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Proceed to summarize the following text: this review deals with the analysis of influences that one system , be it physical , economical , biological or social , for example , can exert over another . in several scientific fields , the finding of the influence network between different systems is crucial . as examples , we can think of gene influence networks @xcite , relations between economical variables @xcite , communication between neurons or the flow of information between different brain regions @xcite , or the human influence on the earth climate @xcite , and many others . the context studied in this report is illustrated in figure [ network : fig ] . for a given system , we have at disposal a number of different measurements . in neuroscience , these can be local field potentials recorded in the brain of an animal ; in solar physics , these can be solar indices measured by sensors onboard some satellite ; in the study of turbulent fluids , these can be the velocity measured at different scales in the fluid ( or can be as in the figure , the wavelet analysis of the velocity at different scales ) . for these different examples , the aim is to find dependencies between the different measurements , and if possible , to give a direction to the dependence . in neuroscience , this will allow to understand how information flows between different areas of the brain ; in solar physics , this will allow to understand the links between indices and their influence on the total solar irradiance received on earth ; in the study of turbulence , this can confirm the directional cascade of energy from large down to small scales . in a graphical modeling approach , each signal is associated to a particular node of a graph , and dependence are represented by edges , directed if a directional dependence exists . the questions addressed in this paper concern the assessment of directional dependence between signals , and thus concern the inference problem of estimating the edge set in the graph of signals considered . climatology and neuroscience were already given as examples by norbert wiener in 1956 @xcite , a paper which inspired econometrist clive granger to develop what is now termed granger causality @xcite . wiener proposed in this paper that a signal @xmath0 causes another time series @xmath1 , if the past of @xmath0 has a strictly positive influence on the quality of prediction of @xmath1 . let us quote wiener @xcite : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ as an application of this , let us consider the case where @xmath2 represents the temperature at 9 a.m. in boston and @xmath3 represents the temperature at the same time in albany . we generally suppose that weather moves from west to east with the rotation of the earth ; the two quantities @xmath4 and its correlate in the other direction will enable us to make a precise statement containing some if this content and then verify whether this statement is true or not . or again , in the study of brain waves we may be able to obtain electroencephalograms more or less corresponding to electrical activity in different part of the brain . here the study of coefficients of causality running both ways and of their analogues for sets of more than two functions @xmath5 may be useful in determining what part of the brain is driving what other part of the brain _ in its normal activity_. " _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ in a wide sense , granger causality can be summed up as a theoretical framework based on conditional independence to assess directional dependencies between time series . it is interesting to note that norbert wiener influenced granger causality , as well as another field dedicated to the analysis of dependencies : information theory . information theory has led to the definition of quantities that measure the uncertainty on variables using probabilistic concepts . furthermore , this has led to the definition of measures of dependence based on the decrease in uncertainty relating to one variable after observing another one . usual information theory is , however , symmetrical . for example , the well - known mutual information rate between two stationary time series is symmetrical under an exchange of the two signals : the mutual information assesses the undirectional dependence . directional dependence analysis viewed as an information - theoretic problem requires the breaking of the usual symmetry of information theory . this was realized in the 1960 s and early 1970 s by hans marko , a german professor of communication . he developed the bidirectional information theory in the markov case @xcite . this theory was later generalized by james massey and gerhard kramer , to what we may now call directed information theory @xcite . _ it is the aim of this report to review the conceptual and theoretical links between granger causality and directed information theory . _ many information - theoretic tools have been designed for the practical implementation of granger causality ideas . we will not show all of the different measures proposed , because they are almost always particular cases of the measures issued from directed information theory . furthermore , some measures might have been proposed in different fields ( and/or at different periods of time ) and have received different names . we will only consider the well - accepted names . this is the case , for example , of ` transfer entropy ' , as coined by schreiber in 2000 @xcite , but which appeared earlier under different names , in different fields , and might be considered under slightly different hypotheses . prior to developing a unified view of the links between granger causality and information theory , we will provide a survey of the literature , concentrating on studies where information theory and granger causality are jointly presented . furthermore , we will not review any practical aspects , nor any detailed applications . in this spirit , this report is different from @xcite , which concentrated on the estimation of information quantities , and where the review is restricted to transfer entropy . for reviews on the analysis of dependencies between systems and for applications of granger causality in neuroscience , we refer to @xcite . we will mention however some important practical points in our conclusions , where we will also discuss some current and future directions of research in the field . we will not debate the meaning of causality or causation . we instead refer to @xcite . however , we must emphasize that granger causality actually measures a statistical dependence between the past of a process and the present of another . in this respect , the word causality in granger causality takes on the usual meaning that a cause occurs _ prior _ its effect . however , nothing in the definitions that we will recall precludes that signal @xmath0 can simultaneously be granger caused by @xmath1 and be a cause of @xmath1 ! this lies in the very close connection between granger causality and the feedback between times series . granger causality is based on the usual concept of conditioning in probability theory , whereas approaches developed for example in @xcite relied on causal calculus and the concept of intervention . in this spirit , intervention is closer to experimental sciences , where we imagine that we can really , for example , freeze some system and measure the influence of this action on another process . it is now well - known that causality in the sense of between random variables can be inferred unambiguously only in restricted cases , such as directed acyclic graph models @xcite . in the granger causality context , there is no such ambiguity and restriction . in his nobel prize lecture in 2003 , clive w. granger mentioned that in 1959 , denis gabor pointed out the work of wiener to him , as a hint to solve some of the difficulties he met in his work . norbert wiener s paper is about the theory of prediction @xcite . at the end of his paper , wiener proposed that prediction theory could be used to define causality between time series . granger further developed this idea , and came up with a definition of causality and testing procedures @xcite . in these studies , the essential stones were laid . granger s causality states that a cause must occur before the effect , and that causality is relative to the knowledge that is available . this last statement deserves some comment . when testing for causality of one variable on another , it is assumed that the cause has information about the effect that is unique to it ; _ i.e. _ this information is unknown to any other variable . obviously , this can not be verified for variables that are not known . therefore , the conclusion drawn in a causal testing procedure is relative to the set of measurements that are available . a conclusion reached based on a set of measurements can be altered if new measurements are taken into account . mention of information theory is also present in the studies of granger . in the restricted case of two gaussian signals , granger already noted the link between what he called the ` causality indices ' and the mutual information ( eq . 5.4 in @xcite ) . furthermore , he already foresaw the generalization to the multivariate case , as he wrote in the same paper : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ `` in the case of @xmath6 variables , similar equations exist if coherence is replaced by partial coherence , and a new concept of partial information is introduced . '' _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ granger s paper in 1969 does not contain much new information , but rather , it gives a refined presentation of the concepts . during the 1970 s , some studies , _ @xcite , appeared that generalized along some of the directions granger s work , and related some of the applications to economics . in the early 1980 s , several studies were published that established the now accepted definitions of granger causality @xcite . these are natural extensions of the ideas built upon prediction , and they rely on conditional independence . finally , the recent studies of dalhaus and eichler allowed the definitions of granger causality graphs @xcite . these studies provide a counterpart of graphical models of multivariate random variables to multivariable stochastic processes . in two studies published in 1982 and 1984 @xcite , geweke , another econometrician , set up a full treatment of granger causality testing for the gaussian case , which included the idea of feedback and instantaneous coupling . in @xcite , the study was restricted to the link between two time series ( possibly multidimensional ) . in this study , geweke defined an index of causality from @xmath0 to @xmath1 ; it is the logarithm of the _ ratio _ of the asymptotic mean square error when predicting @xmath1 from its past only , to the asymptotic mean square error when predicting @xmath1 from its past and from the past of @xmath0 . geweke also defined the same kind of index for instantaneous coupling , and showed , remarkably , that the mutual information rate between @xmath0 and @xmath1 decomposes as the sum of the indices of causality from @xmath0 to @xmath1 and from @xmath1 to @xmath0 with the index of instantaneous coupling . this decomposition was shown in the gaussian case , and it remains valid in any case when the indices of causality are replaced by transfer entropy rates , and the instantaneous coupling index is replaced by an instantaneous information exchange rate . this link between granger causality and directed information theory was further supported by @xcite ( without mention of instantaneous coupling in @xcite ) , and the generalization to the nongaussian case by @xcite ( see also @xcite for related results ) . however , _ prior _ to these recent studies , the generalization of geweke s idea to some general setting was reported in 1987 , in econometry by gouriroux _ @xcite , and in engineering by rissannen&wax @xcite . gouriroux and his co - workers considered a joint markovian representation of the signals , and worked in a decision - theoretic framework . they defined a sequence of nested hypotheses , whether causality was true or not , instantaneous coupling was present or not . they then worked out the decision statistics using the kullback approach to decision theory @xcite , in which discrepancies between hypotheses are measured according to the kullback divergence between the probability measures under the hypotheses involved . in this setting , the decomposition obtained by geweke in the gaussian case was evidently generalised . in @xcite , the approach taken was closer to geweke s study , and it relied on system identification , in which the complexity of the model was taken into account . the probability measures were parameterized , and an information measure that jointly assessed the estimation procedure and the complexity of the model was used when predicting a signal . this allowed geweke s result to be extended to nonlinear modeling ( and hence the nongaussian case ) , and provided an information - theoretic interpretation of the tests . once again , the same kind of decomposition of dependence was obtained by these authors . we will see in section [ infotheory : sec ] that the decomposition holds due to kramers causal conditioning . these studies were limited to the bivariate case @xcite . in the late 1990 s , some studies began to develop in the physics community on influences between dynamical systems . a first route was taken that followed the ideas of dynamic system studies for the prediction of chaotic systems . to determine if one signal influenced another , the idea was to consider each of the signals as measured states of two different dynamic systems , and then to study the master - slave relationships between these two systems ( for examples , see @xcite ) . the dynamics of the systems was built using phase space reconstruction @xcite . the influence of one system on another was then defined by making a prediction of the dynamics in the reconstructed phase space of one of the processes . to our knowledge , the setting was restricted to the bivariate case . a second route , which was also restricted to the bivariate case , was taken and relied on information - theoretic tools . the main contributions were from palu and schreiber @xcite , with further developments appearing some years later @xcite . in these studies , the influence of one process on the other was measured by the discrepancy between the probability measures under the hypotheses of influence or no influence . naturally , the measures defined very much resembled the measures proposed by gouriroux _ et @xcite , and used the concept of conditional mutual information . the measure to assess whether one signal influences the other was termed _ transfer entropy _ by schreiber . its definition was proposed under a markovian assumption , as was exactly done in @xcite . the presentation by palu @xcite was more direct and was not based on a decision - theoretic idea . the measure defined is , however , equivalent to the transfer entropy . interestingly , palu noted in this 2001 paper the closeness of the approach to granger causality , as per the quotation : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ `` the [ latter ] measure can also be understood as an information theoretic formulation of the granger causality concept . '' _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ note that most of these studies considered bivariate analysis , with the notable exception of @xcite , in which the presence of side information ( other measured time series ) was explicitely considered . in parallel with these studies , many others were dedicated to the implementation of granger causality testing in fields as diverse as climatology ( with applications to the controversial questions of global warming ) and neuroscience ; see @xcite , to cite but a few . in a very different field , information theory , the problem of feedback has lead to many questions since the 1950 s . we will not review or cite anything on the problem created by feedback in information theory as this is not within the scope of the present study , but some information can be found in @xcite . instead , we will concentrate on studies that are directly related to the subject of this review . a major breakthrough was achieved by james massey in 1990 in a short conference paper @xcite . following the ( lost ? ) ideas of marko on bidirectional information theory that were developed in the markovian case @xcite , massey re - examined the usual definition of what is called a discrete memoryless channel in information theory , and he showed that the usual definition based on some probabilistic assumptions prohibited the use of feedback . he then clarified the definition of memory and feedback in a communication channel . as a consequence , he showed that in a general channel used with feedback , the usual definition of capacity that relies on mutual information was not adequate . instead , the right measure was shown to be _ directed information _ , an asymmetrical measure of the flow of information . these ideas were further examined by kramer , who introduced the concept of causal conditioning , and who developed the first applications of directed information theory to communication in networks @xcite . after some years , the importance of causal conditioning for the analysis of communication in systems with feedback was realized . many studies were then dedicated to the analysis of the capacity of channels with feedback and the dual problem of rate - distortion theory @xcite . due to the rapid development in the study of networks ( _ e.g. , _ social networks , neural networks ) and of the afferent connectivity problem , more recently many authors made connections between information theory and granger causality @xcite . some of these studies were restricted to the gaussian case , and to the bivariate case . most of these studies did not tackle the problem of instantaneous coupling . furthermore , several authors realized the importance of directed information theory to assess the circulation of information in networks @xcite . tools from directed information theory appear as natural measures to assess granger causality . although granger causality can be considered as a powerful theoretical framework to study influences between signals mathematically , directed information theory provides the measures to test theoretical assertions practically . as already mentioned , these measures are transfer entropy ( and its conditional versions ) , which assesses the dynamical part of granger causality , and instantaneous information exchange ( and its conditional versions ) , which assesses instantaneous coupling . this review is structured here as follows . we will first give an overview of the definitions of granger causality . these are presented in a multivariate setting . we go gradually from weak definitions based on prediction , to strong definitions based on conditional independence . the problem of instantaneous coupling is then discussed , and we show that there are two possible definitions for it . causality graphs ( after eichler @xcite ) provide particular reasons to prefer one of these definitions . section [ infotheory : sec ] introduces an analysis granger causality from an information - theoretic perspective . we insist on the concept of causal conditioning , which is at the root of the relationship studied . section [ links : sec ] then highlights the links . here , we first restate the definitions of granger causality using concepts from directed information theory . then from of a different point of view , we show how conceptual inference approaches lead to the measures defined in directed information theory . the review then closes with a discussion of some of the aspects that we do not present here intentionally , and on some lines for further research . all of the random variables , vectors and signals considered here are defined in a common probability space @xmath7 . they take values either in @xmath8 or @xmath9 , @xmath10 being some strictly positive integer , or they can even take discrete values . as we concentrate on conceptual aspects rather than technical aspects , we assume that the variables considered are well behaved. in particular , we assume finiteness of moments of sufficient order . we assume that continuously valued variables have a measure that is absolutely continuous with respect to the lebesgue measure of the space considered . hence , the existence of probability density functions is assumed . limits are supposed to exist when needed . all of the processes considered in this report are assumed to be stationary . we work with discrete time . a signal will generically be denoted as @xmath11 . this notation stands also for the value of the signal at time @xmath12 . the collection of successive samples of the signal , @xmath13 will be denoted as @xmath14 . often , an initial time will be assumed . this can be 0 , 1 , or @xmath15 . in any case , if we collect all of the sample of the signals from the initial time up to time @xmath16 , we will suppress the lower index and write this collection as @xmath17 . when dealing with multivariate signals , we use a graph - theoretic notation . this will simplify some connections with graphical modeling . let @xmath18 be an index set of finite cardinality @xmath19 . @xmath20 is a @xmath10-dimensional discrete time stationary multivariate process for the probability space considered . for @xmath21 , @xmath22 is the corresponding component of @xmath23 . likewise , for any subset @xmath24 , @xmath25 is the corresponding multivariate process @xmath26 . we say that subsets @xmath27 form a partition of @xmath18 if they are disjoint and if @xmath28 . the information obtained by observing @xmath25 up to time @xmath12 is resumed by the filtration generated by @xmath29 . this is denoted as @xmath30 . furthermore , we will often identify @xmath25 with @xmath31 in the discussion . the probability density functions ( p.d.f . ) or probability mass functions ( p.m.f ) will be denoted by the same notation as @xmath32 . the conditional p.d.f . and p.m.f . are written as @xmath33 . the expected value is denoted as @xmath34,e_x[.]$ ] or @xmath35 $ ] if we want to specify which variable is averaged , or under which probability measure the expected value is evaluated . independence between random variables and vectors @xmath0 and @xmath1 will be denoted as @xmath36 , while conditional independence given @xmath37 will be written as @xmath38 . the early definitions followed the ideas of wiener : a signal @xmath0 causes a signal @xmath1 if the past of @xmath0 helps in the prediction of @xmath1 . implementing this idea requires the performing of the prediction and the quantification of its quality . this leads to a weak , but operational , form of the definitions of granger causality . the idea of improving a prediction is generalized by encoding it into conditional dependence or independence . consider a cost function @xmath39 ( @xmath12 is some appropriate dimension ) , and the associated risk @xmath40 $ ] , where @xmath41 stands for an error term . let a predictor of @xmath42 be defined formally as @xmath43 , where @xmath31 and @xmath44 are subsets of @xmath18 , and @xmath5 is a function between appropriate spaces , chosen to minimize the risk with @xmath45 . solvability may be granted if @xmath5 is restricted to an element of a given class of functions , such as the set of linear functions . let @xmath46 be such a function class . define : @xmath47 \label{risk : eq}\end{aligned}\ ] ] @xmath48 is therefore the optimal risk when making a one - step - ahead prediction of the multivariate signal @xmath49 from the past samples of the multivariate signal @xmath25 . we are now ready to measure the influence of the past of a process on the prediction of another . to be relatively general and to prepare comments on the structure of the graph , this can be done for subsets of @xmath18 . we thus choose @xmath31 and @xmath44 to be two disjoint subsets of @xmath18 , and we define @xmath50 ( we use @xmath51 to mean substraction of a set ) . we study causality from @xmath25 to @xmath49 by measuring the decrease in the quality of the prediction of @xmath42 when excluding the past of @xmath25 . let @xmath52 be the optimal risk obtained for the prediction of @xmath49 from the past of all of the signals grouped in @xmath23 . this risk is compared to @xmath53 , where the past of @xmath25 is omitted . then , for the usual costs functions , we have necessarily : @xmath54 a natural first definition for granger causality is : @xmath25 granger does not cause @xmath49 relative to @xmath18 if and only if @xmath55 this definition of granger causality depends on the cost @xmath56 chosen as well as on the class @xmath46 of the functions considered . usually , a quadratic cost function is chosen , for its simplicity and for its evident physical interpretation ( a measure of the power of the error ) . the choice of the class of functions @xmath46 is crucial . the result of the causality test in definition 1 can change when the class is changed . consider the very simple example of @xmath57 , where @xmath58 and @xmath59 are gaussian independent and identically distributed ( i.i.d . ) sequences that are independent of each other . the covariance between @xmath60 and @xmath58 is zero , and using the quadratic loss and the class of linear functions , we conclude that @xmath1 does not granger cause @xmath0 , because using a linear function of @xmath61 to predict @xmath0 would lead to the same minimal risk as using a linear function of @xmath62 only . however , @xmath58 obviously causes @xmath62 , but in a nonlinear setting . the definition is given using the negative of the proposition . if by using the positive way , @xmath63 , granger proposes to say that @xmath25 is a _ prima facie _ cause of @xmath49 relative to @xmath18 , _ prima facie _ can be translated as at a first glance. this is used to insist that if @xmath18 is enlarged by including other measurements , then the conclusion might be changed . this can be seen as redundant with the mention of the relativity to the observation set @xmath18 , and we therefore do not use this terminology . however , a mention of the relativity to @xmath18 must be used , as modification of this set can alter the conclusion . a very simple example of this situation is the chain @xmath64 , where , for example , @xmath62 is an i.i.d . sequence , @xmath65 , @xmath66 , @xmath67 being independent i.i.d . sequences . relative to @xmath68 , @xmath0 causes @xmath37 if we use the quadratic loss and linear functions of the past samples of @xmath0 ( note here that the predictor @xmath69 must be a function of not only @xmath62 , but also of @xmath70 ) . however , if we include the past samples of @xmath1 and @xmath71 , then the quality of the prediction of @xmath37 does not deteriorate if we do not use past samples of @xmath0 . therefore , @xmath0 does not cause @xmath37 relative to @xmath71 . the advantage of the prediction - based definition is that is leads to operational tests . if the quadratic loss is chosen , working in a parameterized class of functions , such as linear filters or volterra filters , or even working in reproducing kernel hilbert spaces , allows the implementation of the definition @xcite . in such cases , the test needed can be evaluated efficiently from the data . from a theoretical point of view , the quadratic loss can be used to find the optimal function in a much wider class of functions : the measurable functions . in this class , the optimal function for the quadratic loss is widely known to be the conditional expectation @xcite . when predicting @xmath49 from the whole observation set @xmath18 , the optimal predictor is written as @xmath72 $ ] . likewise , elimination of @xmath31 from @xmath18 to study its influence on @xmath44 leads to the predictor @xmath73 $ ] , where @xmath74 . these estimators are of little use , because they are too difficult , or even impossible , to compute . however , they highlight the important of conditional distributions @xmath75 and @xmath76 in the problem of testing whether @xmath25 granger causes @xmath49 relative to @xmath18 or not . the optimal predictors studied above are equal if the conditional probability distributions @xmath75 and @xmath76 are equal . these distributions are identical if and only if @xmath77 and @xmath78 are independent conditionally to @xmath79 . a natural extension of definition 1 relies on the use of conditional independence . once again , let @xmath80 be a partition of @xmath18 . @xmath25 does not granger cause @xmath49 relative to @xmath18 if and only if @xmath81 this definition means that conditionally to the past of @xmath82 , the past of @xmath25 does not bring more information about @xmath77 than is contained in the past of @xmath49 . definition 2 is far more general than definition 1 . if @xmath25 does not granger cause @xmath49 relatively to @xmath18 in the sense of definition 1 , it also does not in the sense of definition 2 . then , definition 2 does not rely on any function class and on any cost function . however , it lacks an inherent operational character : the tools to evaluate conditional independence remain to be defined . the assessment of conditional independence can be achieved using measures of conditional independence , and some of these measures will be the cornerstone to link directed information theory and granger causality . note also that the concept of causality in this definition is again a relative concept , and that adding or deleting data from the observation set @xmath18 might modify the conclusions . the definitions given so far concern the influence of the past of one process on the present of another one . this is one reason that justifies the use of the term causality , when the definitions are actually based on statistical dependence . for an extensive discussion on the differences between causality and statistical dependence , we refer to @xcite . there is another influence between the processes that is not taken into account by definitions 1 and 2 . this influence is referred to as instantaneous causality @xcite . however , we will use our preferred term of instantaneous coupling , specifically to insist that it is not equivalent to a causal link _ per se _ , but actually a statistical dependence relationship . the term contemporaneous conditional independence that is used in @xcite could also be chosen . instantaneous coupling measures the common information between @xmath83 and @xmath77 that is not shared with their past . a definition of instantaneous coupling might then be that @xmath83 and @xmath77 are not instantaneously coupled if @xmath84 . this definition makes perfect sense if the observation set is reduced to @xmath31 and @xmath44 , a situation we refer to as the bivariate case . however , in general , there is also side information @xmath85 , and the definition must include this knowledge . however , this presence of side information then leads to two possible definitions of instantaneous coupling . @xmath25 and @xmath49 are not conditionally instantaneously coupled relative to @xmath18 if and only if @xmath86 , where @xmath87 is a partition of @xmath18 . the second possibility is the following : @xmath25 and @xmath49 are not instantaneously coupled relative to @xmath18 if and only if @xmath88 note that definitions 3 and 4 are symmetrical in @xmath31 and @xmath44 ( the application of bayes theorem ) . the difference between definitions 3 and 4 resides in the conditioning on @xmath89 instead of @xmath90 . if the side information up to time @xmath16 is considered only as in definition 4 , the instantaneous dependence or independence is not conditional on the presence of the remaining nodes in @xmath85 . thus , this coupling is a bivariate instantaneous coupling : it does measure instantaneous dependence ( or independence between @xmath31 and @xmath44 ) without considering the possible instantaneous coupling between either @xmath31 and @xmath85 or @xmath91 and @xmath85 . thus , instantaneous coupling found with definition 4 between @xmath31 and @xmath44 does not preclude the possibility that the coupling is actually due to couplings between @xmath31 and @xmath85 and/or @xmath44 and @xmath85 . inclusion of all of the information up to time @xmath92 in the conditioning variables allows the dependence or independence to be tested between @xmath83 and @xmath77 _ conditionally _ to @xmath93 . we end up here with the same differences as those between correlation and partial correlation , or dependence and conditional independence for random variables . in graphical modeling , the usual graphs are based on conditional independence between variables @xcite . these conditional independence graphs are preferred to independence graphs because of their geometrical properties ( _ e.g. , _ d - separation , @xcite ) , which match the markov properties possibly present in the multivariate distribution they represent . from a physical point of view , conditional independence might be preferable , specifically to eliminate false coupling due to third parties . in this respect , conditional independence is not the panacea , as independent variables can be conditionally dependent . the well - known example is the conditional coupling of independent @xmath0 and @xmath1 by their addition . indeed , even if independent , @xmath0 and @xmath1 are conditionally dependent to @xmath94 . granger causality graphs were defined and studied in @xcite . a causality graph is a mixed graph @xmath95 that encodes granger causality relationships between the components of @xmath23 . the vertex set @xmath18 stores the indexes of the components of @xmath23 . @xmath96 is a set of directed edges beween vertices . a directed edge from @xmath97 to @xmath98 is equivalent to `` @xmath22 granger causes @xmath99 relatively to @xmath18 '' . @xmath100 is a set of undirected edges . an undirected edge between @xmath22 and @xmath99 is equivalent to `` @xmath22 and @xmath99 are ( conditionally if def.4 adopted ) instantaneously coupled '' . interestingly , a granger causality graph may have markov properties ( as in usual graphical models ) reflecting a particular ( spatial ) structure of the joint probability distribution of the whole process @xmath101 @xcite . a taxonomy of markov properties : local , global , block recursive is studied in @xcite , and equivalence between these properties is put forward . more interestingly , these properties are linked with topological properties of the graph . therefore , structural properties of the graphs are equivalent to a particular factorization of the joint probability of the multivariate process . we will not continue on this subject here , but this must be known since it paves the way to more efficient inference methods for granger graphical modeling of multivariate processes ( see a first step in this direction in @xcite ) . directed information theory is a recent extension of information theory , even if its roots go back to the 1960 s and 1970 s and the studies of marko @xcite . the developments began in the late 1990 s , after the _ impetus _ given by james massey in 1990 @xcite . the basic theory was then extended by gerhard kramer @xcite , and then further developed by many authors @xcite to cite a few . we provide here a short review of the essentials of directed information theory . we will , moreover , adopt a presentation close to the spirit of granger causality to highlight the links between granger causality and information theory . we begin by recalling some basics from information theory . then , we describe the information - theoretic approach to study directional dependence between stochastic processes , first in the bivariate case , and then , from section [ sideinfo : ssec ] , for networks , _ i.e. , _ the multivariate case . let @xmath102 $ ] be the entropy of a random vector @xmath78 , the density of which is @xmath103 . let the conditional entropy be defined as @xmath104 $ ] . the mutual information @xmath105 between @xmath78 and @xmath106 is defined as @xcite : @xmath107 where @xmath108 $ ] is the kulback - leibler divergence . @xmath109 is 0 if and only if @xmath110 , and it is positive otherwise . the mutual information effectively measures independence since it is 0 if and only if @xmath78 and @xmath106 are independent random vectors . as @xmath111 , mutual information can not handle directional dependence . let @xmath112 be a third time series . it might be a multivariate process that accounts for side information ( all of the available observations , but @xmath78 and @xmath106 ) . to account for @xmath112 , the conditional mutual information is introduced : @xmath113 \\ & = & d_{kl}\big(p(x_a^n , y_b^n , x_c^n ) || p(x_a^n | x_c^n ) p(y_b^n| x_c^n ) p(x_c^n ) \big)\end{aligned}\ ] ] @xmath114 is zero if and only if @xmath115 and @xmath116 are independent _ conditionally _ to @xmath112 . stated differently , conditional mutual information measures the divergence between the actual observations and those which would be observed under the markov assumption @xmath117 . arrows can be misleading here , as by reversibility of markov chains , the equality above holds also for @xmath118 . this emphasizes how mutual information can not provide answers to the information flow directivity problem . the dependence between the components of the stochastic process @xmath23 is encoded in the full generality by the joint probability distributions @xmath119 . if @xmath18 is partitioned into subsets @xmath27 , studying dependencies between @xmath31 and @xmath44 then requires that @xmath119 is factorized into terms where @xmath25 and @xmath49 appear . for example , as @xmath120 , we can factorize the probability distribution as @xmath121 , which appears to emphasize a link from @xmath31 to @xmath44 . two problems appear , however : first , the presence of @xmath85 perturbs the analysis ( more than this , @xmath31 and @xmath85 have a symmetrical role here ) ; secondly , the factorization does not take into account the arrow of time , as the conditioning is considered over the whole observations up to time @xmath16 . marginalizing @xmath82 out makes it possible to work directly on @xmath122 . however , this eliminates all of the dependence between @xmath31 and @xmath44 that might exist _ via _ @xmath85 , and therefore this might lead to an incorrect assessment of the dependence . as for granger causality , this means that dependence analysis is relative to the observation set . restricting the study to @xmath31 and @xmath44 is what we referred to as the bivariate case , and this allows the basic ideas to be studied . we will therefore present directed information first in the bivariate case , and then turn to the full multivariate case . the second problem is at the root of the measure of directional dependence between stochastic processes . assuming that @xmath123 and @xmath42 are linked by some physical ( _ e.g. , _ biological , economical ) system , it is natural to postulate that their dependence is constrained by causality : if @xmath124 , then an event occurring at some time in @xmath31 will influence @xmath44 later on let us come back to the simple factorization above for the bivariate case . we have @xmath125 , and furthermore : @xmath126 where for @xmath127 , the first term is @xmath128 . the conditional distribution quantifies a directional dependence from @xmath31 to @xmath44 , but it lacks the causality property mentioned above , as @xmath129 quantifies the influence of the whole observation @xmath78 ( past and future of @xmath130 ) on the present @xmath131 knowing its past @xmath132 . the causality principle would require the restriction of the _ prior _ time @xmath130 to the past of @xmath31 only . kramer defined causal conditioning precisely in this sense @xcite . modifying eq . ( [ distcond : eq ] ) accordingly , we end up we the definition of the causal conditional probability distribution : @xmath133 remarkably this provides an alternative factorization of the joint probability . as noted by massey @xcite , @xmath134 can then be factorized as stands for the delayed collections of samples of @xmath49 . if the time origin is finite , 0 or 1 , the first element of the list @xmath135 should be understood as a wild card @xmath136 which does not influence the conditioning . ] : @xmath137 assuming that @xmath25 is the input of a system that creates @xmath49 , @xmath138 characterizes the feedback in the system : each of the factors controls the probability of the input @xmath25 at time @xmath130 conditionally to its past and to the past values of the output @xmath49 . likewise , the term @xmath139 characterizes the direct ( or feedforward ) link in the system . several interesting simple cases occur : * in the absence of feedback in the link from @xmath31 to @xmath44 , there is the following : @xmath140 or equivalently , in terms of entropies , @xmath141 and as a consequence : @xmath142 * likewise , if there is only a feedback term , then @xmath143 and then : @xmath144 * if the link is memoryless , _ i.e. , _ the output @xmath49 does not depend on the past , then : @xmath145 these results allow the question of whether @xmath25 influences @xmath49 to be addressed . if it does , then the joint distribution has the factorization of eq . ( [ factorisation : eq ] ) . however , if @xmath25 does not influence @xmath49 , then @xmath146 , and the factorization of the joint probability distribution simplifies to @xmath147 . kullback divergence between the probability distributions for each case generalizes the definition of mutual information to the directional mutual information : @xmath148 this quantity measures the loss of information when it is incorrectly assumed that @xmath25 does not influence @xmath49 . this was called _ directed information _ by massey @xcite . expanding the kullback divergence allows different forms for the directed information to be obtained : @xmath149 where we define the ` causal conditional entropy ' : @xmath150 \\ & = & \sum_{i=1}^n h\big ( x_b(i ) \big| x_b^{i-1 } , x_a^i \big ) \end{aligned}\ ] ] note that causal conditioning might involve more than one process . this leads to the defining of the causal conditional directed information as : @xmath151 the basic properties of the directed information were studied by massey and kramer @xcite , and some are recalled below . as a kullback divergence , the directed information is always positive or zero . then , simple algebraic manipulation allows the decomposition to be obtained : @xmath152 eq . ( [ decompdi : eq ] ) is fundamental , as it shows how mutual information splits into the sum of a feedforward information flow @xmath153 and a feedback information flow @xmath154 . in the absence of feedback , @xmath155 and @xmath156 . ( [ decompdi : eq ] ) allows the conclusion that the mutual information is always greater than the directed information , as @xmath157 is always positive or zero ( as directed information ) . it is zero if and only if : @xmath158 or equivalently : @xmath159 this situation corresponds to the absence of feedback in the link @xmath124 , whence the fundamental result that the directed information and the mutual information are equal if the channel is free of feedback . this result implies that mutual information over - estimates the directed information between two processes in the presence of feedback . this was thoroughly studied in @xcite , in a communication - theoretic framework . the decomposition of eq . ( [ decompdi : eq ] ) is surprising , as it shows that the mutual information is not the sum of the directed information flowing in both directions . instead , the following decomposition holds : @xmath160 where : @xmath161 this demonstrates that @xmath162 is symmetrical , but is in general not equal to the mutual information , except if and only if @xmath163 . as the term in the sum is the mutual information between the present samples of the two processes conditioned on their joint past values , this measure is a measure of instantaneous dependence . it is indeed symmetrical in @xmath31 and @xmath44 . the term @xmath164 will thus be named the _ instantaneous information exchange _ between @xmath25 and @xmath49 , and will hereafter be denoted as @xmath165 . like directed information , conditional forms of the instantaneous information exchange can be defined , as for example : @xmath166 which quantifies an instantaneous information exchange between @xmath31 and @xmath44 causally conditionally to @xmath85 . entropy and mutual information in general increase linearly with the length @xmath16 of the recorded time series . shannon s information rate for stochastic processes compensates for the linear growth by considering @xmath167 ( if the limit exists ) , where @xmath168 denotes any information measure on the sample @xmath17 of length @xmath16 . for the important class of stationary processes ( see _ e.g. , _ @xcite ) , the entropy rate turns out to be the limit of the conditional entropy : @xmath169 kramer generalized this result for causal conditional entropies @xcite , thus defining the directed information rate for stationary processes as : @xmath170 this result holds also for the instantaneous information exchange rate . note that the proof of the result relies on the positivity of the entropy for discrete valued stochastic processes . for continously valued processes , for which the entropy can be negative , the proof is more involved and requires the methods developed in @xcite , and see also @xcite . as introduced by schreiber in @xcite , _ transfer entropy _ evaluates the deviation of the observed data from a model , assuming the following joint markov property : @xmath171 this leads to the following definition : @xmath172\end{aligned}\ ] ] then @xmath173 if and only if eq . ( [ eq : jointmarkov ] ) is satisfied . although in the original definition , the past of @xmath0 in the conditioning might begin at a different time @xmath174 , for practical reasons @xmath175 is considered . actually , no _ a priori _ information is available about possible delays , and setting @xmath175 allows the transfer entropy to be compared with the directed information . by expressing the transfer entropy as a difference of conditional entropies , we get : @xmath176 for @xmath177 and choosing 1 as the time origin , the identity @xmath178 leads to : @xmath179 for stationary processes , letting @xmath180 and provided the limits exist , for the rates , we obtain : @xmath181 transfer entropy is the part of the directed information that measures the influence of the past of @xmath25 on the present of @xmath49 . however it does not take into account the possible instantaneous dependence of one time series on another , which is handled by directed information . moreover , as defined by schreiber in @xcite , only @xmath182 is considered in @xmath183 , instead of its sum over @xmath130 in the directed information . thus stationarity is implicitly assumed and the transfer entropy has the same meaning as a rate . a sum over delays was considered by palu as a means of reducing errors when estimating the measure @xcite . summing over @xmath16 in eq . ( [ eq : schreibert ] ) , the following decomposition of the directed information is obtained : @xmath184 eq . ( [ dirinfodecomp : eq ] ) establishes that the influence of one process on another can be decomposed into two terms that account for the past and for the instantaneous contributions . moreover , this explains the presence of the term @xmath185 in the r.h.s . ( [ sums2di : eq ] ) : instantaneous information exchange is counted twice in the l.h.s . terms @xmath162 , but only once in the mutual information @xmath186 . this allows eq . ( [ sums2di : eq ] ) to be written in a slightly different form , as : @xmath187 which is very appealing , as it shows how dependence as measured by mutual information decomposes as the sum of the measures of directional dependences and the measure of instantaneous coupling . the preceding developments aimed at the proposing of definitions of the information flow between @xmath25 and @xmath49 ; however , whenever @xmath31 and @xmath44 are connected to other parts of the network , the flow of information between @xmath31 and @xmath44 might be mediated by other members of the network . time series observed on nodes other than @xmath31 and @xmath44 are hereafter referred to as side information . the available side information at time @xmath16 is denoted as @xmath112 , with @xmath27 forming a partition of @xmath18 . then , depending on the type of conditioning ( usual or causal ) two approaches are possible . usual conditioning considers directed information from @xmath31 to @xmath44 that is conditioned on the whole observation @xmath112 . however , this leads to the consideration of causal flows from @xmath31 to @xmath44 that possibly include a flow that goes from @xmath31 to @xmath44 _ via _ @xmath85 in the future ! thus , an alternate definition for conditioning is required . this is given by the definition of eq . ( [ causaldi1:eq ] ) of the causal conditional directed information : @xmath188 does the causal conditional directed information decompose as the sum of a causal conditional transfer entropy and a causal conditional instantaneous information exchange , as it does in the bivariate case ? applying twice the chain rule for conditional mutual information , we obtain : @xmath189 in this equation , @xmath190 is termed the causal conditional transfer entropy. this measures the flow of information from @xmath31 to @xmath44 by taking into account a possible route _ via _ @xmath85 . if the flow of information from @xmath31 to @xmath44 is entirely relayed by @xmath85 , the causal conditional transfer entropy is zero . in this situation , the usual transfer entropy is not zero , indicating the existence of a flow from @xmath31 to @xmath44 . conditioning on @xmath85 allows the examination of whether the route goes through @xmath85 . the term : @xmath191 is the causal conditional information exchange. this measures the conditional instantaneous coupling between @xmath31 and @xmath44 . the term @xmath192 emphasizes the difference between the bivariate and the multivariate cases . this extra term measures an instantaneous coupling and is defined by : @xmath193 an alternate decomposition to eq . ( [ decomp : eq ] ) is : @xmath194 which emphasizes that the extra term comes from : @xmath195 this demonstrates that the definition of the conditional transfer entropy requires conditioning on the past of @xmath85 . if not , the extra term appears and accounts for instantaneous information exchanges between @xmath85 and @xmath44 , due to the addition of the term @xmath196 in the conditioning . this extra term highlights the difference between the two different natures of instantaneous coupling . the first term , @xmath197 describes the intrinsic coupling in the sense that it does not depend on parties other than @xmath85 and @xmath44 . the second coupling term , @xmath198 is relative to the extrinsic coupling , as it measures the instantaneous coupling at time @xmath130 that is created by variables other than @xmath44 and @xmath85 . as discussed in section [ instantcoupl : ssec ] , the second definition for instantaneous coupling considers conditioning on the past of the side information _ only_. causally conditioning on @xmath199 does not modify the results of the bivariate case . in particular , we still get the elegant decomposition : @xmath200 and therefore , the decomposition of eq . ( [ sums2diinst : eq ] ) is generalized to : @xmath201 where : @xmath202 is the causally conditioned mutual information . finally , let us consider that for jointly stationary times series , the causal directed information rate is defined similarly to the bivariate case , as : @xmath203 in this section we have emphasized on kramer s causal conditioning , both for the definition of directed information and for taking into account side information . we have also shown that schreiber s transfer entropy is that part of the directed information that is dedicated to the strict sense of causal information flow ( not accounting for simultaneous coupling ) . the next section more explicitely revisits the links between granger causality and directed information theory . granger causality in its probabilistic form is not operational . in practical situations , for assessing granger causality between time series , we can not use the definition directly . we have to define dedicated tools to assess the conditional independence . we use this inference framework to show the links between information theory and granger causality . we begin by re - expressing granger causality definitions in terms of some measures that arise from directed information theory . therefore , in an inference problem , these measures can be used as tools for inference . however , we show in the following sections that these measures naturally emerge from the more usual statistical inference strategies . in the following , and as above , we use the same partitioning of @xmath18 into the union of disjoint subsets of @xmath31 , @xmath44 and @xmath85 . as anticipated in the presentation of directed information , there are profound links between granger causality and directed information measures . granger causality relies on conditional independence , and it can also be defined using measures of conditional independence . information - theoretic measures appear as natural candidates . recall that two random elements are independent if and only if their mutual information is zero . moreover , two random elements are independent conditionally to a third one if and only if the conditional mutual information is zero . we can reconsider definitions 2 , 3 and 4 and recast them in term of information - theoretic measures . definition 2 stated that @xmath25 does not granger cause @xmath49 relative to @xmath18 if and only if @xmath204 . this can be alternatively rephrased into : @xmath25 does not granger cause @xmath49 relative to @xmath18 if and only if @xmath205 since @xmath206 is equivalent to @xmath207 . otherwise stated , the transfer entropy from @xmath31 to @xmath44 causally conditioned on @xmath85 is zero if and only if @xmath31 does not granger cause @xmath44 relative to @xmath18 . this shows that causal conditional transfer entropy can be used to assess granger causality . likewise , we can give alternative definitions of instantaneous coupling . @xmath25 and @xmath49 are not conditionally instantaneously coupled relative to @xmath18 if and only if @xmath208 , or if and only if the instantaneous information exchange causally conditioned on @xmath85 is zero . the second possible definition of instantaneous coupling is equivalent to : @xmath25 and @xmath49 are not instantaneously coupled relative to @xmath18 if and only if @xmath209 , or if and only if the instantaneous information exchange causally conditioned on the past of @xmath85 is zero . note that in the bivariate case only ( when @xmath85 is not taken into account ) , the directed information @xmath210 summarizes both the granger causality and the coupling , as it decomposes as the sum of the transfer entropy @xmath211 and the instantaneous information exchange @xmath212 . we consider the practical problem of inferring the graph of dependence between the components of a multivariate process . let us assume that we have measured a multivariate process @xmath213 for @xmath214 . we want to study the dependence between each pair of components ( granger causality and instantaneous coupling between any pair of components relative to @xmath18 ) . we can use the result of the preceding section to evaluate the directed information measures on the data . when studying the influence from any subset @xmath31 to any subset @xmath44 , if the measures are zero , then there is no causality ( or no coupling ) ; if they are strictly positive , then @xmath31 granger causes @xmath44 relative to @xmath18 ( or @xmath31 and @xmath44 are coupled relative to @xmath18 ) . this point of view has been adopted in many of the studies that we have already referred to ( _ e.g. _ @xcite ) , and it relies on estimating the measures from the data . we will not review the estimation problem here . however , it is interesting to examine more traditional frameworks for testing granger causality , and to examine how directed information theory naturally emerges from these frameworks . to begin with , we show how the measures defined emerge from a binary hypothesis - testing view of granger causality inference . we then turn to prediction and model - based approaches . we will review how geweke s measures of granger causality in the gaussian case are equivalent to directed information measures . we will then present a more general case adopted by @xcite and based on a model of the data . in the inference problem , we want to determine whether or not @xmath25 granger causes ( is coupled with ) or not @xmath49 relative to @xmath18 . this can be formulated as a binary hypothesis testing problem . for inferring dependencies between @xmath31 and @xmath44 relative to @xmath18 , we can state the problem as follows . assume we observe @xmath215 . then , we want to test : @xmath25 does not granger cause @xmath49 , against @xmath25 causes @xmath49 ; and @xmath25 and @xmath49 are instantaneously coupled against ` @xmath25 are @xmath49 not instantaneously coupled ' . we will refer to the first test as the granger causality test , and to the second one , as the instantaneous coupling test . in the bivariate case , for which the granger causality test indicates : @xmath216 this leads to the testing of different functional forms of the conditional densities of @xmath131 given the past of @xmath25 . the likelihood of the observation under @xmath217 is the full joint probability @xmath218 . under @xmath219 we have @xmath220 and the likelihood reduces to @xmath221 . the log likelihood _ ratio _ for the test is : @xmath222 for example , in the case where the multivariate process is a positive harris recurrent markov chain @xcite , the law of large numbers applies and we have under hypothesis @xmath217 : @xmath223 where @xmath224 is the transfer entropy rate . thus from a practical point of view , as the amount of data increases , we expect the log likelihood _ ratio _ to be close to the transfer entropy rate ( under @xmath217 ) . turning the point of view , this can justify the use of an estimated transfer entropy to assess granger causality . under @xmath219 , @xmath225 converges to @xmath226 , which can be termed ` the lautum transfer entropy rate ' that extends the ` lautum directed information ' defined in @xcite . directed information can be viewed as a measure of the loss of information when assuming @xmath25 does not causally influence @xmath49 when it actually does . likewise , ` lautum directed information ' measures the loss of information when assuming @xmath25 does causally influence @xmath49 , when actually it does not . for testing instantaneous coupling , we will use the following : @xmath227 where under @xmath219 , there is no coupling . then , under @xmath217 and some hypothesis on the data , the likelihood ratio converges almost surely to the information exchange rate @xmath228 . a related encouraging result due to @xcite is the emergence of the directed information in the false - alarm probability error rate . merging the two tests ( [ test1:eq]),([test2:eq ] ) , _ i.e. , _ testing both for causality and coupling , or neither , the test is written as : @xmath229 among the tests with a probability of miss @xmath230 that is lower than some positive value @xmath231 , the best probability of false alarm @xmath232 follows @xmath233 when @xmath183 is large . for the case studied here , this is the so - called stein lemma @xcite . in the multivariate case , there is no such result in the literature . an extension is proposed here . however , this is restricted to the case of instantaneously _ uncoupled _ time series . thus , we assume for the end of this subsection that : @xmath234 which means that there is no instantaneous exchange of information between the three subsets that form a partition of @xmath18 . this assumption has held in most of the recent studies that have applied granger causality tests . it is , however , unrealistic in applications where the dynamics of the processes involved are faster than the sampling period adopted ( see @xcite for a discussion in econometry ) . consider now the problem of testing granger causality of @xmath31 on @xmath44 relative to @xmath18 . the binary hypothesis test is given by : @xmath235 the log likelihood _ ratio _ reads as : @xmath236 again , by assuming that the law of large numbers applies , we can conclude that under @xmath217 @xmath237 this means that the causal conditional transfer entropy rate is the limit of the log likelihood _ ratio _ as the amount of data increases . following definition 1 and focusing on the quadratic risk @xmath238 $ ] , geweke introduced the following indices for the study of gaussian stationary processes @xcite : @xmath239 geweke demonstrated the efficiency of these indices for testing granger causality and instantaneous coupling ( bivariate and multivariate cases ) . furthermore , in the bivariate case , geweke showed that : @xmath240 where @xmath241 is the mutual information rate . this relationship that was already sketched out in @xcite , is nothing but eq . ( [ sums2diinst : eq ] ) . indeed , in the gaussian case , @xmath242 and @xmath243 stem from the knowledge that the entropy rate of a gaussian stationary process is the logarithm of the asymptotic power of the one - step - ahead prediction @xcite . likewise , we can show that @xmath244 and @xmath245 holds . in the multivariate case , conditioning on the past of the side information , _ i.e. _ @xmath199 , in the definition of @xmath246 , a decomposition analagous to eq . ( [ gewedecomp : eq ] ) holds , and is exactly that given by eq . ( [ sums2diinstcond : eq ] ) . in a more general framework , we examine how a model - based approach can be used to test for granger causality , and how directed information comes into play . let us consider a rather general model in which @xmath247 is a multivariate markovian process that statisfies : @xmath248 where @xmath249 is a function belonging to some functional class @xmath46 , and where @xmath250 is a multivariate i.i.d . sequence , the components of which are not necessarily mutually independent . function @xmath251 might ( or might not ) dependon @xmath252 , a multidimensional parameter . this general model includes as a particular case , linear multivariate autoregressive with moving average ( arma ) models , and nonlinear arma models ; @xmath251 can also stand for a function belonging to some reproducing kernel hilbert space , which can be estimated from the data @xcite . using the partition @xmath27 , this model can be written equivalently as : @xmath253 where the functions @xmath254 are the corresponding components of @xmath255 . this relation can be used for inference in a parametric setting : the functional form is assumed to be known and the determination of the function is replaced by the estimation of the parameters @xmath256 . this can also be used in a nonparametric setting , in which case the function @xmath5 is searched for in an appropriate functional space , such as an rkhs associated to a kernel @xcite . in any case , for studying the influence of @xmath25 to @xmath49 relative to @xmath18 , two models are required for @xmath49 : one in which @xmath49 explicitly depends on @xmath25 , and the other one in which @xmath49 does not depend on @xmath25 . in the parametric setting , the two models can be merged into a single model , in such a way that some components of the parameter @xmath257 are , or not , zero , which dependis whether @xmath31 causes @xmath44 or not . the procedure then consists of testing nullity ( or not ) of these components . in the linear gaussian case , this leads to the geweke indices discussed above . in the nonlinear ( nongaussian ) case , the geweke indices can be used to evaluate the prediction in some classes of nonlinear models ( in the minimum mean square error sense ) . in this latter case , the decomposition of the mutual information , eq . ( [ gewedecomp : eq ] ) , has no reason to remain valid . another approach base relies on directly modeling the probability measures . this approach has been used recently to model spiking neurons and to infer granger causality between several neurons working in the class of generalized linear models @xcite . interestingly , the approach has been used either to estimate the directed information @xcite or to design a likelihood ratio test @xcite . suppose we wish to test whether @xmath25 granger causes @xmath49 relative to @xmath18 as a binary hypothesis problem , as in section [ binaryhyp : ssec ] . forgetting the problem of instantaneous coupling , the problem is then to choose between the hypotheses : @xmath258 where the existence of causality is entirely reflected into the parameter @xmath252 . to be more precise , @xmath259 should be seen as a restriction of @xmath260 when its components linked to @xmath25 are set to zero . as a simple example using the model approach discussed above , consider the simple linear gaussian model @xmath261 where @xmath262 is an i.i.d . gaussian sequence , and @xmath263 are multivariate impulse responses of appropriate dimensions . define @xmath264 and @xmath265 . testing for granger causality is then equivalent to testing @xmath266 ; furthermore , the likelihood _ ratio _ can be implemented due to the gaussian assumption . the example developed in @xcite , assumes that the probability that neuron @xmath98 ( @xmath267 ) sends a message at time @xmath268 ( @xmath269 ) to its connected neighbors is given by the conditional probability @xmath270 where @xmath271 is some decision function , the output of which belongs to @xmath272 $ ] , @xmath31 represents the subset of neurons that can send information to @xmath98 , and @xmath273 represents external inputs to @xmath98 . defining this probability for all @xmath274 completely specifies the behavior of the neural network @xmath18 . the problem is a composite hypothesis testing problem , in which parameters defining the likelihoods have to be estimated . it is known that tere is no definitive answer to this problem @xcite . an approach that relies on an estimation of the parameters using maximum likelihood can be used . letting @xmath275 be the space where parameter @xmath252 is searched for and @xmath276 the subspace where @xmath259 lives , then the generalized loglikelihood _ ratio _ test reads : @xmath277 where @xmath278 denotes the maximum likelihood estimator of @xmath252 under hypothesis @xmath130 . in the linear gaussian case , we will recover exactly the measures developed by geweke . in a more general case , and as illustrated in section [ binaryhyp : ssec ] , as the the maximum likelihood estimates are efficient , we can conjecture that the generalized log likelihood _ ratio _ will converge to the causal conditional transfer entropy rate if sufficiently relevant conditions are imposed on the models ( _ e.g. , _ markov processes with recurrent properties ) . this approach was described in @xcite in the bivariate case . granger causality was developed originally in econometrics , and it is now transdisciplinary , with the literature on the subject is widely dispersed . we have tried here to sum up the profound links that exist between granger causality and directed information theory . the key ingredients to build these links are conditional independence and the recently introduced causal conditioning . we have eluded the important question of how to practically use the definitions and measures presented here . some of the measures can be used and implemented easily , especially in the linear gaussian case . in a more general case , different approaches can be taken . the information - theoretic measures can be estimated , or the prediction can be explicitly carried out and the residuals used to assess causality . many studies have been carried out over the last 20 years on the problem of estimation of information - theoretic measures . we refer to @xcite for information on the different ways to estimate information measures . recent studies into the estimation of entropy and/or information measures are @xcite . the recent report by @xcite extensively details and applies transfer entropy in neuroscience using @xmath12-nearest neighbors type of estimators . concerning the applications , important reviews include @xcite , where some of the ideas discussed here are also mentioned , and where practicalities such as the use of surrogate data , for example , are extensively discussed . applications for neuroscience are discussed in @xcite . information - theoretic measures of conditional independence based on kullback divergence were chosen here to illustrate the links between granger causality and ( usual ) directed information theory . other type of divergence could have been chosen ; metrics in probability space could also be useful in the assessing of conditional independence . as an illustration , we refer to the study of fukumizu and co - workers @xcite , where conditional independence was evaluated using the hilbert - schmidt norm of an operator between reproducing kernel hilbert spaces . the operator generalizes the partial covariance between two random vectors given a third one , and is called the conditional covariance operator . furthermore , the hilbert - schmidt norm of conditional covariance operator can be efficiently estimated from data . a related approach is also detailed in @xcite . many important directions can be followed . causality between nonstationary processes has rarely been considered ( see however @xcite for an _ ad - hoc _ approach in neuroscience ) . a very promising methodology is to adopt a graphical modeling way of thinking . the result of @xcite on the structural properties of markov - granger causality graphs can be used to identify such graphs from real datasets . a first step in this direction was proposed by @xcite . assuming that the network under study is a network of sparsely connected nodes and that some markov properties hold , efficient estimation procedures can be designed , as is the case in usual graphical modeling . p.o.a . is supported by a marie curie international outgoing fellowship from the european community . p. o. amblard and o. j. j. michel . sur diffrentes mesures de dpendance causales entre signaux al atoires ( on different measures of causal dependencies between random signals ) . in _ proc . gretsi , dijon , france , sept . _ , 2009 . m. kaminski , m. ding , w. truccolo , and s. bressler . evaluating causal relations in neural systems : granger causality , directed transfer functions and statistical assessment of significance . , 85:145157 , 2001 . c. j. quinn and t. p. coleman and n. kiyavash and n. g.hastopoulos . estimating the directed information to infer causal relationships in ensemble neural spike train recordings , journal of computational neuroscience , 30 : 1744 , 2011

this report reviews the conceptual and theoretical links between granger causality and directed information theory . we begin with a short historical tour of granger causality , concentrating on its closeness to information theory . the definitions of granger causality based on prediction are recalled , and the importance of the observation set is discussed . we present the definitions based on conditional independence . the notion of instantaneous coupling is included in the definitions . the concept of granger causality graphs is discussed . we present directed information theory from the perspective of studies of causal influences between stochastic processes . causal conditioning appears to be the cornerstone for the relation between information theory and granger causality . in the bivariate case , the fundamental measure is the directed information , which decomposes as the sum of the transfer entropies and a term quantifying instantaneous coupling . we show the decomposition of the mutual information into the sums of the transfer entropies and the instantaneous coupling measure , a relation known for the linear gaussian case . we study the multivariate case , showing that the useful decomposition is blurred by instantaneous coupling . the links are further developed by studying how measures based on directed information theory naturally emerge from granger causality inference frameworks as hypothesis testing . * keyword : * granger causality , transfer entropy , information theory , causal conditioning , conditional independence

