to a peak value greater than one . only for @xmath88 s @xmath89 at the peak . indeed , for this time the asymmetry parameter @xmath90 ( see fig . 3 in @xcite ) . then the consistency condition eq . ( [ consit ] ) in order to be satisfied requires a smaller @xmath84 . comparing the results of fig . 2 and 3 one finds a good agreement between the numerical onset of the self - induced flavor conversions triggered by the maa effect and the position of the peak in the @xmath84 function . we now discuss the situation of realistic matter density profiles and a half - isotropic neutrino emission [ case _ ( b ) _ ] . as expected , the flavor instability is strongly suppressed with respect to the previous case without matter . in particular , the @xmath84 function , when not completely vanishing ( as at @xmath91 s ) , would start growing only after the shock front position [ see fig . 1 ] . this is due to the fact that at lower radii the ratio @xmath92 . the @xmath84 function then reaches peak values between 0.5 and 1 only at intermediate times , i.e. @xmath93 s , for which the ratio @xmath94 in the post - shock region . for the other time snapshots @xmath84 it is more suppressed , consistently with a larger value of @xmath78 . comparing these results with the numerical calculation of the @xmath86 in fig . 3 , we realize that the presence of a non - zero @xmath84 is not enough to guarantee the onset of flavor conversions . indeed , for @xmath95 s , the @xmath84 function is too small and dies out too quickly before triggering flavor conversions . for the cases in which flavor conversions occur , i.e. at @xmath96 s , the numerical onset is shifted at larger radii by few hundred km , with respect to the peak of the @xmath84 function . this delay is due to the fact that since the instability is weaker with respect to the case with @xmath85 , the slower rate of growth implies a larger radial distance in order to develop significant effects on the @xmath86 . we checked that a non - zero @xmath84 corresponds to the exponential growth of the off - diagonal components @xmath97 of the flux matrices [ see eq . ( [ eq : phi ] ) ] , while the change of the diagonal components , would occur only at larger radii . it would be interesting to see if the stability analysis can be further developed in order to achieve a better understanding of this dynamics . then we consider the case in which also the flavor - dependent forward - peaked neutrino angular distributions are taken into account [ case _ ( c ) _ ] . we find that the @xmath84 function in this case is completely suppressed . this is consistent with the expectation that the @xmath6-@xmath6 strength is weaker for forward - peaked distributions , making more effective the matter suppression . this result is consistent with the output of the numerical simulations that show for all the considered time snapshots @xmath98 . finally we mention that in @xcite it has been claimed that possible residual scatterings could affect @xmath6 s after the neutrinosphere , producing a small `` neutrino halo '' that would broaden the @xmath6 angular distributions @xcite at @xmath99 km . we checked that the results of the sn simulations we are using have not enough angular resolution to exhibit this feature . however , in order to characterize the possible halo effect , we performed the same analytical estimation presented in @xcite . we repeated the stability analysis including the halo effect in the @xmath6 angular distributions , without finding any change with respect to the results shown here . therefore , we conclude that for our @xmath1 @xmath2 sn model , maa instability is always suppressed by the dense matter effects during the accretion phase . progenitor mass . radial evolution of the @xmath84 function ( in units of km@xmath63 ) at different post - bounce times with @xmath85 for a half - isotropic neutrino emission ( dashed curves ) and in presence of matter effects , with a half - isotropic neutrino emission ( continuous curves ) and with flavor - dependent angular distributions ( dotted curves).,scaledwidth=50.0% ] progenitor mass . radial evolution of the survival probability @xmath86 for electron antineutrinos at different post - bounce times for the maa evolution with @xmath85 for a half - isotropic neutrino emission ( dashed curves ) and in presence of matter effects , with a half - isotropic neutrino emission ( continuous curves ) and with flavor - dependent angular distributions ( dotted curves).,scaledwidth=50.0% ] progenitor mass . radial evolution of the ratio @xmath78 between the matter @xmath62 and neutrino @xmath61 potentials at different post - bounce times.,scaledwidth=50.0% ] in this section we analyze the flavor conversions for the model of @xmath3 @xmath2 o - ne - mg progenitor . in fig . 4 we plot the ratio @xmath79 for the same time snapshots of fig . 9 of @xcite , i.e. in the range [ 0.08,0.25 ] s. we realize that in this case there is no abrupt discontinuity associated with the shock front . indeed , for this low - mass progenitor there is no extended accretion phase , since the explosion succeeds very shortly after the core - bounce . therefore , the shock - front is already beyond the radial range interesting for the flavor conversions . since in this case the matter density of the envelope is low compared to the iron - core progenitors , the electron density profile above the core is very steep . therefore , at @xmath100 few hundred km , the ratio @xmath101 [ for @xmath102~s$ ] it monotonically decreases becoming also smaller than 1 ] suggests that flavor conversions could arise there . in fig . 5 we show the radial evolution of the eigenvalue @xmath84 for the time snapshots shown in fig . 4 , determined from the solution of eq . ( [ consit ] ) . we use the same format of fig . 2 . in fig . 6 we show the corresponding survival probability @xmath86 for electron antineutrinos @xmath87 . starting with the case without matter term , i.e. @xmath85 ( dashed curves ) , we see that the @xmath84 function rapidly becomes larger than 1 , and the peak corresponds to the onset of the flavor conversions in fig . we pass now considering the case with @xmath62 and half - isotropic neutrino angular distributions ( continuous curves ) . we realize that , for @xmath103 s in the region where @xmath84 grows , @xmath104 . therefore , the matter suppression of the instability is never complete . moreover , the rise of @xmath84 is rapid and the position of the peak corresponds to the onset of the flavor conversions seen in fig . conversely , for the other time snapshots ( @xmath105 s ) where @xmath106 , the suppression of the instability is stronger . moreover , the @xmath84 curves are broadened and there is not a clear peak . therefore , one can not easily link a non - zero @xmath84 with the numerical onset of the flavor conversions . in the case of @xmath62 and flavor dependent forward - peaked neutrino angular distributions ( dotted curves ) , as expected we find a stronger suppression in the flavor instability . however , as shown in fig . 1 of @xcite , the angular spectra of different flavors for the 8.8 m@xmath107 sn are significantly less forward - peaked than in the case of the 10.8 m@xmath107 sn . therefore , their effect is less pronounced in this case . in particular , for @xmath108 s , @xmath84 is large enough to trigger flavor conversions . conversely , these are strongly inhibited at @xmath109 s , and completely suppressed for @xmath110 s. finally , we checked also in this case that including a possible halo effect does not change the results of the stability analysis . in conclusion , for our sn model with 8.8 m@xmath107 progenitor mass the matter suppression of the maa instability is not complete at early times . therefore , in principle one would expect interesting time - dependent features in the observable neutrino spectra . progenitor mass . radial evolution of the @xmath84 function ( in units of km@xmath63 ) at different post - bounce times with @xmath85 for a half - isotropic neutrino emission ( dashed curves ) and in presence of matter effects , with a half - isotropic neutrino emission ( continuous curves ) and with flavor - dependent angular distributions ( dotted curves).,scaledwidth=50.0% ] progenitor mass . radial evolution of the survival probability @xmath86 for electron antineutrinos at different post - bounce times for the maa evolution with @xmath85 for a half - isotropic neutrino emission ( dashed curves ) and in presence of matter effects , with a half - isotropic neutrino emission ( continuous curves ) and with flavor - dependent angular distributions ( dotted curves).,scaledwidth=50.0% ] we have performed a dedicated study of the matter suppression of the maa instability , connected with the axial symmetry breaking in the self - induced oscillations , during the accretion phase for two sn models with different progenitor masses . we characterize the sn densities and the neutrino angular spectra with results from recent sn hydrodynamical simulations . we compared the linear stability analysis with the numerical results of the flavor evolution in which we have looked at a local solution of the equations of motions along a specific line of sight . for the case of an iron - core 10.8 m@xmath107 we found that during the accretion phase the dominant matter density strongly suppresses the maa instability . in particular , including realistic forward - peaked @xmath6 angular distributions significantly reduces the strength of the @xmath6-@xmath6 interaction term . as a result , in this case the matter suppression of the self - induced flavor conversions would be complete . in the case of a low - mass o - ne - mg sn with 8.8 m@xmath107 progenitor , where the accretion phase is extremely short , the matter density profile is lower and the @xmath6 angular distributions less forward - peaked than in iron - core models . as a consequence , we found that also with realistic angular distributions flavor conversions would be possible at early times , producing in principle interesting time - dependent modulations . our analysis is complementary to previous studies @xcite , where some of us explored the matter suppression of self - induced flavor conversions in inverted neutrino mass hierarchy , in axial - symmetric models ( see also @xcite ) . the complete suppression of the self - induced effects in both the mass hierarchies for iron - core sne , implies that the neutrino signal during the accretion phase will be processed only by the ordinary mikheyev - smirnov - wolfenstein effect in the outer stellar layers . this effect would allow in principle to distinguish the neutrino mass hierarchy through the study of the rise time of the sn neutrino signal @xcite . the phenomenological importance of our findings motivates further studies with other sn models to confirm the generality of our results . in particular , an accurate characterization of the neutrino angular distributions seems necessary in order to get accurate predictions on the matter suppression . at this regard , in @xcite it has been shown that if perfect cylindrical symmetry in the initial neutrino distributions were given up , then super - fast flavor turnovers could occur . these effects can not be tested within our spherically symmetric sn model . however , recently three - dimensional sn simulations have been carried on , characterizing the neutrino signal during the accretion phase . surprisingly , a lepton - emission asymmetry among different flavor has been found @xcite . in particular , the electron ( anti)neutrino fluxes show a dipole structure , while the @xmath111 are almost spherically symmetric . we plan to investigate in a future work the role of the matter suppression in this flavor configuration , including also the not axisymmetric neutrino and matter angular distributions . self - induced flavor conversions associated with the maa instability would still be possible for o - ne - mg sne during the accretion phase , and possibly for iron core sne during the cooling phase , when the matter term becomes sub - dominant with respect to the neutrino - neutrino interaction term . in these situations , a self - consistent treatment of the neutrino equations of motion considering also the flavor evolution along the transverse direction is still lacking . this would imply passing from an ordinary to a partial differential equation problem , adding a big layer of complication in the solution of the equations of motion . this effort is well motivated by the perspective of getting an accurate characterization of the sn neutrino spectral features that would be observable in the planned large underground neutrino detectors @xcite . we thank t. fischer , g. raffelt , s. sarikas and m. wu for useful discussions . s.c . acknowledges support from the european union through a marie curie fellowship , grant no . piif - ga-2011 - 299861 , and through the itn `` invisibles '' , grant no . pitn - ga-2011 - 289442 . the work of a.m. was supported by the german science foundation ( dfg ) within the collaborative research center 676 `` particles , strings and the early universe . 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it has been recently pointed out that removing the axial symmetry in the `` multi - angle effects '' associated with the neutrino - neutrino interactions for supernova ( sn ) neutrinos , a new multi - azimuthal - angle ( maa ) instability would arise . in particular , for a flux ordering @xmath0 , as expected during the sn accretion phase , this instability occurs in the normal neutrino mass hierarchy . however , during this phase the ordinary matter density can be larger than the neutrino one , suppressing the self - induced conversions . at this regard , we investigate the matter suppression of the maa effects , performing a linearized stability analysis of the neutrino equations of motion , in the presence of realistic sn density profiles . we compare these results with the numerical solution of the sn neutrino non - linear evolution equations . assuming axially symmetric distributions of neutrino momenta we find that the large matter term strongly inhibits the maa effects . in particular , the hindrance becomes stronger including realistic forward - peaked neutrino angular distributions . as a result , in our model for a @xmath1 @xmath2 iron - core sne , maa instability does not trigger any flavor conversion during the accretion phase . instead , for a @xmath3 @xmath2 o - ne - mg core sn model , with lower matter density profile and less forward - peaked angular distributions , flavor conversions are possible also at early times .

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Proceed to summarize the following text: -input multiple - output ( mimo ) techniques offer transmit diversity and high data rates through the use of multiple antennas at both transmitter and receiver sides @xcite-@xcite . a key component of a mimo system is the mimo detector at the receiver , which , in practice , is often the bottleneck for the overall performance and complexity . mimo detectors including sphere decoder and several of its variants @xcite-@xcite achieve near maximum likelihood ( ml ) performance at the cost of high complexity . other well known detectors including zf ( zero forcing ) , mmse ( minimum mean square error ) , and zf - sic ( zf with successive interference cancellation ) detectors @xcite,@xcite are attractive from a complexity view point , but achieve relatively poor performance . for example , the zf - sic detector ( i.e. , the well known v - blast detector with ordering @xcite,@xcite ) does not achieve the full diversity in the system . the mmse - sic detector has been shown to achieve optimal performance @xcite . however , these detectors are prohibitively complex for large number of antennas of the order of tens to hundreds . with small number of antennas , the high capacity potential of mimo is not fully exploited . a key issue with using large number of antennas , however , is the high detection complexities involved . in this paper , we focus on large mimo systems , where by ` _ large _ ' we mean number of transmit and receive antennas of the order of tens to hundreds . such large mimo systems will be of immense interest because of the very high spectral efficiencies possible in such systems . for example , in a v - blast system , increased number of transmit antennas means increased data rate without bandwidth increase . however , major bottlenecks in realizing such large mimo systems include @xmath4 physical placement of large number of antennas in communication terminals , @xmath5 lack of practical low - complexity detectors for such large systems , and @xmath6 channel estimation issues . in this paper , we address the second problem in the above ( i.e. , low - complexity large mimo detection ) . specifically , we present a low - complexity detector for large mimo systems , including v - blast as well as high - rate non - orthogonal space - time block codes ( stbc ) @xcite . the proposed low - complexity detector has its roots in past work on hopfield neural network ( hnn ) based algorithms for image restoration @xcite,@xcite , which are meant to handle large digital images . hnn based image restoration algorithms in @xcite are applied to multiuser detection ( mud ) in cdma systems on awgn channels in @xcite . this detector , referred to as the likelihood ascent search ( las ) detector , essentially searches out a sequence of bit vectors with monotonic likelihood ascent and converges to a fixed point in finite number of steps @xcite . the power of the las detector for cdma lies in @xmath4 its linear average per - bit complexity in number of users , and @xmath5 its ability to perform very close to ml detector for large number of users . taking the cue from las detector s complexity and performance superiority in large systems , we , in this paper , successfully adopt the las detector for large mimo systems and report interesting results . we refer to the proposed detector as mf / zf / mmse - las detector depending on the initial vector used in the algorithm ; mf - las detector uses the matched filter output as the initial vector , and zf - las and mmse - las detectors employ zf and mmse outputs , respectively , as the initial vector . our major findings in this paper are summarized as follows : * in an uncoded v - blast system with bpsk , the proposed detector achieves _ near - exponential diversity _ for hundreds of antennas ( i.e. , achieves near siso awgn performance ) . for example , the proposed detector nearly renders a @xmath7 mimo fading channel into 200 parallel , non - interfering siso awgn channels . the detector achieves this excellent performance with an average per - bit complexity of just @xmath0 , where @xmath1 and @xmath2 denote the number of transmit and receive antennas , respectively . * with an outer turbo code , the proposed detector achieves good coded bit error performance as well . for example , in a 600 transmit and 600 receive antennas v - blast system with a high spectral efficiency of 200 bps / hz ( using bpsk and rate-1/3 turbo code ) , our simulation results show that the proposed detector performs close to within about 4.6 db from the theoretical capacity . we note that performance with such closeness to capacity has not been reported in the literature so far for such large number of antennas using a practical complexity detector . * we have adopted the proposed detector for the low - complexity decoding of large full - rate non - orthogonal stbcs from division algebras ( da ) in @xcite . we decode the @xmath3 full - rate non - orthogonal stbc from da ( which has 256 data symbols in one stbc matrix ) using the proposed detector and show that it performs close to within about 5.5 db from capacity using 4-qam and rate-3/4 turbo code at a spectral efficiency of 24 bps / hz . * we point out that because of the high complexities involved in the decoding of large non - orthogonal stbcs using other known detectors , the ber performance of such high - rate large non - orthogonal stbcs have not been reported in the literature so far . the very fact that we could show the simulated ber performance plots ( both uncoded as well as turbo coded ) for a @xmath3 full - rate non - orthogonal stbc with 256 complex symbols in one stbc matrix in itself is a clear indication of the superior low - complexity attribute of the proposed detector . to our knowledge , this is the first time that simulated ber plots and nearness to capacity results for a full - rate @xmath3 stbc from da are reported in the literature ; this became feasible due to the low - complexity attribute of the proposed detector . * we also illustrate the applicability of the proposed detector in the low - complexity detection of large multicarrier cdma ( mc - cdma ) systems . in large mc - cdma systems with hundreds of users , the proposed detector is shown to achieve near single - user performance , at an average per - bit complexity linear in number of users , which is quite appealing for its use in practical cdma systems . the rest of the paper is organized as follows . in section [ sec2 ] , we present the proposed las detector for v - blast systems and its complexity . the simulated uncoded and coded ber performance of the proposed detector for v - blast is presented in section [ sec3 ] . decoding of non - orthogonal stbcs and ber performance results are presented in section [ sec4 ] . the las detector for mc - cdma and the corresponding ber performance results are presented in section [ sec5 ] . conclusions are presented in section [ sec6 ] . in this section , we present the proposed las detector for v - blast and its complexity . consider a v - blast system with @xmath1 transmit antennas and @xmath2 receive antennas , @xmath8 , where @xmath1 symbols are transmitted from @xmath1 transmit antennas simultaneously . let @xmath9 be the symbol - qam/@xmath10-pam as well . ] transmitted by the @xmath11th transmit antenna . each transmitted symbol goes through the wireless channel to arrive at each of @xmath2 receive antennas . denote the path gain from transmit antenna @xmath11 to receive antenna @xmath12 by @xmath13 . considering a flat - fading mimo channel with rich scattering , the signal received at antenna @xmath12 , denoted by @xmath14 , is given by @xmath15 the @xmath16 , @xmath17 , @xmath18 , are assumed to be i.i.d . complex gaussian r.vs ( i.e. , fade amplitudes are rayleigh distributed ) with zero mean and @xmath19 = e\big[\big(h_{kj}^{q}\big)^2\big]=0.5 $ ] , where @xmath20 and @xmath21 are the real and imaginary parts of @xmath13 . the noise sample at the @xmath12th receive antenna , @xmath22 , is assumed to be complex gaussian with zero mean , and @xmath23 , @xmath24 , are assumed to be independent with @xmath25=n_0=\frac{n_te_s}{\gamma}$ ] , where @xmath26 is the average energy of the transmitted symbols , and @xmath27 is the average received snr per receive antenna @xcite . collecting the received signals from all receive antennas , we write^t$ ] , @xmath28^*$ ] , and @xmath28^h$ ] denote transpose , conjugate , and conjugate transpose operations , respectively . @xmath29 and @xmath30 denote the real and imaginary parts of the complex argument . ] @xmath31 where @xmath32^t$ ] is the @xmath2-length received signal vector , @xmath33^t$ ] is the @xmath1-length transmitted bit vector , * h * denotes the @xmath34 channel matrix with channel coefficients @xmath16 , and @xmath35^t$ ] is the @xmath2-length noise vector . * h * is assumed to be known perfectly at the receiver but not at the transmitter . the proposed las algorithm essentially searches out a sequence of bit vectors until a fixed point is reached ; this sequence is decided based on an update rule . in the v - blast system considered , for ml detection @xcite , the most likely @xmath36 is taken as that @xmath37 which maximizes @xmath38 the likelihood function in ( [ lamda1w ] ) can be written as @xmath39 where @xmath40 _ update criterion in the search procedure : _ let @xmath41 denote the bit vector tested by the las algorithm in the search step @xmath42 . the starting vector @xmath43 can be the output vector from any known detector . when the output vector of the mf detector is taken as the @xmath43 , we call the resulting las detector as the mf - las detector . we define zf - las and mmse - las detectors likewise . given @xmath41 , the algorithm obtains @xmath44 through an update rule until a fixed point is reached . the update is made in such a way that the change in likelihood from step @xmath42 to @xmath45 , denoted by @xmath46 , is positive , i.e. , @xmath47 an expression for the above change in likelihood can be obtained in terms of the gradient of the likelihood function as follows . let @xmath48 denote the gradient of the likelihood function evaluated at @xmath41 , i.e. , @xmath49 where @xmath50 using ( [ 3eq5w ] ) in ( [ deltaw ] ) , we can write @xmath51 now , defining @xmath52 and @xmath4 observing that @xmath53 , @xmath5 adding & subtracting the term @xmath54 to the rhs of ( [ dxw ] ) , and @xmath6 further observing that @xmath55 , we can simplify ( [ dxw ] ) as @xmath56 where @xmath57 now , given @xmath58 , @xmath59 , and @xmath41 , the objective is to obtain @xmath44 from @xmath41 such that @xmath60 in ( [ 3eq8w ] ) is positive . potentially any one or several bits in @xmath41 can be flipped ( i.e. , changed from + 1 to -1 or vice versa ) to get @xmath44 . we refer to the set of bits to be checked for possible flip in a step as a _ check candidate set_. let @xmath61 denote the check candidate set at step @xmath42 . with the above definitions , it can be seen that the likelihood change at step @xmath42 , given by ( [ 3eq8w ] ) , can be written as @xmath62 , \label{3eq12w}\end{aligned}\ ] ] where @xmath63 , @xmath64 , and @xmath65 are the @xmath11th elements of the vectors @xmath41 , @xmath48 , and @xmath66 , respectively . as shown in @xcite for synchronous cdma on awgn , the following update rule can be easily shown to achieve monotonic likelihood ascent ( i.e. , @xmath67 if there is at least one bit flip ) in the v - blast system as well . _ las update algorithm : _ given @xmath68 and an initial bit vector @xmath69 , bits in @xmath41 are updated as per the following update rule : @xmath70 where @xmath71 is a threshold for the @xmath11th bit in the @xmath42th step is taken to be @xmath72 where @xmath73 is the element in the @xmath11th row and @xmath74th column of the matrix @xmath75 . it is noted that different choices can be made to specify the sequence of @xmath76 . one of the simplest sequences correspond to checking one bit in each step for a possible flip , which is termed as a sequential las ( slas ) algorithm with constant threshold , @xmath77 the sequence of @xmath78 in slas can be such that the indices of bits checked in successive steps are chosen circularly or randomly . checking of multiple bits for possible flip is also possible . let @xmath79 denote the set of indices of the bits flipped according to the update rule in ( [ 3eq18w ] ) at step @xmath42 . then the updated bit vector @xmath44 can be written as @xmath80 where @xmath81 is the @xmath74th coordinate vector . using ( [ 3eq20w ] ) in ( [ 3eq9w ] ) , the gradient vector for the next step can be obtained as @xmath82 where @xmath83 denotes the @xmath74th column of the matrix @xmath75 . the las algorithm keeps updating the bits in each step based on the update rule given in ( [ 3eq18w ] ) until @xmath84 for some @xmath85 , in which case @xmath86 is a fixed point , and it is taken as the detected bit vector and the algorithm terminates . in terms of complexity , given an initial vector , the the las operation part alone has an average per - bit complexity of @xmath0 . this can be explained as follows . the complexity involved in the las operation is due to three components : @xmath4 initial computation of @xmath87 in ( [ 3eq9w ] ) , @xmath5 update of @xmath48 in each step as per ( [ 3eq21w ] ) , and @xmath6 the average number of steps required to reach a fixed point . computation of @xmath87 requires the computation of @xmath88 for each mimo fading channel realization @xmath89see eqns . ( [ 3eq9w ] ) , ( [ 3eq10w ] ) , and ( [ 3eq5bw])@xmath90 , which requires a per - bit complexity of order @xmath0 . update of @xmath48 in the @xmath42th step as per ( [ 3eq21w ] ) using sequential las requires a complexity of @xmath91 , and hence a constant per - bit complexity . regarding the complexity component @xmath6 , we obtained the average number of steps required to reach a fixed point for sequential las through simulations . we observed that the average number of steps required is linear in @xmath1 , i.e. , constant per - bit complexity where the constant @xmath92 depends on snr , @xmath1 , @xmath2 , and the initial vector ( see fig . [ fig_bfr ] ) . putting the complexities of @xmath4 , @xmath5 , and @xmath6 in the above together , we see that the average per - bit complexity of las operation alone is @xmath0 . in addition to the above , the initial vector generation also contributes to the overall complexity . the average per - bit complexity of generating initial vectors using mf , zf , and mmse are @xmath93 , @xmath0 , and @xmath0 , respectively . the higher complexity of zf and mmse compared to mf is because of the need to perform matrix inversion operation in zf / mmse . again , putting the complexities of the las part and the initial vector generation part together , we see that the overall average per - bit complexity of the proposed mf / zf / mmse - las detector is @xmath0 . this complexity is an order superior compared to the well known zf - sic detector with ordering , whose per - bit complexity is @xmath94 . in this section , we present the uncoded / coded ber performance of the proposed las detector in v - blast obtained through simulations , and compare with those of other detectors . the las algorithm used is the sequential las with circular checking of bits starting from the first antenna bit . we also quantify how far is the proposed detector s turbo coded ber performance away from the theoretical capacity . the snrs in all the ber performance figures are the average received snr per received antenna , @xmath27 , defined in sec . [ sec2 ] @xcite . _ mf / zf - las performs increasingly better than zf - sic for increasing @xmath95 _ : in fig . [ fig_vb2 ] , we plot the uncoded ber performance of the mf - las , zf - las and zf - sic detectors for v - blast as a function of @xmath95 at an average received snr of 20 db with bpsk . the performance of the mf and zf detectors are also plotted for comparison . from fig . [ fig_vb2 ] , we observe the following : * the ber at @xmath96 is nothing but the siso flat rayleigh fading ber for bpsk , given by @xmath97 $ ] which is equal to @xmath98 for @xmath99 db @xcite . while the performance of mf and zf degrade as @xmath95 is increased , the performance of zf - sic improves for antennas up to @xmath100 , beyond which a flooring effect occurs . this improvement is likely due to the potential diversity in the ordering ( selection ) in zf - sic , whereas the flooring for @xmath101 is likely due to interference being large beyond the cancellation ability of the zf - sic . * the behavior of mf - las and zf - las for increasing @xmath95 are interesting . starting with the mf output as the initial vector , the mf - las always achieves better performance than mf . more interestingly , this improved performance of mf - las compared to that of mf increases remarkably as @xmath95 increases . for example , for @xmath100 , the performance improves by an order in ber ( i.e. , @xmath102 ber for mf versus @xmath103 ber for mf - las ) , whereas for @xmath104 the performance improves by four - orders in ber ( i.e. , @xmath105 ber for mf versus @xmath106 ber for mf - las ) . this is due to the large system effect in the las algorithm which is able to successfully pick up much of the diversity possible in the system . this large system performance superiority of the las is in line with the observations / results reported in @xcite for a large cdma system @xmath89large number of antennas in our case , whereas it was large number of users in @xcite@xmath107 . * while the zf - las performs slightly better than zf - sic for antennas less than 4 , zf - sic performs better than zf - las for antennas in the range 4 to 24 . this is likely because , for antennas less than 4 , the ber of zf is small enough for the las to clean up the zf initial vector into an output vector better than the zf - sic output vector . however , for antennas in the range of 4 to 24 , the ber of zf gets high to an extent that the zf - las is less effective in cleaning the initial vector beyond the diversity performance achieved by the zf - sic . a more interesting observation , however , is that for antennas greater than 25 , the large system effect of zf - las starts showing up . so , in the large system setting ( e.g. , antennas more than 25 in fig . [ fig_vb2 ] ) , the zf - las performs increasingly better than zf - sic for increasing @xmath95 . we found the number of antennas at which the cross - over between zf - sic and zf - las occurs to be different for different snrs . * another observation in fig . [ fig_vb2 ] is that for antennas greater than 50 , mf - las performs better than zf - las . this behavior can be explained by observing the performance comparison between mf and zf detectors given in the same figure . for more than 50 antennas , mf performs slightly better than zf . it is known that zf detector can perform worse than mf detector under high noise / interference conditions @xcite ( here high interference due to large @xmath1 ) . hence , starting with a better initial vector , mf - las performs better than zf - las . ) for v - blast at an average received snr = 20 db . bpsk , @xmath1 bps / hz spectral efficiency . , width=338 ] [ fig_vb2 ] v - blast system . bpsk , 200 bps / hz spectral efficiency . zf - las achieves higher order diversity ( near - exponential diversity ) than zf - sic at a much lesser complexity.,width=319 ] [ fig_vb3 ] _ zf - las outperforms zf - sic in large v - blast systems both in complexity & diversity : _ in fig . [ fig_vb3 ] , we present an interesting comparison of the uncoded ber performance between zf , zf - las and zf - sic , as a function of average snr for a @xmath108 v - blast system . this system being a large system , the zf - las has a huge complexity advantage over zf - sic as pointed out before in sec . [ vb_comp ] . in fact , although we have taken the effort to show the performance of zf - sic at such a large number of antennas like 200 , we had to obtain these simulation points for zf - sic over days of simulation time , whereas the same simulation points for zf - las were obtained in just few hours . this is due to the @xmath94 complexity of zf - sic versus @xmath0 complexity of zf - las , as pointed out in sec . [ vb_comp ] . more interestingly , in addition to this significant complexity advantage , zf - las is able to achieve a much higher order of diversity ( in fact , near - exponential diversity ) in ber performance compared to zf - sic ( which achieves only a little better than first order diversity ) . this is clearly evident from the slopes of the ber curves of zf - las and zf - sic . _ note that the ber curve for zf - las is almost the same as the uncoded ber curve for bpsk on a siso awgn channel , given by @xmath109 @xcite . this means that the proposed detector nearly renders a @xmath108 mimo fading channel into 200 parallel , non - interfering siso awgn channels . _ _ las detector s performance with hundreds of antennas : _ as pointed earlier , obtaining zf - sic results for more than even 50 antennas requires very long simulation run times , which is not the case with zf - las . in fact , we could easily generate ber results for up to 400 antennas for zf - las , which are plotted in fig . [ fig_vb4 ] . the key observations in fig . [ fig_vb4 ] are that @xmath4 the average snr required to achieve a certain ber performance keeps reducing for increasing number of antennas for zf - las , and @xmath5 increasing the number of antennas results in increased orders of diversity achieved ( close to siso awgn performance for 200 and 400 antennas ) . we have also observed from our simulations that for large number of antennas , the las algorithm converges to almost the same near - ml performance regardless of the initial vector chosen . for example , for the case of 200 and 400 antennas in fig . [ fig_vb4 ] , the ber performance achieved by zf - las , mf - las , and mmse - las are almost the same ( although we have not explicitly plotted the ber curves for mf - las and mmse - las in fig . [ fig_vb4 ] ) . so , in such large mimo systems setting , mf - las may be preferred over zf / mmse - las since zf / mmse - las require matrix inverse operation whereas mf - las does not . observation @xmath4 in the above paragraph is explicitly brought out in fig . [ fig_vb5 ] , where we have plotted the average received snr required to achieve a target uncoded ber of @xmath110 as a function of @xmath95 for zf - las and zf - sic . it can be seen that the snr required to achieve @xmath110 with zf - las significantly reduces for increasingly large @xmath95 . for example , the required snr reduces from about 25 db for a siso rayleigh fading channel to about 7 db for a @xmath111 v - blast system using zf - las . as we pointed out in fig . [ fig_vb4 ] , this @xmath111 system performance is almost the same as that of a siso awgn channel where the snr required to achieve @xmath110 ber is also close to 7 db @xcite , i.e. , @xmath112 db . . bpsk , @xmath1 bps / hz spectral efficiency . for large number of antennas ( e.g. , @xmath113 ) , the performance of zf - las , mf - las , and mmse - las are almost the same.,width=326 ] in v - blast for increasing values of @xmath95 . zf - las versus zf - sic . zf - las achieves near siso awgn performance.,width=326 ] for different snrs and initial vectors ( mf , zf , mmse ) . results obtained from simulations.,width=321 ] [ fig_bfr ] in this subsection , we present the turbo coded ber performance of the proposed las detector . we also quantify how far is the proposed detector s performance away from the theoretical capacity . for a @xmath114 mimo system model in sec . [ sec2 ] with perfect channel state information ( csi ) at the receiver , the ergodic capacity is given by @xcite @xmath115 , \label{cap}\end{aligned}\ ] ] where @xmath116 is the @xmath117 identity matrix and @xmath27 is the average snr per receive antenna . we have evaluated the capacity in ( [ cap ] ) for a @xmath118 mimo system through monte - carlo simulations and plotted it as a function of average snr in fig . figure [ fig2 ] shows the simulated ber performance of the proposed las detector for a @xmath118 mimo system with bpsk and rate-1/3 turbo code ( i.e. , spectral efficiency = 200 bps / hz ) . figure [ fig4 ] shows similar performance plots for rate-3/4 turbo code at a spectral efficiency of 450 bps / hz . from the capacity curve in fig . [ fig1 ] , the minimum snrs required at 200 bps / hz and 450 bps / hz spectral efficiencies are -5.4 db and -0.8 db , respectively . the following interesting observations can be made from figs . [ fig2 ] and [ fig4 ] : mimo system with receive csi.,width=312 ] v - blast system . 200 bps / hz spectral efficiency . proposed mf / zf / mmse - las detectors performance is away from capacity by 4.6 db.,width=321 ] v - blast system . 450 bps / hz spectral efficiency . proposed mf / zf / mmse - las detectors performance is away from capacity by 5.6 db.,width=321 ] * in terms of uncoded ber , the performance of mf , zf , and mmse are different , with zf and mmse performing the worst and best , respectively . but the performance of mf - las , zf - las , and mmse - las are almost the same ( near - exponential diversity performance ) with the number of antennas being large ( @xmath119 ) . * with a rate-1/3 turbo code ( fig . [ fig2 ] ) , all the las detectors considered ( i.e. , mf - las , zf - las , mmse - las ) achieve almost the same performance , which is about 4.6 db away from capacity db , where @xmath120 is the turbo code rate , which amounted to a pessimistic prediction of nearness to capacity . here , we have corrected those plotting errors . figures [ fig2 ] , [ fig4 ] , [ fig_qam ] and the nearness to capacity results given in table - i in this paper are the corrected ones . ] ( i.e. , near - vertical fall of coded ber occurs at about -0.8 db ) . turbo coded mf / mmse without las also achieve good performance in this case ( i.e. , less than only 2 db away from turbo coded mf / zf / mmse - las performance ) . this is because the uncoded ber of mf and mmse at around 0 to 2 db snr are small enough for the turbo code to be effective . however , this is not the case with turbo coded zf without las . as can be seen , in the range of snrs shown , the uncoded ber of zf without las is so high ( close to 0.5 ) that the vertical fall of coded ber can happen only at very high snrs , because of which we have not shown the performance of turbo coded zf without las . & at capacity & proposed las & zf & mf & mmse + rate-1/3 , & -5.4 db & -0.8 db & high & 1.2 db & -0.3 db + 200 bps / hz & & & & & + rate-1/2 & -3.2 db & 1.5 db & high & high & 3 db + 300 bps / hz & & & & & + rate-3/4 & -0.8 db & 4.8 db & high & high & high + 450 bps / hz & & & & & + in table i , we summarize the performance of various detectors in terms of their nearness to capacity in a @xmath121 v - blast system using bpsk , and rate-1/3 , rate-1/2 and rate-3/4 turbo codes . from table - i , it can be seen that there is a clear superiority of the proposed mf / zf / mmse - las over mf / mmse without las in terms of coded ber ( nearness to capacity ) when high - rate turbo codes are used . for example , when a rate-3/4 turbo code is used the mf / zf / mmse - las performs to within about 5.6 db from capacity , whereas the performance of rate-3/4 turbo coded mf / mmse without las are much farther away from capacity . _ performance of @xmath10-pam/@xmath10-qam : _ although the las algorithm in sec . [ sec2 ] is presented assuming bpsk , it can be adopted for @xmath10-ary modulation including @xmath10-pam and @xmath10-qam . in the case of bpsk , the elements of the data vector take values from @xmath122 . @xmath10-pam symbols take discrete values from @xmath123 where @xmath124 , @xmath125 , and @xmath10-qam is nothing but quadrature pam . we have adopted the las algorithm for @xmath10-pam/@xmath10-qam and evaluated the performance of the las detector for 4-pam/4-qam and 16-pam/16-qam without and with coding . in @xmath10-pam/@xmath10-qam also , we have observed large system behavior of the proposed detector similar to those presented for bpsk . as an example , in fig . [ fig_qam ] , we present the uncoded and coded performance of the mmse - las detector in a @xmath121 v - blast system for 16-pam/16-qam with rate-1/2 and rate-1/3 turbo codes at spectral efficiencies of 1200 bps / hz and 800 bps / hz , respectively . it can be observed that the las detector achieves performance close to within about 13 db from the theoretical capacity . _ effect of channel estimation errors : _ as we pointed out earlier , another key issue in large mimo systems is channel estimation @xcite,@xcite . we have evaluated the effect of channel estimation errors on the performance of the proposed detector in v - blast by considering an estimation error model , where the estimated channel matrix , @xmath126 , is taken to be @xmath127 where @xmath128 is the estimation error matrix , the entries of which are assumed to be i.i.d . complex gaussian with zero mean and variance @xmath129 . our simulation results showed that in a @xmath108 v - blast system with bpsk , rate-1/2 turbo code and las detection , the coded ber degradation compared to perfect channel estimation is only 0.2 db and 0.6 db for channel estimation error variances of @xmath130 and @xmath131 , respectively . the investigation of estimation algorithms and efficient pilot schemes for accurate channel estimation in large mimo systems as such are important topics for further research . v - blast system for 16-pam and 16-qam with rate-1/2 and rate-1/3 turbo codes.,width=321 ] v - blast with large number of antennas can offer high spectral efficiencies , but it does not provide transmit diversity . on the other hand , well known orthogonal stbcs have the advantages of full transmit diversity and low decoding complexity , but suffer from rate loss for increased number of transmit antennas @xcite,@xcite-@xcite . _ full - rate non - orthogonal stbcs from division algebras ( da ) _ @xcite , on the other hand , are attractive for achieving high spectral efficiencies in addition to achieving full transmit diversity , using large number of transmit antennas . construction of full - rate non - orthogonal stbcs from da for arbitrary number of transmit antennas @xmath42 is given by the matrix in ( 20.a ) at the bottom of this page @xcite . in ( 20.a ) , @xmath132 , @xmath133 , and @xmath134 , @xmath135 are the data symbols from a qam alphabet . note that there are @xmath136 data symbols in one stbc matrix . when @xmath137 and @xmath138 , the stbc in ( 20.a ) achieves full transmit diversity ( under ml decoding ) as well as information - losslessness @xcite . when @xmath139 , the code ceases to be of full - diversity ( fd ) , but continues to be information - lossless ( ill ) @xcite .. \hspace{10 mm } ( \mbox{20.a})\ ] ] ] high spectral efficiencies with large @xmath42 can be achieved using this code construction . for example , with @xmath140 transmit antennas , the @xmath3 stbc from ( 20.a ) with 16-qam and rate-3/4 turbo code achieves a spectral efficiency of 48 bps / hz . this high spectral efficiency is achieved along with the full - diversity of order @xmath141 . however , since these stbcs are non - orthogonal , ml detection gets increasingly impractical for large number of transmit antennas , @xmath42 . consequently , a key challenge in realizing the benefits of these full - rate non - orthogonal stbcs in practice is that of achieving near - ml performance for large number of transmit antennas at low detection complexities . here , we show that near - ml detection of large mimo signals originating from several tens of antennas using full - rate non - orthogonal stbcs is possible at practically affordable low complexities ( using the proposed las detector ) , which is a significant new advancement that has not been reported in the mimo detection literature so far . we have adopted the proposed las detector for the decoding of full - rate non - orthogonal stbcs . in fig . [ fig_n1 ] , we present the uncoded ber of the las detector in decoding @xmath142 full - rate non - orthogonal stbcs from da in ( 20.a ) for @xmath143 , @xmath139 , and 4-qam . it can be observed that as the stbc code size @xmath42 increases , the las performs increasingly better such that it achieves close to siso awgn performance ( within 0.5 db at @xmath110 ber and less ) with the @xmath3 stbc . we point out that due to the high complexities involved in decoding large size stbcs using other known detectors , the ber performance of stbcs with large @xmath42 has not been reported in the literature so far . the very fact that we could show the simulated ber plots ( both uncoded as well as turbo coded ) for a @xmath3 stbc with 256 complex symbols in one stbc matrix in itself is a clear indication of the superior low - complexity attribute of the proposed las detector . to our knowledge , we are the first to report the simulated ber performance of a @xmath3 stbc from da ; this became feasible because of the low - complexity feature of the proposed detector . in addition , the achievement of near siso awgn performance with @xmath3 stbc is a significant result from an implementation view point as well , since 16 antennas can be easily placed in communication terminals of moderate size , which can make large mimo systems practical . in fig . [ fig_n2 ] , we show the coded ber performance of the @xmath3 stbc using different turbo code rates of 1/3 , 1/2 , and 3/4 . with 4-qam , these turbo code rates along with the @xmath3 stbc from da correspond to spectral efficiencies of 10.6 bps / hz , 16 bps / hz and 24 bps / hz , respectively . the minimum snrs required to achieve these capacities are also shown in fig . [ fig_n2 ] . it can be observed that the proposed detector performs to within about 5.5 db of the capacity , which is an impressive result . in all the turbo coded ber plots in this paper , we have used hard decision outputs from the las algorithm . in @xcite , we have proposed a method to generate soft decision outputs from the las algorithm for the individual bits that form the qam / pam symbols . with the proposed soft decision las outputs in @xcite , the coded performance is found to move closer to capacity by an additional 1 to 1.5 db compared to that achieved using hard decision las outputs reported in this paper . full - rate non - orthogonal stbcs from da for @xmath143 . mmse initial vector , 4-qam , @xmath144 . @xmath3 stbc with 256 complex symbols in each stbc matrix achieves close to siso awgn performance . , width=321 ] full - rate non - orthogonal stbc from da . mmse initial vector , 4-qam . rates of turbo codes : 1/3 , 1/2 , 3/4 . proposed las detector performs close to within about 5.5 db from the theoretical capacity . , width=321 ] in this section , we present the proposed las detector for multicarrier cdma , its performance and complexity . consider a @xmath146-user synchronous multicarrier ds - cdma system with @xmath10 subcarriers . let @xmath147 denote the binary data symbol of the @xmath12th user , which is sent in parallel on @xmath10 subcarriers @xcite,@xcite . let @xmath148 denote the number of chips - per - bit in the signature waveforms . it is assumed that the channel is frequency non - selective on each subcarrier and the fading is slow ( assumed constant over one bit interval ) and independent from one subcarrier to the other . let @xmath149^t$]denote the @xmath146-length received signal vector on the @xmath74th subcarrier ; i.e. , @xmath150 is the output of the @xmath12th user s matched filter on the @xmath74th subcarrier . assuming that the inter - carrier interference is negligible , the @xmath146-length received signal vector on the @xmath74th subcarrier @xmath151 can be written in the form @xmath152 where @xmath153 is the @xmath154 cross - correlation matrix on the @xmath74th subcarrier , with its entries @xmath155 s denoting the normalized cross correlation coefficient between the signature waveforms of the @xmath156th and @xmath11th users on the @xmath74th subcarrier . @xmath157 represents the @xmath158 channel matrix , given by @xmath159 where the channel coefficients @xmath160 , @xmath161 , are assumed to be i.i.d . complex gaussian r.vs ( i.e. , fade amplitudes are rayleigh distributed ) with zero mean and @xmath162=e\big[\big(h_{kq}^{(i)}\big)^2\big]=0.5 $ ] , where @xmath163 and @xmath164 are the real and imaginary parts of @xmath160 . the @xmath146-length data vector @xmath37 is given by @xmath165^t,\ ] ] and the @xmath154 diagonal amplitude matrix @xmath166 is given by @xmath167 where @xmath168 denotes the transmit amplitude of the @xmath12th user . the @xmath146-length noise vector @xmath169 is given by @xmath170^t,\ ] ] where @xmath171 denotes the additive noise component of the @xmath12th user on the @xmath74th subcarrier , which is assumed to be complex gaussian with zero mean with @xmath172=\sigma^2 $ ] when @xmath173 and @xmath172=\sigma^2\rho_{kj}^{(i)}$ ] when @xmath174 . we assume that all the channel coefficients are perfectly known at the receiver . we note that once the likelihood function for the mc - cdma system in the above is obtained , it is straightforward to adopt the las algorithm for mc - cdma . accordingly , in the multicarrier system considered , the most likely @xmath36 is taken as that @xmath36 which maximizes @xmath175 the likelihood function in ( [ lamda1 ] ) can be written in a form similar to eqn . ( 4.11 ) in @xcite as @xmath176 where @xmath177 now observing the similarity between ( [ 3eq5 ] ) and ( [ 3eq5w ] ) in sec . [ sec3aa ] , the las algorithm for mc - cdma can be arrived at , along the same lines as that of v - blast in the previous section , with @xmath58 , @xmath59 and @xmath75 replaced by @xmath178 , @xmath179 , and @xmath180 , respectively , with all other notations , definitions , and procedures in the algorithm remaining the same . the complexity of the proposed detector for mc - cdma can be analyzed in a similar manner as done for v - blast in sec . first , given an initial vector , the las operation part alone in mc - cdma has an average per - bit complexity of @xmath181 , which is due to @xmath4 initial computation of @xmath87 in ( [ 3eq9w ] ) , which requires @xmath181 complexity per bit , @xmath5 update of @xmath48 in each step as per ( [ 3eq21w ] ) , which requires @xmath182 complexity for sequential las , and hence constant per - bit complexity , and @xmath6 the average number of steps required to reach a fixed point , which , through simulations , is found to have a constant per - bit complexity . next , the initial vector generation using mmse or zf has a per - bit complexity of @xmath183 for @xmath184 . finally , combining the above complexities involved in the las part and the initial vector generation part , the overall average per - bit complexity of the mmse / zf - las detector for mc - cdma is @xmath183 . the initial vector generation using mf has a per - bit complexity of only @xmath185 . hence , if the mf output is used as the initial vector , then the overall average per - bit complexity of the mf - las is the same as that of the las alone , which is @xmath181 . for large @xmath146 , the performance of mf - las , zf - las , and mmse - las are almost the same ( see fig . [ fig_mc2 ] ) , and hence mf - las is preferred because of its linear complexity in number of users , @xmath146 , for a given @xmath10 . we evaluated the ber performance of the proposed las detector for mc - cdma through simulations . we evaluate the uncoded ber performance of the proposed las detector as a function of average snr , number of users ( @xmath146 ) , number of subcarriers ( @xmath10 ) , and number of chips per bit ( @xmath148 ) . we also evaluate the ber performance as a function of _ loading factor _ , @xmath186 , where , as done in the cdma literature @xcite , we define @xmath187 . we call the system as underloaded when @xmath188 , fully loaded when @xmath189 , and overloaded when @xmath190 . random binary sequences of length @xmath148 are used as the spreading sequences on each subcarrier . in order to make a fair comparison between the performance of mc - cdma systems with different number of subcarriers , we keep the system bandwidth the same by keeping @xmath191 constant . also , in that case we keep the total transmit power to be the same irrespective of the number of subcarriers used . in the simulation plots we show in this section , we have assumed that all users transmit with equal amplitude . the las algorithm used is the slas with circular checking of bits starting from the first user s bit . , @xmath192 , @xmath193 , i.e. , @xmath194.,width=360 ] , for single carrier cdma ( @xmath195 ) in rayleigh fading for a fixed @xmath194 and average snr = 15 db . @xmath148 varied from 15 to 1500.,width=336 ] first , in fig . [ fig_mc1 ] , we present the ber performance of mf / zf - las detectors as a function of average snr in a single carrier ( i.e. , @xmath195 ) _ underloaded system _ , where we consider @xmath194 by taking @xmath192 users and @xmath193 chips per bit . for comparison purposes , we also plot the performance of mf and zf without las . single user ( su ) performance , which corresponds to the case of no multiuser interference ( i.e. , @xmath196 ) , is also shown as a lower bound on the achievable multiuser performance . from fig . [ fig_mc1 ] , we can observe that the performance of mf and zf detectors are far away from the su performance . whereas , the zf - las as well as mf - las detectors almost achieve the su performance . we point out that , like zf detector , other suboptimum detectors including mmse , sic , and pic detectors @xcite also do not achieve near su performance for the considered loading factor of 2/3 , whereas the mf - las detector achieves near su performance , that too at a lesser complexity than these other suboptimum detectors . next , in fig . [ fig_mc2 ] , we show the ber performance of the mf / zf - las detectors for @xmath195 as a function of number of users , @xmath146 , for a fixed value of @xmath194 at an average snr of 15 db . we varied @xmath146 from 10 to 1000 users . su performance is also shown ( as the bottom most horizontal line ) for comparison . it can be seen that , for the fixed value of @xmath194 , both the mf - las as well as the zf - las achieve near su performance ( even in the presence of 1000 users ) , whereas the zf and mf detectors do not achieve the su performance . in fig . [ fig_mc3 ] , we show the ber performance of the mf / zf - las detectors as a function of average snr for different number of subcarriers , namely , @xmath197 , keeping a constant @xmath198 , for a _ fully loaded system _ ( i.e. , @xmath189 ) with @xmath199 . keeping @xmath189 and @xmath199 for all cases means that @xmath4 @xmath200 for @xmath195 , @xmath5 @xmath201 for @xmath202 , and @xmath6 @xmath203 for @xmath204 . the su performance for @xmath195 ( 1st order diversity ) , @xmath202 ( 2nd order diversity ) , and @xmath204 ( 4th order diversity ) are also plotted for comparison . these diversities are essentially due to the frequency diversity effect resulting from multicarrier combining of signals from @xmath10 subcarriers . it is interesting to see that even in a fully loaded system , the mf / zf - las detectors achieve all the frequency diversity possible in the system ( i.e. , mf / zf - las detectors achieve su performance with 1st , 2nd and 4th order diversities for @xmath205 and 4 , respectively ) . on the other hand , zf detector is unable to achieve the frequency diversity in the fully loaded system , and its performance is very poor compared to mf / zf - las detectors . next , in fig . [ fig_mc4 ] , we present the ber performance of zf / mf - las detectors in a mc - cdma system with @xmath204 as a function of loading factor , @xmath186 , where we vary @xmath186 from @xmath206 to 1.5 . we realize this variation in @xmath186 by fixing @xmath207 , @xmath204 , and varying @xmath148 from 300 to 5 . the average snr considered is 8 db . from fig . [ fig_mc4 ] , it can be observed that as @xmath186 increases all detectors loose performance , but the mf / zf - las detectors can offer relatively good performance even at _ overloaded conditions _ of @xmath190 . another observation is that at @xmath190 , mf - las performs slightly better than zf - las . this is because @xmath190 corresponds to a high interference condition , and it is known in mud literature @xcite that zf can perform worse than mf at low snrs and high interference . in such cases , starting with a better performing mf output as the initial vector , mf - las performs better . , @xmath189 , @xmath199 , @xmath198.,width=336 ] , for multicarrier cdma in rayleigh fading . @xmath204 , @xmath207 , @xmath148 varied from 300 to 5 , average snr = 8 db . , width=360 ] further to our present work on the application of mf / zf - las detectors for mc - cdma , several extensions are possible on the practical application of these detectors in cdma systems . two such useful extensions are @xmath4 mf / zf / mmse - las for frequency selective cdma channels with rake combining ; we point out that a similar approach and system model adopted here for mc - cdma is applicable , by taking a view of equivalence between frequency diversity through mc combining and multipath diversity through rake combining , and @xmath5 mf / zf / mmse - las for asynchronous cdma systems , which can be carried out once the system model is appropriately written in a form similar to ( [ eqna ] ) . these two extensions can allow mf / zf - las detectors to be practical in cdma systems ( e.g. , 2 g and 3 g cdma systems ) , with potential for significant gains in system capacity . current approaches to mud considered in practical cdma systems appear to be mainly pic and sic . however , the illustrated fact that mf - las can easily outperform pic / sic detectors both in performance and complexity for large number of users suggests that mf - las can be a powerful mud approach in practical cdma systems . we presented a near - capacity achieving , low - complexity detector for large mimo systems having tens to hundreds of antennas , and showed its uncoded / coded ber performance in the detection of v - blast and in the decoding of full - rate non - orthogonal stbcs from da . the proposed detector was shown to have excellent attributes in terms of both low complexity as well as nearness to theoretical capacity performance , achieving high spectral efficiencies of the order of tens to hundreds of bps / hz . to our knowledge , our reporting of the decoding of a large full - rate non - orthogonal stbc like @xmath3 stbc from da and its ber / nearness to capacity results is for the first time in the literature . we further point out that the proposed detector has good potential for application in practical mimo wireless standards , e.g. , the low - complexity feature of the proposed detector can allow the inclusion of @xmath208 , @xmath209 , @xmath3 non - orthogonal stbcs from da into mimo wireless standards like ieee 802.11n and ieee 802.16e , which , in turn , can achieve higher spectral efficiencies than those are currently possible in these standards . we conclude this paper by pointing to the following remark made by the author of @xcite in its preface in 2005 : _ `` it was just a few years ago , when i started working at at&t labs research , that many would ask ` who would use more than one antenna in a real system ? ' today , such skepticism is gone . '' _ extending this sentiment , we believe large mimo systems would be practical in the future , and the practical feasibility of low - complexity detectors like the one we presented in this paper could be a potential trigger to create wide interest in the implementation of large mimo systems . for example , antenna / rf technologies and channel estimation for large mimo systems could open up as new focus areas . potential large mimo applications include inter - base station / base station controller back - haul connectivity using large mimo links , and wireless iptv . other interesting large mimo applications can be thought of as well . h. vikalo and b. hassibi , `` on the sphere - decoding algorithm ii . generalizations , second - order statistics , and applications to communications , '' _ ieee trans . signal process . 53 , no . 8 , pp . 2819 - 2834 , august 2005 . h. d. zhu , b. farhang - boroujeny , and r .- chen , `` on the performance of sphere decoding and markov chain monte carlo detection methods , '' _ ieee signal proc . letters _ , vol . 669 - 672 , october 2005 . k. higuchi , h. kawai , n. maeda , h. taoka , and m. sawahashi , `` experiments on real - time 1-gb / s packet transmission using mld - based signal detection in mimo - ofdm broadband radio access , '' _ ieee jl . areas in commun . 1141- 1153 , june 2006 . p. w. woliniansky , g. j. foschini , g. d. golden , and r. a. valenzuela , `` v - blast : an architecture for realizing very high data rates over the rich - scattering wireless channel , '' _ proc . 295 - 300 , september - october 1998 . g. d. golden , g. j. foschini , r. a. valenzuela , and p. w. wolniansky , `` detection algorithm and initial laboratory results using v - blast space - time communication architecture , '' _ electron . 14 - 16 , january 1999 . y. sun , `` hopfield neural network based algorithms for image restoration and reconstruction part i : algorithms and simulations , '' _ ieee trans . on signal process . 48 , no . 7 , pp . 2105 - 2118 , july 2000 . k. vishnu vardhan , saif k. mohammed , a. chockalingam , and b. sundar rajan , `` a low - complexity detector for large mimo systems and multicarrier cdma systems , '' _ ieee jsac spl . iss . on multiuser detection for advanced communication systems and networks _ , vol . 26 , no . 473 - 485 , april 2008 . saif k. mohammed , k. vishnu vardhan , a. chockalingam , and b. sundar rajan , `` large mimo systems : a low - complexity detector at high spectral efficiencies '' _ accepted for presentation in ieee icc2008 _ , beijing , china , may 2008 . saif k. mohammed , a. chockalingam , and b. sundar rajan , `` a low - complexity near - ml performance achieving algorithm for large mimo detection , '' _ accepted for presentation in ieee isit2008 _ , toronto , canada , july 2008 . v. tarokh , n. seshadri , and a. r. calderbank , `` space - time codes for high data rate wireless communications : performance criterion and code construction , '' _ ieee trans . inform . theory _ 744 - 765 , march 1998 . b. a. sethuraman , b. s. rajan and v. shashidhar , `` full - diversity , high - rate space - time block codes from division algebras , '' _ ieee trans . inform . theory _ 2596 - 2616 , october 2003 . t. l. marzetta , `` blast training : estimating channel characteristics for high capacity space - time wireless , '' _ proc . 37th annual allerton conf . on communication , control , and computing _ , pp . 958966 , september 1999 s. manohar , v. tikiya , r. annavajjala , and a. chockalingam , `` ber - optimal linear parallel interference cancellation for multicarrier ds - cdma in rayleigh fading , '' _ ieee trans . commun . , _ vol 2560 - 2571 , july 2007 . k. vishnu vardhan was born in andhra pradesh , india . he received the undergraduate degree in electronics and communication engineering from pondicherry university , pondicherry , india , in 2005 . he received the postgraduate degree in telecommunication engineering from indian institute of science , bangalore , india , in 2007 . since july 2007 , he has been with cisco systems ( india ) private limited , bangalore , india . his research interests include multiuser detection and low - complexity detectors for cdma and mimo systems . saif khan mohammed received his b.tech degree in computer science and engineering from the indian institute of technology , new delhi , india , in 1998 . from 1998 to 2000 , he was employed with philips inc . , bangalore , india , as an asic design engineer . from 2000 to 2003 , he worked with ishoni networks inc . , santa clara , ca , as a senior chip architecture engineer . from 2003 to 2007 , he was employed with texas instruments , bangalore as systems and algorithms designer in the wireless systems group . he is currently pursuing his doctoral degree in electrical and communication engineering at the indian institute of science , bangalore , india . his research interests include low - complexity detection , estimation and coding for wireless communications systems . a. chockalingam was born in rajapalayam , tamil nadu , india . he received the b.e . ( honors ) degree in electronics and communication engineering from the p. s. g. college of technology , coimbatore , india , in 1984 , the m.tech degree with specialization in satellite communications from the indian institute of technology , kharagpur , india , in 1985 , and the ph.d . degree in electrical communication engineering ( ece ) from the indian institute of science ( iisc ) , bangalore , india , in 1993 . during 1986 to 1993 , he worked with the transmission r & d division of the indian telephone industries limited , bangalore . from december 1993 to may 1996 , he was a postdoctoral fellow and an assistant project scientist at the department of electrical and computer engineering , university of california , san diego . from may 1996 to december 1998 , he served qualcomm , inc . , san diego , ca , as a staff engineer / manager in the systems engineering group . in december 1998 , he joined the faculty of the department of ece , iisc , bangalore , india , where he is an associate professor , working in the area of wireless communications and networking . dr . chockalingam is a recipient of the swarnajayanti fellowship from the department of science and technology , government of india . he served as an associate editor of the ieee transactions on vehicular technology from may 2003 to april 2007 . he currently serves as an editor of the ieee transactions on wireless communications . he is a fellow of the indian national academy of engineering . b. sundar rajan was born in tamil nadu , india . he received the b.sc . degree in mathematics from madras university , madras , india , the b.tech degree in electronics from madras institute of technology , madras , and the m.tech and ph.d . degrees in electrical engineering from the indian institute of technology , kanpur , india , in 1979 , 1982 , 1984 , and 1989 respectively . he was a faculty member with the department of electrical engineering at the indian institute of technology in delhi , india , from 1990 to 1997 . since 1998 , he has been a professor in the department of electrical communication engineering at the indian institute of science , bangalore , india . his primary research interests include space - time coding for mimo channels , distributed space - time coding and cooperative communication , coding for multiple - access and relay channels , with emphasis on algebraic techniques . rajan is an associate editor of the ieee transactions on information theory , an editor of the ieee transactions on wireless communications , and an editorial board member of international journal of information and coding theory . he served as technical program co - chair of the ieee information theory workshop ( itw02 ) , held in bangalore , in 2002 . he is a fellow of the indian national academy of engineering and recipient of the iete pune center s s.v.c aiya award for telecom education in 2004 . rajan is a member of the american mathematical society .

we consider large mimo systems , where by ` _ large _ ' we mean number of transmit and receive antennas of the order of tens to hundreds . such large mimo systems will be of immense interest because of the very high spectral efficiencies possible in such systems . we present a low - complexity detector which achieves uncoded near - exponential diversity performance for hundreds of antennas ( i.e. , achieves near siso awgn performance in a large mimo fading environment ) with an average per - bit complexity of just @xmath0 , where @xmath1 and @xmath2 denote the number of transmit and receive antennas , respectively . with an outer turbo code , the proposed detector achieves good coded bit error performance as well . for example , in a 600 transmit and 600 receive antennas v - blast system with a high spectral efficiency of 200 bps / hz ( using bpsk and rate-1/3 turbo code ) , our simulation results show that the proposed detector performs close to within about 4.6 db from theoretical capacity . we also adopt the proposed detector for the low - complexity decoding of high - rate non - orthogonal space - time block codes ( stbc ) from division algebras ( da ) . for example , we have decoded the @xmath3 full - rate non - orthogonal stbc from da using the proposed detector and show that it performs close to within about 5.5 db of the capacity using 4-qam and rate-3/4 turbo code at a spectral efficiency of 24 bps / hz . the practical feasibility of the proposed high - performance low - complexity detector could potentially trigger wide interest in the implementation of large mimo systems . we also illustrate the applicability of the proposed detector in the low - complexity detection of large multicarrier cdma ( mc - cdma ) systems . in large mc - cdma systems with hundreds of users , the proposed detector is shown to achieve near single - user performance at an average per - bit complexity linear in number of users , which is quite appealing for its use in practical cdma systems . large mimo systems , v - blast , non - orthogonal stbcs , low - complexity detection , high spectral efficiency , multicarrier cdma .