You are an expert at summarizing long articles. Proceed to summarize the following text: the recent complete dna sequences of many organisms are available to systematically search of genome structure . for the large amount of dna sequences , developing methods for extracting meaningful information is a major challenge for bioinformatics . to understand the one - dimensional symbolic sequences composed of the four letters ` a ' , ` c ' , ` g ' and ` t ' ( or ` u ' ) , some statistical and geometrical methods were developed@xcite . in special , chaos game representation ( cgr)@xcite , which generates a two - dimensional square from a one - dimensional sequence , provides a technique to visualize the composition of dna sequences . the characteristics of cgr images was described as genomic signature , and classification of species in the whole bacteria genome was analyzed by making an euclidean metric between two cgr images@xcite . based on the genomic signature , the distance between two dna sequences depending on the length of nucleotide strings was presented@xcite and the horizontal transfers in prokaryotes and eukaryotes were detected and charaterized@xcite . recently , a one - to - one metric representation of the dna sequences@xcite , which was borrowed from the symbolic dynamics , makes an ordering of subsequences in a plane . suppression of certain nucleotide strings in the dna sequences leads to a self - similarity of pattern seen in the metric representation of dna sequences . self - similarity limits of genomic signatures were determined as an optimal string length for generating the genomic signatures@xcite . moreover , by using the metric representation method , the recurrence plot technique of dna sequences was established and employed to analyze correlation structure of nucleotide strings@xcite . as a eukaryotic organism , yeast is one of the premier industrial microorganisms , because of its essential role in brewing , baking , and fuel alcohol production . in addition , yeast has proven to be an excellent model organism for the study of a variety of biological problems involving the fields of genetics , molecular biology , cell biology and other disciplines within the biomedical and life sciences . in april 1996 , the complete dna sequence of the yeast ( saccharomyces cevevisiae ) genome , consisting of 16 chromosomes with 12 million basepairs , had been released to provide a resource of genome information of a single organism . however , only 43.3% of all 6000 predicted genes in the saccharomyces cerevisiae yeast were functionally characterized when the complete sequence of the yeast genome became available@xcite . moreover , it was found that dna transposable elements have ability to move from place to place and make many copies within the genome via the transposition@xcite . therefore , the yeast complete dna sequence remain a topic to be studied respect to its genome architecture structure in the whole sequence . in this paper , using the metric representation and recurrence plot methods , we analyze global transposable characteristics in the yeast complete dna sequence , i.e. , 16 chromosome sequences . for a given dna sequence @xmath0 ( @xmath1 ) , a plane metric representation is generated by making the correspondence of symbol @xmath2 to number @xmath3 or @xmath4 and calculating values ( @xmath5 , @xmath6 ) of all subsequences @xmath7 ( @xmath8 ) defined as follows @xmath9 where @xmath3 is 0 if @xmath10 or 1 if @xmath11 and @xmath12 is 0 if @xmath13 or 1 if @xmath14 . thus , the one - dimensional symbolic sequence is partitioned into @xmath15 subsequences @xmath16 and mapped in the two - dimensional plane ( @xmath17 ) . subsequences with the same ending @xmath18-nucleotide string , which are labeled by @xmath19 , correspond to points in the zone encoded by the @xmath18-nucleotide string . taking a subsequence @xmath20 , we calculate @xmath21 where @xmath22 is the heaviside function [ @xmath23 , if @xmath24 ; @xmath25 , if @xmath26 and @xmath27 is a subsequence ( @xmath28 ) . when @xmath29 , i.e. , @xmath30 , a point @xmath31 is plotted in a plane . thus , repeating the above process from the beginning of one - dimensional symbolic sequence and shifting forward , we obtain a recurrence plot of the dna sequence . for presenting correlation structure in the recurrence plot plane , a correlation intensity is defined at a given correlation distance @xmath32 @xmath33 the quantity displays the transference of @xmath18-nucleotide strings in the dna sequence . to further determine positions and lengths of the transposable elements , we analyze the recurrent plot plane . since @xmath34 and @xmath27 @xmath35 , the transposable element has the length @xmath18 at least . from the recurrence plot plane , we calculate the maximal value of @xmath36 to satisfy @xmath37 i.e. , @xmath38 and @xmath39 . thus , the transposable element with the correction distance @xmath40 has the length @xmath41 . the transposable element is placed at the position @xmath42 and @xmath43 . the saccharomyces cevevisiae yeast has 16 chromosomes , which are denoted as yeast i to xvi . using the metric representation and recurrence plot methods , we analyze correlation structures of the 16 dna sequences . according to the characteristics of the correlation structures , we summarize the results as follows : \(1 ) the correlation distance has a short period increasing . the yeast i , ix and xi have such characteristics . let me take the yeast i as an example to analyze . fig.1 displays the correlation intensity at different correlation distance @xmath44 with @xmath45 . a local region is magnified in the figure . it is clearly evident that there exist some equidistance parallel lines with a basic correlation distance @xmath46 . ( 4 ) , we determine positions and lengths of the transposable elements in table i , where their lengths are limited in @xmath47 . many nucleotide strings have correlation distance , which is the integral multiple of @xmath48 . they mainly distribute in two local regions of the dna sequence ( 25715 - 26845 ) and ( 204518 - 206554 ) or ( 11.2 - 11.7% ) and ( 88.8 - 89.7% ) expressed as percentages . the yeast ix and xi have similar behaviors . the yeast ix has the basic correlation distance @xmath49 . many nucleotide strings ( @xmath50 ) with the integral multiple of @xmath48 mainly distribute in a local region of the dna sequence ( 391337 - 393583 ) or ( 89.0 - 89.5% ) expressed as percentages . the yeast xi has the basic correlation distance @xmath51 . many nucleotide strings ( @xmath50 ) with the integral multiple of @xmath48 mainly distribute in a local region of the dna sequence ( 647101 - 647783 ) or ( 97.1 - 97.2% ) expressed as percentages . \(2 ) the correlation distance has a long major value and a short period increasing . the yeast ii , v , vii , viii , x , xii , xiii , xiv , xv and xvi have such characteristics . let me take the yeast ii as an example to analyze . fig.2 displays the correlation intensity at different correlation distance @xmath44 with @xmath45 . the maximal correlation intensity appears at the correlation distance @xmath52 . a local region is magnified in the figure . it is clearly evident that there exist some equidistance parallel lines with a basic correlation distance @xmath53 . in table ii , positions and lengths ( @xmath50 ) of the transposable elements are given . the maximal transposable elements mainly distribute in two local regions of the dna sequence ( 221249 - 224565 , 259783 - 263097 ) or ( 27.2 - 27.6% , 31.9- 32.4% ) expressed as percentage . near the positions , there also exist some transposable elements with approximate values for @xmath48 . moreover , many nucleotide strings have correlation distance , which is the integral multiple of @xmath48 . they mainly distribute in a local region of the dna sequence ( 391337 - 393583 ) or ( 89.0 - 89.5% ) expressed as percentages . in the other 9 dna sequences , the yeast v , x , xii , xiii , xiv , xv and xvi have the same basic correlation distance @xmath53 and similar behaviors with different major correlation distance @xmath5449099 , 5584 , 9137 , 12167 , 5566 , 447110 and 45988 , respectively . the yeast vii and viii have different basic correlation distance @xmath55 and 135 , and similar behaviors with the major correlation distance @xmath56 and 1998 , respectively . \(3 ) the correlation distance has a long quasi - period increasing . the yeast iii has such characteristics . 3 displays the coherence intensity at different correlation distance @xmath57 with @xmath45 . the correlation intensity has the maximal value at the correlation distance @xmath58 and two vice - maximal values at the correlation distance @xmath59 and @xmath60 . since @xmath61 , the coherence distance has a quasi - period increasing . a local region is magnified in the figure . these does not exist any clear short period increasing of the correlation distance . using eq . ( 4 ) , we determine positions and lengths ( @xmath50 ) of the transposable elements in table iii . the maximal and vice - maximal transposable elements mainly distribute in local regions of the dna sequence ( 11499 - 13810 , 197402 - 199713 ) , ( 198171 - 199796 , 291794 - 293316 ) and ( 12268 - 12932 , 291794 - 292460 ) or ( 3.6 - 4.4% , 62.6 - 63.6% ) , ( 62.8 - 63.4% , 92.5 - 93.0% ) and ( 3.9 - 4.1% , 92.5 - 92.7% ) expressed as percentage . \(4 ) the correlation distance has a long major value and a long quasi - period and two short period increasing . the yeast iv has such characteristics . 4 displays the coherence intensity at different correlation distance @xmath44 with @xmath45 . the maximal coherence intensity appears at the correlation distance @xmath62 . there also exist three vice - maximal values at the correlation distance @xmath63 , @xmath64 and @xmath65 , which forms a long quasi - period increasing of the correlation distance , i.e. , @xmath66 . a local region is magnified in the figure . it is clearly evident that there exist two short period increasing with @xmath67 and @xmath68 in the correlation distance . in table iv , positions and lengths ( @xmath47 ) of the transposable elements are determined by using eq . ( 4 ) . all correlation distance with the long major value and the long quasi - period and two short period increasing are denoted . the transposable elements with @xmath69 , @xmath70 , @xmath71 , @xmath72 , @xmath73 and @xmath74 mainly distribute in local regions of the dna sequence ( 527570 - 538236 ) , ( 871858 - 876927 , 981207 - 986276 ) , ( 645646 - 651457 , 878346 - 884257 ) , ( 646379 - 651032 , 987600 - 992253 ) , ( 1307733 - 1308591 ) and ( 758135 - 759495 ) or ( 34.4 - 35.1% ) , ( 56.9 - 57.2% , 64.0 - 64.4% ) , ( 42.1 - 42.5% , 57.3 - 57.7% ) , ( 42.2 - 42.5% , 64.4 - 64.8% ) , ( 85.36 - 85.41% ) and ( 49.5 - 49.6% ) expressed as percentages . \(5 ) the dna sequence is hardly relevant . the yeast vi has such characteristics . 5 displays the coherence intensity at different correlation distance @xmath44 with @xmath45 . the maximal coherence intensity appears at the correlation distance @xmath75 . a local region is magnified in the figure . the sequence has not a short period increasing of the coherence distance . in table v , positions and lengths ( @xmath50 ) of the transposable elements are given . only one nucleotide string with the length 337 has the correlation distance @xmath69 . the yeast vi is almost never relevant , so the yeast vi approaches a random sequence . global transposable characteristics in the yeast complete dna sequence is determined by using the metric representation and recurrence plot methods . positions and lengths of all transposable nucleotide strings in the 16 chromosome dna sequences of the yeast are determined . in the form of the correlation distance of nucleotide strings , the fundamental transposable characteristics displays a short period increasing , a long quasi - period increasing , a long major value and hardly relevant . the 16 chromosome sequences are divided into 5 groups , which have one or several of the 4 kinds of the fundamental transposable characteristics . p. j. deschavanne , a. giron , j. vilain , g. fagot , and b. fertil , genomic signature : characterization and classification of species assessed by chaos game representation of sequences . * 16 * ( 1999 ) 1391 .

global transposable characteristics in the complete dna sequence of the saccharomyces cevevisiae yeast is determined by using the metric representation and recurrence plot methods . in the form of the correlation distance of nucleotide strings , 16 chromosome sequences of the yeast , which are divided into 5 groups , display 4 kinds of the fundamental transposable characteristics : a short period increasing , a long quasi - period increasing , a long major value and hardly relevant . * keywords * yeast , dna sequences , coherence structure , metric representation , recurrence plot +

You are an expert at summarizing long articles. Proceed to summarize the following text: structural properties of regularly stepped metal surfaces have been the focus of a broad range of theoretical and experimental studies because of the eminent role they play in technologically important phenomena such as thin film growth , epitaxial layer formation , nanostructuring of material , and catalysis @xcite . according to crystallographic notation , these surfaces are denoted by high miller indices and are called vicinals of their low miller index counterparts ( flat surfaces ) . the presence of arrays of atomic steps separated by flat terraces creates regions of differing local coordination and makes the microscopic structure of a vicinal surface distinct from that of a flat surface . according to smoluchowski s idea of charge smoothing @xcite , for example , electronic charge densities are expected to rearrange in the vicinity of the steps , thereby causing the ion cores to relax to new configurations . the modified electronic structure may also be expected to impact the reactivity and the nature of the force fields in the region around the steps . knowledge of atomic relaxations in the equilibrium positions near the step and kink sites is thus a step towards understanding the novel vibrational and electronic properties of vicinal surfaces . fortunately with advances in atomic scale experimental techniques , there has been a surge in investigations of the structure of vicinal surfaces in recent years . the majority of the experimental data have undoubtedly come @xcite from the low - energy electron diffraction ( leed ) technique which is now capable of detecting changes even for interlayer spacings smaller than 1@xmath0@xcite . for some surfaces the x - ray scattering technique has provided much needed complementary structural data @xcite . an impressive number of theoretical calculations of multilayer relaxations @xcite have also helped in bringing several issues related to the characteristics of vicinal surfaces to the forefront . of particular interest here are experimental and theoretical studies of a set of vicinals of cu(100 ) and cu(111 ) which have addressed the question of the impact of local coordination on the structural and dynamical properties of the surface . in an earlier paper @xcite , a comparative study of the local structural and vibrational properties of cu(211 ) , cu(511 ) , and cu(331 ) was performed using empirical potentials from the embedded atom method ( eam ) @xcite . this study found that the first two surfaces displayed similar local characteristics , while the third surface was somewhat different . an explanation provided for this behavior was the similarity in the local environment of the ( 211 ) and ( 511 ) surfaces of fcc metals ( a combination of ( 100 ) and ( 111 ) , terrace geometry , and step face ) , and its consequent difference from that of the ( 331 ) surface ( a ( 111 ) terrace geometry and a ( 111)-microfacetted step face ) . the issue of the impact of the local geometry was further raised in a joint theoretical and experimental study of the vibrational dynamics of cu(211 ) and cu(511 ) together with those of the kinked surface cu(532 ) @xcite . experimental data from electron energy loss spectroscopy ( eels ) found modes above the bulk band on cu(211 ) but not on cu(511 ) ( or on cu(17,1,1 ) which has the same step geometry as cu(511 ) ) , but theoretical calculations based on eam potentials predicted modes ( slightly ) above the bulk phonon spectrum for each of these surfaces . while the similarity between the calculated structural relaxation patterns of cu(211 ) and cu(511 ) argues in favor of a similarity in the local vibrational dynamics of these two surfaces , the disagreement between the experimental and the theoretical results for the high frequency modes on cu(511 ) ( and cu(17,1,1 ) ) remains unreconciled . for cu(211 ) agreement of the eam based results with available structural data from leed @xcite and with _ ab initio _ calculations for both the structure and the dynamics @xcite provides considerable confidence in its predicted properties . the case of cu(511 ) is not as simple because of lack of calculations based on potentials more accurate than eam , and because of conflicting conclusions from the analysis of experimental data from leed @xcite and x - ray scattering measurements @xcite . the most striking difference in these two sets of data is the relaxation pattern for the second layer which is inwards in leed and outwards in the x - ray data . the oscillatory pattern found in the x - ray data is also in disagreement with the conclusion from a series of previous experimental and theoretical findings on stepped surfaces . based on these studies @xcite , there is a definite symmetry in the relaxation patterns of stepped surfaces . all terrace atoms , save for the corner one , display inward relaxations . the eam based calculations @xcite further predict this oscillatory relaxation pattern to continue into the bulk with a damping in the amplitude @xcite . thus the expected relaxation pattern for the ( 211 ) , ( 511 ) , ( 331 ) surfaces , each with 3-atom wide terraces , would be ( - - + - - + ... ) , although questions have been raised whether cu(331 ) follows this rule @xcite . similarly , the patterns for ( 711 ) and ( 911 ) with , respectively , 4 and 5 atoms on the terrace , would be predicted to be ( - - - + - - - + ... ) and ( - - - - + - - - - + ... ) . the leed data on the first three surfaces follow these predicted trend in relaxations , atleast for the top 3 layers . the very recent leed data @xcite for cu(711 ) also displays the pattern ( - - - + ) for the top layers , in good agreement with eam based predictions . however , a small discrepancy in the sign of the relaxation is found for both cu(711 ) and cu(511 ) , for a particular layer separation ( d@xmath1 for cu(511 ) and d@xmath2 for cu(711 ) ) @xcite , beyond that expected from the error bars . arguably the actual numbers involved in these comparisons are small , but the systematic nature of the discrepancies and the fact that it negates the prediction of a periodicity in the oscillatory relaxation pattern @xcite , raise interesting questions about the complexities of the atomic displacements in these systems . given the above uncertainties arising from experimental observations , it is opportune to carry out more accurate calculations of these relaxation patterns using techniques which are capable of revealing the accompanying changes in the surface electronic structure . it is with this goal in mind that we have carried out _ ab initio _ electronic structure calculations of the surface geometry and interlayer spacing for a set of vicinals of cu(100 ) and cu(111 ) . in addition to cu(211 ) , cu(331 ) and cu(511 ) which are included to address the question of the influence of the local geometry on the structure , we have extended the investigation to cu(711 ) and cu(911 ) to examine the influence of increasing terrace width of the relaxation pattern . of course , for all surfaces comparison with available experimental data is of prime concern . the rest of this report is organized as follows . in section ii , the system geometries are presented together with some computational details . section iii contains the results and their discussion . concluding remarks are presented in section iv . vicinal surfaces can easily be constructed by cutting the crystal at an angle slightly away from the lower - index crystal planes ( i.e. ( 100 ) , ( 111 ) , ( 110 ) ) . for reasons discussed above , we are interested here in the vicinals of the ( 100 ) and ( 111 ) surfaces of fcc metals of which the most tightly packed steps are along the @xmath3 direction . in the case of the ( 111 ) surface , however , the @xmath3 direction is not parallel to any plane of symmetry and there are two different ways of generating monoatomic stepped surfaces . in one type of such vicinals , the step edge has a ( 100)-microfacet , while the other has the ( 111)-microfacet ( these are the so called a and b types , respectively ) . in the standard nomenclature , the vicinals of fcc(111 ) surface with monoatomic steps and ( 100 ) step edges are denoted by @xmath4 , while those with ( 111 ) step edges are labeled as @xmath5 , where @xmath6 is the number of atoms on the terrace . the b - type vicinal cu(331 ) considered here is named accordingly , while the a - type vicinal cu(211 ) seems to be a misnomer . similarly , the vicinals of fcc(100 ) surface consisting of monoatomic step edges with ( 111 ) microfacet are labeled @xmath7 . the cu(111 ) vicinals considered here are created by cutting the crystal at an angle of 19.5@xmath8 and 22@xmath8 away from the ( 111 ) plane towards the [ @xmath9 and [ @xmath10 direction to produce the ( 211 ) and ( 331 ) surfaces , respectively , whereas the three vicinals of ( 100 ) , ( 511 ) , ( 711 ) , and ( 911 ) , are constructed by slicing the crystal at angles of 15.8@xmath8 , 11.4@xmath8 and 8.9@xmath8 , respectively , from the ( 100 ) plane towards the [ 011 ] direction . to facilitate the discussion we have also labeled the atoms that play the dominant role in our calculation @xcite . for the three surfaces consisting of three chains of atoms on the terrace we label them as corner - chain ( cc ) , terrace - chain ( tc1 ) , and step - chain ( sc ) . the chain just underneath the step - chain is called a bulk nearest neighbor chain ( bnn ) . the other two surfaces , cu(711 ) and cu(911 ) , contain , respectively , one and two extra chains of terrace atoms , labeled accordingly as ( tc2 ) and ( tc3 ) . we have taken the @xmath11 and @xmath12 axes to lie in the surface plane , the @xmath11-axis being perpendicular to the step and the @xmath12-axis along the step , and the @xmath13-axis is along the surface normal . in fig . 1 , we display a side view of the ( 511 ) surface with the appropriate labeling of the atoms and interlayer spacing . the _ ab initio _ electronic structure calculations are performed within a pseudopotential approach to density - functional theory in the local density approximation @xcite , numerical implementation of the technique is based on a computer code developed by b. meyer _ et al . _ @xcite the local - density approximation is applied using the hedin - lundqvist form of the exchange - correlation functional @xcite . a norm conserving pseudopotential for cu constructed according to a scheme proposed by hamann - schlueter - chiang @xcite has been used which has already been successfully employed for calculations of the structure and the phonons of low index surfaces of cu @xcite . a mixed basis set is applied to represent the valence states consisting of five d - type local functions at each cu site , smoothly cut off at a radius of 2.3 a.u . , and of plane waves with kinetic energy of 11 ry . the brillouin - zone(bz ) integration was carried out using the special point sampling technique @xcite together with a gaussian broadening of the energy levels of 0.2 ev . for simulating surfaces we used the supercell approach with cells containing 21 to 35 atoms ( 1 atom per layer ) , depending on the surface orientation . the z - dimension of all cells was 47.7155 a.u . the distance between the top and the bottom layer of the slabs were thus 31.2761 a.u . for cu(331 ) , 27.8277 a.u . for cu(211 ) , 26.2340 a.u . for cu(511 ) , 25.7715 a.u . for cu(711 ) , and 25.4390 a.u . for cu(911 ) . structure optimization was carried out until forces on all atoms were smaller than @xmath14 ry / a.u . , which is two orders of magnitude smaller than the forces present on the unrelaxed surfaces . with increasing terrace width , the calculations become increasingly tedious since the reduction in interlayer spacing makes it more difficult to achieve geometries converged to 1% of the interlayer spacing . we also find that results for terraces with ( 100 ) geometry are more sensitive to the number of k - points sampled , as compared to those with ( 111 ) geometry . for the latter case 30 points in the bz are sufficient for the determination of the equilibrium structure , while for surfaces with ( 100 ) terraces at least twice as many points are needed to get converged results . our results for the multilayer relaxations of cu(211 ) , cu(331 ) , cu(511 ) , cu(711 ) and cu(911 ) are summarized in table i. as in previous theoretical studies @xcite of relaxations on stepped cu surfaces , we find changes in interlayer separations , from bulk terminated configurations , to persist on all surfaces for a large number of layers . of course , the concept of layers is different here from that on flat surfaces ( see fig . 1 ) . the first _ n _ layers , for example , are all exposed to the vacuum , where _ n _ is the number of atoms on the terrace . correspondingly , the interlayer separations are small and even large percentage changes in interlayer separations correspond to small numbers in distances . it is important to bear this point in mind when comparing the results for a particular surface either with those for flat surfaces or with the results from other methods . a common feature of all surfaces examined in table i is that all terrace atoms except for the ones in the corner chain ( cc ) undergo significantly large inwards relaxations . the corner atoms are always found to relax outwards . additionally , the atoms in the terrace adjacent to cc exhibit comparatively large inward relaxation whose magnitude maybe larger than that of the step atoms ( sc ) . for example , for cu(511 ) , cu(711 ) , and cu(911 ) changes , respectively , in @xmath15 ( involving tc1 ) , @xmath16 ( involving tc2 ) , and @xmath17 ( involving tc3 ) , are larger than that of @xmath18 . thus , in keeping with schmolkowski s @xcite ideas of charge smooting , the maximum relative change in interlayer separation is focussed around the corner atoms . this is particularly true for the vicinals of cu(100 ) . the situation with the more closepacked surface cu(331 ) is somewhat different , as seen in table i the outward relaxation of the corner atom and the inward relaxation of the preceeding atom on the terrace on cu(331 ) are less than half of that for similar atoms on the other surfaces considered here . incidently , this conclusion is in good agreement with results from previous studies which were based on semi - empirical potentials @xcite . there is , however , a disconcerting difference in the results obtained here from _ ab initio _ electronic structure calculations and those from semi - empirical potentials . an intriguing result for multilayer relaxations of the vicinals of cu(100 ) obtained with eam potentials in ref . @xcite was that the pattern of inward and outward relaxations continued well into the bulk with an expontially decreasing amplitude . thus for cu(711 ) , the relaxation pattern predicted by eam was ( - - - + , - - - + , - - - + , ... ) with an eventual dampimg of the relaxations . the pattern for cu(711 ) from table i is instead ( - - - + , - - + + , - - + + ) . that is our present calculations do not predict a periodically oscillatory relaxation pattern with a decaying amplitude as we move into the bulk . as we shall see , this particular feature is more in agreement with experimental data and help remove the slight discrepancy between experiment and theory presented by the eam result pointed by walter et al . again , it should be recalled that the numbers involved are very small and within the limits of accuracy of _ ab initio _ calculations . in particular , the small numbers for the relaxations of the inner layers of cu(911 ) have to be taken with caution as our convergence criteria for this surface were not as good as that for the others because of the demands on computational resources imposed by a system as large as this one . unlike flat surfaces , vicinal surfaces relax in both @xmath11 and @xmath13 directions , since the existence of steps at the surface leads to broken symmetry in both of these directions . while relaxations along the @xmath13-direction yield characteristic interlayer separations that we have discussed above , those along the @xmath11-direction provide new registries of atoms , as compared to those in the bulk . our calculated percentage intralayer registries for five surfaces are summarized in table ii . as in the observations from eam calculations , the changes in the registries of the atoms are small . it is thus not useful to make a one - to - one comparison with results from semiempirical calculations . however , the changes in registries of the atoms are not inconsequential since they affect the changes in the bond lengths between the atoms in these regions of low coordinations . in table iii , we tabulate our results for the total changes in the distances between the step atoms and their nearest neighbors . for comparison we have included in parenthesis the results obtained earlier for the same quantities with eam potentials @xcite . the largest changes in the bond lengths ( from unrelaxed configurations ) are for those between the step atoms and their bulk nearest neighbour ( bnn ) which lies right below them . the bonds between cc and bnn show small enlargement , while all other bonds in table iii are found to undergo shortening . in figs . 2 and 3 , we have drawn the actual displacements of the atoms on the five surfaces obtained from our calculations . while the size of the arrows are exagerated , it is their relative length and direction that is of consequence . as already noted by durukanoglu et al . @xcite , all atoms in low coordinated sites move to enhance their local coordination . the complex displacement pattern that emerges is thus the net outcome of the competition between the different directions in which the various atoms would like to relax to enhance thein own coordination . for readers who are interested in the exact positions of the atoms on the relaxed surfaces , we have summarized them in table iv . the unusual behavior of cu(331 ) terrace atom is more apparent from this table than the earlier one on changes in the bond lengths . the tc1 atom of cu(331 ) undergoes the least displacement among its counterparts . its displacement is also smaller than that of tc2 ( 1.4 , 0.0 , -2.6 ) on cu(711 ) , and of both tc2 ( 0.8 , 0.0 , -2.0 ) and tc3 ( 0.1 , 0.0 , -2.3 ) on cu(911 ) . the coordinates of the displacements above in paranthesis are in the same units as those in table iv . we now turn to comparisons of the results obtained here for individual surfaces with those available from experimental measurements . in table v we show that for cu(211 ) the salient features in the trends in the relaxation patterns predicted by our calculations are observed in the experimental data . apart from the large inward relaxation of the step atoms , the major change occurs at the corner atom and its adjacent terrace atom . our results are in good agreement with previous dft / lda calculations @xcite , based on the pseudopotential approximation and with results from eam based method . theoretical calculations using the full potential linearized augmented plane wave ( flapw ) method @xcite , however , predict much larger relaxation(-28.4% ) of the step atom than any of the previous theoretical or experimental studies . this brings us to the discussion of ionic relaxation on cu(331 ) in table vi for which also we do not get the large relaxation reported in ref[@xcite ] . the present results for cu(331 ) are , however , in good agreement with the leed data @xcite . the discrepancy with the results from leed for @xmath15 should not be taken too seriously , given an error bar of 4% in the analysis of the leed data . with respect to eam based results @xcite , we find a noticeable difference for d@xmath19 , for which the present results agree better with the leed data and also preserve the predicted relaxation pattern ( - - + ) for the terrace atoms . this trend is inkeeping with what was reported in calculations on al(331 ) @xcite . in trying to reconcile our results with those of geng @xcite , we note that the latter predict an outward displacement of the tc1 atomic chain for both cu(211 ) and cu(331 ) , while we find this not to be the case . as already mentioned , while the changes in the bond lengths of the terrace atoms of cu(331 ) are no different from those of the other surfaces , the displacement of tc1 is strikingly smaller than that of the tc s on other surfaces . the case of multilayer relaxations for cu(511 ) is interesting because of the differences in the published data from leed @xcite and x - ray measurements @xcite . these are displayed in table vii . except for the displacement of the step atoms , for which all results point to a large inward relaxation , the results from x - ray scattering measurments are in disagreement with present results and with those from leed , as well as , from eam calculations . we do not understand the reasons for this disagreement but for the notion that x - ray measurements may be very sensitive to the quality of the crystal surface . it should be noted that the differences with the x - ray results are both qualitative and quantitative , beyond the established error bars in the experiments and calculations . because of the controversy in the experimentally determined multilayer relaxations of cu(511 ) , we have carried out an extensive analysis of the dependence of the theoretical results on the approximations necessary to produce computational feasibility : choice of pseudopotentials , maximum kinetic energy of the plane - waves ( e@xmath20 ) , the number of layers in the supercell , and the number of points used to sample the surface brillouin zone . as for the dependence of the results on the type of pseudopotentials and e@xmath20 , we have carried out calculations with three different pseudopotentials ( on three sets of codes ) to find that the results which lie within 3% of each other . we also find our choice of supercell size to be adequate . there is , however , a strong dependence of the results on the number of bz points sampled . for the case of cu(511 ) this dependence is illustrated in table viii . calculations performed with few points could give erroneous relaxations as signified by the case of d@xmath21 in table viii . an inward relaxation of 1.8% is found with 4 points , while the converged result is 10.7% . convergence in the calculated relaxation is reached once the number of points is increased to 24 and beyond . thus , when comparing results from _ ab initio _ calculations , one has to keep these technical points in mind . unless sufficient checks are made for convergence in the reported values , quantities like equilibrium positions of surface atoms may differ in different calculations and lead to disagreement in the calculated relaxations . it would be worthwhile to clarify whether the differences between our results and those from the flapw method for cu(211 ) and cu(331 ) could be attributed to k - points sampling . finally , we come to the comparison of our results for cu(711 ) with those from experiments ( we are not aware of any data on cu(911 ) , so far ) . the leed data @xcite for this surface has been very carefully analyzed and compared to existing calculations . table ix shows that the _ ab initio _ results obtained here are in excellent agreement with the data , and that the small differences with the eam results that the authors@xcite had noted , is removed by the present calculations . as in the case of cu(511 ) , the largest percentage change in the interlayer spacing is not for d@xmath22 . in this case it is for d@xmath19 which separates cc from tc2 . as before , there is outward relaxation of the spacing between cc and bnn . the fact that relaxations near cc persist on being strong even as the terrace width increases , is interesting in itself . this particular argument has not been made in any previous theoretical result . our calculated values for cu(911 ) further support this argument as the largest percentage change is found for d@xmath23 , the interlayer spacing between cc and tc3 ( in this case ) . while these results are intriguing the main outcome of relaxations that ensue when a surface is created is in the actual displacements of the atoms from their bulk terminated positions to the new equilibrium positions . as already stated , these values are summarized in table v and the related patterns presented in figs . 2 and 3 . obviously , for stepped surfaces there is a complex rearrangement of most terrace atoms . our calculations show that despite this complexity , all terrace atoms except for cc move inwards . in summary , we have performed a comparative study of multilayer and atomic relaxations of five stepped cu surfaces which are vicinals of cu(100 ) and cu(111 ) using _ ab initio _ electronic structure calculations based on density functional theory and non - local , norm - conserving pseudopotentials . the set of three of these surfaces : cu(211 ) , cu(331 ) , and cu(511 ) , provides a comparison of structural changes from bulk termination , for vicinals of similar terrace widths but differing local geometry . the other set consisting of cu(511 ) , cu(711 ) and cu(911 ) offers a comparative study of relaxation patterns with changing terrace width . in each case we find the relaxation of the step atoms to be pronounced inwards and that of the corner atom to be outwards . the other terrace atoms and their nearest neighbors also undergo relaxations following a complex displacement pattern . subsequently , the bond lengths between all the surface atoms and their nearest neighbors change from the bulk terminated values while the bond length between cc and bnn atoms experiences an elongation ( about 1% ) all other surface length shrink anywhere from 1% to 4% . most of our findings are in agreement with previous calculations which were based on semiempirical model potentials except that we do not find the pattern of inward relaxations of sc , tc1 , tc2 etc followed by outward relaxation of cc atoms to continue into inner layers . we also find that the percentage contraction of the spacing between the tc and cc atoms is generally larger than that between sc and the tc atoms . while the actual magnitudes of the changes in the spacing considered here are small , there is a systematic trend in the relaxation pattern which points to significant rearrangements in the electronic charge densities near the sc and cc atoms . by and large our results are in good agreement with available structural data on these surfaces , except for the case of cu(511 ) for which we favor the leed results over those from x - ray scattering meaurements . we believe our results will help settle the issues that have emerged on this particular surface . our systematic examination of five surfaces , also helps address the question wether the relaxations on cu(331 ) are anomalous . the only striking difference between this surface and the others is in the relaxation of tc1 which is very small . otherwise the relaxation pattern and the changes in bond lengths are similar to those on the other surfaces . the main message from these observations is that the important quantity to examine is the displacement pattern of the surface atoms as they relax to their equilibrium positions from their bulk terminated configurations . the deeper question , of course , is the nature of the accompanying changes in the surface electronic structure . it will be interesting to examine the characteristics of the local electronic densities of states in the different regions of low symmetry that are present naturally on the stepped surfaces considered here . we leave this as an exercise for the future . the work of tsr and ak was supported in part by the us national science foundation , grant che-9812397 and by the basic energy research division , department of energy , grant de - fg03 - 97er45650 . tsr also acknowledges the alexander von humboldt foundation for the award of a forschungspreis and thanks her colleagues at the fritz haber institut , berlin and at the forschungszentrum , karlsruhe for their warm hospitality . for a review see k. wandelt , surf . sci . * 251/252 * , 387 ( 1991 ) . r. smoluchowski , phys . rev . * 60 * , 661 ( 1941 ) ; see also m. w. finnis and v. heine , j. phys . f * 4 * , l37 ( 1974 ) . d. l. adams and c. s. sorensen , surf . sci . * 166 * , 495 ( 1986 ) ; f. jona and p. m. marcus , in : j . f. van der veen and m. a. van hove ( eds . ) , the structure of surfaces ii , springer , heidelberg , 1988 , p. 90 , and references therein . zhang , p. j. rous , j. m. maclaren , a. gonis , m. a. van hove , and g. a. somorjai , sur . sci . * 239 * , 103 ( 1990 ) . f. jona , p. m. marcus , e. zanazzi , and m. maglietta , sur . * 6 * , 355 ( 1999 ) , and references therein . d. a. walko and i. k. robinson , phys . b * 59 * , 15446 ( 1999 ) . zang , m. a. van hove , g. a. somorjai , p. j. rous , d. tobin , a. gonis , j. m. maclaren , k. heinz , m. michl , h. lindner , k. muller , m. ehsasi , and j. h. block , phys . lett . * 67 * , 1298 ( 1991 ) . j. s. nelson and p. j. feibelman , phys . lett . * 68 * , 2188 ( 1992 ) . z. j. tian and t. s. rahman , phys . b * 47 * 9752 ( 1993 ) . s. durukanolu , a. kara , and t. s. rahman , phys . b * 55 * , 13 894 ( 1997 ) . sklyadneva , g. g. rusina , and e. v. chulkov , surf . sci . * 416 * , 17 ( 1998 ) . s. m. foiles , m. i. baskes , and m. s. daw , phys . b * 33 * , 7983 ( 1986 ) . a. kara , p. staikov , t. s. rahman , j. radnik , r. biagi , and h. j. ernst , phys . b * 61 * , 5714 ( 2000 ) . th . seyller , r. d. diehl , and f. jona , j. vac . a * 17 * , 1635 ( 1999 ) . c. y. wei , s. p. lewis , e. j. mele , and a. m. rappe , phys . b * 57 * , 10062 ( 1998 ) . m. albrecht , r. blome , h. l. meyerheim , w. moritz , and i. k. robinson , unpublished . y. tian , k .- w . lin , and f. jona , phys . rev b * 62 * , 12844 ( 2000 ) s. walter , h. baier , m. weinelt , k. heinz and th . fauster , phys . b * 63 * , 155407 ( 2001 ) . g. allen and m. lannoo , phys . rev . b * 37 * , 2678 ( 1988 ) and references therein . p. hohenberg and w. kohn , phys . rev . b*136 * , 864 ( 1964 ) . b. meyer , c. elsaesser and m. faehnle , `` fortran90 programm for mixed - basis pseudopotential calculations for crystals '' , max planck institut fuer metallforschung , stuttgart , unpublished . l. hedin and b. i. lundqvist , j.phys.c * 4 * , 2064 ( 1971 ) . d. r. hamann , m. schlueter , and c. chiang , phys.rev.lett . * 43 * , 1494 ( 1979 ) ; g. b. bachelet , d. r. hamann , and m. schlueter , phys.rev.b * 26 * , 4199 ( 1982 ) . monkhorst , and j.d . pack , phys . b * 13 * , 5188 ( 1976 ) . rodach , k .- bohnen , and k. m. ho , surf.sci . * 286 * , 66(1993 ) . w. t. geng and a. j. freeman , phys . b * 64 * , 115401 ( 2001 ) .

we present trends in the multilayer relaxations of several vicinals of cu(100 ) and cu(111 ) of varying terrace widths and geometry . the electronic structure calculations are based on density functional theory in the local density approximation with norm - conserving , non - local pseudopotentials in the mixed basis representation . while relaxations continue for several layers , the major effect concentrates near the step and corner atoms . on all surfaces the step atoms contract inwards , in agreement with experimental findings . additionally , the corner atoms move outwards and the atoms in the adjacent chain undergo large inward relaxation . correspondingly , the largest contraction ( 4% ) is in the bond length between the step atom and its bulk nearest neighbor ( bnn ) , while that between the corner atom and bnn is somewhat enlarged . the surface atoms also display changes in registry of upto 1.5% . our results are in general in good agreement with leed data including the controversial case of cu(511 ) . subtle differences are found with results obtained from semi - empirical potentials . pacs # 61.50.ah , 68.35.bs , 68.47.de

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Proceed to summarize the following text: more than 40 years have passed since the recognition that quasars are at cosmological distances ( schmidt 1963 ; greenstein & mathews 1963 ; mathews & sandage 1963 ) and hence must be powerful energy sources . a combination of arguments based on time variability and energetics strongly supports the view that the activity is produced by the accretion of gas onto supermassive black holes in the centers of galaxies ( e.g. salpeter 1964 ; zeldovich & novikov 1964 ; lynden - bell 1969 ) . however , the mechanism that provides the trigger to fuel quasars remains uncertain . recent discoveries of correlations between masses of black holes in nearby galaxies and either the mass ( magorrian et al . 1998 ) or velocity dispersion ( i.e. the @xmath4-@xmath5 relation : ferrarese & merritt 2000 ; gebhardt et al . 2000 ) of spheroids demonstrate a fundamental link between the growth of supermassive black holes and galaxy formation . however , theoretical evidence connecting the origin of supermassive black holes and quasars to galaxy evolution remains elusive , owing to the complexity of the underlying physics and dynamics . in particular , up to now there has been no comprehensive model to explain the origin and fueling of quasars , or their lifetimes , obscuration , demographics , self - regulation and termination , and dependence on host galaxy properties . observations of the nearby universe suggest that rapid black hole growth may be related to massive flows of gas into the centers of galaxies . infrared ( ir ) luminous galaxies represent the dominant population of objects above @xmath6 locally and it is believed that much of their ir emission is powered by dust reprocessing of radiation from intense nuclear starbursts ( e.g. soifer et al . 1984a , b ; sanders et al . 1986 , 1988a , b ; for a review , see e.g. soifer et al . 1987 ) . at the highest luminosities above @xmath7 , characteristic of ultraluminous infrared galaxies ( ulirgs ) , nearly all the galaxies appear to be in advanced stages of merging ( e.g. allen et al . 1985 ; joseph & wright 1985 ; armus et al . 1987 ; kleinmann et al . 1988 ; for reviews , see sanders & mirabel 1996 and jogee 2004 ) and co observations show that they contain large quantities of gas in their nuclei ( e.g. scoville et al . 1986 ; sargent et al . 1987 , 1989 ) . some ulirgs exhibit `` warm '' ir spectral energy distributions , perhaps indicative of a buried quasar ( e.g. sanders et al . this fact , together with the overlap between bolometric luminosities of ulirgs and quasars , led sanders et al . ( 1988a ) to propose that quasars are the descendents of an infrared luminous phase of galaxy evolution caused by mergers . this scenario is supported by recent x - ray observations which have revealed the presence of two non - thermal point sources near the center of the ulirg ngc6240 @xcite , which are most naturally interpreted as accreting supermassive black holes that are heavily obscured at visual wavelengths ( e.g. gerssen et al . 2004 ; max et al . 2005 ) . hydrodynamic simulations have shown that gas inflows are produced by strong gravitational torques on the gas through tidal forces during mergers involving gas - rich galaxies ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? the resulting dense concentrations of gas in the inner regions of the remnant can sustain star formation at rates high enough and for sufficiently long timescales to account for many properties of ulirgs ( e.g. , * ? ? ? ? * ; * ? ? ? * ) . however , these authors were not able to directly explore the relationship between the gas inflows and quasar activity because their simulations did not include a model for black hole growth and the impact of strong feedback from either star formation or quasars . one of the most fundamental parameters of black hole growth is the quasar lifetime , @xmath8 , which sets the timescale for the most luminous phase of the activity . observations generally constrain quasar lifetimes to the range @xmath9 yr ( for a review , see * ? ? ? these estimates are primarily based on demographic or integral arguments which combine observations of the present - day population of supermassive black holes and accretion by the high - redshift quasar population ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , or incorporate quasars into models of galaxy evolution ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) or reionization of heii ( e.g. sokasian et al . 2002 , 2003 ) . results from clustering in quasar surveys ( e.g. , * ? ? ? * ; * ? ? ? * ) , the proximity effect in the ly@xmath10 forest ( bajtlik , duncan & ostriker 1988 ; haiman & cen 2002 ; yu & lu 2005 ; although such results may still be inconclusive , e.g. croft 2004 ) , and the transverse proximity effect in he ii @xcite similarly suggest lifetimes @xmath11yr . it is not immediately obvious how quasar lifetimes @xmath11yr can be explained in the context of gas inflows triggered by galaxy mergers . the full duration of the starburst phase is set by the timescale during which strong gas inflows are excited , which in turn is determined by the time when significant gravitational torques are exerted on the gas . the numerical simulations of @xcite showed that this occurs for @xmath12 yr , much shorter than typical merger timescales of @xmath13 @xcite , but in good agreement with observed estimates for the gas depletion time in the central regions of ulirgs ( for a discussion , see e.g. barnes & hernquist 1992 ) . if gas - rich mergers are indeed responsible for the origin of quasar activity , as suggested through studies of ulirgs ( e.g. sanders et al . 1988a ) and more directly from observations of quasar hosts ( e.g. stockton 1978 ; heckman et al . 1984 ; stockton & mackenty 1987 ; stockton & ridgway 1991 ; hutchings & neff 1992 ; bahcall et al . 1994 , 1995 ; canalizo & stockton 2001 ) , the factor @xmath14 difference between the gas consumption time in ulirgs and quasar lifetimes must be reconciled . semi - analytical models of supermassive black hole evolution and its correlation with galaxy structure suggest that , beyond a certain threshold , feedback energy expels nearby gas and shuts down the accretion phase @xcite . however , these calculations provide only limits , and in neglecting the dynamics of quasar evolution they do not predict time - dependent effects such as the characteristic lifetime of the accretion phase prior to its self - termination , the fueling rates for black hole accretion , the obscuration of central sources , or the quasar light curve . these quantities are instead usually taken to be independent input parameters of the model . for example , the lifetime is either adopted from observational estimates or assumed to be similar to a characteristic timescale such as the dynamical time of the host galaxy disk or the @xmath15-folding time for eddington - limited black hole growth @xmath16yr for accretion with radiative efficiency @xmath17 and @xmath18 @xcite . efforts to model quasar accretion and feedback in a more self - consistent manner ( e.g. , * ? ? ? * ; * ? ? ? * ) have considered the time - dependent nature of gas inflows feeding accretion and the impact of radiative heating from quasar feedback in a hydrodynamical context . although generally restricted to one - dimensional `` toy models '' which do not include galaxy - galaxy interactions , such modeling has made important progress in predicting the characteristic duty cycles of quasars , in good agreement with e.g. , @xcite , and in demonstrating the importance of radiative feedback on the surrounding medium and quasar evolution ( for a review , see * ? ? ? further , @xcite showed that self regulation by radiative feedback from quasars in this modeling can expel a significant quantity of gas and leave a remnant with properties similar to observed ellipticals and black hole mass corresponding to the observed @xmath19-@xmath5 relation . similarly @xcite showed that incorporating a simple prescription for radiative heating by quasars in the context of cosmological simulations yields populations of elliptical galaxies consistent with observed optical and x - ray properties . recently , @xcite have developed a methodology for incorporating black hole growth and feedback into hydrodynamical simulations of galaxy mergers that includes a multiphase model for star formation and pressurization of the interstellar gas by supernova feedback @xcite . using this approach , we have begun to explore the impact of these processes on galaxy formation and evolution . di matteo et al . ( 2005 ) and springel et al . ( 2005b ) have shown that the gas inflows produced by gravitational torques during a merger both trigger starbursts ( as in the earlier simulations of e.g. mihos & hernquist 1994 , 1996 ) and fuel rapid black hole growth . the growth of the black hole is determined by the gas supply and terminates abruptly when significant gas is expelled owing to the coupling between feedback energy from black hole accretion and the surrounding gas . eventually , as the gas is heated and driven out , the remnant is no longer active because the black hole does not accrete at a high rate , leaving a dead quasar in an ordinary galaxy . the self - regulated nature of the black hole growth explains observed correlations between black hole mass and properties of normal galaxies @xcite , as well as the color distribution of ellipticals @xcite . these results lend support to the view that mergers have played an important role in structuring galaxies , as advocated by e.g. toomre & toomre ( 1972 ) and toomre ( 1977 ) . moreover , @xcite showed that the dynamics of the inflowing gas and its response to the self - regulated growth of the black hole yield a timescale for the strong accretion phase @xmath12 yr , comparable to the full duration of the starburst . during much of this period , the bolometric luminosity of the black hole would exceed the threshold to be classified as a quasar , implying a timescale @xmath20 ( for a milky way mass system ) for the intrinsic quasar phase of the black hole . in @xcite , we employed models for obscuration by gas and dust to show that for most of the accretion lifetime the quasars would be buried ; i.e. heavily obscured by the large gas density powering accretion . eventually , feedback from the accretion energy drives away the gas , creating a brief window in which the central object would be observable as an optical quasar , until accretion levels drop below quasar thresholds . by calculating the effects of obscuration from the simulation of a merger of gas - rich galaxies , we showed that this determines an observable lifetime @xmath21 , in good agreement with observations . here , we extend and further develop implications of the model for quasar evolution proposed in @xcite by analyzing a series of hydrodynamical simulations of galaxy mergers where we vary the masses of the progenitor galaxies , so that they have virial velocities between 80 and @xmath22 . in 2 we describe our series of merger simulations and the calculation of column densities and obscuration , and compare the results obtained with different column density calculations . in 3 we examine the frequency dependence of our results and quantify the differences in lifetimes across the optical and x - ray bands . in 4 we examine the typical column densities along lines - of - sight to the simulated quasars , and compare these to observed distributions from both optical and x - ray surveys . in 5 we show the simulation results for our merger series and compare the observed and intrinsic lifetimes as a function of luminosity for all the simulations . finally , in 6 we discuss our results and the implications of our model for a broad range of quasar studies . the simulations were performed using the gadget-2 code , a new version of the parallel treesph code gadget @xcite . it uses an entropy - conserving formulation @xcite of smoothed particle hydrodynamics ( sph ) , and includes radiative cooling , heating by a uv background ( as described in e.g. katz et al . 1996 ; dav et al . 1999 ) , and a sub - resolution model of a multiphase interstellar medium ( ism ) to describe star formation and supernova feedback @xcite . this sub - resolution model provides an effective equation of state for star - forming gas which includes pressure feedback from supernova heating , and allows us to stably evolve even massive pure gas disks ( see , e.g. springel et al . 2005b ; robertson et al . the methodology of accretion , feedback , and galaxy generation is described in detail in @xcite . in our approach , supermassive black holes ( bhs ) are represented by `` sink '' particles that accrete gas from their local environment , with an accretion rate @xmath23 estimated from the local gas density and sound speed using a bondi - hoyle - lyttleton parameterization with an imposed upper limit equal to the eddington rate . the bolometric luminosity of the bh particle is then @xmath24 , where @xmath25 is the radiative efficiency . we further assume that a small fraction ( @xmath26 ) of @xmath27 couples dynamically to the surrounding gas , and that this feedback is injected into the gas as thermal energy . this fraction is a free parameter , determined in @xcite by fitting to the @xmath19-@xmath5 relation . we do not attempt to resolve the small - scale accretion dynamics near the black hole , but instead assume that the time - averaged accretion can be estimated from the gas properties on the scale of our spatial resolution ( @xmath28pc ) . cccc a1 & 80 & @xmath29 & @xmath30 + a2 & 113 & @xmath31 & @xmath32 + a3 & 160 & @xmath33 & @xmath34 + a4 & 226 & @xmath35 & @xmath36 + a5 & 320 & @xmath37 & @xmath38 + in what follows , we analyze five simulations of colliding disk galaxies , which form a family of structurally similar models with different virial velocity and mass . in each simulation , we generate two stable , isolated disk galaxies , each with an extended dark matter halo with a @xcite profile , motivated by cosmological simulations ( e.g. navarro et al . 1996 ; busha et al . 2004 ) and observations of halo properties ( e.g. rines et al . 2002 , 2002 , 2003 , 2004 ) , an exponential gas disk , and a bulge . the simulations follow the series described in detail in @xcite , with the parameters listed in table [ tbl : sims ] . we denote the simulations a1 , a2 , a3 , a4 , and a5 , with increasing virial velocities of @xmath39 , respectively . note that the self - similarity of these models is broken by the scale - dependent physics of cooling , star formation , and black hole accretion . in @xcite , we describe our analysis of simulation a3 , a fiducial choice with a rotation curve and mass similar to the milky way . the galaxies have mass @xmath40 , with the baryonic disk having mass fraction @xmath41 , the bulge @xmath42 , and the rest of the mass in dark matter with a concentration parameter @xmath43 . the disk - scale length is computed based on an assumed spin parameter @xmath44 , and the scale - length of the bulge is set to @xmath45 times this . we begin our simulation with pure gas disks , which may better correspond to the high - redshift galaxies in which most quasars are observed . each galaxy is initially composed of 168000 dark matter halo particles , 8000 bulge particles , 24000 gaseous disk particles , and one bh particle , with a small initial seed mass of @xmath46 . given these choices , the dark matter , gas , and star particles are all of roughly equal mass , and central cusps in the dark matter and bulge profiles are reasonably well resolved ( see fig 2 . in springel et al . the galaxies are then set to collide in a prograde encounter with zero orbital energy and a small pericenter separation ( @xmath47 for the a3 simulation ) . we calculate the column density between a black hole and a hypothetical observer from the simulation outputs spaced every 10 myr before and after the merger and every 5 myr during the merger of each galaxy pair . the calculation method is described in @xcite , but we review it briefly here . we generate @xmath48 radial lines - of - sight ( rays ) , each with its origin at the black hole particle location and with directions uniformly spaced in solid angle @xmath49 . for each ray , we then begin at the origin , calculate and record the local gas properties using gadget , and then move a distance along the ray @xmath50 , where @xmath51 and @xmath52 is the local sph smoothing length . the process is repeated until a ray is sufficiently far from its origin ( @xmath53 kpc ) . the gas properties along a given ray can then be integrated to give the line - of - sight column density and mean metallicity . we test different values of @xmath54 and find that gas properties along a ray converge rapidly and change smoothly for @xmath55 and smaller . we similarly test different numbers of rays and find that the distribution of line - of - sight properties converges for @xmath53 rays . given the local gas properties , we use the gadget multiphase model of the ism described in @xcite to calculate the local mass fraction in `` hot '' ( diffuse ) and `` cold '' ( molecular and hi cloud core ) phases of dense gas and , assuming pressure equilibrium between the two phases , we obtain the local density of the hot and cold phase gas and the corresponding volume filling factors . the values obtained correspond roughly to the fiducial values of @xcite . given a temperature for the warm , partially ionized component @xmath56 , determined by pressure equilibrium , we further calculate the neutral fraction of this gas , typically @xmath57 . we denote the neutral and total column densities as @xmath58 and @xmath59 , respectively . using only the hot - phase density allows us to place an effective lower limit on the column density along a particular line of sight , as it assumes a ray passes only through the diffuse ism , with @xmath60 of the mass of the dense ism concentrated in cold - phase `` clumps . '' given the small volume filling factor ( @xmath61 ) and cross section of such clouds , we expect that the majority of sightlines will pass only through the `` hot - phase '' component . using @xmath24 , we model the form of the intrinsic quasar continuum sed following @xcite , based on optical through hard x - ray observations ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? this gives a b - band luminosity @xmath62 , where @xmath63 , and we take @xmath64 . we then use a gas - to - dust ratio to determine the extinction along a given line of sight at this frequency . observations suggest that the majority of the population of reddened quasars have reddening curves similar to that of the small magellenic cloud ( smc ; hopkins et al . 2004 ) , which has a gas - to - dust ratio lower than the milky way by approximately the same factor as its metallicity @xcite . we therefore consider both a gas - to - dust ratio equal to that of the milky way , @xmath65 , and a gas - to - dust ratio scaled by metallicity , @xmath66 . for both cases we use the smc - like reddening curve of @xcite . we calculate extinction in x - ray frequencies ( 0.03 - 10 kev ) using the photoelectric absorption cross sections of @xcite and non - relativistic compton scattering cross sections , similarly scaled by metallicity . in determining the column density for photoelectric x - ray absorption , we ignore the inferred ionized fraction of the gas , as it is expected that the inner - shell electrons which dominate the photoelectric absorption edges will be unaffected in the temperature ranges of interest . we do not perform a full radiative transfer calculation , and therefore do not model scattering or re - processing of radiation by dust in the infrared . we quantify the quasar lifetime @xmath8 as in @xcite , as a function of the limiting b - band luminosity @xmath67 . for each sightline , an observed lifetime is determined as the integrated time in the simulation during which the given sightline sees a b - band luminosity above a given threshold , @xmath68 . we consider both the intrinsic lifetime @xmath69 , the total time the intrinsic @xmath70 ( ignoring attenuation ) , and the observed lifetime derived from a particular column density calculation . we note that below 5 myr ( the simulation output frequency in the standard case ) our estimates of @xmath8 become uncertain owing to the effects of quasar variability and our inability to resolve the local small - scale physics of the ism . however , the majority of sightlines see lifetimes in the range @xmath71myr up to @xmath72 , in good agreement with observations suggesting lifetimes @xmath73yr and well above this limit . from the a3 simulation . the dotted lines show the lifetime if attenuation is ignored ( time with observed b - band luminosity @xmath74 ) . the thick solid lines indicate the median observed lifetime , with thin lines giving the 25%-75% contours . lifetimes are calculated using : ( a ) cold - phase ism density , ( b ) hot - phase density ( ignoring metallicity and ionization ) , ( c ) hot - phase density ( including metallicity and ionization ) , ( d ) hot - phase density with artificially low metallicity ( 0.1 times solar ) . [ fig : comparenhcalc],width=326 ] figure [ fig : comparenhcalc ] shows the lifetime obtained in the a3 simulation using different calculations of the column density @xmath58 ( cases @xmath75 ) . the uppermost curve in all cases is the intrinsic lifetime @xmath69 . the lower thick curve is the median observed lifetime calculated using a particular estimate of the column density , with the thin curves representing the 25%-75% inclusion contours . the cold - phase ism density ( @xmath76 ) is large enough for the quasar to be completely extincted out of observable ranges for the duration of the quasar phase , rendering the object observable only during quiescent phases with @xmath77 . we also show the significantly longer lifetimes calculated using the hot - phase ism density , first ignoring corrections for metallicity and ionization ( @xmath78 ; i.e. assuming @xmath79 and @xmath80 ) and then including these effects ( @xmath81 ) , which increases the median lifetime by a factor @xmath82 and significantly reduces the scatter towards shorter lifetimes . finally , we calculate the lifetime using the hot - phase density and an smc - like gas - to - dust ratio , @xmath83 ( @xmath84 ; essentially assuming a metallicity @xmath85 times solar ) , which sets a strong upper limit on the observed lifetime . these lifetimes are still only @xmath82 times as long as the lifetimes calculated using metallicity - weighted neutral column densities , and still @xmath86 the intrinsic lifetime at @xmath72 . we therefore expect that , after accounting for the clumping of most mass in the cold phase of the most dense regions of the ism , the qualitative relation between our observed and intrinsic calculated lifetimes should be relatively insensitive to the details of the column density calculation , with variation by a factor of @xmath87 between lifetimes calculated using different prescriptions for the multiphase ism , ionization , and metallicity effects . we expect the intrinsic quasar lifetime above a given luminosity in a particular band to change with the frequency of that band . the shift in lifetime with frequency is approximately given by an offset in luminosity corresponding to the difference in luminosities at different frequencies in our model quasar spectrum . however , this change is not entirely history - independent , as the model spectrum alters shape with varying luminosity owing to bolometric corrections . more important , our model predicts that the observed lifetime above a given luminosity will change with frequency not only as a result of the shift in intrinsic lifetime , but primarily as a result of varying levels of attenuation at different frequencies . from the a3 simulation , showing the waveband dependence of the lifetime . the thick lines give the lifetime if attenuation is ignored , thin lines indicate the median observed lifetime ( calculated using the metallicity - weighted , hot - phase neutral column density ) . upper panel : from black to blue to red , lifetimes are calculated at 3000 , 4500 , 6000 , 7500 , and 9000 . lower panel : lifetimes are calculated for soft ( @xmath88 , red ) and hard ( @xmath89 , black ) x - ray bands . [ fig : lifetimevsfreq],width=336 ] figure [ fig : lifetimevsfreq ] shows the lifetimes as a function of luminosity for the a3 simulation , for various frequencies in the optical and x - ray bands . we plot lifetimes in the manner of figure [ fig : comparenhcalc ] ; i.e. the total time with @xmath90 ( @xmath91 ) , at the representative optical wavelengths @xmath92 ( colored black to blue to red , respectively ) . we find that the intrinsic lifetime systematically decreases at longer wavelengths ( based on the shape of the quasar sed ) , while the observed lifetime increases ( owing to weakening attenuation ) , with the two approaching one another near infrared wavelengths . at longer wavelengths , we expect the curves to cross , as the observed lifetime in the far ir , owing to dust re - processing of quasar radiation , will be longer than the intrinsic lifetime , but we do not yet incorporate such re - processing into our column density calculations . at very low @xmath93 , lifetimes approximately converge to the total duration of the simulations , as the quasar is above such luminosities throughout almost the entire merger . the lower panel of figure [ fig : lifetimevsfreq ] shows the lifetimes in the soft and hard x - ray bands , defined as @xmath88 and @xmath89 , respectively . we find that the soft x - ray band is heavily attenuated and has much shorter observed than intrinsic lifetimes , primarily owing to photoelectric absorption . however , the hard x - ray band is relatively unaffected , as the hot - phase column densities tend to be well below the compton - thick regime @xmath94 . thus , most quasars obscured in the optical for much of their high accretion rate lifetimes should be observable in hard x - rays . we defer a full treatment of the difference between the attenuation of soft and hard x - rays to a future paper , but note that this difference may account for the slope of the cosmic x - ray background , with a population of obscured quasars at spectral energies @xmath95 a few kev as a natural stage in the evolution of quasar activity . given a calculation of the column density along multiple lines - of - sight to the simulated quasars , we can examine the typical column density distributions at all times and at times in which the quasar is observed above some luminosity threshold . following the analysis in @xcite , we expect the characteristic column densities to rise rapidly during the merger , as gas inflow traces and fuels a rising accretion rate . eventually , column densities rapidly fall in a blowout phase once the black hole has reached a critical mass , creating a window with rapidly changing column densities in which the quasar is observable until accretion rates drop below those necessary to fuel quasar activity . as is clear in figure 2 of @xcite , the dispersion in @xmath59 at any particular time in the simulation is generally small , a factor of @xmath82 in either direction , but typical @xmath59 values can change by an order of magnitude over timescales @xmath96myr . therefore , it is of interest to consider the probability of an observer , viewing the object at a random time , seeing a given column density . from the a3 simulation . the probability of seeing a given @xmath58 is weighted by the time such an @xmath58 is observed along all sightlines . plots are scaled linearly ( left ) and logarithmically ( right ) . the top panel shows all @xmath58 values from all times , the lower panels give the distribution for times where the observed b - band luminosity is above the @xmath67 shown . the smooth curve is the distribution of @xmath58 values from the observed sdss quasar sample of @xcite , who used an absolute @xmath97-band magnitude limit approximately equivalent to @xmath98 , scaled to the appropriate modal @xmath58 . [ fig : nhdistrib],width=355 ] figure [ fig : nhdistrib ] plots the distribution of column densities @xmath58 calculated using the hot - phase metallicity - weighted ism density in the a3 simulation . the probability of seeing a given @xmath58 is calculated as proportional to the total time along all sightlines that such an @xmath58 is observed . the plots are scaled both linearly ( left panels ) and logarithmically ( right panels ) in column density . typical column densities are distributed about @xmath99 , approximately symmetrically in @xmath100 . we further plot the distribution of @xmath58 values requiring that the observed b - band luminosity be above some threshold @xmath67 . the curve shown in the linear plots is the best - fit to the @xmath101 distribution of bright sdss quasars with @xmath102 , from @xcite . the curve has been rescaled in terms of the column density ( inverting our gas - to - dust prescription ) and plotted about a peak ( mode ) @xmath58 ( undetermined in hopkins et al . 2004 ) of @xmath103 . the @xmath97-band absolute magnitude limit imposed in the observed sample , @xmath104 , corresponds approximately to our plotted b - band limit @xmath98 . the agreement between the observed column density distribution and the result of our simulations once the same selection effect is applied is strong evidence in favor of our model for quasar evolution . the distribution as a function of limiting luminosity is a natural consequence of the dynamics of the quasar activity . throughout much of the duration of bright quasar activity , column densities rise to high levels as a result of the same process that feeds accretion , naturally producing a reddened population of quasars ( a red `` tail '' in the quasar color distribution ) , extending to very bright quasars strongly reddened by large @xmath58 . the existence of this extended reddened quasar population in radio and optically selected quasars is well known ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , and we are able to reproduce its distribution not as a distribution of source properties but as a result of the time evolution of quasar phenomena . our estimate of the distribution of @xmath58 for @xmath98 does not account for an additional selection effect , namely that strongly reddened quasars may not be extincted from an observed sample ( if the intrinsic luminosity is large enough ) , but their colors may be significantly reddened to the point where color selection criteria of quasar surveys will not include them . however , this effect would only serve to bring our distribution into better agreement with observations , as it would slightly lower the high-@xmath58 tail . moreover , our estimates of the distribution of @xmath58 allow us to make a directly observable prediction , namely that the column density distribution in optically selected samples with a minimum absolute magnitude or luminosity should broaden in both directions ( to larger and smaller @xmath58 ) as the limiting selection luminosity is decreased ( limiting absolute magnitude increased ) . this is because , at lower luminosities , observers will see both intrinsically bright periods extincted by larger column densities ( broadening the distribution to larger @xmath58 values ) and intrinsically faint periods with small column densities ( broadening the distribution to smaller @xmath58 values ) . the distribution of @xmath59 values in hard x - ray quasar samples , which are much less affected by this extinction ( see 3 ) , should also reflect this trend , and indeed the distribution of @xmath59 values is much broader and flatter , with quasars seen from @xmath105 @xcite . extension of this distribution to the other black hole masses and peak quasar luminosities of our simulations reveals similar qualitative behavior . the characteristic column densities during the obscured , intrinsically bright phases of accretion generally increase with galaxy and black hole mass , but typically by factors @xmath106 across the range of our simulations @xmath107 . additionally , although mean column densities increase somewhat with increasing mass , the functional form of the @xmath58distribution remains similar at each limiting luminosity , implying that a combined population of quasars similar to those we have simulated will match the observed @xmath58 distribution with appropriate selection effects accounted for as in figure [ fig : nhdistrib ] . finally , we note that although we do not see extremely large column densities @xmath108 in the distributions from our simulations , our model does not rule out such values . it is possible that very bright quasars in unusually massive galaxies or quasars in higher - redshift , compact galaxies which we have not simulated may , during peak accretion periods , reach such values in their typical column densities . moreover , as our model assumes @xmath109 of the mass of the densest gas is clumped into cold - phase molecular clouds , a small fraction of sightlines will pass through such clouds and encounter column densities similar to those shown for the cold phase in figure [ fig : comparenhcalc ] , as large as @xmath110 . this also allows a large concentration of mass in sub - resolution obscuring structures , such as an obscuring toroid on scales @xmath111pc , although many of the phenomena such structures are invoked to explain can be accounted for through our model of time - dependent obscuration . as a function of simulation time , with thin contours at 25% and 75% inclusion levels , for the a4 simulation ( @xmath112 ) . solid contours represent the density of the `` hot - phase '' ism , dashed contours the total simulation density . middle panel : bolometric luminosity of the black hole in the simulation , @xmath24 ( thick ) , and ratio of bolometric to eddington luminosity , @xmath113 ( thin ) . values are shown for each simulation timestep . lower panel : observed b - band luminosity with @xmath58 calculated using the median `` hot - phase '' ism density . [ fig : nh.vs.time.a4],width=336 ] , but for the a5 ( @xmath114 ) simulation . [ fig : nh.vs.time.a5],width=336 ] we generalize the results from @xcite and those above ( which use simulation a3 ) to a range of masses , using the simulations a1 , a2 , a4 , and a5 . recall , these simulations are identical except for increasing galaxy mass given by the sequence in virial velocity , @xmath115 for a1-a5 , respectively . the evolution of these simulations would be simply related if not for the scale - dependent physics of cooling , star formation , and black hole accretion and feedback . figures [ fig : nh.vs.time.a4 ] and [ fig : nh.vs.time.a5 ] plot the bolometric luminosity of and column densities to the supermassive black hole as a function of simulation time , for the period shortly before and after the merger in simulations a4 and a5 , respectively . this is presented in the style of figure 2 from hopkins et al . ( 2005a ) , which showed the same for a3 . the median column densities and 25%-75% contours are indicated for the total average simulation density , and the calculated hot - phase density , as described in 2.2 , along with the observed b - band luminosity using the median hot - phase @xmath58 . the final masses of the black holes are also shown , where @xmath116 for simulations a1-a5 , respectively . the evolution seen in simulation a4 is quite similar to that of a3 , with a sharp rise at the time of the merger to very large peak eddington - limited luminosities , for a period @xmath117 , much of which is unobservable in the optical owing to a corresponding rise in @xmath58 . at the final stages of strong accretion , when gas is rapidly being consumed and expelled , the observable luminosity finally rises to peak levels over a much shorter time interval @xmath118 . in the a5 simulation , the period of rapid accretion and large , eddington - limited intrinsic luminosities is significantly longer , @xmath119 , and the peak luminosity is higher , as expected . however , as before , most of this period is obscured by large column densities , and , remarkably , once again only the final stages of intrinsically high luminosity are visible , as gas is expelled and the accretion rate and column density both fall . the same behavior is seen in the a1 and a2 cases , though with much lower peak bolometric luminosities @xmath120 and @xmath121 , respectively . the striking similarity of this process across all our simulations suggests that it is at least qualitatively insensitive to the details of the accretion history and galaxy or black hole masses . from the simulations ( time with observed b - band luminosity @xmath122 ) . results from simulations a1-a5 are shown from top to bottom . solid lines indicate the lifetime if attenuation is ignored . the dashed lines give the median observed lifetime calculated with the metallicity - weighted , hot - phase neutral column density . right panels : same as the left panels , but with @xmath8 defined as a bolometric luminosity @xmath123 , where the observable lifetime is the time above a given ratio @xmath124 with b - band optical depth less than unity . [ fig : lifetime.vs.l],width=336 ] we quantify the resulting quasar lifetimes in figure [ fig : lifetime.vs.l ] , where we plot the quasar lifetime @xmath8as a function of the limiting b - band luminosity @xmath67 . the panels show the lifetimes for simulations a1-a5 , from top to bottom . in each panel , the solid curve shows the intrinsic lifetime @xmath69 ; i.e. the total time the intrinsic @xmath70 . the dashed curves show the integrated time that the observed b - band luminosity meets this criterion , using the median metallicity - weighted hot - phase neutral column density of the simulation ( case @xmath81 of figure [ fig : comparenhcalc ] ) . as demonstrated in figure [ fig : comparenhcalc ] and in @xcite , varying this definition of the column density over realistic ranges can change the intrinsic lifetime by a factor of @xmath87 at a given @xmath67 . we also plot the lifetime as a function of @xmath124 , the ratio of the bolometric luminosity to the eddington luminosity , and find a similar trend . here , we calculate the `` observed '' time above a given ratio @xmath124 as the time above such a ratio with a b - band optical depth less than unity . in all cases , the observed @xmath8 is significantly smaller than the intrinsic lifetime for all @xmath125 , and the ratio @xmath126 decreases with increasing @xmath67 . however , both the intrinsic and observed lifetimes increase systematically at a given @xmath127 with increasing galaxy and black hole mass , and the peak luminosity similarly increases , as expected . in all cases corresponding to quasar - like luminosities , our estimates of observed lifetimes of @xmath118 agree well with observations , with much longer intrinsic lifetimes @xmath128 . a quasar accreting and radiating at the eddington limit , with luminosity increasing exponentially up to some peak luminosity @xmath129 after which accretion shuts off , will spend an integrated time above a given luminosity @xmath130 given by @xmath131 where @xmath132 is the salpeter time , @xmath133yr at the eddington rate with our efficiency @xmath25 , and @xmath129 is given by the eddington luminosity of the final black hole mass . this lifetime agrees well with the plotted lifetimes for simulations a3 , a4 , and a5 at the high - luminosity end . however , simulations a1 and a2 do not radiate at their eddington limits for a long period of time , and the slope of this relation is too shallow in all cases , underpredicting the time that the simulation spends at luminosities @xmath134 . this is because the black holes in the simulations spend significant time at lower eddington ratios both going into and coming out of the quasar stage and in an extended quiescent phase . moreover , the observed lifetimes do not at all correspond to the lifetimes predicted by this simple model . therefore , to simply describe the intrinsic and observed lifetimes as a function of limiting luminosity and final black hole mass ( or peak luminosity @xmath129 ) , we fit the calculated lifetimes @xmath8 to truncated power laws , from @xmath135 to approximately @xmath136 , or where @xmath137myr , whichever occurs first . thus we fit to the form @xmath138 where @xmath139 and @xmath140 . to truncated power laws of the form @xmath141 . upper panel shows the intrinsic ( thick solid line ) and observed ( thin solid line ) lifetimes in the a3 simulation , with dashed lines indicating the corresponding best - fit truncated power laws . the dotted line shows the lifetime expected for eddington - limited accretion up to the final black hole mass . the lower panels plot the power law slopes @xmath10 and normalizations at @xmath135 of the power law fits to the intrinsic ( thick ) and observed ( thin ) lifetimes from the series of simulations a1-a5 , as a function of the final black hole mass in the simulation . the dotted line in the left panel gives a simple linear fit to @xmath10 as a function of the logarithm of the final black hole mass . [ fig : pwrlawfits],width=336 ] figure [ fig : pwrlawfits ] shows the resulting power - law slopes @xmath10 and normalizations @xmath142 of the lifetimes @xmath8 as a function of the final black hole mass ( or equivalently , the peak quasar luminosity given as the eddington luminosity of the final black hole mass ) . the calculated , fitted , and eddington - limit approximated lifetimes are also shown for comparison for the a3 simulation . the lifetimes are well - fitted by power laws in this range , and the discrepancy between the power - law and eddington - limit approximated lifetimes at lower luminosities is clear . the slope of both the intrinsic and observed lifetimes becomes shallower at higher black hole masses , as the lifetime increases for all luminosities and extends to higher peak luminosities , but the slope of the observed lifetime evolves much more rapidly than the slope of the intrinsic lifetime . over the range of our simulations , the intrinsic lifetime slope can be approximated as @xmath143 , or @xmath144 . although the fluctuation in the normalization @xmath142 is large for observed lifetimes , there does not appear to be a systematic trend with final black hole mass , which is expected as by luminosities of this level or lower , the lifetime is dominated by the quiescent - phase lifetime throughout the duration of the simulation . this is especially clear in the normalization of the intrinsic lifetimes , which also show no systematic trend and are all within a factor of @xmath145 two of 2 gyr , the approximate total duration of the simulations . in order to obtain a lower limit to quasar lifetimes at low luminosities , we repeat this analysis , but ignore all times prior to the intrinsically bright quasar phase ( essentially the final merger ) this excludes the luminosity of the black hole as it grows from its small seed mass , and ignores the periods of accretion with high gas fractions early in the merger , giving a lower limit corresponding to an already - large black hole suddenly `` turning on '' in a quasar phase . as expected , the eddington - limited phase of the lifetime is more dominant in this case . however , the pure eddington - limited model still gives lifetimes too short by an order of magnitude at low luminosities . fitting to power laws we find very similar relations to those shown in figure [ fig : pwrlawfits ] , with slightly shallower slopes , @xmath146 for the intrinsic lifetime , and normalization @xmath147gyr . we further fit one additional functional form to the intrinsic lifetimes , in which the lifetime follows the eddington - limited shape for large luminosities , down to some luminosity @xmath148 , and below this luminosity follows a power - law . this allows a continuous description of the lifetime , very accurate at both low and near - peak luminosities . for these fits , we find similar power - law slopes @xmath149 , with @xmath150 nearly constant . we also fit the lifetimes as a function of frequency for the representative visual wavelengths shown in figure [ fig : lifetimevsfreq ] . although , as expected from the figure , the observed lifetime slopes become slightly shallower and the intrinsic lifetime slopes become slightly steeper as the quasar is observed at redder wavelengths , the change with wavelength across this range is very small compared to the change with black hole mass . moreover , the change in normalization @xmath142 across visual wavelengths is consistent with zero . we therefore find that quasar lifetimes can be well approximated by simple power laws , where the power law slope depends on the final black hole mass or peak luminosity of a system , and find significantly longer lifetimes at low luminosities than predicted by simple models of accretion at a constant eddington ratio . furthermore , attempts to de - convolve intrinsic quasar properties and observations as well as semi - analytical attempts to model observable quasar luminosity functions and statistics require an accurate model of quasar lifetimes as a function of luminosity and host system properties . the fits we describe have the advantage of a simple , analytical form which can be applied to a continuum of luminosities and intrinsic quasar properties , while addressing the shortcomings of simpler lifetime models below peak system luminosities . in this paper , we have studied implications of the conclusion of @xcitethat incorporating the effects of obscuration in a galaxy merger simulation gives observed quasar lifetimes of @xmath118 . we extended our analysis to a series of five simulations with virial velocities @xmath151 , producing a range of final black hole masses from @xmath152 , and find qualitatively similar results in all cases and for different determinations of the obscuring column density . the good agreement with observations and the significant difference between observed and much longer intrinsic lifetimes is found in all our simulations . the qualitative evolution of quasars in this model is robust , with the result reproduced across a range of galaxy masses , and using different methods for calculating column densities that can give factors greater than an order of magnitude different attenuations along any given line of sight at a particular time . we therefore expect that quasars will have extended intrinsic lifetimes , much of which are obscured in visual wavelengths . the processes which fuel periods of quasar activity by channeling significant quantities of gas into the central regions of a galaxy will produce this extended intrinsic lifetime of @xmath153yr . however , for a significant fraction of the intrinsic lifetime for quasar activity , this same process will produce large column densities which heavily obscure the quasar and attenuate it well below observable limits in the b - band and other visual wavelengths . eventually , feedback from accretion energy will remove surrounding gas , creating a window in which the quasar is optically observable before accretion rates drop below those needed to maintain quasar luminosities . of course , not all aspects of agn activity are related to mergers . for example , some low redshift quasars ( e.g. bahcall et al . 1996 ) and many seyferts appear to reside in relatively ordinary galaxies . gas expelled by quasar feedback may cool and relax to the central regions of the galaxy , and passive stellar evolution can produce a large fuel supply in winds . in our picture , it is most natural to interpret occurrences of agn activity that are not caused by mergers as arising from normal galaxies being re - activated as gas is sporadically accreted by a black hole created earlier in a bright quasar phase . the principle requirement of this modeling is that such activity should not contribute a large fraction of the black hole mass , to avoid spoiling tight correlations between the black hole and host galaxy properties or overproducing the present - day density of very massive black holes . radiative heating from the final stages of quasar activity may suppress much of this activity ( much of the remaining gas is heated to very high temperatures ; e.g. cox et al . 2005 ) , and the hydrodynamical modeling of @xcite demonstrates that residual accretion from stellar feedback and `` cooling flows '' is dramatically suppressed by even relatively small agn feedback once the black hole has reached a critical mass determined by the @xmath19-@xmath5 relation . further , demographic arguments such as that of @xcite , as well as more detailed analysis by e.g. , @xcite implies that most of the present black hole mass density is accumulated in bright quasar phases , for which our modeling should be most applicable . this remains true in our analysis , and we demonstrate in hopkins et al . ( 2005b , c ) that our modeling can account for the entire quasar luminosity function in different frequencies , although we defer a detailed prediction of the resulting supermassive black hole density and mass function to a future paper . it is important to note , however , that given our prediction of an obscured accretion phase with a duration up to @xmath154 times that of the optically observable phase , a naive application of the @xcite or other demographic arguments would imply a similar increase in the black hole mass density accumulated in obscured quasar phases . but this neglects both the fact that black holes are growing during this phase and the luminosity dependence of the quasar lifetime . a more detailed calculation shows that a large fraction ( @xmath155 in the brightest quasars ) of the quasar mass is accumulated in the optically observable stages of black hole growth when the final @xmath15-folding of exponential black hole growth occurs as the black hole begins to drive strong feedback and soon shuts down accretion . similarly , most of the total black hole radiated energy is emitted in this stage , with a considerable fraction ( @xmath156 ) observable . thus , we expect that corrections to the estimates of black hole growth from demographic arguments based on observations of bright , relatively unobscured quasars will in fact only be of order unity from our modeling , and thus should not conflict with theoretical limits . it is still an interesting and important question , however , to determine from our modeling the implied relic agn density and constraints from the combination of observed luminosity functions and black hole mass functions . we do not expect other considerations , such as the orbital parameters of the merger or the collimation of black hole feedback , to substantially change our results . different orbital separations , orientations , and energies will change the time of interaction , but ultimately our model for quasar evolution depends only on the merger eventually depositing sufficient gas into the central regions of the remnant to fuel rapid black hole growth . after such a process begins , the evolution of the black hole should be determined by the self - regulated mechanisms which terminate black hole growth , as described in @xcite . this will eventually expel gas , rendering the quasar observable and shutting down the accretion phase at the final black hole mass given by the @xmath19-@xmath5 relation . the phase of rapid black hole growth and the expulsion of gas in the final stages of strong accretion occur during the end - stages of the merger and in the very center of the merging cores , when and where the structure of the original orbits should be least important . this view is supported by additional simulations not presented here but shown in e.g. fig . 16 of @xcite , demonstrating that the final black hole mass is relatively insensitive to orbital parameters . the exact mechanism of black hole feedback should similarly not dramatically change this picture , so long as a period of rapid black hole growth occurs , requiring large densities ( which naturally generate obscuring column densities ) , and ultimately terminates via some self - regulated mechanism which expels gas from central regions , rendering the quasar optically observable in the final stages of its life . in particular , collimation of the black hole feedback may significantly increase the energy or momentum input on - axis , `` blowing out '' material along this axis earlier in the quasar lifetime . however , since the black hole is growing exponentially during peak accretion , this should occur at most on the order of a couple of salpeter times ( during which the black hole luminosity grows by an order of magnitude ) before it would in the isotropic case , changing the intrinsic lifetime by only a fraction of its total . the unobscured stage of the quasar lifetime may then begin slightly earlier along on - axis sightlines , but should be of comparable duration as a similar mechanism removes the gas from central regions . in addition , in order for significant black hole growth to cease at the same critical mass as in the isotropic case ( determined by the @xmath4-@xmath5 relation ) , the high - accretion rate phase must end over the same timescale @xmath157 as the `` lead '' with which collimated , on - axis blowout precedes isotropic blowout . the necessity that self - regulation eventually terminates accretion implies that surrounding densities , even off - axis , must decrease , resulting in at most a broadening of the column density distribution as typical column densities off - axis could be somewhat larger than column densities on - axis . for much of this regime , we find that the source should be observable in hard x - ray frequencies , as the attenuation is weaker and column densities generally lie below the compton thick limit . this distinction naturally produces a substantial population of obscured quasars , as a standard phase in the evolution of quasars over their lifetimes . although we defer a detailed calculation of the quasar luminosity function as a function of redshift and in different observed frequencies ( hopkins et al . 2005b ) , we note that such a population can naturally account for both the cosmic x - ray background spectrum and discrepancies between optical and x - ray luminosity functions at various redshifts ( hopkins et al . moreover , optical and x - ray observational samples will be affected rather differently by selection effects and magnitude limits , especially for quasars near their peak luminosities . therefore , the differences in the evolution of optical and x - ray selected samples may be accounted for through the dependence of obscuration on intrinsic luminosity and host galaxy properties . attenuation is also weaker in the infrared , and the large fraction of optical and uv energy absorbed during the obscured phase may be reprocessed by dust , appearing as thermal radiation in the ir . this may render the observable lifetime larger than the intrinsic lifetime at long wavelengths , an effect important for calculating the ir background and estimating source populations . similarly , the total obscuration and obscuration as a function of intrinsic luminosity are important for the evolution of luminous and ultraluminous infrared galaxies , and for estimating the relative energy contributions in the ir of starbursts and active galactic nuclei ( agn ) . the typical column densities in our simulations correspond well to observed column densities of optically selected quasars , once the appropriate observed magnitude limit has been imposed . thus , our model allows us to reproduce the distribution of optically obscured sources above this magnitude limit naturally , as an evolutionary effect of the mechanisms which fuel and regulate quasar growth , without invoking particular distributions of source properties or geometric patterns of obscuration . moreover , the column densities we calculate in our simulations allow us to predict that the distribution of @xmath59 values should become broader as the minimum observable luminosity is decreased , as both faint , quiescent phases with low ( @xmath158 ) column densities and bright , obscured phases with high ( @xmath159 ) column densities become observable . this effect is seen in the broad distribution of @xmath59 values in x - ray samples , which are much less affected by attenuation , but a more detailed analysis including modeling the distribution of quasar properties is needed to reproduce both the optical and x - ray @xmath59 distributions more accurately . we find that the peak luminosities and lifetime above any given luminosity increase systematically with galaxy mass and final black hole mass , although the distinction between observed and intrinsic lifetimes remains significant in all cases . moreover , the ratio of observed to intrinsic lifetimes decreases in all cases with increasing minimum luminosity . we also find that intrinsic lifetimes at luminosities @xmath160 times the peak luminosity are poorly fit by assuming quasars always accrete near the eddington limit , but rather that lifetimes are well - fitted by power laws with a steeper slope . this is a result of sub - eddington accretion rates both before and after the peak accretion phase , even at luminosities significantly above the quiescent steady - state small accretion rates seen at late times in the simulations . this suggests that many quasars seen at low luminosities may be quasars with a large peak luminosity in a significantly sub - eddington phase , although the fraction of quasars observed at high eddington ratios becomes large with increasing luminosity . additionally , these lifetimes imply that , even at high luminosities where growth may be eddington - limited , quasars spend a significant fraction of their lives with intrinsic luminosities well below their peak luminosities . therefore , any observed luminosity function is the convolution of the distribution of quasars with a given peak luminosity ( determined by the final black hole mass and thus the merging galaxy properties ) and a non - trivial light curve ( hopkins et al . it is clear from these calculations that any attempt to theoretically model even the intrinsic luminosity functions of quasars must take into account the functional dependence of the light curve on luminosity and time . in order to apply these models to observed luminosity functions , the dependence of observed luminosity on intrinsic luminosity and the quasar evolution , as well as the observed frequency , must be considered ( hopkins et al . 2005b , c ) . we find that the ratio of observable to intrinsic lifetimes is a strongly decreasing function of the limiting luminosity of observations . modeling this effect is necessary to estimate intrinsic quasar lifetimes from observations , as well as for using theoretically motivated accretion models to predict the quasar luminosity function and space density of present - day supermassive black holes . further , the effects we describe can account for the presence of an obscured population of quasars which are missed by optical , uv , or soft x - ray surveys but may contribute significantly to the cosmic x - ray background . observations of the cosmic x - ray background and comparison of the optical and hard x - ray quasar luminosity functions that indicate the existence of a large obscured population of quasars ( * ? ? ? * and references therein ) , is explained in our picture , because it predicts different observed lifetimes and populations at different frequencies . the scenario we describe also implies that the reprocessing of quasar radiation by dust in surrounding regions can account for observations of luminous and ultraluminous infrared galaxies with merger activity and obscured agn . together with the modeling presented by di matteo et al . ( 2005 ) , springel et al . ( 2005a , b ) , springel & hernquist ( 2005 ) , hopkins et al . ( 2005a , b , c ) , and robertson et al . ( 2005 ) , the results described here motivate the following picture for galaxy formation and evolution . through the hierarchical growth of structure in a cold dark matter universe ( white & rees 1978 ) , mergers between galaxies occur on a regular basis . those involving gas - rich progenitors , which would be increasingly more common towards higher redshifts , produce inflows of gas through gravitational torques @xcite , causing starbursts @xcite like those associated with luminous infrared galaxies ( e.g. sanders & mirabel 1996 ) . the high gas densities triggering these starbursts fuel rapid black hole growth . for most of the period over which black hole growth occurs , optical quasar activity would be buried , but x - rays from the black holes explain the presence of non - thermal point sources in e.g. ngc 6240 @xcite , and reprocessing of most of the black hole energy by surrounding gas and dust can , in principle , account for the spectral energy distributions of `` warm '' ulirgs ( sanders et al . 1988c ) and recent observations of a correlation between quasar obscuration and far - infrared host luminosity ( e.g. , * ? ? * ; * ? ? ? as the black hole mass and radiative output increase , a critical point is reached where feedback energy starts to expel the gas fueling accretion . for a relatively brief period of time , the galaxy would be seen as an optical quasar with a b - band luminosity and lifetime characteristic of observed quasars . this phase of evolution is brief ( @xmath161 yr ) , owing to the explosive nature of the final stages of black hole growth as the gas responds dramatically to the feedback energy from the exponentially evolving black hole . this agn feedback terminates further black hole growth , leaving a remnant that resembles an ordinary galaxy containing a dead quasar and satisfying the @xmath19-@xmath5 relation . modeling both the dependence of quasar lifetime on luminosity , and the complex , time - dependent evolution of quasar obscuration is thus crucial to any observational or theoretical understanding of quasars and quasar host galaxy evolution . this work was supported in part by nsf grants aci 96 - 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based on numerical simulations of gas - rich galaxy mergers , we discuss a model in which quasar activity is tied to the self - regulated growth of supermassive black holes in galaxies . the nuclear inflow of gas attending a galaxy collision triggers a starburst and feeds black hole growth , but for most of the duration of the starburst , the black hole is `` buried '' , being heavily obscured by surrounding gas and dust , limiting the visibility of the quasar , especially at optical and ultraviolet wavelengths . as the black hole grows , feedback energy from accretion heats the gas and eventually expels it in a powerful wind , leaving behind a `` dead quasar '' . in between the buried and dead phases , there is a window in time during which the galaxy would be seen as a luminous quasar . because the black hole mass , radiative output , and distribution of obscuring gas and dust all evolve strongly with time , the duration of this phase of observable quasar activity depends on both the waveband and imposed luminosity threshold . we determine the observed and intrinsic lifetimes as a function of luminosity and frequency , and calculate observable lifetimes @xmath0myr for bright quasars in the optical b - band , in good agreement with empirical estimates and much smaller than our estimated black hole growth timescales @xmath1myr , naturally producing a substantial population of buried quasars . however , the observed and intrinsic energy outputs converge in the ir and hard x - ray bands as attenuation becomes weaker and chances of observation greatly increase . we also obtain the distribution of column densities along sightlines in which the quasar is seen above a given luminosity , and find that our result agrees remarkably well with observed estimates of the column density distribution from the sdss for the appropriate luminosity thresholds . our model reproduces a wide range of quasar phenomena , including observed quasar lifetimes , intrinsic lifetimes , column density distributions , and differences between optical and x - ray samples , having properties consistent with observations across more than five orders of magnitude in bolometric luminosity from @xmath2 ( @xmath3 ) .

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Proceed to summarize the following text: mathematical models are now commonly used in the study of growth of cell tissue . for instance , a wide literature is now available on the study of the tumor growth through mathematical modeling and numerical simulations @xcite . in such models , we may distinguish two kinds of description : either they describe the dynamics of cell population density ( see e.g. @xcite ) , or they consider the geometric motion of the tissue through a free boundary problem of hele - shaw type ( see e.g. @xcite ) . recently the link between both descriptions has been investigated from a mathematical point of view thanks to an incompressible limit @xcite . in this paper , we depart from the simplest cell population model as proposed in @xcite . in this model the dynamics of the cell density is driven by pressure forces and cell multiplication . more precisely , let us denote by @xmath0 the cell density depending on time @xmath1 and position @xmath2 , and by @xmath3 the mechanical pressure . the mechanical pressure depends only on the cell density and is given by a state law @xmath4 . cell proliferation is modelled by a pressure - limited growth function denoted @xmath5 . mechanical pressure generates cells displacement with a velocity whose field @xmath6 is computed thanks to the darcy s law . after normalizing all coefficients , the model reads @xmath7 the choice @xmath8 has been taken in @xcite . this choice allows to recover the well - known porous medium equation for which a lot of nice mathematical properties are now well - established ( see e.g. @xcite ) . the incompressible limit is then obtained by letting @xmath9 going to @xmath10 . however , this state law does not prevent cells to overlap . in fact , it is not possible with this choice to avoid the cell density to take value above @xmath11 ( which corresponds here to the maximal packing density after normalization ) . a convenient way to avoid cells overlapping is to consider a pressure law which becomes singular when the cell density approaches @xmath11 . such type of singularity is encountered , for instance , in the kinetic theory of dense gases where the interaction between molecules is strongly repulsive at very short distance @xcite . similar singular pressure laws have been also considered in @xcite to model collective motion , in @xcite to model the traffic flow , and in @xcite to model crowd motion ( see also the review article @xcite ) . then , in order to fit this non - overlapping constraint , we consider the following simple model of pressure law given by @xmath12 finally , the model under study in this paper reads , for @xmath13 , @xmath14 this system is complemented by an initial data denoted @xmath15 . the aim of this paper is to investigate the incompressible limit of this model , which consists in letting @xmath16 going to @xmath17 in the latter system . at this stage , it is of great importance to observe that from , we may deduce an equation for the pressure by simply multiplying by @xmath18 and using the relation @xmath19 from , @xmath20 formally , we deduce from that when @xmath21 , we expect to have the relation @xmath22 moreover , passing formally to the limit into , it appears clearly that @xmath23 . we deduce from this relation that if we introduce the set @xmath24 , then we obtain a free boundary problem of hele - shaw type : on @xmath25 , we have @xmath26 and @xmath27 , whereas @xmath28 on @xmath29 . thus although the pressure law is different , we expect to recover the same free boundary hele - shaw model as in @xcite . the incompressible limit of the above cell mechanical model for tumor growth with a pressure law given by @xmath30 has been investigated in @xcite and in @xcite when taking into account active motion of cells . in @xcite , the case with viscosity , where the darcy s law is replaced by the brinkman s law , is studied . we mention also the recent works @xcite where the incompressible limit with more general assumptions on the initial data has been investigated . however , in all these mentionned works the pressure law do not prevent the non - overlapping of cells . up to our knowledge , this work is the first attempt to extend the previous result with this constraint , i.e. with a singular pressure law as given by . the outline of the paper is the following . in the next section we give the statement of the main result in theorem [ th1 ] , which is the convergence when @xmath16 goes to @xmath17 of the mechanical model towards the hele - shaw free boundary system . the rest of the paper is devoted to the proof of this result . first , in section [ sec : estim ] we establish some a priori estimate allowing to obtain space compactness . then , section [ sec : tcompact ] is devoted to the study of the time compactness . thanks to compactness results , we can pass to the limit @xmath21 in system in section [ sec : conv ] , up to the extraction of a subsequence . finally the proof of the complementary relation is performed in section [ sec : compl ] . the aim of this paper is to establish the incompressible limit @xmath31 of the cell mechanical model with non - overlapping constraint . before stating our main result , we list the set of assumptions that we use on the growth fonction and on the initial data . for the growth function , we assume @xmath32 } |g'| = \gamma,\\ & \exists\ , p_m>0 , \quad g(p_m)=0 . \end{aligned } \right.\ ] ] the quantity @xmath33 , for which the growth stops , is commonly called the homeostatic pressure @xcite . this set of assumptions on the growth function is quite similar to the one in @xcite , except for the bound on the growth term which is needed here due to the singularity in the pressure law . for the initial data , we assume that there exists @xmath34 such that for all @xmath35 , @xmath36 notice that this set of assumptions imply that @xmath15 is uniformly bounded in @xmath37 . we are now in position to state our main result . [ th1 ] let @xmath38 , @xmath39 . let @xmath5 and @xmath40 satisfy assumptions and respectively . after extraction of subsequences , both the density @xmath41 and the pressure @xmath42 converge strongly in @xmath43 as @xmath44 to the limit @xmath45;l^1(\rr^d))\cap bv(q_t)$ ] and @xmath46;h^1(\rr^d))$ ] , which satisfy @xmath47 and @xmath48 moreover , we have the relation @xmath49 and the complementary relation @xmath50 this result extends the one in @xcite to singular pressure laws with non - overlapping constraint . we notice that we recover the same limit model whose uniqueness has already been stated in ( * ? ? ? * theorem 2.4 ) . although our proof follows the idea in @xcite , several technical difficulties must be overcome due to the singularity of the pressure law . indeed , we first recall that with the choice @xmath30 , equation may be rewritten as the porous medium equation @xmath51 . a lot of estimates are known and well established for this equation ( see @xcite ) , in particular a semiconvexity estimate is used in @xcite which allows to obtain estimate on the time derivative and thus compactness . with our choice of pressure law , should be consider as a fast diffusion equation . thus we have first to state a comparison principle to obtain a priori estimates ( see lemma [ lem : estim ] ) . unlike in @xcite , we may not use a semiconvexity estimate to obtain estimate on the time derivative . to do so , we use regularizing effects ( see section [ sec : tcompact ] ) . then the convergence proof has to be adapted for these new estimates . solving , the dashed line correspond to the constant value @xmath11 . on the left , the pressure law is @xmath52 . on the right , the pressure law is @xmath53 with @xmath54.,title="fig : " ] solving , the dashed line correspond to the constant value @xmath11 . on the left , the pressure law is @xmath52 . on the right , the pressure law is @xmath53 with @xmath54.,title="fig : " ] finally , we illustrate the comparison between the two pressure laws @xmath55 and @xmath56 by a numerical simulation . we display in figure [ fig1 ] the density computed thanks to a discretization with an upwind scheme of . in figure [ fig1]-left , the pressure law is @xmath57 as in with @xmath58 . in figure [ fig1]-right , the pressure law is @xmath53 with @xmath54 . we take @xmath59 as growth function ( which satisfies obviously assumption with @xmath60 ) . the dashed lines in these plots correspond to the constant value @xmath11 . as expected , we observe that the density @xmath61 is bounded by @xmath11 in the case of the pressure law @xmath55 whereas it takes values greater than @xmath11 for the pressure law @xmath56 . this observation illustrates the fact that the choice of the pressure law @xmath56 does not prevent from overlapping . the following lemma establishes the nonnegativity of the density . [ lem : nonneg ] let @xmath62 be a solution to such that @xmath63 and @xmath64 . then , for all @xmath1 , @xmath65 . we have the equation @xmath66 we use the stampaccchia method . we multiply by @xmath67 , then using the notation @xmath68 for the negative part , we get @xmath69 we integrate in space , using assumption , we deduce @xmath70 so , after a time integration @xmath71 with the initial condition @xmath72 , we deduce @xmath73 . in order to use compactness results , we need first to find a priori estimates on the pressure and the density . we first observe that we may rewrite system as , by using , @xmath74 with @xmath75 . [ lem : estim ] let us assume that and hold . let @xmath62 be a solution to . then , for all @xmath38 , we have the uniform bounds in @xmath76 , @xmath77;l^1\cap l^\infty(\rr^d ) ) ; \\ & 0\leq p_\epsilon \leq p_m , \qquad 0\leq n_\epsilon \leq \frac{p_m}{p_m+\epsilon } \leq 1.\end{aligned}\ ] ] more generally , we have the * comparison principle * : if @xmath78 , @xmath79 are respectively subsolution and supersolution to , with initial data @xmath15 , @xmath80 as in and satisfying @xmath81 . then for all @xmath82 , @xmath83 . finally , we have that @xmath84 is uniformly bounded in @xmath85,w^{1,1}(\rr^d))$ ] and @xmath86 is uniformly bounded in @xmath87,w^{1,1}(\rr^d))$ ] . * comparison principle . * let @xmath41 be a subsolution and @xmath88 a supersolution of , we have @xmath89 notice that , since the function @xmath90 is nondecreasing , the sign of @xmath91 is the same as the sign of @xmath92 . moreover , @xmath93 so for @xmath94 and @xmath95 is the positive part , the so - called kato inequality reads @xmath96 . thus multiplying the latter equation by @xmath97 , we obtain @xmath98 from assumption , we have that @xmath5 is nonincreasing . thus , since @xmath99 is increasing , we deduce that the last term of the right hand side is nonpositive . since @xmath5 is uniformly bounded we obtain @xmath100 after an integration over @xmath101 , @xmath102 then , integrating in time , we deduce @xmath103 since we have @xmath104 , we deduce that for all @xmath82 , @xmath105 . * @xmath106 bounds . * we define @xmath107 , such that @xmath108 , then applying the comparison principle with @xmath109 , we deduce , using also the assumption on the initial data that for all @xmath110 , @xmath111 moreover , since @xmath17 is clearly a subsolution to , we also have by the comparison priniciple @xmath112 . since @xmath113 , we have @xmath114 which implies @xmath115 * @xmath116 bound of @xmath117 . * by nonnegativity , after a simple integration in space of equation , we deduce @xmath118 where we use . integrating in time give the @xmath116 bound , @xmath119 then , using @xmath120 by , we get from the bound @xmath121 , which has been proved above , @xmath122 * estimates on the @xmath123 derivative . * we derive equation with respect to @xmath124 for @xmath125 , @xmath126 multiplying by sign@xmath127 , we get @xmath128 we can remark that @xmath129 , so , by the same token as above , we have @xmath130 moreover , @xmath131 , thus @xmath132 . by assumption , we know that @xmath133 we deduce @xmath134 after an integration in time and space , @xmath135 this latter inequality provides us with a uniform bound on the space derivative of @xmath78 in @xmath116 . then @xmath136 we split the integral in two : either @xmath137 and then @xmath138 ; or @xmath139 . @xmath140 where we have used the estimate for the last inequality . then , integrating in time , we deduce , using again the estimate @xmath141 it concludes the proof . the following lemma proves that assuming that the initial data is compactly supported , then the pressure is compactly supported for any time with a control of the growth of the support . [ lem : supp ] under the same assumptions as in theorem [ th1 ] , we have that @xmath142 with @xmath143 , where @xmath144 is the ball of center @xmath17 and radius @xmath145 . using the equation on @xmath42 , @xmath146 let us introduce for @xmath147 , @xmath148 with @xmath149 . then @xmath150 is compactly supported in @xmath151 with @xmath152 we have @xmath153 then , for all @xmath154 $ ] , @xmath155 in other words , @xmath150 is a supersolution for the equation for the pressure . let us show that it implies that @xmath156 . we define @xmath157 . we know that @xmath158 then , on the one hand , multiplying with by @xmath159 we get @xmath160 on the other hand , from , @xmath161 by the comparison principle ( see lemma [ lem : estim ] ) , we have @xmath162 thus , for all @xmath154 $ ] , @xmath163 and @xmath164 is compactly supported in @xmath151 provided we choose @xmath165 large enough such that @xmath166 , which can be done thanks to our assumption on the initial data . since @xmath167 is uniformly bounded in @xmath168 , we may iterate the process on @xmath169 $ ] . after several iterations , we reach the time @xmath170 and prove the result on @xmath171 $ ] . in the following lemma , we state a uniform @xmath172 estimate on the gradient of the pressure . [ lem : l2dp ] under the same assumptions as in theorem [ th1 ] , we have a uniform bound on @xmath174 in @xmath175 . for a given function @xmath176 we have , multiplying by @xmath177 , @xmath178 let @xmath179 be an antiderivative of @xmath176 , we have thanks to an integration by parts @xmath180 we choose @xmath176 such as @xmath181 , i.e. @xmath182 . after straightforward computations , we find @xmath183 and @xmath184 . it gives @xmath185 we integrate in time , using also the expression of @xmath167 in , @xmath186 then , to have a bound on the @xmath172-norm of @xmath174 , it suffices to prove a uniform control on @xmath187 . we have @xmath188 the second term of the right hand side is small when @xmath16 is small thanks to the @xmath116 bound on @xmath78 , thus it is uniformly bounded . using the expression of @xmath167 in , we get @xmath189 then , since @xmath190 and since @xmath191 is uniformly bounded on @xmath192 $ ] , we get @xmath193 we conclude thanks to lemma [ lem : supp ] , which provides a uniform control on the support of @xmath167 . as already noticed in @xcite , regularizing effects , similar to the ones observed for the heat equation @xcite , allow to deduce estimates on the time derivatives . [ lem : regul ] under the assumptions and , the weak solution @xmath194 satisfies the estimates @xmath195 for a large enough ( independent of @xmath16 ) constant @xmath196 . let us denote @xmath197 , the equation on the pressure reads @xmath198 the proof is divided into several steps . we first find a lower bound for @xmath199 by using the comparison principle . then we deduce estimates on the density and on the pressure . + _ 1st step . _ thanks to , we deduce an equation satisfied by @xmath199 . on the one hand , by multiplying by @xmath200 , we deduce , since @xmath5 is decreasing from @xmath201 on the other hand , we have @xmath202 thus , with , we deduce that @xmath203 satisfies @xmath204 by definition of @xmath199 , we have @xmath205 . thus we deduce that @xmath206 where we have used the notation @xmath207 following an idea of @xcite which has been generalized in @xcite , we introduce the function @xmath208 where the function @xmath209 will be defined later such that @xmath210 is a subsolution for . we compute @xmath211 using again equation , we have @xmath212 by definition of @xmath213 in , we deduce with , @xmath214 we may rearrange it into @xmath215 let us choose @xmath216 where @xmath217 is chosen large enough ( independent of @xmath16 ) such that @xmath218 thanks to this choice , we have @xmath219 and @xmath220 finally , we obtain from @xmath221 where we use the fact that by definition we have @xmath222 ( recalling also that @xmath5 is decreasing by assumption ) . thus , by the sub- and super - solution technique , we deduce , using also that @xmath223 + _ 2nd step . _ using again equation , we get from @xmath224 which is the first inequality of lemma [ lem : regul ] . finally , by definition , we have also @xmath225 . thus @xmath226 where we use the definition for the last identity . we conclude easily the proof . thanks to this latter lemma , we may deduce uniform estimates on the time derivative of @xmath78 and @xmath167 . [ estimdtn ] for any @xmath227 , we have that @xmath228 is uniformly bounded in @xmath229;l^1(\rr^d))$ ] and @xmath230 is uniformly bounded in @xmath231\times\rr^d)$ ] . we use the equality @xmath232 , where we recall that @xmath233 denotes the negative part . thus @xmath234 where we have used equation to bound the first term and lemma [ lem : regul ] for the second term . by the same token , we have @xmath235\times\rr^d ) } & = \int_\tau^t \frac{d}{dt } \int_{\rr^d } p_\epsilon \,dx + 2 \int_\tau^t\int_{\rr^d } |\partial_t p_\epsilon|_- \,dx \\ & \leq \|p_\epsilon(t)\|_{l^1(\rr^d ) } + \|p_\epsilon\|_{l^\infty([\tau , t];l^1(\rr^d ) ) } 2 \kappa \ln(t/\tau).\end{aligned}\ ] ] we conclude the proof thanks to the estimates on @xmath78 and @xmath167 in @xmath236 obtained in lemma [ lem : estim ] . this section is devoted to the proof of theorem [ th1 ] apart from the complementary relation which is postponed to the next section . since the sequences @xmath237 and @xmath238 are bounded in @xmath239 , due to lemma [ lem : estim ] and [ estimdtn ] , we may apply the helly theorem and recover strong convergence in @xmath240 , up to an extraction . if we want to extend this local convergence to a global convergence in @xmath43 we need to prove that we can control the mass in an initial strip and in the tail . indeed , let @xmath241 , @xmath242 , @xmath243 @xmath244 since we have strong convergence of @xmath41 in @xmath240 , @xmath245 then we have to control the two other terms in the right hand side . the control of the initial strip comes from the @xmath116 estimate of @xmath61 , @xmath246 for the control of the tail we consider @xmath247 such that @xmath248 , @xmath249 for @xmath250 and @xmath251 for @xmath252 . we define @xmath253 . then @xmath254 where the notation @xmath165 stand for a generic nonnegative constant . moreover , using equation , we deduce @xmath255 then , integrating on @xmath171 $ ] , we get @xmath256 by assumption , since the initial data is uniformly compactly supported , we deduce that the right hand side tends to @xmath17 as @xmath257 goes to @xmath10 and @xmath258 goes to @xmath17 . then @xmath259 is a cauchy sequence in @xmath43 . it implies its convergence in @xmath43 . the convergence of the pressure follows from the same kind of computation . the only difference is for the control of the tail and which is directly given by the estimate @xmath260 therefore , we can extract subsequences and pass to the limit in the equation @xmath261 which implies @xmath262 this is the relation . we can also pass to the limit in the uniform estimate of lemma [ lem : estim ] which provides and @xmath263 . [ [ limit - model . ] ] limit model . + + + + + + + + + + + + we first recall that from , we have @xmath264 we get , @xmath265 thus , the term in the laplacien converges strongly to @xmath266 as @xmath16 goes to @xmath17 . then , thanks to the strong convergence of @xmath78 and @xmath167 , we deduce that in the sense of distribution @xmath267 satisfies . moreover , due to the uniform estimate on @xmath173 in @xmath175 of lemma [ lem : l2dp ] , we can show , by passing into the limit in a product of a weak - strong convergence , that in the sense of distribution @xmath267 satisfies . [ [ time - continuity . ] ] time continuity . + + + + + + + + + + + + + + + + let us define @xmath268 , @xmath269 . for a given @xmath270 , we consider a smooth function @xmath271 on @xmath101 such that @xmath272 , @xmath273 for @xmath250 and @xmath274 for @xmath252 . we have @xmath275 we have @xmath276 with @xmath277 a function which is zero on @xmath278 . thus , as for the control of the tail , for @xmath257 large enough , we have , uniformly for @xmath268 , @xmath279 in addition , we know from lemma [ lem : regul ] ( and the @xmath168 bound on @xmath280 ) that @xmath281 , so @xmath282 . then , since @xmath283 , @xmath284 then , using equation and an integration by parts , we obtain @xmath285 then we can choose @xmath286 close enough such that @xmath287 we conclude that @xmath288 . [ [ initial - trace ] ] initial trace + + + + + + + + + + + + + for any test function @xmath289 , we have from , @xmath290 letting @xmath16 going to @xmath17 , we obtain with , @xmath291 letting @xmath292 we can conclude that @xmath293 . in this section we prove the complementary relation @xmath294 in the weak sense , this identity reads , for any test function @xmath295 , @xmath296 the proof is divided into two steps . + _ 1st step . _ in this first step we prove the inequality @xmath297 in . we start with the pressure equation that we multiply by @xmath298 @xmath299 we multiply by a test function @xmath300 and integrate , @xmath301 where we use an integration by parts for the last identity . from the estimates in lemma [ lem : estim ] , we have @xmath302 we deduce that for any test function @xmath300 , @xmath303 since we have strong convergence of @xmath304 and weak convergence of @xmath305 , we can pass into the limit in the last two term in , @xmath306 now we are looking for the limit of the first term in . we have @xmath307 . by weak convergence of @xmath308 and with jensen inequality ( since @xmath309 is convex ) , @xmath310 thus , we conclude from that @xmath311 which is a first inequality for . + _ _ now we want to show the reverse inequality , i.e. @xmath312 we know that @xmath313 with @xmath314 thanks to the inequality @xmath315 , and the strong convergence @xmath316 , we know that @xmath317 as @xmath21 . because @xmath318 we deduce from lemma [ lem : estim ] that @xmath319;l^{1}(\rr^d ) ) $ ] . it gives us compactness in space but not in time . thus , following the idea of @xcite , we use a regularization process la steklov. let introduce a time regularizing kernel @xmath320 such that @xmath321 . then with the notations @xmath322 , @xmath323 , where the convolution holds only in the time variable , @xmath324 we denote @xmath325 , then @xmath326 since @xmath327 and @xmath328 are uniformly bounded in @xmath329 from lemma [ lem : estim ] , @xmath330 is bounded in @xmath329 and we can extract a converging subsequence , still denoted @xmath331 , converging towards @xmath332 in @xmath333 for @xmath334 fixed . moreover @xmath335 for the second term , we have @xmath344 and @xmath345 . let @xmath346 and @xmath227 the smallest time in its support , we then have for @xmath347 @xmath348 so integrating on @xmath349 @xmath350 then @xmath351 where we use the bound on @xmath352 in lemma [ estimdtn ] . for the third term , since @xmath353 , for any test function @xmath354 as above , @xmath355 dx dt \\ & \qquad = \underset{\eta\to 0}{o}(1 ) . \end{aligned}\ ] ] so for all test function @xmath354 as above , and all @xmath269 , @xmath356 now it remain to pass to the limit @xmath357 in the regularization process . thanks to an integration by parts , @xmath358 from the @xmath172 estimate on @xmath359 ( lemma [ lem : l2dp ] ) and the @xmath236 estimate on @xmath266 ( lemma [ lem : estim ] ) , we deduce that we can pass to the limit @xmath360 and get @xmath361 p. ciarletta , l. foret , m. ben amar , _ the radial growth phase of malignant melanoma : multiphase modelling , numerical simulations and linear stability analysis _ , j. r. soc . interface * 8 * ( 2011 ) , 345368 . j. s. lowengrub , h. b. frieboes , f. jin , y .- l . chuang , x. li , p. macklin , s. m. wise , v. cristini , _ nonlinear modelling of cancer : bridging the gap between cells and tumours _ , nonlinearity ( 2010 ) * 23 * ( 1 ) , r1r91 . c. perrin , e. zatorska , _ free / congested two - phase model from weak solutions to multi - dimensional compressible navier - stokes equations _ , communications in partial differential equations , ( 2015 ) 40:8 , 15581589

a mathematical model for tissue growth is considered . this model describes the dynamics of the density of cells due to pressure forces and proliferation . it is known that such cell population model converges at the incompressible limit towards a hele - shaw type free boundary problem . the novelty of this work is to impose a non - overlapping constraint . this constraint is important to be satisfied in many applications . one way to guarantee this non - overlapping constraint is to choose a singular pressure law . the aim of this paper is to prove that , although the pressure law has a singularity , the incompressible limit leads to the same hele - shaw free boundary problem . nonlinear parabolic equation ; incompressible limit ; free boundary problem ; tissue growth modelling . 35k55 ; 76d27 ; 92c50 .