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Proceed to summarize the following text: a number of studies have reported an anti - correlation between fractional linear polarization and total intensity flux density for extragalactic 1.4 ghz sources ; faint sources were found to be more highly polarized . as a result , the euclidean - normalised differential number - counts of polarized sources have been observed to flatten at linearly polarized flux densities @xmath4 @xmath5 1 mjy to levels greater than those expected from convolving the known total intensity source counts with plausible distributions for fractional polarization @xcite . the flattening suggests that faint polarized sources may exhibit more highly ordered magnetic fields than bright sources , or may instead suggest the emergence of an unexpected faint population . the anti - correlation trend for fractional linear polarization is not observed at higher frequencies ( @xmath6 ghz ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . to investigate possible explanations for the fractional polarization trend seen in previous studies , we have produced the second data release of the australia telescope large area survey ( atlas dr2 ) as described in paper i @xcite of this two paper series . atlas dr2 comprises reprocessed and new 1.4 ghz observations with the australia telescope compact array ( atca ) about the _ chandra _ deep field - south ( cdf - s ; galactic coordinates @xmath7 , @xmath8 ; * ? ? ? * ) and european large area _ infrared space observatory _ survey - south 1 ( elais - s1 ; @xmath9 , @xmath10 ; * ? ? ? * ) regions in total intensity , linear polarization , and circular polarization . the mosaicked multi - pointing survey areas for atlas dr2 are 3.626 deg@xmath11 and 2.766 deg@xmath11 for the cdf - s and elais - s1 regions , respectively , imaged at approximately @xmath12 resolution . typical source detection thresholds are 200 @xmath1jy in total intensity and polarization . in paper i we presented our data reduction and analysis prescriptions for atlas dr2 . we presented a catalogue of components ( discrete regions of radio emission ) comprising 2416 detections in total intensity and 172 independent detections in linear polarization . no components were detected in circular polarization . we presented a catalogue of 2221 sources ( groups of physically associated radio components ; grouping scheme based on total intensity properties alone , as described below ) , of which 130 were found to exhibit linearly polarized emission . we described procedures to account for instrumental and observational effects , including spatial variations in each of image sensitivity , bandwidth smearing with a non - circular beam , and instrumental polarization leakage , clean bias , the division between peak and integrated flux densities for unresolved and resolved components , and noise biases in both total intensity and linear polarization . analytic correction schemes were developed to account for incompleteness in differential component number counts due to resolution and eddington biases . we cross - identified and classified sources according to two schemes , summarized as follows . in the first scheme , described in 6.1 of paper i , we grouped total intensity radio components into sources , associated these with infrared sources from the _ spitzer _ wide - area infrared extragalactic survey ( swire ; * ? ? ? * ) and optical sources from @xcite , then classified them according to whether their energetics were likely to be driven by an active galactic nucleus ( agn ) , star formation ( sf ) within a star - forming galaxy ( sfg ) , or a radio star . due to the limited angular resolution of the atlas data , in paper i we adopted the term _ lobe _ to describe both jets and lobes in sources with radio double or triple morphologies . the term _ core _ was similarly defined in a generic manner to indicate the central component in a radio triple source . under this terminology , a core does not indicate a compact , flat - spectrum region of emission ; restarted agn jets or lobes may contribute or even dominate the emission observed in the regions we have designated as cores . agns were identified using four selection criteria : radio morphologies , 24 @xmath1 m to 1.4 ghz flux density ratios , mid - infrared colours , and optical spectral characteristics . sfgs and stars were identified solely by their optical spectra . of the 2221 atlas dr2 sources , 1169 were classified as agns , 126 as sfgs , and 4 as radio stars . we note that our classification system was biased in favour of agns . as a result , the atlas dr2 data are in general unsuited for statistical comparisons between star formation and agn activity . in the second scheme , described in 6.2 of paper i , we associated linearly polarized components , or polarization upper limits , with total intensity counterparts . in most cases it was possible to match a single linearly polarized component with a single total intensity component , forming a one - to - one match . in other cases this was not possible , due to ambiguities posed by the blending of adjacent components ; for example , a polarized component situated mid - way between two closely - separated total intensity components . in these cases , we formed group associations to avoid biasing measurements of fractional polarization . we classified the polarization total intensity associations according to the following scheme , which we designed to account for differing ( de-)polarized morphologies ( see paper i for graphical examples ) : * _ type 0 _ a one - to - one or group association identified as a lobe of a double or triple radio source . both lobes of the source are clearly polarized , having linearly polarized flux densities within a factor of 3 . ( the ratio between lobe total intensity flux densities was found to be within a factor of 3 for all double or triple atlas dr2 sources . ) * _ types 1/2 _ a one - to - one or group association identified as a lobe of a double or triple radio source that does not meet the criteria for type 0 . a lobe classified as type 1 indicates that the ratio of polarized flux densities between lobes is greater than 3 . a lobe classified as type 2 indicates that the opposing lobe is undetected in polarization and that the polarization ratio may be less than 3 , in which case it is possible that more sensitive observations may lead to re - classification as type 0 . sources with lobes classified as type 1 exhibit asymmetric depolarization in a manner qualitatively consistent with the laing - garrington effect @xcite , where one lobe appears more fractionally polarized than the opposite lobe . * _ type 3 _ a group association representing a source , involving a linearly polarized component situated midway between two total intensity components . it is not clear whether such associations represent two polarized lobes , a polarized lobe adjacent to a depolarized lobe , or a polarized core . * _ type 4 _ an unclassified one - to - one or group association representing a source . * _ type 5 _ a one - to - one association clearly identified as the core of a triple radio source ( where outer lobes are clearly distinct from the core ) . * _ type 6 _ a source comprising two type 0 associations , or a group association representing a non - depolarized double or triple radio source where blended total intensity and linear polarization components have prevented clear subdivision into two type 0 associations . * _ type 7 _ a source comprising one or two type 1 associations . * _ type 8 _ a source comprising one type 2 association . * _ type 9 _ an unpolarized component or source . in this work ( paper ii ) we present the key observational results from atlas dr2 , with particular focus on the nature of faint polarized sources . this paper is organised as follows . in [ ch5:secres ] we present the atlas dr2 source diagnostics resulting from infrared and optical cross - identifications and classifications , diagnostics resulting from polarization total intensity cross - identifications and classifications , differential component number - counts , and our model for the distribution of fractional polarization . in [ ch5:secdisc ] we compare the atlas dr2 differential counts in both total intensity and linear polarization to those from other 1.4 ghz surveys , and discuss asymmetric depolarization of classical double radio sources . we present our conclusions in [ ch5:secconc ] . this paper follows the notation introduced in paper i. we typically denote flux density by @xmath13 , but split into @xmath14 for total intensity and @xmath4 for linearly polarized flux density when needed for clarity . in the following sections we present diagnostics of atlas dr2 sources resulting from the infrared and optical cross - identification and classification schemes described in 6.1 of paper i ( summarised in [ sec:1 ] of this work ) . we focus on three parameter spaces formed by comparing flux densities between different wavelength bands : radio to mid - infrared , mid - infrared colours , and radio to far - infrared . in fig . [ ch5:fig : rnir ] we compare the total intensity 1.4 ghz radio to 3.6 @xmath1 m mid - infrared flux densities for all 2221 atlas dr2 sources , taking into account infrared upper bounds for the 298 radio sources without detected infrared counterparts . the bottom - right panel of fig . [ ch5:fig : rnir ] indicates that the atlas sources classified as stars or sfgs typically exhibit radio flux densities @xmath5 1 mjy . the paucity of atlas sources with @xmath15 @xmath5 0.1 mjy and star or sfg classifications likely represents a selection bias , in which only those sources with relatively bright optical counterparts could be classified spectroscopically . the top - left panel highlights all 130 polarized atlas sources , 12 of which are represented by upper bounds . the paucity of polarized sources with @xmath16 @xmath5 1 mjy is due to the limited sensitivity of our linear polarization data ; fractional polarization trends will be presented in [ ch5:secresidentm ] . in fig . [ ch5:fig : nircc ] we present mid - infrared colour - colour diagrams in which the irac flux density ratios @xmath17 and @xmath18 have been compared for atlas dr2 sources . of the 2221 atlas sources , 988 were detected in all four irac bands , while 935 were detected in only 2 or 3 bands ; the remaining 298 sources were not detected in any band , and have not been shown in fig . [ ch5:fig : nircc ] . regarding the 130 polarized atlas sources , 55 were detected in all four irac bands , 63 were detected in only 2 or 3 bands , and 12 were not detected in any irac band ; thus 118 polarized sources are indicated in fig . [ ch5:fig : nircc ] . the dotted lines in each panel of fig . [ ch5:fig : nircc ] represent the divisions identified through simulations by @xcite . by considering the effects of redshift evolution on the observed mid - infrared colours of three general source classes with spectral characteristics dominated by old - population ( 10 gyr ) starlight , polycyclic aromatic hydrocarbon ( pah ) emission , or a power - law continuum , @xcite identified four regions that could be used to preferentially select different source classes at different redshifts within the @xmath19 range simulated . region 1 was found to preferentially select sources with spectra dominated by continuum emission , likely produced by dust tori associated with agns @xcite , over the full redshift range . region 2 was found to preferentially select pah - dominated sources , indicative of intense star formation , over the full redshift range . region 3 was found to preferentially select sources with spectra dominated by direct stellar light , but only for sources with @xmath20 @xmath5 0.4 . for increasing redshifts , region 3 was found to comprise a mixture of stellar- and pah - dominated sources as the latter migrated from region 2 . however , beyond @xmath20 @xmath21 1.6 , region 3 was found to be largely void of sources following the migrations of both stellar- and pah - dominated sources towards region 4 . region 4 was found to be largely void of sources for @xmath20 @xmath5 0.4 . for increasing redshifts , pah - dominated sources were found to migrate from region 2 into region 4 . for @xmath20 @xmath21 0.9 , stellar - dominated sources from region 3 were also found to migrate into region 4 . @xcite found that at all redshifts , sources dominated by pah emission were located slightly within the boundaries of region 1 . in order to classify as agns only those sources most likely to be such in fig . [ ch5:fig : nircc ] , we constructed the restricted locus indicated by the dashed lines ; we label this region 1r . in paper i we defined this locus ( following * ? ? ? * ) as the union of @xmath22>0 $ ] , @xmath23>0 $ ] , and @xmath22 < 11\log_{{\scriptscriptstyle}10}[s_{5.8\,\mu{\textrm}{\scriptsize m}}/s_{3.6\,\mu{\textrm}{\scriptsize m}}]/9 + 0.3 $ ] . continuum - dominated sources are expected to exhibit power - law spectra , given by the dot - dashed locus in each panel . as noted by @xcite , the spectra of continuum - dominated sources are only expected to exhibit blue irac colours for largely unobscured agns , in cases for which their rest - frame optical wavelengths are redshifted into the mid - infrared band for sources at @xmath24 . thus the atlas sources with blue irac colours in fig . [ ch5:fig : nircc ] are unlikely to be represented by continuum - dominated sources as defined by @xcite . however , this does not imply that a source observed with blue irac colours at @xmath25 can not be an agn , because sources with mid - infrared spectra dominated by old stellar light may yet exhibit stronger signs of agn activity at other wavelengths . the bottom - left panel of fig . [ ch5:fig : nircc ] indicates that atlas sources classified as agns are predominantly located in regions 1r and 3 . the sources classified as agns in region 2 perhaps suggest combinations of star formation and agn activity , or perhaps misclassifications due to the largely statistical nature of our classification system . the upper bounds classified as agns in the top - right panel are consistent with the observed distribution of agns presented in the bottom - left panel . these upper bounds suggest that additional agns are situated in region 4 , though likely in proportion with the additional agns remaining undetected in regions 1 and 3 . the bottom - right panel indicates that atlas sources classified as sfgs are predominantly located in region 2 , as well as between the boundaries of regions 1 and 1r , as expected . a small number of atlas sources classified as sfgs are located in regions 1r and 3 , consistent with the migratory paths of pah - dominated sources . the upper bounds classified as sfgs in the top - right panel are consistent with the majority of sfgs being located in region 2 . all 4 atlas sources classified as stars are located in region 3 . the polarized atlas sources detected in all four irac bands follow the distribution of agns , situated predominantly in regions 1 and 3 in almost equal proportions . the upper bounds for polarized sources presented in the top - right panel are consistent with this finding . the lack of polarized sources in region 2 suggests that the polarized atlas sources observed in region 3 are unlikely to be sfgs with rest - frame colours located in region 2 ( i.e. if sfgs are migrating from region 2 to region 3 with redshift , then a trail of sources would be expected ) . instead , we find two concentrations of polarized atlas sources , highly coincident with the regions of parameter space identified by @xcite in which starlight- and continuum - dominated sources were preferentially located . thus we find that the radio emission from polarized atlas sources is most likely powered by agns , where the active nuclei are embedded within host galaxies with mid - infrared spectra dominated by old - population stellar light ( blue irac colours ) or continuum likely produced by dusty tori ( red irac colours ) . this finding is in general agreement with the results from the elais - north 1 ( elais - n1 ) region presented by both @xcite and @xcite , but with the following two notable exceptions . first , both these works identified radio sources ( both polarized and unpolarized ) that were concentrated in region 3 about @xmath26 , @xmath27 , well beyond the parameter space typically occupied by the three generic source classes investigated by @xcite . @xcite reported that these sources were associated with elliptical galaxies dominated by old - population starlight . however , fig . 11 from @xcite indicates that these sources are located within a region of parameter space occupied by individual stars . it is not clear why the irac colours of so many of the radio sources presented by @xcite and @xcite were found to occupy this region of parameter space , though it is possible that their selection of isophotal flux densities for unresolved infrared sources may have biased their colour ratios ( aperture values are more appropriate for point sources ) . and second , unlike these previous works , we do not find any polarized atlas sources in which the radio emission is likely to be powered by star formation ( i.e. we do not see any polarized sources in region 2 ; cf . * ? ? ? * ) ; we can not conclude that any polarized sources have infrared colours suggestive of significant pah emission ( cf . * ? ? ? the fractional polarization properties of atlas agns and sfgs are described in [ ch5:secresidentm ] and modelled in [ ch5:secressubpi ] . in fig . [ ch5:fig : rfir ] we compare the total intensity 1.4 ghz radio to 24 @xmath1 m infrared flux densities for all atlas dr2 sources , taking into account infrared upper bounds for all radio sources without detected infrared counterparts . as noted in paper i , we use 24 @xmath1 m flux density as a proxy for far - infrared ( fir ) flux density . the bottom - left panel indicates that atlas sources classified as agns are prevalent both away from and on the fir - radio correlation ( frc ) . the presence of a substantial number of agns below the dashed line demonstrates the value of using multiple diagnostic criteria to classify sources ; only sources above the dashed line have been classified as agns using the radio to far - infrared diagnostic . the bottom - right panel indicates that , as expected , atlas sources classified as sfgs typically cluster along the frc and have radio flux densities @xmath5 1 mjy . however , a small number of sources classified as sfgs ( and stars ) are observed with upper bounds clearly located within the agn parameter space . the top - left panel highlights all 130 polarized atlas sources , indicating that each of these was classified as an agn . no polarized stars or sfgs were detected in our data . we now present diagnostics of components , groups , and sources in atlas dr2 , resulting from the linear polarization@xmath28total intensity cross - identification and classification procedures described in 6.2 of paper i ( summarised in [ sec:1 ] of this work ) . in this section we focus on a number of parameter spaces in which we detail relationships between the polarized flux densities , fractional polarizations , classifications , and angular sizes of sources and their constituents . in fig . [ ch5:fig : fracpolraw ] we plot the polarized flux densities and fractional polarizations for all atlas dr2 components , groups , and sources versus their total intensity flux densities , taking into account polarization upper limits . the fractional polarization uncertainties displayed in the lower - left panel were estimated following standard error propagation as @xmath29 fig . [ ch5:fig : fracpolraw ] shows that the polarization upper limits for components and sources are distributed almost identically , the reason being that the majority of unpolarized sources comprise a single component ( relevant statistics are detailed toward the end of this section ) . regarding polarization detections , we find that all components , groups , and sources exhibit @xmath30 . this finding is in contrast to the data from other 1.4 ghz polarization surveys . @xcite found that 1% ( 381/38454 ) of polarized sources in the nrao vla sky survey ( nvss ) exhibited @xmath31 . @xcite and @xcite found that 10% ( 8/83 ) and 7% ( 10/136 ) of polarized sources in the elais - n1 field exhibited @xmath31 , respectively . @xcite found that 10% ( 84/869 ) of polarized sources throughout the two australia telescope low - brightness survey ( atlbs ) fields exhibited @xmath31 . if a population of extragalactic sources with high 1.4 ghz fractional polarizations were to exist , then it would be unexpected for such sources to be detected in the surveys above [ with @xmath32 full - width at half - maximum ( fwhm ) beam sizes ] yet undetected in this work ( with @xmath3 fwhm beam size ) , because the former are more susceptible to both beam and bandwidth depolarization . instead , we attribute the lack of atlas sources with @xmath33 to our careful treatment of local ( rather than global ) root mean square ( rms ) noise estimates , in particular to our employment of blobcat s flood - fill technique for extracting polarized flux densities ( see @xcite for details regarding biases introduced through gaussian fitting ) , and to our statistical classifications of unresolved and resolved components . we found through testing that components with abnormally high levels of fractional polarization ( up to and even beyond 100% ) could be obtained if the features above were not taken into account . in fig . [ ch5:fig : fracpolclass ] we plot the polarized flux densities and fractional polarizations for all atlas dr2 sources only , indicating their infrared / optical classifications . panels in the left column highlight polarized sources with infrared counterparts detected in all four irac bands , or otherwise . we split those detected in all four bands into sources located within or just beyond region 3 in the lower - right panel of fig . [ ch5:fig : nircc ] [ i.e. polarized sources with @xmath34 and those located within or just beyond region 1r [ i.e. @xmath35 . we find no clear distinction between the fractional polarization properties of sources with blue ( region 3 ) or red ( region 1r ) mid - infrared colours . it is possible that the region 3 polarized sources exhibit a larger dispersion in fractional polarization than the region 1r polarized sources ( compare range of observed fractional polarizations in lower - left panel of fig . [ ch5:fig : fracpolclass ] ) , though given our sample size this marginal effect may be attributed to sampling variance . using data from @xcite , @xcite found that polarized sources in region 3 were more highly polarized than those in region 1r ; the atlas data do not support this result . the distributions of upper limits presented in the right - column panels of fig . [ ch5:fig : fracpolclass ] indicate that all sources with @xmath36 have been classified as agns . the fractional polarization upper limits for sources classified as sfgs are not particularly restrictive , as their total intensity flux densities are typically @xmath5 1 mjy . characteristic @xmath37 levels for the sub - millijansky sfg population are @xmath5 @xmath38 . focusing on the lower - left panel of fig . [ ch5:fig : fracpolclass ] , we note that a general observational consequence of the rising distribution of fractional polarization upper limits with decreasing total intensity flux density is that the mean or median fractional polarization of _ detected _ polarized sources will always _ appear _ to increase with decreasing flux density . this increase represents a selection bias ; it is not possible to detect low levels of fractional polarization for the faintest total intensity sources . any changes to the underlying distribution of fractional polarization with decreasing total intensity flux density will be masked , and thus dominated , by this selection bias . therefore , it is not possible to investigate the distribution of fractional polarization at faint flux densities without accounting for polarization non - detections . recently , 1.4 ghz polarimetric studies of the elais - n1 field @xcite and atlbs fields @xcite concluded that their observational data demonstrated an anti - correlation between fractional polarization and total intensity flux density . these studies found that sources with @xmath14 @xmath5 @xmath39 mjy were more highly polarized than stronger sources . however , @xcite did not account for polarization upper limits , leading to their misinterpretation of selection bias as an indication of true anti - correlation . @xcite accounted for selection bias using monte carlo analysis , effectively incorporating polarization upper limits . @xcite accounted for selection bias by comparing samples of sources in bins of polarized flux density rather than total flux density , at sufficient polarized flux densities to neglect upper limits . however , the findings of increased fractional polarization at faint total flux densities by @xcite and @xcite appear to be reliant on the increasing number of sources observed with @xmath33 at these faint levels . for example , both studies reported extreme sources with @xmath40 , but only at faint total intensities . both @xcite and @xcite found that @xmath41 of polarized sources with linearly polarized flux densities @xmath42 mjy ( i.e. a significant proportion of these sources ) exhibited @xmath33 , while no sources with such high levels of fractional polarization were found for @xmath43 mjy . as described earlier , the @xmath33 sources ( and perhaps many with lower @xmath37 ) are likely to reflect rms noise estimation and source extraction errors . the analytic form assumed by @xcite for the distribution of fractional polarization ( which will be described in [ ch5:secressubpi ] ) may have also contributed to their conclusion regarding increased fractional polarization ; spurious conclusions may be obtained if the observed fractional polarization data do not follow the assumed analytic form of the fit . the arguments above suggest that existing evidence for an anti - correlation between fractional polarization and total flux density may not be robust . similar to the studies above , earlier works by and @xcite concluded that nvss @xcite sources exhibited an anti - correlation between fractional linear polarization and total intensity flux density . these analyses in effect incorporated polarization upper bounds ( though not upper _ limits _ ; see * ? ? ? * ) because @xcite recorded a linearly polarized flux density for each nvss source , regardless of the statistical significance of the polarization measurement . to determine the significance of their findings and thus form a conclusion regarding evidence for anti - correlation , which we use to justify our own fractional polarization model presented in [ ch5:secressubpi ] , we need to examine their works in more detail . and @xcite presented fractional polarization distributions for steep- and flat / inverted - spectrum nvss sources in four flux density intervals : @xmath44 , @xmath45 , @xmath46 , and @xmath47 mjy . their distributions are remarkably consistent for @xmath48 , exhibiting a log - normal form with approximately equal dispersion and a peak at @xmath49 . a separate component with a peak at @xmath50 is also present in each distribution , representing sources with polarization dominated by instrumental leakage . we observe that the dispersions of their distributions broaden with increasing flux density , solely due to broadening at @xmath36 . found that the median fractional polarization was larger for the @xmath44 mjy data than for the @xmath47 mjy data , for both steep- and flat - spectrum sources . @xcite found the same result but for steep - spectrum sources only . these results were essentially based on the lack of sources with @xmath51 in the @xmath44 mjy data when compared with the increased presence of such sources at higher flux densities ; a proportional increase in the number of sources with @xmath48 for decreasing flux density was not observed . however , the presence of sources with @xmath51 ( i.e. less than the typical leakage level of @xmath50 ) , and more generally the slight changes in distribution shape observed for @xmath36 between different flux density intervals , may be more appropriately explained by the influence of noise on polarized flux densities rather than by variation in the underlying distribution of fractional polarization . to demonstrate , we first note that the expectation value of @xmath4 for an unpolarized nvss source is given by the mean of a rayleigh distribution ( i.e. a ricean distribution with no underlying polarized signal ) , which is @xmath52 mjy for @xmath53 mjy . this value is also characteristic of the expected observed polarized flux density for a source with true underlying polarized signal @xmath54 @xmath5 @xmath55 ( e.g. see the upper panel of fig . 1 from @xcite ) . thus , a tail of sources with true polarization @xmath54 @xmath5 0.29 mjy will appear in the fractional polarization distribution at @xmath37 @xmath21 0.18% for total intensity sources with @xmath56 mjy , and at @xmath37 @xmath5 0.05% for @xmath57 mjy . these estimates are consistent with the distributions presented by and @xcite ; a tail of sources with @xmath51 was observed for the @xmath47 mjy data but not for the @xmath44 mjy data . we therefore conclude that the results presented by and @xcite do not demonstrate a statistically significant anti - correlation between fractional linear polarization and total intensity flux density . furthermore , we note that the fractional polarization distributions presented in these works are likely to overestimate the population of sources with @xmath36 , even for the @xmath44 mjy data , for the following two reasons . first , all catalogued nvss measurements of polarized flux density were debiased using a modified version of the expectation value for a ricean distribution @xcite . this debiasing scheme is known to impart a significant overcorrection ( i.e. negative bias ) at low snr ( e.g. see ; the relevant scheme is labelled in reference to its application by @xcite ) . thus measurements of fractional polarization obtained using the nvss catalogue are likely to be negatively biased . and second , raw polarization measurements for nvss sources were obtained by interpolation at the total intensity centre position . therefore , polarized flux densities were underestimated for each source in which the spatial peak of polarized emission was located in an adjacent pixel to the total intensity peak . both of these effects could have been largely mitigated by obtaining polarization upper limits for sources , rather than upper bounds . returning to the lower - left panel of fig . [ ch5:fig : fracpolclass ] , we find that the maximum level of fractional polarization exhibited by atlas sources does not appear to be correlated with total intensity flux density . the maximum level appears to be limited to @xmath37 @xmath5 20% for @xmath14 @xmath21 1 mjy , which becomes a strict limit for @xmath58 mjy when accounting for the presence of all upper limits . furthermore , we find @xmath37 @xmath21 0.4% for sources with @xmath14 @xmath21 10 mjy , where sources exhibiting higher levels of fractional polarization significantly outnumber those potentially exhibiting @xmath59 as indicated by the upper limits . in paper i we found that 138 of the total 172 catalogued linearly polarized components exhibited a clear one - to - one match with individual total intensity components . the remaining 34 polarized components required grouping in order to be associated with total intensity counterparts . of the one - to - one associations , we classified 58 as type 0 , 4 as type 1 , 25 as type 2 , 48 as type 4 , and 3 as type 5 . all 3 sources containing type 5 core associations were found to exhibit unpolarized lobes . of the group associations comprising a total of 34 polarized components , 2 groups were classified as type 0 , 14 as type 3 , 1 as type 4 , and 8 as type 6 . there were 29 sources classified as type 6 , 2 as type 7 , and 25 as type 8 . these classifications are catalogued in appendix b of paper i. in fig . [ ch5:fig : fracpoltypes ] we indicate the polarization@xmath28total intensity classifications for all polarized atlas dr2 components , groups , and sources . in the lower - left panel we plot the levels of fractional polarization exhibited by classical double or triple radio sources ( types 68 ) and their individual lobes ( types 02 , respectively ) . we find that sources classified as type 6 , which comprise pairs of roughly equally polarized type 0 lobes , are located throughout most of the populated parameter space . we find that type 7 sources , which comprise pairs of type 1 lobes where one is clearly less polarized than the other , appear to occupy the same parameter space populated by type 6 sources . a selection bias against identifying type 0/1 lobes , and thus type 6/7 sources , is present within the diagonal region of parameter space populated by polarization upper limits ( for visual clarity these limits are not shown in fig . [ ch5:fig : fracpoltypes ] ; see fig . [ ch5:fig : fracpolraw ] ) . type 2 lobes and their parent type 8 sources , which represent ambiguous cases in which it is not possible to differentiate between types 0/1 or 6/7 , are largely confined to this diagonal region . given the observed prevalence of type 6 sources compared with type 7 , it seems likely that more sensitive observations would result in a majority of type 8 sources being reclassified as type 6 . from the lower - right panel of fig . [ ch5:fig : fracpoltypes ] we find that sources classified as type 3 , which exhibit a single polarized component situated midway between two total intensity components , appear to populate the same region of parameter space occupied by type 8 sources . similarly , associations classified as type 5 , which represent cores of triple radio sources , as well as the remaining unclassified sources denoted by type 4 , also appear to be concentrated within the diagonal region of parameter space populated by upper limits . we note that many of the type 4 associations are likely to represent individual type 0 or type 1 lobes of as - yet unassociated multi - component sources , having been erroneously assigned to single - component sources in our catalogue ( note 6.1 of paper i ; statistics regarding polarized multi - component sources are presented below ) . we find that type 5 associations occupy a parameter space consistent with type 6 and type 7 sources . as the latter represent average polarization properties for dual - lobed radio sources , it is possible that type 5 associations also represent dual - lobed structures but with small angular sizes , such as compact steep - spectrum ( css ) sources @xcite . curiously , we found that each of the 3 sources with type 5 cores was found to exhibit unpolarized outer radio lobes . it is possible that the type 5 cores represent restarted agn activity and that the outer lobes are unpolarized because any large - scale magnetic fields within them have dissipated over time since their production during an earlier distinct phase of agn activity . for example , we may be seeing sources similar to the double - double radio galaxy j1835@xmath60620 , though at an earlier stage of evolution where the inner lobes have not yet separated into two separate lobes ( note that fractional polarization levels for the inner lobes of j1835@xmath60620 are higher than for the outer lobes ) . in fig . [ ch5:fig : fracpoltheta ] we plot polarized flux density and fractional polarization versus largest angular size ( las ) for all polarized atlas dr2 sources , highlighted according to morphology and infrared colour . the las for a single - component source is given by its total intensity deconvolved angular size or size upper limit , while the las for a multi - component source is given by the maximum angular separation between its constituent total intensity components . for visual clarity we plot sources with polarization upper limits separately in fig . [ ch5:fig : fracpolthetauls ] , also highlighted according to morphology and infrared colour . note that the apparent anti - correlations between fractional polarization upper limits and angular size upper limits for single - component sources throughout fig . [ ch5:fig : fracpolthetauls ] are spurious ; the restrictiveness of both types of upper limits are intrinsically anti - correlated with total intensity flux density . in fig . [ ch5:fig : fracpoltheta2 ] we again plot polarized flux density and fractional polarization versus las for all polarized sources , but now highlighted according to the polarization@xmath28total intensity classification scheme from 6.2 of paper i. for reference , we note that 1 subtends a linear scale of 1.8 , 3.3 , 6.1 , 8.0 , and 8.5 kpc at redshifts 0.1 , 0.2 , 0.5 , 1.0 , and 2.0 , respectively @xcite , assuming a @xmath61cdm cosmology with parameters @xmath62 km s@xmath63 mpc@xmath63 , @xmath64 , and @xmath65 . following an evolutionary relationship for galaxy sizes given by @xmath66^{-\frac{1}{2 } } \;,\ ] ] and assuming that a typical galaxy has size @xmath67 kpc at @xmath68 ( e.g. * ? ? ? * ) , the corresponding sizes of typical galaxies at the redshifts above are approximately @xcite , may cause observed angular sizes of extended sources to be smaller than true sizes , due to faint source edges . ] 19 , 18 , 16 , 12 , and 7 kpc , respectively , or 10 , 56 , 26 , 15 , and 08 , respectively . we summarise our findings from figs . [ ch5:fig : fracpoltheta][ch5:fig : fracpoltheta2 ] as follows . of the 130 ( 2091 ) polarized ( unpolarized ) sources catalogued in atlas dr2 and presented in fig . [ ch5:fig : fracpoltheta ] ( fig . [ ch5:fig : fracpolthetauls ] ) , 81 ( 74 ) comprise multiple components in total intensity , 40 ( 140 ) comprise a single resolved component in total intensity , and 9 ( 1877 ) comprise a single unresolved component in total intensity . we note that while components observed in linear polarization in atlas dr2 are typically unresolved ( only 29/172 or 17% of polarized components are resolved ; see 5 of paper i ) , 121/130 or 93% of sources exhibiting polarized emission are resolved in total intensity . these statistics support the findings by @xcite that polarized 1.4 ghz sources tend to have structure at arcsecond scales and that , as a consequence , their polarized emission is unlikely to be beamed . combined with our earlier classification from fig . [ ch5:fig : rfir ] of all polarized atlas sources as agns , and our interpretation from fig . [ ch5:fig : fracpoltypes ] that most or all polarized components are associated with agn jets or lobes ( rather than cores ) , the statistics above demonstrate that ( sub-)millijansky polarized sources tend to be extended jet- or lobe - dominated active radio galaxies . this conclusion is supported by the finding from @xcite that polarized sources tend to have steep spectra , which are characteristic of lobes . in fig . [ ch5:fig : fracpolthetauls ] we find that atlas dr2 sources typically have las @xmath5 10 , suggesting that most sources are located at @xmath20 @xmath21 0.2 . this is consistent with the preliminary redshift distributions presented by @xcite and @xcite for atlas dr1 sources ( see also discussion of radio source redshift distribution by * ? ? ? focusing on the panels in the lower - left corners of fig . [ ch5:fig : fracpoltheta ] and fig . [ ch5:fig : fracpolthetauls ] , we find that single- and multi - component sources are distributed approximately equally in fractional polarization space ; their fractional polarization upper limits are not restrictive enough to identify any possible underlying trends . however , having found above that polarized sources are likely to represent lobed galaxies , it is perhaps surprising that we do not find a clear correlation between fractional polarization and las due to beam depolarization . given the @xmath3 resolution of atlas , in general a classical double radio source with dual polarized lobes should exhibit greater fractional polarization than a similar source with smaller las that is observed as a single - component source . a likely explanation may be that a significant number of the polarized single - component sources indicated in fig . [ ch5:fig : fracpoltheta ] are actually individual lobes of as - yet unassociated multi - component sources ( see 6.1 of paper i ) . note that all single - component sources in fig . [ ch5:fig : fracpoltheta ] are classified as type 4 in fig . [ ch5:fig : fracpoltheta2 ] . another potential explanation may be that for dual - lobed sources with small angular size observed as single - component sources , asymmetric depolarization between the lobes @xcite could result in overall source fractional polarization levels similar to those of type 7 sources ( see fig . [ ch5:fig : fracpoltheta2 ] ) , rather than resulting in significantly beam - depolarized ( and thus perhaps unpolarized ) sources overall . the upper limits presented in the left column of fig . [ ch5:fig : fracpolthetauls ] do not reveal any clear underlying trends within or between source classes . the multi - component sources classified as sfgs in fig . [ ch5:fig : fracpolthetauls ] , which are also shown located within the agn parameter space in the lower - right panel of fig . [ ch5:fig : rfir ] , require future study . these may represent composite sources exhibiting both agn and sfg characteristics , for example similar to the ultra - luminous infrared galaxy f001837111 investigated by @xcite or the more general classes of post - starburst quasars ( e.g. * ? ? ? focusing on the right column of fig . [ ch5:fig : fracpoltheta ] , we do not find any angular size distinctions between polarized sources based on their infrared colours . furthermore , we find no underlying trends within the associated upper limit data from the right column of fig . [ ch5:fig : fracpolthetauls ] . focusing on fig . [ ch5:fig : fracpoltheta2 ] , we find that type 6 sources typically extend to greater angular sizes than type 7 sources , though a larger sample size with proportionally fewer type 8 classifications is required to confirm this finding . we also find that each of the 3 polarized cores classified as type 5 are resolved , and that they populate the same region of parameter space as type 4 sources . we present euclidean - normalised differential number - counts derived from the atlas dr2 total intensity and linear polarization component catalogues in fig . [ ch5:fig : countsi ] and fig . [ ch5:fig : countsl ] , respectively , and in tabulated form in appendix a. counts for each bin have been plotted and tabulated at the expected average flux density , which we denote by @xmath69 , as given by equation ( 19 ) from . this value takes into account the number - count slope and becomes important when assigning flux densities for bins containing few counts or with large widths in flux density space ; @xmath69 only equals the bin geometric mean when @xmath70 , where @xmath71 is the slope of the differential number counts @xmath72 . bin widths for all total intensity counts were selected to be a factor of 0.07 dex for @xmath73 mjy , 0.13 dex for @xmath74 mjy , and 0.2 dex otherwise . in linear polarization , bin widths were selected to be a factor of 0.16 dex for @xmath75 mjy , and 0.3 dex otherwise . we removed all bins containing components with visibility area corrections @xmath76 , so as to prevent the number - counts from being dominated by the few components detected in the most sensitive and potentially least - representative regions of the atlas images . ( note that we did not remove individual offending components in order to retain the faintest bins , as this would have led to a bias in their resulting number - counts . ) in total intensity this resulted in the removal of the faintest few bins containing @xmath7730 components from each of the cdf - s component- and bin - corrected datasets , and @xmath7720 components from each of the elais - s1 component- and bin - corrected datasets . the maximum visibility area corrections for any components in the remaining valid cdf - s and elais - s1 bins were @xmath78 and @xmath79 , respectively . in linear polarization , the maximum visibility area corrections for any components in the cdf - s and elais - s1 datasets were @xmath80 and @xmath81 , respectively . as a result , we did not remove any bins in linear polarization . resolution and eddington bias corrections were calculated in 7 of paper i. the former was designed to correct for incompleteness to resolved components with low surface brightness , and for the redistibution of counts between bins resulting from systematic undervaluation of flux densities for components classified as unresolved . the latter was designed to correct for the redistribution of counts between bins due to random measurement errors in the presence of a non - uniformly distributed component population . these bias corrections were calculated in paper i by assuming that the true underlying differential number counts in total intensity were given by the sixth - order empirical fit to the phoenix and first surveys presented by @xcite . this fit , which we denote h03 , is given by @xmath82 = \sum_{j=0}^{6 } a_{j } \left [ \log\left ( \frac{i}{{\textrm}{mjy } } \right)\right]^{j}\,,\ ] ] with @xmath83 , @xmath84 , @xmath85 , @xmath86 , @xmath87 , @xmath88 , and @xmath89 . to illustrate the potential boosting effects of an exaggerated population of faint components , paper i also defined a modified h03 distribution , denoted h03 m , in which a euclidean slope was inserted between 30@xmath28300@xmath1jy , @xmath90 for bias corrections in linear polarization , we modelled the true underlying differential number counts @xmath91 by convolving the total intensity h03 distribution from equation ( [ ch4:eqn : h03 ] ) with a probability distribution for fractional linear polarization @xmath92 , which we denote @xmath93 . the @xmath94 distribution is presented in equation ( [ ch5:eqn : fracpol ] ) in [ ch5:secressubpi ] . the atlas dr2 component counts extend down to a flux density of approximately 140 @xmath1jy in both total intensity and linear polarization . the brightest flux density bins are sparsely sampled because the atlas survey areas are not large enough to include significant numbers of increasingly rare bright components . in both fig . [ ch5:fig : countsi ] and fig . [ ch5:fig : countsl ] we find that the number - counts from the two separate atlas fields are consistent within the errors over their full observed flux density ranges . the impacts of the combined resolution and eddington bias corrections on the number - counts appear to be relatively minor . in total intensity , the two corrections largely cancel each other out , while in linear polarization the resolution bias corrections dominate . in both total intensity and linear polarization , the combined corrections affect the underlying visibility area corrected counts by a factor of @xmath5 0.5 , and do not affect the counts for @xmath13 @xmath21 3 mjy . we find that differences between the two independent eddington bias correction schemes are largely negligible for both the total intensity and linear polarization number - counts , providing confidence in these approaches . in fig . [ ch5:fig : countsi ] we find that the total intensity counts closely follow the h03 model within a factor of @xmath95 , though the atlas counts may begin to systematically drop below the h03 model for @xmath13 @xmath5 @xmath96 mjy . it is likely that the drop is caused by residual incompleteness in our resolution bias corrections , in turn caused by uncertainties regarding our assumed true angular size distribution for @xmath97 as discussed in 7.1 of paper i. however , we note that if we assume that the model presented in fig . 19 of paper i is the best representation of the true angular size distribution ( without any flux density scaling ) , then the faintest bins at @xmath98 @xmath1jy only require an additional correction factor of at most approximately @xmath6030% . the faintest bins are therefore consistent with the h03 model . as we do not find any systematic divergence between the atlas total intensity counts and the h03 model at the faintest flux densities ( when accounting for the suspected residual resolution bias described above ) , we confirm that the h03 model is suitable for predicting 1.4 ghz component counts ( and source counts as described below ) down to at least @xmath0 @xmath1jy in surveys with resolution fwhm @xmath3 . should we have found a systematic divergence , it would have indicated that our predicted eddington bias corrections were unrealistic , and that in turn the h03 model underpinning these corrections formed an increasingly poor representation of the true number - counts for decreasing flux density . under this hypothetical situation , an iterative approach would have been required in order to correctly identify an input true number - count model so as to bring about convergence with the fully corrected observed counts . in 7.2 of paper i we predicted the levels of eddington bias that would be present within the observed atlas counts if the true counts were given by the h03 or h03 m models [ the latter model contains a larger population of components with @xmath99 mjy than the former ; see equation ( [ ch4:eqn : h03 m ] ) ] . we predicted that the h03 m model would induce significantly greater eddington bias at @xmath99 mjy than the h03 model ( see fig . 23 in paper i ) . therefore , if the h03 model was used to predict the observed eddington bias when in fact the h03 m model best represented the true counts , then the observed counts would exhibit significant positive residual eddington bias ; if vice versa , the residual bias would be negative . given that we do not observe a systematic rise ( or fall ) at faint flux densities in the fully corrected atlas counts ( again accounting for the suspected residual resolution bias described above ) , we conclude that the h03 m model is not supported by the atlas data . we note that the resolution bias corrections applied in this work are practically insensitive to changes between the h03 and h03 m models . this is because for any given flux density bin , the resolution bias corrections are unaffected by the assumed form of the number - count distribution at fainter flux densities . therefore , assuming that our resolution bias corrections are appropriate to begin with , we can focus on eddington bias alone in order to draw the conclusions described above . below a flux density of @xmath100 mjy , we expect the atlas total intensity component counts to be dominated by single - component sources , with negligible contributions from components within multi - component sources . while we are unable to explicitly quantify this expectation given present data , we note that conservatively @xmath101 of all 2416 atlas components reside within multi - component sources ( this fraction takes into account the number of components estimated to reside within as - yet unassociated multi - component sources ; see 6.1 of paper i ) . we expect that most of these multi - component sources represent frii sources , which are known to dominate the source counts at flux densities @xmath21 10 mjy and which diminish significantly below @xmath102 mjy ( e.g. * ? ? ? * ) . at sub - mjy levels , radio sources in general are expected to have angular sizes @xmath103 ; these are likely to be observed as single - component sources in atlas . therefore , we conclude that the atlas component counts may act as a suitable proxy for source counts at sub - mjy levels . we note that our characterisation of the faint component / source population using the h03 model in this work has relied on the similar resolutions of the atlas and phoenix surveys . should these resolutions have differed significantly , so too would have the properties of their observed components . @xcite obtained their model by using a sixth - order fit to the observed component counts from the phoenix survey , supplemented at @xmath104 mjy by source counts from the first survey @xcite . the h03 model was thus intended to characterise source counts at all flux densities , despite being derived from a component catalogue at faint flux densities . for @xmath104 mjy , the atlas total intensity component counts follow the h03 model and thus the first source counts . we explain this correspondence as follows by first presenting results that examine how source and component counts are expected to differ . given that frii sources dominate the source counts above @xmath2 mjy and that these sources are likely to comprise multiple components within a survey such as atlas , we expect the differential counts for sources to rise and extend to brighter flux densities than those for components . to roughly illustrate this behaviour and examine the difference between source and component counts in general , we considered an idealised scenario in which all sources were assumed to comprise two identical components , each with half the flux density of their parent . for illustrative purposes we assumed that the component count distribution was given by the h03 model . to derive the idealised differential source counts , we integrated the differential component counts to obtain integral component counts , divided these integral counts by two , doubled the flux density scale , and differentiated . for completeness , we also derived differential source counts in linear polarization by following a similar procedure , where the relevant differential component counts were assumed to follow the @xmath93 model . we present the resulting total intensity and linear polarization source counts in fig . [ ch5:fig : ratiosc ] . we find that the predicted source counts remain within @xmath105 of the component counts across the flux density ranges probed by the atlas data in total intensity ( @xmath14 @xmath5 1 jy ) and linear polarization ( @xmath4 @xmath5 100 mjy ) . ( separately , while not shown , we note that the integral counts for both components and sources within our rudimentary model are very similar , for both total intensity and linear polarization . ) as expected , at bright flux densities the component counts drop below the source counts , though these drops occur at brighter flux densities than relevant to the atlas data . note that in reality , the differential source and component counts are likely to overlap more closely than presented in fig . [ ch5:fig : ratiosc ] because of the presence of single - component sources . thus we conclude that for surveys with resolution fwhm @xmath3 similar to phoenix and atlas , the h03 model may be used to characterize both component and source counts in total intensity for @xmath13 @xmath5 1 jy . we conjecture that , as modelled above , the h03 model characterises component rather than source counts at all flux densities , including at @xmath106 jy . to justify this claim , we note that components in the first survey were only grouped into multi - component sources if they were located within 50@xcite . from fig . [ ch5:fig : fracpoltheta ] of this work we can see that a cutoff of 50 is likely to be too small to capture sources with the most widely - separated components , which are also likely to be the brightest sources . in addition , flux densities for extended first components are likely to be underestimated due to insensitivity to extended emission . therefore , the first source counts are likely to be deficient at the brightest flux densities . incidentally , the first source counts and thus the h03 model appear to form a suitable hybrid distribution for describing component counts at all flux densities in surveys with resolution fwhm @xmath3 such as atlas . we may therefore conclude that the @xmath93 model is suitable for characterising component counts in linear polarization at all flux densities , not just at @xmath4 @xmath5 100 mjy where differences between polarized component and source counts are likely to diminish as shown in fig . [ ch5:fig : ratiosc ] . if the h03 model were to better represent source counts rather than component counts at @xmath107 jy , then the polarized counts resulting from convolution with @xmath108 would reside ambiguously between a component and source count distribution for @xmath4 @xmath21 5 mjy . thus it would be inappropriate to estimate integral component or source counts from the @xmath93 ( or indeed h03 ) model ; this point is relevant to results presented shortly . in fig . [ ch5:fig : countsl ] we find that the atlas linear polarization component counts steadily decline with decreasing flux density , as generally predicted by all four models displayed in the background . the solid curve displays our assumed true component count model , namely @xmath93 , which we used to derive the corrections for resolution and eddington bias . the fully corrected atlas counts closely follow this model within statistical error , indicating consistency between the model , the corrections , and the observational data . each of the four background models in fig . [ ch5:fig : countsl ] were calculated by convolving the h03 model with a fractional polarization distribution . we note that these convolutions are only appropriate because , as described above , the h03 model appears to appropriately characterise the total intensity component counts at all flux densities relevant to atlas . in [ ch5:secressubpi ] we describe each of the fractional polarization distributions underlying the four background models , and compare their abilities to predict the atlas polarized counts and polarization data in general . the number of polarized components expected per square degree at or brighter than a given flux density , as constrained by the observed atlas component counts , can be estimated by integrating the @xmath93 polarized count distribution ( the solid curve in fig . [ ch5:fig : countsl ] ) . the resulting integral component counts are displayed in fig . [ ch5:fig:17 ] . we estimate that the sky density of polarized components for @xmath109 @xmath1jy is 30 deg@xmath110 , for @xmath111 @xmath1jy it is 50 deg@xmath110 , and for @xmath112 @xmath1jy it is 90 deg@xmath110 . if we make the rudimentary assumption described earlier regarding fig . [ ch5:fig : ratiosc ] that every polarized component belongs to a dual - component source with double the flux density , we can estimate the integral source count distribution ; this is displayed alongside the integral component count distribution in fig . [ ch5:fig:17 ] . we thus estimate that the sky density of polarized sources for @xmath109 @xmath1jy is @xmath113 deg@xmath110 , and for @xmath111 @xmath1jy it is @xmath114 deg@xmath110 . we expect that these integral source count estimates are accurate to within 10% , even if a more suitable model incorporating polarized single - component sources is utilised . in this section we present a model to describe the distribution of fractional polarization for agn sources and their components / groups observed at 1.4 ghz in surveys with resolution fwhm @xmath21 10 , as constrained by the atlas dr2 data . there appears to be a significant overlap between the fractional polarization properties of all classification types representing both components / groups and sources in fig . [ ch5:fig : fracpoltypes ] . taking into account the presence of upper limits ( see fig . [ ch5:fig : fracpolraw ] ) , we find that typical levels of fractional polarization are concentrated between 0.4% and 20% , regardless of whether the focus is on sources or on their constituent components / groups . given this apparent overlap , we assume for simplicity that the distribution of fractional polarization for both components / groups and sources can be modelled using the same pdf , which we denote by @xmath94 . before presenting our model for this distribution , we note three caveats . first , following our conclusions presented in [ ch5:secresidentm ] regarding potential correlation of the distribution of fractional polarization with total flux density , we assume that @xmath94 is independent of total intensity flux density . this assumption may not be suitable for @xmath14 @xmath5 @xmath39 mjy for which our atlas data become sparse . second , our model for @xmath94 may only be relevant for surveys with resolution fwhm @xmath21 10 . surveys with finer resolution may encounter less beam depolarization across components , and thus recover higher average levels of fractional polarization ( in 5 of paper i we found that @xmath115 of polarized atlas components were resolved ) . we note that surveys with coarser resolution will incur increased blending between components within multi - component sources , resulting in a greater number of low-@xmath37 sources than observed for atlas due to enhanced beam depolarization . and third , given that all polarized components in atlas dr2 are associated with agns , we restrict our model for @xmath94 to the characterisation of agns , rather than the characterisation of all radio sources including sfgs and individual stars . we do not attempt to differentiate between different types of agns or their components within our model , i.e. fri / frii / radio quiet / core / lobe . we discuss fractional polarization levels for sfgs in [ ch5:secdiscsfg ] . we modelled @xmath94 by qualitatively fitting two independent sets of atlas data : ( i ) the fractional polarizations of components , groups , and sources displayed in fig . [ ch5:fig : fracpolraw ] , importantly taking into account upper limits , and ( ii ) the differential number - counts for polarized components displayed in fig . [ ch5:fig : countsl ] . we obtained a concordance fit to these data by modelling @xmath94 using a log - normal distribution , @xmath116^{{\scriptscriptstyle}2 } } { 2 \sigma_{{\scriptscriptstyle}10}^{{\scriptscriptstyle}2}}\bigg\ } \,,\ ] ] where the parameters @xmath117 and @xmath118 are the median fractional polarization and scale parameter , respectively , given by best - fit values @xmath119 and @xmath120 . the fit given by equation ( [ ch5:eqn : fracpol ] ) is consistent with the result obtained by analysing the fractional polarization data alone , using the product - limit estimator @xcite as implemented within the survival package in the r environment . the mean level of fractional polarization for the distribution in equation ( [ ch5:eqn : fracpol ] ) is given by @xmath121 , which equates to @xmath122 . for values of @xmath117 or @xmath118 larger than the best - fit values above , we found that the @xmath93 model predicted differential counts in excess of the observed atlas counts . for smaller values , the predicted counts were deficient . we plot equation ( [ ch5:eqn : fracpol ] ) in fig . [ ch5:fig : pimodels ] . for comparison we also plot the 1.4 ghz fractional polarization distributions proposed by @xcite , @xcite , and @xcite . for clarity we explicitly document each of these distributions , as follows . @xcite investigated the distribution of fractional polarization for nvss sources with @xmath123 mjy , which they fit using the following quasi log - normal form , @xmath124^{{\scriptscriptstyle}2 } } { 2 \sigma_{{\textrm}{\tiny b04}}^{{\scriptscriptstyle}2}}\bigg\ } \,,\ ] ] where @xmath125 @xmath126 and where @xmath127 . the median and mean fractional polarization levels of the @xmath128 distribution are 2.1% and 3.3% , respectively . similarly , @xcite investigated the distribution of fractional polarization for nvss sources with @xmath129 mjy , which they fit using the following monotonic form , @xmath130^{{\scriptscriptstyle}-1 } + b_{{\textrm}{\tiny t04}}\right\}\ ] ] where @xmath131 , @xmath132 and where we have included a correction factor of 1.32 to ensure that the distribution is normalised . the median and mean fractional polarization levels of the @xmath133 distribution are 2.1% and 2.7% , respectively . @xcite fit the distribution of fractional polarization for sources with @xmath134 mjy in the elais - n1 field by modifying a gram - charlier series of type a ( e.g. * ? ? ? * ) , resulting in the following monotonic form , @xmath135\bigg\ } & \\ \hspace{4 cm } \textrm{if $ i<30$~mjy}\\ f_{{\textrm}{\tiny b04}}\left(\pi\right ) & \\ \hspace{4 cm } \textrm{if $ i\ge30$~mjy}\,,\\ \end{array } \right.\ ] ] where @xmath136 , @xmath137 , and where we have included a correction factor of 11.06 to ensure that the distribution is normalised . for @xmath138 mjy , @xcite found that the elais - n1 data were consistent with the @xmath128 distribution from equation ( [ ch5:eqn : fracpolb04 ] ) . the median and mean fractional polarization levels of the @xmath139 distribution for @xmath134 mjy are 4.8% and 6.0% , respectively . the four curves presented in fig . [ ch5:fig : pimodels ] are replicated in figs . [ ch5:fig : fracpolraw][ch5:fig : fracpoltheta ] and fig . [ ch5:fig : fracpoltheta2 ] . the four curves are also presented in fig.s [ ch5:fig : countsl ] and [ ch5:fig : countsl2 ] following convolution with the h03 differential count model . in fig . [ ch5:fig : countsl2 ] we find that the fractional polarization distributions proposed by @xcite , @xcite , and @xcite are in general agreement with the observed atlas polarized number counts . the models predict polarized counts that are within a factor of 5 of each other , and they all pass within a few standard errors of the atlas data points . however , we find that these three distributions are incompatible with the observed distribution of fractional polarization for atlas components , groups , sources , and in particular upper limits as presented in fig . [ ch5:fig : fracpolraw ] . the extended tails below @xmath36 for the distributions proposed by @xcite and @xcite are likely to reflect the various systematic biases we described earlier in [ ch5:secresidentm ] regarding the nvss data . polarized flux densities for nvss sources were recorded regardless of whether or not the measurements met statistical criteria for formal detection . if upper limits were calculated for the nvss data following a similar procedure to that described for the atlas data in 6.2 of paper i , then we suspect that far fewer detections strictly implying @xmath36 would have been made . we note that the @xmath108 model proposed in this work peaks at @xmath140 , which is consistent with the nvss data for @xmath48 from and @xcite . the extended tail below @xmath36 in the @xcite model reflects their assumption that the distribution peaks at @xmath141 and declines monotonically with increasing @xmath37 . the atlas dr2 data do not support this assumption . as noted earlier , a caveat of the @xmath108 model is that it may not be suitable for @xmath14 @xmath5 @xmath39 mjy , because the upper limits presented in fig . [ ch5:fig : fracpolraw ] do not constrain the behaviour of the true fractional polarization distribution for low values of @xmath37 . however , given that the maximum level of fractional polarization exhibited by atlas components and sources appears to be limited to @xmath37 @xmath5 @xmath142 , and given that this limit appears to be uncorrelated with flux density down to at least @xmath100 mjy ( see comments regarding fig . [ ch5:fig : fracpolclass ] in [ ch5:secresidentm ] ) , we may draw tentative conclusions regarding the true distribution of fractional polarization for @xmath143 @xmath5 @xmath14 @xmath5 @xmath39 mjy . the atlas dr2 data are consistent with 3 general alternatives . first , the @xmath108 distribution may remain unchanged for @xmath144 mjy . second , for decreasing @xmath14 , the mean of @xmath108 may decrease while its dispersion increases so as to maintain an approximately constant level of fractional polarization for outliers with large @xmath37 . and third , for decreasing @xmath14 , the mean of @xmath108 may increase while its dispersion decreases . more sensitive observations are required to distinguish between these alternatives . in fig . [ ch5:fig : ratiosc ] of [ ch5:secressubcnts ] we demonstrated that differences between differential number - counts of components and sources within a survey such as atlas are likely to be negligible below @xmath100 jy in total intensity , and below @xmath0 mjy in linear polarization . we may therefore directly compare the atlas dr2 component counts with source counts from the literature in both total intensity and linear polarization . we present these comparisons in the following two sections . in fig . [ ch5:fig : countsi2 ] we compare the atlas dr2 bin - corrected total intensity component counts ( from fig . [ ch5:fig : countsi ] or tabulated data from appendix a ) with source counts from other 1.4 ghz surveys of comparable sensitivity . these include the b1301@xmath603034 field @xcite , the hdf - n , lockman hole , and elais - n2 fields @xcite , the ssa13 field @xcite , the j1046@xmath605901 field ( * ? ? ? * hereafter om08 ) , a revised survey of the lockman hole @xcite , the goods - n field @xcite , the cdf - s field observed with the vla @xcite , and the atlbs fields with counts at @xmath145 mjy @xcite and deeper counts @xcite . at high flux densities the source counts are dominated by luminous radio galaxies and quasars . the flattening of the source counts below 1 mjy is produced by the emerging dominance of a population of sources comprised of radio - quiet agns ( agns lacking significant jets and dominated in the radio band by non - thermal emission ) , low - power radio - loud agns , and star forming galaxies @xcite . the extent to which the source counts flatten is somewhat controversial because counts from deep surveys appear to exhibit a large degree of scatter , for example as seen in fig . [ ch5:fig : countsi2 ] where there is a factor of 2 variation in the counts below 1 mjy . measurements at 3 ghz from the absolute radiometer for cosmology , astrophysics , and diffuse emission ( arcade ) 2 balloon - borne experiment have indicated a temperature for the radio background about five times that previously expected from known populations of radio sources @xcite , which if not due to a residual calibration error @xcite suggest the presence of a new population of faint ( @xmath146 @xmath1jy at 1.4 ghz ) or diffuse ( few mpc in extent ) extragalactic sources @xcite . some studies have attributed the large scatter in the faint counts to cosmic variance , namely to intrinsic differences between survey fields caused by large scale structure ( e.g. * ? ? ? * ) . however , significant differences in the counts for fields observed in separate studies , such as the goods - n field ( located within the hdf - n field ) or the lockman hole ( see fig . [ ch5:fig : countsi2 ] ) , indicate that data processing and calibration errors may be entirely responsible for the scatter ( e.g. * ? ? ? by considering the consistent power - law form of the angular correlation function for both nvss and first sources obtained by @xcite and , @xcite estimated the cosmic variance for millijansky radio sources to be @xmath147 where @xmath148 is the survey area in square degrees . the total variance for each source count bin containing @xmath149 sources is then given by @xmath150 , which includes poisson chance . for a survey with @xmath151 deg@xmath11 and @xmath152 , cosmic variance contributes @xmath153 to the total rms uncertainty for each bin ( this is consistent with a similar estimate presented by @xcite and a more detailed analysis by @xcite ) . the clustering behaviour of sub - millijansky sources is likely to be similar to that of millijansky sources , or perhaps even less clustered @xcite , in which case the cosmic variance contribution estimated above represents a conservative upper limit . the error bars for many of the faintest counts in fig . [ ch5:fig : countsi2 ] require enlargement by factors much larger than @xmath154 to become consistent with each other within a few standard errors . our experience in constructing source counts for atlas suggests to us that there are a large number of data processing procedures that , if not carefully implemented , could easily give rise to significant systematic biases of order the observed scatter in the faint counts . we therefore agree with previous conclusions in the literature that the observed scatter in the sub - millijansky counts is likely to be significantly affected by data processing differences between surveys . the atlas data support the h03 model down to @xmath0 @xmath1jy and rule out any flattening above this level ; flattening similar to the h03 m model is ruled out by a lack of residual eddington bias . however , the dr2 data are not sensitive enough to support or refute the general trend of flattening reported by deeper surveys . recently , @xcite used the probability of deflection technique [ @xmath155 ; @xcite ] and a spectral index conversion to investigate the behaviour of the 1.4 ghz source counts at @xmath156 @xmath1jy within a confusion - limited observation of the om08 j1046@xmath605901 field at 3 ghz . by combining the results from a similar @xmath155 analysis performed by mc85 , @xcite ruled out any flattening or an upturn in the 1.4 ghz euclidean counts between 2 @xmath1jy and 100 @xmath1jy , such as that reported by om08 or proposed to account for the arcade 2 results @xcite . in fig . [ ch5:fig : countsl2 ] we compare the atlas dr2 bin - corrected linear polarization component counts ( from fig . [ ch5:fig : countsl ] or tabulated data from appendix a ) with the 1.4 ghz polarized source counts from the nvss @xcite and the elais - n1 field ( @xcite ; deeper counts from @xcite ) . the atlas dr2 counts improve upon the @xcite study by a factor of @xmath157 in sensitivity . the observed number - counts from the @xcite , @xcite , and @xcite studies are in general agreement with the atlas counts , though the atlas data do not exhibit flattening at faint levels that might otherwise lead to suggestions of increasing levels of fractional polarization with decreasing flux density or perhaps the emergence of a new source population . the flattening of the data from these studies are unlikely to be real , but rather probably reflective of spurious populations of sources with abnormally high levels of fractional polarization as described earlier in [ ch5:secresidentm ] . this explains the difficulty encountered by @xcite in attempting to model the flattening . similarly , despite the apparent consistency between the observed counts and the various predicted curves displayed in fig . [ ch5:fig : countsl2 ] , in [ ch5:secressubpi ] we found that the fractional polarization distributions presented by @xcite , @xcite , and @xcite were inconsistent with the atlas data and therefore unlikely to be suitable for population modelling . the flux density range over which the @xmath93 model extends in fig . [ ch5:fig : countsl2 ] corresponds to the brightest regions of the total intensity counts , in which luminous radio galaxies and quasars dominate . this is consistent with our independent conclusion from [ ch5:secresidentm ] that the polarized sources contributing to the atlas counts tend to be fri / ii radio galaxies , and with our earlier findings regarding the infrared colours of polarized sources from [ ch5:secres:2 ] . to fully confirm this picture , luminosity functions for polarized sources of different classifications need to be constructed ( e.g. fri / frii / radio - quiet ) , which can then be compared with theory ( e.g. * ? ? ? very recently , while we were finalising this manuscript for resubmission , @xcite published a similar study of faint polarized sources . @xcite presented 1.4 ghz linearly polarized integral ( not differential ) source counts from the goods - n field , observed with the vla at @xmath158 resolution . their polarized counts extend to 20 @xmath1jy , an order of magnitude deeper than our atlas dr2 results . qualitatively , their results are consistent with our main finding that the fractional polarization levels of faint sources are not anti - correlated with total flux density . quantitatively , however , their results are discrepant with ours . @xcite predict that surveys with @xmath159 resolution will observe a polarized source density of 22 deg@xmath110 for @xmath111 @xmath1jy ; this is a factor of 2 lower than the directly observed counts presented in this work . explain that it is difficult to directly compare polarized source counts from surveys with @xmath158 and @xmath159 resolutions , but they consider a factor of 2 difference to be optimistically large ( this factor was used to form their @xmath159 resolution prediction ) . given that @xcite use peak surface brightness measurements as a proxy for flux density irrespective of source angular size , that they do not attempt to correct their data for effects such as resolution bias , and that they do not present total intensity counts with similar processing as for their linear polarization counts , it is difficult to assess the robustness of their results here . detailed assessment is beyond the scope of this work , and is better suited to future studies when results from other deep polarization surveys or modelling efforts become available . we did not detect any polarized sfgs in this work . the fractional polarization upper limits for individual sfgs presented in the lower - right panel of fig . [ ch5:fig : fracpolclass ] indicate that characteristic @xmath37 levels for the sub - millijansky ( @xmath14 @xmath21 100 @xmath1jy ) sfg population are likely to be typically less than @xmath160 . given that sfgs are only expected to begin contributing significantly to the total intensity source counts at @xmath14 @xmath5 100 @xmath1jy ( e.g. see fig . [ ch5:fig : fracpolclass ] ; see also * ? ? ? * ; * ? ? ? * ) , the limit above indicates that the @xmath93 model is unlikely to be affected by the presence of sfgs unless @xmath4 @xmath5 60 @xmath1jy . our limit of @xmath161 is consistent with the fractional polarization distribution for 1.4 ghz sfgs predicted by @xcite ; see panel ( b ) of fig . 6 from their work . @xcite modelled the integrated polarized emission of spiral galaxies , finding typical fractional polarization levels of @xmath162 with overall mean level @xmath163 . the @xmath93 number - counts predicted in this work are therefore likely to represent an upper limit to the true polarized number - counts at @xmath1jy levels , due to the diminished mean level of fractional polarization for faint radio sources with respect to the @xmath108 model . more sensitive observations are required to detect polarized emission from faint sfgs and to quantify their polarization properties . @xcite and @xcite discovered that double radio sources depolarize less rapidly with increasing wavelength on the side with the brighter ( or only ) radio jet than on the opposite side , providing strong evidence that the apparent one - sided nature of jets in otherwise symmetric radio galaxies and quasars is caused by relativistic beaming . this ` laing - garrington ' effect is typically interpreted as being caused by orientation - induced path - length differences through a foreground , turbulent , magnetised intragroup or intracluster medium which surrounds the entire radio source , where the approaching side is seen through less of this medium . however , this interpretation is not unambiguous . the asymmetric depolarization effect may be contaminated or even dominated by depolarization internal to the lobes @xcite , a sheath mixing layer at the interface where relativistic and thermal plasmas meet @xcite , draping of undisturbed intracluster magnetic fields over the surface of a lobe expanding subsonically @xcite or supersonically @xcite , or by intrinsic asymmetries in local environment which act separately or in addition to orientation - induced depolarization @xcite . in reality , it is likely that each of the mechanisms above may contribute , requiring a ` unification scheme ' to predict which will dominate for any given source . for example , @xcite describes an emerging picture that differentiates between properties expected for fri and frii sources , and that includes an inner depolarization region associated with shells of dense thermal plasma around the radio jets in addition to the undisturbed intergalactic medium surrounding the source . however , this picture does not yet include variations on the general orientation - induced depolarization effect due to source environment asymmetries , such as the correlation between lobe properties and optical line emission described by @xcite . if the laing - garrington effect is caused predominantly by source orientation , rather than asymmetries in source environment , then we expect the fractional surface density of sources exhibiting asymmetric depolarization in a volume - limited sample to approximately relate to the fraction of randomly - oriented sources with @xmath13 @xmath21 10 mjy that are pointed towards earth . we justify this expected relationship by noting that frii sources are dominated by un - beamed lobe emission rather than jet emission which may be beamed , frii sources dominate fri sources in flux - limited samples for @xmath13 @xmath21 10 mjy @xcite , and the median redshift for frii sources is @xmath164 with relatively small scatter @xcite such that a flux - limited sample may crudely approximate a volume - limited sample . for a jet lying within an angle @xmath165 to the line of sight , the fraction of randomly - oriented sources pointed towards earth is @xmath166 . orientation schemes predict that the transition from quasars ( typically beamed ) to radio galaxies ( typically not beamed ) is expected to occur at @xmath167 @xcite . the critical angle to induce asymmetric depolarization in a double radio source ( quasar or radio galaxy ) is likely to be similar ( e.g. * ? ? ? * ) ; here we estimate this angle as ranging between @xmath168 and @xmath169 , implying fractional surface densities of @xmath170 amongst the general double radio source population . this is , of course , a crude model , not least because intracluster magnetic field strengths at @xmath171 ( i.e. the expected median redshift for double radio sources ; * ? ? ? * ) are only expected to be a few percent of their @xmath68 values . a magnetised cluster atmosphere is clearly a prerequisite for depolarization , though a separate depolarizing medium close to the radio jets as described by @xcite may null this point . as far as we are aware , no census of asymmetric depolarization has been performed for radio sources in a blind survey ; studies to date have typically compiled samples of targeted observations ( e.g. * ? ? ? * ; * ? ? ? the atlas data are suitable for this purpose . we note that it is difficult to estimate what the fractional surface density of asymmetrically depolarized sources might be if environmental asymmetries were to dominate , rather than orientation ( significant merger activity is certainly expected in clusters ; e.g. * ? ? ? therefore , we simply focus on whether the atlas data are consistent with an orientation scheme or not . to avoid selection effects relating to visibility area and the detectability of sources with low fractional polarization , we selected only the 40 atlas sources with total intensity @xmath172 mjy . we expect that each of these sources is a radio - loud agn with dual - lobe structure dominated by un - beamed lobe emission ( though not all need exhibit multiple components due to viewing angle and source size ) . the breakdown of these 40 sources according to the polarization@xmath28total intensity classification scheme ( see [ sec:1 ] ) is as follows . there was 1 polarized source classified as type 3 ( midway polarized ) , though it is unclear if this demonstrates a depolarization asymmetry or not . there were 7 polarized single - component sources classified as type 4 . in an attempt to account for the possibility that many of these ` unclassified ' sources represent as - yet unassociated lobes of multi - component sources ( see comments in [ ch5:secresidentm ] ) , we assumed that perhaps only 1 of the detected type 4 sources was likely to truly represent a polarized single - component source . we have interpreted this as a dual - lobed asymmetrically - depolarized source with jet axis close to the line of sight , such that only a single component is effectively seen . there were 15 polarized dual - lobed sources classified as type 6 ( no asymmetric depolarization ) . there were 2 polarized dual - lobed sources classified as type 7 ( clear indication of asymmetric depolarization ) ; for reference , these sources were displayed earlier in fig . 17 of paper i. there were 4 polarized dual - lobed sources classified as type 8 ( unclear whether asymmetrically depolarized or not ) . we assumed that approximately one quarter of these sources ( i.e. 1 source ) would likely demonstrate asymmetric depolarization if more sensitive observations were obtained . this assumption is consistent with the finding from fig . [ ch5:fig : fracpoltheta2 ] that most type 8 sources have lass reflective of type 6 sources , rather than the smaller lass observed for the type 7 sources . finally , our sample included 11 unpolarized sources ( type 9 ) , each with fractional polarization upper limits below 1% ( see fig . [ ch5:fig : fracpolclass ] ) . we do not interpret these sources as being asymmetrically depolarized . we note that while it is possible that some of the 11 unpolarized sources represent unassociated lobes of multi - component sources , at least some of them must be truly isolated , single - component sources with @xmath36 . for example , the brightest unpolarized source displayed in fig . [ ch5:fig : fracpolclass ] is the source c3 , which is barely - resolved in atlas dr2 ( deconvolved angular size @xmath173 ) and has a @xmath174 ghz spectral index of @xmath175 . this source is therefore consistent with identification as a cso @xcite ; csos are known to exhibit flat radio spectra @xcite and low fractional polarization due to strong depolarization @xcite . the statistics above suggest that between @xmath176 and @xmath177 of the 40 atlas dr2 sources in our flux density limited sample exhibit depolarization asymmetry , i.e. , @xmath178 . this fraction falls within the theoretical range estimated above , demonstrating that the laing - garrington effect appears consistent with orientation dependence , at least within the rudimentary confines of our analysis . future high - resolution polarization studies are clearly required to form more robust conclusions . in this work we have presented results and discussion for atlas dr2 . our key results are summarised as follows . for convenience we use the term ` millijansky ' loosely below to indicate flux densities in the range @xmath179 mjy . a. radio emission from polarized millijansky sources is most likely powered by agns , where the active nuclei are embedded within host galaxies with mid - infrared spectra dominated by old - population ( 10 gyr ) starlight or continuum produced by dusty tori . we find no evidence for polarized sfgs or individual stars to the sensitivity limits of our data - all polarized atlas sources are classified as agns . b. the atlas data indicate that fractional polarization levels for sources with starlight - dominated mid - infrared hosts and those with continuum - dominated mid - infrared hosts are similar . c. the morphologies and angular sizes of polarized atlas components and sources are consistent with the interpretation that polarized emission in millijansky sources originates from the jets or lobes of extended agns , where coherent large - scale magnetic fields are likely to be present . we find that the majority of polarized atlas sources are resolved in total intensity , even though the majority of components in linear polarization are unresolved . this is consistent with the interpretation that large - scale magnetic fields that do not completely beam depolarize are present in these sources , despite the relatively poor resolutions of the atlas data . d. we do not find any components or sources with fractional polarization levels greater than 24% , in contrast with previous studies of faint polarized sources . we attribute this finding to our improved data analysis procedures . e. the atlas data are consistent with a distribution of fractional polarization at 1.4 ghz that is independent of flux density down to @xmath180 mjy , and perhaps even down to 1 mjy when considering the upper envelope of the distribution . this result is in contrast to the findings from previous deep 1.4 ghz polarization surveys ( with the very recent exception of * ? ? ? * ) , and is consistent with results at higher frequencies ( @xmath6 ghz ) . the anti - correlation observed in previous 1.4 ghz studies is due to two effects : a selection bias , and spurious high fractional polarization detections . both of these effects can become more prevalent at faint total flux densities . we find that components and sources can be characterised using the same distribution of fractional linear polarization , with a median level of 4% . we have presented a new lognormal model to describe the distribution of fractional polarization for 1.4 ghz components and sources , specific to agns , in surveys with resolution fwhms @xmath3 . f. no polarized sfgs were detected in atlas dr2 down to the linear polarization detection threshold of @xmath181 @xmath1jy . the atlas data constrain typical fractional polarization levels for the @xmath14 @xmath21 100 @xmath1jy sfg population to be @xmath161 . g. differences between differential number - counts of components and of sources in 1.4 ghz surveys with resolution fwhm @xmath3 are not likely to be significant ( @xmath5 20% ) at millijansky levels . h. the atlas total intensity differential source counts do not exhibit any unexpected flattening down to the survey limit @xmath182jy . i. the atlas linearly polarized differential component counts do not exhibit any flattening below @xmath100 mjy , unlike previous findings which have led to suggestions of increasing levels of fractional polarization with decreasing flux density or the emergence of a new source population . the polarized counts down to @xmath0 @xmath1jy are consistent with being drawn from the total intensity counts at flux densities where luminous fr - type radio galaxies and quasars dominate . j. constrained by the atlas data , we estimate that the surface density of linearly polarized components in a 1.4 ghz survey with resolution fwhm @xmath3 is 50 deg@xmath110 for @xmath183 @xmath1jy , and 90 deg@xmath110 for @xmath184 @xmath1jy . we estimate that the surface density for polarized sources is @xmath185 deg@xmath110 for @xmath186 @xmath1jy , assuming that most polarized components belong to dual - component sources ( e.g. fr - type ) at these flux densities . k. we find that the statistics of atlas sources exhibiting asymmetric depolarization are consistent with the interpretation that the laing - garrington effect is due predominantly to source orientation within a surrounding magnetoionic medium . to our knowledge , this work represents the first attempt to investigate asymmetric depolarization in a blind survey . we thank walter max - moerbeck for insightful discussions . we thank the anonymous referee for helpful comments that led to the improvement of this manuscript . c. a. h. acknowledges the support of an australian postgraduate award , a csiro oce scholarship , and a jansky fellowship from the national radio astronomy observatory . b. m. g. and r. p. n. acknowledge the support of the australian research council centre of excellence for all - 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gonzlez e. , toffolatti l. , gonzlez - nuevo j. , de zotti g. , 2004 , mnras , 349 , 1267 tucci m. , toffolatti l. , 2012 , adast , 2012 , 624987 urry c. m. , padovani p. , 1995 , pasp , 107 , 803 van der marel r. p. , franx m. , 1993 , apj , 407 , 525 vernstrom t. , scott d. , wall j. v. , 2011 , mnras , 415 , 3641 white r. l. , becker r. h. , helfand d. j. , gregg m. d. , 1997 , apj , 475 , 479 wilman r. j. , et al . , 2008 , mnras , 388 , 1335 windhorst r. a. , van heerde g. m. , katgert p. , 1984 , a&as , 58 , 1 wright e. l. , 2006 , pasp , 118 , 1711 xu h. , li h. , collins d. c. , li s. , norman m. l. , 2009 , apj , 698 , l14 zinn p .- c . , middelberg e. , norris r. p. , hales c. a. , mao m. y. , randall k. e. , 2012 , a&a , 544 , a38 this appendix presents 1.4 ghz euclidean - normalised differential number - counts , in tabulated form , derived from the atlas dr2 total intensity and linear polarization component catalogues from appendix a of paper i. the tabulated results have been organised as follows according to emission type , eddington bias correction scheme , and atlas field . tables [ tbl : countsicdfsie ] and [ tbl : countsielaisie ] ( each with 7 columns ) present total intensity number - counts for the cdf - s and elais - s1 fields , respectively , using ` component - corrected ' data whereby individual components were deboosted prior to the application of visibility area and resolution bias corrections . similarly , tables [ tbl : countsicdfs ] and [ tbl : countsielais ] ( each with 9 columns ) present total intensity number - counts for the two atlas fields , but now using ` bin - corrected ' data whereby non - deboosted components were corrected for visibility area , resolution bias , and eddington bias . tables [ tbl : countslcdfsie ] and [ tbl : countslelaisie ] ( each with 11 columns ) present linear polarization number - counts for the cdf - s and elais - s1 fields , respectively , using component - corrected data . tables [ tbl : countslcdfs ] and [ tbl : countslelais ] ( each with 17 columns ) present bin - corrected linear polarization number - counts for the cdf - s and elais - s1 fields , respectively . columns for the tables above are organised as follows . tables describing component - corrected total intensity data give for each bin the flux density range ( @xmath187 ) , expected average flux density ( @xmath69 ) , raw number of deboosted components ( @xmath188 ) , effective number of deboosted components following visibility area correction only ( @xmath189 ) , effective number of deboosted components following both visibility area and resolution bias corrections ( @xmath190 ) , euclidean - normalised counts following visibility area correction only ( @xmath191 ) , and euclidean - normalised counts following both visibility area and resolution bias corrections ( @xmath192 ) . columns for tables describing bin - corrected total intensity data are similar , but without the superscript d which indicates use of deboosted component data . the bin - corrected tables contain two additional columns : effective number of components following combined visibility area , resolution bias , and eddington bias corrections ( @xmath193 ) , and an associated column for their euclidean - normalised counts ( @xmath194 ) . the tables describing linear polarization data are similar to those for total intensity data , but with additional columns cataloguing the effective number of components or euclidean - normalised counts resulting from the resolution bias corrections associated with the lower ( -r ) or upper ( + r ) bounds described in 7.1 of paper i and displayed in fig . 22 of paper i. thus the additional columns have been assigned descriptors with superscripts v - r , v+r , v - re , and v+re . errors associated with the fully corrected euclidean - normalised counts in the figures and tables above are @xmath195 poissonian and were calculated following @xcite .

this is the second of two papers describing the second data release ( dr2 ) of the australia telescope large area survey ( atlas ) at 1.4 ghz . in paper i we detailed our data reduction and analysis procedures , and presented catalogues of components ( discrete regions of radio emission ) and sources ( groups of physically associated radio components ) . in this paper we present our key observational results . we find that the 1.4 ghz euclidean normalised differential number counts for atlas components exhibit monotonic declines in both total intensity and linear polarization from millijansky levels down to the survey limit of @xmath0 @xmath1jy . we discuss the parameter space in which component counts may suitably proxy source counts . we do not detect any components or sources with fractional polarization levels greater than 24% . the atlas data are consistent with a lognormal distribution of fractional polarization with median level 4% that is independent of flux density down to total intensity @xmath2 mjy and perhaps even 1 mjy . each of these findings are in contrast to previous studies ; we attribute these new results to improved data analysis procedures . we find that polarized emission from 1.4 ghz millijansky sources originates from the jets or lobes of extended sources that are powered by an active galactic nucleus , consistent with previous findings in the literature . we provide estimates for the sky density of linearly polarized components and sources in 1.4 ghz surveys with @xmath3 resolution . [ firstpage ] polarization radio continuum : galaxies surveys .