You are an expert at summarizing long articles. Proceed to summarize the following text: planetary atmospheres and oceans are strongly turbulent media . however , highly ordered coherent structures arise in a process of self - organization , and dominate the dynamics on slow temporal and large spatial scales . + the spontaneous appearance of coherent structures is a characteristic of two - dimensional fluid flows . the basic underlying structure of these flows is linked to the existence of two quadratic , positive definite invariants , energy and enstrophy . in spectra of two - dimensional turbulence one observes two different cascades associated with these conserved quantities ; a direct enstrophy cascade towards small spatial scales , and an indirect energy cascade towards larger spatial scales . it is the latter which gives rise to vortex merging leading to larger and larger vortices . in this letter we view the problem as one of weakly nonlinear hydrodynamic stability rather than turbulence phenomenology . + most vortices are monopolar , but dipole and even tripole vortices can also appear spontaneously . monopole vortices are mainly created by shear flow instabilities whereas dipole vortices typically appear when some additional forcing is applied to the flow . + the richness and complexity of two - dimensional flows and the simultaneous presence of motion on very different temporal and spatial scales makes a direct analysis of the basic equations of motion very difficult . the presence of rotation reinforces the two - dimensional character in accordance with the taylor - proudman theorem , but rotation can also introduce baroclinic instability . the latter is a three - dimensional feature and , thus supports a direct energy cascade towards small scales . hence , the dynamics is determined by competing two - dimensional and three - dimensional processes @xcite . to study vortices in geophysical fluid dynamics the primitive equations are further reduced by approximations which allow to focus on temporal and spatial length scales of vortices @xcite . if additionally baroclinic processes are excluded a further simplification can be made . the dynamically important variable is the so called potential vorticity @xmath0 . the resulting quasigeostrophic barotropic vorticity equation @xmath1 describes large scale motion on a slow time scale . here @xmath2 is the stream function , @xmath3 describes the ambient rotation of the planet and @xmath4 is the fluid depth . this equation was first derived by charney @xcite , and then independently by obukhov @xcite . in the context of low - frequency drift waves in magnetized plasmas equation ( [ charney ] ) is known as the hasegawa - mima equation @xcite . this equation has been the mathematical starting point for much of the research done on coherent structures and vortices . it supports so called modons which are localized soliton - like coherent solutions . exact modon solutions were obtained by larichev & reznik @xcite for a stationary double - vortex solution which is antisymmetric in longitude . extensions to more general solutions have been made @xcite , and the spherical geometry of planets has been incorporated @xcite . however , modons have the drawback that the potential vorticity is not a smooth function of the stream function , but may be multivalued . therefore interest has grown in low - dimensional models , although a rigorous proof of existence of a low - dimensional attractor in quasigeostrophic systems is still an unsolved problem . strictly speaking , one can only define a `` slowest invariant manifold '' @xcite , since the small - scale events , i.e. the high - frequency and high - wavenumber processes , enlarge the hausdorff dimension for the attractor without any convergence @xcite . nevertheless , in order to understand better the particular mechanisms involved in the formation and dynamics of vortices in geophysical fluid dynamics , it is useful to perform asymptotic techniques to derive reduced amplitude equations of the basic quasigeostrophic equations in a multiple scale analysis and study the derived model evolution - equations . the basic idea is that coherent vortices may be identified with solitary wave solutions of generic nonlinear dispersive wave equations . + most research has been done in the frame work of the korteweg - de vries equation or in the framework of the boussinesq equation @xcite . while these models were helpful in describing and identifying mechanisms for atmospheric blocking , cyclogenesis , meandering of oceanic streams and the persistence of the great red spot in the jovian atmosphere , they are all one - dimensional models with their obvious limitations . + in this letter we will extend weakly nonlinear , long wave multiple scale analysis to two dimensions and derive the zakharov - kuznetsov equation @xmath5 the zakharov - kuznetsov ( zk ) equation @xcite is one of two well - studied canonical two - dimensional extensions of the korteweg - de vries equation @xcite ; the other being the kadomtsev - petviashvilli ( kp ) equation @xcite . in contrast to the kp - equation , the zk - equation has so far never been derived in a geophysical fluid dynamics context . for a derivation of the kp equations for internal waves , see @xcite . whereas the kp - equation is valid in isotropic situations , the zk - equation is valid in anisotropic settings which is exactly the case for rotating fluids where the differential longitudinal dependence of the rotation rate causes anisotropy between the meridional and the longitudinal directions . moreover , in contrast to the kp - equation the zk - equation supports stable lump solitary waves . this makes the zk - equation a very attractive model equation for the study of vortices in geophsyical flows . + the letter is organized as follows . in section 2 we set up the barotropic vorticity equation and the mean flow configurations under consideration . in the beginning of section 3 we will give a simple heuristic scaling argument based on the linearized barotropic vorticity equation to motivate why the zk - equation is the generic two - dimensional nonlinear wave equation . in the remainder of section 3 we will derive the zk - equation in an asymptotic multiple scale analysis . section 4 concludes the letter with a discussion and an outlook on further research . we shall use a non - dimensional coordinate system , based on a typical horizontal length scale @xmath6 , a typical vertical scale @xmath7 , and typical coriolis parameter @xmath8 . a typical velocity @xmath9 is taken to be the maximum of the mean current velocity and the time scale is given by @xmath10 . if we separate the meridional meanflow @xmath11 from the perturbation pressure fields @xmath12 and use the boussinesq approximation , we obtain the following equation for the non - dimensional perturbation pressure field @xcite @xmath13 where @xmath14 with froude number @xmath15 and the jacobian defined by @xmath16 . we investigate a channel flow with a storm track superimposed on a constant meanflow @xmath17 confined at @xmath18 ( see fig.1 ) . the storm tracks may have a critical layer where @xmath19 . important is , as we will see , the non - vanishing slope at at least one boundary of the localized storm track . the boundary conditions are @xmath20 at @xmath21 , and we require that the jet forms a transport barrier to the flow . before we consider the weakly nonlinear , long wave approximation , we motivate our approach by looking at the linearization of equation ( [ qgp ] ) which yields @xmath22 in terms of @xmath23 we obtain the dispersion relation @xmath24 provided that the meanflow @xmath25 is constant . if we focus on dynamics on the spatial and temporal time scales @xmath26 , i.e. small @xmath27 , and @xmath28 this is suggestive of the coupled zakharov - kuznetsov equation @xmath29 the reason why we expect an equation of the zakharov - kuznetsov type instead of the usual kadomtsev - petviashvilli type mostly encountered in fluid systems is the anisotropic character of ( [ qgp ] ) caused by the @xmath30-effect . we consider weakly nonlinear waves riding on a background meanflow . the meanflow consists of a constant part @xmath17 and a strong but narrow jetstream ( see fig . the narrow storm track is located on a short meridional scale @xmath31 . in the outer region the problem ( [ qgp ] ) can be reduced to the linear problem ( [ lin1 ] ) with constant meanflow @xmath17 . in the interior the structure of the storm track does not allow for sinusoidal wave solutions but instead we will derive a nonlinear wave equation . in order for the nonlinear wave equation which is valid only in the inner region where the storm tracks are nonuniform , the inner solution has to be matched to the outer sinusoidal solution . in the outer region where the meanflow is uniform and constant , ( [ qgp ] ) reduces to the simple linear equation ( [ lin1 ] ) for the streamfunction with constant coefficients @xcite . + there the solution of the streamfunction can be written as @xmath32 where @xmath33 is the meridional wave number and is determined by the dispersion relation of the linearized model ( [ lin1 ] ) . in the interior of the storm track on the small scale @xmath31 , the meanflow is not constant . we shall study weakly nonlinear long waves . we introduce the following scales , @xmath34 where @xmath35 is a small parameter , the inverse of which measures the large scales of the disturbance . next , we rescale the parameters @xmath36 and @xmath37 . the scaling of the froude numbers implies that our model is valid for situations where the internal rossby radius of deformation is of the order of the long horizontal scale . further , the scaling of @xmath30 implies that @xmath38 at the lowest order . the boundary conditions we use are @xmath39 at @xmath21 and @xmath40 at @xmath18 which simply states that there is no transport of fluid across the jet - stream . + substituting this scaling into equation ( [ qgp ] ) yields , @xmath41 counting the orders of @xmath35 we obtain to the lowest order , @xmath42 @xmath43 where @xmath44\ ; .\end{aligned}\ ] ] we look for an amplitude equation , ie we want to write @xmath45 and seek an evolution equation for the slowly varying amplitude @xmath46 . we easily find @xmath47 where @xmath48 is a constant of integration . if we allow for zero meanflow within the narrow jet region we need to impose @xmath49 . we summarize the solution of equation ( [ ord_0 ] ) @xmath50 the meridional structure on the small scale @xmath31 of @xmath2 is entirely determined by the mean currents at the leading order . + at the next order , @xmath51 , we obtain a linear inhomogeneous equation for @xmath52 , @xmath53 which can be written , using ( [ varphi0 ] ) , as @xmath54 this equation is again solved by the method of variation of parameters and we obtain @xmath55 with @xmath56 we note that the higher order term @xmath52 is slaved to the @xmath57 term and the dynamics of the corresponding amplitude equation which will be derived shortly . for the same reasons as above we set @xmath58 and obtain @xmath59 the @xmath60 terms give us an evolution equation for the amplitude @xmath61 . we obtain @xmath62 which , using ( [ psi0 ] ) , can be written as @xmath63 where @xmath64 to assure boundedness of the solutions of ( [ 116 ] ) we have to require a solvability condition in form of a fredholm alternative . + the homogeneous adjoint problem to equation ( [ 116 ] ) may be written as @xmath65 with @xmath66 where we have used the boundary conditions @xmath67 at @xmath18 . the adjoint eigenvalue problem ( [ adjoint ] ) has one trivial constant kernel mode @xmath68 and one nontrivial , namely @xmath69 the nontrivial kernel mode @xmath70 has to be discarded because it does not satisfy the boundary condition . to see this note that the fredholm alternative for the elliptic operator [ lop ] together with the boundary condition @xmath67 is equivalent to one for the operator @xmath71-u_y\partial_y\psi - u_{yy}\partial_y\psi$ ] with the boundary condition @xmath72 at @xmath18 . also note that a non - zero kernel mode @xmath73 is only consistent with the boundary conditions if @xmath74 . the solvability condition is thus given by the trivial constant kernel mode @xmath75 on substituting the expressions ( [ psi0 ] ) with @xmath76 and ( [ gn ] ) we obtain the desired amplitude equation for @xmath61 , @xmath77 where @xmath78_{-l}^l \ ; , \nonumber \\ i\xi & = & \int_{-l}^l u^2 \ ; dy \ ; , \nonumber\\ i\zeta & = & \left [ u^2(1-y - l ) \right]_{-l}^l\ ; , \nonumber\\ i \mu & = & - \left [ u_{y}^2 \right]_{-l}^l + \left [ uu_{yy } \right]_{-l}^l \ ; , \nonumber\\ i\delta & = & -\int_{-l}^l \beta u \ ; dy \ ; .\end{aligned}\ ] ] we note that due to the last term of ( [ gn ] ) the zakharov - kuznetsov equation is inhomogeneous in the sense @xmath79 . it is pertinent to mention that a nonzero @xmath80 requires a nonzero mean flow at at least one of the boundaries of the storm track . the coefficients of the nonlinear terms @xmath81 require a non - vanishing slope at at least one boundary . the slope @xmath82 and also @xmath83 at the boundaries of the jet @xmath18 may be determined from a given meanflow configuration by averaging over a very short region , say @xmath84 , where a sudden change of the constant mean flow @xmath17 to the jet occurs . at the lowest order the inner solution ( [ varphi0 ] ) with @xmath49 and the outer solution ( [ psi_out ] ) have to be matched . the outer solution has been derived on the large scale @xmath85 whereas the inner solution and its associated amplitude equation , the zakharov - kuznetsov equation ( [ zk11 ] ) , were derived on the short scale @xmath31 . henceforth we need to require that the asymptotic limit of the outer solutions for @xmath86 coincides with the asymptotic limit of the inner solution for @xmath87 . the limit of the inner solution is @xmath88 . the limit of the outer solution is @xmath89 . hence we find @xmath90 , which extends the dynamics of the zakharov - kuznetsov equations to the outer region . we have derived the nonlinear dispersive zakharov - kuznetsov equation from the quasigeostrophic barotropic vorticity equation . it is well known that the zk - equation , although it is not integrable by means of the inverse scattering transform , supports a family of steady - shape stable lump solitary waves , moving at an arbitrary velocity @xcite . these may help to describe two - dimensional coherent structures such as atmospheric blocking events , long lived eddies in the ocean or coherent structures in the jovian atmosphere such as the great red spot . the model is from an analytical point of view easier to treat than the full barotropic quasigeostrophic equation and its solutions do not exhibit multivalued potential vorticity - stream function relationships as modons do . + geophysical flow on large scales is widely accepted to be conservative . this allows for hamiltonian descriptions of the flow on large scales . our model also exhibits a hamiltonian structure . note that the momentum @xmath91 and the hamiltonian with the hamiltonian density @xmath92 are conserved . + we have assumed a meridional meanflow @xmath11 which consists of a constant part @xmath17 and a narrow localized storm track . note that the jet stream may also be a narrow interface between two regions of meanflow with opposite flow direction . such persistent shear layers exist between the zones and belts in the jovian atmosphere . + analysis of the solutions of ( [ zk11 ] ) is planned . their stability has to be numerically tested within the zakharov - kuznetsov system . the zk - equation has been derived using asymptotic techniques and is as such an asymptotic limit to the barotropic quasigeostrophic vorticity equation . however , it is not clear that the same is true for the solutions . the solutions of the zakharov - kuznetsov equations do not necessarily have to be asymptotically close to the solutions of the full quasigeostrophic system . this is due to the lack of a centre manifold as discussed in the introduction . in further work we will test the approximation of the solution numerically by taking solutions of the zk - equation and testing their dynamics in the full quasigeostrophic system . + i would like to thank tom bridges , daniel daners , roger grimshaw , charlie macaskill , marcel oliver , dmitry pelinovsky , victor shrira and vladimir zeitlin for valuable discussions .

we study the dynamics of two - dimensional coherent structures in planetary atmospheres and oceans . we derive the zakharov - kuznetsov equation for large scale motion from the barotropic quasigeostrophic equation in a weakly nonlinear , long wave approximation . we consider coherent structures emerging out of an instability caused by a narrow jet - like meanflow . we use multiple scale analysis combined with asymptotic matching . pacs numbers : 47.35.+i ; 47.32.-y ; 92.10.-c ; 92.60.-e keywords : solitary waves ; zakharov - kuznetsov equation ; barotropic quasigeostrophic equation

You are an expert at summarizing long articles. Proceed to summarize the following text: brightest cluster galaxies ( bcgs ) are luminous ellipticals found near the centres of galaxy clusters and groups . they are further subdivided into giant ellipticals ( ge ) , d galaxies which have shallower surface brightness profiles than ges , and finally cd galaxies which have extended low surface brightness envelopes ( mathews , morgan & schmidt 1964 ; schombert 1987 , 1988 ) . bcgs are often associated with powerful radio sources ( mathews _ et al . _ 1964 ) , and extended cluster x - ray emission ( jones _ et al . _ 1979 ) . in regular clusters , bcgs are invariably at the centre of the cluster potential , as defined by the cluster x - ray emission , galaxy distributions , and strong / weak gravitational lensing ( e.g. miralda - escude 1995 ) , and their velocities are usually close to the mean cluster velocity . bcgs are often observed to have extremely high globular cluster specific frequencies ( e.g. harris , pritchet , & mcclure 1995 ; bridges _ et al . _ 1996 ) . the origin of bcg halos is one of the most puzzling aspects of galaxy formation , and it is unclear whether they are formed at the same time as the galaxy proper , as proposed by merritt ( 1984 ) , or are accreted later . if accreted later they might form via dissipational processes , such as star formation in cooling flows ( fabian & nulsen 1977 ) , or by purely stellar dynamical processes , such as galactic cannibalism ( hausman & ostriker 1978 ; mcglynn & ostriker 1980 ) , accretion of tidally stripped material ( richstone 1976 ) or `` galaxy harassment '' ( moore _ et al . _ 1996 ) . for a review of bcg formation , see tremaine ( 1990 ) ; see also garijo , athanassoula , & garcia - gomez ( 1997 ) , and dubinski ( 1998 ) . the kinematics and metallicities of bcg halos provide important clues as to how they form . if they form in a dissipational process , then we might expect a low velocity dispersion , as the energy of the material from which the stars form has been radiated away . if bcgs form in a dissipative collapse , we would expect to see metallicity gradients in the halo ( though metallicity gradients are not unique to dissipative formation models ) . if the halos form out of material from other cluster galaxies via dissipationless processes , then we might expect the halos to reflect the kinematics of the cluster as a whole , with high velocity dispersion and little net rotation ; metallicity gradients would be shallower if bcgs experience significantly more mergers than other cluster galaxies , since mergers are expected to weaken metallicity gradients . if the halos are primordial , and formed in a process involving violent relaxation , then the kinematics will reflect the underlying mass distribution . while there is considerable information on the central kinematics of ellipticals , there have been few studies of the halo kinematics at very large radii from optical spectroscopy . most such studies have concentrated on normal ellipticals , and have found that the velocity dispersion profiles are either flat or decreasing with radius ( e.g. saglia _ et al . _ 1993 ; carollo _ et al . _ 1995 ; statler , smecker - hane , & cecil 1996 ) . similar results have been found in the even fewer studies of bcgs ( e.g. carter _ et al . _ 1981 , 1985 ; tonry 1984 , 1985 ; heckman _ et al . _ 1985 ; fisher , illingworth , & franx 1995 ) . to date , one cd ( ic 1101 in abell 2029 : dressler 1979 ; fisher _ et al . _ 1995 ; sembach & tonry 1996 ) , and two dumbell galaxies ( abell 3266 : carter _ et al . _ 1985 , and ic 2082 : carter _ et al . _ 1981 ) , have been found to have _ rising _ dispersion profiles . an alternative approach to the study of the kinematics of the outer halos of cd and other galaxies is the use of globular clusters and planetary nebulae as tracers . because of the geometric constraints of current multi - slit and fibre fed spectrographs , this technique can be used only in the outer halos , and because of magnitude limits can not be applied at distances beyond two or three times the distance of the virgo cluster . nevertheless for m87 ( mould _ et al . _ 1990 ; cohen & ryzhov 1997 ) and ngc 1399 ( grillmair _ et al . _ 1994 ; kissler - patig _ et al . _ 1998 ; minniti _ et al . _ 1998 ) , globular cluster velocities have shown conclusively that the m / l ratio increases outwards in the halo , and thus that there are substantial dark matter halos in both galaxies . ( 1994 ) confirm this result for ngc 1399 for a sample of planetary nebulae . interestingly , cohen & ryzhov ( 1997 ) find that the globular cluster velocity dispersion in m87 rises outwards , and in ngc 1399 the velocity dispersions of the planetary nebulae and globular clusters match smoothly onto the velocity dispersion of the fornax cluster itself . these bcgs at least seem to merge smoothly into their host clusters , and the bcg and cluster dynamics seem closely linked ( see discussion in freeman 1997 ) . the evidence from the distribution of lensed arcs in more distant clusters points to the dark matter being more centrally concentrated than the x - ray gas ( miralda - escude & babul 1995 ) . in this case the dark matter will affect the dynamics of the cd halo significantly . we have embarked on a program to obtain very deep optical spectroscopy of bcgs . here we present long - slit spectra for ngc 6166 , ngc 6173 , and ngc 6086 , obtained at the 2.5 m int in la palma in may 1996 . the properties of these galaxies and their host clusters are listed in table [ tab1 ] . ngc 6166 is a luminous cd and a classic multiple - nucleus galaxy ( though the ` nuclei ' are probably not bound to ngc 6166 ) , centrally located in a rich cluster with a large cooling flow ( @xmath9 160 m@xmath10/yr : allen & fabian 1997 ) , and a regular , symmetrical x - ray appearance ( buote & canizares 1996 ) . abell 2197 is 1.3 degrees north of abell 2199 ( gregory & thompson 1984 ) , and is a sparser cluster with an irregular x - ray morphology dominated by two main concentrations ( muriel , bohringer , & voges 1996 ) . ngc 6173 is a d galaxy found at the centre of one of these concentrations ( subclusters ) , and appears to have a significant peculiar velocity ( i.e. relative to the cluster mean velocity ) . abell 2162 is probably best described as a poor cluster or compact group ; we have not been able to find any x - ray data for this cluster . however , ngc 6086 is classified as a cd galaxy based on its surface brightness profile , and does not have a peculiar velocity . thus , our three galaxies / clusters span a wide range of properties . [ tab1 ] cccccccccc galaxy / cluster & galaxy & m@xmath11 & @xmath2 & v@xmath12 & v@xmath13 & @xmath14 & bm & r & x - ray + & type & & & ( km s@xmath0 ) & ( km s@xmath0 ) & & & + + & & & our & sample & & & & + + n6166/a2199 & cd & @xmath1522.7 & 326 & 9293 & 9063 & 823 & i & 2 & cf , reg + n6173/a2197 & ge & @xmath1522.4 & 261 & 8800 & 9134 & 550 & iii & 1 & irr + n6086/a2162 & cd & @xmath1521.8 & 304 & 9547 & 9795 & 302 & ii - iii & 0 & ? + + + ic 1101/a2029 & d & @xmath1523.5 & 359 & 23399 & 23169 & 1436 & i & 2 & cf , reg + sersic 40 - 6/a3266 & cd & & 327 & 17914 & 17802 & 1186 & i - ii & 2 & cf , reg + ic2082/as0463 & d & & 265&12051&12035&844 & i - ii & 0 & ? + [ tab1 ] ngc 6166 has previously been observed spectroscopically by tonry ( 1985 ) and fisher _ et al . _ ( 1995 ) with 2 and 3 position angles respectively ( each include the major axis ) . unfortunately , both studies only extend to 10@xmath1520 arcsec from the galaxy centre , and over this limited range both find a flat or slightly decreasing velocity dispersion profile ( we show below that this is consistent with our data , which extend much further out ) . fisher _ et al . _ find that ngc 6166 does not have significant rotation along either the major or minor axes out to 20 arcsec radius . we have only been able to find central velocity dispersions and line strengths for the other two galaxies ( e.g. burstein _ et al . _ 1987 ; mcelroy 1995 ) . table [ tab2 ] contains details of the observations . spectra are in each case taken with the slit aligned with the major axis to maximise the signal to noise ratio . the spectra include the mg and fe absorption lines , as well as [ oiii ] and [ ni ] emission lines ( for ngc 6166 only ) . we also obtained template spectra of 28 stars , with spectral type ranging from g6 to m1 , and luminosity class iii , iv and v. figure [ fig : censpec ] shows the central spectra for each of the three galaxies ; for each spectrum , the best - fitting stellar template is also shown ( see below ) , as well as the bandpass definitions for 2 g , 52 , and 53e . preliminary data reduction was done using standard figaro routines , and followed the procedures of carter , thomson and hau ( 1998 ) . this involves debiassing , flatfielding and wavelength calibrating the data . a slope in the spectrum due to detector rotation was corrected using the s - distortion correction routines in figaro . the template spectra were re - binned to zero velocity , using their published radial velocities , mainly from wilson ( 1953 ) . [ tab2 ] cc + telescope & isaac newton telescope ( 2.5 m ) + instrument & intermediate dispersion spectrograph + observing dates & 1016/06/1996 + camera & 235 mm + grating & r1200y + dispersion & 35 / mm + detector & tektronix 1024 ccd + resolution & 2 + spectral coverage & 49305730 + slit width & 3.0 arcsec + average seeing & 1.3 arcsec + slit pa & 41/140/0 deg@xmath16 + exposure time & 18.75/6.25/6.33 hrs@xmath16 + @xmath16 for ngc 6166/6173/6086 respectively . [ fig1 ] following van der marel & franx ( 1993 ) , the line - of - sight velocity distributions ( losvds ) are modelled as gaussians with small hermite deviations , parameterised as mean velocity @xmath17 , dispersion @xmath18 , and hermite moments @xmath19 of order @xmath20 . the parameters are recovered by the program kinematics written by h - w . rix ; for more details please see van der marel & franx ( 1993 ) , rix & white ( 1992 ) , and hau , carter , & balcells ( 1998 ) . the gauss - hermite moments 3 and 4 are closely related to the skewness and kurtosis of the losvd respectively , and are therefore easy to visualise . large 3 with opposite sign to the mean velocity is typical of rapidly - rotating systems , whilst for spherical two - integral models , 4 is proportional to the velocity dispersion anisotropy @xmath21 ( van der marel & franx 1993 ) . the higher moments are more difficult to interpret . however , they are useful for identifying structures in the residuals of the model fit to the data . we measure and plot gauss - hermite parameters up to 6 . the observed wavelength range is well known to give template mismatch problems ( van der marel & franx 1993 ) . in order to minimize template mismatch , an optimal stellar template is employed for the direct fitting . this is estimated as the best linear combination of the 28 template spectra which when broadened with a gaussian profile minimizes @xmath22 , the difference between the combination and the galaxy spectrum , after an initial guess of the velocity for each template and the dispersion , ( rix & white 1992 ) . this optimal template is then used for recovering the losvd for the galaxy spectrum , modelled up to 6 . this procedure is done for each of the individual spectra in each galaxy , thus reducing radial template mismatch due to abundance gradients . the galaxy nucleus is located by a gaussian fit to the light profile of the central few pixels . monte carlo experiments show that 2500 counts / pixel or higher gives satisfactory results for the direct fitting ( see appendix a ) . our procedure allows the determination of an individual optimal composite template for each data point , to ensure that this procedure does not introduce systematic effects into our profiles we repeated the analysis for ngc 6166 using the composite template for the central spectrum for all data points . there are no systematic differences in zeropoint or overall trend in any of the kinematic values derived . subjectively , we estimate that the scatter is higher , justifying our decision to use individually determined optimal templates . emission lines could affect the measurement of absorption - line indices if they fall within one of the bandpasses , and also can potentially introduce systematics in the absorption line profile analysis . significant [ oiii ] and [ ni ] emission are observed in the nuclear region @xmath23 of ngc 6166 , but not in ngc 6173 or ngc 6086 . thus , for ngc 6166 , pixels within a wavelength difference corresponding to 3@xmath18 from an emission line are excluded from the line - profile fitting . these regions are : @xmath24@xmath25 ( [ oiii ] ) , @xmath26@xmath27 ( [ ni ] ) , and @xmath28@xmath29 ( strong sky line ) . experimentation shows that excluding a few pixels has a negligible effect on the recovery of the losvd as kinematics are measured over the entire spectrum . for spectra more than 11 arcsec from the nucleus where no emission is present , only the region affected by the strong sky line is excluded , allowing fitting over the entire spectrum . another difficulty with ngc 6166 is the presence of companion nucleus b , which is at a pa of 62with respect to the central cd , and contributes significantly to the light in our slit between @xmath9 5@xmath1520 arcsec from the galaxy centre along the ne side . kinematics was not designed to recover two losvds , so we restrict our analysis to the sw side of the galaxy . the parameters recovered by the gauss - hermite analysis are presented in figures [ fig : hermplot](a ) , ( b ) , and ( c ) , for ngc 6166 , ngc 6173 , and ngc 6086 respectively . _ central velocity dispersions _ we can compare our central velocity dispersions , @xmath2 = 320 @xmath30 5 , 295 @xmath30 5 , and 325 @xmath30 8 km s@xmath0 for ngc 6166 , 6173 , and 6086 respectively , with previous determinations . ngc 6166 has been the best - studied of the three : malumuth & kirshner ( 1981 ) , tonry ( 1985 ) , burstein _ et al . _ ( 1987 ) , and fisher _ et al . _ ( 1995 ) , find @xmath2= 350 @xmath30 35 , 303 , 326 , and 295 @xmath30 6 km s@xmath0 respectively . for ngc 6173 , burstein _ et al . ( 1987 ) and mcelroy ( 1995 ) find @xmath2= 261 and 240 km s@xmath0 respectively , both values lower than ours . for ngc 6086 , burstein _ et al . _ ( 1987 ) and mcelroy ( 1995 ) find @xmath2= 304 and 325 km s@xmath0 respectively . overall , the agreement is good , and the present data are of much higher quality than any previous data . _ velocity dispersion profiles _ from figures [ fig : hermplot](b ) and ( c ) , we see that both ngc 6173 and ngc 6086 have velocity dispersion profiles that decline outwards . such behavior is found in most normal ellipticals . in contrast , figure [ fig : hermplot](a ) shows that the velocity dispersion profile of ngc 6166 _ increases _ from @xmath9 325 km s@xmath0 at the centre to @xmath9 450 km s@xmath0 at 35 arcsec along the major axis . tonry ( 1985 ) and fisher _ et al . _ ( 1995 ) both have limited dispersion profiles for ngc 6166 , and find a flat and slightly decreasing dispersion profile respectively , out to @xmath9 20 arcsec along the major axis . this is consistent with what we find , since our dispersion only starts rising significantly beyond @xmath9 20 arcsec . in appendix a we analyse possible systematic errors introduced into the kinematic parameters by noise and imperfect sky subtraction . we find that there is a bias towards low velocity dispersion when the signal - to - noise is poor , but this does not significantly affect the points plotted in figure 2(a - c ) . from figure [ fig : hermplot](a ) , ngc 6166 shows modest major - axis rotation , and a weighted least - squares fit yields a slope of 1.07 @xmath30 0.12 km s@xmath0 arcsec@xmath0 , or a rotation of @xmath9 45 km s@xmath0 at 40 arcsec . tonry ( 1985 ) found a rotation amounting to 4 km s@xmath0 arcsec@xmath0 at pas of 62and 102 , while fisher et al ( 1995 ) find no significant rotation at pas of 31 , 35and 125 . ngc 6173 and ngc 6086 both show evidence of rotation in the core , decoupled from the kinematic properties further out . in ngc 6173 ( figure [ fig : hermplot](b ) ) , a linear least - squares fit gives a velocity gradient of @xmath152.56 @xmath30 0.27 km s@xmath0 arcsec@xmath0 for @xmath31 arcsec , with no significant rotation outside this , and in ngc 6086 ( figure [ fig : hermplot](c ) ) , similar fits give a velocity gradient of 0.86 @xmath30 0.48 km s@xmath0 arcsec@xmath0 for @xmath32 arcsec and @xmath151.33 @xmath30 0.13 km s@xmath0 arcsec@xmath0 for @xmath33 arcsec . although the core rotation is only marginally significant , it does appear to be counter - rotating . ngc 6173 is a shell galaxy , and both the shells and higher core rotation could be a result of a merger or interaction . however , in both galaxies , there is no significant asymmetry in the line profiles ( i.e. changes in @xmath34 ) associated with the peculiar kinematics , as expected from classical kdcs like ic 1459 or ngc 2865 ( hau , carter , & balcells 1998 ) . also , statler ( 1991 ) proposes that kdcs might also arise from streaming in a triaxial potential , without a merger / interaction . the parameter @xmath1 measures symmetric deviations from a gaussian profile and is related to the velocity anisotropy . @xmath1 is significantly positive , with no radial dependence , in all of our galaxies . although it is not straightforward to translate @xmath1 to the velocity anisotropy parameter @xmath21 ( @xmath35 ) , positive values , at least outside the nucleus , generally indicate a bias towards radial orbits ( gerhard 1993 ; gerhard _ et al . _ 1998 , rix _ the most important result from the line - profile analysis is that there is no change in @xmath1 associated with the rising dispersion in ngc 6166 , indicating that the increase in velocity dispersion is not associated with a change in velocity anisotropy towards tangential orbits . @xmath1 has a magitude of 0.10 , 0.05 & 0.05 in ngc 6166 , 6173 & 6086 respectively . template mismatch can introduce systematics in @xmath1 by either increasing or decreasing its magnitude ; see , for example , van der marel _ ( 1994 ) , and hau , carter , & balcells ( 1998 ) . the @xmath22 fits in ngc 6173 & ngc 6086 appear to be near perfect . inspection of the residuals for ngc 6166 shows that the fits are less perfect than for the other two galaxies , but are satisfactory . we feel that template mismatch is not a major problem for all three galaxies as the optimal template is estimated from 28 template stars . in order to estimate the effect of template mismatch on the absolute value of @xmath1 ( and other kinematic parameters ) , in appendix a we derive the kinematic parameters for ngc 6166 for a range of individual templates . none of these templates fit the galaxy data as well as the optimal template . given an extreme range of templates , we find zero point differences of up to 0.1 in @xmath1 , @xmath36 and @xmath37 ; 0.2 in @xmath34 , and 50 km / s in velocity dispersion . however these differences are largely in the zero points for stars at the extreme end of the range of spectral types of the observed templates ( g6 iii and m1 iii ) . in particular low @xmath1 is found only when an m1 iii template is used , this template does not fit the data well and contributes little to the optimal composite templates . although template mismatch may in principle shift the zero - point in @xmath1 , it can not explain the significantly positive @xmath1 seen in our data . in appendix a we also analyse the effect of noise on the recovery of @xmath1 when the magnitude of the real value of @xmath1 is large . we find that there are no systematic biases unless @xmath38 , when @xmath39 is systematically underestimated at low signal to noise . thus the only possible bias in our results would be to underestimate @xmath1 in the outer regions of ngc 6166 , although there is no evidence that this has happened , we can not absolutely rule it out . van der marel & franx ( 1993 ) give formulae which relate @xmath1 to @xmath21 , but they are model - dependent . to better quantify the anisotropy parameter @xmath21 , we need to solve for the @xmath40 ratio using all of the available kinematic and photometric data . such techniques have been developed by rix _ ( 1997 ) , using schwarzchild s method of populating orbits , and by dehnen ( 1995 ) for the axisymmetric case in order to model m32 . in summary , the rising velocity dispersion profile in ngc 6166 is not due to a change in velocity anisotropy , but rather , reflects an increasing @xmath40 ratio . this indicates that ngc 6166 possesses a massive dark halo . significant [ oiii ] and [ ni ] emission is observed in the centre of ngc 6166 . in this section we present the line - of - sight velocity distribution of the ionized gas inferred by the profile of the [ oiii ] and [ ni ] emission lines , obtained directly from the residual map of the gauss - hermite analysis , described in the previous section . this method is very successful and is more objective than fitting the baseline by hand . the results are plotted in figure [ fig : gaskin ] , with the stellar rotation curve from figure [ fig : hermplot](a ) over - plotted . we see that the [ oiii ] and [ ni ] emission is very concentrated to the galaxy centre , with spatial fwhm of 1.9 and 3.0 arcsec respectively . the spatial fwhm of template stars is about 2.3 arcsec , thus the [ oiii ] emission region is unresolved but that of the [ ni ] may be marginally resolved . at the galaxy centre , the [ oiii ] and [ ni ] have velocity dispersions @xmath18 of @xmath41 and @xmath42 km s@xmath0 respectively , which is very similar to the central stellar velocity dispersion ( figure 2a ) . in this section the absorption line indices are presented for the three galaxies . mg@xmath7 , 52 , and 53e are extracted from individual , de - redshifted spectra following the definitions and recipes of faber ( 1985 ) , and adopting the passband definitions of worthey ( 1994 ) ; see appendix b for more details . we first discuss the local mg@xmath43fe@xmath44 relations for each galaxy , then present the mg@xmath7 radial profiles . we discuss the significance of these results in our conclusions . as discussed in appendix b , there is some uncertainty regarding the zero - point of our measured indices in the lick / ids scale , due to the fact that only one star with lick indices ( hr 5227 ) has been observed , and also that the grating settings for the galaxy and the template star observations were different . however recent observations by trager _ ( 1998 ) for the nucleii of ngc 6166 and ngc 6086 do agree within their admittedly rather large error bars . we also find that our 2 g measurements agree well with those measured by cardiel _ ( 1997 ) , if both are not zero - point corrected ( figure [ fig : uscardiel ] ) . as most of our discussion is not based on the absolute values of the indices , throughout this section we present line indices without zero - point corrections . in figure [ fig : fevsmg2 ] , we plot @xmath45fe@xmath44 [ @xmath46 ( 52 + 53e)/2 ) ] against 2 g for the three galaxies . all three galaxies appear to have [ mg / fe ] ratios higher than the model values with solar [ mg / fe ] abundance , but there are uncertainties in our zero - points . notice further from figure [ fig : fevsmg2 ] that ngc 6086 and ngc 6166 are more ` enhanced ' than ngc 6173 . this is interesting , given that the first two galaxies are centrally - located cds , while ngc 6173 is best described as a ge in a subcluster in abell 2197 . similarly , m87 is offset from the non - bcgs in the data of davies , sadler , & peletier . hau ( 1998 ) has pointed out that in galaxies of high velocity dispersion , a positive value of @xmath1 , such as might result from radial anisotropy , leads to more absorption being lost from the passband that the fe indices are measured from , and thus to an artificially high value of [ mg / fe ] . this systematic error might account for as much as a third of the offset between ngc 6166 and ngc 6173 . however we still find that our galaxies with cd morphology have higher [ mg / fe ] compared with ngc 6173 , which is the only normal bright elliptical in our sample . if further observations confirm that ngc 6173 is typical ( despite uncertainty in the relative zero - points , the data of davies _ et al . _ ( 1993 ) support this ) , then there is a difference between cds and normal bright ellipticals . bender ( 1996 ) uses the offset of [ mg / fe ] to higher values in ellipticals when compared with spirals as an argument against the idea that ellipticals form from the mergers of objects similar to _ present - day _ spirals ; any such argument is clearly stronger for cds . bender ( 1996 ) and davies ( 1996 ) feel that the most plausible explanation for the non - solar [ mg / fe ] ratios in ellipticals is a short burst of star formation , @xmath45 1@xmath152 gyr . this would produce a population enriched in mg from rapid type ii supernovae from massive stars , while few stars would be formed from fe - enriched gas formed in longer - lived type ia supernovae . see section 6 for further discussion on these topics . we first compare our mg@xmath7 profile for ngc 6166 with that of cardiel _ et al . _ ( 1997 ) in figure [ fig : uscardiel ] . we have only plotted the sw side of the major axis because of contamination from the secondary nucleus on the ne side . neither set of data has had any zeropoint corrections applied ( see appendix b ) . out to @xmath9 15 arcsec , the agreement is very good . beyond 15 arcsec , our 2 g is significantly higher than that of cardiel _ et al . _ the source of this discrepancy is unclear , however our signal - to - noise is higher than that of cardiel _ et al . _ their three outermost points have 2 g values @xmath45 0.2 , lower than any of the ellipticals studied by davies _ these extreme values seem unlikely , and we are confident of the reliability of our 2 g profiles . the sensitivity of our line strength measurements to sky subtraction errors needs to be considered . the sky spectrum that we subtract is averaged over many spatial pixels and is relatively low noise . the only feature in the sky spectrum are the night sky emission line at 5577 and ( much weaker ) 5199 , apart from this the sky is well represented by a constant continuum . at these redshifts the [ ni ] line at 5199 does not fall in the 2 g band or its continuum bands , and so can be neglected . therefore the sky in the region of all of the line and continuum bandpasses can be represented by a constant , and the error in this constant can be determined from the maximum residuals seen in the [ oi ] line at 5577 in the sky subtracted spectra , and the ratio of the peak intensity in this line to the continuum in the sky spectra . inspection of the sky line residuals in the outer regions of our profile show that sky subtraction errors do not exceed 1% of the signal level , thus systematic errors in our 2 g values will not exceed 0.01 mag . in figure [ fig : mg2prof ] , we present the major axis profiles of 2 g , for ngc 6166 , 6173 , and 6086 . outside 35 arcsec , the 2 g profiles are well - fit by power - laws of slope ( d(2g)/d(log @xmath47 ) ) @xmath150.080 , @xmath150.059 , and @xmath150.073 for ngc 6166 , 6086 and 6173 respectively ( inside 3 arcsec , seeing effects are likely to be important ; the flattening of 2 g in ngc 6166 within a larger radius of @xmath9 5 arcsec may be due to star formation inside the emission - line region in this galaxy ) . these slopes are fairly similar to each other , and are also similar to the average 2 g slope of @xmath150.058 found by cardiel _ ( 1997 ) for eight bcgs without cooling flows or optical emission lines . gorgas , efstathiou , & aragon - salamanca ( 1990 ) also measured shallower gradients for two bcgs ( @xmath150.051 and @xmath150.021 for the cds in 0559 - 40 and pks 2354 - 35 ) . however , examination of their figure 9 shows that these gradients are based on only three points , and the outermost point in each case has large uncertainty . cardiel _ et al . _ ( 1997 ) find that the 2 g slopes are shallower in bcgs with optical line - emission , inside the emission - line region ; our data for ngc 6166 are consistent with this . _ attribute this flattening to star formation from cooling flow gas near the galaxy centre . this makes comparison of bcg 2 g data difficult , unless one is careful to compare slopes outside emission - line regions in all cases . we can also compare our 2 g slopes with those of non - bcg ellipticals . couture & hardy ( 1988 ) found a mean 2 g gradient of @xmath150.053 @xmath30 0.015 for six early - type galaxies ; davies , sadler , & peletier ( 1993 ) find a mean slope of @xmath150.059 @xmath30 0.022 for 13 normal ellipticals ; gorgas _ et al . _ ( 1990 ) obtain a mean slope of @xmath150.058 @xmath30 0.027 for 16 ellipticals ; davidge ( 1992 ) found a steeper gradient of @xmath150.081 @xmath30 0.01 for 11 ellipticals . gonzalez & gorgas ( 1998 ; see also gonzalez & gorgas 1997 ) find a mean 2 g gradient of @xmath150.055 @xmath30 0.025 for 109 early - type galaxies . thus , our three bcgs have slopes consistent with those found in non - bcg ellipticals . it seems likely that these bcgs have not experienced significantly more mergers than normal ellipticals , as mergers are expected to dilute abundance gradients ( white 1980 ) . _ our main results are : _ * ( 1 ) : * we find small rotation along the major axes of ngc 6166 , ngc 6173 , and n6086 , with v@xmath48 @xmath49 50 km s@xmath0 at the outermost observed major axis radii ( 3040 arcsec ) in the three galaxies ; the corresponding v/@xmath18 ranges from 0.020.18 . this small rotation is consistent with the nearly complete lack of rotation found near the centres of a sample of 13 bcgs by fisher _ this is consistent with merger models of bcg formation , but data at larger radii are urgently needed , since numerical simulations predict that mergers should contain significant amounts of angular momentum beyond 2@xmath153 r@xmath50 ( e.g. hernquist 1993 ) . ngc 6173 and ngc 6086 have larger velocity gradients near the galaxy centres , and ngc 6086 appears to have a counter - rotating core . these cores may arise from a previous merger / interaction ( supported by the shells seen in ngc 6173 ) , or may be due to streaming in a triaxial potential ; further observations at other position angles will be needed to decide between these two alternatives ( statler 1991 ) . * ( 2 ) : * ngc 6086 and ngc 6173 have slowly declining velocity dispersion profiles . in contrast , ngc 6166 has a rising velocity dispersion : @xmath18 increases from @xmath9 325 km s@xmath0 at the centre to @xmath9 440 km s@xmath0 at 35 . * ( 3 ) : * the gauss - hermite 4 moment is positive and roughly constant at 0.05 at all radii along the major axis of ngc 6173 and ngc 6086 , and 0.1 along the major axis of ngc 6166 . template mismatch is unlikely to account for all of this offset , and we conclude that there is possibly radial anisotropy present . we will confirm and quantify this suggestion with detailed modelling in a later paper . * ( 4 ) : * we find [ oiii ] and [ ni ] emission concentrated within 5 arcseconds ( 3h@xmath0 kpc ) of the centre of ngc 6166 , with a central velocity dispersion of @xmath9 350 km s@xmath0 , similar to the central stellar velocity dispersion . the extent of the [ oiii ] emission region is unresolved , but that of the [ ni ] may be marginally resolved . * ( 5 ) : * we find that [ mg / fe ] is larger in ngc 6166 and ngc 6086 than in ngc 6173 . although positive @xmath1 can lead to systematic errors in [ mg / fe ] , the magnitude of the offset appears too large to be explained by such an error . the difference in [ mg / fe ] between galaxies with cd morphology and normal bright ellipticals , if confirmed by further observations , would present a difficulty for models in which the cds were formed from mergers of normal galaxies of any type . * ( 6 ) : * the 2 g gradients outside 35 arcsec range from @xmath150.06 to @xmath150.08 for the three galaxies . these slopes are consistent with those found for bcgs without cooling flows or emission lines by cardiel _ ( 1997 ) , and for non - bcg ellipticals . more data are needed to definitively compare bcgs with non - bcg ellipticals in this regard . we defer to a forthcoming paper a detailed dynamical modelling of these three galaxies , but we note that ( 2 ) and ( 3 ) together imply that the m / l ratio increases outwards in the ngc 6166 halo , and likely in the other two galaxies as well . further support for a dm halo in ngc 6166 comes from x - ray data , e.g. buote & canizares ( 1996 ) . do these results favour one bcg formation model over others ? there are five main scenarios for bcg formation : * ( i ) * deposition of cooling flow gas from the icm ( e.g. fabian 1994 ) ; * ( ii ) * cannibalism of other cluster galaxies ( ostriker & tremaine 1975 ) ; * ( iii ) * tidal stripping of other cluster galaxies , either by the cluster tidal field or by the bcg itself ( richstone 1976 ; moore _ et al . _ 1996 ) ; * ( iv ) * earlier origin in smaller groups / subclusters , which later merged to form the present - day cluster ( merritt 1984 , 1985 ; tremaine 1990 ) ; and * ( v ) * dissipative collapse models ( larson 1975 ; carlberg 1984a , b ) . these scenarios are not necessarily mutually exclusive ; for instance , many authors have suggested that mergers / cannibalism create the central body of the cd , while tidal stripping accounts for their extended halos . dubinski ( 1998 ) has run an impressive simulation of cd formation , using a large cosmological n - body simulation of cluster collapse with hierarchical merging . his ` cluster ' has a mass of 1.0 @xmath51 10@xmath52 m@xmath10 and a line - of - sight velocity dispersion of 550 km s@xmath0 , and he notes that it would be classified as a poor cluster or large group by observers . he finds that the final central giant elliptical has a small rotation of @xmath9 50 km s@xmath0 , is radially anisotropic in the outer regions , is aligned in position angle with the ` cluster ' galaxy distribution , and has a slowly declining major - axis velocity dispersion profile . these properties agree with those observed in many bcgs , including our sample described in this paper . all three of our galaxies have little or no rotation , and have significantly positive @xmath1 values which generally indicate radial anisotropy . ngc 6086 and ngc 6173 have velocity dispersion profiles similar to dubinski s remnant , and ngc 6173 and ngc 6166 are aligned ( to within 20 degrees ) of their host cluster / subcluster galaxy distributions and x - ray isophotes ( carter & metcalfe 1980 ; dixon , godwin , & peach 1989 ; muriel , bohringer , & voges 1996 ; buote & canizares 1996 ) . however , more such simulations are needed to see if , among other things , remnants with increasing dispersion profiles and extended halos can be created . garijo , athanassoula , & garcia - gomez ( 1997 ) have run a larger set of simulations of cd formation , with a variety of initial conditions . some of the simulations produce objects with extended halos and radial anisotropy , and all of the final galaxies have declining velocity dispersion profiles . note , however , that these simulations were run with all of the mass initially bound to galaxies , probably not physically realistic . the radial anisotropy found by both authors is a result of the ` cosmological ' mergers that are being modelled , where the initial filamentary structure means that merging galaxies enter the central object on mainly radial orbits . further constraints can be placed on formation models if one considers the stellar abundance gradients . our three galaxies all show modest gradients , comparable with normal bright ellipticals ( e.g. davies , sadler , & peletier 1993 ) . these gradients are not as steep as predicted by simple dissipational collapse models ( e.g. larson 1975 ) , however they are steeper than we would expect from hierarchical dissipationless merger models , which predict no gradients . it is not clear what the net effect of gaseous mergers will be , since on the one hand mergers are expected to dilute existing abundance gradients ( white 1980 ) , while on the other hand central star formation induced by the merger could augment any gradient ( see for instance mihos & hernquist 1994 ) . in any case the gradients point to a formation history not too different from that of normal bright ellipticals . the [ mg / fe ] ratio in ngc 6166 and ngc 6086 , and the difference between these two bcgs and ngc 6173 , provide important new clues as to the differences between cds and normal bright ellipticals . if this difference is confirmed by further observations , this is a second property , alongside the globular cluster specific frequency , in which ellipticals differ from spirals and in which cds are even more extreme . fisher _ et al . _ ( 1995 ) and bender ( 1996 ) find that [ mg / fe ] is correlated with velocity dispersion , so perhaps the large [ mg / fe ] offsets seen in our bcgs are simply the continuation of this correlation to the largest measured velocity dispersions . the trend of increasing [ mg / fe ] we see in our three bcgs ( figure 4 ) does agree with their relative velocity dispersions . a dissipative process is most likely to give rise to these differences , although mergers as a trigger for such a process can not be ruled out ; for instance zepf and ashman ( 1993 ) interpret the bimodal colour ( and hence metallicity ) distributions for globular clusters in some elliptical galaxies , including the bcgs m87 and ngc 1399 , as evidence for a second , merger induced , epoch of cluster formation . if we had data for ngc 6166 and ngc 6173 alone , it would be tempting to draw a clear distinction between the two galaxies . ngc 6166 has a rising velocity dispersion , and enhanced [ mg / fe ] , a high globular cluster specific frequency ( bridges _ et al . _ 1996 ) and an x ray cooling flow . ngc 6173 has none of these things , yet is a galaxy with a similar luminosity and surface brightness profile ( carter 1977 ) . ngc 6086 appears to have a mixture of properties : it has a high [ mg / fe ] , yet does not have a rising dispersion profile . it is important to establish which of these properties depend upon the environment of a galaxy , and similar observations of galaxies with cd morphology in low density environments ( such as ngc 4839 ; oemler 1976 ) are required . it is not clear how to interpret the rising velocity dispersion profile that we observe in ngc 6166 . one would expect a rising dispersion profile under any model of bcg formation , since under simple energy arguments the bcg dispersion profile should eventually join that of the cluster as a whole ( as is the case for ngc 1399/fornax ; see section 1 ) . however , we note that the three bcgs with increasing dispersions , and where x ray data for the cluster exist ( ngc 6166 , ic 1101 , abell 3266 ) are all in rich clusters with smooth x - 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[ fig : n6166monte ] , where the left , centre and right panels correspond to @xmath64 400 , 450 & @xmath57 respectively . for @xmath65 , the mean velocity @xmath17 can be determined to within @xmath66 at distances less than @xmath67 from the nucleus . the scatter increases to @xmath68 at @xmath69 , @xmath70 at @xmath71 , and @xmath72 at @xmath73 . in addition beyond @xmath61 there is a small systematic offset of @xmath74 towards higher velocity . for broader losvds , the behavior is similar , except that the recovering accuracy is lower . if the velocity dispersion is @xmath75 , it can be determined to within @xmath76 at distances less than @xmath67 from the nucleus . its scatter increases with distance . in addition , as the contribution of the sky increases with decreasing @xmath77 , the recovered value is systematically lower . the offset is about @xmath78 at @xmath69 , @xmath79 at @xmath71 , and @xmath80 at @xmath73 . the behavior is similar if the losvd is broader , except that the magnitude of the systematic offset is larger . at the extreme , the recovered value for a gaussian losvd with @xmath81 at @xmath73 is @xmath82 lower . in general , the gauss - hermite moments 3 and 4 can be recovered reasonably well if @xmath83 , to within @xmath84 at distances up to @xmath61 . their scatter increases with @xmath18 . no obvious systematics are introduced in 3 even at the outermost points , or when @xmath18 is increased to @xmath57 . a small positive offset of order @xmath85 is introduced in 4 for data points beyond @xmath61 , and this tends to increase with @xmath18 . the gauss - hermite moments 5 and 6 can be recovered well to within 0.01 in the inner @xmath59 , and to @xmath86 at @xmath87 . again , the scatter increases with @xmath18 . in addition a significantly positive offset of @xmath88 is introduced to the @xmath89 at the outer points if sigma is increased beyond @xmath90 . the conclusions from this exercise are that at distances beyond @xmath91 , where summation of more than 4 spectra are required , systematics start to creep in as the sky contribution increases . at such distances , the recovered velocity dispersions are systematically lower . this shows that we are probably unable to measure losvds broader than @xmath75 beyond these distances , and the low ( @xmath92 ) velocity dispersions beyond @xmath53 actually measured for ngc 6166 are entirely consistent with expectation . for these reasons , we measure @xmath17 up to @xmath63 , and @xmath18 , 3 , 4 , h5 & h6 up to @xmath93 from the nucleus . for ngc 6173 , losvds with @xmath94 and @xmath95 , 250 , & @xmath96 are modelled . the artificial spectra are scaled to continuum levels of 17000 , 3600 & 2000 counts per pixel , corresponding to distances of 0 , 6 , & @xmath97 from the nucleus respectively . beyond @xmath98 summation is required to achieve 2000 counts per pixel . the number of spectra to be summed is 4 , 8 & 16 for distances of approximately @xmath99 , @xmath100 and @xmath101 from the nucleus respectively . the results of the experiments are plotted in fig . [ fig : n6173monte ] , where the left , centre and right panels correspond to @xmath64200 , 250 & @xmath96 respectively . for @xmath102 , the mean velocity @xmath17 can be determined to within @xmath103 at distances less than @xmath104 from the nucleus . the scatter increases to @xmath68 at @xmath105 and @xmath106 at @xmath107 . in addition beyond @xmath100 there is a small systematic offset of @xmath108 . for broader losvds , the behavior is similar , except that the magnitude of the scatter and the systematic offset is slightly higher . if the velocity dispersion is @xmath109 , it can be determined to within @xmath110 at distances less than @xmath104 from the nucleus . its scatter increases with distance . in addition , as the contribution of the sky increases with decreasing @xmath77 , the recovered value is systematically lower . the offset is about @xmath111 at @xmath112 , @xmath113 at @xmath114 , and @xmath115 at @xmath107 . the magnitude of the systematic offset is larger if the losvd is broader , and smaller if the losvd is narrower . at the extreme , the recovered value for a gaussian losvd with @xmath116 at @xmath107 is @xmath117 lower . in general , the gauss - hermite moments 3 and 4 can be recovered reasonably well if the @xmath77 is high . if @xmath118 , they can be determined to within @xmath86 at distances up to @xmath98 . their scatter increases with @xmath18 and sky contribution . a small positive offset in 3 is introduced with increasing sky contribution . no significant systematics are introduced in 4 , except for the outermost points at @xmath119 , where 4 tends to be lower . the gauss - hermite moments 5 and 6 can be recovered well only near the galaxy centre ( @xmath120 ) . the scatter increases with @xmath18 and increasing sky contribution . if @xmath121 , there are no obvious systematics in 5 with increasing sky contribution . however , at distances larger than @xmath99 5 is scattered upwards if @xmath102 , and downwards if @xmath116 . 6 tends to be scattered downwards at distances larger than @xmath99 for all @xmath18 . the tests for ngc 6173 should also be applicable to ngc 6086 , as both galaxies have similar velocity dispersions , and are at similar distances . for ngc 6086 , the continuum levels are 14600 and 2000 counts per pixel at @xmath122 and @xmath67 . beyond @xmath67 summation is required . the number of spectra to be summed is 4 , 6 & 15 for distances of approximately @xmath123 , @xmath59 and @xmath100 from the nucleus respectively . the conclusions from this exercise are that at distances beyond @xmath124 , where summation of more than 4 spectra are required , systematics start to creep in as the sky contribution increases . at such distances , the recovered velocity dispersions are systematically lower , but the problem is not as severe as in ngc 6166 . this shows that the fall in @xmath18 beyond @xmath54 is probably more gradual in reality . the significance of the monte carlo tests is that the @xmath18 in ngc 6173 is consistent with being flat or falling beyond @xmath54 , but inconsistent with rising outwards . we measure @xmath17 , @xmath18 & 3 up to @xmath53 , 4 up to @xmath125 , and 5 & 6 up to @xmath99 from the nucleus . the tests described above show that the systematic errors on the recovery of @xmath1 caused by noise are small in the case of @xmath1 = 0 . it is important also to ask what other values of @xmath1 can be recovered by our techniques . for ngc 6166 we model losvds with @xmath126 ans @xmath65 , and @xmath127 - 0.10 , -0.05 , 0.00 , + 0.05 , + 0.10 and + 0.15 respectively , and the results are plotted in fig . [ fig : h4test ] . the systematic bias on @xmath18 at low @xmath77 is marginally worse for positive @xmath1 ( second panel up in each column ) , and the main result of this exercise is that there is a systematic underestimation of @xmath39 for @xmath128 . our measurement of high @xmath1 in ngc 6166 could possibly still be a systematic underestimate . our kinematic parameters are all determined using optimal best fit templates constructed from 28 stellar spectra . to demonstrate that this procedure does not itself introduce systematic effects into the kinematic parameters , here we repeat the analysis for ngc 6166 using individual template stars . moreover to demonstrate the effect of template choice on the derived parameters we have carried out this exercise with templates with a wide range of spectral type . in fig . [ fig : templatetest ] we plot the kinematic parameters for ngc 6166 derived from the five individual templates : hr7555 ( g6iii , open squares ) ; hr4864 ( g7v , filled triangles ) ; hr6674 ( k0iii , crosses ) ; hr4676 ( k4iii , open circles ) and hr8057 ( m1iii , thin rectangles ) . the trends in the derived parameters are identical to those in fig . [ fig : hermplot ] , although there are differences in the zero points which depend upon the template choice . the difference in the velocity zero points is not real , it is because this procedure determines the velocity relative to the first star in the list of templates , if there is only one it depends upon the radial velocity of that star . for @xmath18 and all of the hermite terms the m1iii star hr8057 gives a very different zero point , the dispersions are so much lower that many of them are off the bottom of the dispersion panel in fig . [ fig : templatetest ] . however this star is a very poor fit to the galaxy spectra and does not contribute much to the optimal composite templates . for the other templates , @xmath1 is tightly defined around + 0.10 , suggesting that template mismatch errors do not contribute to our large positive values of @xmath1 . on the other hand there are large zero point differences in @xmath34 between the g and k star templates , to get reliable values of @xmath34 it is important that the templates used fit well . the velocity dispersion zero points differ with a total spread of 50 @xmath129 ( again neglecting hr8057 ) , so the uncertainty of the zero point of the velocity dispersion profile due to template mismatch might be estimated at 25 @xmath129 . in this appendix we describe the procedures for obtaining absorption line indices in the lick system . line indices are extracted from individual , de - redshifted spectra following the procedures outlined in worthey & ottaviani ( 1997 ) , adopting the passband definitions of worthey _ et al . _ the blue - ward continuum bandpass for 2 g is redefined to be 4931.625@xmath154957.625 instead of 4895.125@xmath154957.625 , because the spectra of the template stars only start from about 4931 . the difference between the mg index 2g@xmath130 under this alternative definition , and the lick 2 g index is tiny ( see below ) . the molecular band 2g@xmath130 is measured in pseudo magnitudes , whilst the atomic absorption line indices 52 and 53e are measured as equivalent widths in angstroms . lccc feature & type & central bandpass & continuum bandpasses + & & ( ) & ( ) + + 2g@xmath130&molecular band & 5154.1255196.625 & 4931.6254957.625 , 5301.1255366.125 + 2 g & molecular band & 5154.1255196.625 & 4895.1254957.625 , 5301.1255366.125 + 52 & atomic absorption line & 5245.6505285.650 & 5233.1505248.150 , 5285.6505318.150 + 53e & atomic absorption line & 5312.1255352.125 & 5304.6255315.875 , 5353.3755363.375 + first , our spectra are broadened to a resolution of @xmath131 fwhm , in common with the lick / ids resolution in this part of the spectrum . in ngc 6166 , there is significant [ ni ] emission at @xmath132 . as the [ ni ] emission line is halfway inside the 2 g passband , its flux can artificially lower the 2 g by @xmath133 at the nucleus . therefore the flux contribution between rest wavelengths 5183.125 @xmath134 and 5196.526 @xmath134 is measured from the residuals of the kinematic fitting and subtracted off the flux in the 2 g passband . the indices are then corrected to a zero dispersion system by applying correction factors . these are estimated by broadening the optimal stellar template of ngc 6166 to different velocity dispersions , and are summarized in table [ tab : n6166dispcor ] . the correction for 2 g is insignificant even at large dispersions , whilst 53e requires a rather large correction . these corrections are not sensitive to the choice of template used to determine these corrections , in the worst case ( 53e at high velocity dispersion ) the total range in the factor determined using a range of templates is 0.034 , implying an error in the factor as determined from the best fit template of at most 0.01 . cccc @xmath18 & 2 g & 52 & 53e + ( @xmath129 ) & & & + + 160 & 0.998 & 0.938 & 0.895 + 180 & 0.998 & 0.924 & 0.868 + 200 & 0.997 & 0.909 & 0.842 + 220 & 0.997 & 0.895 & 0.815 + 240 & 0.997 & 0.879 & 0.785 + 260 & 0.996 & 0.864 & 0.756 + 280 & 0.996 & 0.849 & 0.727 + 300 & 0.995 & 0.834 & 0.697 + 320 & 0.995 & 0.819 & 0.668 + 340 & 0.994 & 0.804 & 0.640 + 360 & 0.993 & 0.788 & 0.612 + 380 & 0.991 & 0.773 & 0.586 + 400 & 0.990 & 0.758 & 0.560 + 420 & 0.988 & 0.742 & 0.536 + 440 & 0.987 & 0.726 & 0.512 + to convert 2g@xmath130 to 2 g , a scaling relationship is required . this is obtained by comparing the 2 g indices for 9 template stars measured under the lick and our alternative definition . the template stars were observed on the aat in march 1996 using the rgo spectrograph , and their spectra span the wavelength range 4850 5610 @xmath134 , and are broadened to @xmath131 fwhm resolution . the fact that these spectra are taken with a completely different instrument and telescope does not pose a problem , as the only variable is the change in bandpass definition . the 2 g indices under worthey s definition are plotted against those under the alternative definition in fig . [ fig : mglickvsalt ] , and the following relationship is obtained by a linear least - squares fit : @xmath135 because the changes introduced by redefining the continuum bandpass are only 0.5% for 2 g , they are unlikely to be a major source of error . because our spectra are not flux calibrated , there might be a zero point difference between our measured indices and the lick values , due to the curvature of the response function . of our standard stars , only one , hr 5277 has been measured on the lick system . worthey ( private communication ) ascribes low weight to the observations of this particular star , and in any case using observations of a standard star to determine the zero points for galaxies is subject to systematic error , due to the substantial redshifts of our galaxies , and the corresponding difference in the grating settings for the observations . thus we are unable to tie our measurements to the lick system directly . on the other hand cardiel _ ( 1997 ) have measurements which we , and they , believe to be on the lick system . figure [ fig : uscardiel ] demonstrates good agreement between the 2 g values in the region in which the data of cardiel _ et al . _ are reliable , and thus we believe that the zero point offset between our values and the lick system is small , at least in 2 g . ( 1998 ) publish lick indices for a 1.4 x 4 arcsecond aperture centred on the nucleii of ngc 6086 and ngc 6166 . for ngc 6086 they find @xmath136 ; @xmath137 ; @xmath138 . for ngc 6166 they find @xmath139 ; @xmath140 ; @xmath141 . these are consistent with our results , although the error bars on their iron line equivalent widths are disappointingly large . although some uncertainty remains as to the zero point of the iron line indices , it seems that our indices are within 0.01 magnitudes of the lick system in the case of 2 g , and within 0.3 angstrom in the case of the combined iron line index .