You are an expert at summarizing long articles. Proceed to summarize the following text: the main aim of the author in this paper is to present a new tool which can make easier the study of the real dynamics of families of iterative methods which depends on a certain parameter or even the study of an iterative method applied to a uniparametric family of polynomials . this tool can be modified in order to extend , amongst other ones , to methods which needs two approximations as for example , secant - type methods , modified newton s method , etc . the dynamics of iterative methods used for solving nonlinear equations in complex plane has been studied recently by many authors @xcite . there exists a belief that real dynamics is included in the complex dynamics but this is not true at all . for example in the real dynamics one can proof the monotone convergence which does not exist in the complex plane , there exists also asymptotes in the real dynamics but in the complex one that concept has no sense or the point @xmath0 in the complex plane can be studied as another point but in the real line it is not possible . as a consequence real dynamics is not contained in the complex dynamics and both must be studied separately . taking into account that distinction , many authors @xcite have begun to study it since few years ago . if one focus the attention on families of iterative methods studied in the complex plane , parameter spaces have given rise to methods whose dynamics are not well - known . these parameter spaces consist on studying the orbits of the free critical points associating each point of the plane with a complex value of the parameter . several authors @xcite have studied really interesting dynamical and parameter planes in which they have found some anomalies such as convergence to @xmath1-cycles , convergence to @xmath2 , or even chaotical behavior . in the real line , there exists tools such as feigenbaum diagrams or lyapunov exponents that allow us to study what happens with a concrete point , but it is really hard to study each point in a separate way . this is the main motivation of the author to present the new tool called _ the convergence plane_. _ the convergence plane _ is obtained by associating each point of the plane with a value of the starting point and a value of the parameter . that is , _ the convergence plane _ is based on taking the vertical axis as the value of the parameter and the horizontal axis as the starting point , so every point in the plane represents an initial estimation and a member of the family . if one draws a straight horizontal line in a concrete value of the parameter , the dynamical behavior , for that value , for every starting point is on that line . on the other hand , if the straight line is vertical , the dynamical behavior for that starting point and every value of the parameter is on that line , this is the information that gives feigenbaum diagrams or lyapunov exponents , so both tools are included . the rest of the paper is organized as follows : in section [ s2 ] the algorithm of _ the convergence plane _ is shown and in section [ s3 ] an example of _ the convergence plane _ associated to the damped newton s method applied to the polynomial @xmath3 is provided in order to validate the tool . finally , the conclusion are shown in the concluding section [ s4 ] . as it is said in the introduction each point of the plane corresponds to a starting point and a value of the parameter , in other words the pair @xmath4 represents that the study is developed using @xmath5 as the starting point and @xmath6 as the value of the parameter . the algorithm of this new tool is the following one : * the fixed points of the method must be computed . * then , a color is assigned to each fixed point . * moreover , the region in which one wants to study the family , @xmath7 , the maximum number of iterations , @xmath8 , and the tolerance , @xmath9 , are prefixed . * then , a grid of @xmath10 points in @xmath7 of the initial points and values of the parameter must be chosen . * if after @xmath8 iterations of the family with @xmath6 as the value of the parameter , the point @xmath5 does not converge to any of the fixed points that point must be black . * if one is interested on representing the convergence to any cycle or other behavior such as convergence to extraneous fixed points , divergence etc . , there exists the possibility of assigning a color to that behaviors too . once _ the convergence plane _ has been computed it is easy to distinguish the pairs @xmath11 for which the element of the family is convergent to any of the roots using @xmath5 as a starting point . so this tool provides a global vision about what points converges and shows what are the best choices of the parameters to ensure the greatest basin of attraction . moreover , it can be used also as a tool that show how the basins of attraction of a family changes with the value of the parameter or even to study the convergence of methods which depend on 2 points such as , for example , secant - type methods . in this moment , the author is going to explain how figure [ fig1 ] and figure [ fig2 ] of this paper were generated . to do this , it is shown the mathematica programs that has been used which are a modification of the ones that appears in @xcite . in concrete , in the example the region will be @xmath12\times [ 2,2.26]$ ] ( the season of taking that values is in @xcite and will appear in section 3 ) . moreover , a number of iterations @xmath13 , a tolerance of @xmath14 and a grid of @xmath15 have been taken . first of all , the function , the roots and the procedure that identifies the root to which converge the iterations have been defined as .... f[x_]:=x^3-x ; rootf[1]=-1 ; rootf[2]=0 ; rootf[3]=1 ; rootposition = compile[{{z,_real } } , which [ abs[z - rootf[1 ] ] < 10.0^(-6 ) , 1 , abs[z - rootf[2 ] ] < 10.0^(-6 ) , 2 , abs[z - rootf[3 ] ] < 10.0^(-6 ) , 3 , abs[z ] > 10.0^(3 ) , 11 , true , 0 ] , { { rootf[_],_real } } ] .... then , the iteration method is the following .... iternewtonlamda = compile[{{x,_real},{k,_real}},x - k*f[x]/f'[x ] ] .... the algorithm used to show if the iteration of _ itermethod _ of a point converges to a root or a cycle is the following .... itercoloralgorithm[itermethod_,xx_,yy_,lim _ ] : = block[{z , z2,kk , ct , r } , z = xx ; kk = yy ; ct = 0;r = rootposition[z ] ; while[(r==0 ) & & ( ct < lim),++ct ; z = itermethod[z , kk ] ; r = rootposition[z ] ; ] ; if[r==0,z2= itermethod[z , kk];z2= itermethod[z2,kk ] ; if[abs[z2-z]<10^(-6),r=4,z2=itermethod[z2,kk ] ; if[abs[z2-z]<10^(-6),r=5,z2=itermethod[z2,kk ] ; if[abs[z2-z]<10^(-6),r=6,z2=itermethod[z2,kk ] ; if[abs[z2-z]<10^(-6),r=7,z2=itermethod[z2,kk ] ; if[abs[z2-z]<10^(-6),r=8,z2=itermethod[z2,kk ] ; if[abs[z2-z]<10^(-6),r=9,z2=itermethod[z2,kk ] ; if[abs[z2-z]<10^(-6),r=10,z2=itermethod[z2,kk ] ; ] ] ] ] ] ] ] ] if[head[r]==which , r = 0 ] ; ( * " which " unevaluated * ) return[n[r+ct/(lim+0.001 ) ] ] ] .... the palette of colors used is defined as .... convergencecolor[p _ ] : = switch[integerpart[11p ] , 11 , cmykcolor[0.0,0.0,0.0,0.0],(*white * ) 10 , cmykcolor[0.0,1.0,1.0,0.8],(*dark red * ) 9 , cmykcolor[0.0,0.0,1.0,0.5],(*dark yellow * ) 8 , cmykcolor[1.0,0.0,1.0,0.5],(*dark green * ) 7 , cmykcolor[1.0,0.0,0.0,0.5],(*dark blue * ) 6 , cmykcolor[0.0,0.5,1.0,0.0],(*orange * ) 5 , cmykcolor[1.0,0.0,1.0,0.0],(*green * ) 4 , cmykcolor[0.0,1.0,1.0,0.0],(*red * ) 3 , cmykcolor[0.0,0.0,1.0,0.0],(*yellow * ) 2 , cmykcolor[0.0,1.0,0.0,0.0],(*magenta * ) 1 , cmykcolor[1.0,0.0,0.0,0.0],(*cyan * ) 0 , cmykcolor[0.0,0.0,0.0,1.0](*black * ) ] .... the function used by the author to plot the convergence plane is the following .... plotconvergenceplane[itermethod_,points _ ] : = densityplot[itercoloralgorithm[itermethod , x , k , limiterations ] , { x , xxmin , xxmax } , { k , kkmin , kkmax } , plotrange->{0,11 } , plotpoints->points , mesh->false , colorfunction->convergencecolor ] .... then , a graphic is obtained in this way ( notice that we avoid overflow and underflow errors and other errors by means of using the instruction _ off _ ) : .... numberpoints = 1024 ; limiterations = 4000 ; xxmin = -2.0 ; xxmax = 2 ; kkmin = 0.0 ; kkmax = 2.6 ; off[general::ovfl ] ; off[general::unfl ] ; off[infinity::indet ] off[compiledfunction::cccx ] ; off[compiledfunction::cfn ] ; off[compiledfunction::cfcx ] ; off[compiledfunction::cfex ] ; off[compiledfunction::crcx ] ; off[compiledfunction::cfse ] ; off[compiledfunction::ilsm ] ; off[compiledfunction::cfsa ] ; plotconvergenceplane[iternewtonlamda , numberpoints ] .... to show the goodness of _ the convergence plane _ we are going to study the dynamics of the damped newton s method which has the following form @xmath16 where @xmath17 is a polynomial with real coefficients and we take @xmath18 . in concrete , in @xcite , the author has made an extensive and deep study about the real dynamics of the damped newton s method applied to polynomials of degrees 2 , 3 , 4 and 5 . in order to proof that the tool works properly we are going to apply the iterative method to find the real roots of a cubic polynomial the scaling theorem @xcite allows up to suitable change of coordinates , to reduce the study of the dynamics of iterations of general maps , to the study of specific families of iterations of simpler maps . specifically , the study of cubic polynomials reduce to the study of @xmath19 , @xmath20 @xmath3 and the uniparametric family @xmath21 . in @xcite the author shows the real dynamics of the damped newton s method applied to @xmath3 are not easy and that there exists some cycles and chaotic behavior of the iterations of some points by means of using lyapunov exponents , feigenbaum diagrams and analytical techniques . we are going to use the new tool _ the convergence plane _ to study the dynamics of the damped newton s method applied to @xmath22 . we denote the three roots of @xmath22 as @xmath23 , @xmath24 and @xmath25 . before , applying the new tool we have to see what information give to us the lyapunov exponents and feigenbaum diagrams . these two tools gives information only about the point which is being iterated . in concrete , the lyapunov exponent is defined , for an orbit @xmath26 as @xmath27 the applicability of this tool resides in the following result . @xcite an orbit @xmath28 is chaotic if and only if the following conditions hold : * the orbit is not asymptotically periodic . * @xmath29 . as a consequence , this tool shows which orbits are chaotical or not . on the other hand , the feigenbaum diagrams shows if the orbit of a point converges to a cycle , to a point , if it is chaotic or it diverges . in figure [ fff1 ] it is shown the results obtained using the both two tools . fftt.pngfeigenbaum diagram taking into account that a brief study has been made in @xcite and the idea of not making this paper very long we are going to focus on the interval @xmath30 which is sufficient to show the powerful of the new tool . in figure [ ff1 ] we can see both tools centered on that interval . and we distinguish three different zones . fft.pngfeigenbaum diagram there exists 3 clear zones of different behavior . in the first one which corresponds with the interval @xmath31 ( see left side of the figure [ ff2 ] ) , we see the zone in which the iterations converge to the fixed points of the damped newton s method or equivalently to the roots of the polynomial @xmath22 . the second zone , shown in the center of figure [ ff2 ] is where it appears cycles of different orders , in concrete cycles of order @xmath32 , @xmath33 and @xmath34 . the third zone , shown in the right side of figure [ ff2 ] is where we found chaotical behavior or convergence to different cycles of periods different to @xmath35 . ll2.png@xmath36$]ll3.png@xmath37$]ff2.png@xmath36$]ff3.png@xmath37 $ ] now if we use _ the convergence plane _ in the region @xmath12\times[0,2.6]$ ] ( see figure [ fig1 ] ) the goodness of this tool is going to be proof . in this case , we use the program _ mathematica 5.0 _ ( the code appears in section 2.1 ) as in @xcite , with tolerance @xmath38 , a maximum of 4000 iterations and the following palette of colors : * cian , if the iterations converge to root @xmath39 , magenta , if the iterations converge to root @xmath40 and yellow , if the iterations converge to root @xmath41 . * red , if the iterations converge to a @xmath32-cycle , green , if converge to a @xmath42-cycle , orange , if converge to a @xmath33-cycle , dark blue , if converge to a @xmath43-cycle , dark green , if to a @xmath44-cycle , dark yellow , if converge to a @xmath45-cycle and dark red if the iterations converge to a @xmath34-cycle . * white , if the iterations diverge to @xmath2 . * black , in other case . on the region @xmath46\times[0,2.6]$].,title="fig:",scaledwidth=70.0% ] + again we distinguish 3 clear zones with different dynamical behavior , which corresponds with the 3 zones using feiganbaum diagrams and lyapunov exponents . the first one , shown in the left hand of figure [ fig2 ] corresponds to the interval @xmath47 $ ] and it is the zone in which every point ( except the poles and its preimages ) , converges to any of the roots of the polynomial or equivalently to any fixed point of the damped newton s method . additionally , in this zone the tool gives the idea of how the basins of attraction changes with the value of @xmath48 . for example , the basin of @xmath24 is getting lower when the value of @xmath48 is closer to two , an the basin of the other two roots are getting bigger as the parameter is closer to @xmath32 . furthermore , it it shown that the julia set gets more intricate when the value of the parameter increases until @xmath32 . the second zone , see the center of figure [ fig2 ] , corresponds with the zone in which there exists convergence to cycles of orders @xmath32 , @xmath33 and @xmath34 . and the third zone , shown in the right hand of the figure [ fig2 ] , the zone in which there exists chaotical behavior and convergence to cycles of order different to @xmath35 . in concrete in the interval @xmath49 appears cycles of order @xmath43 , @xmath44 and @xmath45 . summarizing , the author has shown the helpfulness of this new tool using it to study the behavior of the damped newton s method applied to a cubic polynomial . the conclusions drawn this study are : the damped newton s method is a good method as a root - finding algorithm if @xmath50 $ ] , if @xmath51 $ ] , the iteration can converge to cycles , diverge to infinity , etc . ; the basin of attraction of @xmath24 increases when the damping factor decreases , and the other basin increases with the parameter ; the julia set associated with the damped newton s method applied to @xmath3 is more intricate as the damping factor increases to @xmath32 . all these conclusions coincide with the ones given in @xcite . p2.png@xmath36$]p3.png@xmath37 $ ] in the present paper , the author has presented a new tool that allows to study the convergence of a family of iterative methods , taking into account every initial point . _ the convergence plane _ , includes the information given by tools such as lyapunov exponents and feigenbaum diagrams , indeed , the new tool provides more information because it considers each initial point of the real line and every value of the parameter . moreover , this technique can be used to choose the best value of the parameter that ensure the convergence zone as large as possible and shows how the basins of attraction changes with the value of the parameter . on the other hand , this technique can be modified in order to : study methods applied to polynomials which depends on a parameter , to other kind of methods such as two - point methods ( for example secant - type methods ) , methods applied to non - differentiable functions , etc . therefore , this new tool can be used in a large amount of situations , making the study of real dynamics easier , deeper and in a more compact way .

in this paper , the author presents a new tool , called _ the convergence plane _ , that allows to study the real dynamics of iterative methods whose iterations depends on one parameter in an easy and compact way . this tool can be used , inter alia , to find the elements of a family that have good convergence properties and discard the bad ones or to see how the basins of attraction changes along the elements of the family . to show the applicability of the tool an example of the dynamics of the damped newton s method applied to a cubic polynomial is presented . + * keywords : * real dynamics , nonlinear equations , the convergence plane , iterative methods , basins of attraction . + + ngel alberto magren + universidad de la rioja + departamento de matemticas y computacin + 26002 logroo , la rioja , spain + alberto.magrenan@gmail.com +

You are an expert at summarizing long articles. Proceed to summarize the following text: research in spintronics and carbon nanotubes sparked renewed interests in the effects of spin - orbit coupling on quantum transport @xcite . one of the remarkable features of non - interacting disordered electron systems with spin - orbit coupling is the existence of a metallic phase in two dimensions ( 2d ) @xcite . more exactly the 2d metallic phase appears in systems with symplectic symmetry , i.e. , in systems with time reversal symmetry but without spin rotation symmetry . such systems are an exception to the prediction of abrahams _ et al . _ that there is no metallic phase in 2d disordered electron systems @xcite . recently some 2d metals have been discovered experimentally @xcite . although the mechanism inducing the 2d metal might be different from spin - orbit coupling , we think that detailed understanding of the 2d symplectic systems gives some insights on general transport properties of 2d metals . a scaling theory for the symplectic systems @xcite can be described by the @xmath0 function for the conductance @xmath1 ( in units of @xmath2 ) @xmath3 where @xmath4 is the linear size of a system . ( the conductance is actually a distributed quantity . see refs . @xcite for discussions on the scaling hypothesis of the conductance . ) because of the so - called anti - localization effect , the @xmath0 function shows non - monotonic behavior in the symplectic systems . the asymptotic behavior in the strongly metallic region in 2d is conjectured to be @xmath5 this indicates that there is a metallic phase in 2d . in the scaling theory for the symplectic systems in 2d , however , there is a long standing problem for more than two decades @xcite : the logarithmic divergence of the conductance in the 2d metallic phase in the limit @xmath6 indicated by eq . ( [ eq : betaasymptotic ] ) . since the conductivity and the conductance are the same quantity in 2d , this predicts a perfect conductivity , @xmath7 , in spite of the system being disordered . this is in contrast to 3d disordered metals , in which the conductivity converges to a finite value . according to the renormalization group theory of continuous phase transitions , the scaling hypothesis is valid when the correlation length is much larger than all other microscopic lengths . the scaling argument which deduced ( [ eq : betaasymptotic ] ) , however , is based on the weak localization theory in which disorder is supposed to be weak . in such a weakly disordered metallic phase this condition on the correlation length may not be satisfied . therefore , a numerical check of the scaling hypothesis in the 2d metallic region is desirable . in this paper , we report a numerical check of the scaling hypothesis in the 2d metallic region . we also report numerical calculation of the landauer conductance in 2d , which is a direct check of the prediction of a perfect conductivity . last , numerical simulations of transport through a sierpinski carpet are reported . among the models with symplectic symmetry , we employ the su(2 ) model @xcite , @xmath8 the random potential @xmath9 is distributed with box distribution in the range @xmath10 $ ] . random spin - orbit coupling is included in the nearest neighbor hoping term . we parameterize the hopping matrix as @xmath11 and we distribute @xmath12 and @xmath13 with uniform probability in the range @xmath14 , and @xmath15 according to the probability density @xmath16 in the range @xmath17 $ ] . in the actual simulations @xmath18 s in one direction are set to the unit matrix with the aid of the local su(2 ) gauge transformation . the su(2 ) model has an advantage that corrections to scaling arising from irrelevant variables are smaller than other models @xcite , so it is quite a useful model when we study universal aspects of anderson localization and the anderson transition . we analyze the quasi-1d localization length @xmath19 on a quasi-1d strip with width @xmath4 @xcite . periodic boundary conditions are imposed in the transverse direction . the scaling hypothesis implies that the dimensionless quantity @xmath20 obeys @xmath21 where @xmath22 is the 2d correlation ( localization ) length which depends on @xmath23 and the fermi energy @xmath24 . the subscript @xmath25 indicates the metallic and insulating phases . we have calculated @xmath26 for sizes @xmath27 $ ] with an accuracy from @xmath28 to @xmath29 . the ensemble transfer matrix method has been used @xcite . in addition to the data at @xmath30 analyzed in ref . @xcite , we have also accumulated data at @xmath31 in @xmath32 $ ] . from the numerical data the correlation length @xmath22 at each pair of @xmath33 and the scaling function @xmath34 are estimated using numerical fit described in ref . the quality of the fit is assessed by the goodness of fit probability @xmath35 . to eliminate the ambiguity of the absolute scale of @xmath22 , we set @xmath36 at @xmath37 . figure [ fig : metalsps1 ] demonstrates the single parameter scaling in the metallic region . all data fall on a common scaling curve when @xmath26 is plotted as a function of @xmath38 , indicating the validity of the scaling hypothesis . in a strongly metallic region ( @xmath39 ) , the data are well fitted to @xmath40 as shown in fig . [ fig : metalsps2 ] . in ref . @xcite , the @xmath0 function for @xmath26 from the metallic to the insulating limits was estimated . the estimate has not changed significantly when we add data for @xmath31 . is @xmath41 . ] is @xmath42 . ] in last section and in ref . @xcite , the scaling hypothesis has been checked numerically . this supports the scaling theory for the 2d symplectic systems . furthermore , the logarithmic increase of @xmath26 in the strongly metallic region gives the impression that the conductance may also increase logarithmically . in this section , we study the size dependence of the mean conductance directly . we now attach two perfect leads to the square sample with linear size @xmath4 and calculate the two terminal landauer conductance @xmath1 ( in units of @xmath2 ) with the transfer matrix method @xcite . a fixed boundary condition is imposed in the direction transverse to the current . the hopping matrices in the transverse direction are set to the unit matrix , and the fermi energy @xmath30 . we have accumulated 10000 samples for @xmath43 $ ] and 5000 samples for @xmath44 . as shown in fig . [ fig:2dmeang ] , the mean conductance @xmath45 in the strongly metallic region does show the logarithmic increase @xmath46 in agreement with the prediction of the scaling theory @xcite . the prefactor of the logarithmic term , which is an important parameter for the scaling in the strongly metallic region , is also consistent with it . note that a perfect conductivity does not mean perfect transmission . the transmission probability per channel rather goes to zero as @xmath4 increases , so most of incoming electrons are reflected . it is left for future to understand physics behind this possible perfect conductivity in 2d . vs. system size @xmath4 in the 2d metallic region . the solid lines are a guide to the eye only . for a reference , @xmath47 is indicated . ] recently we have obtained numerical results which suggest that an anderson transition occurs even below 2d in the presence of spin - orbit coupling @xcite . this is based on numerical simulations of electrons on a sierpinski carpet sc(@xmath48 ) , where @xmath49 is the generation number . in the limit @xmath50 , sc(@xmath48 ) becomes a true fractal whose spectral dimension is @xmath51 @xcite . here we study the size dependence of the landauer conductance through the sierpinski carpet in a delocalized phase . we have attached two perfect leads to sc(@xmath48 ) and have calculated the conductance with the recursive green s function method @xcite . a fixed boundary condition is imposed in the transverse direction . the hopping matrices in the transverse direction are set to the unit matrix . we set the fermi energy @xmath52 and the disorder @xmath53 where relatively large mean conductance is obtained . the number of samples is 10000 for @xmath54 and 500 for @xmath55 . figure [ fig : sc51meang ] shows the mean conductance @xmath56 as a function of the linear size @xmath57 . the mean conductance increases with @xmath4 , possibly indicating a delocalized phase . it also indicates that the increase is slower than that in 2d . it is an open problem whether the conductance in the possible delocalized phase of the sierpinski carpet diverges .

electron transport phenomena in disordered electron systems with spin - orbit coupling in two dimensions and below are studied numerically . the scaling hypothesis is checked by analyzing the scaling of the quasi-1d localization length . a logarithmic increase of the mean conductance is also confirmed . these support the theoretical prediction that the two dimensional metal in systems with spin - orbit coupling has a perfect conductivity . transport through a sierpinski carpet is also reported . , , quantum transport , spin - orbit coupling , symplectic class , scaling theory , perfect conductivity

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Proceed to summarize the following text: the last cosmological and astrophysical data of large scale structure , the observations of type ia and cosmic microwave background radiation have demonstrated that currently there is an acceleration expansion phase in the universe @xcite . the cosmic expansion can be well described by a negative pressure so - called dark energy ( de ) . the simplest candidate for de is the cosmological constant . however , the cosmological constant suffers from the fine - tuning and the cosmic coincidence problems @xcite,@xcite,@xcite . therefore , to avoid these problems , different models for dark energy have been proposed such as quintessence , k - essence , tachyon @xcite , phantom , quintom @xcite , and the quantum gravity models , as well as holographic and new agegraphic models @xcite,@xcite . the tachyon model as a scalar field model arises in particle physics and string theory . thus , it can be considered as one of the potential candidates to describe the nature of the de . on the other hand , the problem of structure formation in the universe is a very important issue in theoretical cosmology . a simple model of structure formation is the spherical collapse model . the spherical collapse model was presented by gunn and gott @xcite . this model studies the evolution of growth of overdense structures with respect to the dynamics of scale factor or cosmic redshift . the dynamics of overdense structures depends on the dynamics of the background hubble flow and expansion of the universe . in the frame of general relativity , the spherical collapse model has been studied @xcite,@xcite,@xcite . in this paper , we study the spherical collapse and the evolution of spherical overdensities in the framework of tachyon scalar field model and compare the results with the results of einstein - de sitter ( eds ) and @xmath8-cold dark matter ( @xmath7 ) models . the action of tachyon scalar field over a cosmological background is given by @xcite @xmath9 where @xmath10 and @xmath11 are the tachyon scalar field and tachyon potential , respectively , and @xmath12 is the friedmann - robertson - walker ( frw ) metric having the cosmic time @xmath13 dependent scale factor @xmath14 . for a homogeneous field , the equation of motion is obtained as @xmath15 where the symbols @xmath16 and @xmath17 denote the derivatives with respect to @xmath13 and @xmath10 , respectively , and @xmath18 is called the hubble parameter . in the flat frw universe , the energy density @xmath19 and the pressure @xmath20 of the tachyon field read as @xmath21 @xmath22 for the pressureless matter and tachyon scalar field matter , the friedmann equation is given by @xmath23 where @xmath19 is the energy density of tachyon scalar field , and @xmath24 is the density of pressureless matter . we suppose that there is no interaction between @xmath19 and @xmath24 , so the continuity equations are given separately by @xmath25 @xmath26 using eqs . ( [ rho ] ) and ( [ p ] ) and also @xmath27 , the equation of state parameter ( eos ) for tachyon scalar field is obtained as @xmath28 the requirement for a real @xmath19 results in @xmath29 according to which @xmath30 should vary as @xmath31 . the fractional energy densities are defined by @xmath32 @xmath33 taking time derivative of eq . ( [ omega ] ) and using eq . ( [ dot ] ) yields @xmath34.\label{domega}\ ] ] also , taking time derivative of eq . ( [ friedman ] ) and using eqs . ( [ dot ] ) and ( [ dotm ] ) yields @xmath35 using eq . ( [ hdot ] ) and inserting eq . ( [ domega ] ) , we obtain @xmath36 here , the prime is the derivative with respect to @xmath37 where @xmath38 and z is the cosmic redshift . using @xmath39 and eq . ( [ eos ] ) , one finds @xmath40 the differential equation for the evolution of dimensionless hubble parameter , @xmath41 , in tachyon scalar field model , is obtained by using eqs . ( [ dot ] ) , ( [ dotm ] ) , ( [ eos ] ) and ( [ hdot ] ) as follows @xmath42.\label{ddf}\ ] ] the cosmological acceleration equation is also given by @xmath43 using eqs . ( [ rho ] ) and ( [ eos ] ) and inserting in eq . ( [ ddd ] ) , the acceleration equation reads as @xmath44 which can be converted to obtain the tachyon scalar field kinetic energy , using eqs . ( [ rho ] ) , ( [ friedman ] ) , and @xmath45 , as follows @xmath46.\label{phidot}\ ] ] with no loss of generality , we suppose the power law behavior @xmath47 and use @xmath48 to obtain @xmath49.\label{phi2}\ ] ] inserting ( [ phi2 ] ) in eqs . ( [ eos ] ) , ( [ df ] ) , ( [ ddf ] ) we can get the evolution of eos parameter @xmath50 , the density parameter of dark energy @xmath51 , and the dimensionless hubble parameter @xmath52 in tachyon scalar field model as a function of cosmic redshift . in figure ( 1 ) , assuming the present values @xmath53 , @xmath54 and @xmath55 , we have shown the evolution of eos parameter , the evolution of density parameter and the evolution of dimensionless hubble parameter of tachyon scalar field model with respect to the redshift parameter @xmath56 for @xmath57 . the evolution of dark energy density parameter of tachyon scalar field model with respect to the redshift parameter @xmath56 . , respectively . the dotted line shows the @xmath7 model.,title="fig:",width=192 ] the evolution of dimensionless hubble parameter in tachyon scalar field model and in the @xmath58 model . , respectively . the dotted line shows the @xmath7 model.,title="fig:",width=192 ] in this section , we study the linear growth of perturbation of non relativistic dust matter by computing the evolution of growth factor @xmath59 in tachyon scalar field model , and then compare it with the evolution of growth factor in eds and @xmath7 models . the differential equation for the evolution of the growth factor @xmath59 is given by @xcite,@xcite,@xcite @xmath60 in order to study the linear growth in tachyon scalar field model , using eqs.([df ] ) , ( [ ddf ] ) and ( [ phi2 ] ) , we solve numerically eq.([ga ] ) . also , we solve numerically eq.([ga ] ) for the eds model and the @xmath7 model . , respectively . the dotted line indicates the @xmath7 model and the dashed line represent the @xmath61 model . , title="fig:",width=192 ] in figure ( 2 ) , we have plotted the evolution of growth factor @xmath59 with respect to the scale factor . at first , namely for small scale factors , the growth factor in the tachyon scalar field model is larger than those of eds and @xmath7 models . however , for rather larger scale factors , the growth factor in the tachyon scalar field model becomes smaller than the eds model while it is still larger enough than that of @xmath7 model . this means that , at the beginning , the tachyon scalar field model predicts structure formation more efficient than eds and @xmath7 models . for later times , however , the structure formation in the tachyon scalar field model is dropped behind that of eds model , whereas it precedes the structure formation in the @xmath7 model . the structure formation is described by a non - linear differential equation for the evolution of the matter perturbation @xmath62 in a matter dominated universe @xcite,@xcite . in @xcite this differential equation was generalized to the case of evolution of @xmath62 in a universe including a dark energy component . now , we consider the non - linear differential equation which is given by @xcite @xmath63 where @xmath64 denotes the derivative with respect to @xmath65 . in the linear regime , we have @xmath66 in both cases , we consider the initial conditions @xmath67 , @xmath68 and @xmath69 @xcite . ( b)the non - linear growth of density perturbation@xmath62 in terms of@xmath65 . , respectively . the dotted line indicates the @xmath7 model and the dashed line indicates the eds model.,title="fig:",width=192 ] the figure ( 3-a ) shows that the linear growth factor in the tachyon scalar field model is larger than those of eds and the @xmath7 models , and the figure ( 3-b ) shows that the non - linear growth factor in the tachyon scalar field model is larger than the eds model . as time passes , the perturbation is growing and one can no longer use the linear regime . at this stage , the radius of perturbation region becomes maximal @xmath72 and the perturbation stops growing . this condition is called turn - around which points to the epoch when the grows of perturbation decouples from the hubble flow of the homogenous background . after the turn - around the perturbation starts contracting . for a perfect spherical symmetry and perfect pressureless matter , the perturbation would collapse to a single point becoming infinitely dense . since there is hardly any perfect spherical symmetric overdensity in the universe , the perturbation does not collapse to a single point and finally a virialized object of a certain finite size in equilibrium state is formed that is called halo . we call ( @xmath73 , @xmath74 ) and ( @xmath75 , @xmath76 ) as the redshift and radius corresponding to virialization and the turn - around epochs , respectively . now , we peruse two characterising quantities of the spherical collapse model for the tachyon scalar field model : the virial overdensity @xmath71 and the linear overdensity parameter @xmath70 . we consider a spherical overdense region with matter density @xmath77 in a surrounding universe described by its background dynamics and density @xmath78 . the virial overdensity @xmath71 is defined by @xcite @xmath79 which is a function of scale factor and redshift . we can rewrite the virial overdenty @xmath71 as follows @xcite @xmath80 where @xmath81 @xmath82 here , @xmath83 is the virial radius which is given by @xcite @xmath84 where @xmath85 and @xmath86 are the ( wang - steinhardt ) ws parameters @xmath87 @xmath88 now , we discuss the results obtained for the linear overdensity parameter and the virial overdensity for the models introduced in this work . the figure ( [ t2 ] ) shows the time evolution of linear overdensity , @xmath89 in terms of a function of the collapse redshift for the @xmath7 model , the eds model and the tachyon scalar field model . in the eds model , @xmath90 is independent of the redshift , hence it has a constant value i.e. @xmath91 . in the @xmath7 model , @xmath90 is smaller than 1.686 but the time evolution of the linear overdensity approaches the value of the eds model at high redshifts . , in terms of a function of the collapse redshift for the @xmath7 model , the eds model and the tachyon scalar field model . the blue , green and red lines represent the tachyon scalar field model for @xmath57 , respectively . the dotted line indicates the @xmath7 model and the dashed line indicates the eds model . , title="fig:",width=192 ] in fact , at high redshifts we have a matter dominated universe ( dust matter ) , but at lower redshifts we have a dark energy dominated universe , thus the structure formation must occur earlier . in the tachyon scalar field model , @xmath90 drives more slowly than the @xmath7 and the eds models because in fig . ( 1 ) , the hubble parameter in the tachyon scalar field model is larger than that of @xmath7 model . in terms of the collapse redshift @xmath73 for the @xmath7 model , the eds model and the tachyon scalar field model . the blue , green and red lines represent the tachyon scalar field model for @xmath57 , respectively . the dotted line indicates the @xmath7 model and the dashed line indicates the eds model . , title="fig:",width=192 ] in the figure ( [ u2 ] ) , we represent @xmath2 in terms of @xmath73 for the @xmath7 model , the eds model and the tachyon scalar field model . in the eds model , @xmath2 is independent of the redshift , thus it has a constant value i.e. @xmath92 . in the @xmath7 model , @xmath2 is smaller than 0.5 but it approaches the value of the eds model at high redshifts . in the tachyon scalar field model , @xmath2 drives more slowly than the @xmath7 and the eds models but its value approaches the value of the eds model at high redshifts . therefore , we can conclude that the size of structures in the @xmath7 model is larger than the tachyon scalar field model . \(b ) the variation of @xmath4 - @xmath73 for the @xmath7 model , the eds model and the tachyon scalar field model . , respectively . the dotted line indicates the @xmath7 model and the dashed line indicates the eds model . , title="fig:",width=192 ] in the figure ( 6-a ) , we represent @xmath3 in terms of @xmath73 for the @xmath7 model , the eds model and the tachyon scalar field model . in the eds model , @xmath3 is independent of redshift thus it has a constant value i.e. @xmath93 . in the @xmath7 model , @xmath3 is larger than 5.6 but its value approaches the value of the eds model in terms of high redshifts . in the tachyon scalar field model , @xmath3 drives faster than the @xmath7 model and the eds model but its value approaches the value of the eds model in terms of high redshifts . therefore , we can conclude that in the tachyon scalar field model the overdense spherical regions in terms of @xmath73 are denser than the @xmath7 model and the eds model . in the figure ( 6-b ) , we show the virial overdensity @xmath4 in terms of @xmath73 for the @xmath7 model , the eds model and the tachyon scalar field model . in the eds model , @xmath4 is independent of redshift , thus it has a constant value , @xmath94 . in the @xmath7 model , @xmath4 is more slowly than 178 , but its value approaches the value of the eds model in terms of high redshifts . in the tachyon scalar field model , @xmath4 drives faster than the @xmath7 model and its value approaches the value of the eds model in terms of high redshifts . the evolution of virial overdensity parameter @xmath4 is the main quantity for the halo size . therefore , we can conclude that in the tachyon scalar field model the halo size is larger than the eds model and the @xmath7 model . the average comoving number density of halos of mass m is given by @xcite,@xcite @xmath95 where @xmath96 and @xmath77 are the multiplicity function and the background density , respectively and @xmath97 is given by @xmath98 here @xmath99 is the r.m.s of the mass fluctuation in sphere of mass m. we can use the formula given by @xcite @xmath100 where @xmath101 is the mass variance of the overdensity on the scale of @xmath102 , @xmath103 and @xmath104 are the mass and the radius inside a sphere . also , @xmath105 is given by @xmath106 where @xmath107 is the linear growth factor , @xmath108 and @xmath109,\label{2}\ ] ] where @xmath110 . ( [ 1 ] ) and ( [ 2 ] ) have a validation range @xcite . they express that the fitting formula predicts higher values of the variance for @xmath111 and the fitting formula predicts lower values of the variance for @xmath112 . following @xcite , we apply st mass function @xmath113\exp(-\frac{b\nu}{2}).\label{st}\ ] ] here @xmath114 , @xmath115 and @xmath116 . we use the mass function introduced by del popolo ( po mass function)@xcite @xmath117\nonumber\\ \exp\big[-0.4019 b\nu\big(1+\frac{0.5526}{(b\nu)^{0.585}}+\frac{0.02}{(b\nu)^{0.4}}\big)^{2}\big],\end{aligned}\ ] ] where @xmath118 . also , we use the mass function ( yny mass function ) presented in @xcite @xmath119\nu^{(\frac{d}{2 } ) } \exp \big[-(b\sqrt{\frac{\nu}{2}})^{2}\big].\label{y}\ ] ] here @xmath120 , @xmath121 , @xmath122 and @xmath123 . now , we represent the evolution of the st mass function with respect to @xmath124 ( @xmath125 in figure ( 7 ) for the tachyon scalar field model and the @xmath7 model . we can see that the evolution of st mass function with respect to @xmath124 is the same for tachyon scalar field and the @xmath7 models in the @xmath126 case , but it is different for tachyon scalar field and @xmath7 models in the @xmath127 case . ( b)the evolution of the mass function with respect to @xmath124 for tachyon scalar field model and @xmath7 model in the case @xmath127 . , respectively . the dotted line indicates the @xmath7 model . , title="fig:",width=192 ] now , using eqs . ( [ n ] ) and ( [ st ] ) , for the tachyon scalar field model and the @xmath7 model , we obtain the average comoving number density of halos of mass @xmath128 in the cases @xmath129 . in the figure ( [ jop ] ) , we can see explicitly the differences for the cases @xmath126 and @xmath127 . we can see that difference of the number densities of halo objects is negligible for small objects in the case @xmath127 . therefore , we can obtain the number density of halo objects for high mass , and we find that the number of objects per unit mass is increasing for high mass in the tachyon scalar field model . the evolution @xmath130$]- @xmath124 for the tachyon scalar field model and the @xmath7 model in the case @xmath127 . , respectively . the dotted line indicates the @xmath7 model . , title="fig:",width=192 ] also , using eqs . ( [ st ] ) , ( [ del ] ) and ( [ y ] ) , we can compare the various mass functions at @xmath131 in figure ( [ jop11 ] ) . we can see that the po mass function is larger than st mass function and yny mass function for all mass scales . in this paper , we have studied the evolution of spherical overdensities in tachyon scalar field model by assuming the scale factor behavior as @xmath0 . we have shown the evolution of the eos parameter , the evolution of the density parameter and the evolution of the dimensionless hubble parameter of tachyon scalar field model with respect to a function of @xmath56 for @xmath57 . we saw that at early times of the scale factor evolution , the growth factor in the tachyon scalar field model is faster than the eds and @xmath7 models . so , we concluded that the tachyon scalar field model makes the structure formation to occur sooner , in comparison to other models . at later times , however , we showed that the growth factor in the tachyon scalar field model is more slower than that of eds model . also , in the eds model , @xmath90 is independent of the redshift and thus it has a constant value @xmath91 . in the @xmath7 model , @xmath90 is smaller than 1.686 , but the time evolution of the linear overdensity approaches the value of eds model at high redshifts . in fact , at high redshifts , we have a matter dominated universe ( dust matter ) , but at lower redshifts we have a dark energy dominated universe , thus the structure formation occurs earlier . in the tachyon scalar field model , @xmath90 is driven more slower than that of @xmath7 and eds models , because in figure ( 1 ) , the hubble parameter in the tachyon scalar field model is larger than that of @xmath7 model . also , we have shown that in the eds model , @xmath2 is independent of the redshift , thus it has a constant value i.e. @xmath92 . moreover , the size of structures in the @xmath7 model was larger than that of tachyon scalar field model . in the eds model , @xmath3 is independent of the redshift , hence it has a constant value i.e. @xmath93 . we saw that in tachyon scalar field model , the overdense spherical regions with respect to @xmath73 were denser than those of @xmath7 and eds models . in the eds model , @xmath4 is independent of the redshift , so it has a constant value , @xmath94 . the evolution of virial overdensity parameter @xmath4 is the main quantity for the halo size . therefore , we find that in the tachyon scalar field model the halo size is larger than the eds and @xmath7 models . finally , we saw that the evolution of the st mass function with respect to @xmath124 is the same for tachyon scalar field and @xmath7 models in the @xmath126 case , but it is not the same for tachyon scalar field model and the @xmath7 model in the @xmath127 case . also , the evolution of the number density with respect to @xmath124 is the same for the tachyon scalar field and @xmath7 models in the @xmath126 case , but its evolution is not the same for the tachyon scalar field and @xmath7 models in the @xmath127 case . the difference of number densities of halo objects is negligible for small objects in the case @xmath127 case . therefore , in obtaining the number density of halo objects for high mass , we find that the number of objects per unit mass is increasing for high mass in the tachyon scalar field model . we would like to thank m. malekjani for giving us useful comments that helped us to improve the scientific content of the manuscript . l. r. , batista . r. c. , liberato . l. , rosenfeld . r. , 2007 , jcapp , 11 , 12 . r. j. , cole . s. , efstathiou . g. , kaiser . n. 1991 , apj , 379 , 440 . f. 1994 , apj , 433 , 1 . 2000 , nature 404 , 955 ; 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we study the tachyon scalar field model in flat frw cosmology with the scale factor behavior @xmath0 . we consider the spherical collapse model and investigate the effects of the tachyon scalar field on the structure formation in flat frw universe . we calculate @xmath1 , @xmath2 , @xmath3 , @xmath4 , @xmath5 $ ] and @xmath6 $ ] for the tachyon scalar field model and compare the results with the results of eds model and @xmath7 model . [ firstpage ] tachyon scalar field ; spherical over - densities .

You are an expert at summarizing long articles. Proceed to summarize the following text: there are currently several exciting proposals to use the ( 001 ) surface of silicon for the construction of atomic - scale electronic devices , including single electron transistors @xcite , ultra - dense memories @xcite and quantum computers @xcite . however , since any random charge or spin defects in the vicinity of these devices could potentially destroy their operation , a thorough understanding of the nature of crystalline defects on this surface is essential . the si(001 ) surface was first observed in real space at atomic resolution using scanning tunneling microscopy ( stm ) by tromp _ _ et . al.__@xcite in 1985 . in this study they observed the surface consisted of rows of `` bean - shaped '' protrusions which were interpreted as tunneling from the @xmath1-bonds of surface si dimers , thereby establishing the dimer model as the correct model for this surface . since then , stm has been instrumental in further elucidating the characteristics of this surface , and in particular atomic - scale defects present on the surface@xcite . the simplest defect of the si(001 ) surface is the single dimer vacancy defect ( 1-dv ) , shown schematically in figs . [ def1](a ) and [ def1](b ) . this defect consists of the absence of a single dimer from the surface and can either expose four second - layer atoms ( fig . [ def1](a ) ) or form a more stable structure where rebonding of the second - layer atoms occurs @xcite as shown in fig . [ def1](b ) . while the rebonded 1-dv strains the bonds of its neighboring dimers it also results in a lowering of the number of surface dangling bonds and has been found to be more stable than the nonbonded structure . @xcite single dimer vacancy defects can also cluster to form larger defects such as the double dimer vacancy defect ( 2-dv ) and the triple dimer vacancy defect ( 3-dv ) . more complex clusters also form , the most commonly observed@xcite example is the 1 + 2-dv consisting of a 1-dv and a 2-dv separated by a single surface dimer , the so - called `` split - off dimer '' . the accepted structure of the 1 + 2-dv , as proposed by wang _ et . based on total energy calculations,@xcite is shown in fig . [ def1](c ) and consists of a rebonded 1-dv ( left ) , a split - off dimer , and a 2-dv with a rebonding atom ( right ) . recently we have observed another dv complex that contains a split - off dimer , called the 1 + 1-dv , which consists of a rebonded 1-dv and a nonbonded 1-dv separated by a split - off dimer , as shown in fig . [ def1](d ) . here we present a detailed investigation of dv defect complexes that contain split - off dimers . using high - resolution , low - bias stm we observe that split - off dimers appear as well - resolved pairs of protrusions under imaging conditions where normal si dimers appear as single `` bean - shaped '' protrusions . we show that this difference arises from an absence of the expected @xmath1-bonding between the two atoms of the split - off dimer but instead the formation of @xmath1-bonds between the split - off dimer atoms and second layer atoms . electron charge density plots obtained using first principles calculations support this interpretation . we observe an intensity enhancement surrounding some split - off dimer defect complexes in our stm images and thereby discuss the local strain induced in the formation of these defects . finally , we present a model for a previously unreported triangular - shaped split - off dimer defect complex that exists at s@xmath2-type step edges . experiments were performed in two separate but identical variable temperature stm systems ( omicron vt - stm ) . the base pressure of the ultra - high vacuum ( uhv ) chamber was @xmath3 mbar . phosphorus doped @xmath4 and @xmath5 @xmath6 wafers , orientated towards the [ 001 ] direction were used . these wafers were cleaved into @xmath7 mm@xmath8 sized samples , mounted in sample holders , and then transferred into the uhv chamber . wafers and samples were handled using ceramic tweezers and mounted in tantalum / molybdenum / ceramic sample holders to avoid contamination from metals such as ni and w. sample preparation@xcite was performed in vacuum without prior _ ex - situ _ treatment by outgassing overnight at 850 k using a resistive heater element , followed by flashing to 1400 k by passing a direct current through the sample . after flashing , the samples were cooled slowly ( @xmath9 k / s ) from 1150 k to room temperature . the sample preparation procedure outlined above routinely produced samples with very low surface defect densities . however , the density of defects , including split - off dimer defects , was found to increase over time with repeated sample preparation and stm imaging , as reported previously.@xcite it is known that split - off dimer defects are induced on the si(001 ) surface by the presence of metal contamination such as ni , @xcite and w @xcite . the appearance of these defects in our samples therefore points to a build up of metal contamination , either ni from in - vacuum stainless steel parts , or more likely w contamination from the stm tip . after using an old w stm tip to scratch a @xmath10 1 mm line on a si(001 ) sample in vacuum and then reflashing , the concentration of split - off dimer defects on the surface was found to have dramatically increased , confirming the stm tip as the source of the metal contamination . figure [ sods ] shows an stm image of a si(001 ) surface containing a @xmath10 10% coverage of split - off dimer defects . the majority of the defects in this image can be identified as 1 + 2-dvs , however , two 1 + 1-dvs are also present , as indicated . the most striking feature of this image is the difference in appearance of the split - off dimers in contrast to the surrounding normal surface dimers . each split - off dimer in this image appears as a double - lobed protrusion , while the surrounding normal si dimers each appear as a single `` bean - shaped '' protrusion , as expected at this tunneling bias . @xcite line profiles taken across a 1 + 2-dv both parallel and perpendicular to the dimer row direction are shown in fig . [ sods](b ) . the line profile parallel to the dimer row direction agrees with previously reported profiles over 1 + 2-dvs and fits well with the accepted structure , @xcite as shown by the overlayed ball and stick model . the line profile taken perpendicular to the dimer row direction , however , clearly shows that the split - off dimer of this defect is separated into two protrusions while the neighboring si dimers are single protrusions . this is the first recognition and explanation of split - off dimers appearing as double - lobed protrusions . 1 surface with split - off dimer defects is shown in ( a ) . tunneling conditions for this image were @xmath11 v sample bias and 0.8 na tunnel current . line profiles are taken across a single 1 + 2-dv both parallel , x x@xmath12 ( b ) , and perpendicular , y y@xmath12 ( c ) , to the dimer row direction , as indicated in ( a ) . the schematic ( d ) is a top view ball and stick model of a 1 + 2-dv with the approximate positions of @xmath1-bonds indicated by shaded ellipses . ] to understand why split - off dimers appear as double - lobed protrusions we must consider the structure of these defects shown in figs . [ def1](c ) and [ def1](d ) . normally si(001 ) surface dimers appear as `` bean - shaped '' protrusions in stm images because the dangling bonds of each si dimer atom mix to form a @xmath1-bond between the two dimer atoms . however , if we examine the split - off dimer structure closely ( figs . [ def1](c ) and [ def1](d ) ) we see that unlike normal surface dimers , the split - off dimer has two nearest neighbor second layer atoms that each have a dangling bond . the separation distance between the split - off dimer atoms and these second layer atoms is sufficiently close to allow the formation of @xmath1-bonds . the resulting four - atom structure can therefore be referred to as a _ tetramer_. we propose that the four dangling bonds of the split - off dimer tetramer interact primarily along the backbonds between the split - off dimer atoms and the second layer atoms to form @xmath1-bonds down the backbonds , as drawn schematically in fig . [ sods](c ) . these two spatially separated @xmath1-bonds therefore lead to the double - lobed appearance of the split - off dimers under low bias filled - state tunneling conditions , which we confirm in section [ theory1 ] with charge density calculations . in an attempt to fully characterize the appearance of these split - off dimers in stm images , we have performed a series of experiments observing split - off dimers with changing stm sample bias . figure [ sodv ] summarizes our results , showing images where a 1 + 2-dv and a 1 + 1-dv located next to each other are observed at four different sample biases two filled - state images and two empty - state images . in the filled - state image of fig . [ sodv](a ) we see that at @xmath13 v the split - off dimer of both the 1 + 2-dv and the 1 + 1-dv appear as double - lobed protrusions similar to those in fig . [ sods](a ) . however , when the filled - state bias is increased in magnitude to @xmath14 v , fig . [ sodv](b ) , the split - off dimers become single - protrusions and appear very similar to the surrounding normal si surface dimers . this is because as the bias magnitude is increased towards @xmath14 v , the dimer @xmath15-bond and bulk states contribute increasingly to the tunneling current @xcite and the image of the split - off dimer reverts to the bean - shaped protrusion in the same manner as normal surface si dimers . in both of the empty - state images , figs . [ sodv](c ) and [ sodv](d ) , acquired at + 0.8 v and + 2 v , respectively , the appearance of the split - off dimers is very similar to that of the surrounding normal surface dimers . this is because under empty - state tunneling conditions electrons tunnel into the @xmath16-antibonding orbitals of the dimers , resulting in the normal si dimers appearing as double - lobed protrusions . @xcite it is therefore only under low bias magnitude filled - state tunneling conditions that split - off dimers appear significantly different to the surrounding normal si surface dimers . v , ( b ) @xmath14 v , ( c ) @xmath17 v , ( d ) @xmath18 v. ] another noticeable feature of figs . [ sods](a ) and [ sodv](a ) is the enhanced brightness of the 1 + 1-dv compared to the 1 + 2-dv . this is a reproducible effect that we attribute to an increased amount of surface strain induced by the 1 + 1-dv . figure [ strain ] shows a series of adjacent defects forming a short vacancy line channel in the surface . this channel is composed of individual 1-dv , 3-dv , 1 + 2-dv , and 1 + 1-dv defects ( see figure caption ) . in the filled - state image , fig . [ strain](a ) , there is a clear brightening of the dimers on one end of the 1 + 1-dvs and the dimers on both ends of the 1-dv , which is not present for the 1 + 2-dvs . in the empty - state image of the line of defect complexes , fig . [ strain](b ) , we notice that there is a darkening of the same dimers that are enhanced in the filled - state image . v , @xmath19 v , 0.15 na ) of a short chain of dvs in a si(001 ) surface . the individual defects are ( from top left to bottom right ) : 1 + 1-dv , 1 + 1-dv , 1 + 2-dv , 3-dv , 1 + 2-dv , 1 + 2-dv , 1 + 1-dv , 1-dv , and 1 + 2-dv . note the strain - induced brightening of the 1-dv and 1 + 1-dvs in the filled - state ( a ) and the corresponding darkening in the empty - state ( b ) ] owen _ et . _ , @xcite have shown using low bias stm and first principles calculations , that the dimers neighboring a rebonded 1-dv are enhanced in low bias filled - state stm images due to the strain induced by the defect shifting the surface states upwards in energy toward the fermi energy . this effect can be seen for the 1-dv in fig . [ strain](a ) , where the neighboring dimers in the same row as the 1-dv are enhanced in intensity , with the magnitude of the enhancement decaying with distance from the 1-dv . a very similar enhancement can be seen around the 1 + 1-dv sites in this image , with the split - off dimer in particular appearing much brighter than the surrounding normal surface dimers . however , for the 1 + 1-dv only the dimers on one end of the defect are enhanced in intensity while the dimers on the other end of the defect are not . this observation can be readily explained since the 1 + 1-dv is composed of a rebonded 1-dv adjacent to a nonbonded 1-dv ( fig . [ def1](d ) ) and owen _ et . @xcite have shown that while the rebonded 1-dv results in strain - induced image enhancement , the nonbonded 1-dv does not . the observation of an asymmetric strain - induced enhancement of the 1 + 1-dv in fig . [ strain](a ) can therefore be taken as an experimental confirmation of the structure of this defect ( fig . [ def1](d ) ) and the first application of the method of owen _ et . @xcite for identifying strain in more complex surface defect structures . the fact that the 1 + 2-dv causes no enhancement of its neighboring dimers over the surrounding normal surface dimers suggests that the 1 + 2-dv , unlike the 1-dv and 1 + 1-dvs , does not increase the strain of the surface . this at first seems strange , since the 1 + 2-dv involves a rebonded 1-dv similar to the 1 + 1-dv structure . however , wang _ et . _ @xcite have shown , using total energy calculations , that the junction formed between the 1-dv and the 2-dv to create the 1 + 2-dv releases the surface strain that is present when these two defects exist separately from one another . the stm data that we have presented here is therefore the first experimental verification of this calculation . the fact that both the 1-dv and the 1 + 1-dv show local enhancement due to strain , while the 1 + 2-dv does not , indicates that the 1 + 2-dv structure induces less local strain than the 1-dv . in their paper , owen _ et . do not present empty state stm images , nor do they consider empty states in their tight binding calculations . in fig . [ strain](b ) , we show an empty state image of the same line of defects shown in fig . [ strain](a ) . interestingly , in this empty state image the dimers that were enhanced in brightness surrounding the 1-dv and 1 + 1-dvs in the filled - state image are less bright than the surrounding si dimers in the empty - state image . this suggests that the strain associated with these defects causes the lowest unoccupied molecular orbital ( lumo ) of the adjacent dimers to also shift higher in energy , away from the fermi energy . to confirm the interpretation of our stm images , we have performed first principles electronic structure calculations of both the 1 + 2-dv and 1 + 1-dv complexes using the car - parrinello molecular dynamics program . @xcite valence electrons were described using goedecker pseudopotentials @xcite expanded in a basis set of plane waves with an energy cutoff of 18 rydbergs and the exchange - correlation functional was of the blyp form . @xcite slab calculations contained between 124 and 128 si atoms in a @xmath20 @xmath21 supercell , corresponding to six layers of vacuum in the @xmath22-direction , and all calculations were performed with gamma point sampling of the brillouin zone only . a reference calculation was performed with no surface vacancies and assuming the @xmath23 structure in which the dimers buckled alternately along the row . a single 256 atom calculation with a duplication along the y - axis confirmed that the effect of dispersion across the rows is minor as has been noted elsewhere . @xcite both zero temperature geometry optimization and high temperature molecular dynamics calculations were used to explore a variety of surface and second - layer bonding configurations for the 1 + 2-dv and 1 + 1-dv . the results confirm the configurations in figs . [ def1](c ) and [ def1](d ) are the lowest energy geometries of both defect complexes . the dimers are drawn symmetric in these schematics , however , the true minimum energy structure at zero temperature involved charge - transfer buckling of the si dimers . it is well known that at room temperature the barrier is sufficiently small for the dimers to flip - flop between the two equivalent configurations . @xcite our calculations show that the split - off dimer tetramer also has two symmetrically equivalent buckling configurations , with charge transfer between the atoms of the tetramer buckling adjacent atoms in alternate directions . by analogy with the normal dimers we can expect room - temperature stm measurements of the tetramer to image the average of the two configurations . the chemical potential was determined from a 512-atom bulk calculation , which yielded a formation energy of 0.85 ev for the 1 + 2-dv , similar to the value of 0.65 ev computed by wang _ at . al_. @xcite the 1 + 1-dv formation energy has not been previously reported , and we found it to be 1.13 ev . we note that this value is high , but this is consistent with the rarity of observation of the 1 + 1-dv in stm experiments . in fig . [ 1 + 2-dv ] we present a series of calculated electron density slices through various regions of the 1 + 2-dv marked by ( a ) , ( b ) , ( c ) , and ( d ) in the ball and stick schematic . the charge density shown in the figure is the sum of the occupied kohn - sham orbitals within 0.25 ev of the highest occupied molecular orbital ( homo ) . taking into account the @xmath24 ev surface band gap of si(001 ) and the n - type doping of the experimental samples , these states correspond approximately to the accessible states for a @xmath25 v sample bias and can therefore be directly compared to the experimental data in fig . [ sodv](a ) , which was acquired with a @xmath13 v sample bias . -bonding as inferred from the electron density ( see text ) . each electron density plot is an average of both buckling configurations , and the atomic positions and bonds are shown as black balls and sticks . the slices are ( a ) rebonded 1-dv edge dimer , ( b ) split - off dimer , ( c ) split - off dimer backbonds , ( d ) 2-dv edge dimer . ] the four charge density slices in fig . [ 1 + 2-dv ] show : fig . [ 1 + 2-dv](a ) the 1-dv edge dimer , fig . [ 1 + 2-dv](b ) the split - off dimer , fig . [ 1 + 2-dv](c ) the backbond of the split - off dimer , and fig . [ 1 + 2-dv](d ) the 2-dv edge dimer , as indicated schematically in fig . [ 1 + 2-dv](e ) . the charge densities of both buckling configurations of the dimers and backbond atoms are averaged , and the positions of the dimer and tetramer atoms are shown superimposed in both buckling configurations . in the case of the backbonds , the two configurations are not coincident , and so the atoms and bonds are shown in projection onto the plane in fig . [ 1 + 2-dv](c ) . the 1-dv edge dimer in fig . [ 1 + 2-dv](a ) shows a clear three - lobed character with significant overlap between the up - atom charge density of the two buckling orientations , and a single lobe beneath the plane of the surface at the mid - point of the dimer . density functional calculations by hata _ et . al . _ @xcite and tight - binding green s function calculations by pollman _ et @xcite have separately identified this three - lobed feature as being characteristic of @xmath1-bonding in flip - flop dimers on the silicon surface , and we can therefore take this three - lobed feature as a signature of @xmath1-bonding in this work . the backbond of the split - off dimer in fig . [ 1 + 2-dv](c ) connects a first - layer atom to a second - layer atom and also shows a three - lobed structure . by analogy with the surface dimer in fig . [ 1 + 2-dv](a ) we characterize this bond as having @xmath1-character and have indicated this by the shaded ellipse ( c ) shown in fig . [ 1 + 2-dv](e ) . the split - off dimer itself in fig . [ 1 + 2-dv](b ) , however , does not exhibit three - lobed character . instead , the split - off dimer has four lobes ; two located above the up - atoms of the dimer in each buckling configuration , and a second pair of spatially separated lobes beneath the bond . the calculations thus show that @xmath1-bonding occurs down the backbonds of the split - off dimer , but not across the dimer itself . the absence of the @xmath1-bond across the split - off dimer correlates with the double - protrusions observed in the stm images . finally , we also consider the charge density of the 2-dv edge dimer , fig . [ 1 + 2-dv](d ) , and note that it also exhibits three - lobed character , indicative of @xmath1-bonding . this gives the 2-dv edge dimer a bean - shaped appearance in the stm image , as for the 1-dv dimer in fig . [ 1 + 2-dv](a ) . a similar situation exists for the 1 + 1-dv charge density slices shown in fig . [ 1 + 1-dv ] the first three charge density slices , figs . [ 1 + 1-dv](a ) [ 1 + 1-dv](c ) , are analogous to the slices for the 1 + 2-dv as was the case for the 1 + 2-dv , the rebonded 1-dv edge dimer , fig . [ 1 + 1-dv](a ) and the split - off dimer backbonds , fig . [ 1 + 1-dv](c ) exhibit three - lobed @xmath1-like character , while the split - off dimer , fig . [ 1 + 1-dv](b ) exhibits four - lobed character , consistent with an end - on view of @xmath1-bonding down the backbonds . finally , another slice is presented in fig . [ 1 + 1-dv](d ) , which is through the nonbonded 1-dv edge dimer as indicated schematically in fig . [ 1 + 1-dv](e ) . it can be seen that the nonbonded 1-dv edge dimer appears quite different to the charge density slices discussed so far . in particular , we notice that the nonbonded 1-dv edge dimer has a much reduced charge density compared to the other slices , fig . [ 1 + 1-dv](a ) [ 1 + 1-dv](c ) . examination of the structure identifies strain as the characteristic that differentiates the dimer in fig . [ 1 + 1-dv](d ) from the other dimers . since the dimer in fig . [ 1 + 1-dv](d ) is part of a tetramer , one might expect its appearance to resemble the split - off dimer which is also part of the tetramer shown figs . [ 1 + 1-dv](b ) and [ 1 + 1-dv](c ) . however , a detailed examination of the simulated structure reveals that the nonbonded 1-dv tetramer is relaxed , since there is one adjacent dimer present , while the split - off tetramer is highly strained because of the rebonding in the second - layer . since the nonbonded 1-dv tetramer is much less strained , its occupied states lie further from the fermi level , explaining the charge reduction observed in calculations in fig . [ 1 + 1-dv](d ) . as discussed in ref . , the minimum energy arrangement of the electrons in a tetramer is one where the @xmath1-states are delocalized across the four atoms , to form three bonding segments , as indicated by the ellipses in fig . [ 1 + 1-dv](e ) . the charge density slice of fig . [ 1 + 1-dv](d ) is consistent with such an arrangement where the charge density is shared between @xmath1-like bonds on both backbonds and across the dimer atoms . we conclude that this charge density arrangement forms for the nonbonded 1-dv tetramer because it is allowed to relax . in the case of the split - off dimer , the tetramer is constrained by the rebonding and instead forms a higher energy configuration in which the @xmath1-bonds conjugate to form two @xmath1-bonds down its backbonds . -bonding as inferred from the electron density ( see text ) . each electron density plot is an average of both buckling configurations , and the atomic positions and bonds are shown as black balls and sticks . the slices are ( a ) rebonded 1-dv edge dimer , ( b ) split - off dimer , ( c ) split - off dimer backbonds , ( d ) nonbonded 1-dv edge dimer . ] having presented a detailed understanding of the electronic structure of previously observed split - off dimer defects in the si(001 ) surface using both stm and first - principles calculations , we now turn our attention to elucidating the structure of a previously unreported split - off dimer defect . in figs . [ triangular](a ) and [ triangular](b ) we show filled- and empty - state stm images of dv defects at a single - layer s@xmath2-type step edge . at the top of these images white arrows indicate are three defects known as s@xmath2-dvs , which are rebonded 1-dvs at the step edge , which leave a single split - off dimer as the last dimer before the lower terrace begins . @xcite as was the case for the 1 + 1-dv and 1 + 2-dv , the split - off dimers in s@xmath2-dvs appear as double - lobed protrusions under low - bias filled - state imaging conditions , fig . [ triangular](a ) . at the bottom of fig . [ triangular](a ) two similar dv complexes can be observed , as indicated by black arrows , however these defects have a third protrusion giving them a triangular appearance . in empty - state imaging , fig . [ triangular](b ) , however , the additional third feature is not present . these triangular - shaped defects have not been reported on the si(001 ) surface before and most likely arise due to the presence of w contamination . 1.2 v ) of dv defects at an s@xmath2-type step edge . white arrows indicate s@xmath2-dvs , @xcite while black arrows point to a previously unreported defect that exhibits a third protrusion in the filled - state giving it a triangular appearance . we propose the structure ( c ) as a model for this defect . calculated charge density slices at a constant @xmath22-height for the dashed region of ( c ) are shown in ( d ) and ( e ) ( for kohn - sham orbitals summed over 0.45 ev below the homo and 0.45 ev above the lumo , respectively ) . these contour slices are in good agreement with the stm images in ( a ) and ( b ) , in particular predicting the correct spacing of 6.4 between the split - off dimer and third protrusion and also the disappearance of the third protrusion in the empty - state . the horizontal tic - marks in ( d ) and ( e ) indicate the dimer positions on the defect - free surface . ] our proposed structural model of the triangular - shaped defects in fig . [ triangular](a ) is shown in fig . [ triangular](c ) . this model consists of a nonbonded 1-dv defect at an s@xmath2-type step edge , followed by a rebonded split - off dimer and a bound si monomer . swartzentruber has previously observed si monomers on the si(001 ) surface using high - resolution stm after depositing a few percent of a monolayer of si atoms to the surface . @xcite these monomers were bound at rebonded s@xmath2-type step edges , confirming the minimum energy binding position predicted by first principles calculations . the binding position of the monomer in our proposed structure , fig . [ triangular](c ) , is essentially the same position observed by swartzentruber , with the difference being the presence of the dv defect adjacent to the step edge . swartzentruber also observed that the si monomers bound at s@xmath2-type step edges were visible in one bias polarity ( empty - state ) but invisible in the other ( filled - state ) . our images reveal a similar effect , however the feature we observe appears in filled - state images while being invisible in empty - state images . we have performed first - principles calculations to produce charge density contours for our proposed structure . figure 7(d ) shows a constant @xmath22-height contour slice taken 1.2 above the monomer for occupied kohn - sham orbitals within 0.45 ev of the homo . we see in this charge density contour slice the two lobes expected for the split - off dimer as well as a third lobe due to the bound monomer . moreover , the distance between the split - off dimer lobes and the monomer lobe is 6.4 in agreement with the separation seen in the stm image . in fig . [ triangular](e ) we show an empty - state slice taken at the same @xmath22-height and summed over kohn - sham orbitals up to 0.45 ev above the lumo . in this contour the double lobe of the split - off dimer is still present but the monomer lobe is significantly lessened in intensity . the results of our first - principles calculations therefore give good agreement between our proposed structure and the observed defect . the presence of the split - off dimer must therefore be responsible for the reversal of the filled- and empty - state monomer characteristics when compared to those observed for monomers bound to rebonded s@xmath2-type step edges . we have investigated split - off dimers on the si(001)2@xmath261 surface using high resolution stm and first principles calculations . we find that split - off dimers form @xmath1-bonds with second layer atoms which gives them a double - lobed appearance in low bias filled - state stm images . we apply the method of owen _ et . @xcite for identifying local areas of increased surface strain to dimer vacancy defect complexes and thereby present the first experimental confirmation of the predicted strain relief offered by the 1 + 2-dv . finally , we have presented a previously unreported triangular - shaped defect on the si(001 ) surface and a proposed model for this structure involving a bound si monomer .

dimer vacancy ( dv ) defect complexes in the si(001)@xmath0 surface were investigated using high - resolution scanning tunneling microscopy and first principles calculations . we find that under low bias filled - state tunneling conditions , isolated ` split - off ' dimers in these defect complexes are imaged as pairs of protrusions while the surrounding si surface dimers appear as the usual `` bean - shaped '' protrusions . we attribute this to the formation of @xmath1-bonds between the two atoms of the split - off dimer and second layer atoms , and present charge density plots to support this assignment . we observe a local brightness enhancement due to strain for different dv complexes and provide the first experimental confirmation of an earlier prediction that the 1 + 2-dv induces less surface strain than other dv complexes . finally , we present a previously unreported triangular shaped split - off dimer defect complex that exists at s@xmath2-type step edges , and propose a structure for this defect involving a bound si monomer .

You are an expert at summarizing long articles. Proceed to summarize the following text: the structure and heating of the solar corona , as well as the acceleration of the solar wind , are influenced by the structure and topology of the large - scale coronal magnetic field . on this basis , the existence of a planet at a distance of 0.1 au or less @xcite , with a strong internal magnetic field is expected to have a significant effect on the stellar magnetosphere , which is controlled by the magnetic field structure @xcite . in recent years , some signatures of this star - planet interaction ( spi ) have been observed . shkolnik et al . @xcite have reported on modulations in the ca ii k emission line , an indicator for chromospheric activity . they find enhancements in the line intensity that have the same period as the planetary orbital motion , though sometimes with a significant non - zero phase - shift . the cause is deemed magnetic and not tidal because of the lack of an equivalent hot spot offset in phase by @xmath0 . in addition , a statistical survey of the x - ray fluxes from stars with close - in planets has found them enhanced by a 30 - 400% on average over typical fluxes from similar stars with planets that are not close - in @xcite . direct x - ray observations of the hd 179949 system @xcite showed that the spi contributed @xmath130% to the emission at a mean temperature of @xmath1 1 kev . some analytical and semi - empirical arguments have been advanced to explain these observations . one posits that particles are accelerated along magnetic field lines that connect the star and planet , creating hot spots where they hit the chromospheric layer @xcite . as a result , hot spots are observed generally in phase with the planetary orbit , but with the capacity to have large offsets , depending on the exact structure of the magnetic field between the star and planet . another shows that transition of field lines from a high - helicity state to a linear force - free state is energetically adequate to power the enhanced intensities @xcite . the detailed behavior of the dynamical interaction of coronal and wind plasma with two magnetic field systems is , however , very difficult to realize with idealized models . the problem properly requires simultaneous descriptions of both the stellar and the planetary magnetospheres , the planetary orbital motion , and often asynchronous stellar rotation , together with a self - consistent stellar wind solution . here we describe an initial simulation of the magnetic star - planet interaction . we use idealized test cases to study the fundamental changes in the steady - state coronal structure due to the presence of the planet and its magnetic field . the dynamical interaction due to the planetary orbital motion is captured in an indirect manner . the numerical simulation has been performed using the university of michigan solar corona ( sc ) model @xcite , which is based on the bats - r - us global mhd code @xcite and is part of the space weather modeling framework ( swmf ) @xcite . the model solves the set of magnetohydrodynamic equations on a cartesian grid using adaptive mesh refinement ( amr ) technology . this model has been extensively validated for the solar corona using coronal observations and in - situ solar wind measurements taken at 1 au @xcite . we assume that the particular physical description of the coronal heating and wind acceleration is not crucial to study the change in the existing coronal structure due to the planet . it is important to mention that we use a _ global _ model for the corona that can not reproduce realistic chromospheric emission due to heating of coronal loops . we also do not fully describe the observed coronal heating , since for example no input from magnetic reconnection or loop footpoint motion is included . thus , while we adopt the physical parameters of some real systems in the modeling , we do not expect the models to fully reproduce all aspects of observations ( in particular , details of the temperature and level of the emissions ) at this point . the full physical description of the model and its limitations can be found in @xcite we performed several different numerical simulations , of which we highlight two here . case a : both the stellar and planetary magnetic fields are perfectly aligned dipoles . we set the stellar polar field to be @xmath2 and the planetary polar field to be antiparallel at @xmath3 ( i.e. , opposite to the stellar dipole ) . the planetary magnetic field is weaker than jupiter s , and follows the assumption that hot - jupiters are expected ( but not required ) to have lower spin rates due to tidal locking , and thus have weaker magnetic fields @xcite . we note that a simulation in which the planetary dipole was set to be in the same direction with the stellar dipole resulted in a quantitatively similar solution as in this case . case b : the planetary magnetic field is a perfect dipole and the stellar magnetic field is driven by solar magnetic synoptic map ( magnetogram ) . this map contains measurements of the photospheric radial magnetic field taken during solar maximum ( carrington rotation cr2010 , very active sun ) . the use of a magnetic synoptic map enables us to generate a realistic , sun - like , three - dimensional magnetic field . in case a , we mimic the relative motion between the planet and the background plasma by fixing the planet and rotating the star and the coronal plasma in the inertial frame . this way , the planet orbits the star backwards in the frame rotating with the star . this is done due to the fact that the actual orbital motion of the planet requires time - dependent boundary conditions . we plan to implement this technical improvement in future simulations . for the sake of definiteness , we partially match the parameters of the system to the observed parameters of hd 179949 @xcite , which is an f8v type star . we use the following stellar parameters : @xmath4 , @xmath5 , and stellar rotation period of @xmath6 . in the hd 179949 system , the planet is located at a distance of 0.045 au ( 9.65 stellar radii ) , and a @xmath7 phase - lead of a chromospheric hot spot is observed . the planetary parameter @xmath8 has not been used here . in case b , we fix the planet relative to the star and run the simulation in the frame rotating with the star . in this case we use solar parameters except for the planetary properties , which are the same as in case a. this case represents a steady - state , large scale interaction of a sun - like star with with a tidally - locked extrasolar planet ( e.g. , @xmath9 boo ; @xcite ) located at the same distance as before . in both simulations , the boundary condition for the planetary plasma number density and temperature were @xmath10 and @xmath11 respectively @xcite . the stellar boundary conditions were @xmath12 and @xmath13 respectively , based on previous simulation of the solar corona @xcite . to further aide interpretation of the results , we performed two additional simulations as a reference , identical to the cases above , but with the planet removed ; i.e. , considering the star with just the @xmath2 dipolar field , and with the cr2010 magnetogram . in each simulation , the set of mhd equations is solved until convergence . the end result is a three - dimensional , steady - state solution for the particular system that includes all the mhd variables ( density , pressure , velocity , and magnetic field ) . since the mhd solution contains the values for @xmath14 and @xmath15 at each spatial cell , we can perform the line - of - sight integration to obtain the predicted x - ray emissions for a particular view angle . the integration takes in to account cells in front of the star but omits cells behind it . we repeat this procedure for different view angles to mimic the predicted x - ray flux as the system rotates . the x - ray flux , @xmath16 , is calculated as the line - of - sight integral @xmath17 . here @xmath18 is the line - of - sight depth and we have used a piecewise linear approximation to the radiative loss @xmath19 for a plasma with solar photospheric abundances @xcite . simulation results are illustrated in figure [ fig : f1 ] . top and middle panels show the three - dimensional solutions excluding and including the planet , respectively . bottom panels show the difference in x - ray flux for temperature range of @xmath20 with and without the planet . left panels show results of case a , while right panels show results of case b. considering first case a , we note a key difference between simulations with and without the planet . in the former , magnetic field lines are conspicuously brought in toward the planet and are constricted due to the presence of the planetary magnetosphere . this has a palpable influence on the coronal electron number density , @xmath14 , which now increases azimuthally approaching the star - planet line to form an x - ray bright spot facing the planet . this solution is qualitatively very similar in case b , where there is also a clear longitudinal concentration of the plasma density , and consequently the x - ray flux ( which is proportional to @xmath21 ) . while we can not simulate chromospheric emission with our current models , its surface intensity distribution on the sun follows closely those regions of the disk that are brighter at euv and x - ray wavelengths . the results of these simulations are , then , fully consistent with the observed location of chromospheric hot spots seen in - phase with the planetary orbit . the longitudinal brightening effect is also clear in figure [ fig : f2 ] , where the line - of - sight x - ray flux originating at different plasma temperatures is shown as a function of viewing angle for the simulations including planets . in the idealized dipole field case , the emission is much more intense when the corona is viewed from the direction of the planet than from the opposite direction when the star hides the brighter parts of the corona that form the hot spot in phase with the planet ( a difference of 25 - 35% in observed x - ray flux ) . the simulation based on the realistic , complex magnetic field results in a hot spot ( 15 - 30% difference in x - ray flux ) shifted by about @xmath7 relative to the star - planet line . this suggests that the phase shifts between hot spots and planetary orbital phase seen in spi observations are probably due to the complexity of the stellar coronal magnetic fields and the consequent complexity of the magnetic connectivity between star and the planet . also of interest is the magnitude of the x - ray flux enhancement seen compared to the case where there is no planet . in the ideal dipole field case , we find enhancements of at least 10% in base emission and as much as @xmath180% relative to the no - planet case , and a contrast between minimum and maximum of 20 - 30% . these numbers are consistent with the observed signatures of spi found in x - ray observations . when solutions driven by realistic magnetogram data are considered , we find that enhancements in base emission by a factor of 10 are possible . the simulation with realistic , complex magnetic field results in much higher density enhancement and closing of coronal loops compared to the dipolar case . the mhd model , however , does not take into account the detailed physics of smaller scale x - ray emitting coronal loops , and the magnitude of the predicted x - ray enhancements should be considered approximate . the density enhancement in the closed - field zone provides a vital medium which can be heated to produce the spi effect . the compression alone produces significant increase in radiative loss at low temperatures ; localized dynamical effects , not modeled here , such as megnetic reconnection due to the planetary orbital motion , and particle acceleration then likely further heat the plasma to the observed @xmath22 1 kev temperatures @xcite . in conclusion , we find that a dominant physical effect creating observable time - variable spi signatures is that the existence of the planet and its magnetosphere , close to the star , prevents the expansion of the stellar coronal magnetic field and the acceleration of the stellar wind . the pressure gradient is not as large as it would be in the absence of the planet , so the coronal field lines that would be opened by the wind remain closed and the plasma in these loops do not escape . this effect alone reproduces three observable feature : 1 ) enhancement of total x - ray flux , 2 ) appearance of coronal hot spots , 3 ) phase - shift of the hot spots from star - planet line . the density enhancement results in low temperature coronal heating . we will further develop the model to include the planetary orbital motion in order to capture more dynamical effects of spi . this work has been inspired by initial study performed by noe lugaz . we thank an unknown referee for his / hers useful comments and ruth murray - clay for useful discussion . oc is supported by nsf - shine atm-0823592 grant , nasa - lwstrt grant nng05gm44 g . jjd and vlk were funded by nasa contract nas8 - 39073 to the _ chandra x - ray center_. simulation results were obtained using the space weather modelling framework , developed by the canter for space environment modelling , at the university of michigan with funding support from nasa ess , nasa esto - ct , nsf kdi , and dod muri . , g. , sokolov , i. v. , gombosi , t. i. , chesney , d. r. , clauer , c. r. , de zeeuw , d. l. , hansen , k. c. , kane , k. j. , manchester , w. b. , oehmke , r. c. , powell , k. g. , ridley , a. j. , roussev , i. i. , stout , q. f. , volberg , o. , wolf , r. a. , sazykin , s. , chan , a. , yu , b. , & kta , j. 2005 , journal of geophysical research ( space physics ) , 110 , 12226

since the first discovery of an extrasolar planetary system more than a decade ago , hundreds more have been discovered . surprisingly , many of these systems harbor jupiter - class gas giants located close to the central star , at distances of 0.1 au or less . observations of chromospheric hot spots that rotate in phase with the planetary orbit , and elevated stellar x - ray luminosities , suggest that these close - in planets significantly affect the structure of the outer atmosphere of the star through interactions between the stellar magnetic field and the planetary magnetosphere . here we carry out the first detailed three - dimensional magnetohydrohynamics ( mhd ) simulation containing the two magnetic bodies and explore the consequences of such interactions on the steady - state coronal structure . the simulations reproduce the observable features of 1 ) increase in the total x - ray luminosity , 2 ) appearance of coronal hot spots , and 3 ) phase shift of these spots with respect to the direction of the planet . the proximate cause of these is an increase in the density of coronal plasma in the direction of the planet , which prevents the corona from expanding and leaking away this plasma via a stellar wind . the simulations produce significant low temperature heating . by including dynamical effects , such as the planetary orbital motion , the simulation should better reproduce the observed coronal heating .