we present kinematic parameters and absorption line strengths for three brightest cluster galaxies , ngc 6166 , ngc 6173 and ngc 6086 . we find that ngc 6166 has a velocity dispersion profile which rises beyond 20 arcsec from the nucleus , with a halo velocity dispersion in excess of 400 km s@xmath0 . all three galaxies show a positive and constant @xmath1 hermite moment . the rising velocity dispersion profile in ngc 6166 thus indicates an increasing mass - to - light ratio . rotation is low in all three galaxies , and ngc 6173 and ngc 6086 show possible kinematically decoupled cores . all three galaxies have 2 g gradients similar to those found in normal bright ellipticals , which are not steep enough to support simple dissipative collapse models , but these could be accompanied by dissipationless mergers which would tend to dilute the abundance gradients . the [ mg / fe ] ratios in ngc 6166 and ngc 6086 are higher than that in ngc 6173 , and if ngc 6173 is typical of normal bright ellipticals , this suggests that cds can not form from late mergers of normal galaxies . # 1#1 0@xmath2 52fe@xmath3 53efe@xmath4 54efe@xmath5 50fe@xmath6 2gmg@xmath7 2mg@xmath7 1mg@xmath8 galaxies : normal galaxies : elliptical and lenticular , cd

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Proceed to summarize the following text: since the discovery of q0957 + 561 @xcite , about 80 gravitationally lensed quasars have been discovered @xcite . lensed quasars are not only intriguing phenomena but also have become indispensable astronomical tools , including probes of the cosmological parameters and the structure of galaxies ( e.g. , * ? ? ? * ; * ? ? ? in particular , the abundance of gravitational lenses in a well - defined source sample can be used to constrain dark energy ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . unfortunately , the largest existing survey , the cosmic lens all - sky survey ( class ; * ? ? ? * ; * ? ? ? * ) , contains only 22 lensed radio sources ( with a well - defined statistical sample of 13 lenses ) discovered from @xmath6 radio sources , which is still insufficient to place tight constraints on dark energy models . the sloan digital sky survey ( sdss ; * ? ? ? * ) should lead to a significantly larger lens sample for attacking the dark energy problem . sdss is expected to identify @xmath7 quasars spectroscopically ( e.g. , * ? ? ? * ) and @xmath8 quasars photometrically ( e.g. , * ? ? ? * ) , which should lead to a sample of over @xmath9 lensed quasars given a standard lensing probability of @xmath10 @xcite . indeed , 12 new lensed quasars have been discovered from the sdss quasars so far @xcite , in addition to recovering 5 previously known lensed quasars @xcite . we can presently construct a well - defined statistical sample of 16 lensed sdss quasars , but there remain many promising sdss lensed quasar candidates for which the necessary follow - up observations are incomplete . in this paper , we report on the discovery of two more gravitationally lensed quasars , sdss j080623.70 + 200631.9 ( sdss j0806 + 2006 ) and sdss j135306.35 + 113804.7 ( sdss j1353 + 1138 ) . we present imaging and spectroscopic follow - up observations with the university of hawaii 2.2-meter ( uh88 ) telescope , the w. m. keck observatory s keck i and ii telescopes , and the magellan consortium s landon clay 6.5-m ( lc6.5 m ) telescope . we model the systems to check that their geometries are consistent with the lensing hypothesis . the structure of this paper is as follows . we describe our lens candidate selections from the sdss data in [ sec : sdss ] . the follow - up observations and mass modeling of sdss j0806 + 2006 and sdss j1353 + 1138 are presented in [ sec:0806 ] and [ sec:1353 ] , respectively . we finally present a summary and give a conclusion in [ sec : conc ] . throughout the paper we assume a cosmological model with the matter density @xmath11 , cosmological constant @xmath12 , and hubble constant @xmath13 @xcite . the sdss is conducting a photometric and spectroscopic survey of 10,000 square degrees of the sky approximately centered on the north galactic cap using the dedicated wide - field ( @xmath14 field of view ) 2.5-m telescope @xcite at the apache point observatory in new mexico , usa . photometric observations @xcite are made in five optical filters @xcite . after automated data processing by the photometric pipeline @xcite , quasar and galaxy candidates are selected by the spectroscopic target selection algorithms @xcite . spectra of these candidates are obtained according to the tiling algorithm of @xcite using a multi - fiber spectrograph covering 3800 to 9200 at a resolution of r@xmath15 . the data are very homogeneous , with an astrometric accuracy better than about @xmath16 rms per coordinate @xcite and photometric zeropoint errors less than about 0.03 magnitude over the entire survey area @xcite . the data have been released continuously to the public @xcite . we selected the two objects , sdss j0806 + 2006 and sdss j1353 + 1138 , as lensed quasar candidates from the sdss spectroscopic quasar sample with the same algorithm used for the discovery of most sdss lensed quasars ( e.g. , * ? ? ? * ) . specifically , the algorithm uses the sdss image parameters dev_l , exp_l and star_l for the likelihood that a source can be modeled as a de vaucouleurs profile , an exponential disk or a point source to quantify the structure of each quasar . lensed quasars should be modeled poorly by all three profiles , so quasars with small values for all three likelihoods are good lens candidates ( see * ? ? ? * for more details ) . the sdss @xmath17-band images of sdss j0806 + 2006 and sdss j1353 + 1138 are shown in figure [ fig : field08061353 ] . both systems are clearly resolved , making them excellent lens candidates since their spectra are clearly those of @xmath18 quasars . for sdss j0806 + 2006 , the total magnitudes ( within @xmath19 aperture radius ) of the system are @xmath20 , @xmath21 , @xmath22 , @xmath23 , and @xmath24 , in @xmath25 , @xmath26 , @xmath27 , @xmath17 , and @xmath28 , respectively . for sdss j1353 + 1138 , the total magnitudes ( @xmath29 , within @xmath30 aperture radius ) are @xmath31 , @xmath32 , @xmath33 , @xmath34 , and @xmath35 , respectively . the redshifts of sdss j0806 + 2006 and sdss j1353 + 1138 measured from their sdss spectra are @xmath36 and @xmath37 , respectively . from the sdss spectra , with a fiber aperture of @xmath38 , we know that the components of the candidates can not have greatly dissimilar spectra . however , we can not resolve the spectra of the individual components . thus , while the sdss data are sufficient to identify these objects as lens candidates , additional observations are needed to confirm them as lensed quasars . deeper and higher resolution images are needed to confirm the existence of multiple quasar images and to search for the lens galaxies , and spatially resolved spectra are needed to confirm that the quasar images have identical redshifts . we present this evidence for sdss j0806 + 2006 in 3 and for sdss j1353 + 1138 in 4 . we obtained @xmath39 , @xmath40 and @xmath41-band images of sdss j0806 + 2006 using the 8k mosaic ccd camera at the uh88 telescope , on 2004 december 16 . the pixel scale of the 8k mosaic ccd camera is 0235 @xmath42 . the exposure time was 360 sec for each band . the typical seeing in the exposures was @xmath43 , about a half of the typical seeing size of the sdss ( @xmath44 ) . bias - subtracted and flat - field corrected images are shown in the upper panels of figure [ fig : uh88_0806 ] . we also obtained an @xmath45-band image using the quick infrared camera ( quirc ) of the uh88 telescope on 2005 february 21 and a @xmath46-band image using the near infrared camera ( nirc ; * ? ? ? * ) of the keck i telescope on 2005 april 24 . the pixel scale of quirc is @xmath47 @xmath42 , and that of the nirc is @xmath48 @xmath42 . the seeing was @xmath43 in the quirc observation , and it was @xmath49 in the nirc observation . the total exposure time was 720 sec and 900 sec for @xmath45-band imaging and @xmath46-band imaging , respectively . the @xmath45-band image is shown in figure [ fig : uh88_0806 ] , and the @xmath46-band image is shown in figure [ fig : keck_0806 ] . all the images ( @xmath50 ) clearly show two stellar components ; we name these two stellar components a ( eastern component ) and b ( western component ) , with a being the brighter component . to determine if there are any extended objects in between the stellar components , we subtracted point spread functions ( psfs ) from the raw @xmath51 images , adopting a nearby star as a template for the psf . the results are shown in the lower panels of figure [ fig : uh88_0806 ] ; there is clearly residual flux between components a and b in all four images . this object ( named g ) is quite red , with colors of @xmath52 and @xmath53 that are similar to those of an early - type galaxy at @xmath54 @xcite . such a redshift is consistent with the redshift of a absorption line system in the spectra of the quasars ( see [ sec:0806spc ] ) and an estimate based on the faber - jackson relation ( see [ sec:0806model ] ) . therefore , we conclude that component g is the lens galaxy . in the higher resolution nirc @xmath46-band image ( left panel of figure [ fig : keck_0806 ] ) , we can identify component g between components a and b even before psf subtraction . we fit this image using galfit @xcite to find that g is well - modeled by a de vaucouleurs profile of ellipticity @xmath55 and major axis position angle @xmath56 . the astrometry and photometry of components a , b and g are summarized in table [ tab:0806 ] . the optical images were calibrated using the standard star pg 0231 + 051 @xcite and the @xmath45-band image was calibrated using the standard star fs 21 @xcite . we lack a photometric standard stars for the nirc @xmath46-band image , thus the data were used only for astrometry and the flux ratio constraints in the lens models . when we fit the nirc @xmath46-band image using a model consisting of the two quasar images and the lens galaxy , we find a quasar flux ratio of 0.53 as compared to the mean flux ratio of 0.67 ( derived from fitting only the psf models ) for the optical bands that could be created by a modest amount of dust extinction or chromatic microlensing . we report the astrometry from using galfit to simultaneously fit components a , b and g in the nirc @xmath46-band image . the angular separation of components a and b is @xmath57 . a spectroscopic observation of sdss j0806 + 2006 was conducted with the keck ii telescope on 2005 april 12 in @xmath58 seeing . we used the echellette mode of the echellette spectrograph and imager ( esi ; * ? ? ? * ; * ? ? ? * ) with the mit - ll 2048@xmath594096 ccd camera . the spectral range was 3900 to 11,000 at a spectral resolution of @xmath60 ( r@xmath6127000 ) . the exposure time was 900 sec . the @xmath62-wide slit was oriented to observe components a and b simultaneously . the spectra of each component was extracted using the standard method ( summing the fluxes in a window around the position of each component and subtracting the sky using neighboring windows on either side of the trace ) . we show the binned spectra in figure [ fig : spec0806 ] . both spectra have the and emission lines at the same wavelength , with an estimated velocity difference of @xmath63 km s@xmath64 for the emission line . we summarize the redshifts calculated from these emission lines in table [ tab:0806em ] . in addition , the spectral energy distributions ( seds ) are also very similar . the ratio of the spectra , which is shown in figure [ fig : spec0806 ] , is almost constant ( @xmath65 ) for a wide range of wavelengths , and it is consistent with the mean flux ratio of the optical images ( 0.67 ) . note , however , that the emission line flux ratios appear to differ slightly from that of the continuum , suggesting that there is some microlensing of the continuum by the stars in the lens galaxy . in addition to the quasar emission lines , we found a strong absorption line system at @xmath66 in both spectra ( see the inset of figure [ fig : spec0806 ] ) . we also found h and k ( @xmath67 ) and absorption lines ( @xmath68 ) at @xmath66 in the spectra . the fact that the redshift of this absorption line system is close to the estimated redshift ( see [ sec:0806img ] ) of the probable lens galaxy ( component g ) and that absorption lines are frequently associated with galaxies ( e.g. , * ? ? ? * ) suggests that the absorption is due to the lens galaxy . if this is the case , the lens redshift should be @xmath69 ; we adopt this value for the mass models presented in the next subsection . to explore the lensing hypothesis further , we modeled the system using the two standard mass models : a singular isothermal sphere with an external shear ( sis+shear ) model , and a singular isothermal ellipsoid ( sie ) model . both models have eight parameters : the einstein radius @xmath70 , the shear @xmath71 or ellipticity @xmath72 and its position angle ( @xmath73 or @xmath74 ) , the position of the lens galaxy , and the position and flux of the source quasar . we have only eight constraints ( the positions of a , b and g , and the fluxes of a and b ) , so we should be able to fit the data perfectly . we should , however , be able to do so with sensible values for the shear or ellipticity of the models . we used the component positions from table [ tab:0806 ] and the flux ratio of 0.53 derived from the galfit model of the nirc @xmath46-band image including the two stellar components and the lens galaxy . lensmodel _ @xcite to determine the model parameters , with the results summarized in table [ table : model0806 ] . the required ellipticities of @xmath75 or @xmath76 are typical of other lensed systems and roughly consistent with that of the lens galaxy ( @xmath55 ) , but there is a significant misalignment between the position angle of the major axis of the models ( @xmath77 ) and that of the lens galaxy ( @xmath78 ) . this might imply that the system is affected by a strong external perturbation @xcite . indeed , there are at least four galaxies within a 16@xmath79 radius of the lens ( named g14 , with the nearest galaxy only 4@xmath79 from the lens ; see table [ tab:0806gal ] ) , so the position angle of the models may be a compromise between that of the lens galaxy and the shear induced by g1 . finally , these models predict that the time delay between the images is @xmath80days . we can estimate the redshift of the lens galaxy based on the faber - jackson relation @xcite . from the table 3 of @xcite , we estimate that the lens magnitude should be @xmath81 for @xmath2 , @xmath82 , and assuming @xmath83 . while this is somewhat brighter than the observed @xmath40-band magnitude of galaxy g ( @xmath84 ) , it is roughly consistent given the @xmath85 mag scatter @xcite . the rough agreement of this estimate further supports for a lens galaxy redshift of @xmath69 . we obtained @xmath39 , @xmath40 , and @xmath41-band images of sdss j1353 + 1138 using the 8k mosaic ccd camera of the uh88 telescope on 2004 may 25 in @xmath58 seeing . the exposure times were 120 sec for @xmath39 and 180 sec for @xmath40 and @xmath41 , respectively . we also acquired an @xmath45-band image on the uh88 telescope using quirc on 2005 february 21 . the exposure time was 720 sec , and the seeing was @xmath43 . the images , shown in the upper panels of figure [ fig : uh88_1353 ] , clearly show two stellar components , which we named a ( northern and brighter component ) and b ( southern component ) . although it is not obvious in the raw @xmath45-band image of figure [ fig : uh88_1353 ] , an extended object can be seen between components a and b even before the psf subtraction . we also obtained @xmath26 and @xmath17-band images using the magellan instant camera ( magic ) at the lc6.5 m telescope on 2005 april 15 . the pixel scale of magic is @xmath86 @xmath42 . the seeing was @xmath87 , and the exposure time was 360 sec for @xmath26-band and 480 sec for @xmath17-band . after subtracting psf models for the two stellar components , we clearly detect an extended red object positioned between them in all the residual images . these residual images are shown in figure [ fig : uh88_1353 ] and [ fig : mag_1353 ] for the uh88 and magellan data , respectively . we identify this extended object , which we denote as g , with the lens galaxy . we fit the @xmath17-band image using galfit to find that the lens galaxy is well - modeled with a de vaucouleurs profile of ellipticity @xmath88 and major axis position angle @xmath89 . in addition to component g , there is an additional object which we named component c near image a. it is most easily seen in the @xmath26-band image after subtracting components a and b ( see the middle panel of figure [ fig : mag_1353 ] ) , but it can also be seen at the same position in the @xmath17-band image . there are four possible interpretations of component c. first , it could be a chance superposition of a foreground star . it is well fit by the psf ( see the bottom panels of figure [ fig : mag_1353 ] ) , and our lens models require only components a , b and g to reproduce the system with reasonable parameter values ( see [ sec:1353model ] ) . second , it could be an object related to the lens galaxy . however , lens models with significant extra mass at the position of component c generally fit the data very badly . third , it could be a third image of the quasar . this seems unlikely since its colors are very different from components a and b ( see table [ tab:1353 ] ) . moreover , identifying it as a quasar image makes little sense as a lens geometry since it is in the wrong place to be a central , odd image ( which should lie between components b and g ) or to be part of a four - image system in which one of the a / b / c components is an unresolved image pair . fourth , it could be emission from the host galaxy of the source quasar . probably only the first possibility is permitted , although we can not completely exclude the other possibilities . we summarize astrometry and photometry of components a , b , c and g in table [ tab:1353 ] . the optical images were calibrated using the standard stars pg 1528 + 062 @xcite and sa 107 - 351 @xcite , and the @xmath45-band image was calibrated using the standard star fs 21 @xcite . the flux ratio of components a and b changes significantly with wavelength , from a mean flux ratio in the optical of @xmath90 to an @xmath45-band flux ratio of 0.54 . the angular separation of components a and b is @xmath91 . we used esi on the keck ii telescope to obtain spectra of the two components of sdss j1353 + 1138 on 2005 april 12 . the instrumental set up was the same as in [ sec:0806spc ] and we used an exposure time of 600 sec . the binned spectra are shown in figure [ fig : spec1353 ] . both components have and emission lines at identical wavelengths , with a velocity difference of only @xmath92 km s@xmath64 for the emission line . we summarize the redshifts derived from the emission lines in table [ tab:1353em ] . the flux ratio of the spectra is nearly constant ( @xmath93 ) and consistent with that in the optical images ( @xmath94 ) . there is , however , a slight increase in the ratio redwards of about 5300 that may be due to contamination of the spectrum of component b by emission from the lens galaxy . assuming this feature is due to the 4000 break of the lens galaxy , it implies a lens redshift of @xmath95 . since the colors of the lens galaxy of @xmath96 , @xmath97 , and @xmath98 are roughly consistent with those of an early - type galaxy at @xmath99 @xcite , we adopt @xmath100 for mass modeling . the quasar spectra also contain strong absorption line systems at @xmath101 , @xmath102 , and @xmath103 , with their absorption lines . these are unlikely to be associated with the lens galaxy , given the large difference from the lens redshift estimated from the colors , and also given the fact that many ( unlensed ) quasars show absorption . we note that an absorption line due to a @xmath104 lens is undetectable in our spectra because it would lie beyond the atmospheric cutoff . we modeled sdss j1353 + 1138 in the same way as in [ sec:0806model ] . the positions of the quasars and lens galaxy are taken from table [ table : model1353 ] ( neglecting component c ) . for the flux ratio of components a and b , we adopt @xmath105 , which was derived from the galfit model of the magic @xmath17-band image . the results are summarized in table [ table : model1353 ] . the derived shear and ellipticity of @xmath106 and @xmath107 , are again typical for a lensed quasar system . the derived ellipticity ( 0.15 ) appears to be significantly smaller than the ellipticity of the observed light profile ( @xmath108 ) , but this difference is commonly seen in the lensed quasar systems @xcite . however , in addition to the disagreement of the ellipticity , there is a significant misalignment between the position angle of the models ( @xmath109 ) and that of the lens galaxy ( @xmath89 ) . while this might be due to external perturbations , there are no nearby galaxies that can easily explain the misalignment . it could be due to component c , but we find that simple tests of lens models with mass located at the position of component c generally fail to fit the data . the time delay between a and b is predicted to be @xmath110days , assuming the lens redshift of @xmath100 . as in [ sec:0806model ] we also estimated the lens redshift based on the faber - jackson relation . again we used table 3 of @xcite , @xmath3 , and @xmath111 to estimate that the lens magnitude should be @xmath112 , assuming @xmath113 . this estimate is in rough agreement with the observed @xmath40-band magnitude of galaxy g ( @xmath114 ) , again supporting for a lens redshift of @xmath115 . we report the discovery of two doubly - imaged quasar lenses , sdss j0806 + 2006 and sdss 1353 + 1138 . both were selected from the sdss spectroscopic quasar sample as lensed quasar candidates and confirmed in subsequent imaging and spectroscopic observations . sdss j0806 + 2006 consists of two @xmath0 quasar images separated by @xmath2 lensed by a galaxy at @xmath116 . the lens galaxy is closer to the fainter image as expected , and its redshift , as suggested by its magnitude , colors , and the presence of a absorption feature , is @xmath117 . several nearby galaxies may perturb this system and indicate that the lens galaxy is part of a small group . sdss 1353 + 1138 consists of two @xmath1 quasar images separated by @xmath3 with a lens galaxy at @xmath118 . the redshift of the lens galaxy is estimated based on its magnitude , colors , and the spectral flux ratio between the two quasar images . there is an additional component , which we have labeled c , superposed on this system whose nature is presently unexplained . observations using the _ hubble space telescope _ are probably required to clarify its role in the lensed quasar system . n. i. and m. o. are supported by jsps through jsps research fellowship for young scientists . a portion of this work was also performed under the auspices of the u.s . department of energy , national nuclear security administration by the university of california , lawrence livermore national laboratory under contract no . w-7405-eng-48 . some of the data presented herein were obtained at the w.m . keck observatory , which is operated as a scientific partnership among the california institute of technology , the university of california and the national aeronautics and space administration . the observatory was made possible by the generous financial support of the w.m . keck foundation . funding for the creation and distribution of the sdss archive has been provided by the alfred p. sloan foundation , the participating institutions , the national aeronautics and space administration , the national science foundation , the u.s . department of energy , the japanese monbukagakusho , and the max planck society . the sdss web site is http://www.sdss.org/. the sdss is managed by the astrophysical research consortium ( arc ) for the participating institutions . the participating institutions are the university of chicago , fermilab , the institute for advanced study , the japan participation group , the johns hopkins university , the korean scientist group , los alamos national laboratory , the max - planck - institute for astronomy ( mpia ) , the max - planck - institute for astrophysics ( mpa ) , new mexico state university , university of pittsburgh , university of portsmouth , princeton university , the united states naval observatory , and the university of washington . kochanek , c. s. , schneider , p. , wambsganss , j. , 2004 , part 2 of gravitational lensing : strong , weak & micro , proceedings of the 33rd saas - fee advanced course , g. meylan , p. jetzer & p. north , eds . ( springer - verlag : berlin ) lupton , r. , gunn , j. e. , ivezi , z. , knapp , g. r. , kent , s. , & yasuda , n. 2001 , in asp conf . 238 , astronomical data analysis software and systems x , ed . f. r. harnden , jr . , f. a. primini , and h. e. payne ( san francisco : astr . soc . pac . ) , p. 269 ( astro - ph/0101420 ) crrrrrr a & 0.000@xmath1190.005 & 0.000@xmath1190.005 & 19.23@xmath1190.01 & 18.93@xmath1190.01 & 18.54@xmath1190.01 & 16.87@xmath1190.01 + b & @xmath1201.136@xmath1190.007 & @xmath1200.823@xmath1190.007 & 19.82@xmath1190.02 & 19.36@xmath1190.02 & 18.84@xmath1190.01 & 17.34@xmath1190.02 + g & @xmath1200.811@xmath1190.015 & @xmath1200.574@xmath1190.015 & 22.27@xmath1190.07 & 21.20@xmath1190.04 & 20.16@xmath1190.03 & 17.90@xmath1190.05 + crrrrrrrr a & 0.000@xmath1190.003 & 0.000@xmath1190.004 & 17.08@xmath1190.01 & 16.74@xmath1190.01 & 16.44@xmath1190.01 & 15.16@xmath1190.01 & 16.91@xmath1190.01 & 16.77@xmath1190.01 + b & @xmath1200.267@xmath1190.003 & @xmath1201.380@xmath1190.004 & 18.63@xmath1190.02 & 17.63@xmath1190.01 & 17.43@xmath1190.01 & 15.83@xmath1190.02 & 18.50@xmath1190.01 & 17.88@xmath1190.01 + c & 0.107@xmath1190.044 & @xmath1200.358@xmath1190.044 & & & & & 20.96@xmath1190.06 & 21.67@xmath1190.10 + g & @xmath1200.255@xmath1190.008 & @xmath1201.041@xmath1190.008 & 19.06@xmath1190.03 & 18.80@xmath1190.02 & 17.80@xmath1190.01 & 16.16@xmath1190.02 & 19.84@xmath1190.01 & 18.77@xmath1190.01 +

we report the discoveries of two , two - image gravitationally lensed quasars selected from the sloan digital sky survey : sdss j0806 + 2006 at @xmath0 and sdss j1353 + 1138 at @xmath1 with image separations of @xmath2 and @xmath3 respectively . spectroscopic and optical / near - infrared imaging follow - up observations show that the quasar images have identical redshifts and possess extended objects between the images that are likely to be lens galaxies at @xmath4 in sdss j0806 + 2006 and @xmath5 in sdss j1353 + 1138 . the field of sdss j0806 + 2006 contains several nearby galaxies that may significantly perturb the system , and sdss j1353 + 1138 has an extra component near its einstein ring that is probably a foreground star . simple mass models with reasonable parameters reproduce the quasar positions and fluxes of both systems .

You are an expert at summarizing long articles. Proceed to summarize the following text: the dimensionality of the 115 materials , cerhin@xmath1 , ceirin@xmath1 , and cecoin@xmath1 , appears to be related to their superconducting transition temperature . the material with the highest t@xmath2 , cecoin@xmath0 , has the most 2d - like fermi surface ( fs ) of the three . @xcite cerhin@xmath0 has a high t@xmath2 ( @xmath32.1 k ) , but only under a pressure of @xmath316 kbar . at ambient pressures , cerhin@xmath0 is an anti - ferromagnet . the fs of cerhin@xmath0 was the subject of one of our recent publications.@xcite in order to confirm the link between the superconducting state and fs dimensionality , the fs as a function of pressure in cerhin@xmath0 should be measured . if the fs becomes more 2d - like as the critical pressure is approached , then this will be evidence for making a connection . in these materials it seems that superconductivity does not appear until the overlap between the _ f _ electron wavefunctions is sufficient to allow band - like behavior . measurements of the fs as a function of pressure should show this increasing overlap as a change in topography . here we present measurements up to 7.9 kbar , about half the critical pressure for cerhin@xmath1 . we have designed and built small pressure cells , capable of running in a dilution refrigerator and in a rotator . measuring torque inside a pressure cell is impossible , so we have made small compensated pickup coils which fit into the cell . each coil has four to five thousand turns . the filling factor approaches unity because we are able to situate the coil along with the sample inside the cell . a small coil is wound on the exterior of the cell to provide an ac modulation of the applied field . we have measured the fs of cerhin@xmath0 under several pressures . at each pressure we measure fs frequencies and their amplitude dependence as a function of temperature . from this we can extract information about how the effective mass of the quasiparticles is changing as the pressure is increased . the figures show the fourier spectra of cerhin@xmath0 under @xmath37.9 kbar . the crystal was oriented so that the a - b axis plane is perpendicular to the applied field . at @xmath37.9 kbar and at ambient pressures ( measured in the pressure cell prior to pressurization ) reveals little that is suggestive of change . ] we show the 7.9 kbar data compared with two sets of data taken at ambient pressure . in fig . [ highfft ] the fs at 7.9 kbar is compared with the ambient data taken with a torque cantilever ( the same data reported in @xcite ) . because the modulation field for the ac measurements ( in the pressure cell ) was so small , the lowest frequencies can be ignored . notice that the 1411 t ( f@xmath4 , the designation given in ref . @xcite ) and 1845 t peaks are reproduced exactly in the ambient and the pressure data sets . the 1845 t peak was not included in ref . @xcite because of its small amplitude in ambient pressure torque measurements . the 3600 t ( f@xmath5 ) and 6120 t ( f@xmath6 ) peaks are present in both data sets ; however , the f@xmath5 appears to have split and the f@xmath6 appears to have shifted down in frequency . such changes could be explained as slight differences of sample alignment with respect to the applied field between the torque measurement and the pressure cell measurement . three other frequencies , 2076 t , 2710 t , and 4613 t , emerge in the pressure data which are close to to some reported in ref . @xcite to be observed only at the lowest temperatures ( 25 mk ) . all but the first of these frequencies are seen also in ambient pressure data taken with the sample in the pressure cell prior to pressurization as shown in fig . [ lowfft ] . thus , assuming the differences in frequency between the torque measurements and pressure cell measurements are due to differences in alignment , we can make frequency assignments that follow ref . @xcite ( also shown in fig . [ lowfft ] ) . the relative increase in amplitude with increasing pressure of these three peaks could be a result of the increase of the coupling factor between the sample and the coil as the two are compressed together . the lack of any clear differences in the fs up to 7.9 kbar suggests that if the fs changes , then such change is not a linear function of pressure . nor is there a compelling reason to think that it should be a linear function . possibly , at some pressure closer the the critical pressure , the transition to _ f _ electron itinerate behavior will take place leading to more noticable changes in the fs . the fs of cerhin@xmath1 appears to remain topographically stable under the application of pressure up to 7.9 kbar . additional measurements which approach the critical pressure ( @xmath316 kbar ) are of prime importance . this work was performed at the national high magnetic field laboratory , which is supported by nsf cooperative agreement no . dmr-9527035 and by the state of florida . work at los alamos was performed under the auspices of the u. s. dept . of energy . donavan hall , e.c . palm , t.p . murphy , s.w . tozer , eliza miller - ricci , lydia peabody , charis quay huei li , u. alver , r.g . goodrich , j.l . sarrao , p.g . pagliuso , j. m. wills , and z.fisk . b _ * 64 * , 064506 ( 2001 ) , cond - mat/0011395

measurements of the de haas - van alphen effect have been carried out on the heavy fermion anti - ferromagnet cerhin@xmath0 at temperatures between 25 mk and 500 mk under pressure . we present some preliminary results of our measurements to track the evolution of the fermi surface as the pressure induced superconducting transition is approached . , , , , , de haas - van alphen ; heavy fermions ; superconductivity ; high pressure

You are an expert at summarizing long articles. Proceed to summarize the following text: considerable uncertainties remain as to what controls the apparent link between the activity of active galactic nuclei ( agn ) during the time that a black hole is fed and star formation . the accretion disk surrounding a super - massive black hole ( smbh ) emits highly energetic radiation and particles , and can form powerful winds and/or collimated , relativistic jets ( e.g. , * ? ? ? . the radiation and outflows might then affect the interstellar medium , triggering star formation which might be detectable as an excess of blue light from the central regions of galaxies . alternatively , the activity of the agn may be sparked by specific events in the galaxy s past . for example , there is morphological evidence that activity in radio galaxies might be triggered by mergers and galaxy interactions ( e.g. , * ? ? ? * ) , which in turn could contribute to central blue light through enhanced star formation . in either case , we might expect an observable association between agn activity , for which in this work we use radio loudness as a tracer , with bluer stellar continuum in the central regions of the galaxy . previous work has hinted at such a relationship . for example , @xcite examined the central light of 30 radio galaxies as compared to 30 normal galaxies from the molongo reference catalogue and found excess blue light in the inner regions of the radio galaxies as compared with the control sample . this central excess is unrelated to the conventional colour gradient of the broader distribution of stars . @xcite studied colour variations within galaxies from the sloan digital sky survey ( sdss ; * ? ? ? * ) dr7 , finding marginally steeper colour gradients in massive galaxies with nuclear activity and concluded that this is due to a higher fraction of young stars in their central regions . in the present paper we expand the work of @xcite , by combining optical information from sdss with the first and nvss radio surveys @xcite to construct a local ( @xmath2 ) sample of powerful radio - loud agn host galaxies ( r - agn ) , and a control sample of ` normal ' early - type ellipticals . we define @xmath10 as the ratio of sdss @xmath6 to @xmath12 band de vaucouleurs effective radii and compare its value between the samples . the paper is structured as follows . in section [ sec : surveys ] we briefly describe the surveys used to construct our samples . separation into the radio - loud and control samples is discussed in section [ sec : sampleselection ] . section [ sec : analysis ] outlines the maximum likelihood analysis used to compare the average values of @xmath10 for the two samples . in section [ sec : discussion ] we present a discussion of our findings in the context of radio - agn fuelling mechanisms and the properties of the host galaxies . throughout , we use a flat @xmath13cdm cosmology with @xmath14 and @xmath15 . we adopt @xmath16 kms@xmath1mpc@xmath1 . sdss is an imaging and spectroscopic survey covering @xmath1710,000 deg@xmath18 , primarily in the northern hemisphere . its 7th data release ( sdss - dr7 ; * ? ? ? * ) contains over 350 million entries , of which about one million are confirmed galaxies with spectroscopic follow up . in the photometric sample , flux densities are measured simultaneously in five broadband filters ( @xmath19 ) , with effective wavelengths @xmath20 , @xmath21 , @xmath22 , @xmath23 and @xmath24 @xcite . all magnitudes are given on the ab@xmath25 system @xcite , and are determined using petrosian apertures ( see * ? ? ? * ; * ? ? ? * for a complete definition of the petrosian system ) . in common with @xcite , we refer to sdss measured magnitudes as @xmath26 and @xmath27 due to the uncertainty in the absolute calibration of the sdss photometric system , which has been assessed at @xmath280.03mag @xcite . for a complete guide to the photometric system , see @xcite , @xcite and @xcite . we use the main galaxy survey spectroscopic sample ( mgs , * ? ? ? the mgs algorithm selected extended sources brighter than @xmath29mag with petrosian half - light surface brightness @xmath30mag arcsec@xmath31 , providing @xmath17 90 galaxies deg@xmath31 @xcite . candidate galaxies were then observed spectroscopically . @xmath32 of the resulting redshifts have velocity errors @xmath3330kms@xmath1 . the survey is unbiased , except for galaxies with close companions . around @xmath34 of galaxies satisfying the photometric target criteria were not included for spectroscopic follow up due to a companion galaxy within the 55 minimum fibre separation , although some of these galaxies have been subsequently observed . the mgs contains @xmath17680,000 spectroscopically confirmed galaxies . the full target selection is described in @xcite . we corrected for galactic extinction using the sdss ` reddening ' parameter , which was derived from maps of the infrared emission from dust across the sky in accordance with @xcite . we applied @xmath35-corrections using the sdss - derived ` kcorr ' parameter as detailed in @xcite . typical corrections @xmath36 were @xmath37 , @xmath38 , and @xmath39mag . the first survey @xcite utilized the vla ( at 1.4ghz in b array ) to map the radio sky over @xmath179000deg@xmath18 in the northern hemisphere and in a @xmath40 wide strip along the celestial equator . it has @xmath41deg@xmath18 overlap with sdss , contains @xmath42 sources deg@xmath31 at the 1mjy survey threshold and reaches an rms sensitivity of @xmath170.15mjybeam@xmath1 . in this configuration , the vla has a synthesized beam of 5.4 fwhm , providing accurate flux densities for small - scale radio structures , but underestimating the flux densities of sources extended to several arcminutes . at the 1mjy survey threshold , individual sources have 90% confidence astrometric errors @xmath43 . we adopted a mean spectral index of @xmath44 ( where @xmath45 ) and obtained rest - frame 1.4ghz power densities by applying k - corrections assuming the usual form @xmath46 where @xmath47 is the luminosity distance . the nrao vla sky survey ( nvss , * ? ? ? * ) mapped the radio sky ( at 1.4ghz in d array ) north of @xmath48 declination . the survey is complete down to a point source flux density of @xmath172.5mjy . the synthesized beam of 45 is much larger than first , providing more accurate flux measurements for highly extended sources . astrometric accuracy ranges from 1 for bright sources to 7 for the faintest detections . the entire survey contains over 1.8 million unique sources brighter than 2.5mjy . @xmath35-corrections are applied as in [ subsec : firstsurvey ] . we initially derived a first - sdss sample of radio galaxies and a comparison control sample with no nearby detectable radio counterpart . figure [ fig : idlsep ] shows the distribution of angular separations between sdss objects and their nearest first counterpart . we find that at @xmath175 the distribution becomes dominated by random matches . hence we adopt a match criterion of 2 angular separation to avoid such false matches . our cross - matched catalogue contains @xmath49 galaxies and we denote this as our preliminary ` radio ' sample . all unmatched sources are initial candidates for our control sample of radio - quiet galaxies . we define our efficiency as the fraction of matches in our radio sample which are physically real ( including contamination from line of sight false matches , which can not be accounted for here ) . in order to evaluate the number of random false matches in our sample , we offset the ra and dec of the first sources by @xmath50 and re - matched to our set of @xmath51 sdss galaxies , selecting objects within a 2 radius . we found on average @xmath52 random matches , corresponding to 0.3% contamination of the first - sdss match sample ( @xmath5399% efficiency , figure [ fig : idlsep ] , red dotted line ) . and sdss is shown as a red dotted line . this indicates the number of likely false matches . the net distribution of ` real ' matches is shown by the blue dashed line , and our 2 cut indicates 99.9% efficiency of the matched sample . ] we also need to know the completeness of our matching criteria - the fraction of correct matches we recover . a factor affecting this is the possible exclusion of large lobe - dominated radio sources @xcite . both lobes will be included in first but may be excluded in the cross - matched subset for lobe - core distances @xmath532 , if the core is weak . we estimated @xmath178@xmath54 of real matches between the sdss subset and first that are excluded from our radio sample , by assuming the radio lobes are outside a 2 radius from the optical core , but within a 5 radius ( see figure [ fig : idlsep ] ) . our matching algorithm efficiency and completeness are similar to @xcite , who match first to sdss - dr6 positions within a 2 radius for sources with @xmath55 2 ( 95% efficiency and 98% completeness ) . however , we could not account for sources with first lobes @xmath56 from the optical core or the population of first double - lobed sources with no detectable radio core . @xcite cross - correlated all first sources with sdss , then identified potential double - lobed radio sources with undetected cores and accepted all matches with offsets @xmath57 . ] , estimating these contribute less than @xmath58 of all radio sources * estimate @xmath59 of their radio - loud agn sample had no first detection ) . @xcite estimated the first catalogue to be @xmath60 complete at 2mjy and @xmath61 complete down to the survey limit of 1mjy . figure [ fig : radiopower_z ] shows integrated radio power against redshift for the preliminary radio sample ( 25,931 sources ) . the solid black lines trace the 1mjy and 2mjy peak flux density thresholds , the dotted and dashed lines show cuts we applied in redshift and radio power ( see [ subsec : redshiftrange ] and [ sec : radioagn ] ) , and the red points are the resultant selection of radio sources . it is noted that for sources that are not well - described by an elliptical gaussian model , the integrated flux density as derived by first may be an inaccurate measure of the true value . such sources with radio powers corresponding to flux densities below the 1mjy threshold can be seen in figure [ fig : radiopower_z ] . we note that although the two cuts improve the sample completeness ( all red points are above 1mjy ) , many of our sources are below the 2mjy line , and so our sample completeness can not be @xmath62 . combining this with our estimation of completeness in our selection criteria ( @xmath63 ) , we estimate the actual completeness of our preliminary radio sample to be @xmath64 for sources brighter than 1mjy . the incompleteness is dominated by statistical effects rather than the @xmath58 effect of sources missing due to weak cores . and the dotted line denotes the radio power cut imposed in [ sec : radioagn ] . the red points are the remaining radio sample after these cuts are applied.,scaledwidth=45.0% ] to produce a preliminary control sample , we select sources from the sdss subset which fall within the same spatial region as first ( @xmath65 deg@xmath18 ) . we selected all objects from this subset which are not matched to a first source within 2 of their optical core , providing @xmath17625,664 radio - quiet galaxies . the efficiency of this sample , based on figure [ fig : idlsep ] is @xmath66 ( i.e. , @xmath67 contamination by radio sources detectable at the first flux density limit ) and the sample completeness is @xmath68 . for first sources substantially larger than @xmath69 some flux density is resolved out , leading to an increase in the survey threshold for extended objects and flux density underestimates for larger objects @xcite . our first - sdss radio sample is large due to the number of first candidate sources near the 1mjy survey limit . its limitations are potential flux density underestimates , and incompleteness to extended radio sources , notably extended , double - lobed sources with no detectable radio core within several arcseconds of its optical counterpart . to attempt to quantify these limitations , we created a complementary sample of radio - loud agn host galaxies using both nvss and first . @xcite note that the 45 resolution of nvss is large enough that @xmath70 of all radio sources are contained within a single component , allowing for a higher sample completeness . however , nvss is less deep than first and consequently the matched sample contains fewer galaxies . the nvss - first - sdss matched sample therefore does not benefit from such good statistics . we used the unified radio catalogue constructed by @xcite ( see sample ` c ' in their table 8) , selecting sources which are detected by both first and nvss , matched to within 25 . this selection yields a radio flux density limited ( @xmath71mjy ) catalogue ( hereafter first - nvss ) containing 141,881 sources . we cross - matched these to the @xmath72 mgs sources adopting the same 2 match criterion used in [ subsec : crosscorrelation ] . our cross - matched catalogue contains 5,719 galaxies and we denote this as the preliminary ` comparison radio ' ( hereafter cr ) sample . integrated flux densities as derived by @xcite are adopted in our analysis . we estimated our matching criterion of 2 to be @xmath11 complete , and @xmath11 efficient ( 8 random matches were found when the first - nvss ra and dec were offset by @xmath50 ) . the first - nvss catalogue used to create the cr sample is 99@xmath54 complete and matched with @xmath73 efficiency ( see table 2 of * ? ? ? we therefore estimated the comparison radio sample to be @xmath11 complete and @xmath74 efficient ( see table [ tab : comeff ] ) . 5 of the 5,719 galaxies in the cr sample reside in the control sample derived in [ subsec : crosscorrelation ] and these were removed . 4,935 of the cr sample ( 86@xmath54 ) are also in the first - sdss sample . in the following sections ( [ subsec : redshiftrange]-[subsec : redshiftdist ] ) we discuss the secondary selection criteria applied to the first - sdss sample . the cr sample follows the same path , and a summary of the final cr sample properties are presented alongside those of the final radio sample in [ subsec : crprop ] . we imposed a redshift cut of @xmath75 0.18 to ensure that the galaxy spatial structure could be well examined . for @xmath76 , redshift is not a reliable distance indicator . figure [ fig : lumredradio ] shows less than 2% of matched sources have redshift @xmath77 . the mgs magnitude limit of @xmath7817.77 corresponds to l@xmath79whz@xmath1 at @xmath80 , and our sample should be complete to this optical luminosity density . we restricted the sdss - derived redshift confidence ( zconfidence ) to be greater than @xmath60 , which cuts out 12% of objects from the preliminary radio sample and 10% of objects from the preliminary control sample . after redshift selection , @xmath81 radio - quiet galaxies remained in the control sample , @xmath1770% of the parent mgs , whilst the radio sample contained @xmath82 galaxies . the average error in redshift for both samples is @xmath83 and the redshift distributions of these two samples were indistinguishable at the 1% level on the basis of a two - tailed k - s test . p@xmath84 luminosity bins . we cut at @xmath2 , providing a high sample completeness ( @xmath85 ) down to p@xmath86whz@xmath1 . ] @xcite suggested that typical radio - loud agn have powers above @xmath87whz@xmath1 at 1.4ghz and @xcite shows that p@xmath88whz@xmath1 separates the spiral starburst population from the agn - e / s0 population in local galaxy fields . we therefore restrict the radio sample to galaxies harbouring radio sources with p@xmath89whz@xmath1 , which selects 5,119 galaxies ( 30% ) from the radio sample . figure [ fig : lumredradio ] shows that this threshold in radio power is well matched to our redshift selection . we name this set of 5,119 galaxies the ` radio - loud ' sample . the mean redshift of the radio sample increased from @xmath90 to @xmath91 when the radio power cut was applied . agn can be split on the basis of large - scale radio structure into fri type , where radio brightness decreases outwards from the centre and frii type , with edge - brightened lobes and hot spots @xcite . friis are more luminous p@xmath92whz@xmath1 and are rare in a low - redshift sample such as ours ( figure [ fig : lumredradio ] ) , since this 178mhz power corresponds to p@xmath93whz@xmath1 for a typical spectral index of @xmath44 . in unified agn models ( e.g , * * ) the appearance of the central black hole and associated continuum of an agn differ only in the viewing angle at which it is observed . sources viewed face on ( type 1 ) show broad emission lines that are absent in those observed edge on and where the broad emission line region is obscured by a dusty torus ( type 2 ) . type 1 agn are excluded from this study , as the optical continuum can be dominated by relativistically boosted non - thermal emission , which may overwhelm measurements of the host galaxy s properties . the sdss spectral classification pipeline automatically flags and excludes quasars from the mgs , but we chose to verify its reliability and the relative numbers of type 1 agn remaining in our sample . we followed the method outlined by @xcite to identify broad line emission . h@xmath94 or h@xmath95 emission lines exceeding 1000kms@xmath1 ( fwhm ) with a @xmath96 and h@xmath94/h@xmath95 ew @xmath56 were classified as broad line . galaxies with both broad h@xmath94 and h@xmath95 emission were classified as type 1 agn . 3 galaxies of the 5,119 radio - loud sample have broad h@xmath94 and h@xmath95 emission lines and were removed from the sample . 184 galaxies ( 4@xmath54 ) of the remaining 5,116 radio - loud sample have broad h@xmath94 emission , but do not have broad @xmath97 emission . @xcite classified these objects as type ` 1.9 ' agn , which have substantial but not complete obscuration of the central continuum source . @xcite determine that the contribution to the observed continuum is not significant in these sources , so we retain these within our sample . the control sample contained 10 type 1 agn and 1835 type ` 1.9 ' agn . we removed the type 1 agn to leave 499,408 galaxies within the control sample . type 2 agn have narrow permitted and forbidden lines and their stellar continuum is often similar to normal starforming galaxies . @xcite show a tight correlation between far infra - red luminosity ( indicative of star formation ) and p@xmath84 . this will cause a level of contamination by star - forming galaxies if p@xmath98 is used as the sole tracer of galaxies hosting a radio - loud agn . we should therefore remove the small subset of radio - loud galaxies in which the radio power arises from star formation ( sf ) and not from an agn . agn separation from sf galaxies in the local universe can be achieved via optical emission line ratio diagnostics ( * ? ? ? * herein bpt ) . emission - line ratios probe the ionizing source : for agn , non - thermal continuum from the accretion disc around a black hole and in star - forming galaxies ( sfgs ) , photoionization via hot massive stars . however , @xcite find no correlation between a galaxy being radio - loud and whether it is optically classified as an agn . in agreement with this , we found no correlation0.06 or less between the radio - flux and optical emission - line flux ratios for our radio galaxy sample . ] between radio flux at 1.4ghz and the optical line ratios [ nii]/h@xmath94 or [ oiii]/h@xmath95 in our sample . hence , a substantial fraction of radio - loud agn would not be selected using bpt diagnostics , and were we to apply them to the radio sample it would be biased towards radio galaxies with particularly strong optical emission lines . we instead remove galaxies which are strongly identified as non - agn , i.e. star - forming . despite this method leaving a small fraction of star - forming galaxies which are faint in optical line emission , all radio - loud agn , whether optically bright or otherwise , will remain in the sample . this decrease in efficiency of the sample is preferential to a drastic decrease in completeness . a similar problem was identified by @xcite , who defined a sample of radio - loud agn from the 2dfgrs catalogue . they found approximately half of the sample have absorption spectra similar to those of inactive giant ellipticals , and therefore would be mostly missed by optical agn emission - line selection . in classifying galaxies as agn or star - forming , we utilized the demarcation criterion of @xcite @xmath99}/\mathrm{h}\beta ) > 0.61/\{\mathrm{log}([\mathrm{nii}]/\mathrm{h}\alpha ) - 0.05\ } + 1.3 % \\\ ] ] plotted as the dashed line on figure [ fig : bpt_sf ] . ) indicates the demarcation given by @xcite between optical agn ( above the line ) and sf galaxies ( below the line ) . 766 galaxies plotted have emission line ratios with @xmath5 significance . the density of these galaxies is shown by contours , 85% of the 766 galaxies lie above the line as optically classified agn . the remaining 15% ( 118 galaxies ) lie below the curve ( red circles ) and we classified these as star - forming . we also show the maximum uncertainty for a point on the kauffmann separator with s / n of 3 ( dotted lines ) . 69 objects detected at s / n@xmath100 ( black crosses ) lie between the kauffmann separator and the upper maximum uncertainty line . ] figure [ fig : bpt_sf ] shows the standard line ratio diagnostics for galaxies from the radio - loud sample with all four emission lines catalogued in the sdss ( grey crosses , 1,069 objects of the 5,116 ) . 766 of these 1,069 galaxies have both optical emission - line ratios at s / n @xmath101 and their density on the bpt diagram is shown as contours . 118 ( 11% ) lie below the demarcation line and are marked with red circles ( 112 out of these 118 have all four emission - lines with s / n @xmath102 ) . we removed these 118 optically selected sf galaxies from the radio - loud sample , to leave a sample of 4,998 predominantly radio - loud agn hosts . we then estimated the residual contamination expected in the radio agn sample from star - forming galaxies by plotting the maximum uncertainty for a point on the kauffmann separator with s / n of 3 ( fig [ fig : bpt_sf ] , dotted lines ) . 69 objects detected at s / n@xmath100 ( black crosses ) lie between the kauffmann separator and the upper maximum uncertainty line . therefore the contamination expected in the radio agn sample from star - forming galaxies is @xmath103 . our radio sample contains predominantly fri galaxies , which are usually hosted by giant elliptical galaxies and on average have weak or no optical nuclear emission lines ( * ? ? ? * and references therein ) . within our sample of radio - loud agn , 79% of objects do not have optical emission line fluxes . we estimate @xmath104 contamination of non - agn ( e.g. sfg ) if galaxies without sdss emission lines are similar to galaxies with bright line emission . 5 galaxies without emission line fluxes @xmath53 3@xmath105 lie below the demarcation line in figure [ fig : bpt_sf ] and are potentially star forming , but without reliable line information we retain these within our sample . as discussed by @xcite , a potential shortfall of spectral classification of emission - line radio - loud agn is that emission - line agn activity is often accompanied by star formation ( e.g. , * ? ? ? . this star formation will give rise to radio emission , even if the agn itself is radio quiet . for these sources , the optical spectrum could still be dominated by a ( radio - quiet ) agn leading to classification as an emission - line radio - loud agn . @xcite ( hereafter m10 ) derive a population of sources matched from all nvss galaxies in the unified radio catalog @xcite , the sdss - mgs and iras data . the matched sources are divided into star - forming , composite and agn using standard bpt diagnostics . star formation rates are derived via broad - band spectral fitting to the nuv - nir sdss photometry , and the _ average _ fractional star formation / agn contribution to the radio power is estimated ( see their table 2 ) . they find that in 203 composite galaxies , 81.3@xmath54 of the total radio power is due to star formation . the variation of this fraction with radio power is not specified . following m10 , we defined 206 composite galaxies in our own radio sample ( 27@xmath54 of the emission - line galaxies confirmed at s / n @xmath101 ) , using the diagnostics of @xcite and @xcite . if the average fractional contribution is independent of total radio luminosity , then we estimated an upper limit of @xmath106 of our radio sample may have radio power boosted by star formation but possess a radio - quiet agn . however , @xmath107 of the composite sample defined by mori have log[p@xmath108(whz@xmath1 ) ] @xmath109 . @xcite find sf galaxies tend to have median log[p@xmath108(whz@xmath1 ) ] = 22.13 , whereas agn have a median log[p@xmath108(whz@xmath1 ) ] = 23.04 . therefore , our luminosity cut will have significantly reduced the numbers of galaxies where star - formation is the principal contributer to the total radio power , and we expect far less than @xmath110 contamination from radio - quiet , optically - loud agn . we also can not account for the population of ` composite ' radio - loud host galaxies with ongoing star formation that have been lost from our sample through this selection technique . @xcite estimate that @xmath111 of local radio sources may have a starforming spectrum but have radio flux densities dominated by a radio - loud agn . as low - redshift radio - loud agn are hosted predominantly by elliptical galaxies , we attempted to constrain the control sample to contain only early - type galaxies . the colour distributions of galaxies have been shown to be highly bimodal @xcite . figure [ fig : colourcolour ] shows a colour - colour plot in @xmath112 * -@xmath7 * against @xmath8 * -@xmath112 * as explored by @xcite who define an optimal colour separator98% completeness and @xmath11383% reliability . ] between early and late types of @xmath114mag . early - types ( e , s0 , sa ) populate the upper right region , and late types ( sb , sc , irr ) occupy the lower left . contours ) . the dashed line is @xmath114mag from @xcite and defines a morphological colour separator . 80% of the galaxy zoo - mgs sample lie above the line . the @xmath12 , @xmath115 and @xmath6 magnitudes are derived from sdss columns : petrocounts @xmath116 reddening @xmath116 kcorr . ] in order to test the completeness of this demarcation , we used the 62,190 galaxies visually classified as ellipticals in galaxy zoo data @xcite . we cross - matched these to the control sample , selecting objects within 1 radius . 34,606 matches were found , and 27,569 of these lie above @xmath8 * -@xmath7*@xmath532.22mag . therefore , this demarcation gives @xmath117 reliability ( fig [ fig : colourcolour ] , contours ) , suggesting @xmath118 of the early - type galaxy population may be excluded through applying this criterion . 147,275 visually classified ` spiral ' galaxies from the galaxy zoo data are also present in our control sample . 22% of these lie above @xmath8 * -@xmath7*@xmath532.22 , i.e. we expect @xmath119 contamination in the early - type sample . of the 499,408 galaxies in the control sample , @xmath120 were classified as early - types , using the @xcite colour selection , to leave 199,391 early - type galaxies in the control sample , morphologically selected in colour space at @xmath121 completeness and @xmath122 efficiency based on galaxy zoo visual analysis . we explore the 4,998 radio - loud agn selected galaxies in colour space . figure [ fig : radiocc ] shows the radio - loud agn are predominately early types , with @xmath123 lying above @xmath8 * -@xmath7*@xmath532.22 , i.e. the radio - loud agn sources are predominantly hosted by ellipticals as defined in colour space . the distribution in colour space is similar to that of the control sample , despite the agn colours possibly influencing the light . in agreement with this result , @xcite found radio - selected agn have a high incidence of being hosted by early - type galaxies . the 118 galaxies we flagged as star forming and removed from the radio - loud sample in [ sec : sf ] are shown as crosses in figure [ fig : radiocc ] , and 98% of these sit below the line . we select the radio - loud agn above the colour cut , to give 3,516 ` early - type ' radio - loud agn hosts . mag , 4 equally spaced levels , max density contour = 400/bin ) , and sf galaxies as defined in [ sec : sf ] ( crosses ) the dashed line is @xmath114 from @xcite and defines a morphological colour separator . ] at this stage we re - examined the redshift distributions of the samples , using the ks test . the probability of the control sample and the radio - loud agn being drawn from the same parent sample in redshift was @xmath124 : the distributions in redshift differ ( figure [ fig : idl_zscale_lum ] ) . evolutionary differences in host galaxy properties should be small over this entire redshift range , however we chose to re - sample the control sample redshift distribution to match that of the radio - loud agn sample . we also examined the intrinsic @xmath6 band luminosity of the two samples . figure [ fig : idl_lum_zscale ] shows the radio sample is globally brighter in @xmath6 than the control sample . it has been well established that radio - loud agn are hosted preferentially in the brightest and most massive elliptical galaxies @xcite . since we wanted to test whether radio - loud agn harbour an excess of blue emission in their centres , we chose to re - sample the control sample to have the same @xmath6 band optical properties so as to avoid any possible correlation of colour gradient with galaxy luminosity . we generated a matched control sample by selecting the 10 galaxies from the whole sample that lie closest to each radio - loud agn in magnitude - redshift space . the final control sample contains 35,160 galaxies , whose magnitude and redshift distributions are shown as dotted lines in figure [ fig : idl_scaled ] . this final selection should allow the distribution of observables in the radio and control samples to be compared directly . the criteria applied to the ` radio ' sample in [ subsec : redshiftrange]-[subsec : redshiftdist ] were also applied to the 5,719 objects in the cr sample ( see [ subsec : crossnvssfirst ] ) . we imposed a redshift cut of @xmath2 and restricted the redshift confidence parameter given by sdss to @xmath74 , leaving 3,727 sources . we examined the cr sample radio powers based on nvss and first . figure [ fig : lumfn ] shows the nvss and first luminosities for the 3,727 cr sources ( grey crosses ) . those with p@xmath125whz@xmath1 are highlighted by blue squares ( 50@xmath54 ) . we restricted the cr sample to galaxies harbouring radio sources with p@xmath125whz@xmath1 , which selects 1,847 ( 50@xmath54 ) of the sample . of those 1,847 sources , 251 have first derived powers @xmath126whz@xmath1 , and were therefore excluded in the radio sample selection in [ sec : radioagn ] . whz@xmath1 are highlighted by blue squares ( 1,847 ) . of those , 251 sources with p@xmath127whz@xmath1 are shown as red crosses . these are the sources excluded in the radio sample s luminosity cut ( [ sec : radioagn ] ) . ] we removed 2 type 1 agn and 48 star - forming galaxies confirmed at s / n @xmath128 from the cr sample , leaving 1,797 radio - loud narrow - line agn hosts . 4@xmath54 of the sample are classified as type ` 1.9 ' agn , with substaintial but not full obscuration of the central source . we applied the strateva et al . colour cut ( @xmath8 * -@xmath7*@xmath532.22mag ) detailed in [ subsec : colour ] to select 1,295 early - type galaxies from the cr sample . the control sample was cross - matched with the first - nvss catalogue to within 2and the 5 galaxies found were removed to ensure the comparison control sample is comprised solely of ` radio - quiet ' agn elliptical hosts . 199,386 galaxies remain in the cr - control sample . we selected the 10 galaxies from the control sample that lie closest to each cr radio - loud agn in magnitude - redshift space . the final cr - control sample contains 12,950 sources , matched to the cr sample in @xmath6-band magnitude and redshift distributions . * radio - loud ` early type ' agn ( r - agn ) - 3,516 sources * after matching first to the mgs ( @xmath3mag , @xmath2 ) within 2 , we classified as radio - loud agn hosts with p@xmath89whz@xmath1 . type 1 agn exhibiting both broad h@xmath94 and h@xmath95 emission lines were removed from the sample . optical emission line ratios ( if confirmed at 3@xmath105 ) were used to remove star - forming galaxies using the demarcation of @xcite . we then created a subset with @xmath12-@xmath129mag i.e. ` early - types ' to remain consistent with the colour cut applied to the control sample . * control sample - 35,160 sources * we removed all galaxies from sdss - mgs ( @xmath3mag , @xmath2 ) with first and nvss counterparts . type 1 agn exhibiting both broad h@xmath94 and h@xmath95 emission lines were removed from the sample . galaxies were constrained in colour to bias towards early - type morphology where a comparison with results of the galaxy zoo program suggests we have @xmath130 completeness and @xmath131 efficiency . the resultant sample comprises radio - quiet , ` normal ' , predominantly elliptical galaxies . we then selected a subsample to match the redshift and optical @xmath6-band magnitude distributions of the r - agn sample , thus removing any redshift bias and potential correlation of colour gradient with galaxy luminosity . * comparison radio ( cr ) - 1,295 sources * we cross - matched the first - nvss catalogue derived by @xcite with the mgs ( @xmath3mag , @xmath2 ) to within 2 , and classified radio - loud agn hosts as those sources exhibiting p@xmath132whz@xmath1 . type 1 agn exhibiting both broad h@xmath94 and h@xmath95 emission lines were removed from the sample . optical emission line ratios ( if confirmed at 3@xmath105 ) were used to remove star - forming galaxies using the demarcation of @xcite . we then created a subset with @xmath12-@xmath1332.22mag i.e. ` early - types ' to remain consistent with the colour cut applied to the control sample . * cr - control sample - 12,950 sources * all selection criteria as above for the control sample , we also removed 5 galaxies within the control sample that were also present in the cr sample . we then selected a subsample to match the redshift and optical @xmath6-band magnitude distributions of the cr sample . we therefore have two radio - loud agn samples , both with a comparison control sample . the cr sample is derived using both radio catalogues , and is a brighter radio sample ( the nvss flux density completeness limit is @xmath134mjy ) for comparison with our deeper ( @xmath135mjy ) , larger but more incomplete r - agn sample derived solely from the first catalogue . the surface brightness profile of an early - type galaxy can be described by the srsic equation @xmath136}\ ] ] where @xmath137 is the effective radius ( scale length ) and @xmath138 is the corresponding effective surface brightness . @xmath139 is chosen so @xmath137 contains half the light in the galaxy . n @xmath140 4 for bright ellipticals decreasing to n @xmath140 2 as luminosity decreases @xcite . sdss chooses to fix @xmath141 , fitting the ` de vaucouleurs profile ' , truncating the profile beyond 7@xmath137 . the model fitted by sdss has an arbitrary axis ratio and position angle and is convolved with a double - gaussian representation of the psf . the fitting then yields , among other properties ; the effective radius , @xmath137 and error , @xmath142 . in general , the error for the @xmath8 model is greater than other bands . it is noted that sdss s fitting algorithm generates some weak discretization of model parameters , especially in @xmath143 and @xmath144 . objects with scale lengths lying in discrete bands in @xmath137 tend to have poorer goodness of fit . this is not included in the error estimates , @xmath142 , which were derived from count statistics on the image . in our work with the @xmath137 values , we consider @xmath142 rather than dev_l ( the likelihood of the model fit ) due to our reluctance to remove data with poor de vaucouleurs @xmath8 band fitting . a poor goodness of fit to the devaucouleurs profile may be indicative of a cuspy @xmath8 band and removing such objects could bias results . sdss denotes the fracdev parameter as the fraction of the total flux contributed by the de vaucouleurs component in a linear combination of a de vaucouleurs and an exponential model to find the best fit . it is noted that the likelihood values in the @xmath6-band are intrinsically poor ( all samples have mean values @xmath145 ) , but the likelihood ratios still pick out reliable best - fit parameters . the fracdev parameter ( @xmath146 hereafter ) is correlated with the srsic index ; n=1 corresponds to @xmath146=0 , n=4 corresponds to @xmath146=1 ( * ? ? ? * and references therein ) . following @xcite , galaxies with @xmath147 ( @xmath148n@xmath149 ) are labeled de / ex " galaxies and galaxies with @xmath146@xmath150 ( n@xmath151 ) are labeled de " galaxies . we chose not to remove objects with @xmath146@xmath152 , but instead created subsets of each sample with @xmath146@xmath153 for comparison . we investigate the colour structure of agn host galaxies principally through the distribution of @xmath10 , which we define as @xmath154 , which is the ratio of @xmath7 to @xmath8 de vaucouleurs effective radii . this should avoid any intrinsic scale differences in @xmath7 and @xmath8 between individual galaxies . the distributions of @xmath10 for the r - agn and control sample are shown in figure [ fig : dist1 ] . both samples have a few ( @xmath155 ) strong outliers ( @xmath156 ) which significantly disturb the mean . the medians of the r - agn and control distributions are 0.837 and 0.793 respectively . using a two - tailed kolmogorov - smirnov test we find the probability of the control and the r - agn sample being drawn from the same distribution to be small ( @xmath124 ) . similarly , the cr sample s @xmath10 distribution also shows a higher median compared to the cr - control sample ( 0.851 and 0.797 respectively ) . the ks test also finds the probability of the cr and cr - control sample being drawn from the same distribution in @xmath10 is vanishingly small ( @xmath124 ) . for the radio - loud agn ( r - agn ) sample ( solid ) and the control sample ( red dashed).,scaledwidth=45.0% ] the fractional errors in individual values of @xmath144 are relatively large , which makes proving an intrinsic difference between the radio - loud agn samples and their respective control sample distributions challenging . we used a maximum likelihood analysis to compare the average values of @xmath10 for the agn and control samples , taking into account measurement errors according to @xcite and @xcite , in order to quantify the difference between the distributions of @xmath10 . under the assumption that both the agn and control samples have normally - distributed intrinsic values of @xmath10 , then the population mean , @xmath157 , and intrinsic dispersion , @xmath105 , of either can be determined from individual values @xmath158 by minimizing the function @xmath159 } . \label{eq : smin}\ ] ] figure [ fig : rsig ] shows the results of minimization of eq [ eq : smin ] across @xmath157 and @xmath105 parameter space , for the two samples . the 90% and 99% joint - confidence contours for @xmath157 and @xmath105 are given by s@xmath160s for @xmath161s@xmath162 and 9.21 respectively . table [ tab : smin ] shows the best fit @xmath163 planes for both distributions . a clear separation between radio - loud ` early type ' agn and ` normal ' control galaxies can be seen for both the r - agn and cr samples , and we conclude the two are not drawn from the same population at @xmath164 confidence . we applied the cut @xmath165 in the @xmath6-band , and re - examine the results for both radio - agn samples . this cut will ensure the reliability of the de vaucouleur s scale lengths used to derive @xmath10 , and this subset is used as a comparison to the full samples to verify whether galaxies with shapes that differ significantly from the de vaucouleurs profile perturb our main result . table [ tab : smin ] also shows the fraction of each sample with ` good ' de vaucouleurs @xmath6-band fits and their @xmath157 values ( see figure [ fig : rsigf ] ) . although the values do not significantly differ from the full sample , the confidence contours now overlap in the smaller , but more complete , cr sample . however , the r - agn and control samples are still seen to come from distinct populations . l ccccc sample & @xmath157 & @xmath105 & @xmath165 & @xmath166 & @xmath167 + [ 0.5ex ] r - agn & 0.76@xmath1680.02 & 0.23@xmath1680.01 & 53@xmath54 & 0.77@xmath1680.02 & 0.25@xmath1680.02 + control & 0.719@xmath1680.005 & 0.206@xmath1680.003 & 49@xmath54 & 0.722@xmath1680.006 & 0.208@xmath1680.005 + cr & 0.78@xmath1680.02 & 0.20@xmath1680.02 & 44@xmath54 & 0.77@xmath1680.03 & 0.19@xmath1680.03 + cr - control & 0.731@xmath1680.007 & 0.204@xmath1680.006 & 49@xmath54 & 0.730@xmath1680.010 & 0.206@xmath1680.007 + [ 0.5ex ] our results confirm that radio - loud agn are hosted by brighter , bigger galaxies on average ( see figure [ fig : idl_scaled ] ) . our technique has shown the presence of an agn is associated with an increase of the scale size of a galaxy in red light relative to blue light . this suggests that radio galaxies have a bluer central bulge and/or more diffusely distributed red light compared to their radio - quiet counterparts . with the analysis so far , we can not tell whether this difference is from agn driven or diffuse star - like light within the bulge , or more distributed red light . the cr sample is also significantly different in @xmath157 relative to the cr - control sample , so that our r - agn sample s lower completeness in extended sources does not significantly affect the result . the cr sample dispersion is smaller than that of the r - agn sample , whereas all four control samples have values of @xmath105 that are similar . we applied a flux cut of @xmath169mjy to the r - agn sample ( 61@xmath54 ) , and found @xmath170 and @xmath171 . the increased scatter about @xmath10 is caused by fainter radio sources , which tend to be more heterogenous , and/or have underestimates of errors on @xmath137 values in more distant objects . the subset of cr galaxies constrained to have good @xmath6-band de vaucouleurs fits ( @xmath165 ) does not contain enough radio galaxies to show strongly a significant separation from the cr - control sample ( see figure [ fig : rsigf](b ) ) . we present discussion predominantly on the r - agn and control sample , which display a clear separation in both figure [ fig : rsig](a ) and [ fig : rsigf](a ) . we concluded the two are not drawn from the same population at @xmath164 confidence . this sample is incomplete to radio sources with lobes @xmath172 and the population of galaxies with a weak radio agn core but powerful lobes . however it is large due to the depth of the first survey . the cr results for all figures in the discussion are similar to the results presented for the r - agn sample . we find no significant deviation between the two in any subsequent parameter examined , except in @xmath105 , as discussed above . for the cr sample . we found no correlation ( @xmath11 certainty ) between the two.,scaledwidth=45.0% ] ( * ? ? ? * hereafter mkm99 ) used a complementary ratio ] to demonstrate the ubiquity of inner blue components in a sample of only 30 radio galaxies and 30 control galaxies , and argued that the blue light is due to star formation associated with the presence of a radio source ( e.g. * ? ? ? * ) . figure 4 of mkm99 ( power at 408mhz vs log [ @xmath173 ) shows a trend for more powerful radio galaxies to have steeper colour gradients as indicated by the ratio of scale lengths . in contrast to this , when considering only the brightest ( p@xmath174whz@xmath1 ) radio sources in the cr sample ( for which the powers are more accurate ) we find no correlation between radio power and @xmath10 . we also find no correlation over the full range of radio powers and @xmath10 as shown in figure [ fig : lumr ] : the spearman s rank correlation result is @xmath175 or less between p@xmath84 and @xmath10 . we infer that a high value of @xmath10 ( and possible blue central bulge ) is not associated with the power of the radio source . there are several possible scenarios to be considered to explain the larger @xmath10 in the r - agn sample . central point - like @xmath12 light may be indicative of a blue agn . we investigated the nature of the r - agn excess @xmath10 , using the goodness of the devaucouleurs fit . the sdss devaucouleurs fitting procedure returns the likelihood ( between @xmath176 and @xmath177 ) associated with the model from the @xmath178 fit . if the @xmath12-band goodness of fit for the r - agn sample approaches 1 for low @xmath10 and 0 for high @xmath10 then this is indicative of an agn / core component ( as the agn point - like component would perturb a devaucouleurs fit ) driving a blue excess and high @xmath10 . if no structure is seen in @xmath10 vs devaucouleurs @xmath12-band goodness of fit in either sample , then any blue excess is more likely driven by some sort of diffuse starlight in the central bulge . the @xmath12-band data for both the r - agn and control samples are generally well fit by a devaucouleurs profile ( the r - agn / control median likelihood values are 0.86 and 0.83 respectively ) , and there is no trend for the @xmath12-band or @xmath6-band fits to be worse at large @xmath10 in the r - agn than the control sample . we can conclude that the high @xmath10 in the r - agn sample is not due to any point - like agn component . it is generally agreed that agn feedback regulates the supply of cold gas for star formation by supplying energy to the ism and igm . some radio - loud star - forming agn show a connection between the suppression of star formation and the strength of the radio jets , by heating and expelling the surrounding gas @xcite . @xcite looked at a sample of low redshift sdss early - type galaxies for which late - time star formation is being quenched . they found that molecular gas disappears less than 100myr after the onset of accretion onto the central black hole . these galaxies were not associated with radio jets , but show that low - luminosity agn episodes are sufficient to suppress residual star - formation in early - type galaxies . -band and @xmath6-band scale radii . there is a higher fraction of blue cores in the r - agn sample as compared to the control sample at @xmath179kpc . this plot does not demonstrate the correlation between the red and blue scale radii ( see figure [ fig : rdev ] ) . the measurement errors are larger in the @xmath12 band which may account for the higher variability in the residuals.,scaledwidth=45.0% ] the residual fractional number density distribution of scale lengths are shown in figure [ fig : residual ] , for the control sample subtracted from the r - agn sample . there is a clear excess of blue cores ( @xmath180kpc ) in the r - agn sample as compared to the control sample . there are also more small @xmath181 in the r - agn sample , ranging from @xmath182kpc . this plot does not demonstrate the correlation between the red and blue emission , which is shown in figure [ fig : rdev ] , but it does demonstrate that the presence of a radio agn seems not to have suppressed star - formation in the central regions of its host galaxy . and @xmath181 for the r - agn ( solid ) and control sample ( red dashed ) . the dotted - dashed line is the locus @xmath183 , galaxies above this line become bluer inward . contours show the fractional number densities of each sample with levels at 0.004 , 0.008 and 0.011 . there is a larger fraction of the r - agn sample above the line , possibly denoting a tendency to become bluer inward more often than the control sample.,scaledwidth=45.0% ] figure [ fig : rdev ] plots @xmath184 against @xmath181 for the r - agn sample ( black ) and the control sample ( red dashed ) . the dotted - dashed line denotes the locus @xmath185 as defined by the control sample ( see table [ tab : smin ] ) . at @xmath186 , the surface brightness in @xmath12 increases more rapidly toward the centre of a galaxy than in @xmath6 as compared to the mean of the control sample , i.e. half the blue light from the galaxy is contained in a smaller region than half the red light , implying the galaxy becomes bluer inward . the distribution of number densities in figure [ fig : rdev ] shows radio galaxies appear to become bluer inwards more often than the control galaxies ( 66% of the r - agn sample are above the locus @xmath187 , compared to 60% of the control sample ) . there are fewer blue smaller cores in the control sample as compared with the r - agn sample ( see also figure [ fig : residual ] ) , implying there may be a blue excess near the agn core for some of the radio galaxies . however , the higher @xmath10 values of the r - agn sample also seems to be driven by a higher scale length in @xmath6 , seen in the difference between the inner two contour levels of each sample ( see also [ subsubsec : redenv ] ) , so we can not definitively conclude that the increased @xmath10 in the r - agn sample is due to star formation within the central few kpc . star - formation within the central few kpc would contradict feedback models which predict the suppression of star formation near an agn . @xcite found that radio sources in massive hosts are re - triggered more frequently than their less massive counterparts , suggesting that the onset of an agn quiescent phase is due to fuel depletion . agn activity is therefore promoted by an increase in gas in the centres of the galaxies , which may imply a link between the agn radio phase and star formation in the bulge ( @xmath188kpc ) of the host galaxy . it has been suggested that agn activity and a major episode of star formation in radio - loud galaxies is triggered by the accretion of gas during major mergers and/or tidal interactions . however agn activity is initiated later in the merger event than the starburst ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? our spectroscopic selection rules out major merger events ( see [ subsubsec : mgs ] ) , but residual star formation may still be present on global scales . observationally , there is a strong link between agn and starbursts ( * ? ? ? * and references therein ) . @xcite found powerful optical agn ( as classified by the strength of the [ o ] emission @xmath189}}>10 ^ 7 l_{\odot}$ ] ) predominantly reside in ` young bulges ' . recent star formation can provide up to @xmath190 of the optical / uv continuum in radio galaxies at low and intermediate z e.g. ( e.g. , * ? ? ? * ; * ? ? ? we found a higher fraction of r - agn galaxies have @xmath191kpc as compared to the control sample ( see figure [ fig : residual ] ) , perhaps indicative of star formation in the outskirts of the bulge , fuelled by gas expelled from the central regions by the agn . an alternative interpretation of a larger @xmath10 is that the r - agn sample has more distributed red light . radio - loud ` early - type ' galaxies predominantly reside in elliptical galaxies , which are well known to be redder than late - types ( e.g. , * ? ? ? diffuse red emission would cause a larger devaucouleurs scale radius in the @xmath6 band and an increased @xmath10 . figure [ fig : rdev ] suggests the increased @xmath10 in the r - agn sample may also be attributed to more diffuse red light ; between @xmath192kpc , @xmath181 is on average higher for the radio population . this is also seen in the lower panel of figure [ fig : residual ] . our results show a difference in the distributions of red and blue light in r - agn and normal ` early - type ' galaxy populations , though a high value of @xmath10 is not a property of every active galaxy . we found the higher @xmath10 in the r - agn sample is contributed to by a higher fraction of radio galaxies harbouring small blue cores but also an increase in the numbers of radio galaxies with more diffuse red light as compared to the control sample . given that star formation proceeds over a longer timescale than radio activity , this disfavours the idea that all galaxies undergo short bursts of radio activity , but rather implies that a subset have the predisposition to become radio - loud . we cross - matched low - redshift ( @xmath2 ) data from the sdss mgs ( @xmath3mag ) and first to within 2 , deriving a radio sample of galaxies at @xmath11 efficiency and @xmath193 completeness . radio luminosities were in the range @xmath0whz@xmath1 . type 1 agn were removed from the sample ( identified via @xmath194 and @xmath97 broad line characteristics ) , 4@xmath54 of the sample are classified as type ` 1.9 ' agn , with substantial but not full obscuration of the central source . contamination from sfgs ( identified via optical emission line ratios ) in the radio sample is expected at @xmath103 . a control sample was defined from sdss sources with no match to a first source within 2 of their optical core , providing a sample of @xmath11 efficiency and @xmath68 completeness . at 80% reliability , the demarcation @xmath4mag selected ` early - type ' galaxies in both samples . samples were matched in @xmath6-band magnitude and redshift distributions and final sample sizes were 3,516 radio - loud agn galaxies ( r - agn ) and 35,160 control galaxies . we also created a complementary flux - limited sample through cross - matching with nvss ( the cr sample ) . the same cuts were applied to derive a radio - loud ` early - type ' agn sample , except we used the more all - encompassing nvss flux estimates to cut in radio luminosity , thereby retaining the populations of galaxies with weak / no core radio emission but bright , extended radio lobes . this sample had higher completeness for comparison with the r - agn sample . a control sample was defined from the sdss sources with no match to a nvss - first source within 2 of their optical cores . we further considered only sources where fracdev(f ) @xmath195 to restrict all four samples to having good @xmath6-band devaucouleurs fits . we investigated the colour structure of agn host galaxies through @xmath10 , the ratio of @xmath6 to @xmath12 de vaucouleurs effective radii and used maximum likelihood analysis to quantify the degree of difference in the distribution of @xmath10 between samples . we concluded the radio ( r - agn / cr ) samples are not drawn from the same population as their radio - faint control samples at @xmath196 confidence : the presence of an agn increases the scale size of a galaxy in red light relative to blue light , on average . our result does not appear to be driven by the presence of blue agn in the radio - loud samples since the goodness of the de vaucouleurs fits does not become worse as @xmath10 increases . we found no structure in @xmath10 vs @xmath12-band goodness of fit in either radio sample . we found an excess of blue cores in radio - loud galaxies as compared to radio - quiet , ` early - type ' galaxies , implying the increased @xmath10 may be due to star formation in the central few kpc , in contrast with feedback models which predict the suppression of star formation near an agn . spectroscopic selection of our samples rules out major merger events , as starbursts can be triggered by the accretion of gas / tidal interactions . we also found radio agn hosts to have larger red scale lengths in relation to their blue light and note this to be a contributing factor in an increased @xmath10 . we can not definitively discern whether a small blue core or larger distribution of red light is the driving factor in this result . given the longer timescale for star formation than radio activity , our results imply a subset of galaxies have the predisposition to become radio - loud , rather than all galaxies undergoing bursts of radio activity at some stage in their lifetimes . em gratefully acknowledges support from the uk science and technology facilities council and thanks luke davies and james price for helpful comments . we thank the anonymous referee for helpful and insightful comments that have resulted in an improved analysis . in undertaking this research , we made extensive use of the topcat software @xcite . funding for the sdss has been provided by the alfred p. sloan foundation , the participating institutions , the national science foundation , the u.s . department of energy , the national aeronautics and space administration , the japanese monbukagakusho , the max planck society , and the higher education funding council for england . this research makes use of the nvss and first radio surveys , carried out using the national radio astronomy observatory very large array : nrao is operated by associated universities inc . , under cooperative agreement with the national science foundation . w. , fragile c. , croft s. , de vries w. , anninos p. , murray s. , 2004 , in storchi - bergmann t. , ho l. c. , schmitt h. r. , eds , the interplay among black holes , stars and ism in galactic nuclei vol . 222 of iau symposium , jet - induced star formation : good news from big , bad black holes . pp 485488 d. m. , 1989 , in j. hunt & b. battrick ed . , two topics in x - ray astronomy , volume 1 : x ray binaries . volume 2 : agn and the x ray background vol . 296 of esa special publication , population properties of agn in the soft x - ray band .

we construct a sample of 3,516 radio - loud host galaxies of active galactic nuclei ( agn ) from the optical sloan digital sky survey ( sdss ) and faint images of the radio sky at twenty cm ( first ) . these have 1.4ghz luminosities in the range @xmath0whz@xmath1 , span redshifts @xmath2 , are brighter than @xmath3mag and are constrained to ` early - type ' morphology in colour space ( @xmath4mag ) . optical emission line ratios ( at @xmath5 ) are used to remove type 1 agn and star - forming galaxies from the radio sample using bpt diagnostics . for comparison , we select a sample of 35,160 radio - quiet galaxies with the same @xmath6-band magnitude - redshift distribution as the radio sample . we also create comparison radio and control samples derived by adding the nrao vla sky survey ( nvss ) to quantify the effect of completeness on our results . we investigate the effective radii of the surface brightness profiles in the sdss @xmath7 and @xmath8 bands in order to quantify any excess of blue colour in the inner region of radio galaxies . we define a ratio @xmath9 and use maximum likelihood analysis to compare the average value of @xmath10 and its intrinsic dispersion between both samples . @xmath10 is larger for the radio - loud agn sample as compared to its control counterpart , and we conclude that the two samples are not drawn from the same population at @xmath11 significance . given that star formation proceeds over a longer time than radio activity , the difference suggests that a subset of galaxies has the predisposition to become radio loud . we discuss host galaxy features that cause the presence of a radio - loud agn to increase the scale size of a galaxy in red relative to blue light , including excess central blue emission , point - like blue emission from the agn itself , and/or diffuse red emission . we favour an explanation that arises from the stellar rather than the agn light . [ firstpage ]

You are an expert at summarizing long articles. Proceed to summarize the following text: strong electronic interactions are known to play a central role in disordered solids , of which coulomb glasses are a canonical example . the lack of metallic screening on the insulating side of the metal - insulator transition ( mit ) enables long - range coulomb interactions @xcite . efros and shklovskii ( es ) , following the original considerations for the non - interacting fermi glass case of mott@xcite , derived a form for the t = 0 k photon assisted frequency dependent conductivity describing the crossover from interacting coulomb glass - like behavior to fermi glass - like behavior@xcite . these derivations were based on a theory of resonant absorption@xcite and take into account the mean coulomb interaction between two sites forming a resonant pair @xmath1 , where @xmath2 $ ] is the most probable hop distance between pairs and @xmath3 is the dielectric constant . the real part of the es crossover form for the frequency dependent conductivity is : @xmath4^{4}[\hbar\omega + u(r_{w } ) ] \label{eq : esxover}\ ] ] where @xmath5 is a constant of order one , @xmath6 is the non - interacting single particle density of states ( dos ) , @xmath7 is the pre - factor of the overlap integral and @xmath8 is the localization length . the concentration dependent localization length is predicted to diverge as @xmath9 as the mit is approached , where @xmath10 is the dopant concentration , @xmath11 is the critical dopant concentration of the mit ( @xmath12 in si : p@xcite ) and @xmath13 is the localization length exponent . neglecting logarithmic factors , eq . ( [ eq : esxover ] ) predicts a gradual crossover from linear to quadratic behavior as the incident photon energy exceeds the interaction energy of a typical charge excitation . for the case where the photon energy , @xmath14 , one recovers the quadratic frequency dependence , plus logarithmic corrections , that mott originally derived for the non - interacting fermi glass case@xcite . in the opposite limit , @xmath15 the conductivity shows an approximately linear dependence on frequency , plus logarithmic corrections , and the material is called a coulomb glass . we should note that eq . ( [ eq : esxover ] ) was derived for the case where @xmath16 , the coulomb gap width . however a quasi - linear dependence ( albeit with a different pre - factor ) and an eventual crossover to mott s non - interacting quadratic law is still expected even for the case where @xmath17 . @xmath18 there is a lack of experimental evidence to either corroborate or disprove eq . ( [ eq : esxover ] ) due to the difficulties associated with performing frequency dependent measurements in the so - called quantum limit , i.e. @xmath0 , but at small enough photon energies so as to not be exciting charge carriers to the conduction band . moreover , in order to study the possible crossover from mott to es type behavior , one must measure across a broad enough bandwidth centered about the characteristic crossover energy scale for instance the coulomb interaction energy @xmath19 or the coulomb gap width@xcite , @xmath20 . there have been some very recent experiments that have attempted to address these issues . m. lee et al . found that for concentrations close to the mit the expected linear to quadratic crossover occurs , but is much sharper than predicted @xcite . they proposed that this sharp crossover was controlled not by the average interaction strength @xmath19 as in eq.([eq : esxover ] ) @xcite , but instead by a sharp feature in the density of states , i.e. the coulomb gap @xcite . they postulated that this coulomb gap was not the single particle one measured in tunneling , but rather a smaller `` dressed '' or renormalized coulomb gap that governs transport . there is some evidence from dc transport that such a feature exists , at least close to the mit @xcite . nominally uncompensated n - type silicon samples were obtained from recticon enterprises inc . a czochralski method grown boule with a phosphorous gradient along its length was cut into 1 mm thick discs . room temperature resistivity was measured using an ade 6035 gauge and the dopant concentration calibrated using the thurber scale@xcite . the si : p samples discussed here span a range from 39% to 69% , stated as a percentage of the sample s dopant concentration to the critical concentration at the mit . a number of samples were measured before and after etching with a @xmath21 solution ; this resulted in no difference in the results . in the millimeter spectral range , 80 ghz to 1000 ghz , backward wave oscillators ( bwo ) were employed as coherent sources in a transmission configuration@xcite . the transmitted power through the si : p samples as a function of frequency was recorded . for plane waves normally incident on a material , resonances occur whenever the thickness of the material is an integer number of half wavelengths . both components of the complex conductivity can be uniquely determined for each resonance . the real part of the conductivity was evaluated at microwave frequencies from the measured loss of highly sensitive resonant cavities at 35 and 60 ghz via the perturbation method . this is a common technique and is described in the literature@xcite . the conductivity as determined from the resonant cavity data was normalized to the dc conductivity at higher temperatures , at above approximately 25 k. the resonant cavity data confirmed the linear dependence on frequency of the real part of the complex conductivity into the microwave regime for the samples closest to critical . in fig . [ armitagecg1 ] , we show the t@xmath220 frequency dependent conductivity for two samples . this data , representative of all samples in our range , shows an approximately linear dependence at low frequencies and then a sharp crossover to an approximately quadratic behavior at higher frequencies . this is the qualitatively expected behavior from eq . ( [ eq : esxover ] ) . however , as seen by the overlayed fits , eq . ( [ eq : esxover ] ) provides only a rough guide . the solid lines are linear and quadratic fits to the low frequency and high frequency data respectively . the dotted line is a fit to the form of the es crossover function achieved by summing the separately determined linear and quadratic fits . as can be seen , the crossover between linear and quadratic portions is much more abrupt than the es function predicts . the dashed line is a fit using the same method as ref . @xcite , namely forcing the linear portion to pass through the low frequency data , as well as the origin and leaving the pre - factor of the quadratic term as a free variable . the fit is not satisfactory in either case . a sharp crossover as such is observed over our entire doping range and has been observed previously in an analogous system , si : b , for samples closer to the mit @xcite . note , that a linear dependence is seen in the imaginary part of the conductivity @xmath23 over the whole measured frequency and doping range @xcite . this is consistent with theoretical predictions @xcite . because our data spans a large range of concentrations , the doping dependence of the crossover energy scale can be analyzed to see whether its dependence is consistent with other energy scales , e.g. the coulomb interaction energy @xmath19 or the coulomb gap width @xmath20 as @xmath24 ref . recall that the coulomb interaction energy between two sites forming a resonant pair is @xmath25 which is dependent on concentration via the dielectric constant ( measured , but not shown ) and the localization length dependent most probable hop distance . by equating the crossover energy scale to the expected functional form for this coulomb interaction energy we are able to determine the magnitude of the localization length and its scaling exponent . with an appropriate pre - factor in the overlap integral @xcite , @xmath26 for the expression for the most probable hop distance term , @xmath27 , we get a localization length dependence of @xmath28 with a magnitude of 21.2 , 19.9 , 20.1 , 14.5 , 14.3 and 13.0 nm for the 69% , 62% , 56% , 50% , 45% and 39% samples respectively . the localization length exponent is close to unity , the value originally predicted by mcmillan in his scaling theory of the mit @xcite , and the magnitude of the localization length is reasonable . due to the fact that we obtain reasonable estimates for the relevant physical parameters over the whole doping range , we do not favor the previous speculation that it is in fact the coulomb gap energy that creates the sharp crossover and hence sets its energy scale@xcite . the approximately linear power law seen in the coulomb glass regime at low @xmath29 in fig . [ armitagecg1 ] of the conductivity can be expressed with the imaginary part as a simple kramer - kronig compatible form , @xmath30 in order to determine the power @xmath31 one can take the ratio of @xmath32 versus @xmath33 ( with the frequency as a variable ) . the power @xmath31 is given by , @xmath34 as determined from eq . 3 . the dashed line through the nbsi data is a guide to the eye . the bottom panel shows the divergence of the prefactor of the real part of the conductivity , and the dotted line is a simple power law fit.,width=264 ] fig . [ armitagecg2]a shows the ratio mentioned above of the imaginary to the real part of the dielectric constant for si : p . similar data from amorphous nbsi is included for comparison purposes@xcite . we note that this ratio for si : p is large and essentially constant across the entire doping range . from eq . ( [ eq : esingap ] ) , one expects @xmath35 to be approximately equal to @xmath32 to within a factor of 2 - 5 ( with a reasonable estimate for @xmath7 ) because @xmath36 . applied to si : p , the theory correctly predict a linear correspondence between @xmath33 and @xmath32 , but incorrectly predicts the proportionality by at least a factor of thirty . the proportionality is near the predicted value for nbsi , but has a dependence on the doping concentration which is presumably due to entering the quantum critical ( qc ) regime as discussed below . here we have used the susceptibility @xmath37 of the dopant electrons ( i.e. with the background dielectric constant of silicon subtracted ) in the expression for the magnitude of the imaginary component of the conductivity in eq . [ eq : alpha ] . the middle panel in fig . [ armitagecg2 ] shows the power @xmath31 as determined by eq . ( [ eq : alpha ] ) . the values for si : p are approximately equal to , but slightly less than one , consistent with fig . [ armitagecg1 ] . this indicates that the prefactor of the real and imaginary components of the complex conductivity have the same concentration dependence . the situation is different for nbsi . near the mit , @xmath38 is expected to cross over to the qc dynamics @xcite , i.e. @xmath39 when @xmath8 , the localization length , is of the same scale as @xmath40 , the dephasing length ( the characteristic frequency dependent length scale ) @xcite . this should be a smooth crossover and therefore looking at a fixed window of frequencies , a continuous change from @xmath41 is expected , similar to that measured for nbsi shown in the middle panel of fig [ armitagecg2 ] . setting the relations for localization length and dephasing length equal @xcite , one finds the crossover condition for the frequency in terms of the normalized concentration , @xmath42 where @xmath43 is the dynamic exponent . as the prefactor can vary from system to system , the fact that we see an @xmath44 across our entire doping range in si : p , but an @xmath31 that approaches @xmath45 in nbsi indicates that the critical regime in si : p is much narrower and out of our experimental window . this is consistent with simple dimensional arguments@xcite that show the crossover should be inversely proportional to the dopant density of states . the much smaller dopant density in si : p vs. nbsi ( a factor of @xmath46 ) is consistent with a narrower qc regime as compared to nbsi . the bottom panel in fig . [ armitagecg2 ] shows magnitude of the real part of the conductivity as the mit is approached . this demonstrates that the prefactor a can be written as a function of the normalized concentration , i.e. @xmath47 for si : p . in @xmath48 interacting systems , the effects of correlations become simpler as one goes to lower energies and/or lower temperatures . the canonical example of this is a fermi liquid where at t=0 and @xmath29=0 one recovers the non - interacting theory , but with parameters that are substantially modified ( renormalized ) from the free electron ones . the coulomb glass is a fundamentally important example in solid state physics because it belongs to a class of systems where this does not occur and the non - interacting functional forms are not recovered at asymptotically low energies . as predicted in the crossover function eq . [ eq : esxover ] , we have observed in si : p that the non - interacting functional form is recovered in the high - frequency limit and the low - frequency response shows interactions . within the theory this is a result of the additional internal excitation structure of a resonant pair caused by interactions and the fact that any one pair can be thought of as a distinct entity , well separated in energy from other spatially nearby pairs because of the large disorder induced energy spread . this internal structure enables the excitation of pairs relatively deep within the fermi sea and changes the factor in the initial state phase space from @xmath29 to @xmath49 . in contrast , in amorphous nbsi a smooth crossover is observed as the mit is approached from a linear frequency dependence to a power law characterized by an exponent smaller than unity . this is consistent with the eventual frequency dependence @xmath50 expected from quantum critical scaling arguments . no crossover to @xmath51 was observed in nbsi . since it is the exponent @xmath31 that distills the important physics ( it indicates the phase space of initial states for @xmath31 = 1 or 2 and the presence of critical dynamics for @xmath52 ) , we propose that one can classify the electrodynamics of electron glasses based on their @xmath31 value . a schematic showing the parameter space for @xmath31 is shown in fig . [ armitagecg3 ] . here one has @xmath31 s close to @xmath45 near the mit . there is a smooth crossover to coulomb glass - like @xmath53 at lower doping levels and an intervening non - interacting fermi glass regime at even higher energies and lower dopings . we expect that these general considerations are valid , despite the fact that some of the parameter space may not be accessible in certain systems . for instance , one may begin to excite structural modes at energies high enough to see @xmath51 behavior in nbsi . for dopings not close enough to critical in si : p , excitation to the conduction band may be observed before critical dynamics are . for @xmath54 . note that the boundaries drawn on the plot are smooth crossovers and not sharp onsets . a classification based on @xmath31 gives a taxonomy for the electrodynamics of electron glasses.,width=264 ] in summary , we have observed a crossover in the frequency dependence of the conductivity from coulomb glass - like behavior to fermi glass - like behavior across our entire range of doping concentrations in si : p . the existence of a crossover is consistent with theoretical predictions , but it is sharper than predicted . the fact that we see the same functional form over the whole doping range ( even deep into the insulating regime , where eq . [ eq : esxover ] is expected to be more valid ) shows that the nature of the low energy charge excitations is qualitatively the same over the whole doping range ; the inadequacy of eq . [ eq : esxover ] in describing the frequency dependent conductivity quantitatively is not limited to concentrations close to critical . in the amorphous semiconductor nbsi , we observe a gradual crossover from @xmath53 coulomb glass - like behavior to quantum critical - like dynamics . this comparison allows us to obtain a general classification scheme for the electrodynamics of electron glasses based on the exponent of the frequency dependence @xmath31 . we expect that this classification or ` taxonomy ' will be valid even when certain regimes are not experimentally accessible . we wish to thank phu tran for assisting with the cavity measurements and barakat alavi for assisting with the sample preparation . we would also like to thank steve kivelson and boris shklovskii for helpful conversations . this research was supported by the national science foundation grant dmr-0102405 .

we report measurements of the real and imaginary parts of the ac conductivity in the quantum limit , @xmath0 of insulating nominally uncompensated n - type silicon . the observed frequency dependence shows evidence for a crossover from interacting coulomb glass - like behavior at lower energies to non - interacting fermi glass - like behavior at higher energies across a broad doping range . the crossover is sharper than predicted and can not be described by any existing theories . despite this , the measured crossover energy can be compared to the theoretically predicted coulomb interaction energy and reasonable estimates of the localization length obtained from it . based on a comparison with the amorphous semiconductor nbsi , we obtain a general classification scheme for electrodynamics of electron glasses .

You are an expert at summarizing long articles. Proceed to summarize the following text: the discovery of quark - hadron duality in electron - nucleon interactions @xcite rises a natural question if the same phenomenon can be seen also in neutrino - nucleon scattering . unfortunately the available data is not yet precise enough to discuss the problem on the experimental level . at present the only possibility is to analyze the existing theoretical models of resonance production . this approach was adopted by sato and lee @xcite who discovered that for their model of @xmath4 production the quark - hadron duality is seen : the resonance peaks of structure functions calculated at @xmath5 gev@xmath3 slide along the dis structure functions ( with cteq6 pdfs ) calculated at @xmath6 gev@xmath3 , both as functions of the nachtmann variable . in this contribution we wish to report an investigation of the rein - sehgal ( rs ) model @xcite of resonance production . the rs model is used by almost all monte carlo generators of neutrino interactions and understanding its properties is of practical value . the model describes resonance production in the region of hadronic invariant mass @xmath7 below @xmath8 gev by summing contributions from 18 resonances . it contains also a non - resonant background fine tuned in order to get a good agreement with the existing single pion production ( spp ) data . we calculate the structure functions @xmath9 as they are defined by the rs model @xcite . they turn out to be linear combinations of cross sections for three polarization states of the intermediate @xmath7 boson . the aim of the rs model is to describe only spp channels and the elasticity coefficients are used in order to select exclusive spp channels from the overall resonance production cross section . the non - resonant background is adjusted only for spp channels . the analysis of duality requires the knowledge of the inclusive cross section in the resonance region . in the spirit of the rs model one should add contributions from inelastic channels taking care of complicated interference patterns . in order to make the analysis simpler we introduce 1@xmath10 functions which are defined as probabilities that in the given region of the kinematically allowed space the final state is that of spp . we calculate them numerically for each spp channel separately using the lund fragmentation and hadronization routines . all the details of our approach can be found in @xcite . functions defined in eq . [ 1pion_def ] with resonance elasticity factors taken from pdg @xcite . in the case of @xmath11-neutron scattering we show the sum of two functions corresponding to @xmath12 and @xmath13 reactions [ elasticity],width=302 ] it turns out that @xmath15 do not depend on @xmath16 . their plots for cc @xmath17 channels are presented in fig . [ elasticity ] . in the case of neutron we show the sum of 1@xmath10 functions corresponding to two exclusive spp channels . on the same plot we show also the available data on elasticity of @xmath18 and @xmath4 resonances . we see that 1@xmath10 functions provide satisfactory average description . we select the physically relevant region in the @xmath16 variable as @xmath28 gev@xmath3 by looking at double differential cross section @xmath29 for neutrino energy in a few gev region , see fig . 2 @xcite . having in mind mc generators of events the quark - hadron duality should ensure the smooth passage of the cross section for res and dis contributions in the relevant kinematical transition region . structure function for cc neutrino reactions on proton , neutron and isoscalar target . @xmath1 for rs model is calculated at @xmath30 gev@xmath3 and for dis at @xmath6 gev@xmath3 . [ slizganie],width=302 ] in fig . [ slizganie ] we show @xmath1 structure functions for cc reactions defined by the rs model for three values of @xmath16 : @xmath31 gev@xmath3 and also dis @xmath1 structure functions calculated at @xmath32 gev@xmath3 . the targets are from the top to the bottom : proton , neutron and their average isoscalar one . the plots for the dis part are done with grv94 pdfs @xcite . rs structure functions are yet not re - scaled by 1@xmath10 functions . similar plots for @xmath1 at @xmath33 gev@xmath3 were shown before in @xcite . we see that in the case of proton and isoscalar target typical manifestation of local quark - hadron duality : the @xmath4 peak slides along the dis curve . the duality does not seem to apply neither to other resonances nor to neutron target even in the case of the prominent @xmath4 resonance . in fig . [ 1pion_in_action ] we show the role played by 1@xmath10 functions . on the same plot both rein - sehgal and 1@xmath10 function re - scaled res @xmath1 structure functions for cc reaction on neutron are presented . we see that the modifications apply mostly to the region of @xmath7 close to 2 gev . [ ratios ] presents the functions defined in eq . [ ratio_q2=const ] for three different integration regions i.e. for three definitions of the resonance region : @xmath34 where @xmath35 gev . we see that the choice @xmath36 gev makes the functions slowly varying in the wide region of @xmath16 . the behavior for small values of @xmath37 gev@xmath3 is very different in agreement with the predictions made in @xcite . for the proton target the best choice is @xmath36 gev while for other two targets it is preferable to choice the resonance region as more confined at the price of significant variations with @xmath16 . the choice @xmath36 gev is a natural one for the rs model . but it is suggested in @xcite that the rs model underestimates the cross section for @xmath38 gev . we conclude that with the choice @xmath36 gev the duality holds very well for the proton and badly for other targets . it seems to be difficult to have duality for all the targets simultaneously . p. lipari , _ calculation of the neutrino cross section . open problems , lines of research _ a talk given at _ third international workshop on neutrino - nucleus interactions in the few - gev region _ , assergi , march 17 - 21 , 2004 . a. bodek , _ nuint02 conference summary : modelling quasi - elastic , resonance and inelastic neutrino and electron scattering on nucleons and nuclei _ , a talk at _ second international workshop on neutrino - nucleus interactions in the few - gev region _ , irvine , dec 12 - 15 , 2002 .

quark - hadron duality in neutrino - nucleon reactions is investigated under the assumption that cross sections in the resonance region are given by the rein - sehgal model . the quantitative analysis of the duality is done by means of appropriate integrals of the structure functions in the nachtmann variable . we conclude that with the definition of the resonance region @xmath0 gev ) the duality holds for neutrino - proton reaction @xmath1 structure function for @xmath2 gev@xmath3 and it is absent for neutrino - neutron reaction .