You are an expert at summarizing long articles. Proceed to summarize the following text: eddy currents are induced electric currents in a conducting material that result when either the object moves through a nonuniform magnetic field or is stationary but subject to a time - changing magnetic field . they were first observed by franois arago@xcite but it was not until michael faraday s discovery of induction@xcite that the mechanism of this phenomenon was understood . a pioneering experiment to quantify the heat dissipated by the eddy currents , and hence connect electromagnetism to thermodynamics , is von waltenhofen s pendulum@xcite [ figure [ fig : schematic](a)(c ) ] . the bob of the pendulum is a non ferromagnetic conducting plate which swings between the poles of an electromagnet , generating circulating eddy currents . the dissipative eddy currents tend to oppose the motion of the bob , and hence damp the pendulum . the experiment is an example of magnetic braking : when the magnet is turned off , the pendulum oscillates as a classic pendulum , damped only by air resistance and frictional forces in the hinge ; when the magnetic field is switched on , the pendulum comes to a rapid stop . von waltenhofen s pendulum is a popular demonstration in high school@xcite and introductory undergraduate@xcite classes because it provides a striking visual illustration of the invisible eddy currents . it is easy to set up , and gives the instructor a chance to discuss real - world applications such as magnetic braking in roller coasters , hybrid cars and the detection of flaws in conductive materials . moreover , by using different shaped plates as bobs , some of the characteristics of eddy currents can be explored : if the plate is now substituted with one where the slits do not reach the edges [ figure 1(d)(ii ) ] , the damping is only mildly altered . if a solid plate is replaced by a comb - like plate with slits cut into it [ figure 1(d)(iii)-(iv ) ] the damping effect is greatly attenuated . students infer from this that the eddy currents must be localized to some region of the plate . typically , however , the relationship between the shape of the plate and the damping is considered only qualitatively due to the complexity of the calculations required , which require advanced mathematical techniques introduced in higher - level electromagnetism classes . in this paper , therefore , we apply some of these strategies to predict the damping behavior of conducting plates with different geometries . we investigate four plate shapes , illustrated in figure 1(d ) : ( i ) a square plate , ( ii ) a plate with holes that do not reach the edge , ( iii ) a plate with with two slots , ( iv ) another with four slots . the present work was primarily performed during a project - based graduate electrodynamics class with the objective of giving students a real , tractable example of the utility of their calculations . it was also an opportunity to address the challenges of connecting theory to experiment , even where the fundamental theory is uncontroversial . the paper is organized as follows : in section ii , the form of the damping for the present experiment is derived first in terms of a simplified model suitable for undergraduate classes and the effect of these damping terms on the simple pendulum is explored . in section iii , we present a more sophisticated model derived from maxwell s equations in the magneto - quasistatic approximation . within this framework , the rate of power dissipation for each plate is estimated using a conformal mapping technique and a finite element solution . in section iv , the motion of the pendulum with each of the four plates is determined experimentally and compared with the predictions of each of the theoretical models in section v. in section vi , the pedagogical context of this work is discussed . brief conclusions are presented in section vii . schematic of von waltenhofen s pendulum . ( a ) an aluminium plate swings between the pole pieces of an electromagnet . as it passes through the fringing field , eddy currents are induced in the plate that dissipate energy and damp the pendulum . ( b ) as the plate passes through the center of the pole pieces , the rate of change in flux through the plate vanishes and so the eddy current momentarily vanishes . ( c ) as the pendulum swings back , the eddy current reverses direction . ( d ) various plate geometries under investigation in the present work ( to scale ) . ] the damped simple pendulum is , in the presence of dissipative forces , described by the equation , @xmath0 where @xmath1 is the rayleigh dissipation function , @xmath2 is the natural frequency of the pendulum and @xmath3 is a generalized coordinate . in the absence of the magnetic field , a stokes - like drag is assumed , @xmath4 arising from friction in the bearing and air resistance . in this section and the next , the form of the dissipation function in the presence of the magnetic field is derived , first for a simplified model suitable for introductory classes and second a more physically correct model based on the magneto - quasistatic approximation . the remaining theory section considers the effect of these terms on the solutions of ( [ eq : pendulumeqn ] ) . to develop the simplified model , we make the gross approximation that the eddy currents are completely localized to the boundary of the plate . then , the plate may be replaced by a thin closed loop of wire of equivalent shape with area @xmath5 , length @xmath6 , resistivity @xmath7 and with a normal unit vector @xmath8 . the magnetic flux passing through the loop is , @xmath9 where @xmath10 is the mean value of the magnetic field over the plate when it is located at a particular position @xmath11 . the emf induced around the loop is , by the faraday - lenz law and using the chain rule , @xmath12 the power dissipated in the loop is precisely the dissipation function desired and readily calculated , @xmath13 which , using @xmath14 , becomes , @xmath15 where the highlighted term represents the `` geometric factor '' arising from the shape of the plate . the dissipation function ( [ eq : rayleighdissipationfunction ] ) has the form of a position - dependent stokes drag . it is now necessary to solve the pendulum equation including the new dissipation terms ( [ eq : rayleighdissipationfunction ] ) , @xmath16\dot{q}=0,\ ] ] for which the form of the magnetic field must be known . here , the coordinate system is oriented as shown in figure 1 such that the plate normal @xmath8 is parallel to the @xmath17 axis , so only @xmath18 is required ; the origin is chosen to lie in the middle of the pole pieces . an approximation made in other studies@xcite is that the magnetic field is constant between the pole pieces and zero everywhere else , i.e. that , @xmath19 where @xmath20 is the width of the pole pieces . if the plate is square of side @xmath6 , the average value of the field over the surface of the plate when it is centered at @xmath11 along the @xmath21-axis may be determined , @xmath22 for the ansatz ( [ eq : bansatz ] ) , the flux through the plate increases ( decreases ) linearly as it enters ( leaves ) the field . depending on the size of the plate , and the amplitude of the motion , the plate may also experience portions of the motion where there is no change in flux . if the plate is the same size as the pole pieces @xmath23 , and never fully leaves the field , i.e. @xmath24 , then @xmath25 and the classical damped pendulum is recovered , but with a modified damping constant , @xmath26\dot{x}=0.\label{eq : equationofmotion}\ ] ] in order to gauge the validity of the ( [ eq : bansatz ] ) for the pendulum we calculated the spatial dependence of the magnetic field due to the pole pieces in the magnetostatic formulation , using a scalar potential function defined @xmath27 and subject to laplace s equation @xmath28 . this was solved using the finite element software ` flexpde ` for square pole pieces of width @xmath29 cm and separation @xmath30 cm . dirichlet boundary conditions were imposed on the pole pieces . a plot of @xmath18 along the horizontal centerline of the midplane of the pole pieces is displayed in fig . [ fig : fieldprofile](a ) , showing as expected that the field is constant between the pole pieces with the fringing field decaying rapidly outside . from this calculated profile , the flux @xmath31 through a square plate placed at different horizontal positions was also calculated ; the result is displayed in fig . [ fig : fieldprofile](b ) . notice that the flux depends linearly on position for @xmath32 cm , suggesting that the equation of motion ( [ eq : equationofmotion ] ) is likely to be valid for moderate amplitudes ; however around @xmath33 the flux depends quadratically on @xmath21 , suggesting that ( [ eq : equationofmotion ] ) is likely to be inaccurate for small amplitudes . ( a ) field strength as a function of distance @xmath21 through the center of the pole pieces . ( b ) magnetic flux through a square plate as a function of the position of the plate @xmath21 . ] since the eddy current is not strictly confined to the edges of the plate , it is necessary to develop a more sophisticated model by solving maxwell s equations @xmath34 together with the continuity equation @xmath35 to determine the true current density in the plate . here , the _ magneto - quasistatic _ approximation@xcite has been made , i.e. the maxwell correction to ampere s law is neglected . this is justified because the timescale of variation of the magnetic field over the surface of the plate is sufficiently long that the eddy current produced does not significantly alter the magnetic field in turn . to perform the analysis , consider a thin metal plate of uniform conductivity @xmath36 and thickness @xmath37 and where the shape of the plate is arbitrary . it is assumed that ohm s law holds for the plate , so @xmath38 in the frame of the laboratory , since the pendulum is rigid , the position of the plate may be described by the single generalized coordinate @xmath3 in eq . [ eq : pendulumeqn ] . as the pendulum swings , the plate passes through regions of different magnetic field strength . viewed from the point of view of the plate , the magnetic field across its surface changes as a function of time . this may be described by defining a local coordinate system @xmath39 relative to the plate such that the plate lies in the @xmath40 plane , and considering the magnetic field in this frame , @xmath41 we note that the reference frame @xmath39 is _ not _ an inertial frame as the pendulum undergoes periodic acceleration during the course of its motion ; however we neglect these effects as the pendulum velocity @xmath42 . by faraday s law , an electric field and hence an eddy current is induced in the plate : if @xmath43 the skin depth of the material it may be assumed that @xmath44 is oriented in the @xmath45 plane and does not vary significantly over the thickness of the plate ; hence the problem is quasi - two - dimensional . from equations ( [ eq : maxwell ] ) and ( [ eq : ohm ] ) , it is a standard textbook problem to show that the quantities @xmath44 , @xmath46 , @xmath47 and @xmath48 all obey the diffusion equation , @xmath49 since @xmath50 , the continuity equation implies that @xmath48 must have no divergence and hence can be written using a stream function , _ @xmath51 _ by inserting ( [ eq : jstreamfunction ] ) into ( [ eq : diffusioneq ] ) , together with the given applied magnetic field into maxwell s equations , it may be shown that the stream function obeys poisson s equation @xmath52 where @xmath53 here is the 2d laplacian , @xmath54 is the magnetic field strength and we defined a source term @xmath55 . since @xmath48 must lie tangentially to the edge of the plate , @xmath56 obeys a dirichlet boundary condition @xmath57 on all edges of the plate . from the solution for @xmath56 , the instantaneous rate of power dissipation by the plate is then calculated , @xmath58 where the integral is to be taken over the plate . inserting the stream function ( [ eq : jstreamfunction ] ) into ( [ eq : powerdissipation ] ) , integrating by parts and using ( [ eq : poisson ] ) , we obtain @xmath59 a formal solution to ( [ eq : poisson ] ) may be constructed using the green function for the laplacian , @xmath60 inserting this into ( [ eq : powerdissipationintermediate ] ) and rearranging we obtain the dissipation function , @xmath61\zeta(\mathbf{x}')\ da'\\ & = & -d\sigma b_{0}^{2}\dot{q}^{2}\int\int\zeta(\mathbf{x}',q)\zeta(\mathbf{x}'',q)g(\mathbf{x}',\mathbf{x}'')\ da'\ da''.\label{eq : powerdissipation-2}\end{aligned}\ ] ] since the source function @xmath62 depends on the position of the pendulum @xmath11 , this has the form @xmath63 where we define a local dissipation rate @xmath64 . to determine an effective damping coefficient , the calculations proceed as follows : in subsection [ sub : evaluating - the - fringe ] the form of @xmath62 is determined , then the positional dependence of the drag term is evaluated using a conformal mapping technique in subsection [ sub : conformal - mapping - solution ] , and also by solving ( [ eq : poisson ] ) with ` flexpde ` in subsection [ sub : finite - element - simulations ] ; finally , the effect of the position dependent damping on the motion of the pendulum is considered in the subsection [ sub : effect - on - the ] . before evaluating the power dissipation , it is first necessary to determine the functional form of the magnetic fringe field that the plate experiences as it moves between the pole pieces and hence the function @xmath65 . this could also be obtained from a finite element calculation as was done for the simplified model in section ( [ sec : simplified - model ] ) . we made the approximation that only the fringe fields from the sides of the pole pieces are significant in the damping , and the vertical extent of the pole pieces are neglected ; the formulation above remains sufficiently general that these details could easily be included in a more sophisticated calculation of @xmath65 . conformal mapping @xmath66 from ( a ) a portion of the strip @xmath67 to ( b ) the depicted region in the @xmath17 plane . places where the integrand in ( [ eq : scfringe ] ) vanishes are indicated by red circles ; these introduce corners into the map . contours of constant @xmath68 ( dashed lines ) and @xmath69 ( solid lines ) and their projection are indicated corresponding to equipotentials and field lines respectively . inset : plots of the field profile @xmath70 and its spatial derivative @xmath71 calculated from the conformal mapping . , width=288 ] within this approximation , the problem is two dimensional and hence can be solved using conformal mapping , an important technique for solving electrostatics problems . for a full discussion we refer the reader to @xcite and @xcite . briefly , however : a map from a domain to its image is called _ conformal _ if angles measured locally around a point in the source domain are preserved under the mapping . a key property of such maps is that if a function that is a solution to laplace s equation is constructed on some domain , then the projection of that function under the action of the mapping will _ also _ obey laplace s equation . boundary value electrostatics problems in two dimensions can hence often be solved using this method by constructing a solution to laplace s equation with appropriate boundary conditions in some simple domain and then finding a conformal map to the domain of interest . the powerful riemann mapping theorem guarantees such a map exists if the domains are simply connected ; mappings between multiply - connected domains may also exist if certain compatibility criteria are met@xcite . a rich source of such maps are analytic complex functions @xmath66 that map a domain in the complex plane @xmath72 to an image in the plane @xmath73 . the required mapping to find the fringe field is , @xmath74^{1/2}\ d\xi.\label{eq : scfringe}\ ] ] as shown in figure [ fig : cmfringe ] , this maps the strip defined by @xmath75 in the @xmath72 plane onto the depicted region in the @xmath76 plane ; the resulting shape resembles an overhead view of the edge of two parallel pole pieces . the integrand in ( [ eq : scfringe ] ) vanishes at @xmath77 and @xmath78 . the effect of these poles is to introduce corners in the map ; the angle of these is controlled by the strength of the pole , here @xmath79 . the factor of @xmath80 ) is required to ensure that the boundary of the @xmath81 part of the strip remains horizontal while the @xmath82 half is mapped to the vertical boundaries . the integral can be performed analytically , @xmath83 where the overall prefactor is chosen to ensure that the @xmath81 half of the strip is mapped onto @xmath84 . the magnetic field strength can be determined using this map as follows : in the space between the pole pieces @xmath85 and hence the magnetic field can be constructed from a scalar potential @xmath86 subject to laplace s equation @xmath87 . on the boundary , @xmath46 must be perpendicular to the pole pieces so these must be surfaces of constant @xmath88 . in the @xmath89-plane , the system resembles a parallel plate capacitor with the solution @xmath90 where @xmath5 and @xmath91 are arbitrary constants . since laplace s equation is invariant under the conformal mapping , this solution remains valid when projected into the @xmath17-plane : the lines of constant @xmath92 are equipotentials and lines of constant @xmath93 are the field lines as shown in figure [ fig : cmfringe ] . the magnetic field can be found from @xmath94 ; this can be expressed in complex form in the @xmath89-plane as @xmath95@xcite . evaluating @xmath66 and @xmath96 along the center of the strip @xmath97 yields @xmath98 along the midplane ; the @xmath21 derivative of @xmath99 required for the pendulum motion is evaluated from these using the chain rule , @xmath100\nonumber \\ b_{y}^{0 } & = & \left(1+e^{2\pi t}\right)^{-1/2}\nonumber \\ \frac{\partial b_{y}^{0}}{\partial x } & = & -\frac{1}{4}\pi\text{sech}(\pi t)^{2}.\label{eq : fringefield}\end{aligned}\ ] ] having tabulated values of * @xmath101 * using the above implicit expressions , a final form for @xmath65 was assembled from two appropriately translated and rescaled copies of @xmath101 as shown in the inset of figure [ fig : cmfringe ] ; this was used in subsequent calculations as an ` interpolatingfunction ` in _ mathematica _ , or exported as a table for use in ` flexpde ` . for the plates and pole pieces considered , it is actually the case that the plates were slightly larger than the width of the pole pieces ; while this could easily be accommodated by a simple rescaling of the solution ( [ eq : fringefield ] ) , it was found in practice not to affect the dissipation enough to affect the quality of the fit . ( a ) map from the unit circle to a paddle with a single slit of variable geometry . ( b ) map from a rectangular domain to the slit . the position of poles in the canonical domains is indicated with red circles ; these become vertices in the image domain . ( c ) two maps relevant to the experiment : i ) map from a circle to a square ; ii ) map from a rectangular domain to a square with two slitsnote significant crowding in the central spar . ] conformal mapping can be also used to evaluate the power dissipation of the plate as it passes through the fringing field . the strategy here is to find a mapping from a simple canonical domain on which the green function is known to one that approximates the shape of the plates ; then the integral ( [ eq : powerdissipation-2 ] ) can be computed . as will be seen , the choice of canonical domain is important to render the problem tractable . we therefore examined two choices : first , the mapping from the unit circle ( @xmath89-plane ) to a polygon of interest ( @xmath17-plane ) defined by an ordered set of vertices @xmath102 is given by the schwarz - christoffel formula@xcite , @xmath103 where @xmath104 is the exterior turning angle of the polygon at vertex @xmath105 and the @xmath106 are the points on the unit circle that are mapped to the vertices of the polygon @xmath102 . the constants @xmath5 and @xmath107 are translation and scaling factors respectively . while the set of parameters @xmath108 can be determined directly from the polygon , the positions of the parameters @xmath109 must be determined numerically@xcite . @xmath5 and @xmath107 are then determined to rotate and scale the result onto the polygon of interest . some representative results for a single slit are shown in fig . [ fig : conformalmapping-2](a ) . due to the symmetry of the problem , the position of three poles must be determined numerically with the remaining vertices provided by reflection . the second choice of canonical domain is a rectangular portion of the semi - infinite strip , @xmath110 , @xmath111 . a mapping onto the shape of interest is given by , @xmath112\right)^{-\alpha_{i}}\ d\xi\label{eq : scstripmap}\ ] ] where similarly @xmath106 are the position of the poles which produce the required corners in the image domain and @xmath104 are the turning angles ; this formula is a more general version of equation ( [ eq : scfringe ] ) used to determine the fringing field . results for the single slit are shown in fig . [ fig : conformalmapping-2](b ) . note that this map is approximate only : the left and right sides of the rectangle are mapped to the bottom of the paddlethe side that the slit is onbut their image under ( [ eq : scstripmap ] ) is not a straight line . hence , only two poles @xmath109mapped to the interior vertices of the slitneed be determined numerically ; the third parameter is the width of the rectangle @xmath113 . despite the approximate nature of the map , the distortion of the lower boundary is very small . these parameters were found by minimizing an objective function that was constructed from the @xmath114 norm of the difference between the ratios of the sides of the polygon and their desired values ; a notebook to do so is provided as supplementary material . robust numerical codes including ` scpack ` and the sc toolbox@xcite are available for more complex shapes . comparing the two choices of canonical domain in figure [ fig : conformalmapping-2 ] , an important difference is apparent . the poles in the maps from the unit circle tend to lie very close to one another , becoming in some cases indistinguishable in the figure , though they remain numerically distinct . as a consequence of this phenomenon , known as _ _ crowding__@xcite , very small regions adjacent to these crowded poles are mapped to large regions in the image domain . crowding is problematic , because it makes numerically finding the @xmath106 challenging ; it will also cause problems in evaluating the power dissipation . notice that using the rectangular source domain eliminates the crowding problem as the poles remain well - separated . in fig . [ fig : conformalmapping-2](c ) , we show two maps relevant to the experiment . first , the map from the circle to the square is famously @xmath115 an example with two slits that closely matches the experimental plate is shown in fig . [ fig : conformalmapping-2](c)(ii ) , constructed by introducing additional poles into one side of the one - slit rectangle maps above . unfortunately , the crowding phenomenon is visible in this map : a very small piece of the rectangle is mapped to the central spar of the plate . as will be seen later , this will reduce the accuracy of the power dissipation calculation when this spar coincides with the edge of the pole pieces . while it is possible to find maps with crowding localized to different portions of the plate@xcite , it can not be alleviated entirely if more than one slit is present . hence , we did not pursue maps for plates with more than 2 slits . turning , then , to the issue of evaluating the power dissipated by the induced current , we initially tried to use the eigenfunction expansion of the green function , @xmath116 which , inserted into the integral in ( [ eq : powerdissipation-2 ] ) allows one to obtain , @xmath117^{2}}{\lambda_{n}}\label{eq : conformalintegral}\end{aligned}\ ] ] the 2d integral in ( [ eq : ci1 ] ) can then be performed in the @xmath89-plane using the normalized eigenfunctions of the unit disk and the mapping @xmath66 , @xmath118 unfortunately , we found that the series ( [ eq : conformalintegral ] ) converges very slowly except for the square map ( [ eq : squaremap ] ) . the problem is further compounded for the unit circle by the fact that the integrand in ( [ eq : conformal2d ] ) is very sharply peaked and localized to the boundary in many cases due to crowding . for these reasons , we abandoned this approach and the unit circle maps as viable means to evaluate the dissipation . the power dissipation integral ( [ eq : powerdissipation-2 ] ) can be re - expressed in the @xmath89-plane as , @xmath119 in order to evaluate this integral , one requires a numerically efficient representation of the green function . unfortunately , the green s function for a rectangle can not be expressed in closed form , requiring expensive evaluation of elliptic integrals . recently , however , renewed interest in green function methods has yielded rapidly converging series representations@xcite . briefly , the strategy is to start from the eigenfunction expansion of @xmath120 ( [ eq : eigenfunctionexpansion ] ) on the rectangle and separate the double summation into two pieces : one that can be summed analytically and a second that remains to be performed numerically . fortuitously , the analytical part contains the inherent singularity ; the remaining numerical sum is a smooth function requiring few terms to converge to machine precision . equation ( 7 ) of @xcite presents an expression for the green function in a square ; we derived an equivalent expression for the rectangle @xmath110 , @xmath111 of interest , @xmath121 where @xmath72 and @xmath122 consistent with our earlier definitions . inserting ( [ eq : rapidgreenfunction ] ) , ( [ eq : scstripmap ] ) and @xmath65 derived in the previous section into ( [ eq : dissipation4d ] ) , we evaluated the local power dissipation rate @xmath64 as a function of @xmath11 for a variety of plate geometries . a notebook to perform these calculations is presented as supplementary material ; we defer consideration of the results until the subsequent section , in which the same function is calculated by finite element simulations . ( a ) dissipation rate as a function of position and ( b ) i - iv corresponding current density profiles obtained numerically for the various shapes from ` flexpde ` . ] finite element analysis is a commonly used numerical technique for solving partial differential equations in systems too complicated for analytical solution . here , the pde of interest is , as shown in the previous sections , poisson s equation with a spatially dependent source term ( [ eq : poisson ] ) . this must be solved to obtain the stream function @xmath123 for each of the paddle shapes of interest and at each point @xmath11 in the motion of the paddle . suitable input files for ` flexpde ` for each plate are included as supplementary material . each script contains field variable and parameter definitions , a definition of the pde , a description of the appropriate domain boundary and specification of the boundary conditions , and a list of the outputs desired , i.e. @xmath56 , @xmath48 and the value of the integral ( [ eq : powerdissipation-2 ] ) . a relative error of @xmath124 in the absolute @xmath56 was requested and used by ` flexpde ` to guide adaptive refinement . for each plate , the boundary condition on the exterior is simply @xmath57 , which forces the stream function to lie tangentially to the edge of the plate . the plate with holes requires special treatment : around each of the holes @xmath56 should be constant , i.e. the hole should be an equipotential surface , but the correct value is not _ a priori _ known . a way to obtain the correct solution is to exploit an electrostatic analogy , modifying poisson s equation by introducing a permittivity @xmath125 , @xmath126 the computational domain is extended so that @xmath56 is also defined inside the holes . different values of @xmath125 are used for the interior of the plate @xmath127 and the holes @xmath128 . as the ratio @xmath129 , the solution converges to the desired solution where @xmath56 constant inside the holes . for the purposes of calculation , @xmath127 is chosen arbitrarily to be @xmath130 and @xmath128 is successively increased ; the value of ( [ eq : powerdissipation-2 ] ) converged to a relative error of @xmath131 for @xmath132 . plots of the stream function @xmath56 and the corresponding current densities @xmath48 resulting from the calculations are displayed in figure [ fig : flexpde ] , together with a plot of the dissipation rate @xmath64 . comparison of finite element ( solid lines ) and conformal mapping ( points ) approaches to calculating the local rate of dissipation @xmath64 . ] it is now possible to compare the conformal mapping and finite element methods for this problem . to do so , we used both methods to calculate the dissipation rate @xmath64 for the plate geometries displayed in fig . [ fig : conformalmapping-2](b ) and ( c ) , i.e. those for which we were able to find appropropriate mappings . to perform the integral ( [ eq : dissipation4d ] ) , we used _ mathematica s _ ` nintegrate ` with the ` adaptivemontecarlo ` method and @xmath133 evaluation points . the calculation for each plate took @xmath134 minutes for ` flexpde ` and @xmath135 minutes for the conformal mapping approach on the same computer ( apple macbook air 11 late 2014 ) . results for the various plates are shown in fig . ( [ fig : flexpdecomparison ] ) . the agreement is very good for the single slit geometries , with the position and size of extrema reproduced well by both methods . for the two slit geometry , however , the conformal mapping method performs less well : the outer maxima are reproduced well , but the central one is not . this corresponds to the positioning of the plate where the edge of the pole piece coincides the central spar , where as discussed in section [ sub : conformal - mapping - solution ] the mapping is crowded . clearly from this comparison , the method of choice appears to be finite elements , since the results are less noisy and all plate shapes can be simulated using the method . having formulated both models , we turn to the effect of the dissipation on the motion of the pendulum . for both the simplified wire model of section [ sec : simplified - model ] and the magneto - quasistatic model of section [ sec : magneto - quasistatic - model ] , the equation of motion has the form , @xmath136 for the wire model , @xmath137\ ] ] as read off from eq . ( [ eq : equationofmotion ] ) ; for the magneto - quasistatic model the stokes drag is supplemented by the position - dependent dissipation rate , @xmath138 as is well - known , the equation of motion ( [ eq : generalequationofmotion ] ) with constant @xmath139 can be solved analytically yielding , @xmath140 where @xmath141 and @xmath142 are amplitude and offset parameters , @xmath56 is a phase and @xmath143 . hence , the motion remains oscillatory for @xmath144 . if the plate is centered with respect to the pole pieces , @xmath145 ; @xmath146 and @xmath56 similarly represent initial conditions that are experimentally controllable . rather than fit to the entire trajectory , it is convenient to look only at the extreme points where the pendulum is stationary and @xmath147 . inserting this into ( [ eq : solutionlinear ] ) and rearranging , we obtain @xmath148 showing that the values of the position of the pendulum at successive stationary points should obey a linear relation on a plot of @xmath149 versus @xmath150 . for the magneto - quasistatic model , analytical solution of ( [ eq : generalequationofmotion ] ) is impossible and hence the equation must be solved numerically . this was performed in _ mathematica_. because the strategy of looking at successive extrema on log scales remains a valuable fitting strategy even if the equation of motion is solved numerically , we additionally incorporated an extremum - finding routine to identify the stationary points from the calculated trajectory . with these simplified analytical and numerical magneto - quasistatic models , we are now ready to compare them with experimental data . ( a ) photograph of the experimental setup , designed as a classroom demonstration for introductory physics . ( b ) detail showing placement of camera and screen for imaging . ( c ) photograph of paddle shapes used . , width=316 ] in order to test the predictions of the calculations performed in previous sections , we performed an experiment to observe the motion of a pendulum with the different paddles attached and fit the models described in previous sections to the trajectories obtained . the magnetic brake apparatus used , shown in figure [ fig : app ] , was designed as a classroom demonstration for introductory physics classes . the apparatus consists of an electromagnet with flat pole pieces of adjustable separation , powered by a 12v car battery . a potential divider and ammeter were incorporated into the circuit so that the current through the magnet could be changed and measured . the pole pieces are @xmath151 cm and for the present experiment were separated by @xmath152 cm . the pendulum hangs from a supporting frame above the pole pieces and is made from a solid rod of length 39.5 cm to which different paddles of interest can be affixed with screws . four plates were studied , each of side 11.43 cm by 11.43 cm with different arrangements of holes or slits as shown earlier in figure [ fig : schematic](d ) . students collected videos of the motion of the pendulum using a cellphone camera , and in a later iteration a digital camera was used to achieve higher image quality and frame rate . although data logging hardware could also be used where available , the video approach was adopted as it is inexpensive , required no additional apparatus beyond the existing demonstration and is easily reproduced by others . a white piece of paper with a scale bar was attached to the pole piece behind the camera to provide a uniform background and facilitate quantitative analysis of the resulting videos . the camera was positioned so that the lens was aligned with the rod at rest ; its position was marked with chalk so that it could be easily replaced after each trial . the plates were released from the same point outside of the magnet , determined by a ruler placed on top of the pole pieces . the plates were released when the camera started recording and data was collected until the pendulum came to a rest . two sample frames from one of the movies are shown in figure [ fig : reslice](a ) . for each plate , 5 videos were collected with @xmath153 , @xmath154 , @xmath155 , @xmath156 and @xmath157 current flowing through the circuit respectively . selected videos were taken more than once to verify repeatability . these values obviously depend on the details of the circuit , but were chosen as follows : for the lower bound , a value was sought such that there was a visible difference between the motion of the plate with weakest damping ( the one with 4 slits ) in the on and off states . for the upper bound , we found a current level such that the plate with the strongest damping gave 2 - 3 oscillations before coming to a stop , i.e. we are in the oscillatory regime of @xmath144 in equation ( [ eq : generalequationofmotion ] ) . ( a ) sample frames from a video of the pendulum s motion . from this movie , the row of pixels indicated with the dashed line is taken from each frame and composited to produce the resliced image ( b ) . ] having collected the dataset of videos of the pendulum with different plates , the trajectories were extracted from the videos . this was done in two steps : first , the open - source image processing program ` imagej ` @xcite was used to `` reslice '' the movie , converting it from a stack of images of the @xmath158 plane at different time points @xmath150 to a new stack of images of @xmath159 at different vertical positions @xmath160 . an sample output is displayed in fig . [ fig : reslice](b ) , showing the damped motion of the pendulum as a function of time for the row of the movie indicated by the dashed line in fig . [ fig : reslice](a ) . the second step in the data extraction was performed by a custom program written in _ , included as supplementary material . this first subtracted the background , then for each time point located the position of the bob by finding the centroid of the pixels with intensity above a certain value . the result of this process was to yield for each movie a list of @xmath159 representing the trajectory of the pendulum . using the rulers and known frame rate of the camera ( 30fps ) , the list was calibrated to physical coordinates ; the program also subtracts the equilibrium position of the pendulum from the @xmath21 position . to facilitate comparison with the models , the extremum - finding routine mentioned in the previous section was used to identify a list of stationary points @xmath161 from each trajectory . a constant was subtracted from the times such that for each trajectory the first extremum was set to @xmath162 . having obtained a set of experimental trajectories , we now fit both models to the data . beginning with the simplified wire model , the analytical solution in section [ sub : effect - on - the ] implies that for each individual trajectory the extrema should follow a straight line on semi - log scales with slope @xmath163 from the set of results displayed as points in fig . [ fig : fit ] , it is seen that this is generally the case , though some deviations are apparent . in particular , the motion starts to decay more rapidly once @xmath164 ; we therefore excluded this data from our fit . for each plate , several values of current @xmath165 were used to take a succession of trajectories . although it is obvious that @xmath166 , we did not explicitly measure the magnetic field and therefore absorb this unknown constant of proportionality together with @xmath7 the effective resistivity of the wire into a single fitting parameter @xmath167 . this fitting parameter could , of course , in principle be calculated or measured separately , but the purpose of fitting here is to establish whether the model is consistent with the results for all plates simultaneously . the slopes extracted from the ensemble of plates with different @xmath168 ( calculated values are shown in table [ tab : fitsimplified ] ) at different applied currents @xmath165 should therefore universally be described by , @xmath169 estimates for the initial position of each trajectory @xmath141 were obtained by fitting a straight line to the extrema individually on semi - log scales . we then estimated @xmath170 for each plate from the @xmath171 trajectory for each plate ; the results are shown in table [ tab : fitsimplified ] . two pairs of plates seem to cluster around similar values ; these were collected at different times indicating that the natural damping of the apparatus , perhaps lubricant in the hinge , changed between them . .[tab : fitsimplified]geometric parameters and fitted @xmath167 , @xmath170 for each plate using the simplified model . [ cols="^,^,^,^",options="header " , ] results from the universal least - squares one - parameter fit for @xmath167 are shown in fig . [ fig : fit](a ) . as can be seen , the fit is nt bad and follows many of the trajectories quite well . that said , there are many discrepancies . we therefore estimated @xmath167 from each individual plate using the data at different applied field . while the details of these are not shown here , the quality of fit was very much better . the resultant values of @xmath167 are displayed in table [ tab : fitsimplified ] . in particular , it is not surprising that the plate with 4 holes is such an outlier as the wire model can not account for this geometry . values for the other plates are the same order of magnitude , which together with fig . [ fig : fit](a ) demonstrates that the simplified model gives reasonable estimates for the influence of plate geometry even though it is based on quite egregious approximations . in the same spirit , we assume that the damping term in ( [ eq : generalequationofmotion ] ) for the magneto - quasistatic model can similarly be described by a universal constant @xmath167 , @xmath172 for this model , as well as @xmath141 , @xmath170 , @xmath167 , @xmath173 and @xmath165 , it is also necessary to specify the natural angular frequency @xmath174 ; the value from the @xmath171 trajectory for each plate is used since @xmath175 for the weak damping in the absence of the magnetic field and hence @xmath176 . the results of the one - parameter universal fit for @xmath167 are displayed in [ fig : fit](b ) . the quality of fit is significantly better , especially for large amplitudes and early times , and more consistent between plates than for the simple model , implying that the magneto - quasistatic model yields good predictions of the effect of plate geometry . of particular importance is the fact that the fit for the 4 hole plate is no worse than that of the other plates , suggesting that these features have been handled correctly by the model . nonetheless , there remains an unexplained phenomenon in the data , i.e the anomalous increase in damping once the amplitude of the motion drops below @xmath177 . this is not predicted by either model . the simplified model predicts a constant rate of damping , while the magneto - quasistatic model predicts a crossover from magnetically dominated damping at large amplitudes to stokes dominated damping at small amplitudes . we tried adjusting the magneto - quasistatic model in various waysoffsetting the position of the plate with respect to the pole pieces and changing the size of the plate relative to the pole piecesbut found no similar effect for any reasonable parameters for the experiment . clearly , the stokes form of the non - magnetic damping is most likely at fault , because the source of damping is likely to be primarily the hinge rather than air resistance ; further analysis should be performed to determine the origin of this anomaly . the scientific results presented in this paper were obtained in part by students in spring 2013 as part of the electromagnetic theory ii class at tufts university , the second course in the e&m sequence in the graduate program at tufts ; two students from the class are co - authors on the paper ( cw and jw ) having spent a considerable time subsequently acquiring additional data and performing calculations . in this section , the pedagogical context of these activities is briefly documented in the hope that it is of use to others . the idea of incorporating ( simple ) experiments in class was motivated by an observation that key scientific skills expected of graduate studentsconnecting theory and experiment ; article reading ; journal selection and article preparationare not included explicitly in the curriculum . the project was spread over the second half of the semester as follows : 1 . a prepatory homework where students derived the magneto - quasistatic model [ sec : magneto - quasistatic - model ] . two @xmath178 hour class periods where students worked in groups . the students were divided into groups of four and worked on : i ) conformal mapping analysis , ii ) finite element analysis and iii ) experiments . the instructor provided scaffolding activities to help students learn flexpde and the schwarz - christoffel mapping technique . these activities continued subsequently outside of class . 3 . a reading activity whereby students had to identify possible journals for publication of this work ; this was followed by a 15 minute discussion of journal selection in class . 4 . a just - in - time teaching@xcite - like pre - class activity in which students had to identify the structural elements in a typical paper and construct an outline ; these were passed round anonymously for peer feedback . a homework activity in which students prepared an introduction for the paper . students were asked to read each other s work and identify strong features as well as possible modifications as a pre - class activity . this was followed by a discussion in class . each group collaboratively produced a draft section for the final paper as part of a homework ; these were combined by tja into a coherent document and extensively rewritten during the second iteration of experiments . the focus of the class and restricted time available necessitated some design trade - offs . for example , we chose to use an existing commercial software package , ` flexpde ` , as opposed to implementing a custom finite element program , for several reasons : first , the scale and complexity of problems solvable with a specialist code is much greater than with a simple custom - written program and hence should be applicable to situations encountered in student s research projects . second , the program contains advanced features such as adaptive refinementallocation of grid points based on estimated errorthat are time - consuming to implement . third , by removing the focus of the exercise from programmatic implementation , time can be spent on understanding how to properly interpret the output using fundamental principles of numerical analysis such as order , error estimation and convergence . while the project - based approach is well - grounded in a theory of learning ( constructivism ) , due to the small @xmath179 size of the class , it is difficult to rigorously demonstrate the effectiveness of the activity . future iterations would strongly benefit from pre and post testing . even so , the outcomes of the project indicate that students made considerable progress towards learning these challenging scientific skills . moreover , the class was very highly rated in the formal feedback mechanism . it is hoped that , in future , physics education researchers might attempt a thorough analysis of this and related approaches at the graduate level . the present work has provided a careful , quantitative analysis of a familiar classroom demonstration , the van waltenhofen pendulum . a magneto - quasistatic model has been shown to successfully predict the relative damping of a selection of plates with different geometry using conformal mapping and finite elements as tools to carry out the necessary calculations . as a by - product , a number of valuable resources for more general use , including plots of the current distribution during the course of the motion of the pendulum , have been produced . additionally , a greatly simplified model , suitable for use in introductory physics classes , has been formulated that could be used to make the classroom demonstration more quantitative . _ the authors wish to thank students from the tufts electromagnetic theory ii class in spring 2013 , the tufts department of physics & astronomy for providing equipment for the project , and badel mbanga and chris burke for helpful discussions . the authors contributed to the paper as follows : cw , jw and pw obtained the experimental data used in the paper ; conformal mapping solutions were obtained by jw and tja ; finite element simulations were performed by tja based on initial results obtained by students in the class ; the paper was written by tja from sections submitted by each group and revised collectively . _ 13ifxundefined [ 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 10.1098/rstl.1825.0023 [ * * , ( ) ] link:\doibase 10.1098/rstl.1832.0006 [ * * , ( ) ] @noop ( ) @noop ( ) @noop * * , ( ) @noop _ _ ( , ) @noop _ _ ( , ) @noop _ _ , cambridge monographs on applied and computational mathematics ( , ) @noop * * , ( ) @noop ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ , ed . 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we quantitatively analyze a familiar classroom demonstration , van waltenhofen s eddy current pendulum , to predict the damping effect for a variety of plate geometries from first principles . results from conformal mapping , finite element simulations and a simplified model suitable for introductory classes are compared with experiments .