You are an expert at summarizing long articles. Proceed to summarize the following text: oscillatory integrals play an important role in the theory of pseudodifferential operators . they are also a useful tool in mathematical physics , in particular in quantum field theory , where they are used to give meaning to formal fourier integrals in the sense of distributions . for phase functions which are homogeneous of order one , this also leads to a characterization of the wave front set of the resulting distribution , as it is known to be contained in the manifold of stationary phase of the phase function . in these applications , the restriction to phase functions that are homogeneous of order one is often obstructive . in many cases , this restriction can be overcome by shifting a part of the would - be phase function to the symbol , cf . example [ ex : delta+ ] below . however , such a shift is not always possible , for instance if the would - be phase function contains terms of order greater than one . such phase functions are present in the twisted convolutions that occur in quantum field theory on moyal space , cf . examples [ ex : ncqft ] and [ ex : ncqftb ] below . up to now , a rigorous definition of these twisted convolution integrals could be given only in special cases and in such a way that the information on the wave front set is lost . thus , it is highly desirable to generalize the notion of oscillatory integrals to encompass also phase functions that are not homogeneous of order one . such generalizations were proposed by several authors . however , to the best of our knowledge , the wave front sets of the resulting distributions were not considered , except for one very special case . we comment on these settings below , cf . remark [ rem : asadafujiwara ] . it is shown here that the restriction to phase functions that are homogeneous of order one can indeed be weakened , without losing information on the wave front set . the generalization introduced here not only allows for inhomogeneous phase functions , but also for phase functions that are symbols of any positive order . however , one has to impose a condition that generalizes the usual nondegeneracy requirement . it is also shown that the wave front sets of the distributions thus obtained are contained in a set that generalizes the notion of the manifold of stationary phase . we conclude with a discussion of some applications . throughout , we use the following notation : for an open set @xmath0 , @xmath1 means that @xmath2 is a compact subset of @xmath0 . @xmath3 stands for @xmath4 . for a subset @xmath5 , @xmath6 stands for the projection on the first component @xmath7 denotes the @xmath8 times continuously differentiable functions supported on @xmath0 and @xmath9 the set of elements of @xmath7 with compact support in @xmath0 . the dual space of @xmath10 is denoted by @xmath11 and @xmath12 . the pairing of a distribution @xmath13 and a test function @xmath14 is denoted by @xmath15 . the dot @xmath16 stands for the scalar product on @xmath17 . @xmath18 denotes the angle between two vectors @xmath19 . as usual , cf . @xcite , we define a symbol as follows : let @xmath20 be an open set . a function @xmath21 is called a _ symbol of order @xmath22 _ if for each @xmath1 and multiindices @xmath23 , we have @xmath24 the set of all such functions , equipped with these seminorms will be denoted by @xmath25 . furthermore , we denote @xmath26 and @xmath27 . for simplicity , we restrict ourselves to these symbols . the generalization to the symbols @xmath28 is straightforward . one then has to restrict to @xmath29 , @xmath30 , where @xmath31 is the order of the phase function introduced below . also the generalization to asymptotic symbols as discussed in @xcite is straightforward . the following proposition is a straightforward consequence of the definition of @xmath25 : [ prop : cont ] the maps @xmath32 , @xmath33 and the multiplication @xmath34 are continuous . the following proposition is proven in ( * ? ? ? 1.7 ) : [ prop : dense ] if @xmath35 , then @xmath36 is dense in @xmath25 for the topology of @xmath37 . now we introduce our new definition of a phase function . [ def : phase ] a _ phase function _ of order @xmath31 on @xmath38 is a function @xmath39 such that 1 . @xmath40 is a symbol of order @xmath41 . [ enum : phase ] for each @xmath1 there are positive @xmath42 such that @xmath43 [ rem : phase ] condition [ enum : phase ] generalizes the usual nondegeneracy requirement and ensures that @xmath40 oscillates rapidly enough for large @xmath44 . in particular it means that @xmath40 is not a symbol of order less than @xmath31 . it also means that one can choose @xmath45 such that @xmath46 is well - defined and a symbol of order @xmath47 . here @xmath48 can be chosen such that @xmath49 is compact for each @xmath50 . [ rem : asadafujiwara ] our definition of a phase function is a generalization of a definition introduced by hrmander ( * ? ? ? 2.3 ) in the context of pseudodifferential operators . he considered phase functions of order 1 ( in the nomenclature introduced above ) and characterized the singular support of the resulting distribution , but not its wave front set . our characterization of the singular support ( cf . corollary [ cor : m ] ) coincides with the one given by hrmander ( * ? ? ? inhomogeneous phase functions were also considered by asada and fujiwara @xcite in the context of pseudodifferential operators on @xmath51 . in their setting , @xmath52 , @xmath53 and there must be a positive constant @xmath54 such that @xmath55 furthermore , all the entries of this matrix ( and their derivatives ) are required to be bounded . thus , the phase function is asymptotically at least of order 1 and at most of order 2 . the admissible amplitudes are at most of order 0 . the wave front set of such operators on @xmath51 is not considered by asada and fujiwara . the same applies to the works of boulkhemair @xcite and ruzhansky and sugimoto @xcite , who work in a similar context . coriasco @xcite considered a special case of hrmander s framework , where again @xmath52 , @xmath53 and @xmath56 with @xmath57 , a subset of the symbols of order 1 . furthermore , he imposed growth conditions on @xmath58 that are more restrictive than condition [ enum : phase ] . the resulting operators on @xmath59 can then be extended to operators on @xmath60 . if a further condition analogous to is imposed , then also the wave front set , which is there defined via @xmath61-microregularity , can be characterized ( at least implicitly , by the change of the wave front set under the action of the operator ) . [ prop : diff ] if @xmath40 is a phase function of order @xmath31 and @xmath62 and there is a @xmath63 such that @xmath64 , then @xmath65 and the map @xmath66 is continuous . for @xmath67 we have @xmath68 so that @xmath69 is continuous . differentiation gives @xmath70 the expression in curly brackets is a symbol of order @xmath71 . with the same argument as before one can thus differentiate @xmath72 times . we formulate the main theorem of this section analogously to ( * ? ? ? * thm . the proof is a straightforward generalization of the proof given there . [ thm : osc ] let @xmath73 be a phase function of order @xmath31 on @xmath38 . then there is a unique way of defining @xmath74 for @xmath75 such that @xmath76 coincides with when @xmath62 for some @xmath67 and such that , for all @xmath22 , the map @xmath77 is continuous . moreover , if @xmath63 and @xmath78 , then the map @xmath79 is continuous . to prove this , we need the following lemma : [ lemma : v ] let @xmath40 be a phase function of order @xmath31 on @xmath38 . then there exist @xmath80 , @xmath81 and @xmath82 such that for the differential operator @xmath83 with adjoint @xmath84 we have @xmath85 furthermore , @xmath86 is a continuous map from @xmath25 to @xmath87 . we choose @xmath88 as in definition [ def : phase ] and @xmath48 as in remark [ rem : phase ] . we set @xmath89 then we have @xmath90 it is easy to see that @xmath91 , @xmath92 and @xmath93 are symbols in the required way . the last statement follows from proposition [ prop : cont ] . the uniqueness is a consequence of proposition [ prop : dense ] . for @xmath94 and @xmath95 , we have , with @xmath86 as in lemma [ lemma : v ] , @xmath96 { \mathrm{d}}^nx { \mathrm{d}}^s\theta,\end{aligned}\ ] ] for any @xmath63 and thus @xmath97 \rvert } { \mathrm{d}}^nx { \mathrm{d}}^s\theta.\ ] ] now the multiplication @xmath98 is continuous . thus , if @xmath62 , then @xmath99 $ ] is a symbol of order @xmath100 and in particular we have , for each @xmath1 , @xmath101 \rvert } ( 1+{\lvert \theta \rvert})^{p\mu - m } \leq f_{p , k}(a ) \sum_{{\lvert \alpha \rvert } \leq p } \sup_{x \in k } { \lvert d^\alpha f \rvert},\ ] ] where @xmath102 is a seminorm on @xmath25 . for @xmath62 , we may thus choose @xmath72 such that @xmath103 and define @xmath104 { \mathrm{d}}^nx { \mathrm{d}}^s\theta.\ ] ] as the sum on the r.h.s . of is a seminorm on @xmath105 , and due to the continuity properties discussed above , the map @xmath106 is continuous . for @xmath107 , we have @xmath108 by . thus , we can unambiguously define @xmath109 . we may now further characterize the distributions that result from a generalized oscillatory integral . [ def : sp ] let @xmath40 be a phase function of order @xmath31 on @xmath38 . we define @xmath110 we call @xmath111 the _ asymptotic manifold of stationary phase_. by definition , it is conic . [ lemma : sp ] @xmath112 is a closed conic subset of @xmath113 . @xmath111 is a closed subset of @xmath114 . from the definition of @xmath112 it follows that if @xmath115 , then @xmath116 for all @xmath117 , so @xmath112 is conic . we now show that @xmath118 is open in @xmath113 . let @xmath119 be such that there are positive @xmath42 such that @xmath120 we set @xmath121 . by taylor s theorem we have @xmath122 where @xmath123 and @xmath124 fulfill the bounds @xmath125 here @xmath126 are chosen such that @xmath127 , @xmath128 . furthermore , we restrict to @xmath129 . then we may use that @xmath13 is a symbol of order @xmath130 to conclude that there are positive constants @xmath131 , which are bounded for @xmath132 , for which @xmath133 holds . as the zeroth order term in the taylor expansion grows faster than @xmath134 for a fixed positive constant @xmath54 , we can make @xmath135 so small that @xmath136 grows faster than @xmath137 for some positive @xmath138 . thus , @xmath112 is closed in @xmath113 . in order to prove the closedness of @xmath111 , we first note that if @xmath139 , then by the above there is a neighborhood of @xmath140 that does not intersect @xmath141 . thus , it suffices to show that for @xmath142 , @xmath143 there is a neighborhood that does not intersect @xmath111 . by condition [ enum : phase ] of definition [ def : phase ] , there must be positive constants @xmath42 such that @xmath144 by the same argument as above , such a bound holds true in a conic neighborhood @xmath145 of @xmath146 . by the definition of @xmath111 , there are posititve @xmath147 such that @xmath148 we now want to show that one can choose a conic neighborhood @xmath149 of @xmath119 , contained in @xmath145 , such that an analogous bound holds , i.e. , there are positive @xmath147 so that @xmath150 by the above construction , @xmath151 grows as @xmath152 in @xmath145 . the deviations that occur by varying @xmath50 and @xmath44 also scale as @xmath152 , as @xmath40 is a symbol of order @xmath31 . recalling @xmath153 and again using taylor s theorem , one shows that by making @xmath149 small enough , one still retains an inequality of the form . by suitably restricting @xmath149 in @xmath0 , we can ensure that for @xmath154 we have @xmath155 whenever @xmath115 . then no new direction @xmath44 for which we would have to check the bound can appear while varying @xmath50 in @xmath156 . given , it is clear that we can also take a conic neighborhood @xmath157 of @xmath158 , by tilting it by angles less than @xmath159 . choosing @xmath160 gives a neighborhood of @xmath161 that does not intersect @xmath111 . [ prop : m ] if the support of the symbol @xmath162 does not intersect @xmath112 , then @xmath76 is smooth . we choose a neighborhood @xmath163 of @xmath164 whose closure does not intersect @xmath112 . we choose a smooth function @xmath48 that is equal to one in a neighborhood of @xmath112 and vanishes on @xmath165 . we choose another smooth function @xmath166 on @xmath38 with support in @xmath163 which is identical to one whenever @xmath167 for @xmath168 . by definition of @xmath112 , the set @xmath169 is bounded , so we can choose @xmath166 such that @xmath170 is compact for each @xmath50 . then we define @xmath171 by the definition of @xmath112 and @xmath48 , we have @xmath80 and @xmath82 . with these definitions , we have @xmath172 here we used that @xmath48 and @xmath166 have nonoverlapping supports . as @xmath173 and @xmath86 differentiates only w.r.t . @xmath44 , we thus have @xmath174 for arbitrary integer @xmath72 . as @xmath86 maps symbols of order @xmath22 to symbols of order @xmath175 , @xmath76 is smooth by proposition [ prop : diff ] . [ cor : m ] the singular support of @xmath76 is contained in @xmath141 . [ thm : sp ] the wave front set of @xmath76 is contained in @xmath111 . [ lemma : sp1 ] let @xmath176 , @xmath143 . then there is a conic neighborhood @xmath145 of @xmath119 , a conic neighborhood @xmath86 of @xmath158 and positive constants @xmath177 such that @xmath178 furthermore , there is a positive @xmath138 such that @xmath179 condition is fulfilled for @xmath119 by condition [ enum : phase ] of definition [ def : phase ] . that it is also fulfilled in a neighborhood of @xmath119 can be shown analogously to the proof of the closedness of @xmath112 in lemma [ lemma : sp ] . condition is fulfilled for @xmath180 . that it is also fulfilled in a neighborhood of @xmath180 can again be shown as in lemma [ lemma : sp ] . in order to prove the last statement , we note that by we have @xmath181 where @xmath182 has length @xmath183 and lies on the cone with angle @xmath159 around @xmath8 ( see the figure , where @xmath184 is denoted by @xmath72 ) . for the distance of @xmath8 and @xmath185 we have the bound ( see the dashed lines in the figure ) @xmath186 using , we then obtain the above statement . by corollary [ cor : m ] , it suffices to consider points @xmath187 . let @xmath143 . due to proposition [ prop : m ] , we may assume that @xmath162 is supported in an arbitrarily small closed conic neighborhood @xmath145 of @xmath112 . we thus need to show that there is a @xmath188 , identically one near @xmath140 and a conic neighborhood @xmath86 of @xmath158 such that for each @xmath189 there is a seminorm @xmath190 on @xmath25 such that @xmath191 for all @xmath162 supported in @xmath145 . as in the proof of theorem [ thm : osc ] , it suffices to construct such a bound for @xmath94 and then make use of proposition [ prop : dense ] . let @xmath192 be as in lemma [ lemma : sp1 ] . choose @xmath193 that is identically one near @xmath140 and whose support is contained in @xmath194 . choose a @xmath195 that is identical to one on @xmath196 . now we set @xmath197 then for @xmath198 we have @xmath199 now we choose @xmath200 , identical to one near @xmath140 and with support in @xmath201 . we also choose @xmath202 which is identically one on @xmath203 . we consider @xmath204 . then by proposition [ prop : diff ] , the second term yields a smooth function , so that the above bound is fulfilled . it remains to consider @xmath205 { \mathrm{d}}^n x { \mathrm{d}}^s \theta \rvert } \\ & \leq \int { \lvert v_k^p[\psi(x ) ( 1-\xi(\theta ) ) a(x , \theta ) ] \rvert } { \mathrm{d}}^n x { \mathrm{d}}^s \theta.\end{aligned}\ ] ] here we used that @xmath206 is identically one on the support of @xmath207 . by lemma [ lemma : sp1 ] , one now has @xmath208 \rvert } \leq c_{m , p}(a ) ( 1+{\lvert \theta \rvert})^m ( { \lvert \theta \rvert}^\mu + { \lvert k \rvert})^{-p},\ ] ] where @xmath209 is a seminorm on @xmath25 . by using @xmath210 one can make the integral convergent and assure by choosing @xmath72 large enough . [ ex : delta+ ] we consider the two - point function @xmath211 of a free massive scalar relativistic field , where one has @xmath212 and @xmath213 with @xmath214 here , we use the notation @xmath215 , with @xmath216 . note that @xmath40 is not homogeneous . in @xcite this problem is circumvented by using @xmath217 as phase function and multiplying the symbol with the function @xmath218 . it is then no longer a symbol ( as it is not smooth in @xmath219 ) , so one has to allow for so - called asymptotic symbols . furthermore , one has to show that the multiplication with such a term does not spoil the fall - off properties , in particular that differentiation w.r.t . @xmath44 lowers the order . in the present approach , this is not necessary . @xmath40 is a phase function in the sense of definition [ def : phase ] and therefore , by theorem [ thm : osc ] , it defines an oscillatory integral for every symbol @xmath162 . in order to find the wave front set , we compute @xmath220 it is easy to see that its modulus is bounded from below by a positive constant unless @xmath221 or @xmath222 and @xmath223 . thus , we have @xmath224 furthermore , @xmath225 for large @xmath44 , this behaves as @xmath226 where the remainder term @xmath227 scales as @xmath228 . thus , only in the directions @xmath229 the bound on the angle of @xmath8 and @xmath230 can not be fulfilled . hence , we obtain the well - known result . in that convention , the sign of the zeroth component in the cotangent bundle has to be reversed . ] @xmath231 a variant of this example is obtained by considering phase functions of the form @xmath232 with @xmath14 a positive function that is a symbol of order @xmath233 . such expressions occur for example in quantum field theory on the moyal plane with hyperbolic signature and signify a distortion of the dispersion relations , cf . the above trick to put @xmath234 into the symbol still works , but then the symbol will be of type @xmath235 , where @xmath236 is original type of the symbol . it is straightforward to check that still defines a phase function of order 1 in the sense defined here , and that its stationary phase is as above . if the function @xmath14 in is a symbol of order @xmath237 , of corresponds to solutions of the hyperbolic wave equation @xmath238 . the modification suggested here means that the underlying pde is no longer hyperbolic . ] then the shift of @xmath234 to the symbol is not possible , as this would no longer give a symbol of type @xmath239 . thus , a treatment of in the context of phase functions that are homogeneous of order 1 is not possible . however , one can still interpret as a phase function of order @xmath240 and easily computes @xmath241 [ ex : ncqft ] in quantum field theory on the moyal plane of even dimension @xmath242 with euclidean signature , one frequently finds phase funtions of the form @xmath243 here @xmath244 is some real antisymmetric @xmath245 matrix of maximal rank @xmath242 . the above is clearly a symbol of order 2 , and we have @xmath246 as @xmath244 has rank @xmath242 , condition [ enum : phase ] of definition [ def : phase ] is fulfilled . from the above it follows that @xmath247 and thus also @xmath248 , so that the resulting distributions are smooth . we note that up to now such integrals could only be treated in the so - called adiabatic limit @xcite . but then one loses the information about the singular behaviour in position space , contrary to the present case , where the wave front set is known completely . [ ex : ncqftb ] in quantum field theory on the moyal plane with hyperbolic signature , one frequently finds phase functions of the form @xmath249 where @xmath250 with @xmath251 as in and @xmath244 as in example [ ex : ncqft ] . the above is a symbol of order 2 , but it is not a phase function as defined here , as can most easily be seen in the case @xmath252 . then with @xmath253 one obtains @xmath254 if the signs of @xmath255 and @xmath256 coincide , then the above derivatives tend to a constant as a function of @xmath44 , so that condition [ enum : phase ] of definition [ def : phase ] is not fulfilled . the rigourous treatment of such integrals is an open problem , which we plan to address in future work . it is a pleasure to thank dorothea bahns for helpful discussions and her detailed comments on the manuscript . i would also like to thank ingo witt for valuable comments . this work was supported by the german research foundation ( deutsche forschungsgemeinschaft ( dfg ) ) through the institutional strategy of the university of gttingen . d. bahns , s. doplicher , k. fredenhagen and g. piacitelli , _ field theory on noncommutative spacetimes : quasiplanar wick products _ , phys . d * 71 * , 025022 ( 2005 ) . c. dscher and j. zahn , _ dispersion relations in the noncommutative @xmath258 and wess - zumino model in the yang - feldman formalism _ , annales henri poincare * 10 * , 35 ( 2009 ) . r. wulkenhaar , _ field theories on deformed spaces _ , j. geom . phys . * 56 * , 108 ( 2006 ) . c. dscher , _ yang - feldman formalism on noncommutative minkowski space _ , ph.d . thesis , hamburg ( 2006 ) .

a generalized notion of oscillatory integrals that allows for inhomogeneous phase functions of arbitrary positive order is introduced . the wave front set of the resulting distributions is characterized in a way that generalizes the well - known result for phase functions that are homogeneous of order one .

You are an expert at summarizing long articles. Proceed to summarize the following text: the recent high - precision proper motion ( pm ) measurements of the l / smc determined by ( * ? ? ? * kallivayalil et al . ( 2006a , 2006b - hereafter k06a and k06b ; see also these proceedings ) ) imply that the magellanic clouds are moving @xmath0100 km / s faster than previously estimated and now approach the escape velocity of the milky way ( mw ) . * besla et al . ( 2007 ) ) ( hereafter b07 ) re - examined the orbital history of the clouds using the new pms and a @xmath3cdm - motivated mw model and found that the l / smc are either on their first passage about the mw or , if the mass of the mw is @xmath4 , that their orbital period and apogalacticon distance are a factor of three larger than previously estimated . this means that models of the magellanic stream ( ms ) need to reconcile the fact that although the efficient removal of material via tides and/or ram pressure requires multiple pericentric passages through regions of high gas density , the pms imply that the clouds did not pass through perigalacticon during the past @xmath55 gyr ( this is true even if a high mass mw model is adopted ) . while the most dramatic consequence of the new pms is the limit they place on the interaction timescale of the clouds with the mw , there are a number of other equally disconcerting implications : the relative velocity between the clouds has increased such that only a small fraction of the orbits within the pm error space allow for stable binary l / smc orbits ( k06b , b07 ) ; the velocity gradient along the orbit is much steeper than that observed along the ms ; and the past orbits are not co - located with the ms on the plane of the sky ( b07 ) . in these proceedings the listed factors are further explored and used to argue that the ms is not a tidal tail . ) k06b pm error space for the smc ( where the mean value is indicated by the triangle ) . each corresponds to a unique 3d velocity vector and is color coded by the number of times the separation between the clouds reaches a minimum within a hubble time . the circled dot indicates the gn96 pm for the smc and the asterisk corresponds to the mean of the ( * ? ? ? * piatek et al . ( 2008 ) ) ( p08 ) re - analysis of the k06b data - neither correspond to long - lived binary states . the clouds are modeled as plummer potentials with masses of @xmath6 and @xmath7 and the mw is modeled as a nfw halo with a total mass of @xmath8 as described in b07 . the lmc is assumed to be moving with the mean k06a pm ( v=378 km / s ) . the black square represents a solution for the smc s pm that allows for close passages between the clouds at characteristic timescales ( see fig . [ fig2 ] ) and is our fiducial case.,width=307 ] and assuming a mass ratio of 10:1 between the l / smc . the separation reaches a minimum at @xmath0300 myr and @xmath01.5 gyr in the past , corresponding to the formation times for the bridge and the ms . , width=307 ] doubt concerning the binarity of the clouds is particularly troubling , as a recent chance encounter between dwarf galaxies in the mw s halo is improbable if they did not have a common origin . to address this issue , ten thousand points were randomly drawn from the smc pm error space ( k06b ) , each corresponding to a unique velocity vector and orbit ( fig . [ fig1 ] ) . bound orbits are identified and color coded based on the number of times the separation between the clouds reaches a minimum , assuming a mass ratio of 10:1 between the l / smc ( although the mass ratio is not well constrained ) . orbits with only one close encounter ( like for the smc pm determined in the re - analysis of the k06b data by ( * ? ? ? * piatek et al . 2008 ) , hereafter p08 ) are not stable binary systems . the new lmc pm also implies that orbits where the smc traces the ms on the plane of the sky ( like that chosen by ( * ? ? ? * gardiner & noguchi 1996 ) , hereafter gn96 ) are no longer binary orbits . it is clear from fig . [ fig1 ] that stable binary orbits exist within 1@xmath1 of the mean k06b value - however , in all cases the smc s orbit about the lmc is highly eccentric ( fig . [ fig2 ] ) , which differs markedly from the conventional view that the smc is in a circular orbit about the lmc ( gn96 , ( * ? ? ? * gardiner et al . 1994 ) ) . it should also be noted that the likelihood of finding a binary l / smc system that is stable against the tidal force exerted by the mw decreases if the mw s mass is increased . we further require that the last close encounter between the clouds occurred @xmath0300 myr ago , corresponding to the formation timescale of the magellanic bridge ( ( * ? ? ? * harris 2007 ) ) , and that a second close encounter occurs @xmath01.5 gyr ago , a timeframe conventionally adopted for the formation of the ms ( gn96 ) . a case that also satisfies these constraints is indicated in fig . [ fig1 ] by the black square and will be referred to as our fiducial smc orbit . the corresponding orbital evolution of the smc about the lmc is plotted in fig . [ fig2 ] : the new pms are not in conflict with known observational constraints on the mutual interaction history of the clouds . this provides an important consistency check for the k06a , b pms : if the measurements suffered from some unknown systematics , it would be unlikely for binary orbits to exist within the error space . the spatial location of the fiducial orbit on the plane of sky and the line - of - sight velocity gradient along it are compared to the observed properties of the ms . the gn96 orbits were a priori chosen to trace both the spatial location and velocity structure of the ms , but this is an assumption . indeed , from fig . [ fig3 ] , the lmc s orbit using the new pm is found to be offset from the ms ( indicated by the gn96 orbits ) by roughly @xmath9 . the offset arises because the north component of the lmc s pm vector as defined by k06a , the re - analysis by p08 , _ and _ the weighted average of all pm measurements prior to 2002 ( ( * ? ? ? * van der marel et al . 2002 ) ) , is not consistent with 0 ( which was the assumption made by gn96 ) : this result is thus independent of the mw model ( b07 ) . furthermore , the smc must have a similar tangential motion as the lmc in order to maintain a binary state , meaning that our fiducial smc orbit deviates even further from the ms than that of the lmc . in addition , the line - of - sight velocity gradient along the lmc s orbit is found to be significantly steeper than that of the ms , reaching velocities @xmath0200 km / s larger than that observed at the same position along the ms ( fig . [ fig4 ] ) . in the sky . the fiducial orbits deviate markedly from the current location of the ms , which is traced by the gn96 orbits.,width=307 ] the offset and steep velocity gradient are unexplainable in a tidal model . while tidal tails may deviate from their progenitor s orbit , they remain confined to the orbital plane ( ( * ? ? ? * choi et al . 2007 ) ) : since the clouds are in a polar orbit no deviation is expected in projection in a tidal model . furthermore , material in tails is accelerated by the gravitational field of the progenitor - however , to explain the observed velocity gradient the opposite needs to occur . coupling these factors to a first passage scenario strongly suggests that , while tides likely help shape the stream ( e.g. the leading arm feature ) , hydrodynamic processes are the _ primary _ mechanism for the removal of material from the clouds and for shaping its velocity structure ( e.g. via gas drag ) . ) with respect to the local standard of rest are plotted as a function of magellanic longitude ( l ) ( see b07 , fig . 20 ) along the lmc s orbit . the orbital velocity corresponding to the mean pm for the b07 fiducial(isothermal sphere ) mw model is indicated by the solid red(dashed red ) line . the best and worst @xmath10 fits to the ms data within the pm error space are indicated by the arrows and the blue lines . the black line indicates the hi data of the ms from ( * ? ? ? * putman et al . ( 2003 ) ) ( p03 ) . the velocity gradients along the new orbits in both the isothermal sphere and fiducial nfw mw models are significantly steeper than the gn96 results ( magenta line ) , which were contrived to trace the velocity data.,width=307 ] the main difficulty for ram pressure stripping in a first passage scenario is the low halo gas densities at large galactocentric distances . the efficiency of stripping may be improved if material is given an additional kick by stellar feedback ( e.g. ( * ? ? ? * nidever et al 2008 ) , ( * ? ? ? * olano 2004 ) ) or if the lmc initially possessed an extended disk of hi like those observed in isolated dirrs - note that the latter is not a viable initial condition if the lmc were not on its first passage and the former may violate metallicity constraints on the ms which indicate the ms is metal poor ( ( * ? ? ? * gibson et al . 2000 ) ) . if ram pressure stripping is efficient , the offset may occur naturally : ( * ? ? ? * roediger & brggen ( 2006 ) ) have shown that material can be removed asymmetrically from gaseous disks that are inclined relative to their line of motion ( the lmc s disk is inclined by @xmath11 ) and caution that tails do not always indicate the direction of motion of the galaxy . these authors considered ram pressure stripping in the context of massive galaxies in cluster environments . we are currently conducting simulations of the formation of the ms via the ram pressure stripping of the clouds , assuming they initially entered the mw system with extended gaseous disks . [ fig5 ] illustrates the proposed mechanism at work : here the lmc has been moving at 380 km / s through a box of gas at a uniform temperature of @xmath12 k and density of @xmath13 /@xmath14 for 300 myr . once the material is removed beyond the lmc s tidal radius , the mw s tidal force may then be able to stretch the material to its full extent - but now since the material is removed asymmetrically it will not trace the orbit in projection . . the left panel is the face - on view and the right is edge - on : in all cases the lmc is moving to the left at 380 km / s and is inclined 30@xmath15 relative to its direction of motion . the snapshots indicated the evolution of the gas surface density after 0.3 gyr . the face - on projection illustrates how material is preferentially removed from the side of the disk rotating in - line with the ram pressure wind . in the edge - on projection , material from the leading edge lags behind that removed from the trailing edge . , title="fig:",width=249 ] . the left panel is the face - on view and the right is edge - on : in all cases the lmc is moving to the left at 380 km / s and is inclined 30@xmath15 relative to its direction of motion . the snapshots indicated the evolution of the gas surface density after 0.3 gyr . the face - on projection illustrates how material is preferentially removed from the side of the disk rotating in - line with the ram pressure wind . in the edge - on projection , material from the leading edge lags behind that removed from the trailing edge . , title="fig:",width=249 ] the new pms have dramatic implications for phenomenological studies of the clouds that assume they have undergone multiple pericentric passages about the mw and/or that the smc is in a circular orbit about the lmc . the orbits deviate spatially from the current location of the ms on the plane of the sky and the velocity gradient along the orbit is much steeper than that observed . these results effectively rule out a purely tidal model for the ms and lend support for hydrodynamical models , such as ram pressure stripping . the offset further suggests that the clouds have travelled across the little explored region between ra 21@xmath2 and 23@xmath2 ( i.e. the region spanned by the blue lines in fig . [ fig3 ] ) . * putman et al . ( 2003 ) ) detected diffuse hi in that region that follow similar velocity gradients as the main stream ( their fig . 7 ) , but otherwise material in that region has been largely ignored by observers and theorists alike . the offset orbits suggest that the ms may be significantly more extended than previously believed and further observations along the region of sky they trace are warranted .

the _ hst _ proper motion ( pm ) measurements of the clouds have severe implications for their interaction history with the milky way ( mw ) and with each other . the clouds are likely on their first passage about the mw and the smc s orbit about the lmc is better described as quasi - periodic rather than circular . binary l / smc orbits that satisfy observational constraints on their mutual interaction history ( e.g. the formation of the magellanic bridge during a collision between the clouds @xmath0300 myr ago ) can be located within 1@xmath1 of the mean pms . however , these binary orbits are not co - located with the magellanic stream ( ms ) when projected on the plane of the sky and the line - of - sight velocity gradient along the lmc s orbit is significantly steeper than that along the ms . these combined results ultimately rule out a purely tidal origin for the ms : tides are ineffective without multiple pericentric passages and can neither decrease the velocity gradient nor explain the offset stream in a polar orbit configuration . alternatively , ram pressure stripping of an extended gaseous disk may naturally explain the deviation . the offset also suggests that observations of the little - explored region between ra 21@xmath2 and 23@xmath2 are crucial for characterizing the full extent of the ms .

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Proceed to summarize the following text: ultracold samples of bi - alkali polar molecules have been created very recently in their ground electronic @xmath7 , vibrational @xmath8 , and rotational @xmath9 states @xcite . this is a promising step before achieving bose - einstein condensates or degenerate fermi gases of polar molecules , provided that further evaporative cooling is efficient . for this purpose , elastic collision rates must be much faster than inelastic quenching rates . this issue is somewhat problematic for the bi - alkali molecules recently created , since they are subject to quenching via chemical reactions . if a reaction should occur , the products are no longer trapped . for alkali dimers that possess electric dipole moments , elastic scattering appears to be quite favorable , since elastic scattering rates are expected to scale with the fourth power of the dipole moment @xcite . inelastic collisions of polar species can originate from two distinct sources . the long - range dipole - dipole interaction itself is anisotropic and can cause dipole orientations to be lost . this kind of loss generally leads to high inelastic rates , and is regarded as the reason why electrostatic trapping of polar molecules is likely not feasible @xcite . moreover , these collisions also scale as the fourth power of dipole moment in the ultracold limit @xcite , meaning that the ratio of elastic to inelastic rates does not in general improve at higher electric fields . this sort of loss can be prevented by trapping the molecules in optical dipole traps . more serious is the possibility that collisions are quenched by chemical reactions . chemical reaction rates are known to be potentially quite high even at ultracold temperatures @xcite . indeed , for collision energies above the bethe wigner threshold regime , it appears that many quenching rates , chemical or otherwise , of barrierless systems are well described by applying langevin s classical model @xcite . in this model the molecules must surmount a centrifugal barrier to pass close enough to react , but are assumed to react with unit probability when they do so . this model has adequately described several cold molecule quantum dynamics calculations @xcite . within the bethe wigner limit , scattering can be described by an elegant quantum defect theory ( qdt ) approach @xcite . this approach makes explicit the dominant role of long - range forces in controlling how likely the molecules are to approach close to one another . consequently , quenching rate constants can often be written in an analytic form that contains a small number of parameters that characterize short - range physics such as chemical reaction probability . for processes in which the quenching probability is close to unity , the qdt theory provides remarkably accurate quenching rates @xcite . for dipoles , however , the full qdt theory remains to be formulated . in this article we combine two powerful ideas suppression of collisions due to long - range physics , and high - probability quenching inelastic collisions for those that are not suppressed to derive simple estimates for inelastic / reactive scattering rates for ultracold fermionic dipoles . the theory arrives at remarkably simple expressions of collision rates , without the need for the full machinery of close - coupling calculations . strikingly , the model shows that quenching collisions scale as the _ sixth _ power of the dipole moment for ultracold @xmath3wave collisions . on the one hand , this implies a tremendous degree of control over chemical reactions by simply varying an electric field , complementing alternative proposals for electric field control of molecule - molecule @xcite or atom - molecule @xcite chemistry . on the other hand , it also implies that evaporative cooling of polar molecules may become more difficult as the field is increased . in section ii , we formulate the theoretical model for three dimensional collisions . in section iii , we apply this model to pure two dimensional collisions and conclude in section iv . in the following , quantities are expressed in s.i . units , unless explicitly stated otherwise . atomic units ( a.u . ) are obtained by setting @xmath10 . in quantum mechanics , the quenching cross section of a pair of colliding molecules ( or any particles ) of reduced mass @xmath11 for a given collision energy @xmath12 and a partial wave @xmath13 is given by @xmath14 where @xmath15 is the transition matrix element of the quenching process , @xmath16 represents the quenching probability , and the factor @xmath17 represents symmetrization requirements for identical particles @xcite . if the two colliding molecules are in different internal quantum states ( distinguishable molecules ) , @xmath18 and if the two colliding molecules are in the same internal quantum state ( indistinguishable molecules ) , @xmath19 . the total quenching cross section of a pair of molecules is @xmath20 . the quenching rate coefficient of a pair of molecules for a given temperature @xmath21 ( collisional event rate ) is given by @xmath22 where @xmath23 } \end{aligned}\ ] ] is the maxwell boltzmann distribution for the relative velocities for a given temperature and @xmath24 is the maxwell boltzmann constant . the total quenching rate coefficient of a pair of molecules is @xmath25 . to avoid confusion , we will also write the corresponding rate equation for collisions between distinguishable and indistinguishable molecules . first , we consider collisions between two distinguishable molecules in quantum states @xmath26 and @xmath27 ( @xmath18 in eq . ) . during a time @xmath28 , where @xmath29 is the time of a quenching collisional event , the number of molecules @xmath30 lost in each collision is one and the number of molecules @xmath31 lost in each collision is one . then @xmath32 and @xmath33 . the volume per colliding pairs of molecules is @xmath34 , where @xmath35 stands for the volume of the gas . during the time @xmath29 , the quenching collisional event is associated with a volume @xmath36 . by definition of @xmath29 , this volume should be equal to the one occupied by just one colliding pair of molecules . then we get @xmath37 . the rate equation for the number of molecule @xmath30 or @xmath31 is then given by @xmath38 if @xmath39 and @xmath40 are the densities of molecule @xmath26 and @xmath27 in the gas , then @xmath41 we consider now the case of collisions between two indistinguishable molecules ( @xmath19 in eq . ) . during the time @xmath28 the number of molecules @xmath42 lost in each collision is two . then we get @xmath43 . the volume per colliding pairs of molecules is @xmath44 where we have taken into account the indistinguishability of the molecules . for the same reason explained above , the volume associated with the collisional event during the time @xmath29 should be equal to the volume occupied by just one colliding pair of molecules . and then @xmath45 . the rate equation for the number of molecule @xmath42 is then given by @xmath46 if @xmath47 and @xmath48 , then @xmath49 [ t ] as a function of the intermolecular separation @xmath50 . @xmath51 and @xmath52 denote the height and the position of the centrifugal barrier . [ spag - fig ] , width=302 ] we consider the case of two identical ultracold fermionic polar molecules , as has been achieved very recently for krb dimers @xcite in their ro - vibronic ( @xmath53 ) ground state . under these circumstances , because of fermi exchange symmetry , the relative orbital angular momentum quantum number @xmath54 between the two molecules must take odd values @xmath55 . these molecules are polar molecules and can be controlled by an electric field @xmath56 . in the usual basis set of partial waves @xmath57 , the long - range behavior of two colliding polar molecules in a presence of an electric field is governed by an interaction potential matrix whose elements are @xmath58 where @xmath50 denotes the distance between the two molecules . the diagonal elements represent effective potentials for the colliding molecules and the non - diagonal elements represent couplings between them . the coefficient @xmath59 is the van der waals coefficient , assumed to be isotropic in the present treatment . the @xmath60 is the term corresponding to the electric dipole - dipole interaction expressed in the partial wave basis set @xmath61 between two polarized molecules in the electric field direction , with @xmath62 , where @xmath63 is the induced electric dipole moment , and @xmath64 represent the relative orientation between the molecules . in the basis set of partial waves , @xmath65 takes the form @xmath66 the large bracket symbols denote the usual 3@xmath67 coefficients . the coefficient @xmath68 is introduced to simplify further notations . the combination between repulsive and attractive terms in the effective potentials ( diagonal terms ) of eq . generate a potential barrier of height @xmath51 which is plotted schematically in fig . [ spag - fig ] . the height of this barrier plays a crucial role as it can prevent the molecule from accessing the short range region where reactive chemistry occurs . the quantum threshold ( qt ) model consists of two conditions . first , for @xmath69 , we use the bethe wigner threshold laws @xcite for ultracold scattering . second , we use the classical capture model ( langevin model ) @xcite to estimate the probability of quenching for @xmath70 . a classical capture model is indiferent to collision energies @xmath69 since the barrier prevents the molecules from coming close together . in real - life quantum scattering , collisions do occur at these energies due to quantum tunneling , and they are the ones relevant to ultracold collisions . moreover , collisions in this energy regime are dictated by the the bethe wigner quantum threshold laws . for quenching collisions , the threshold laws @xcite state that @xmath71 . for @xmath72 , a classical capture model will guarantee to deliver the molecule pair to small values of @xmath50 , where chemical reactions are likely to occur with unit probability ( see fig . [ spag - fig ] ) . following this classical argument , we will assume that when @xmath73 , the quenching probability reaches unitarity @xmath74 . using this assumption together with the quantum threshold laws , the qt quenching tunneling probability below the barrier can be written as @xmath75 consequently , the quenching cross section and rate coefficient are approximated by @xmath76 for @xmath69 . the qt model has the simple and intuitive advantage of showing how the cross sections and rate coefficients scale with the height of the entrance centrifugal barrier . for @xmath72 , it is easy to find the corresponding expression of the cross section in eq . by setting @xmath74 . the cross section @xmath77 will reach the unitarity limit at @xmath72 . it is also easy to find the corresponding expression of the rate coefficient in eq . . the qt model is general for any collision between two particles provided that there is a barrier in the entrance collision channel and that chemical reactions occur with near unit probability at short range . the only information on short range chemistry is that chemical reactions occur at full and unit probability and the only information on long range physics is provided by the height of the entrance barrier @xmath51 . the qt model describes the background scattering process , it does not take into account scattering resonances . note that the model will not be appropriate in the present form for barrierless ( @xmath78wave ) collisions since @xmath79 . for this particular type of collisions that do not possess a centrifugal barrier , the qdt theory can be usefully applied @xcite . the present form of the qt model does not take into account the anisotropy of the intermolecular potential at intermediate range and/or the electronic and nuclear spin structure of the molecular complex but remains suitable as far as the entrance centrifugal barrier takes place at long range . the qt model will have to be modified if longer range interactions takes place . for example , collisions between @xmath9 and @xmath80 polar molecules can have long range interactions between hyperfine states due to dipolar and hyperfine couplings @xcite . however , for collisions between rotationless @xmath9 polar molecules , the hyperfine couplings are weak and the qt model can be applied without further modifications . [ h ] k + @xmath81k@xmath82 as a function of the collision energy for the partial wave @xmath83 : ( i ) calculated with a full quantum calculation ( solid lines ) , reproduced from ref . @xcite ( ii ) using the qt model ( dashed lines ) ( iii ) fitting the qt model ( dotted line ) , using @xmath84 in eq . . [ crossk3-fig ] , width=302 ] [ h ] k + @xmath81k@xmath82 as a function of the collision energy for the partial wave @xmath4 : comparison between the full quantum calculation ( solid line ) and the qt model ( dashed line ) . the fitted qt model appears as a dotted line . the height of the barrier @xmath51 and the corrected height @xmath85 ( @xmath86 ) appear as vertical lines . [ probak3-fig ] , width=302 ] in the absence of an electric field in eq . , the long range potential reduces to a diagonal term in the basis set of partial waves . the position and height of the barrier are given by @xmath87 we can insert eq . in eq . to get analytical forms of the quenching cross section or rate coefficient . for two indistinguishable fermionic polar molecules at ultracold temperatures when @xmath4 , we get @xmath88 in eq . , we used the fact that @xmath89 in three dimensions . note that to get the overall contribution for a given @xmath54 , we have to multiply eq . by the degeneracy factor @xmath90 corresponding to all values of @xmath91 . we can get similar expressions for any partial wave @xmath54 . to test the validity of the model , we compare in fig . [ crossk3-fig ] the quenching cross sections of @xmath81k@xmath92 + @xmath81k@xmath93 collisions as a function of the collision energy for the partial waves @xmath83 : ( i ) calculated in ref . @xcite with a full quantum time - independent close - coupling calculation based on hyperspherical democratic coordinates @xcite and the full potential energy surface of k@xmath94 ( solid lines ) ( ii ) using the simple qt model ( dashed lines ) with a value of @xmath95 a.u . given in ref . @xcite ( 1 a.u . = 1 @xmath96 where @xmath97 is the hartree energy and @xmath98 is the bohr radius ) . in this example , the qt model provides an upper limit to the cross sections . this is due to the fact that the quenching cross section does not reach a maximum value at the height of the barrier @xmath51 , but rather at somewhat higher energy , say @xmath85 , with @xmath99 ( see ref . @xcite ) . for all partial waves , there is a worse agreement for collision energies in the vicinity of the height of the barrier where the passage from the ultralow regime to the unitarity limit is smoother than for the qt model . this smoother passage is visible in fig . [ probak3-fig ] for the full quantum @xmath4 quenching probability compared to the qt model , which has a sharp corner in the vicinity of @xmath51 . to account for more flexibility in the qt model , one can replace @xmath51 in eq . by @xmath85 ( @xmath99 ) , and use the coefficient @xmath100 as a fitting parameter to reproduce either full quantum calculations or experimental observed data . alternatively , we can correct the qt quenching tunneling probability with an overall factor @xmath101 , @xmath102 with @xmath103 . @xmath104 can be interpreted as the quenching probability reached at the height of the barrier @xmath51 in the qt model , rather than the rough full unit probability ( @xmath105 ) . as an example , we find that @xmath106 reproduces the quantum @xmath4 partial wave cross section for @xmath81k + @xmath81k@xmath82 ( dotted line in fig . [ crossk3-fig ] and fig . [ probak3-fig ] ) . this yields a maximum quenching probability of @xmath107 instead of 1 . in other words , the qt model is only a factor of @xmath108 higher than the full quantum calculation for @xmath81k + @xmath81k@xmath82 collisions at ultralow energies . given the fact that full quantum calculations are computationally demanding @xcite and impossible at the present time for alkali molecule - molecule collisions , the accuracy of the qt model is satisfactory and can be a quick and powerful alternative way to estimate orders of magnitude for the scattering observables . besides , agreement between the qt model with experimental data or full quantum calculations is expected to be satisfactory for collisions involving alkali species , because it is likely that short range quenching couplings will dominate and lead to high quenching probability in the region where the two particles are close together @xcite . very recently , eq . of the qt model has been applied for the evaluation of ultracold chemical quenching rate of collisions of two @xmath109k@xmath110rb molecules in the same internal quantum state , and provided good agreement with the experimental data @xcite . [ h ] as a function of the induced dipole moment @xmath0 for the partial waves @xmath111 ( red curves ) and @xmath112 ( blue curves ) . [ barriers - fig ] , width=302 ] [ h ] k@xmath110rb molecules as a function of the induced electric dipole moment for @xmath4 and for a temperature of @xmath113 nk ( black curves ) . the rates have been calculated using the barrier heights of fig . [ barriers - fig ] . the red lines represent the @xmath111 partial wave contribution . the blue lines represent the sum of @xmath114 and @xmath115 partial wave contributions . the dashed lines represent the rates calculated with the diabatic barriers while the solid lines with the adiabatic barriers ( see text for detail ) . the total , @xmath116 and @xmath117 curves have been indicated in the left hand side . [ rate - num - fig ] , width=302 ] in the presence of an electric field in eq . , the long - range interaction potential matrix is no more diagonal and couplings between different values of @xmath54 occur . @xmath91 is still a good quantum number . a first approximation ( diabatic approximation ) consists of neglecting these couplings and using only the diagonal elements of the diabatic matrix directly . then one can find numerically for which @xmath50 the centrifugal barriers are maximum and evaluate the height of the diabatic barriers @xmath118 . this is repeated for all values of the induced dipole moment @xmath0 . a second approximation ( adiabatic approximation ) is to diagonalize this matrix ( including the non - diagonal coupling terms ) for each @xmath50 and again find the maximum of the centrifugal barriers to get the height of the adiabatic barriers @xmath119 . as an example , we compute these barrier heights for @xmath109k@xmath110rb@xmath120@xmath109k@xmath110rb collisions , using a value of @xmath121 a.u . @xcite . we plot in fig . [ barriers - fig ] the heights of the diabatic ( dashed lines ) and adiabatic ( solid lines ) barriers for the quantum numbers @xmath116 ( red curves ) and @xmath117 ( blue curves ) . the adiabatic barriers have been calculated using five partial waves @xmath122 in eq . . the effect of the couplings can be clearly seen in this figure by comparing diabatic and adiabatic barriers . especially for the @xmath117 case for @xmath123 d ( 1 d = 1 debye = @xmath124 c.m ) , couplings with higher partial waves make the adiabatic barrier decrease as the dipole increases while the diabatic barrier continues to increase . using these heights of the barriers , we use eq . to plot in fig . [ rate - num - fig ] the total quenching rate coefficients ( black curves ) as a function of @xmath0 for two indistinguishable fermionic polar @xmath109k@xmath110rb molecules in the same quantum state for @xmath4 and at a typical experimental temperature of @xmath113 nk @xcite . for @xmath113 nk , the mean collision energy @xmath125 nk , and the maximum dipole moment for which @xmath126 nk ( that is for which eq . does not apply anymore ) is around @xmath127 d ( see fig . [ barriers - fig ] ) . the dashed curves correspond to rates calculated with the diabatic approximation while the solid curves correspond to rates calculated with the adiabatic approximation . the @xmath116 contribution is plotted in red and the contribution of @xmath128 and @xmath129 is plotted in blue . the rates highly reflect the behavior of the centrifugal barriers in the entrance collision channel . when the barrier increases with the dipole , it prevents the molecules from getting close together and the quenching rates decreases . when the barrier decreases , the tunneling probability is increased allowing the molecules to get close together , and the quenching rates increases . [ t ] but we use the analytical expressions for the rates ( see text for detail ) . the total , @xmath116 and @xmath117 curves have been indicated in the left hand side . the individual analytical curves have been indicated in the right hand side by roman numbers . [ rate - anal - fig ] , width=302 ] in order to have an intuitive sense of how the chemical quenching rate scales with the induced dipole moment ( and the electric field ) , we evaluate analytical expressions of the barriers and the rates as it has been done in the previous section for a zero electric field . the analytical expression of the height of the diabatic barrier @xmath118 is complicated by the occurrence of two distinct long - range potentials in the diagonal matrix term of eq . . we circumvent this difficulty by looking in the two limits where one dominates over the other . for small electric fields , we use the zero electric field limit discussed in the preceding section by setting @xmath130 . for larger electric fields we ignore the @xmath59 coefficient in eq . if the electric dipole - dipole interaction is attractive ( positive @xmath65 ) . we ignore the centrifugal term in eq . if the electric dipole - dipole interaction is repulsive ( negative @xmath65 ) . these two cases are discussed below . in between , to accommodate the transition between the low - field and high - field limit , we will simply add the rate coefficients derived in the two limiting cases . for positive @xmath65 coefficients in eq . , @xmath131 is attractive in eq . . for @xmath4 partial waves for example , this occurs when @xmath116 , which favors an attractive orientation of dipoles . we consider @xmath132 in eq . . in this case , the position and height of the barrier are given by @xmath133 the position of the barrier in eq . has to be in the region where eq . is satisfied . this happens for suitably large dipole moments , @xmath134 where @xmath135 the subscript _ a _ stands for the attractive interaction . for two indistinguishable fermionic polar @xmath109k@xmath110rb molecules , and for @xmath4 and @xmath116 , @xmath136 and we get @xmath137 d. the threshold laws for quenching collisions in an electric field are the same as in the zero - field limit . consequently , the quenching cross sections and rate coefficients behaves as in eq . except that @xmath51 is given now by eq . and varies with @xmath0 . we can insert eq . in eq . to get the corresponding analytical expressions . for a partial wave @xmath2 , the quenching rate scales as @xmath1 . for two indistinguishable fermionic polar molecules at ultracold temperatures when @xmath4 and @xmath116 , we get @xmath138 thus the @xmath111 quenching rate increases as @xmath139 . this is a more rapid dependence on dipole moment than for purely long - range dipolar relaxation in dipolar gases @xcite . for negative @xmath65 coefficients in eq . , @xmath131 is repulsive in eq . . for @xmath4 partial waves for example , this occurs when @xmath140 , which favors a repulsive orientation of dipoles . we consider @xmath141 in eq . . the long - range potential again experiences a barrier , but now it is generated by the balance between the repulsive dipole potential at large @xmath50 , and the attractive van der waals potential at somewhat smaller @xmath50 . in this case , the position and height of this barrier are given by @xmath142 for this approximation to hold , the position of the barrier in eq . has to be in the region where eq . is satisfied . this requires that @xmath143 where @xmath144 the subscript _ _ stands for the repulsive interaction . for two indistinguishable fermionic polar @xmath109k@xmath110rb molecules , and for @xmath4 and @xmath145 or @xmath129 , @xmath146 and we get @xmath147 d. we can replace eq . in eq . to get the corresponding analytical expressions . for a partial wave @xmath2 , the quenching processes scale as @xmath148 . for two indistinguishable fermionic polar molecules at ultracold temperatures when @xmath4 and @xmath117 , we get @xmath149 the @xmath112 quenching rate decreases as @xmath150 as the electric field grows . these analytical expressions use the diabatic barriers . if we consider that at large @xmath0 , the total rate is mostly given by the @xmath116 contribution ( we neglect the @xmath117 contributions at large @xmath0 ) , one can have an analytical expression using the adiabatic barrier . if we take into account the couplings between @xmath111 and @xmath151 , we can diagonalize analytically the @xmath152 matrix in eq . . it can be shown that for each dipole moment @xmath0 , the coupling with @xmath153 lower the diabatic barrier of @xmath154 by a factor of 0.76 at the position of the barrier , to give rise to the adiabatic barrier . inserting this correction of the barrier in eq . , this yields a correction of @xmath155 for @xmath156 . the difference between diabatic and adiabatic calculations can be already seen in fig . [ barriers - fig ] and fig . [ rate - num - fig ] for the numerical barriers at large dipole moment . in fig . [ rate - anal - fig ] the black curve corresponds to the total quenching rate coefficient as a function of @xmath0 for two indistinguishable fermionic polar @xmath109k@xmath110rb molecules in the same quantum state for @xmath4 and at a temperature of @xmath113 nk . the analytical expressions i , ii , iii , iv , v ( green thin lines ) correspond respectively to eq . , 2 @xmath157 eq . , eq . , 2 @xmath157 eq . , 1.51 @xmath157 eq . . the curves iii and iv are for the diabatic barriers , while curve v is to account for the adiabatic barrier . the red dashed line ( i+iii ) represents the @xmath111 partial wave contribution for the diabatic barriers while the blue dashed line ( ii+iv ) represents the sum of @xmath114 and @xmath115 partial wave contributions for the diabatic barriers . the analytical sum i+ii+iii+iv is represented as a black dashed line . to account for the adiabatic barriers we assume that the correction for the total rate comes only from the @xmath111 partial wave , and we replace i+iii by i+v ( red solid line ) . the analytical sum i+ii+v+iv is represented as a black solid line . neglecting the @xmath150 contribution at larger @xmath0 , the analytical @xmath3wave quenching rate ( taking into account the adiabatic barriers ) is given by the simple expression @xmath158 @xmath159 ( @xmath160 ) is the quenching probability reached at the height of the barrier in the qt model for the zero ( non - zero ) electric field regime . the qt model assumes that @xmath161 but become fitting parameters ( @xmath162 ) when compared with full quantum calculations or experimental data . the limiting value @xmath163 d ( @xmath164 d ) , where the @xmath5 ( @xmath150 ) behavior begins , has also been indicated with an arrow . it turns out that the total rates for @xmath4 calculated analytically ( for both the use of diabatic and adiabatic barriers ) are very similar to the numerical ones of fig . [ rate - num - fig ] ( 10 % difference at most , around @xmath165 ) . however , the sub - components @xmath111 and @xmath112 have different behaviors . for example the numerical @xmath111 ( @xmath112 ) component starts to increase ( decrease ) at earlier dipole moment ( typically at 0.02 d ) than their analytical analogs ( typically after 0.06 d ) . the use of the simple analytical expressions ( using the diabatic or adiabatic barriers ) can be useful to estimate the total rate coefficients , while the numerical ones are prefered to estimate the @xmath111 and @xmath112 individual rates . [ t ] for the partial waves @xmath166 in two dimensions . the green thin curves represent the analytical eq . ( constant ) and eq . ( @xmath167 ) . the dashed black curve is the sum of them . the solid black curve is the height of the barrier in eq . . [ barriers2d - fig ] , width=302 ] [ h ] k@xmath110rb molecules as a function of the induced electric dipole moment for the @xmath168 and @xmath169 components at a temperature of @xmath113 nk . the dashed lines represent the rate using analytical expressions while the solid line represents the rate using the numerical expression ( see text for detail ) . the individual analytical curves have been indicated in the right hand side by roman numbers . [ 2d - fig ] , width=302 ] in three dimensional collisions , the quenching loss is largely due to incident partial waves with angular momentum projection @xmath116 , emphasizing head - to - tail orientations of pairs of dipoles . these are the kind of collisions that are largely suppressed in traps with a pancake geometry , with the dipole polarization axis orthogonal to the plane of the pancake @xcite . if these collisions can be removed , then it is likely that increasing the electric field will suppress quenching collisions , making evaporative cooling possible . if we assume an ideal pancake trap that confines the particles to move strictly on a plane , one can apply the present model to estimate the behavior of the quenching processes . we assume that the molecules are polarized along the electric field axis , perpendicular to the two dimensional plane . in this case , the long range potential is given by @xmath170 where @xmath171 stands for the distance between 2 particles in a two dimensional plane , @xmath172 stands for the angular momentum projection on the electric field axis . the last term comes from the repulsive dipole - dipole interaction when the dipoles are pointing along the electric field and approach each other side by side . the height of this barrier has been plotted as a function of @xmath0 in fig . [ barriers2d - fig ] ( black solid line ) . at ultralow energy and large molecular separation , the bethe wigner laws for quenching processes depend only on the long - range repulsive centrifugal term @xmath173 @xcite . the repulsive centrifugal terms are different in eq . and eq . . as the repulsive centrifugal term in eq . leads to the threshold laws in eq . , the replacement @xmath174 ( that is @xmath175 ) in eq . leads to @xmath176 where @xmath51 denotes the height of the centrifugal barrier in two dimensions . this result requires that the centrifugal potential is repulsive , i.e. , that @xmath177 . for @xmath178 the threshold law exhibits instead a logarithmic divergence @xcite . in two dimensions , quenching cross sections and rate coefficients have respectivelly units of length and length squared per unit of time , and are given by @xcite @xmath179 within this model , it follows that the quenching cross section and rate coefficient for @xmath180 are given by @xmath181 the energy dependence is in agreement with the one found in ref . @xcite . in eq . , @xmath182 for the zero - electric field regime and @xmath183 for the non - zero electric field regime . the height of these barriers has been reported in fig . [ barriers2d - fig ] ( green thin lines ) . these results imply that for @xmath166 the quenching processes within this model will be independent of the dipole moment in the zero electric field regime , where @xmath184 and will scale as @xmath6 in the non - zero electric field regime , where @xmath185 we use the fact that @xmath186 in two dimensions . the non - zero electric field regime is reached when @xmath187 where @xmath188 for two indistinguishable fermionic polar @xmath109k@xmath110rb molecules , and for @xmath166 , we get @xmath189 d. the behavior of the quenching rate ( black lines ) is shown in fig . [ 2d - fig ] for two indistinguishable fermionic polar @xmath109k@xmath110rb molecules as a function of the induced electric dipole moment for @xmath168 and @xmath169 components at a temperature of @xmath113 nk . the dashed line represents the analytical rate which is the sum of the analytical expression vi corresponding to @xmath190 eq . and analytical expression vii corresponding to @xmath190 eq . . the solid line represents the rate using the general expression eq . and the numerical height of the barrier calculated in eq . . the limiting value @xmath191 d , where the @xmath6 behavior for the quenching rate begins , has also been indicated with an arrow . the difference between the numerical calculation and the analytical expression reflects the difference in the calculation of the height of the barrier , already seen in fig . [ barriers2d - fig ] . the numerical calculation is more exact , while the other is analytical . however , at large @xmath0 , the numerical rate tends to the analytical @xmath6 behavior . the quenching rate decreases rapidly as the dipole moment increases and this may be promising for efficient evaporative cooling of polar molecules since the elastic rate is expected to grow with increasing dipole moment @xcite . we have proposed a simple model which combines quantum threshold laws and a classical capture model to determine analytical expressions of the chemical quenching cross section and rate coefficient as a function of the collision energy or the temperature . we also provide an estimate as a function of the induced electric dipole moment @xmath0 in the presence of an electric field . we found that the quenching rates of two ultracold indistinguishable fermionic polar molecules grows as the sixth power of @xmath0 . for weaker electric field , quenching processes are independent of the induced electric dipole moment . prospects for two dimensional collisions have been discussed using this model and we predict that the quenching rate will decrease as the inverse of the fourth power of @xmath0 . this fact may be useful for efficient evaporative cooling of polar molecules . this model provides a general and comprehensive picture of ultracold collisions in electric fields . preliminary data suggest that this model gives good agreement with experimental chemical rates for three dimensional collisions in an electric field @xcite . we acknowledge the financial support of nist , the nsf , and an afosr muri grant . we thank k .- k . ni , s. ospelkaus , d. wang , m. h. g. de miranda , b. neyenhuis , p. s. julienne , j. ye , and d. s. jin for helpful discussions .

we use the quantum threshold laws combined with a classical capture model to provide an analytical estimate of the chemical quenching cross sections and rate coefficients of two colliding particles at ultralow temperatures . we apply this quantum threshold model ( qt model ) to indistinguishable fermionic polar molecules in an electric field . at ultracold temperatures and in weak electric fields , the cross sections and rate coefficients depend only weakly on the electric dipole moment @xmath0 induced by the electric field . in stronger electric fields , the quenching processes scale as @xmath1 where @xmath2 is the orbital angular momentum quantum number between the two colliding particles . for @xmath3wave collisions ( @xmath4 ) of indistinguishable fermionic polar molecules at ultracold temperatures , the quenching rate thus scales as @xmath5 . we also apply this model to pure two dimensional collisions and find that chemical rates vanish as @xmath6 for ultracold indistinguishable fermions . this model provides a quick and intuitive way to estimate chemical rate coefficients of reactions occuring with high probability . = cmr7