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Proceed to summarize the following text: particles , magnetic fields , and radiation all contribute to the propulsion of relativistic jets , though the relative contribution of each is still an open matter of debate . in certain situations , however , the mechanism responsible for launching the jet operates simultaneously with the release of a large amount of energy in the form of radiation , making it likely that photons dominate the bulk energetics . this scenario occurs , for example , during the super - eddington phase of jetted tidal disruption events ( tdes ) , such as _ swift _ j1644 + 57 @xcite and _ swift _ j2058 + 05 @xcite . radiation - dominated jets should also be present in the collapsar picture of long gamma - ray bursts ( grbs ; @xcite ) , where the energy released in the form of gamma - rays is ultimately derived from accretion onto a black hole , the associated accretion luminosity exceeding the eddington limit of the hole by more than ten orders of magnitude . in both of these cases , the propagation of the radiation - dominated jet is modulated by the presence of a radiation pressure - supported environment ; for super - eddington tdes , this environment is in the form of a highly inflated , quasi - spherical torus of fallback debris @xcite , while a `` cocoon '' of shocked jet material @xcite and the overlying stellar envelope itself @xcite serve as the confining medium for grbs . in a companion paper ( @xcite , hereafter paper i ) , we presented a model that describes the viscous interaction of a radiation - dominated , relativistic jet with its surrounding medium . in that analysis we treated the jet and its surroundings as two separate fluids , interacting with one another via small anisotropies in the comoving radiation field that are explicitly accounted for in the equations of radiation hydrodynamics in the viscous limit . this two - stream approximation , in agreement with the non - relativistic analysis of @xcite , demonstrates the manner in which the shear between the two fluids carves out a region of low density material within the boundary layer between them . we also deduced the dependence of the boundary layer thickness on the asymptotic properties of the jet and the ambient medium . these models also show , however , that the contact discontinuity separating the jet and its surroundings , necessary for maintaining their respective identities , results in the likely unphysical vanishing of the density of scatterers along that surface of separation . the contact discontinuity also prevents the jet from entraining ambient material ; since the jet in the two - stream model we considered had an infinite amount of momentum , any entrainment or lack thereof is formally inconsequential to the evolution of the system . however , realistic jets those with finite extent will almost certainly engulf more material as they expand into their surroundings ; because the total amount of momentum in the system , which is realistically finite , must be conserved , that entrainment will then result in an overall deceleration of the outflow that can not be captured in the two - stream model . in view of these unphysical properties of the two - stream treatment the vanishing of the mass density of scatterers along the contact discontinuity and the lack of entrainment we present here an alternative boundary layer scenario to describe the interaction of a relativistic , radiation - dominated jet with its surroundings . this `` free - streaming jet '' model , which has a well - known counterpart in the non - relativistic , incompressible limit ( see chapter 10 , section 12 of @xcite ) , assumes that the jet is injected through a narrow opening into a static , homogenous medium , and that far enough from that opening the entire system can be modeled as a single fluid that is independent of the details at the injection point . by considering the jet and the ambient medium as one fluid we obviate the need for a contact discontinuity , which we demonstrate allows the density of scatterers to remain finite throughout the flow and for the jet to entrain material . in section 2 of this paper we present the equations of radiation hydrodynamics in the viscous limit . section 3 uses those equations to analyze the free - streaming jet model , and we demonstrate the existence of approximate self - similar solutions in the limit that the interaction between the jet and the ambient medium is concentrated in a thin boundary layer . in section 4 we discuss the implications of our model and make comparisons to the two - stream scenario , and in section 5 we conclude and consider the application of the free - streaming jet model to super - eddington tdes , grbs , and other astronomical sources . when changes in fluid quantities over the mean free path of a photon are small , radiation behaves like an effective viscosity and transfers momentum and energy between neighboring fluid elements . the precise form of the viscosity can be determined by investigating the general relativistic boltzmann equation , which was recently done by @xcite for the case where thomson scattering dominates the interactions between the photons and scatterers in the fluid rest frame . in this limit , they found that the relativistic equations of radiation hydrodynamics for a cold gas ( gas pressure much less than the gas rest mass density and radiation pressure ) are ( see their equation ( 49 ) ; see also paper i ) : @xmath0}+\frac{1}{3}g^{\mu\nu}\partial_{\mu}e ' \\ -\frac{8}{27}\nabla_{\mu}\bigg{[}\frac{e'}{\rho'\kappa}\pi^{\mu\sigma}\pi^{\nu\beta}\bigg{(}\nabla_{\sigma}u_{\beta}+\nabla_{\beta}u_{\sigma}+g_{\beta\sigma}\nabla_{\alpha}u^{\alpha}\bigg{)}\bigg { ] } \\ -\frac{1}{3}\nabla_{\mu}\bigg{[}\frac{e'}{\rho'\kappa}\bigg{(}\pi^{\mu\sigma}u^{\nu}+\pi^{\nu\sigma}u^{\mu}\bigg{)}\bigg{(}4u^{\beta}\nabla_{\beta}u_{\sigma}+\partial_{\sigma}\ln{}e'\bigg{)}\bigg { ] } = 0 \label{radhydroco}.\end{gathered}\ ] ] here the speed of light has been set to one , greek indices range from 0 3 , @xmath1 is the fluid rest frame mass density of scatterers , @xmath2 is the fluid rest frame radiation energy density , @xmath3 is the scattering opacity ( in units of @xmath4 g@xmath5 ) , @xmath6 is the metric of the spacetime , @xmath7 is the covariant derivative , @xmath8 is the four - velocity of the flow , and @xmath9 is the projection tensor . the einstein summation convention has been adopted here , meaning that repeated upper and lower indices imply summation . this equation also shows , in agreement with previous findings @xcite , that the coefficient of dynamic viscosity , @xmath10 , for an optically - thick , radiation - dominated gas is @xmath11 we will also write down the gas energy equation , obtained by contracting equation with the four - velocity @xmath12 , which gives ( see equation ( 50 ) of @xcite ) @xmath13 } \\ + \frac{8}{27}\frac{e'}{\rho'\kappa}\bigg{(}\nabla_{\sigma}u_{\beta}+\nabla_{\beta}u_{\sigma}+g_{\sigma\beta}\nabla_{\alpha}u^{\alpha}\bigg{)}\pi^{\mu\sigma}\nabla_{\mu}u^{\beta } \\ + \frac{1}{3}\pi^{\mu\sigma}\nabla_{\mu}\bigg{[}\frac{e'}{\rho'\kappa}\bigg{(}4u^{\beta}\nabla_{\beta}u_{\sigma}+\partial_{\sigma}\ln{e'}\bigg{)}\bigg { ] } \\ + \frac{1}{3}\frac{e'}{\rho'\kappa}\bigg{(}4u^{\beta}\nabla_{\beta}u_{\sigma}+\partial_{\sigma}\ln{e'}\bigg{)}\bigg{(}2u^{\mu}\nabla_{\mu}u^{\sigma}+u^{\sigma}\nabla_{\mu}u^{\mu}\bigg { ) } \label{gasenergyco}.\end{gathered}\ ] ] to close the system , we require that the normalization of the four - velocity be upheld and that particle flux be conserved : @xmath14 @xmath15 } = 0 \label{masscont}.\ ] ] equations and constitute six linearly independent equations for the six unknowns @xmath8 , @xmath2 , and @xmath1 . in addition to the energy density of the radiation , @xmath2 , one can also calculate the number density of photons by requiring that the number flux , @xmath16 , be conserved . one can show @xcite that the equation @xmath17 becomes , in the viscous limit , @xmath18 } \\ = \nabla_{\mu}\bigg{[}\frac{1}{\rho'\kappa}\bigg{(}\frac{10}{9}n'u^{\mu}\nabla_{\sigma}u^{\sigma}+n'u^{\alpha}\nabla_{\alpha}u^{\mu}+\frac{1}{3}\pi^{\mu\sigma}\nabla_{\sigma}n'\bigg{)}\bigg { ] } \label{fluxeq},\end{gathered}\ ] ] where @xmath19 is the rest - frame number density of photons . once we solve the equations of radiation hydrodynamics for the four - velocity of the fluid and the mass density of scatterers , we can solve equation for the number flux of photons . the goal of the next two sections is to apply equations and to the boundary layers established between fast - moving jets and their ambient media . for a more thorough discussion of the nature of the equations of radiation hydrodynamics in the viscous limit , we refer the reader to @xcite . in this section we consider the problem where a narrow stream of material is continuously injected into a plane - parallel , static , ambient medium , known as the free - streaming jet problem . if the reynolds number of the outflow is high , the transition between the stream of material and the external environment will be confined to a thin layer , permitting the use of a boundary layer approximation . the basic setup is similar to that of the two - stream problem ( see paper i ) , with the motion of the 2-d , plane - parallel injected stream predominantly along the @xmath20-direction , the majority of the variation along @xmath21 , no variation or velocity in @xmath22 , and the point of injection at @xmath23 . now , however , there is no contact discontinuity between the stream and the ambient environment , meaning that the entire system is considered a single fluid . we therefore have less freedom in prescribing the asymptotic characteristics of the jet and the environment ; however , this configuration permits mixing between the two media , allowing the outflow to entrain material , which almost certainly occurs in realistic jets . we can reduce the complexity of equations and by assuming that the interaction between the jet and the ambient medium takes place over a thin boundary layer of thickness @xmath24 , thin in the sense that @xmath25 is a small number when @xmath26 is a typical length along the jet . in this case , @xmath27 scales in an identical fashion to that derived for the two - stream boundary layer , which can be determined by comparing leading - order terms in the boundary layer thickness to the inviscid terms in the gas energy equation . setting the left - hand side of equation to @xmath28 , we find , as in paper i , @xmath29 here @xmath30 and @xmath31 are the jet velocity and lorentz factor , respectively , measured at some characteristic length along the jet axis @xmath32 , and @xmath1 is a characteristic density of scatterers throughout the outflow . due to the fact that the boundary layer thickness is the same , the boundary layer equations governing the outflow are identical to those found in paper i , which can be compactly written as @xmath33}+\frac{1}{3}g^{\mu\nu}\partial_{\mu}e ' = \frac{8}{27}\frac{\partial}{\partial{y}}\bigg{[}\frac{e'}{\rho'\kappa}\frac{\partial{u^{\nu}}}{\partial{y}}\bigg { ] } \label{bleqs},\ ] ] @xmath15 } = 0.\ ] ] in paper i we dealt with the @xmath34 and @xmath35 components of equation and its contraction with @xmath12 the gas energy equation . for the free - streaming jet problem , however , it will be more convenient to deal with the @xmath34 component of equation , the gas energy equation , and the contraction of equation with the projection tensor @xmath36 , which , as we will see , is the relativistic , viscous counterpart of the bernoulli equation . performing a few manipulations , we find that these equations become , respectively , @xmath37 @xmath38}+\frac{1}{3}e'\nabla_{\mu}u^{\mu } = \frac{8}{27}\frac{e'}{\rho'\kappa}\bigg{(}\frac{\partial{s}}{\partial{y}}\bigg{)}^2 \label{gasen1},\ ] ] @xmath39 } \label{zmom1},\ ] ] @xmath15 } = 0 \label{masscont1}.\ ] ] where @xmath40 the first of these shows that the radiation energy density , and consequently the pressure , is constant across the boundary layer . in the inviscid limit , equation can be transformed to give @xmath41 } = 0,\ ] ] which , as we mentioned , is the relativistic generalization of the bernoulli equation . we will also assume that the ambient energy density is independent of @xmath20 , i.e. , that @xmath42 , with @xmath43 a constant . because it will be convenient , we will change variables from @xmath21 to @xmath44 , where @xmath44 is given by @xmath45 which is related to the optical depth across the boundary layer as measured from the axis . in terms of this variable , equations and become , respectively , @xmath46 @xmath47 where we have used the assumption that @xmath42 . it should also be noted that the @xmath21-component of the convective derivative is now with respect to @xmath44 , not @xmath21 , i.e. , @xmath48 in the ensuing section we will seek self - similar solutions to these equations . we can immediately solve the continuity equation by introducing the stream function @xmath49 via @xmath50 @xmath51 where the factor of @xmath3 ensures that @xmath49 remains dimensionless . in paper i we showed that there exist self - similar solutions for @xmath49 , the velocity , the comoving density of scatterers and comoving density of photons in terms of the variable @xmath52 , where @xmath53 . one difference between the two - stream problem and this type of outflow , however , is that we expect the jet to expand into the ambient medium , entraining material in the process . therefore , as we look farther along the @xmath20-direction , the amount of inertia predominantly in the form of radiation for the systems that we are considering contained in the flow will increase . owing to the conservation of momentum , the @xmath20-component of the velocity should thus be a decreasing function of @xmath20 . in light of this observation , we will assume that the @xmath20-component of the four - velocity scales as @xmath54 where @xmath32 is a characteristic length scale in the @xmath20-direction , @xmath55 is a positive constant , for brevity we defined @xmath56 , and @xmath57 is a function of the self - similar variable @xmath58 ; @xmath59 is the characteristic boundary layer thickness in terms of @xmath44 , which , from equations and in the previous subsection , is given by @xmath60 , or @xmath61 we are most interested in the behavior of the properties of the outflow when the velocities are relativistic , as this is the limit that is most applicable to sources of astronomical interest , and in order for the self - similar nature of our solutions to be upheld , the dependence of our self - similar functions on the bulk properties of the outflow , such as the @xmath20-dependence of the jet lorentz factor along the axis , should be minimal . a fully self - similar solution is likely impossible here , as the speed of light plays a role in setting a finite scale factor for our solutions and becomes problematic when we try to connect the ultra- and non - relativistic regions of the outflow . we will show , however , that the non - self - similarity of our solutions only affects a small region of the outflow . in paper i , we showed that the self - similarity of the comoving density of scatterers , @xmath1 , was approximately satisfied , i.e. , we could find solutions with @xmath62 . for the case at hand , then , this assumption implies that @xmath63 is independent of @xmath64 . this means , however , that the observer - frame density of scatterers , given by @xmath65 , can be made arbitrarily large ( as @xmath63 is independent of @xmath64 in the self - similar limit ) , which is a nonsensical result . motivated by this reasoning , we conclude that the comoving density of scatterers is unlikely to be independent of @xmath64 . on the contrary , a more reasonable approximation is that the _ observer_-frame density of scatterers varies self - similarly . investigating equations and , we see that the @xmath1-dependent quantity that enters both is the combination @xmath66 . our expectation that the observer - frame density varies self - similarly then prompts the assumption @xmath67 where @xmath68 , @xmath69 being the density of scatterers in the ambient medium . ( note that we could have simply let @xmath70 , but equation , which is merely a change of variables from the initial assignment @xmath70 , will allow equations and to be written in a more compact form . ) we can determine the value of @xmath55 , which controls how rapidly the flow decelerates due to entrainment , by integrating the momentum flux , @xmath71 for a radiation - dominated system , over the entire boundary layer , and requiring that the result be independent of @xmath20 . this is equivalent to requiring that the total momentum contained in the outflow be conserved . integrating @xmath72 from @xmath73 to @xmath74 , using equations and , taking the ultrarelativistic limit and changing variables from @xmath21 to @xmath75 , we find @xmath76 since the integral in this equation is a constant that is greater than zero , we find that @xmath77 if the momentum flux is conserved . with this value of @xmath55 , the @xmath20-component of the four - velocity scales as @xmath78 and the self - similar variable @xmath75 is given by @xmath79 we then find from equation that the stream function must satisfy @xmath80 equation then gives the @xmath21-component of the four - velocity : @xmath81 } \label{vysim},\ ] ] where , both here and in future equations , a subscripted @xmath75 on the functions @xmath57 and @xmath63 denotes differentiation with respect to @xmath75 and the number of subscripts indicates the number of derivatives , i.e. , @xmath82 , @xmath83 , etc . inserting equations , and into equations and , changing variables from @xmath44 to @xmath75 and performing a bit of algebra , we find that they become , respectively , @xmath84 @xmath85 now , the solutions we seek for @xmath57 and @xmath63 should depend only on @xmath75 , as otherwise our assumption of the self - similarity of those functions breaks down . because of the complicated dependence of @xmath86 on @xmath87 , we see that the solutions will not be self - similar for arbitrary @xmath88 . however , in the ultrarelativistic limit , for which @xmath89 , we see that @xmath90 , and it is apparent that the solutions are indeed self - similar , i.e. , the @xmath64 dependence of equations and drops out . as these equations are third order in @xmath57 and first order in @xmath63 , we require four boundary conditions to solve them numerically . since @xmath87 is given by equation and it is assumed that @xmath91 along the axis , our first boundary condition is given by @xmath92 . for the second , we note that the jet axis is the streamline along which the @xmath21-component of the velocity is zero . from equation , then , we find @xmath93 ( we will see that @xmath94 at @xmath95 ) . the third boundary condition is obtained by noting that the @xmath20-component of the velocity should approach zero as we proceed into the ambient medium , so that , from equation , we find @xmath96 . finally , the density of scatterers should approach that of the ambient medium far from the jet center . equation then gives @xmath97 . as we noted , the truly self - similar limit of equations and is obtained by setting @xmath98 , but we encounter an issue with this scaling when we consider the boundary conditions on our flow in the ambient medium . in particular , we expect the outflow velocity to approach zero for @xmath99 ; when the velocity becomes subrelativistic , however , @xmath100 , when in actuality the @xmath101 limit of equation is @xmath102 . thus , we see that taking the ultrarelativistic limit of equations and will not result in the solutions matching the correct boundary conditions far from the jet center . to correct this problem , we will allow the function @xmath86 to take on its full form and treat @xmath64 as a constant , which will break the self - similarity of our solutions . however , since our boundary conditions do not depend on the value of @xmath64 , the functions @xmath57 and @xmath63 themselves should be largely independent of that parameter . we therefore expect the assumption of self - similarity to be upheld in the ultrarelativistic ( @xmath103 ) and the non - relativistic ( @xmath99 ) limits of our solutions , with small deviations from self - similarity in the trans - relativistic regime of the outflow . , which is the normalized @xmath20-component of the four - velocity , with @xmath75 for @xmath104 ( blue , solid curve ) , @xmath105 ( purple , dot - dashed curve ) , and @xmath106 ( red , dashed curve ) . as expected , the curves are all coincident when @xmath103 and @xmath99 , with a non - self - similar transition ( one that depends on @xmath64 ) in between those two limits.,width=336 ] , which is approximately the inverse of the lab - frame density , plotted with respect to @xmath75 for the same set of @xmath64 chosen in figure [ fig : fpsims ] . as was true for @xmath107 , @xmath63 is approximately self - similar close to and far from the jet , with the deviation from self - similarity in the trans - relativistic region being apparent but small.,width=336 ] to exemplify this point , figures [ fig : fpsims ] and [ fig : gsims ] show , respectively , the variation of @xmath107 the normalized @xmath20-component of the four - velocity and @xmath63 approximately the inverse of the observer - frame density in terms of the parameter @xmath75 for @xmath108 and 25 . as we anticipated , the functions for different @xmath64 are indistinguishable for @xmath103 and @xmath99 , meaning that the self - similarity is nearly exact in those regions . in between those limits , where the flow is trans - relativistic , the deviations from self - similarity are apparent , though they remain small . keeping the full form of @xmath86 in equations and therefore preserves well the self - similarity of our solutions and provides a reasonable interpolation between the relativistic and non - relativistic regions of the flow . the solution for @xmath63 approaches infinity as we near the axis , which can be seen by investigating the small-@xmath75 behavior of equation . specifically , letting @xmath109 , @xmath110 , @xmath111 , and @xmath112 , equation can be written as @xmath113 where @xmath114 is the second derivative of @xmath57 evaluated at @xmath95 . this equation can be integrated , and we find @xmath115 @xmath116 being a constant of integration . since it is roughly proportional to @xmath117 , the value of @xmath63 can not be negative , meaning that @xmath116 must be greater than zero . therefore , the asymptotic , @xmath103 behavior of @xmath63 is @xmath118 . in addition to the mass density of scatterers and the velocity , we can also calculate the number density of photons , @xmath19 , throughout the boundary layer . as we showed in paper i , the equation of photon number conservation becomes , to lowest order in the boundary layer thickness , @xmath119 performing a few manipulations , this equation becomes @xmath120 as was true for the density of scatterers , we expect the observer frame number density of photons to vary approximately self - similarly , which will be true if we let @xmath121 where @xmath122 is the number density of photons in the ambient medium and @xmath123 is a dimensionless function . inserting this ansatz into equation gives @xmath124 for the boundary conditions on @xmath123 , we first require that the number density of photons approach that of the ambient medium in the @xmath99 limit . equation then gives @xmath125 . for the second condition , return to equation , integrate both sides from @xmath73 to @xmath74 , and require that the derivative of @xmath126 vanish in both of those limits . doing so yields @xmath127 the right - hand side can be determined by returning to the continuity equation , integrating from @xmath73 to @xmath74 and performing a few manipulations to show that @xmath128 , where @xmath129 is the function @xmath57 evaluated at infinity . using the definition of @xmath19 in terms of @xmath123 , we find that equation becomes @xmath130 which serves as our second boundary condition on @xmath123 . this integral states that the increase in the number flux of photons occurs at a rate provided by the influx of material at infinity . the equations derived in this section were all written in terms of the variable @xmath75 , which is itself a function of @xmath63 via equations and . to write the solutions in terms of the physical parameter @xmath21 , we can return to equation , differentiate both sides with respect to @xmath21 , rearrange the resulting equation and integrate to yield @xmath131 where @xmath132 is a dummy variable of integration . once we calculate the functions @xmath133 and @xmath134 , this relation can be integrated and solved numerically to yield @xmath135 . this expression also shows that @xmath136 which we can use in equation to give @xmath137 equation also confirms that @xmath94 when @xmath95 , which we used in order to determine the boundary condition @xmath93 . in this section we plot solutions for the outflow velocity , the density of scatterers and the density of photons for various values of @xmath138 and @xmath64 . as was mentioned in the previous subsection , the physical self - similar variable against which we would like to plot our solutions is given by @xmath139 . however , as is apparent from equation , the definition of @xmath139 depends on @xmath64 . therefore , if we are comparing , for example , the outflow velocity of two systems with differing @xmath64 , we must incorporate the @xmath64 dependence in @xmath139 so that the range of physical space that we consider for each solution is the same . for this reason , in this section we will plot our solutions as functions of the variable @xmath140 -component of the four - velocity ( @xmath107 ) for @xmath141 and @xmath104 , 10 , and 25 ( the solid , blue curve , the dot - dashed , purple curve , and the dashed , red curve , respectively ) , which , for @xmath142 , correspond to @xmath143 and 25 . we see that the width of the boundary layer is nearly unchanged as we alter the value of @xmath64 . , width=336 ] . for all solutions the number density of scatterers approaches zero as we near the center of the jet . we see that the average number density of scatterers within the boundary layer is lower for larger lorentz factors.,width=336 ] used in figure [ fig : fpplotsgamjet ] . the photon number density closely follows that of the scatterers.,width=336 ] -component of the three - velocity normalized by @xmath144 ( see equation ) for the same set of parameters used in figure [ fig : fpplotsgamjet ] . for positive @xmath145 , each solution is initially positive and reaches a relative maximum before approaching a negative constant , which shows that the flow expands outwards near the center of the jet and entrains ambient material far from the axis . , width=336 ] figure [ fig : fpplotsgamjet ] shows the solution for @xmath107 , the normalized @xmath20-component of the four - velocity , for @xmath104 , 10 , and 25 . since @xmath146 , these values of @xmath64 scale approximately linearly with the lorentz factor until @xmath147 . the outflow velocity is maximized at the origin and decays as we move farther into the ambient medium . we see that the width of the boundary layer , loosely defined as the value of @xmath145 at which @xmath107 is some fraction of its central value , is nearly unchanged as we modify @xmath64 . the average value of the normalized @xmath20-component of the velocity is also slightly larger for larger @xmath64 . figure [ fig : rhoplotsgamjet ] demonstrates how the normalized , fluid - frame mass density of scatterers varies as we traverse the boundary layer for the same set of parameters chosen in figure [ fig : fpplotsgamjet ] . since @xmath63 approaches infinity as we near the origin , the mass density of scatterers , related to @xmath63 by equation , equals zero at the origin for all of the solutions , meaning that the center of the jet is evacuated of massive particles . the average comoving density of scatterers across the boundary layer is also lower for larger @xmath64 , in accordance with equation . in figure [ fig : hplotsgamjet ] we plot the normalized number density of photons for the same set of @xmath64 . it is evident that the density of photons closely follows the density of scatterers throughout the boundary layer . we see , however , that the density of photons stays above and below the density of scatterers as we move toward and away from the center of the jet , respectively . the photon density also remains finite at the center of the jet . the @xmath21-component of the three - velocity normalized by @xmath148 is illustrated in figure [ fig : vyplotsgamjet ] for the same set of @xmath64 used in figure [ fig : fpplotsgamjet ] . for @xmath149 , each solution initially has a positive @xmath150 , which shows that the jet material expands away from the axis . the @xmath21-component of the three - velocity then reaches a relative maximum , one which increases slightly for larger @xmath88 , before approaching a negative , constant value . this behavior is then inverted for negative @xmath75 . because the transverse velocity approaches a negative constant for @xmath151 and a positive constant for @xmath152 , we see that the jet entrains material from the ambient medium . , which is the normalized @xmath20-component of the four - velocity , for @xmath105 and @xmath153 , 1 , and 10 , which correspond to the blue , solid curve , the purple , dot - dashed curve , and the red , dashed curve , respectively . increasing the value of @xmath138 , we see , has little effect on the solution , while decreasing @xmath138 drastically widens the boundary layer.,width=336 ] chosen in figure [ fig : fpplotsmujet ] . the mean value of the density decreases as @xmath138 decreases.,width=336 ] chosen in figure [ fig : fpplotsmujet ] . this figure demonstrates , as we saw in figure [ fig : hplotsgamjet ] , that the density of photons tracks that of the scatterers.,width=336 ] -component of the three - velocity , normalized by @xmath148 , for the same set of @xmath138 chosen in figure [ fig : fpplotsmujet ] . we see that the relative maximum increases for smaller @xmath138.,width=336 ] figures [ fig : fpplotsmujet ] [ fig : vyplotsmujet ] illustrate how our solutions depend on @xmath138 . as is apparent , changing the value of @xmath138 does not drastically alter the qualitative aspects of the functions . we do see , however , that decreasing @xmath138 from 1 to 0.1 results in a large increase in the boundary layer thickness ; on the contrary , changing @xmath138 from 1 to 10 results in only a slight narrowing of its thickness . it is also evident that a smaller @xmath138 compared to 1 results in a lower average value of the density throughout the boundary layer and a larger peak in the transverse velocity @xmath150 . figures [ fig : fpplotsgamjet ] [ fig : vyplotsgamjet ] demonstrate that the width of the boundary layer is nearly independent of @xmath64 , which is the result of the scaling of our boundary layer thickness @xmath27 . specifically , note from equation that @xmath27 is given by @xmath154 . since our ansatz posited that @xmath155 , with @xmath63 a dimensionless function of order unity , and @xmath156 for @xmath157 , the boundary layer thickness is roughly independent of @xmath64 . figures [ fig : fpplotsmujet ] [ fig : vyplotsmujet ] illustrate that a value of @xmath158 causes the boundary layer thickness to increase dramatically compared to @xmath141 , while setting @xmath159 causes only a slight narrowing of the width compared to @xmath141 . this dependence is due to the fact that the viscous heating , which increases the specific entropy by decreasing the density of scatterers ( at fixed pressure ) , is most efficient when the flow is compressible , as is evident from the gas energy equation . the fluid only becomes compressible , however , when the flow is supersonic , and we can show @xcite that the sound speed of a radiation - dominated gas is @xmath160 when @xmath161 , the sound speed reduces to @xmath162 , and the location at which the flow becomes transsonic extends farther into the ambient medium , widening the boundary layer . conversely , when @xmath163 , the sound speed approaches a constant @xmath164 , which results in only a slight narrowing of the boundary layer . equation shows that the energy density of the radiation , and hence the pressure , is constant across the jet , which is a statement of the causal connectedness of the boundary layer . in order for this equation to remain valid , then , we require that the transverse sound crossing time over the boundary layer thickness @xmath24 be less than the time it takes the fluid to traverse the distance @xmath26 . since the transverse sound speed is given @xmath165 , where @xmath166 is given by equation , this requirement yields the inequality @xmath167 . once this inequality is no longer satisfied , equation does not hold , and we must include more terms in equations to account for the gradients in the energy density of the radiation . as we noted in section 3 , the density approaches zero as we near the axis of the jet . physically , this effect arises from the fact that , for @xmath168 , the lorentz factor grows unbounded and the boundary layer thickness goes to zero . therefore , the center of the jet originates along a curve of infinite shear and , consequently , infinite entropy . since the specific entropy scales as @xmath169 and @xmath2 is constant across the boundary layer , we see that the density of scatterers must equal zero at the center of the jet . from figures [ fig : vyplotsgamjet ] and [ fig : vyplotsmujet ] , we see that the @xmath21-component of the velocity initially causes the outflow to expand into its surroundings . when @xmath170 becomes large , however , the directionality of @xmath150 reverses toward the jet . the outflow therefore entrains material from the ambient medium not only by expanding in the transverse direction , but also by dragging material in from the environment . interestingly , the @xmath21-component of the velocity approaches a non - zero , constant value as we move into the external medium . this behavior arises from the fact that the jet is removing material from the system at a rate @xmath171 , showing that amount of mass excavated from the envelope increases as we move along @xmath20 . therefore , in order to maintain a steady - state , we require a constant influx of material at infinity that can resupply the amount lost due to the jet . this interpretation is substantiated by integrating the continuity equation from @xmath73 to @xmath172 , which shows that the @xmath21-component of the velocity at infinity scales as @xmath173 . the factor of @xmath174 arises from the fact that the mass loss rate scales as @xmath175 ; therefore , if one could decrease the mass loss rate to one that was constant in @xmath20 , then the @xmath21-component of the velocity would vanish at infinity . likewise , if one could create a scenario in which @xmath176 scaled as a _ negative _ power of @xmath20 , then the value of @xmath177 would maintain an efflux of material at infinity to keep the system from amassing inertia towards the center of the jet ( thus violating the steady - state assumption ) . the free - streaming jet solutions analyzed in this paper are only valid when the lorentz factor is relativistic , i.e. , when @xmath178 . when @xmath179 , @xmath180 , @xmath181 , and we can show that conservation of momentum along the @xmath20-axis results in the scaling @xmath182 , @xmath183 , which agrees with the results of the incompressible , non - relativistic theory ( in chapter 10 , section 12 of @xcite , see their discussion at the top of page 383 ) . therefore , once @xmath184 , the outflow will undergo a non - self - similar transition from the solution presented here to its non - relativistic counterpart . the free - streaming jet model has observational consequences . for example , the decrease in the density along the jet axis means that observers looking down the barrel of the jet see farther into the outflow . because @xmath185 and @xmath186 , where @xmath187 is the lab - frame radiation energy density , this means that those observers see a higher energy density of photons . also , those same observers , using the fact that the lab - frame radiation number density is @xmath188 , see an energy per photon of @xmath189 . therefore , not only do the on - axis observers see a more lorentz - boosted spectrum because they can see deeper into the outflow , their spectrum is also hardened from the fact that @xmath19 is minimized along the axis of the jet ( where @xmath190 in the figures of section 3 ) . paper i analyzed how the presence of a radiation pressure - supported envelope affected the propagation of a radiation - dominated , relativistic jet through the two - stream approximation . this approximation treats the jet and its surroundings as semi - infinite , separate fluids , a contact discontinuity serving as the surface of separation between the two . in contrast , the free - streaming jet solution presented here treats the entire system the jet and its surroundings as a single fluid . there is thus no formal distinction between `` jet material '' and `` ambient material , '' meaning that no contact discontinuity exists in the system . by comparing figure 1 of paper i and figure [ fig : fpplotsgamjet ] of the previous section , we see that the @xmath20-component of the four - velocity at a fixed @xmath20 behaves similarly between the two models . namely , the solution starts at some `` jet '' velocity , which corresponds to @xmath172 in the two - stream model and to @xmath191 in the jet model , and smoothly transitions to a velocity of approximately zero over the extent of a few boundary layer thicknesses . however , the full spatial dependence of the velocity , one that includes variation in the @xmath20-direction , differs drastically between the two models : while the two - stream jet maintains a constant @xmath88 along @xmath20 , we found here that the @xmath20-component of the four - velocity scales as @xmath192 , implying that the overall velocity of the jet slows as we look farther down @xmath20 . this behavior arises from the fact that the free - streaming jet can entrain material , this entrainment causing an increase in the inertia contained in the outflow and a resultant decrease in its velocity . the general behavior of the number densities of scatterers and photons between the models is also similar , which can be understood by comparing figures 2 and 3 of paper i to figures [ fig : rhoplotsgamjet ] and [ fig : hplotsgamjet ] of section 3 , respectively . in particular , both densities decrease within the boundary layer separating the outflow and its surroundings , asymptotically approaching their jet values as @xmath193 in the two - stream model and as @xmath194 in the free - streaming jet model . likewise , each approaches its ambient value as @xmath195 in the two - stream model and as @xmath196 in the free - streaming jet model . although the gross properties of both are similar , one striking difference arises , however , in the behavior of the scatterer density : in the two - stream model , the existence of the contact discontinuity causes the function @xmath63 , and hence the density @xmath1 , to vanish within the boundary layer ( it vanishes specifically at @xmath191 the location of the contact discontinuity ) . conversely , @xmath1 remains non - zero throughout the boundary layer that connects the jet and its surroundings in the model presented here , and only as we near the center of the jet does the density of scatterers go to zero . the presence of a contact discontinuity thus results in the likely non - physical vanishing of the massive particles within the boundary layer . finally , the scaling of the boundary layer thickness itself differs between the two models . in the two - stream case , we found that @xmath197 ( see equation ( 15 ) of paper i ) . therefore , as one looks farther down the @xmath20-axis , the boundary layer that develops between the jet and the ambient environment extends into both media at a rate proportional to @xmath198 . contrarily , equation shows that the free - streaming jet boundary layer expands into its surroundings as @xmath199 ( though this transitions to @xmath200 in the non - relativistic limit ; see the discussion at the top of page 383 in chapter 10 , section 12 of @xcite ) . the free - streaming jet boundary layer thus expands more rapidly than does the two - stream solution . employing the equations of radiation hydrodynamics in the viscous limit , which are applicable as long as changes in fluid quantities are small over the mean free path of a photon , we analyzed the dynamics of a relativistic , free - streaming jet under the boundary layer approximation . this approximation , which should be upheld in jets with transverse optical depths substantially greater than one , states that variations in the properties of the outflow are confined to a thin layer of width @xmath27 , and it allowed us to transform the full set of equations into a set of greatly simplified boundary layer equations . perhaps the biggest difference between the two - stream solutions , presented and analyzed in paper i , and the free - streaming jet solutions presented here is in the distinction between the jet and the ambient medium . in the former , the two are considered as distinct , interacting entities , which allows one to specify separately their asymptotic properties . the latter approach , on the other hand , considers the whole configuration as a single fluid . because one has more freedom in specifying the properties of the outflow , the two - stream solution has the added benefit of being able to treat scenarios in which the properties of the jet and the ambient medium differ significantly . however , maintaining the distinction between the two media necessitates the existence of a surface of contact between the two across which no fluid can flow , meaning that the jet can not entrain ambient material . furthermore , as was demonstrated in paper i , this boundary condition results in the density formally vanishing at the interface , which is likely non - physical . by treating the outflow and the environment as one fluid , we demonstrated that entrainment does occur in the radiation - viscous , free - streaming jet solution , which causes the @xmath20-component of the four - velocity of the jet to slow as @xmath201 ( this power - law , however , may differ if the adopted symmetry is azimuthal as opposed to planar ; see below ) . these solutions also show that the comoving densities of scatterers and photons decrease dramatically within the boundary layer , with the density of scatterers approaching zero as one nears the center of the jet . therefore , because observers that look `` down the barrel of the jet '' can see farther into the outflow , they see a more lorentz - boosted energy density than those that view the outflow off - axis . they also observe a higher energy per photon , given by @xmath202 , both because @xmath19 is minimized along the axis and because @xmath203 , the observed lorentz factor , is larger . such features are in qualitative agreement with the event _ swift _ j1644 + 57 , where such an observer orientation is invoked to explain the x - ray emission @xcite . in addition to super - eddington tdes and long grbs ( as well as the relatively new class of `` ultra - long '' grbs ; @xcite ) the two applications considered in the introduction the free - streaming jet model could be applied to other sources . as mentioned in paper i , this model may also be relevant to microquasars @xcite ( particularly those that fall in the class of ulx s ; @xcite ) such as the object ss 433 ( @xcite ; see @xcite for an application of the two - stream model to this source ) . additionally , a jetted quasi - star a protogalactic gas cloud supported by a supercritically accreting black hole @xcite provides another situation in which a radiation - dominated jet propagates alongside a radiation pressure - supported envelope . the free - streaming jet solution presented here is limited to describing plane - parallel , two - dimensional systems . one consequence of this assumption is that the entrainment of material , which slows down the jet as @xmath185 , happens effectively in one dimension . if , on the other hand , one imposed azimuthal symmetry and described the system in terms of spherical coordinates @xmath204 , which is likely more relevant for realistic jetted systems , the entrainment would occur in two dimensions . other conditions being equal , the slowing of the jet along the axis would then be more pronounced , i.e. , the lorentz factor would scale as @xmath205 with @xmath206 . the other main assumption of the free - streaming jet model presented here is that the energy density of the ambient medium is independent of @xmath20 , i.e. , @xmath42 . if a pressure gradient were present , this force would tend to accelerate ( or , in principle , decelerate ) the jet material , offsetting the power - law scaling @xmath185 . in fact , one can imagine that if the pressure gradient were strong enough , it could indeed reverse the overall slowing of the jet and cause the outflow to accelerate . if the energy density scaled as @xmath207 , which is relevant for super - eddington tdes as long as the constant @xmath208 satisfies @xmath209 @xcite , this situation could be actualized near the launch point of the jet . in an ensuing paper , we plan to compare more quantitatively the predictions made by the free - streaming jet model presented here and the observations of _ swift _ j1644 + 57 . we will also extend our analysis to incorporate a spherical geometry as azimuthal symmetry is almost certainly more relevant for this and other systems than the planar symmetry adopted here as well as radially - dependent ambient pressure and density profiles . arav n. , begelman m.c . , 1992 , apj , 401 , 125 arav n. , begelman m.c . , 1993 , apj , 413 , 700 begelman m.c . , king a.r . , pringle j.e . , 2006a , mnras , 370 , 399 begelman m.c . , rossi e.m . , armitage p.j . , 2008 , mnras , 387 , 1649 begelman m.c . , volonteri m. , rees m.j . , 2006b , mnras , 370 , 289 blandford r.d . , jaroszyski m. , kumar s. , 1985 , mnras , 215 , 667 burrows d.n . , kennea j.a . , ghisellini g. , et al . , 2011 , nature , 476 , 421 cenko s.b . , krimm h.a . , horesh a. , et al . , 2012 , apj , 753 , 77 coughlin e.r . , begelman m.c . , 2014a , apj , 781 , 82 coughlin e.r . , begelman m.c . , 2014b , apj , 797 , 103 coughlin e.r . , begelman m.c . , 2015 , apj , accepted ( paper i ) czerny b. , janiuk a. , sikora m. , et al . , apj , 2012 , 755 , l15 fabrika s. , 2004 , asprv , 12 , 1 fender r.p . , belloni t.m . , gallo e. , 2004 , mnras , 355 , 1105 king a.r . , davies m.b . , ward m.j . , et al . , 2001 , apj , 552 , l109 kundu p.k . , cohen i.m . , 2008 , _ fluid mechanics _ , academic press levan a.j . , tanvir n.r , fruchter a.s . , et al . , 2014 , apj , 781 , 13 loeb a. , laor a. , 1992 , apj , 384 , 115 lpez - cmara d. , morsony b.j . , begelman m.c . , lazzati d. , 2013 , apj , 767 , 19 macfadyen a.i . , woosley s.e . , 1999 , apj , 524 , 262 matzner c.d . , 2003 , mnras , 345 , 575 mszros p. , rees m.j . , 1993 , apj , 405 , 278 morsony b.j . , lazzati d. , begelman m.c . , 2007 , apj , 665 , 569 piran t. , 2004 , rvmp , 76 , 1143 rees m.j . , mszros p. , 1994 , apj , 430 , l93 woosley s.e . , 1993 , apj , 405 , 273 woosley s.e . , heger a. , 2006 , apj , 637 , 914 zauderer b.a . , berger e. , soderberg a.m. , et al . , 2011 , nature , 476 , 425

we analyze the interaction of a radiation - dominated jet and its surroundings using the equations of radiation hydrodynamics in the viscous limit . in a previous paper we considered the two - stream scenario , which treats the jet and its surroundings as distinct media interacting through radiation viscous forces . here we present an alternative boundary layer model , known as the free - streaming jet model where a narrow stream of fluid is injected into a static medium and present solutions where the flow is ultrarelativistic and the boundary layer is dominated by radiation . it is shown that these jets entrain material from their surroundings and that their cores have a lower density of scatterers and a harder spectrum of photons , leading to observational consequences for lines of sight that look `` down the barrel of the jet . '' these jetted outflow models may be applicable to the jets produced during long gamma - ray bursts and super - eddington phases of tidal disruption events .

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Proceed to summarize the following text: starting with the seminal work of unruh @xcite , it has now been recognised for 30 years that a uniformly accelerated observer in minkowski space sees minkowski vacuum as a thermal state in temperature @xmath0 , where @xmath1 is the magnitude of the proper acceleration . this result is of interest already in its own right within flat spacetime quantum field theory , and it has been confirmed by a number of methods @xcite . for relativists , the result is of particular interest because of its close mathematical similarity to the thermal properties of quantum fields in stationary black hole spacetimes @xcite . a conceptually concrete way to address quantum effects in accelerated motion is to analyse a particle detector coupled to the quantum field . for the uniformly accelerated motion , a subtlety in such an analysis arises from the fact that the motion is _ stationary _ , that is , the orbit of a timelike killing vector . because of stationarity , the first - order perturbation theory transition probability over the whole trajectory is infinite , owing to the infinite total proper time . however , this probability can be formally factorised into the product of the total proper time and a finite remainder , and the remainder can by stationarity be interpreted as the transition rate per unit proper time @xcite . this regularisation prescription can be extended from stationary trajectories in minkowski space to curved spacetime @xcite , both for the unruh - dewitt monopole detector @xcite and a variety of its generalisations . a recent review can be found in @xcite . for nonstationary motions the transition rate need not be constant along the detector s trajectory , and a regularisation that relies on stationarity is no longer available . the first message of this talk is : * for an unruh - dewitt monopole detector , the instantaneous transition rate is ill - defined without regularisation . this observation appears to have been first made by schlicht @xcite , who showed that a conventional @xmath2 regularisation yields a lorentz - noninvariant transition rate for uniformly accelerated motion . we discuss the mathematical reason for this phenomenon and show that the @xmath2 regularisation leads to a lorentz - noninvariant transition rate for _ every _ non - inertial trajectory . the conventional @xmath2 prescription does therefore not provide a physically acceptable regularisation for the instantaneous transition rate . schlicht proposed to regularise the transition rate in arbitrary motion by making the detector spatially extended in its instantaneous rest frame , with a spatial sensitivity profile that has a certain fixed shape but depends on a size parameter , and then letting the size parameter approach zero @xcite . he showed that this regularisation yields the expected planckian spectrum for uniform acceleration , and he analysed a selection of nonstationary trajectories via mainly numerical methods . we show that schlicht s regularisation yields a well - defined transition rate for every trajectory satisfying certain technical conditions , and we express the result as a manifestly finite integral formula that no longer involves regulators or limits . for the stationary trajectories the result agrees with that obtained in @xcite , and for nonstationary trajectories we extract asymptotic results that appear physically reasonable . the second message of this talk thus is : * a spatial sensitivity profile is a viable regulator for the instantaneous transition rate . the rest of the talk will put some flesh on these messages . the main conclusions rely on a particular choice of the spatial profile function , but in section [ sec : shape ? ] we present some evidence suggesting that that the zero size limit may be insensitive to the detailed form of the profile . the talk is based on @xcite , where further detail can be found . we work in four - dimensional minkowski spacetime with metric signature @xmath3 and in units in which @xmath4 . boldface letters denote spatial three - vectors and sans - serif letters spacetime four - vectors , and a square of a spatial vector ( respectively spacetime vector ) is understood in the sense of the euclidean ( minkowskian ) scalar product . consider a pointlike detector that moves in four - dimensional minkowski space on the world line @xmath5 , where @xmath6 is the proper time . we take the detector to have two quantum states , denoted by @xmath7 and @xmath8 , which are eigenstates of the detector internal hamiltonian @xmath9 with the respective eigenvalues @xmath10 and @xmath11 , @xmath12 . the detector is coupled to the real , massless scalar field @xmath13 with the interaction hamiltonian @xmath14 where @xmath15 is a coupling constant and @xmath16 is the detector s monopole moment operator , evolving in the heisenberg picture under @xmath9 . @xmath17 is a switching function , which specifies how the interaction is switched on and off by an external agent . suppose first that the switching function is smooth and has compact support , so that the initial and final states can be described in terms of the uncoupled system . if before the interaction the field is in the minkowski vacuum @xmath18 and the detector in the state @xmath7 , the first - order perturbation theory probability of finding the detector in the state @xmath8 after the interaction is @xcite @xmath19 where the response function @xmath20 is given by @xmath21 and the correlation function @xmath22 is the pull - back of the wightman distribution , @xmath23 as @xmath22 is a well - defined distribution on @xmath24 @xcite , the transition probability given by ( [ eq : total - probability])([eq : w ] ) is well defined . suppose then that no friendly neighbourhood external agent is available to switch the interaction off before we observe the detector . we wish to ask : _ what is the probability of finding the detector in the state @xmath8 while the interaction is still switched on ? _ this is arguably the question encountered in a practical measurement where one looks at an ensemble of accelerated detectors ( say , atoms or ions ) at a given moment of time and counts what fraction of the detectors are in an excited state . an attempt to answer this question within the detector model ( [ eq : hint ] ) would be to introduce in the switching function a sharp cutoff , @xmath25 , where @xmath6 is the proper time at which the detector is read and @xmath26 is the heaviside function . if we further push the switch - on to the asymptotic past , this would mean making in ( [ eq : total - probability])([eq : w ] ) the replacement @xmath27 . formal manipulations then yield for the @xmath6-derivative of the response function the expression @xmath28 @xmath29 differs from the instantaneous transition rate only by a multiplicative constant that is independent of the trajectory , and we shall from now on suppress this constant . the problem with these manipulations is that formula ( [ defexcitation - sharp - infty ] ) is ambiguous , since @xmath22 is a distribution with a singularity at the coincidence limit and the integration range has a sharp boundary at this singularity . to see that the problem is significant , suppose we go to a specific lorentz frame , @xmath30 , replace the wightman distribution by its conventional @xmath2 regularisation , @xmath31 and take the limit @xmath32 after performing the integral in ( [ defexcitation - sharp - infty ] ) . assuming that the trajectory is sufficiently differentiable and has suitable falloff properties in the distant past , the result is @xcite @xmath33 \ , \label{resultado1-noninvariant}\end{aligned}\ ] ] where @xmath34 . the last term in ( [ resultado1-noninvariant ] ) vanishes for inertial trajectories but is lorentz - noninvariant wherever the proper acceleration is nonzero . in the usual distributional setting of integrating against smooth test functions , the functions ( [ tradwightman - eps ] ) duly converge to the lorentz - invariant wightman distribution as @xmath35 @xcite , but the instantaneous transition rate ( [ defexcitation - sharp - infty ] ) falls outside this setting because of the sharp switch - off and retains a lorentz - noninvariant piece even in the limit @xmath36 . * moral * : the instantaneous transition rate ( [ defexcitation - sharp - infty ] ) is ill - defined as it stands and needs to be regularised . schlicht @xcite proposed to regularise the transition rate ( [ defexcitation - sharp - infty ] ) by giving the detector a spatial sensitivity profile that is rigid in the detector s instantaneous rest frame . this idea can be motivated by the observation that real material systems ( say , atoms or ions ) are not pointlike . technically , schlicht s proposal is to replace the field operator in the interaction hamiltonian ( [ eq : hint ] ) by a spatially smeared field operator , @xmath37 where @xmath38 are three unit vectors that together with the velocity @xmath39 form an orthonormal tetrad , fermi - walker transported along the trajectory . the four quantities @xmath40 are thus fermi - walker coordinates in a neighbourhood of the trajectory @xcite . the profile function @xmath41 is assumed to be non - negative and to integrate to unity , and @xmath42 is a positive parameter that determines the characteristic size of the smeared detector . when @xmath43 is chosen to be the lorentzian function , @xmath44 schlicht showed that @xmath22 in ( [ defexcitation - sharp - infty ] ) gets replaced by @xmath45 ^ 2 } \ , \ ] ] where the unprimed and primed quantities are evaluated respectively at @xmath6 and @xmath46 . note that @xmath47 ( [ corrschlicht ] ) is manifestly lorentz covariant . schlicht further showed that the @xmath48 limit yields the planckian spectrum for the uniformly accelerated motion , thus agreeing with the regularisation that relies on stationarity @xcite . he also examined the @xmath48 limit for a number of other trajectories , with physically reasonable results . schlicht s results have been generalised by p. langlois @xcite to a variety of situations , including minkowski space in an arbitrary number of dimensions , quotients of minkowski space under discrete isometry groups , the massive scalar field , the massless dirac field and certain curved spacetimes . langlois also observed that an alternative way to arrive at @xmath47 ( [ corrschlicht ] ) is to regularise the mode sum expression for the wightman function by an exponential frequency cut - off in the detector s instantaneous rest frame , rather than in a fixed lorentz frame . when the regularised correlation function ( [ corrschlicht ] ) is substituted in ( [ defexcitation - sharp - infty ] ) , the existence of an @xmath32 limit is not obvious for an arbitrary trajectory since @xmath42 appears under the integral . however , for trajectories that are sufficiently differentiable and have suitable falloff properties in the distant past , the limit exists and equals @xcite @xmath49 since the integrand in ( [ resultado1-infty ] ) remains finite at @xmath50 and since @xmath51 , formula ( [ resultado1-infty ] ) is manifestly well - defined . formula ( [ resultado1-infty ] ) gives the transition rate as split into its odd and even parts in @xmath11 . another useful split is into the inertial part and the noninertial correction , as introduced for stationary trajectories in @xcite . this can be accomplished by a suitable addition and a subtraction in the integrand , with the result @xmath52 the first term in ( [ inermasacc ] ) is the transition rate of a detector in inertial motion , and the integral term is thus the correction due to acceleration . as the correction is even in @xmath11 , we see that the acceleration induces excitations and de - excitations with the same probability . note that the correction term in ( [ inermasacc ] ) is nonvanishing for _ every _ noninertial trajectory . note also that inversion of the cosine transform in ( [ inermasacc ] ) shows that @xmath53 fully determines @xmath54 as a function of @xmath55 and @xmath6 . from ( [ inermasacc ] ) it follows that @xmath56 has a large @xmath57 expansion that proceeds in inverse powers of @xmath58 , with coefficients given by @xmath6-derivatives of @xmath5 . in the leading order we obtain @xmath59 which shows that for a generic trajectory the first correction to the inertial response is of order @xmath60 . a case - by - case analysis of all stationary trajectories shows that the transition rate ( [ resultado1-infty ] ) for them agrees with that obtained with the regularisation that relies on stationarity @xcite . in particular , in the special case of uniform acceleration of magnitude @xmath1 we have the planckian spectrum , @xmath61 as an example of nonstationary motion , consider a detector that moves in a timelike plane with the proper acceleration @xmath62 , where @xmath1 is a positive constant . in the distant past the trajectory is asymptotically inertial , and we obtain the transition rate @xmath63 where the @xmath64-term holds uniformly in @xmath11 . in the distant future the trajectory has asymptotically uniform acceleration of magnitude @xmath1 , and we obtain the transition rate @xmath65 where @xmath66 stands for a term that goes to zero as @xmath67 . the first term in ( [ eq : as - unruh - future ] ) is the transition rate ( [ eq : planck ] ) in uniform acceleration . the asymptotics thus agrees with what one would expect on physical grounds , both in the future and in the past . the above results rely on the choice ( [ lorentzian ] ) for the profile function . while all sufficiently regular profile functions are known to yield the same @xmath48 limit for inertial motion @xcite , it is at present not known to what extent the @xmath48 limit might depend on the profile function for more general motions . there is however a modified notion of spatial smearing in which we have been able to establish a result on profile - independence . for positive @xmath42 , the transition rate with this modified smearing reads @xmath68 where @xmath69 equations ( [ eq : smearedfdot - def ] ) and ( [ eq : g - def ] ) would follow from ( [ defexcitation - sharp - infty ] ) with the replacement ( [ eq : smeared - phi ] ) if it were known that the interchange of the @xmath70 and @xmath71 integrals is valid in a sense in which @xmath72 ( [ eq : g - def ] ) contains no distribution with support at @xmath73 . while we do not know whether the interchange can be justified in this sense , we shall take equations ( [ eq : smearedfdot - def ] ) and ( [ eq : g - def ] ) as a _ definition _ of a detector model in their own right , arguing that this model captures at least some of the effects of the spatial smearing of section [ sec : profile ] . now , if the trajectory is real analytic and satisfies suitable falloff conditions in the distant past , and if the profile function @xmath43 is smooth and has compact support , it can be shown @xcite that @xmath74 is well defined by ( [ eq : smearedfdot - def ] ) and ( [ eq : g - def ] ) for sufficiently small @xmath42 , and the @xmath48 limit exists and is given by ( [ resultado1-infty ] ) . as this limit agrees with that obtained with the lorentzian profile function ( [ lorentzian ] ) ( which is not of compact support ) , we suspect that the equivalence of the two models of spatial smearing could be established for at least some classes of profile functions . we have shown that regularising the transition rate of an accelerated unruh - dewitt detector on minkowski space by a spatial profile is a mathematically well - defined procedure and yields physically viable predictions in a number of situations . for the lorentz - function spatial profile ( [ lorentzian ] ) the zero size limit could be computed explicitly , leading to the transition rate ( [ resultado1-infty ] ) . for other spatial profiles the results remain to some extent inconclusive but they suggest that the zero - size limit may not be sensitive to the details of the profile . we re - emphasise that the need for a spatial smearing arose because we chose to address the instantaneous transition rate _ while the interaction continues to be switched on _ , rather than the total excitation probability after the interaction has been smoothly switched on and off by an external agent . it would be of interest to examine in comparison a pointlike detector whose smooth switching function is allowed to approach the step - function : might there exist limiting prescriptions that reproduce the effects of spatial smearing ? if the detector is turned on sharply at the finite proper time @xmath75 , the transition rate formula ( [ resultado1-infty ] ) is replaced by @xcite @xmath76 which is asymptotically proportional to @xmath77 as @xmath78 . the total transition probability , obtained by integrating the transition rate ( [ resultado1-conc ] ) , is therefore infinite , owing to the violent switch - on event , regardless how small the coupling constant in the interaction hamiltonian is . for the stationary trajectories the transition rate ( [ resultado1-infty ] ) of a detector switched on in the asymptotic past is constant in time , and the total transition probability is again infinite , now owing to the infinite amount of time elapsed in the past . in these situations one may therefore have reason to view our results , all of which were obtained within first - order perturbation theory , as suspect . however , in situations where the detector is switched on in the asymptotic past of infinite proper time and the total probability of excitation ( @xmath79 ) is finite , the first - order perturbation theory result should be reliable at least for the excitation rate , although the total probability of de - excitation ( @xmath80 ) then still diverges . this situation occurs for the asymptotically inertial trajectory discussed in section [ sec : examples ] , and we expect it to occur whenever the proper acceleration vanishes sufficiently fast in the distant past . it would be interesting to investigate to what extent our results can be generalised to the variety of situations to which schlicht s lorentzian profile detector was generalised in @xcite . for example , do the formulas ( [ resultado1-infty ] ) and ( [ resultado1-conc ] ) generalise to spacetime dimensions other than four , and if yes , what is the form of the subtraction term ? does the clean separation of the spectrum into its even and odd parts continue ? further , to what extent can the notion of spatial profile be employed to regularise the transition rate in a curved spacetime , presumably reproducing known results for stationary trajectories @xcite but also allowing nonstationary motion ? in particular , might there be a connection with the regularisation prescriptions of the classical self - force problem @xcite ? finally , would a nonperturbative treatment be feasible ? jl thanks the organisers of the neb xii meeting for the invitation to present this work and their kind hospitality , pierre martinetti for discussions during the meeting and the british council for travel support . jl acknowledges the hospitality and financial support of the isaac newton institute programme global problems in mathematical relativity " and the perimeter institute for theoretical physics and thanks howard j. magnuson for hospitality during the preparation of the manuscript . as was supported by an epsrc dorothy hodgkin research award to the university of nottingham . 99 poisson e 2005 in : _ proceedings of the 17th international conference on general relativity and gravitation _ ( dublin , ireland , july 1823 , 2004 ) , edited by florides p , nolan b and ottewill a ( singapore : world scientific ) ( _ preprint _ gr - qc/0410127 )

the instantaneous transition rate of an arbitrarily accelerated unruh - dewitt particle detector on four - dimensional minkowski space is ill defined without regularisation . we show that schlicht s regularisation as the zero - size limit of a lorentz - function spatial profile yields a manifestly well - defined transition rate with physically reasonable asymptotic properties . in the special case of stationary trajectories , including uniform acceleration , we recover the results that have been previously obtained by a regularisation that relies on the stationarity . finally , we discuss evidence for the conjecture that the zero - size limit of the transition rate is independent of the detector profile .

You are an expert at summarizing long articles. Proceed to summarize the following text: the transfer of a quantum state from one place to another is an important task in quantum information processing . a quantum state prepared by one party needs to be measured by another party at a distance . long distance communication between two parties , for example , in quantum key distribution @xcite , can be realized by means of photons . in this case , photons have the advantage that they have an extremely small interaction with the environment and also travel long distances quickly through optical fibers or empty space . however , for short distance communication , such as connecting distinct quantum processors or registers inside a quantum computer @xcite , conditions and requirements are different @xcite . recently , quantum communication through spin chains has been intensively investigated for this purpose @xcite . the primary scheme is that one quantum state is produced at one end of the chain ; it evolves naturally under spin chain dynamics ; and at some time @xmath0 we receive the state at the other end @xcite . for more complex systems , christandl et al . suggested a perfect state transfer algorithm which can transfer an arbitrary quantum state between two ends of a spin chain @xcite , or a more complex spin network christandl / etal:05 . in addition , researchers have investigated measurement - assisted optimal quantum communication by a single chain burgarth / etal:06 and parallel chain @xcite , the entanglement transfer through a heisenberg xy chain @xcite and parallel spin chains @xcite , enhancement of state transfer with energy current @xcite and entanglement transfer by phase control @xcite . however , due to the complexity of the problem researchers usually consider cases where the magnetization ( _ z _ component of the total spin @xmath2 ) is a conserved quantity , which means that @xmath3=0 $ ] . for the cases that the hamiltonian does not satisfy above conditions , l. amico et al . studied the dynamics of entanglement and found that the anisotropy of the hamiltonian has an evident effect on the evolution of entanglement @xcite . they calculated the entanglement using the out - of - equilibrium correlation functions , but did not investigate state transfer in these systems . the anisotropy and magnetic field effects on the entanglement transfer in two parallel heisenberg spin chains was later investigated in ref . @xcite . for the simple cases where each chain only has two spins , it was determined that perfect entanglement transfer can be realized between spin pairs by adjusting the magnetic field strength and the anisotropy parameter . in this paper , we study quantum communication in an anisotropic heisenberg xy chain with small number of particles @xmath4 and an arbitrary initial state . following the scenario of earlier work on state transfer through spin chains , we encode a state to be transferred at the first site of the chain and , without external control , let the chain freely evolve . after time @xmath0 the state is to be readout at the @xmath5 site of the chain . but different from the other cases with @xmath6=0 $ ] , the quantum communication channel , the spin chain , not only transfers the state , but also generates entanglement . this paper is organized as follows : in part two we will give our model hamiltonian and calculate the time dependence of the one - site correlation function which will be used in the expression of the fidelity and tangle . in part three , we discuss the time evolution of transmission fidelity and purity in a short chain for three regimes : strong - field , weak - field and intermediate regime . the final part is devoted to the conclusions . the hamiltonian of the anisotropic heisenberg xy chain in a uniform transverse magnetic field @xmath7 is given by @xcite@xmath8where @xmath9 is the @xmath10 spin operator ( @xmath11 ) at site @xmath12 , @xmath13 and @xmath14 are the anisotropic exchange interaction constants , and @xmath7 is the transverse magnetic field . we assume periodic boundary conditions , so that the @xmath15 site is identified with the @xmath16 site . the standard procedure used to solve eq . ( [ eq : anish ] ) is to transform the spin operators @xmath9 into fermionic operators via the jordan - wigner(j - w ) transformation @xmath17where @xmath18 are one - dimensional spinless fermions annihilation operators and @xmath19 @xmath20 . it will be convenient to introduce operators @xmath21 and @xmath22 which fulfill the anti - commutation relations @xmath23 in terms of these operators , the j - w transformation reads @xmath24using the j - w transformation , eq . ( [ eq : anish ] ) becomes the bi - linear form @xmath25 + h\left ( c_{i}^{\dagger } c_{i}-\frac{1}{2}\right ) \right\}. \label{eq : hc}\]]where @xmath26 and @xmath27 is the anisotropy parameter . the limiting values , @xmath28 and @xmath29 correspond to the isotropic and ising chain , respectively . ( [ eq : hc ] ) can be diagonalized by the transformation amico / etal:04,lamico / etal:04 @xmath30where @xmath31the hamiltonian in eq . ( [ eq : hc ] ) then becomes @xmath32where the wave number @xmath33 with @xmath34 now that the _ xy _ hamiltonian has been diagonalized , we will calculate the evolution of the operators @xmath35 , which will be used to obtain the final state . from eq . ( [ eq : eta ] ) and its inverse @xmath36 where @xmath37 @xmath38 . then from eqs . ( [ eq : eta ] ) and ( [ eq : coft ] ) @xmath39 , \notag \\ c_{j}^{\dagger } ( t ) & = & \sum_{l}[\overset{\thicksim } { b}_{lj}^{\dagger } ( t)c_{l}+\overset{\thicksim } { a}_{lj}^{\dagger } ( t)c_{l}^{\dagger } ] , \end{aligned}\ ] ] where @xmath40 , \notag \\ \overset{\thicksim } { b}_{lj}(t)&=&\frac{2}{n}\sum_{k}e^{ik(l - j)}\alpha _ { k}\beta _ { k}\sin \lambda_{k}t.\end{aligned}\ ] ] for @xmath28 , and @xmath41 the evolution of the creation operator is @xmath42 in this case the @xmath43-component of the total spin @xmath44 commutes with the hamiltonian and is a conserved quantity . now we seek to transfer a quantum state from the site @xmath45 to @xmath46 . first we assume that all the spins of the system are initially in spin down states . then we encode the state @xmath47 at the first spin of the chain . the initial state of the whole system is then @xmath48 , where @xmath49 denotes the state with all the spins down . for simplicity , @xmath50 and @xmath51 are taken to be real with @xmath52 . now our task is to calculate the fidelity of transmission of a state from the first spin to the @xmath5 spin of the chain . the reduced density matrix of the @xmath5 spin of the chain can be constructed using the operator expansion for the density matrix of a system on @xmath4 spin-1/2 particles in terms of tensor products of pauli matrices . the reduced density matrix is @xcite @xmath53 where @xmath54 means @xmath55 the fidelity between the received state @xmath56 and the initial state @xmath57 is defined by @xmath58 which is @xmath59 so if the parameters @xmath60 , the fidelity @xmath61 , which only depends on the value of @xmath62 . another quantity we want to calculate is the purity ( also known as the tangle or one - tangle ) , which provides a measure of the entanglement between the spin at one site and the rest of sites in the chain . the purity is often expressed as @xmath63 , but can also be expressed as @xmath64=4\det [ \rho ^{(1)}],\ ] ] where @xmath65 is the one - site reduced density matrix , eq . ( eq : dmat ) . this is referred to as the one - tangle @xcite , which was apparently motivated by the tangle defined in ref . note that eq . ( [ eq : tangle ] ) is gives a valid measure of entanglement when the whole system is in a pure state and given there is only one parameter for a any such measure for a two - state system , this is as good as any other . however , if the system is in a mixed state , one must use some other measure of entanglement . the one - tangle , or purity , is connected to the von neumann entropy of the reduced density matrix through the relation @xmath66=h(\frac{1}{2}(1+\sqrt{1-\tau \lbrack \rho ^{(1)}]})),\ ] ] where @xmath67 . from eq . ( [ eq : dmat ] ) , the tangle / purity can be written as @xmath68=4\det [ \rho ^{(1)}]=1 - 4(\left\langle s_{r}^{x}\right\rangle ^{2}+\left\langle s_{r}^{y}\right\rangle ^{2}+\left\langle s_{r}^{z}\right\rangle ^{2}),\ ] ] where the components @xmath69 are the components of the bloch vector . now we will calculate @xmath70 ( @xmath11 ) for use in eqs . ( [ eq : f ] ) and ( [ eq : tao ] ) . @xmath71 however , note that @xmath72 . using wick s theorem , @xmath73 for the simple case , @xmath74 , @xmath75@xmath76 the value of @xmath77 can be obtained from the above expression by replacing @xmath78 and @xmath79 . now we need to calculate _ contractions _ of two field operators @xmath80 , @xmath81 , @xmath82 , @xmath83 . from eqs . ( eq : cs)- ( [ eq : coeffsc ] ) , we get @xmath84\sin ^{2}\lambda _ { k}t \notag \\ & & + \frac{2}{n}\sum_{k}\alpha _ { k}\beta _ { k}\cos k(j - m)\sin 2\lambda _ { k}t , \\ \left\langle \mathbf{0}\right\vert a_{j}(t)a_{m}(t)\left\vert \mathbf{0}\right\rangle & = & \delta _ { jm } \notag \\ & & -\frac{4i}{n}\sum_{k}[\alpha _ { k}\beta _ { k}(1 - 2\alpha _ { k}^{2})\cos k(j - m ) \notag \\ & & -2\alpha _ { k}^{2}\beta _ { k}^{2}\sin k(j - m)]\sin ^{2}\lambda _ { k}t \notag \\ & & + \frac{2i}{n}\sum_{k}\alpha _ { k}\beta _ { k}\sin k(j - m)\sin 2\lambda _ { k}t , \\ \left\langle \mathbf{0}\right\vert b_{j}(t)b_{m}(t)\left\vert \mathbf{0 } \right\rangle & = & -\delta _ { jm } \notag \\ & & -\frac{4i}{n}\sum_{k}[\alpha _ { k}\beta _ { k}(1 - 2\alpha _ { k}^{2})\cos k(j - m ) \notag \\ & & + 2\alpha_{k}^{2}\beta _ { k}^{2}\sin k(j - m)]\sin ^{2}\lambda _ { k}t \notag \\ & & + \frac{2i}{n}\sum_{k}\alpha _ { k}\beta _ { k}\sin k(j - m)\sin 2\lambda _ { k}t.\label{eq : sxbb}\end{aligned}\ ] ] and @xmath85 , \\ \left\langle \mathbf{0}\right\vert b_{j}(t)c_{1}^{\dag } + c_{1}b_{j}(t)\left\vert \mathbf{0}\right\rangle & = & \frac{-2i}{n}\sum_{k}[(1 - 2\alpha _ { k}^{2})\cos k(1-j)\sin \lambda _ { k}t \notag \\ & & + \sin k(1-j)\cos \lambda _ { k}t \notag \\ & & + 2\alpha _ { k}\beta _ { k}\sin k(1-j)\sin \lambda _ { k}t].\end{aligned}\ ] ] now we can calculate @xmath86@xmath87.\ ] ] note that , since @xmath88 then @xmath89 = @xmath90 = @xmath91 @xcite , @xmath92.\ ] ] now , from eq . ( [ eq : sx ] ) , @xmath93 for the case @xmath28 , @xmath94 , we find that @xmath95,\;\;\;\;\;\ ; \left\langle s_{r}^{z}\right\rangle = \beta ^{2}\left\vert \overset{\thicksim } { a}_{1r}\right\vert ^{2}-\frac{1}{2}.\ ] ] which agrees with bose s case @xcite . the @xmath96 gives the transmission amplitude @xmath97 of an excitation ( the @xmath98 state ) from the @xmath99 to the @xmath100 spin . when @xmath60 , the fidelity is @xmath101}.\ ] ] which agrees with ref . we will now investigate the performance of a spin chain for which there exists some anisotropy parameter @xmath1 and the @xmath43-component of the total spin is not a conserved quantity . we first perform numerical calculations of the fidelity @xmath102 for a particular , and small @xmath4 value and discuss the variations with changes in physical properties of the system . as stated above , after some time @xmath0 , the state at the @xmath5 spin will be measured , where @xmath103 for @xmath4 even and @xmath104 for @xmath4 odd ) , i.e. the sending and receiving positions are at opposite sites on the chain with periodic boundary conditions . clearly the complexity of the computation grows with the value of @xmath46 because the expansion of @xmath105 has @xmath106 terms . so for simplicity , we take @xmath107 as an example . we will also choose a particular state ( @xmath108 ) to analyze . however , it is important to note that we have examined several states and have found these trends typical ; they exhibit similar variations , only the maximal values of the fidelity and purity / tangle are different . in each case the final state eq . ( [ eq : dmat ] ) has a time dependence described by eqs . ( [ eq : sxx ] ) and ( [ eq : sz ] ) . the state to be transferred is encoded at the first site of the chain . then as time evolves the state propagates to the @xmath5 site . the time evolution of the operators @xmath109 can be expressed as a product of all contractions of the two operators @xmath110 and @xmath111 , which are superpositions of many different states in the basis spanned by the jordan - wigner fermions with definite momenta @xcite . this is clearly more complex than the isotropic case ( @xmath28 ) where the state transfer can be characterized in terms of their dispersion @xcite . is plotted versus time @xmath0 and anisotropic parameter @xmath112 for ( a ) strong - field regime , @xmath113 @xmath114 . ( b ) weak - field regime , @xmath115.(c ) intermediate case , @xmath116 . the number of sites @xmath4 is 5 , the input state is encoded in the first site and output state is at the @xmath117 site , @xmath118,@xmath119.,title="fig : " ] is plotted versus time @xmath0 and anisotropic parameter @xmath112 for ( a ) strong - field regime , @xmath113 @xmath114 . ( b ) weak - field regime , @xmath115.(c ) intermediate case , @xmath116 . the number of sites @xmath4 is 5 , the input state is encoded in the first site and output state is at the @xmath117 site , @xmath118,@xmath119.,title="fig : " ] is plotted versus time @xmath0 and anisotropic parameter @xmath112 for ( a ) strong - field regime , @xmath113 @xmath114 . ( b ) weak - field regime , @xmath115.(c ) intermediate case , @xmath116 . the number of sites @xmath4 is 5 , the input state is encoded in the first site and output state is at the @xmath117 site , @xmath118,@xmath119.,title="fig : " ] the fidelity @xmath102 as a function of time @xmath0 and the anisotropic parameter @xmath120 are shown in fig . [ fig:1 ] with different parameter values for @xmath121 and @xmath7 . first in the strong - field regime ( fig . [ fig:1](a ) ) , @xmath122 , the dispersion is given by @xmath123 $ ] . the effect of the anisotropy can be neglected if we only consider the term which is first order in @xmath124 . the parameters in eq . ( [ eq : afa ] ) become @xmath125 and @xmath126 in eq . ( [ eq : coeffsc ] ) . so the contractions of two field operators in eqs . ( [ eq : sx])- ( [ eq : sxbb ] ) are negligible for @xmath127 which indicates that using only a uniform magnetic field ( global interaction ) can not cause a uncorrelated state to become correlated . in fig . [ fig:1](a ) , it is clear that the anisotropy does not significantly affect the fidelity , but the presence of the cosine term in the first order of the dispersion produces the observed oscillation of fidelity with time @xmath0 . in the weak - field regime @xmath128 ( fig . [ fig:1](b ) ) , the dispersion can be written as @xmath129^{1/2}$ ] . in this case , the anisotropy has pronounced effects on the fidelity . we see that increasing the anisotropy does not always decrease the fidelity . for certain values of the parameters @xmath120 and @xmath0 , the fidelity is greater , which corresponds to a constructive interference . for example , there are peaks with @xmath130 such as ( @xmath131 ) and ( @xmath132 ) that exhibit this behavior . with increasing @xmath120 and @xmath0 , the frequency of oscillation of @xmath102 becomes greater . the higher - frequency oscillation can be attributed to the fact that with increasing @xmath133 the initial state differs more from the true ground state , and therefore , with the evolution of time , it exhibits fluctuations , even near the ground state . the intermediate regime @xmath134 , which is the most interesting , is shown in fig . [ fig:1](c ) . for relatively short times @xmath135 the anisotropy does not have a pronounced effect on the fidelity . the oscillation of @xmath102 with time @xmath0 is relatively slow in this case compared to the cases in both the strong- and weak - field regimes , fig . [ fig:1](a ) and ( b ) respectively . when the strength of the exchange interaction @xmath136 or the magnetic field @xmath7 dominate , the oscillation of @xmath102 with time @xmath0 is greater . but for intermediate regimes , the competition between @xmath136 and @xmath7 gives a smaller oscillation frequency for @xmath102 . furthermore , the high - fidelity peaks are fairly broad and therefore correspond to values of the fidelity which are stable under perturbation of the parameters . in summary , realistic quantum communication devices will require larger fidelities in a shorter times . one may well have expected a significant oscillation of fidelity with time and anisotropy in the intermediate regime . however , this is not the case , but rather the intermediate regime has the highest fidelities in the shortest amount of time with the more stable values for the fidelity . so if one wants to achieve the highest fidelity under our assumed conditions , the intermediate regime is , surprisingly , the best choice . site as a function of time @xmath0 and @xmath137 . the initial state of the system is @xmath138 . ( a ) strong - field regime , @xmath139 . ( b ) weak - field regime , @xmath115 . ( c ) intermediate regime , @xmath116 . , title="fig : " ] site as a function of time @xmath0 and @xmath137 . the initial state of the system is @xmath138 . ( a ) strong - field regime , @xmath139 . ( b ) weak - field regime , @xmath115 . ( c ) intermediate regime , @xmath116 . , title="fig : " ] site as a function of time @xmath0 and @xmath137 . the initial state of the system is @xmath138 . ( a ) strong - field regime , @xmath139 . ( b ) weak - field regime , @xmath115 . ( c ) intermediate regime , @xmath116 . , title="fig : " ] it is clear that the initial state of the system is an unentangled state . however , with the evolution of time , entanglement is generated in the chain . in order to quantify this change , we have calculated the purity , or tangle , at the @xmath140 site which measures the entanglement between the @xmath140 site and all the other sites in the chain . the tangle at @xmath140 site as a function of time @xmath0 and anisotropy @xmath120 is plotted in fig . [ fig:2 ] and fig . [ fig:3 ] with different initial states @xmath141 and vacuum state , respectively . from fig . [ fig:2 ] , for small anisotropy ( @xmath142 ) , the tangle is negligibly small . in the strong - field regime , from fig . [ fig:3](a ) , the anisotropy does not have a significant affect on the tangle similar to behaviour of the fidelity as seen in fig . [ fig:2](a ) . also , the strong - field tangle is relatively small compared to the weak - field regime and the oscillation of tangle with time @xmath0 is suppressed . furthermore , the time for the tangle reach its first peak in the strong - field regime is at @xmath143 while for the intermediate regime it is @xmath144 . in the weak - field regime , which is plotted in fig . [ fig:2](b ) and fig . [ fig:3](b ) , we see that a stronger exchange interaction even with a stronger anisotropy can generate more entanglement at certain times . for example , at @xmath145 and @xmath146 , @xmath147 . in the intermediate regime , the behaviour of the tangle is similar to the weak - field regime . with increasing anisotropy , the tangle increases . however , even in the @xmath145 case , it does not reach its maximal value @xmath147 . also , as @xmath124 increases , the tangle oscillates more rapidly , but different from the effect of anisotropy on fidelity in the intermediate case , ( see fig . [ fig:2](b ) ) the tangle increases with increasing @xmath1 . this is an effect of entanglement dynamically generated from the ground state and depends on the anisotropy . when the initial state is the ground state , the system will still generate entanglement , which exists only in the @xmath130 case . this is due to the double spin - flip operator terms @xmath148 in eq . ( [ eq : hc ] ) . in this case , @xmath149 , and @xmath150=1 - 4\left\langle \mathbf{0}\right\vert s_{r}^{z}(t)\left\vert \mathbf{0}\right\rangle ^{2}$ ] . unlike fig . [ fig:2 ] , the one - tangle always equals zero when @xmath28 since the vacuum state is the ground state of the system . and the @xmath151 spin will always remain in the spin - down state and will never be entangled with the other spins . for the two initial states of the system , fig . fig:2(b)(c ) and fig . [ fig:3](b)(c ) , we find that the time evolution of the one - tangle shows similar behavior with increasing @xmath120 , which means the one - tangle is not sensitive to the initial state of the system when a strong anisotropy is present in the case of weak - field and intermediate regimes . this shows that the anisotropy parameter aides in the generation of entanglement in the spin chain . except that the initial state is the vacuum state , @xmath152.,title="fig : " ] except that the initial state is the vacuum state , @xmath152.,title="fig : " ] except that the initial state is the vacuum state , @xmath152.,title="fig : " ] in conclusion , we have investigated quantum state transfer through an anisotropic heisenberg xy model in a transverse field and also entanglement generation in that same system . the interest in these problems stems from the possible use of spin chains as a communication channels . we expect that a realistic ferromagnetic material would have some anisotropy and have shown that this anisotropy can have significant effects on the fidelity of state transfer as well as the entanglement in the chain . specifically , we have calculated the fidelity and the one - tangle , or purity , for three different cases : a weak external magnetic field , an intermediate regime , and strong external magnetic field . we found that in all three cases a relatively high fidelity can be obtained for certain times and vales of the anisotropy . however , in the intermediate regime , the oscillation of the fidelity with time is fairly low , and the peaks fairly broad . furthermore , a fairly high fidelity is achieved in a relatively shorter time for the intermediate regime . this would imply a more reliable output , in a shorter time , when some anisotropy is present , compared to when there is none . thus the intermediate regime presents some interesting and somewhat surprising results for state transfer and is also the best choice for reliable transfer . we have also calculated the one - tangle , or the purity of a typical one - particle state in the chain . we began with a pure initial state and found that the stronger the anisotropy and exchange interaction , the more entanglement will be generated . this indicates that anisotropy also aides in the production of entanglement in the chain . this material is based upon work supported by the national science foundation under grant no . 0545798 to msb . zmw thanks the scholarship awarded by the china scholarship council(csc ) . we gratefully acknowledge c. allen bishop for helpful discussions .

we study quantum communication through an anisotropic heisenberg xy chain in a transverse magnetic field . we find that for some time @xmath0 and anisotropy parameter @xmath1 , one can transfer a state with a relatively high fidelity . in the strong - field regime , the anisotropy does not significantly affect the fidelity while in the weak - field regime the affect is quite pronounced . the most interesting case is the the intermediate regime where the oscillation of the fidelity with time is low and the high - fidelity peaks are relatively broad . this would , in principle , allow for quantum communication in realistic circumstances . moreover , we calculate the purity , or tangle , as a measure of the entanglement between one spin and all the other spins in the chain and find that the stronger the anisotropy and exchange interaction , the more entanglement will be generated for a given time . quantum state transfer , entanglement 03.67hk,03.65ud,75.10jm

You are an expert at summarizing long articles. Proceed to summarize the following text: this paper is the third of a series presenting results from studies of the qsos discovered in the apm survey for z 4 quasars . a study of the evolution of lyman limit absorption systems over the redshift range 0.04 @xmath28 z @xmath28 4.7 was presented in storrie - lombardi ( 1994 ) [ paper i ] . the intermediate resolution ( 5 ) qso spectra and the survey for high redshift damped absorbers are presented in storrie - lombardi ( 1996 ) [ paper ii ] . the evolution of the cosmological mass density of neutral gas at high redshift and the implications for galaxy formation theories are discussed in storrie - lombardi , mcmahon & irwin ( 1996 ) [ paper iv ] . in separate papers we will describe the intrinsic properties of the qsos and studies of the forest clouds at high redshift . a high resolution study of the forest region in a redshift @xmath29 qso has been completed by williger ( 1994 ) . how and when galaxies formed are questions at the forefront of work in observational cosmology . absorption systems detected in quasar spectra provide the means to study galaxy formation and evolution up to redshifts of approximately five , back to when the universe was less than 10 percent of its present age . surveys for absorption features have several advantages over trying to directly detect galaxies at high redshift . much shorter exposure times are required because the qsos are relatively bright ( r@xmath30 18 - 19.5 ) and the large equivalent width systems are easily detected in the spectra . this provides good absorption candidates to follow up with higher resolution spectra . the redshift and column density can be accurately determined from the wavelength of the absorption system and the line profile . this is far easier and more reliable than trying to directly get a spectrum of a very faint high redshift galaxy . while the baryonic content of spiral galaxies that are observed in the present epoch is concentrated in stars , in the past this must have been in the form of gas . the principal gaseous component in spirals is neutral hydrogen which has led to surveys for absorbers detected by the damped lines they produce ( wolfe 1986 , hereafter wtsc ; lanzetta 1991 , hereafter lwtlmh ; lanzetta , wolfe & turnshek 1995 , hereafter lwt ; wolfe 1995 ; paper ii ) . though damped systems are observationally very rare objects with @xmath3140 confirmed examples known , the mass per unit comoving volume they contain is roughly comparable to the mass density of baryonic matter in present - day spirals , ie . a major constituent of the universe ( wolfe 1987 , lwt ) . their metal abundances are much lower than galactic values ( pettini , boksenberg & hunstead 1990 ; rauch 1990 ; pettini 1994 ) and they are characterised by low molecular content and low , but not negligible , dust content ( fall , pei & mcmahon 1989 ; pei , fall & bechtold 1991 ; pettini 1994 ) , features consistent with an early phase of galactic evolution . they may be the progenitors of spiral galaxies like our own and are clearly important for the study of the formation and evolution of galaxies . they have been detected across a very large redshift range z@xmath30[0.5,4.5 ] providing the means to pinpoint the epoch of formation of disk galaxies and study their evolution . eleven candidate damped absorption systems out of 32 measured features were identified in 27 spectra of the mainly non - bal quasars from the apm z 4 survey ( paper ii ) . the eleven candidates cover the redshift range @xmath0 ( 8 with @xmath1 ) and have estimated column densities @xmath2 10@xmath3 atoms @xmath4 . in this paper the qso br1144@xmath60723 with a candidate absorber at z@xmath53.26 is removed from further consideration in the sample . it has been observed with the anglo - australian telescope at high resolution and the damped candidate has been found to be all absorption at z@xmath54.0 ( r. hunstead , private communication ) . high resolution echelle spectra ( 0.8 fwhm ) were obtained by s. dodorico as part of the eso key programme studying high redshift quasars for four of the qsos in the apm sample ( bri0952@xmath60115 , br1033@xmath60327 , bri1108@xmath60747 , br1202@xmath60725 ) . the signal - to - noise ratio in bri0952@xmath60115 was very poor but the other spectra have been used to confirm two features as damped with another falling just below the log @xmath2 20.3 threshold . we have discovered the highest redshift damped absorber known at z@xmath54.383 in qso br1202@xmath60725 . the confirmation of the absorption systems is discussed in section 2 . these data have been combined with data from previous surveys ( wtsc , lwtlmh , and lwt ) and the results for the lyman limit systems obtained in paper i to study the column density distribution for @xmath12@xmath7 17.2 and redshift evolution of these systems for 0.008 @xmath8 z @xmath8 4.7 . numerous authors have studied the distribution of column densities , @xmath32 , for absorption lines . the first determination was by carswell ( 1984 ) for lines with @xmath33 @xmath34 atoms @xmath4 . they found @xmath35 ( @xmath36 ) . damped absorption ( dla ) systems comprise the high column density tail of neutral hydrogen absorbers with column densities of @xmath37 atoms @xmath4 . they dominate the baryonic mass contributed by . when damped systems are included in the column density distribution function for a single power law fit the exponent is @xmath38 = 1.41.7 ( tytler 1987 ; petitjean 1993 and references therein ) . assuming the baryonic mass is proportional to the column density and takes the form @xmath39 for the column density distribution function , the mass contribution from the damped systems can be estimated as @xmath40.\end{aligned}\ ] ] @xmath41 one problem with the power law representation is that if @xmath42 , as all current estimates indicate , then the total mass in damped systems diverges unless an upper bound to the hi column density is assumed . for example , if we take 20.3 log @xmath43 22 , the fractional contribution to the total hi mass for damped systems , @xmath44 , is then @xmath45 for @xmath46 and @xmath47 for @xmath48 . however , there is no a priori reason for assuming this upper limit and hence there is no strict upper bound to any estimate of the total hi mass in damped systems . an alternative parameterisation using a gamma function to describe the hi column density distribution was adopted by pei & fall and provides an elegant solution to the diverging mass problem . we discuss these points in more detail in section 3 and the redshift evolution of the absorbers in section 4 . echelle spectra of four qsos were obtained in march , 1993 by s. dodorico as part of an eso key programme studying high redshift quasars . they were taken at la silla with the 3.5 m ntt telescope using the emmi instrument in echelle mode using a 2048@xmath492048 pixel loral ccd as the detector . a slit of 15 `` in length was used and generally the slit width was 1.2 '' . two grating setups were used , one covering 4700 - 8300 and the other covering 5800 - 9500 with a resolution of @xmath3140 km s@xmath50 ( 1 ) . [ see giallongo 1994 for more details . ] the observations are summarised in table [ t_obs_eso ] . .eso observations march , 1993 [ cols= " < , < , > , < , < " , ] a single power law fit to the combined data set described above of the form in equation [ fnzeqn ] with @xmath51 results in @xmath52 and @xmath53 with similar values for @xmath54 ( @xmath55 ) . the quoted error for the normalisation constant @xmath56 is large because for a small change in the value of @xmath38 , @xmath56 can change by 2 orders of magnitude . these results are in good agreement with the results found by lwtlmh ( @xmath57 , log k@xmath58 ) and are plotted for the entire data set in figure [ f_dlafn_all ] . the data is binned for display purposes only with the vertical error bars plotted at the mean column density for each bin . we will see in the next section that a single power law is not a good fit to the data . as shown in paper i for the lyman - limit system evolution , the arbitrary binning of the data for presentation in differential plots includes a subjective component that can mask exactly what is happening in the underlying data . a cumulative distribution plot is far better at revealing the true nature of the distribution and this approach is examined now . the @xmath12 of the cumulative number of damped systems detected versus @xmath12 is plotted in figure [ f_dlacum](a ) . a point for the expected number of lyman - limit systems that would be detected down to @xmath12 @xmath59 is shown with a circled star . this is calculated by integrating the number density per unit redshift ( @xmath60 ) over the redshift path covered by the @xmath61 qsos in the dla sample , @xmath62 @xmath63 it is obvious from figure 3(a ) that a power law will not fit the entire column density range 17.2 @xmath28 log @xmath28 22 . a kolmogorov - smirnov ( k - s ) yields a probability of less than @xmath64 that the fit represents the underlying data set . in figures [ f_dlacum](b - d ) the same distribution is overplotted with single power law fits for different values of @xmath38 that were fit to the graph by eye . ( b ) shows that @xmath65 will fit from the lyman limit column density through the damped systems with @xmath12 @xmath66 , a flatter slope than the canonical 1.5 - 1.7 range . ( c ) shows that @xmath67 fits the damped distribution with @xmath68 @xmath69 well but does not describe the high or low column density tails of the distribution . ( d ) shows a fit to the sharp drop off in numbers for damped systems with @xmath12 21.3 . this can be expected from looking at the estimated column densities for the damped systems in table [ t_dla ] or by looking at the spectra . there are not a lot of heavily damped systems . clearly the results for a single power law fit depend critically on the range of column densities included . this characteristic can explain much of the variation in the results previously seen by various authors that were summarised in section 3.1 . to qualitatively study the redshift evolution of the column density distribution of the damped systems the cumulative distribution shown in figure [ f_dlacum ] has been split in half at redshift 2.5 and each set plotted individually ( figure [ f_zsplit ] ) . the damped systems with @xmath70 are shown by the solid line and the absorbers with @xmath71 are shown by the dashed line . the higher redshift absorbers appear to have a slightly flatter slope up to @xmath12 @xmath521 and then a sharper drop in the number of very high column density systems , though a k - s test shows that this is not a statistically significant difference . the evolution with redshift in the slope of the column density distribution is also apparent when looking at the differential @xmath72 . lwt plotted this in 3 redshift bins z@xmath5[0.008,1.5],[1.5,2.5 ] , and [ 2.5,3.5 ] . in the highest redshift bin there was a flattening of the column density distribution slope towards higher column densities . in figure [ f_dlafn ] we have plotted our combined data set with this same binning with the addition of one higher redshift bin z@xmath5[3.5,4.7 ] . the flattening of the distribution function towards higher column density systems in the z@xmath5[2.5,3.5 ] bin in the lwt data is no longer pronounced . the most striking feature is the steepness of the distribution in the highest redshift bin . it is not just steeper due to a decrease in the highest column density systems ( log @xmath721 ) , but there is also an increase in the number of lower column density systems relative to the other bins . even if 15 - 20% of the candidate systems with log @xmath3020.3 turn out not to be damped when observed at higher resolution , as we expect , this result still holds there are two strong motivating factors to find an alternative model for describing the hi column density distribution . first , as shown in section 3.4 , there is direct evidence for an apparent variation in the power law slope as a function of @xmath43 . this implies that a higher order functional form other than a power law is needed to describe the column density distribution . second , as noted in the introduction , with a power law model the integral mass contained within damped ly@xmath73 systems is divergent for realistic values of @xmath38 . this in turn means that it is impossible to assign a formal upper limit to any estimate of the neutral gas content of the early universe . consequently , following pei and fall ( 1995 ) , we have chosen to model the data with a gamma distribution of the form @xmath74 where @xmath75 is the characteristic number of absorbing systems at the column density @xmath76 , and @xmath76 is a parameter defining the turnover , or ` knee ' , in the number distribution . both @xmath75 and @xmath76 may in general vary with redshift but for the moment we treat them as constants . this functional form is similar to the schechter luminosity function ( schechter 1976 ) . for @xmath77 the gamma function tends to the same form as the single power law , @xmath35 ; whilst for @xmath78 @xmath76 , the exponential term begins to dominate . we can understand how a gamma function might provide a better description of the damped ly@xmath73 data by considering the differential logarithmic slope which is given by @xmath79 as the column density approaches @xmath76 the slope begins to steepen and rapidly turns over at higher column densities , qualitatively similar to what we observe in figures [ f_dlacum ] and [ f_zsplit ] . furthermore , the integral hi over the column density distribution ( equation [ himass1 ] ) for @xmath80 is now given by @xmath81 where @xmath82 denotes the standard gamma function . this function is bounded if @xmath83 . the maximum likelihood technique outlined in appendix a can readily be modified to incorporate this form . we note that the likelihood solution can be found over a two - dimensional grid of pairs of values of @xmath76 and @xmath38 , since the constant @xmath75 can be directly computed using the constraint @xmath84 where m is the total number of observed systems . this is computationally much less intensive than doing a 3-d grid search . the results of a single functional fit to the entire dataset are log @xmath15 , @xmath16 , and @xmath17 . the log - likelihood function results with confidence contours are shown in figure [ f_gamconf](a ) . the best fit is overplotted on the differential form of @xmath72 in figure [ f_gamconf](b ) and on the cumulative distribution in figure [ f_gamconf](c ) . ( the single power law form of @xmath72 was shown fitted to the same data in figures [ f_dlafn_all ] and [ f_dlacum ] ) . the differential form of the plots ( figures [ f_dlafn_all ] and [ f_gamconf](b ) ) show little difference between the single power law and @xmath85-distribution fits . when displayed with the cumulative number of absorbers in figure [ f_gamconf](c ) , the @xmath85-distribution now clearly fits the entire data set with column densities log @xmath2 20.3 . if the expected number of lyman - limit systems are included in the fit , the results are log @xmath86 , @xmath87 , and @xmath88 . this also provides a reasonable fit to the data as shown in figure [ f_gamconf2](a - c ) . differential evolution in the number density of damped absorbers has been described by lwt and wolfe ( 1995 ) . while the change in number density per unit redshift is consistent with no intrinsic evolution of the absorbers over the range 0 @xmath8 z @xmath8 3.5 , they find that the systems with log @xmath89 disappear at a much faster rate from z@xmath53.5 to z@xmath50 than does the population of damped absorbers as a whole . we now examine the redshift evolution of the damped absorbers in our combined data set by determining the number density of absorbers per unit redshift , @xmath90 . in a standard friedmann universe for absorbers with cross section @xmath91 and number density @xmath92 per unit comoving volume @xmath93 it is customary to represent the number density as a power law of the form @xmath94 where @xmath95 . this yields @xmath25 = 1 for @xmath96 = 0 and @xmath25 = 1/2 for @xmath96 = 1/2 for the case of no evolution with redshift in the product of the number density and cross section of the absorbers @xcite . a maximum likelihood fit to the data yields @xmath97 which is consistent with no intrinsic evolution even though the value of @xmath25 is similar to that found for the lyman limit systems where evolution is detected at a significant level ( paper i ; stengler - larrea 1995 ) . the log - likelihood function for @xmath25 and @xmath98 with @xmath768.3% and @xmath795.5% confidence contours is plotted in figure [ f_maxz ] . we also find redshift evolution in the higher column density systems but with a decline in @xmath26 for z@xmath73.5 . these results are displayed in figure [ f_dndz ] . the entire data set is plotted as dashed lines with the above fit . the results for only the absorbers with log n(hi)@xmath99 are shown as solid lines . figure [ f_zhi ] shows hi column density versus redshift , and the paucity of absorbers with log @xmath7 21 at z @xmath7 4 is apparent . three qsos from apm survey have been observed at 0.8 resolution . two have damped systems with confirmed hi column densities of @xmath2 10@xmath3 atoms @xmath4 , with a third absorber falling just below this threshold . we have discovered the highest redshift damped absorber known at z@xmath54.383 in qso br1202@xmath60725 . the two systems with @xmath2 10@xmath3 atoms @xmath4 , and remaining nine candidate damped absorbers from the apm survey have been combined with data from previous surveys to study the column density distribution and number density evolution for absorbers with @xmath2 17.2 . if the hi column density distribution function is fit with a power law , @xmath10 , we find evidence for breaks in the power law , flattening for 17.2 @xmath28 log 21 and steepening for log 21.2 . the column density distribution function for the data with log@xmath220.3 is better fit with the @xmath85-distribution form @xmath14 with log @xmath15 , @xmath16 , and @xmath17 . for the number density evolution of the damped absorbers ( log @xmath2 20.3 ) over the redshift range 0.008 @xmath8 z @xmath8 4.7 we find the best fit of a single power law form for @xmath18 yields @xmath23 and @xmath24 . this is consistent with no intrinsic evolution in the absorbers even though the value of @xmath25 is similar to that found for the lyman limit systems where evolution is detected at a significant level . evolution is evident in the highest column density absorbers with the incidence of systems with log n(hi)@xmath221 decreasing for z 3.5 . * acknowledgments * we would like to thank bob carswell for providing software for and assistance with the data reduction and profile fitting of the spectra . lsl acknowledges support from an isaac newton studentship , the cambridge overseas trust , and a university of california president s postdoctoral fellowship . rgm acknowledges the support of the royal society . , peebles p.j.e . , 1969 , apj , 156 , l7 r.f . , morton d.c . , smith m.g . , stockton a.n . , turnshek d.a . , weymann r.j . , 1984 , apj , 278 , 486 a.j . , 1994 , ph.d . thesis . cambridge university s.m . , pei y.c . , mcmahon r.g . , 1989 , apj , 341 , l5 e. , dodorico s. , fontana a. , mcmahon r.g . , savaglio s. , cristiani s. , molaro p. , trevese d. , 1994 , apj , 425 , l1 k.m . , wolfe a.m. , turnshek d.a . , 1995 , apj , 440 , 435 k.m . , wolfe a.m. , turnshek d.a . , lu l. , mcmahon r.g . , hazard c. , 1991 , apjs , 77 , 1 lu l , sargent w.l.w . , womble d.s . , barlow t.a . , 1996 , apj , 457 , l1 y.c . , fall s.m . , bechtold j. , 1991 , apj,378 , 6 y.c . , fall s.m . , 1995 , apj , 454 , 69 p. , webb j.k . , rauch m. , carswell r.f . , lanzetta k.m . , 1993 , mnras , 262 , 499 m. , boksenberg a. , hunstead r.w . , 1990 , apj , 348 , 48 m. , smith l.j . , hunstead r.w . , king d.l . , 1994 , apj , 426 , 79 m. , carswell r.f . , robertson j.g . , shaver p.a . , webb j.k . , 1990 , mnras , 242 , 698 w.l.w . , steidel c.c . , boksenberg a. , 1989 , apjs , 79 , 703 w.l.w . , young p.t . , boksenberg a. , tytler d. , 1980 , apjs , 42 , 41 p. , 1976 , apj , 203 , 297 p. , press w.h . , 1976 , apj , 203 , 557 stengler - larrea e.a . , , 1995 , apj , 444 , 64 l.j . , mcmahon r.g . , irwin m.j . , hazard c. , 1994 , apj , 427 , l13 ( paper i ) l.j . , mcmahon r.g . , irwin m.j . , hazard c. , 1996 , apj , 468 , 121 ( paper ii ) l.j . , mcmahon r.g . , irwin m.j . , 1996 , mnras , in press ( paper iv ) d. , 1987 , apj , 321 , 49 e.j . , williger g.m . , baldwin j.a . , carswell r.f . , hazard c. , mcmahon r.g . , 1996 , a&a , in press g.m . , baldwin j.a . , carswell r.f . , cooke a.j . , hazard c. , irwin m.j . , mcmahon r.g . , storrie - lombardi l.j . , 1994 , apj , 428 , 574 a.m. , 1987 , proc.phil.trans.roy.soc . , 320 , 503 a.m. , turnshek d.a . , smith h.e . , cohen r.d . , 1986 , apjs , 61 , 249 a.m. , lanzetta k.m . , foltz c.b . , chaffee f.h . , 1995 , apj , 454 , 698 using equation [ fnzeqn ] for the column density distribution function , the damped absorbers will be found randomly distributed according to this function along the qso line - of - sight in @xmath100 space . if the space is divided into @xmath101 cells each of volume @xmath102 , the expected number of points in cell @xmath103 is given by @xmath104 the probability of observing @xmath105 points in cell @xmath103 is @xmath106 the likelihood function for qso@xmath107 taking the product over all the cells is then @xmath108 if the volume of each cell @xmath102 becomes very small such that there is either 1 or 0 points in each cell , @xmath109 then the likelihood can be rewritten separating out the terms for full and empty cells . for @xmath110 empty cells @xmath111 @xmath112 full cells @xmath113 taking the @xmath12 of the likelihood function gives @xmath114 @xmath115 ( schechter & press 1976 ) . ignoring the constant terms , in the limit where @xmath116 this becomes @xmath117 \end{aligned}\ ] ] @xmath118 to get the overall log likelihood for @xmath61 qsos we evaluate the integrals in equation [ dlal4eqn ] and additively combine the @xmath12 l s resulting in @xmath119 \end{aligned}\ ] ] @xmath120 where @xmath121 is the number of detected dlas in qso@xmath122 and @xmath123 is the minimum column density .

eleven candidate damped absorption systems were identified in twenty - seven spectra of the quasars from the apm z 4 survey covering the redshift range @xmath0 ( 8 with @xmath1 ) . high resolution echelle spectra ( 0.8 fwhm ) have been obtained for three quasars , including two of the highest redshift objects in the survey . two damped systems have confirmed hi column densities of @xmath2 10@xmath3 atoms @xmath4 , with a third falling just below this threshold . we have discovered the highest redshift damped absorber known at z@xmath54.383 in qso br1202@xmath60725 . the apm qsos provide a substantial increase in the redshift path available for damped surveys for z@xmath73 . we combine this high redshift sample with other quasar samples covering the redshift range 0.008 @xmath8 z @xmath8 4.7 to study the redshift evolution and the column density distribution function for absorbers with log @xmath9 . in the column density distribution @xmath10 we find evidence for breaks in the power law , flattening for @xmath1121 and steepening for @xmath12@xmath13 . the breaks are more pronounced at higher redshift . the column density distribution function for the data with log @xmath220.3 is better fit with the form @xmath14 with log @xmath15 , @xmath16 , and @xmath17 . we have studied the evolution of the number density per unit redshift of the damped systems by fitting the sample with the customary power law @xmath18 . for a population with no intrinsic evolution in the product of the absorption cross - section and comoving spatial number density this will give @xmath19 ( @xmath20 ) or @xmath21 ( @xmath22 ) . the best maximum likelihood fit for a single power law is @xmath23 and @xmath24 , consistent with no intrinsic evolution even though the value of @xmath25 is also consistent with that found for the lyman limit systems where evolution is detected at a significant level . however , redshift evolution is evident in the higher column density systems with an apparent decline in @xmath26 for z@xmath73.5 . 0q@xmath27 1 2 2oi+siii . ^m . ^s . . cosmology galaxies : evolution galaxies : formation quasars : absorption lines quasars : individual ( br1033@xmath60327 , bri1108@xmath60747 , br1202@xmath60725 )