You are an expert at summarizing long articles. Proceed to summarize the following text: over the last two decades , the study of driven diffusive systems kls , sz , vp , schutz has formed a significant branch in the general pursuit to understand non - equilibrium statistical mechanics . the underlying dynamics violates detailed balance , so that the long - time behavior is governed by a true _ non - equilibrium _ steady state ( ness ) . highly unexpected and non - trivial properties are manifested , even in one - dimensional systems with purely local dynamics . a paradigmatic model in the last class is the totally asymmetric simple exclusion process ( tasep ) krug , derrida92,dehp , s1993,derrida , schutz . characterized by open boundaries and particle transport , it displays the key physical signatures of a system driven far from equilibrium . in particular , its ness develops a complex phase diagram controlled by the interactions of the system with its environment . while an understanding of these properties is of fundamental theoretical interest , the tasep and its generalizations have also acquired fame as models for practical problems in traffic flow @xcite and biological transport @xcite . in this paper , we study a simple variant of the basic tasep , by introducing a global constraint on the available number of particles . deferring the motivations for such a model to the next paragraphs , let us briefly summarize the essentials . each site of a one - dimensional lattice is either empty or occupied by a single particle . following random sequential dynamics , the particles enter the lattice at one end ( e.g. , the `` left '' edge ) with a given rate @xmath0 , hop to the right with rate @xmath1 ( scaled to unity ) , subject to an excluded volume constraint , and exit at the far end with a rate @xmath2 . in the standard version of the model , both @xmath3 and @xmath2 are constant rates , independent of the number of particles on the lattice . thus , we may regard the lattice as being coupled to a reservoir or a pool with an arbitrarily large number of particles . here , we consider a tasep with a _ finite _ supply of particles and report our numerical and analytical findings . a fixed number of particles , @xmath4 , is shared between the lattice and the reservoir , in such a way that the entrance rate , @xmath0 , depends on the number in the pool : @xmath5 . as a result , if more particles are found on the lattice , the number of available particles in the pool is lowered , leading to a corresponding decrease of @xmath0 . since the number of particles on the tasep lattice , @xmath6 , feeds back into @xmath0 through the constraint @xmath7 constant @xmath8 , we will refer to our model as `` a constrained tasep . '' for simplicity , we assume the reservoir to be so large that particles exiting the lattice are unaffected by @xmath5 , leaving @xmath2 unchanged . our focus here is to explore the consequences of limited resources : how will the phase diagram and the properties of the phases be affected as the total number of available particles is reduced ? this behavior mimics the limited availability of resources required for a given physical or biological process . for example , in protein synthesis , ribosomes bind near the start sequence of a messenger rna , in order to translate the genetic information encoded on the rna into the associated protein . ribosomes are large molecular motors which are assembled out of several basic units , and numerous ribosomes can be bound to the same , or other , mrnas , so that multiple proteins are translated in parallel . when a protein is completed , the ribosome is disassembled and recycled into the cytoplasm . under conditions of rapid cell growth , ribosomes or their constituents can find themselves in short supply , so that a self - limitation of translation , mimicked via a modification of @xmath0 , can occur . in a previous study @xcite , a different aspect of ribosome recyling was considered . this work focused on the enhancement of the ribosome concentration at the initiation site , as a result of diffusion of the ribosome subunits from the termination site ( of the same mrna ) . due to the spatial proximity of initiation and termination sites , @xmath0 is effectively increased . by contrast , our investigation of constrained resources leads to an effective _ reduction _ of @xmath0 . in the context of traffic models , our problem corresponds to a generalization of the parking garage problem @xcite . here , one special site ( the `` parking garage '' , or reservoir ) is introduced into a tasep on a ring ( lattice with periodic boundary conditions ) and , for this site only , the occupancy is unlimited . particles ( `` cars '' ) jump into the garage with unit rate ( corresponding to @xmath9 ) , irrespective of its occupancy . particles exit the garage with rate @xmath0 , provided the site following the garage is empty . as we will see presently , this is a special case of our more general model . we begin with a description of our model and its observables of interest . next , we outline results from simulations and compare them to a simple theory which builds on exact analytic results for the standard tasep . we find excellent agreement for nearly all ( @xmath0,@xmath2 ) and conclude with some comments and open questions . the standard tasep is defined on a one - dimensional lattice of length @xmath10 , with sites labeled by @xmath11 , @xmath12 . each site can be empty or occupied by a single particle , reflected through a set of occupation numbers @xmath13 which take the values @xmath14 or @xmath15 . the boundaries of the lattice are open . particles hop from a reservoir onto the lattice with a rate @xmath3 . once on the lattice , particles will hop to the nearest - neighbor site on the right , provided it is empty . once a particle reaches the end of the lattice , it hops back into the reservoir with rate @xmath2 . the dynamics is random sequential , leading to fluctuations in the local occupations as well as in the total number of particles , defined via @xmath16 the overall density is given by @xmath17 . ensemble averages are denoted by @xmath18 . three distinct phases can be identified and are displayed in an @xmath19 phase diagram ( see fig . [ fig : phase_diagram ] ) : a low density ( ld ) , a high density ( hd ) , and a maximum current ( mc ) phase . the phases are distinguished , respectively , by their average densities : @xmath20 , @xmath21 , and @xmath22 . the corresponding stationary currents are given by @xmath23 , @xmath24 , and @xmath25 . in all of these expressions , finite - size corrections have been neglected . because of particle - hole symmetry , all aspects of the hd and ld phases are related . the phase boundaries between the hd and mc phases and between the ld and mc phases mark continuous transitions . the line separating the hd and ld phases , @xmath26 , is a coexistence line . here , the system consists of a region with low density ( @xmath27 ) followed by one with high density ( @xmath28 ) , connected by a microscopically sharp `` shock . '' this shock diffuses freely between the ends of the system , so that the _ average _ density profile is linear . often this is referred to as the `` shock phase '' ( sp ) . the system of interest here differs from the standard tasep in one important way : the number of particles in the reservoir , @xmath5 , is finite , and we choose the on - rate @xmath0 to depend on @xmath5 as follows . so as to distinguish this varying rate from the @xmath0 of the ordinary tasep , we denote the former as the _ effective _ on - rate @xmath29 and write @xmath30 where @xmath31 is a function that satisfies three conditions : ( i ) @xmath32 , ( ii ) @xmath33 , and ( iii ) @xmath31 is monotonically increasing . the first of these conditions is self - evident ; the second simply connects our model to the standard one with unlimited resources ; the last is just common sense . in this notation , the `` parking garage '' problem @xcite is characterized by @xmath34 where @xmath35 is the heaviside step function . here , we wish to model , say , a cell with finite number of ribosomes , so that it is natural to assume that a smaller number of particles in the pool results in a lower on - rate , etc . , and that @xmath36 would be a smoother function . specifically , for the simulations shown below we choose @xmath37 where @xmath38 provides the scale for crossover to saturation . the detailed choice of @xmath38 is unimportant , but it is convenient to make it extensive in @xmath10 . for easier comparison , we choose @xmath38 to be the average ( stationary ) density of the unconstrained tasep defined by the parameters @xmath39 : for example , if @xmath40 ( hd phase , ordinarily ) , we set @xmath38 to @xmath41 . for the sp , we arbitrarily choose @xmath42 . to summarize , the control parameters of our model are the same as the ordinary tasep ( @xmath43 ) plus the total number of particles , @xmath4 ( i.e. , @xmath44 ) . to characterize our system in steady state , we measure the average density @xmath45 , the average ( local ) density profile @xmath46 , and the average current @xmath47 . for the last quantity , we record the total number of particles entering and leaving the chain over the run ( thereby improving statistics ) and divide the average of these by the length of the run . in our monte carlo simulations , we pick randomly from the @xmath6 particles on the lattice and one additional _ virtual _ particle . if a lattice particle is selected , we attempt to update its position as in ordinary tasep ; if the virtual particle is selected , we attempt to place a new particle on the first site of the chain . @xmath48 attempts constitute one monte carlo step ( mcs ) . initially , the chain is empty , i.e. , @xmath49 , @xmath50 . typically , the first @xmath51 ( @xmath52 for sp ) mcs are discarded , to ensure that the system has reached steady state , before data are taken . then , configurational observables are recorded every @xmath53 mcs and averaged until the run terminates . the typical length of each run is @xmath52 ( @xmath54 for sp ) mcs . longer runs are required to obtain low noise data for density profiles and currents . statistical errors were estimated through visual inspections of time series and through multiple runs , to ensure the reproducibility of the data . we simulated system sizes in the range of @xmath55 to @xmath56 , with most data taken for @xmath57 . to obtain the phase diagram , we simulated more than a dozen @xmath39 pairs . here , we report results for four points : @xmath58 , @xmath59 , @xmath60 , and @xmath61 . since these correspond , respectively , to ld , hd , mc , and sp in a standard tasep , we will use these letters to refer to the four cases studied . in this section , we summarize our simulation results . since only the on - rate depends on the feedback , the behavior of our system is no longer governed by particle - hole symmetry . as a result , the simplest case is the first in the above , i.e. , `` ld '' . both @xmath62 and @xmath47 increase with @xmath4 in an expected fashion . the next case , mc , is already slightly more interesting : both @xmath62 and @xmath47 increase until the effective on - rate reaches 1/2 , after which both become constant . the end result is a pronounced kink in @xmath62 . the hd case provides an even more interesting scenario : the system properties range through _ three _ regions as @xmath4 is increased . finally , the sp provides the most unexpected behavior . in the next section , we turn to a simple theoretical description of our findings . _ the ld case . _ here , we set @xmath63 , @xmath64 , and vary the total number of particles , @xmath4 , in the system . with @xmath36 given by eqn ( [ alpha - mcs ] ) , our simulation results for @xmath57 and @xmath65 are shown in fig . fig : ld . as expected , we see that , for large @xmath4 , the system density @xmath66 approaches the value for the standard tasep for @xmath67 ( here , @xmath68 ) , namely , @xmath69 ( which is @xmath70 in our case ) . more interestingly , for @xmath71 , we find a reduced density , @xmath72 , a signal that the system is responding to the limited availability of particles from the reservoir . in the limit @xmath73 , @xmath62 naturally vanishes , with a predictable slope ( see below ) . the current @xmath47 in this `` phase '' also decreases monotonically with decreasing @xmath4 , from its asymptotic limit @xmath74 ( given by @xmath75 ) . and current @xmath47 vs @xmath76 for an ld case ( @xmath57 , @xmath77 , and @xmath78 ) . open circles and diamonds are from the analysis in section 4.,scaledwidth=35.0% ] _ the mc case . _ this domain of the phase diagram is characterized by @xmath79 and @xmath80 in the standard tasep . [ fig : mc ] shows the averages @xmath62 and @xmath47 , as @xmath4 varies , for @xmath81 . the former displays a single kink , just below @xmath82 , accompanied by a rather smooth crossover in the current . as we will see below , this kink is associated with a crossover from an mc - like behavior to an ld - like behavior , as @xmath4 drops below the value required to sustain a density @xmath83 on the chain . and current @xmath47 vs @xmath76 for an mc case ( @xmath57 , @xmath84 , and @xmath78 ) . open circles and diamonds are from the analysis in section 4 . ] and current @xmath47 vs @xmath76 for an hd case ( @xmath57 , @xmath84 , and @xmath85 ) . open circles and diamonds are from the analysis in section 4 . the predictions from domain wall theory ( not shown ) are similiar . ] _ the hd case . _ here , we set @xmath86 and @xmath87 . fig : hd shows our results for @xmath88 . as expected , for sufficiently high @xmath4 , the system settles into the density and current associated with the hd phase of the standard tasep : @xmath89 , and @xmath90 . as @xmath4 is reduced , to the extent that the chain is prevented from sustaining a density of @xmath28 , the naive expectation is that a crossover to an ld - like behavior appears . remarkably , however , fig . [ fig : hd ] shows not just two , but _ three _ distinct regimes , separated by two pronounced `` kinks '' at @xmath91 and @xmath92 . as fig . [ fig : hd_zoom ] illustrates , these kinks become sharper with increasing @xmath10 . looking for potentially critical behavior , we measured the variance of @xmath93 near the kinks and compared it to its values deep within each regime . to our surprise , at least for the @xmath57 system , the variance was not noticeably larger at the kinks than at other locations . simulations at larger values of @xmath10 would be required to settle this issue with more certainty . the current @xmath47 is also displayed in fig . [ fig : hd ] . for low values of @xmath94 , @xmath47 increases in an expected way . around @xmath95 , it shows a distinct kink and then becomes essentially constant , with a value of @xmath96 . this is close to @xmath75 , the value of @xmath47 for infinite @xmath10 . since the current appears to have already reached the `` asymptotic '' value here , it is not too surprising that there is no sign of a `` second kink '' ( near @xmath97 , where @xmath62 has another one ) . below , we offer a theoretical description for these findings . , near @xmath98 , for the hd case in fig . 4:@xmath99 , @xmath100 , and @xmath101 . ] _ the sp case . _ defined in the standard tasep by @xmath26 , this line of first - order phase transitions separates the hd and ld phases . though the _ average _ density is 1/2 , its fluctuations are large . in particular , since a shock ( which separates a region with density @xmath28 from one with @xmath27 ) diffuses freely between the ends of the system , the probability @xmath102 of finding the system with a density @xmath62 approaches a flat distribution : @xmath103 , in the limit @xmath104 . of course , finite - size effects will smooth out the steps at @xmath28 and @xmath27 . imposing finite resources , we may expect that @xmath62 increases simply with @xmath4 as in the ld case , perhaps rising more slowly to the asymptotic value of 1/2 . instead , its properties are much more subtle . as illustrated in fig . [ fig : sp ] ( for @xmath105 ) , @xmath62 increases with @xmath76 * * , * * slows , speeds up , and slows down again . the behavior in the low @xmath4 regime is intuitively understandable , since there are too few particles to sustain any phase other than ld . the large @xmath4 regime is also expected . in the crossover regime ( @xmath106 ) , an intriguing `` shoulder '' develops . though this behavior is accounted for by the domain wall theory @xcite ( see below ) , we have found no simple and intuitive way to understand it . perhaps the presence of large fluctuations , as the system gets closer to the sp , smooths out the two kinks observed in the hd case . to gain more insight , we investigated @xmath102 . [ fig : sp - hist ] shows particle density histograms ( gathered from 50k data points and plotted with arbitrary normalization here ) , for several choices of @xmath76 . for low @xmath76 ( e.g. , 0.5 in the figure ) , our system is dominated by the lack of resources , so that this distribution is essentially gaussian , quite similar to those measured deep within the ld phase . as @xmath76 increases , the system enters the crossover regime and @xmath102 becomes quite asymmetric , as a detailed fitting of the @xmath107 case shows . at the other extreme , this distribution is indeed relatively flat . however , the true asymptotic regime is reached only for very large @xmath76 . thus , the slope for even the @xmath108 case ( cf . fig . [ fig : sp - hist ] ) is discernably non - zero . throughout the intermediate regime , the structure of @xmath109 is more interesting . each histogram has the cross - section of a `` lean - to '' ( a shack with a slanted roof ) . a more detailed inspection shows that the `` roof '' is predominantly a ( slowly ) decaying exponential . nearly identical to @xmath2 . ] this picture is consistent with the one from domain wall theory @xcite , as will be discussed below . it is this slow crossover - from well - defined gaussians to a completely flat distribution - which accounts for the interesting structure in the rise of @xmath62 with @xmath94 . finally , the behavior of the current is also sensitive to the slow crossover , although it displays no interesting structures like those in @xmath110 . comparing fig . [ fig : sp ] with the earlier ones , we see that it saturates at values of @xmath76 which are 2 to 3 times higher than in the other cases . and current @xmath47 vs @xmath76 for an sp case ( @xmath57 , @xmath77 , and @xmath85 ) . open circles are from the domain wall theory in section 4 . open diamonds are the current computed from eqn ( [ ld - th ] ) . ] for @xmath57 , @xmath111 , and @xmath85 . ( for ease of comparisons , the histograms with solid lines only are scaled down by 2 . ) legend shows various values of @xmath76 . ] in the steady state of a standard tasep , the overall density of particles is controlled only by the entrance and exit rates . it is natural to expect that these relations would also hold in our case , except that the stationary density needs to be determined from the ( variable ) entrance rate in a self - consistent way . in the following , we pursue this approach and compare it to the simulation results reported in the preceding section . _ the ld case . _ again , we begin with a choice of @xmath112 such that the system is in an ld phase , with @xmath113 . invoking the exact expression for the density of the standard tasep , namely , @xmath69 , we write the density of particles in the resource - limited case as @xmath114 . with @xmath115 , this becomes an implicit equation for @xmath62 : @xmath116 which can be solved for given @xmath0 and @xmath4 . specifically , since @xmath117 , the solution will lie between @xmath14 and @xmath3 . the upper bound will be reached if @xmath118 so that the right hand side of eqn ( [ ld - th ] ) remains essentially constant as a function of @xmath62 . for any finite value of @xmath76 , the right hand side is a monotonically decreasing function of @xmath62 so that ( i ) eqn ( [ ld - th ] ) always has a solution , and ( ii ) the solution decreases monotonically with @xmath76 . the resulting density , and the associated current , @xmath119 , are shown as open circles and diamonds , respectively , in fig . [ fig : ld ] , for the choice of @xmath120 given in eqn ( [ alpha - mcs ] ) . we see , in particular , that for small values of @xmath76 , we may use @xmath121 whence @xmath122 . clearly , the agreement with the simulation data is impressive . _ the mc case . _ after these considerations , the analysis of the mc phase is quite straightforward . for large @xmath4 , the density saturates at @xmath123 , with @xmath124 . for small @xmath4 , the system is ld - like and follows eqn ( [ ld - th ] ) . the transition occurs when @xmath29 is forced to drop below @xmath125 , i.e. , at the intersection of eqn ( [ ld - th ] ) and @xmath126 . here , @xmath4 assumes the critical value of @xmath127 , given by @xmath128 the current follows the behavior of the density . _ the hd case . _ next , we turn to the high - density phase , with @xmath129 and @xmath87 . if the number of available particles is very large , we may again invoke the exact relation for the density of the standard tasep , namely @xmath130 this reflects the constant density observed in the simulations . clearly , this can not be sustained as the number of available particles is reduced , since the effective @xmath0 of the system is also reduced . a transition should be expected when it reaches @xmath2 . @xmath131 . translating @xmath2 @xmath132 into an equation for the system density @xmath62 , we arrive at @xmath133 solving for @xmath62 , and setting it to the one from eqn ( [ hd - th1 ] ) , we find the critical value of the total particle number for this transition : @xmath134 . ( the subscript , @xmath135 , helps to distinguish it from the lower transition point . ) for the specific @xmath36 used in simulations , this results in @xmath136 . \label{hd - c2}\ ] ] as we see , @xmath137 \simeq 1.01 $ ] is in excellent agreement with the `` second kink '' shown in fig . [ fig : hd ] . at the opposite extreme , the resources are very low , so that @xmath29 is significantly less than @xmath2 , and the system is pushed into ld - like behavior . for such low @xmath4 , the density follows eqn ( [ ld - th ] ) . in this regime , @xmath36 is well approximated by a linear function , so that @xmath138 , i.e. , @xmath139 of course , eqn ( [ ld - th ] ) is also not `` sustainable , '' in the sense that @xmath140 for large @xmath4 instead of the correct limit : @xmath141 . the critical value of @xmath4 again occurs when @xmath29 reaches @xmath2 , i.e. , when the density given by eqn ( [ ld - th ] ) reaches @xmath2 . the result is @xmath142 again , for the case shown in fig . [ fig : hd ] , there is excellent agreement between the data and this prediction : @xmath143 . between @xmath144 and @xmath145 , eqn ( [ hd - th1a ] ) provides a _ linear _ dependence of @xmath62 on @xmath76 @xmath146 with _ unit _ slope . again , with _ no _ fitting parameters , this theoretical prediction agrees remarkably well with the observations . in this regime , the reservoir occupation ( and so @xmath29 ) remains constant , while the change in @xmath4 is balanced by the change in total lattice population : @xmath147 . since @xmath148 ( on the average ) , the current remains at the _ constant _ value @xmath149 throughout . while this `` three - piece '' approach is both intuitively appealing and surprisingly successful in reproducing simulation data , it begs the question : is there a unified theory that is just as successful with predictions ? the answer is affirmative , namely , the domain wall theory dw , sa . since this theory is absolutely indispensable for providing good predictions for the next case , we will discuss its details in the next subsection . _ the sp case . _ here , monte carlo simulations presented us with unexpected phenomena : the `` shoulder '' in fig . [ fig : sp ] for @xmath62 . the simple approach taken above succeeds only in predicting the behavior for the extreme values of @xmath76 . dominated by the limited resources , the solution of @xmath150 fits the data very well for @xmath151 . at the other extreme ( @xmath152 ) , the system is necessarily in the sp , where @xmath126 on the average . in the crossover regime , the understanding of the structure in @xmath153 requires a more sophisticated theory @xcite . based on a biased random walk of the shock ( i.e. , domain wall ) , santen and appert @xcite derived the average occupation for a _ finite _ tasep : @xmath154 where @xmath155 , and @xmath156 is the ratio of the diffusion constants to the right / left . to apply to our sp case , we simply set @xmath0 to be @xmath157 in this equation . that sets up a relationship between @xmath62 and @xmath94 which can be computed numerically . the agreement between this theory and data is quite respectable ( fig . [ fig : sp ] ) , though not as spectacular as in the other cases . not surprisingly , most of the disagreement occurs in the crossover region . we believe that the main difficulty lies with the large fluctuations of the domain wall , feeding back in non - trivial ways to the on - rate . for another perspective on the successes and limitations of this theory , we can compare @xmath102 above with the predicted @xmath158 , the probability of finding the domain wall to be at site @xmath11 . the latter is , for the steady state , a pure exponential @xcite : @xmath159 . since our focus here is the dependence on @xmath62 , @xmath158 can be translated into @xmath102 , via the relation @xmath160 . we find that the _ predicted _ @xmath102 is proportional to @xmath161 , and nonzero only in the interval @xmath162 $ ] . our simulations show several non - trivial deviations from this theoretical prediction , however . ( i ) the slopes in the exponent are systematically steeper than predicted . ( ii ) quadratic terms in @xmath163 are far from negligible . ( iii ) the tails of @xmath102 extend substantially outside the range @xmath164 $ ] . in other words , there is room for improvements to the domain wall theory . another intriguing phenomenon , not shown here explicitly , is the complex dependence of this crossover on @xmath10 . in particular , both simulations and theory show that the `` shoulder '' completely disappears for small @xmath10 . for example , in an @xmath165 run , @xmath153 rises linearly for @xmath166 and simply `` slows down '' into the asymptotic value of @xmath125 for @xmath167 . for large @xmath10 , in contrast , there exists a subtle competition between the factors @xmath168 and @xmath169 in eqn ( [ < n > ] ) . the result is that the crossover region gets longer with larger @xmath10 , producing challenges at the simulations front as well . of course , these difficulties are hardly surprising , since the complexities here are undoubtedly due to large fluctuations associated with the infrared scales . further studies into this issue , especially more detailed investigations of the lean - to s in @xmath102 , should provide insight for a better understanding of such fluctuations . to conclude this section , we briefly note that we also simulated an alternate form of the on - rate , namely , @xmath170 $ ] . we will not present any details here , but the agreement of simulation data and analytic results for the three regimes ( hd , ld , and mc ) is equally convincing . however , the details of the crossover regime in sp are likely to be quite sensitive to functional form of @xmath36 . in this paper , we studied a tasep with open boundary conditions . in contrast to the standard model , where the particles are supplied by an infinite reservoir , the total number of available particles in our model , @xmath4 , is _ fixed_. mimicking limited resources , the available particles are shared between the reservoir and the system itself : if more particles bind to the chain , the reservoir is depleted , and vice versa . we account for this situation through an effective on - rate , @xmath29 , which depends on @xmath5 , the number of particles in the pool , and hence , on @xmath6 , the number on the chain . specifically , we mainly used eqn ( [ alpha - mcs ] ) , @xmath171 , here . as the number of available particles is reduced from infinity , the system crosses over from the asymptotic behavior of the standard tasep to different behaviors dominated by limited resources . if the corresponding standard model ( i.e. , @xmath172 limit ) lies in the ld phase , there are no surprises in either @xmath62 , the average particle density on the chain , or @xmath47 , the average current : both are proportional to @xmath4 for @xmath173 , crossing gently over to the asymptotic values much like the @xmath174 function . for the mc phase , there is already a sublety : both @xmath62 and @xmath47 increase monotonically and then , at a characteristic @xmath29 ( @xmath175 ) become constant . @xmath62 displays a notable kink there . for the other two cases ( hd and sp ) , @xmath62 exhibits an additional , intermediate regime ( though the behavior of @xmath47 shows little hint of this regime ) . in the hd case , @xmath62 is linear in @xmath76 with _ unit _ slope ; only the intercept depends on @xmath176 . here , @xmath5 remains constant , as any increase in @xmath76 is `` absorbed '' by the chain ( through a shift in the domain wall ) . this behavior may be regarded in the same light as an equilibrium system with phase co - existence . considering , e.g. , the pressure _ vs. _ volume isotherm for a liquid - gas system below criticality , the pressure remains constant over some range , as any increase in volume is absorbed by a shift in the liquid - to - gas ratio . the most interesting and challenging case is the sp . in the intermediate regime , a `` shoulder '' appears in @xmath153 . in addition , this structure displays a non - trivial dependence on @xmath10 . although the predictions of domain wall theory agree reasonably well with mc data , there are small ( @xmath177 ) systematic discrepancies . we also investigated histograms of particle densities , which cross over from sharply peaked gaussians for @xmath173 to a flat distribution ( between @xmath178 and @xmath179 ) for @xmath180 . in the intermediate regime , this distribution displays a much richer behavior than a truncated exponential the result of domain wall theory . several questions remain open for further study . first , are the transitions between the three regimes in hd associated with true thermodynamic singularities ? we find that the changes in slope accompanying these `` crossovers '' become sharper with increasing chain length @xmath10 . while this speaks for a true transition , the absence of large fluctuations is somewhat surprising . clearly , a more careful analysis is required to explore these questions more fully . second , though domain wall theory is reasonably successful for @xmath181 in the sp case , we would like to find improvements of this approach , so as to account for the crossover regime better . arriving at a good understanding of @xmath102 and the large fluctuations would be desirable . progress along these lines may also help us to develop an intuitive picture for the complex @xmath10 dependence . finally , we are mindful of one of the motivations of this study , namely , finite resources in biological systems , e.g. , ribosomes ( and aa - trna s ) for modeling protein synthesis in a cell @xcite . for that application , the above tasep needs to be generalized , to include exclusion at a distance ( particles covering more than one site ) and inhomogeneous hopping rates . how these systems are affected by finite resources will pose many interesting new challenges , especially on the theoretical frontier . _ acknowledgements . _ we thank travis merritt for preliminary simulation results and meesoon ha and marcel den nijs for helpful discussions . this work was supported in part by the national science foundation through dmr-0414122 and dmr-0705152 .

the totally asymmetric simple exclusion process ( tasep ) is a well studied example of far - from - equilibrium dynamics . here , we consider a tasep with open boundaries but impose a global constraint on the total number of particles . in other words , the boundary reservoirs and the system must share a finite supply of particles . using simulations and analytic arguments , we obtain the average particle density and current of the system , as a function of the boundary rates and the total number of particles . our findings are relevant to biological transport problems if the availability of molecular motors becomes a rate - limiting factor . _ keywords _ : non - equilibrium statistical physics , totally asymmetric exclusion process , biological transport

You are an expert at summarizing long articles. Proceed to summarize the following text: the relative proximity of the andromeda galaxy ( m31 ) and the global perspective from our external vantage point make m31 an excellent laboratory for studying the stellar halos of large galaxies . resolved stellar maps of m31 s halo , assembled over the past decade , have revealed highly complex inhomogeneities , the most striking of which is the giant southern stream ( gss ) , extending @xmath1 away from m31 s center in the southeast direction ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * hereafter i07 ) and falling towards m31 s center with relative radial velocities as high as @xmath2 @xcite . other significant morphological and kinematic features in the m31 halo include stellar shelves ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * hereafter / ) as well as the recently discovered minor axis `` streams '' . the gss is especially notable because it offers an opportunity to precisely measure m31 s potential ( * ? ? ? * ; * ? ? ? * hereafter ) and provides a view into the most significant local group galaxy disruption in the last gyr . models detailing the formation of the gss agree remarkably well with most aspects of the observations , and suggest the progenitor had a stellar mass of @xmath3 ( * ? ? ? our kinematic analysis in f07 finds that seemingly unrelated features like the `` northeast shelf '' and less prominent `` western shelf '' are also the result of the same disruption process , a conclusion supported by independent studies of their stellar populations @xcite . the observed gss s most striking point of contrast with the models is its asymmetry in the transverse direction . as shown both with photometric samples @xcite and spectroscopic surveys , its stellar distribution is sharply truncated on the ne side and falls off much more slowly on the sw side . in addition , the current models do not address the observed stellar population gradients within the gss ( * ? ? ? * ; * ? ? ? * i07 ) . in this letter , we show that this structure in the gss can be accounted for if the progenitor hosted a cold , rotating stellar disk , unlike the simple spherical progenitors used in previous simulations . surprisingly , we find that the disruption of a disk galaxy can also give rise to features similar to the recently discovered arc - like minor - axis `` streams '' , leading to the tantalizing possibility that most of the major inhomogeneities observed in the m31 halo are tidal debris from the same galaxy that caused the gss . in section 2 , we briefly describe our model for the progenitor and our @xmath4-body study of its tidal disruption . in section 3 , we show results from these simulations , focusing on the transverse density profile of the gss , the metallicity gradient , and arc - like structures that overlap the minor axis . section 4 summarizes our conclusions . our simulations are based on the methods worked out in our earlier papers : @xcite , , and . we use the orbit and potential from table 1 of and their spherical plummer model to represent a non - rotating progenitor . for runs with a disk progenitor , we use the same initial position and velocity , but substitute a different initial structure of the satellite . briefly , our disk models assume the satellite is composed of a bulge and rotating disk of stars . for simplicity we assume that the dark matter associated with the galaxy has been tidally stripped before the encounter modeled here . we use a hot exponential @xmath5 disk with mass @xmath6 , radial scale length @xmath7 , and vertical scale height @xmath8 . we add to this a hernquist bulge of mass @xmath9 and scale length @xmath8 . we initialize both components with the package zeno written by josh barnes . we evolve the satellite in m31 s potential starting from 12 evenly spaced orientations of the disk . from this we select two models displaying particularly good agreement with observational features , referred to here as disks a and b. in a forthcoming paper we will conduct a more systematic survey of possible initial conditions and quantify the debris structure in detail ( fardal et al . , in preparation ) . the remaining details of the simulations are the same as those in and . we set the satellite in motion inbound and slightly past apocenter , minimizing initial transients from m31 s tidal forces . we run the simulations with the multistepping tree code pkdgrav @xcite . our simulations include the self - gravity of the progenitor satellite , but ignore dynamical friction , perturbations from the other m31 satellites , and the history of the progenitor prior to the orbit that produces the gss , as justified in our earlier papers . we stop the simulations 840 myr into the run , which is approximately 650 myr past the initial pericenter . this time was chosen in to give a reasonable match to the debris structure around m31 , including the radii of the `` shells '' on the e and w sides . figures [ fig.skymaps]b,d show surface density maps based on the disk a and plummer models , respectively . both models reproduce the main feature of a stream extending to the se . they also reproduce the observed line - of - sight distances and velocities along the gss . however , the transverse distribution of gss stars is strikingly different between the two models disk a displays a much sharper ne edge . the observed star - count maps @xcite are not directly comparable since they contain both non - gss - related m31 components and non - m31 contaminants and are not explicitly calibrated to stellar surface density , but the morphology of the gss in these maps appears much closer to our disk model . comparison of the minor - axis contamination to the observations of . the deimos masks ( rectangles ) are grouped into inner minor - axis masks , outer minor - axis masks , and a single mask ( f135 ) offset from the minor axis . m31 s center is at ( 0,0 ) . the inset plots for each group show the ratio @xmath10 of the strength of the gss component to the peak of the gss at the same @xmath11 . @xmath10 is measured as discussed in the text for disks a and b and the plummer model 840 myr into the runs . the observational estimates and @xmath12 error bars from gilbert et al . ( in preparation ) are plotted as horizontal solid and dotted lines . the plummer model clearly contributes too much debris on the minor axis . , width=288 ] figure [ fig.skymaps]d shows that the plummer model results in a large amount of stars spilling over as far as the se minor axis , located to the ne of the gss . when compared their keck / deimos spectroscopic data near m31 s se minor axis to this model , they noted much less spillover from the gss than predicted by the model . gilbert et al . ( in preparation ) has quantified this by dividing the number of stars moving with gss - like velocities on the minor axis to those in the gss core at the same projected radius @xmath11 . for the nine innermost deimos masks on the minor axis combined , this ratio @xmath13 ; for the three outermost masks on the minor axis , @xmath14 ; and for the mask f135 located somewhat nearer the gss , they find a likely detection of gss material with @xmath15 . in figure [ fig.fieldbars ] , we compare the density of gss stars in all three @xmath4-body models to these results . we have selected `` gss '' particles by defining the trend of radial velocity @xmath16 with @xmath11 and then taking stars that fall within @xmath17 of this velocity in the given field . we also restrict the particles to those actually in the gss s `` shell '' . we then repeat the procedure for a control field located at the peak of the gss at the same @xmath11 , using a smaller interval @xmath18 as the gss core has a sharper velocity distribution . clearly the two disk models are in better accord with the observations than the plummer model . the sharper ne edge and smaller minor - axis contamination of the disk models thus imply that the progenitor was rapidly rotating . we will explore this argument in more detail in fardal et al . ( 2008 , in preparation ) . the mean color of gss rgb stars is observed to vary in the transverse direction : the gss is significantly broader in blue than in red stars @xcite . this is probably due to a metallicity gradient . quantified the metallicity distribution in two gss - dominated regions , one in the center of the gss and one in a less dense `` cocoon '' region to the sw , and showed that the latter has a lower mean metallicity . disk galaxies , of course , tend to have metallicity gradients . therefore it is interesting to see how a plausible gradient in our disk progenitor translates to the metallicity pattern on the sky . we use a simple parametric model to produce a plausible metallicity gradient in our initial disk model . we first find the specific orbital energy @xmath19 of each particle . we then assign it a metallicity using @xmath20}= a_z + b_z \ , \log_{10 } \left [ -e_i/(50 { \,\mbox{km}\,\mbox{s}^{-1}})^2 \right]$ ] , setting @xmath21 and @xmath22 to agree with the results of as explained below . this produces the metallicity gradient seen in figure [ fig.metals]a . observational results for the stars in the small disk galaxy m33 are also plotted , with the radius for both galaxies normalized by the disk scale length ; the metallicity pattern of our disk model agrees quite well . the gss progenitor should perhaps be lower in metallicity than m33 by a few tenths of dex due to its lower inferred mass , but the photometric metallicity measurements probably have systematic uncertainties at this level in any case . figure [ fig.skymaps]e shows the sky view of the resulting model metallicity pattern . the gradient along the stream is very weak , but the mean metallicity along the denser central part is clearly higher than in the broad wing to the sw , similar to the pattern seen in m31 s gss . using s figure 27 , we estimate the `` core '' and `` cocoon '' regions ( at @xmath23 kpc ) have mean metallicities of @xmath20}= -0.54 $ ] and @xmath24 , respectively ( mean @xmath20}=-0.51 $ ] was obtained at the gss sharp ne edge by @xcite ) . figure [ fig.skymaps]e shows `` broad wing '' and `` central gss '' boxes chosen at a similar radius , but better matching the slightly different model stream position . once we set @xmath25 and @xmath26 , the metallicities in these boxes are also @xmath27 and @xmath24 . the bare fact we can match two metallicities with two parameters is not in itself meaningful , but it is significant that the magnitude _ and sign _ of our initial metal gradient are very reasonable ( fig . [ fig.metals]a ) . figure [ fig.metals]b shows that within each box there is a wide range of metallicities ; the distributions in appear somewhat broader , but given measurement errors and the contributions from other halo components this is not surprising . using their megacam photometric survey of m31 s halo , found multiple surface density ridges along the minor axis which they called `` streams '' . streams c and d ( the two closest to m31 ) form a pair of curving ridges at slightly different orientations , which appear to merge as they approach the survey boundary ( see their fig . stream c appears to be slightly broader than stream d , and slightly more metal - rich , though not as metal - rich as the gss core / cocoon . from i07 s figure 33 we estimate the mean metallicity of streams c and d to be @xmath28 and @xmath29 respectively . @xcite suggested these `` streams '' might be shell features from a satellite disruption , similar to the event that created the gss but from a different progenitor . while studying our overall sample of runs based on 12 disk orientations , we noticed one ( disk b ) containing two curious `` arcs '' crossing the minor axis . these arcs are clearest at the step 680 myr into the run shown in figure [ fig.skymaps]c . morphologically , the two arcs somewhat resemble streams c and d , with a fatter southern arc nearly merging into a sharper northern arc . like the observed `` streams '' , neither arc crosses the gss to the sw . compared to the observed arcs , the simulated arcs are significantly further from m31 s center . as figure [ fig.skymaps]f shows , the simulated arcs are significantly less rich in metals than the gss . using the same metallicity model as for disk a and the regions defined by boxes in this figure , the mean @xmath20}$ ] is @xmath30 for the southern arc and @xmath31 for the northern arc . thus there is considerable if inconclusive evidence that these arcs are close analogues of the `` streams '' in . in our model , these two arcs originate from the outer regions of the disk , and are sharp mainly because of the relatively cold velocity field of the disk . both arcs consist of material that takes a path around m31 s center nearly opposite to the bulk of the progenitor , explaining why they lie so far from the gss . the large size of our disk is thus crucial ; a compact progenitor resembling m32 , for example , would be unable to produce similar arcs . the southern arc consists of a group of particles sharing nearly the same energy , and come from fairly far out in the progenitor s disk . the northern arc consists of particles that lie even further out ( explaining its lower metallicity on average ) , which form a tidal tail during the interaction with m31 . we can not yet explore the full parameter space of the collision for the presence and properties of these arc - like features . however , we did conduct a few additional runs with changes to the disk mass , radius , and orientation of disk b , finding the arcs were sensitive to the exact input parameters . thus we will require more theoretical investigation as well as more observational constraints to determine whether the arcs explain some of the minor - axis streams , or are merely a fortuitous similarity . if the arcs are shown to be related to the gss , they will be a very solid argument for the disk nature of the progenitor , and will place strong constraints on the parameters of the collision . in summary , several strands of observational evidence suggest that the gss originated from a progenitor with a strong sense of rotation , such as a disk galaxy . the transverse density profile of the gss is more easily produced by a rotating satellite . the observed decline in mean metallicity from the central core of the gss to its `` cocoon '' to the sw suggests that the progenitor had a strong radial metallicity gradient , of the sort found mainly in disk galaxies . furthermore , several observed arcs lying across the minor axis in m31 have very suggestive analogues in one of our runs . if shown to be related to the gss in the manner suggested by our model , these features would be definite confirmation of a disk - like progenitor . the notion of a disk galaxy progenitor is somewhat at odds with age measurements of the gss , which suggests little star formation during the last 4 gyr @xcite . however , the fields used to infer this were placed in the central , metal - rich part of the gss ; it is possible that the progenitor had an age gradient as well as a metallicity gradient , with the older stars on the inside . age measurements in the gss cocoon would therefore be interesting . it is also possible that the gss progenitor was more similar to an s0 galaxy than a spiral , perhaps due to stripping of its gas in an earlier phase of its encounter with m31 . many papers have used metallicity to assess the relationship among various m31 disk and halo features . our suggestion that the gss progenitor had a strong metallicity gradient means that metallicity can no longer be used as a reliable fingerprint of origin . this obviously complicates the forensic reconstruction of m31 s merger history . despite this , the rapidly growing database on m31 halo structure is a fascinating puzzle , offering unique insight into the life of a typical disk galaxy and the death of its unfortunate former companions . guhathakurta , p. , et al.2005 , arxiv preprint ( astro - ph/0502366 ) guhathakurta , p. , et al.2006 , aj , 131 , 2497 ibata , r. , irwin , m. j. , ferguson , a. m. n. , lewis , g. , & tanvir , n. 2001 , nature , 412 , 49 ibata , r. , chapman , s. , ferguson , a. m. n. , irwin , m. , lewis , g. , & mcconnachie , a. 2004 , mnras , 351 , 117 ibata , r. , martin , n. f. , irwin , m. , chapman , s. , ferguson , a. m. n. , lewis , g. f. , & mcconnachie , a. w. 2007 , , 671 , 1591 ( i07 )

the halo region of m31 exhibits a startling level of stellar inhomogeneities , the most prominent of which is the `` giant southern stream '' . our previous analysis indicates that this stream , as well as several other observed features , are products of the tidal disruption of a _ single _ satellite galaxy with stellar mass @xmath0 less than 1 gyr ago . here we show that the specific observed morphology of the stream and halo debris favors a cold , rotating , disk - like progenitor over a dynamically hot , non - rotating one . these observed characteristics include the asymmetric distribution of stars along the stream cross - section and its metal - rich core / metal - poor sheath structure . we find that a disk - like progenitor can also give rise to arc - like features on the minor axis at certain orbital phases that resemble the recently discovered minor - axis `` streams '' , even reproducing the lower observed metallicity of these streams . though interpreted by the discoverers as new , independent tidal streams , our analysis suggests that these minor - axis streams may alternatively arise from the progenitor of the giant southern stream . overall , our study points the way to a more complete reconstruction of the stream progenitor and its merger with m31 , based on the emerging picture that most of the major inhomogeneities observed in the m31 halo share a common origin with the giant stream .

You are an expert at summarizing long articles. Proceed to summarize the following text: an epidemic working through a population , cascading electrical power failures , product adoption , and the spread of a mutant gene are all examples of diffusion processes that can happen in multi - agent systems structured as complex networks . these network processes have been studied in a variety of disciplines , including computer science @xcite , biology @xcite , sociology @xcite , economics @xcite , and physics @xcite . much existing work in this area is based on pre - existing models in sociology and economics in particular the work of @xcite . however , recent examinations of social networks both analysis of large data sets and experimental have indicated that there may be additional factors to consider that are not taken into account by these models . these include the attributes of nodes and edges , competing diffusion processes , and time . in this paper , we outline seven design criteria ( section [ sec : criteria ] ) for such a framework and introduce @xmath0 ( section [ prelimsec ] ) , which is to the best of our knowledge the first logical language for modeling diffusion in complex networks that meets these criteria . @xmath0 is a rule - based framework ( inspired by logic programming ) that can richly express how agents adopt or fail to adopt certain behaviors , and how these behaviors cascade through a network . we also introduce fixed - point based algorithms that allow for the calculation of the result of the diffusion process in section [ fpsec ] . note that these algorithms are proven not only to be correct , but also to run in polynomial time . hence , our approach can not only better express many aspects of cascades in complex networks , but it can do so in a reasonable amount of time . we conclude by discussing applications of @xmath0 in section [ sec : learn ] . _ proofs of all results stated in this paper can be found in the appendix . _ we begin by identifying a set of criteria that we believe a framework for reasoning about cascades in complex networks should satisfy . * 1 . multiply labeled and weighted nodes and edges . * many existing frameworks for studying diffusion in complex networks assume that there is only one type of vertex that may become `` active '' @xcite or may `` mutate '' @xcite and only one possible relationship between nodes . in reality , nodes and edges often have different properties . for instance , labels on edges can be used to differentiate between strong and weak ties ( edge types ) a concept that is well studied @xcite . recently , such attributes of nodes have been shown to impact influence in a network @xcite . * 2 . explicit representation of time . * most work in the literature assumes static models , with the exception of the recent developments in @xcite , which assume the existence of a timestamped log referring to actions taken in the network in order to learn how nodes influence each other . though @xcite tackles the problem of predicting the time at which a certain node will take an action , the authors make several simplifying assumptions such as monotonicity of probability functions , probabilistic independence , sub - modularity and , most importantly for this criterion , a modeling of time solely based on temporal decay of influence . we seek a richer model of temporal relationships between conditions in the network structure , the current state of the cascades in process , and how influence propagates . * non - markovian temporal relationships . * apart from time being explicitly represented , the temporal dependencies should be able to span multiple units of time . hence , the `` memoryless '' mode of a standard markov process , where only the information of the current state is required , is insufficient . here , we strive to create a framework where dependencies can be from other earlier time steps . this issue has been previously studied with respect to more general logic programming frameworks such as @xcite , but to our knowledge has not been applied to social networks . * representation of uncertainty . * as in practice it is not always possible to judge the attributes of all individuals in a network , an element of uncertainty must be included . however , in connection with point 7 , this should not be at the expense of tractability . for instance , the probabilistic models of @xcite are normally addressed with simulation ( and hence do not scale well ) as the computation of the expected number of activated nodes is a @xmath1-hard problem @xcite . * competing cascades . * often , in real - world situations there will be competing cascading processes . for example , in evolutionary graph theory @xcite , `` mutants '' and `` residents '' compete for nodes in the network the success of one hinges on the failure of the other . . non - monotonic cascades . * in much existing work on cascades in complex networks , the number of nodes attaining a certain property at each time step can only increase . however , if we allow for competing cascades in the same model , we can not have such a strong restriction as the success of one cascade may come at the expense of another . * 7 . tractability . * the social networks of interest in today s data mining problems often have millions of nodes . it is reasonable to expect that soon billion - node networks will be commonplace . any framework for dealing with these problems must be solvable in a reasonable amount of time and offer areas for practical improvement for further scalability . the above criteria can be summarized as the desire to design the most expressive language for network cascades possible while still allowing computation of the outcome of a diffusion process to be completed in a tractable amount of time . as a comparison , let us briefly describe some relevant related work . perhaps the best known general model for representing diffusion in complex networks is the independent cascade / linear threshold ( ic / lt ) model of @xcite . however , although this framework was shown to be capable of expressing a wide variety of sociological models , it assumes the markov property and does not allow for the representation of multiple attributes on vertices and edges . a more recent framework , social network optimization problems ( snops ) @xcite uses logic programming to allow for the representation of attributes , but this framework does not allow for competing processes or non - monotonic cascades . a related logic programming framework , competitive diffusion ( cd ) @xcite allows for competitive diffusion and non - monotonic processes but does not explicitly represent time and also makes markovian assumptions . further , we also note that the semantics of cd yields a `` most probable interpretation '' that is not a unique solution . hence , a given model in that framework can lead to multiple and possibly contradictory , outcomes to a cascade ( this problem is avoided in @xmath0 ) . another popular class of models is evolutionary graph theory ( egt ) @xcite , which is highly related to the voter model ( vm ) @xcite . although this framework allows for competing processes and non - monotonic diffusion , it also makes markovian assumptions while not explicitly representing time . further , determining the outcome of a cascade in those models is np - hard , while determining the outcome in @xmath0 can be accomplished in polynomial time . table [ rwtab ] lists how these models compare to @xmath0 when considering our design criteria . [ cols="<,^,^,^,^,^,^",options="header " , ] we define a @xmath0 interpretation as follows . a @xmath0 interpretation @xmath2 is a mapping of natural numbers in the interval @xmath3 $ ] to network interpretations , i.e. , @xmath4 . let @xmath5 be the set of all possible interpretations . first , we define what it means for an interpretation to satisfy a fact and a rule . an interpretation @xmath2 * satisfies * @xmath0 fact @xmath6 $ ] , written @xmath7 $ ] , iff @xmath8 $ ] , @xmath9 . [ satfactex ] consider interpretation @xmath10 , where @xmath11 ( from example [ niex ] ) , and @xmath0 facts @xmath12 and @xmath13 from example [ ex3 ] . in this case , @xmath14 and @xmath15 . for non - fluent facts , we introduce the notion of strict satisfaction , which enforces the bound in the interpretation to be set to exactly what the fact dictates . interpretation @xmath2 * strictly satisfies * @xmath0 fact @xmath16 $ ] iff @xmath8 $ ] , @xmath17 . next , we define what it means for an interpretation to satisfy an integrity constraint . an interpretation @xmath2 * satisfies * integrity constraint @xmath18 iff for all @xmath19 and @xmath20 , @xmath21 . before we define what it means for an interpretation to satisfy a rule , we require two auxiliary definitions that are used to define the bound enforced on a label by a given rule , and the set of time points that are affected by a rule . [ bounddef ] for a given rule @xmath22 , node @xmath23 , and network interpretation @xmath24 , @xmath25 @xmath26 where @xmath27 @xmath28 and @xmath29 @xmath30 intuitively , the bound returned by the function depends on the influence function and the number of qualifying and eligible nodes that influence it . for interpretation @xmath2 , node @xmath23 , and rule @xmath22 , the _ target time set _ of @xmath31 is defined as follows : @xmath32 \ ; | \ ; i(t-\delta t)(v ) \models f\big\}\ ] ] we also extend this definition to a program @xmath33 , for a given @xmath20 and @xmath34 , as follows ; @xmath35 @xmath36 \ ; | \ ; ( \<l,\bnd\>,c):[t_1,t_2 ] \in p\big\}\ ] ] @xmath37 we can now define satisfaction of a rule by an interpretation . an interpretation @xmath2 * satisfies * a rule @xmath38 iff for all @xmath39 and @xmath40 it holds that @xmath41 [ satrule ] let @xmath10 be the interpretation from example [ satfactex ] . suppose that @xmath42\ > \in i(1)(5)$ ] . in this case , let @xmath44 be equivalent to @xmath10 except that we have @xmath45\ > \in i_2(1)(3)$ ] . in this case , @xmath46 . we now define satisfaction of programs , and introduce _ canonical interpretations _ , in which time points that are not `` targets '' retain information from the last time step . for interpretation @xmath2 and program @xmath33 : @xmath2 is a * model * for @xmath33 iff it satisfies all rules , integrity constraints , and fluent facts in that program , strictly satisfies all non - fluent facts in the program , and for all @xmath47 @xmath20 and @xmath48 , @xmath49\ > \in i(c)(t)$ ] . @xmath2 is a * canonical model * for @xmath33 iff it satisfies all rules , integrity constraints , and fluent facts in @xmath33 , strictly satisfies all non - fluent facts in @xmath33 , and for all @xmath47 @xmath50 and @xmath48 , @xmath49\ > \in i(c)(t)$ ] when @xmath51 and @xmath52 where @xmath53 , otherwise . [ canonex ] following from previous examples , if we consider interpretation @xmath10 and program @xmath54 , we have that @xmath55\>$ ] must be in @xmath56 in order for @xmath10 to be canonical . in this section we discuss consistency and entailment in @xmath0 programs , and explore the use of minimal models towards computing answers to these problems . a @xmath0 program @xmath33 is ( canonically ) consistent iff there exists a ( canonical ) model @xmath2 of @xmath33 . a @xmath0 program @xmath33 ( canonically ) entails @xmath0 fact @xmath57 iff for all ( canonical ) models @xmath2 of @xmath33 , it holds that @xmath58 . now we define an ordering over models and define the concept of minimal model . we then show that if we can find a minimal model then we can answer consistency , entailment , and tight entailment queries . to do so , we first define a pre - order over interpretations . given interpretations @xmath59 we say @xmath60 if and only if for all @xmath61 if there exists @xmath62 then there must exist @xmath63 s.t . @xmath64 . next , we define an equivalence relation for interpretations denoted with @xmath65 ; we will use the notation @xmath66 $ ] for the set of all interpretations equivalent to @xmath2 w.r.t . this allows us to define a partial ordering . two interpretations @xmath67 are * equivalent * ( written @xmath68 ) iff for all @xmath69 , @xmath70 iff @xmath71 . given classes of interpretations @xmath66,[i']$ ] that are equivalent w.r.t . @xmath65 , we say that @xmath66 $ ] precedes @xmath72 $ ] , written @xmath66 \sqsubseteq [ i']$ ] , iff @xmath60 . the partial ordering is clearly reflexive , antisymmetric , and transitive . note that we will often use @xmath73 as shorthand for @xmath66 \sqsubseteq [ i']$ ] . we define two special interpretations , @xmath74 and @xmath75 , such that @xmath76 , @xmath77 and there exists network atom @xmath78 . clearly , no other interpretation can be below @xmath74 as the @xmath79 $ ] bound on all network atoms for each time step and each component is @xmath80 $ ] ; similarly , no other interpretation is above @xmath75 , since for any interpretation @xmath2 for which there exists @xmath81 where @xmath82 , we have @xmath83 . we can prove ( see the full version of the paper for details ) that with @xmath75 and @xmath74 , @xmath84 is a complete lattice . we can now arrive at the notion of _ minimal model _ for a @xmath0 program . given program @xmath33 , the minimal model of @xmath33 is a ( canonical ) interpretation @xmath2 s.t . @xmath70 and for all ( canonical ) interpretation @xmath85 s.t . @xmath71 , we have that @xmath73 . suppose we have some algorithm @xmath86 that takes as input a program @xmath33 and returns an interpretation @xmath2 ( where @xmath2 does not necessarily satisfy @xmath33 ) s.t . for all @xmath85 where @xmath71 , @xmath73 . it is easy to show that if @xmath87 then @xmath33 is consistent . likewise , if @xmath88 then @xmath33 is inconsistent , as all models must then have a tighter weight bound for the network atoms than an invalid interpretation ( hence , making such an interpretation invalid as well ) . clearly , any such algorithm @xmath86 would provide a sound and complete answer to the consistency problem . likewise , if we consider the entailment problem , it is easy to show that for fact @xmath89 $ ] , @xmath33 ( canonically ) entails @xmath57 iff the minimal model of @xmath33 ( canonically ) satisfies @xmath57 . this is because for minimal model @xmath90 of @xmath33 , for any time @xmath91 $ ] , if @xmath92 then there is network atom @xmath93 s.t . we note that for any other interpretation @xmath2 of @xmath33 with @xmath94 we have that @xmath95 . hence , having a minimal model allows us to solve any entailment query . we can think of a minimal model of a @xmath0 program as the outcome of a diffusion process in a multi - agent system modeled as a complex network . hence , a question such as `` how many agents will adopt the product with a weight of at least @xmath96 in two months ? '' can be easily answered once the minimal model is obtained . in this section we introduce a fixed - point operator that produces the non - canonical minimal model of a @xmath0 program in polynomial time . this is followed by an algorithm to find a canonical minimal model also in polynomial time . first , we introduce three preliminary definitions . for a given @xmath0 program @xmath33 , @xmath20 , @xmath34 , and @xmath97 we define function @xmath98 @xmath99\in p\textit { s.t . } t \in [ t_1,t_2]}\bnd\ ] ] [ ibound ] for a given @xmath0 program @xmath33 , @xmath20 , @xmath34 , and @xmath97 we define function @xmath100 @xmath101 given @xmath0 program @xmath33 , interpretation @xmath2 , @xmath39 , @xmath34 , and @xmath97 , we define @xmath102 @xmath103 we can now introduce the operator . for a given @xmath0 program @xmath33 , we define the operator @xmath104 as follows : for a given @xmath2 , for each @xmath19 , @xmath105 , and @xmath34 , add @xmath106 to @xmath107 where @xmath108 is defined as : @xmath109 where @xmath110 . it is easy to show that @xmath111 can be computed in polynomial time ( the proof is in the full version ) . next , we introduce notation for repeated applications of @xmath111 . given natural number @xmath112 , interpretation @xmath2 , and program @xmath33 , we define @xmath113 , the multiple applications of @xmath111 , as follows : @xmath114 we can prove that the iterated @xmath111 operator converges after a polynomial number of applications : [ gammapolyconverge ] given interpretation @xmath2 and program @xmath33 , there exists a natural number @xmath115 s.t.@xmath116 , and @xmath117 where @xmath118 is the maximum in - degree in the network . for a given vertex @xmath119 , we will use the notation @xmath120 to denote the number of incoming neighbors ( of any edge type ) . first note that for a given @xmath121 and @xmath34 , a given rule @xmath122 can tighten the bound on a network atom formed with @xmath123 no more than @xmath124 times . at each application of @xmath111 , at least one network atom must tighten . hence , as there are only @xmath125 tightenings possible , this is also the bound on the number of applications of @xmath111 . in the following , we will use the notation @xmath126 to denote the iterated application of @xmath111 after a number of steps sufficient for convergence ; theorem [ gammapolyconverge ] means that we can efficiently compute @xmath126 . we also note that as a single application of @xmath111 can be computed in polynomial time , this implies that we can find a minimal model of a @xmath0 program in polynomial time . we now prove the correctness of the operator . we do this first by proving a key lemma that , when combined with a claim showing that for consistent program @xmath33 , @xmath126 is a model of @xmath33 , tells us that @xmath126 is a minimal model for @xmath33 . following directly from this , we have that @xmath33 is inconsistent iff @xmath127 . [ boundlemma ] if @xmath70 and @xmath128 then @xmath129 . [ minmodelgamma ] if program @xmath33 is consistent then @xmath126 is a minimal model for @xmath33 . these results , when taken together , prove that tight entailment and consistency problems for @xmath0 can be solved in polynomial time , which is precisely what we set out to accomplish as part of our desiderata described in section [ sec : criteria ] . next , we develop an algorithm for the _ canonical _ versions of consistency and tight entailment , and show that we can bound the running time of the algorithm with a polynomial . we also note that subsequent runs of the convergence of @xmath111 will likely complete quicker in practice , as the initial interpretation is the last interpretation calculated ( cf . line [ recalcline ] ) . we also show that the interpretation produced by the algorithm is a canonical minimal model . following from that , a program is inconsistent iff the algorithm returns @xmath75 . program @xmath33 interpretation @xmath2 [ bigfor ] [ condline ] [ vlloop][insideforbegin ] [ insideforend ] [ recalcline ] [ newcurfree ] @xmath2 algorithm performs no more than @xmath130 calculations of the convergence of @xmath111 . [ canonsound ] if @xmath33 is consistent , then @xmath131 is the minimal canonical model of @xmath33 . in this section , we will briefly discuss work in progress on how can be applied in real world settings . it is widely acknowledged that modeling influence in multi - agent systems ( most usefully modeled as complex networks ) is highly desirable for many practical problems as varied as viral marketing , prevention of drug use , vaccination , and power plant failure . though programs are a rich model to work with , the acquisition of rules is the principal hurdle to overcome ; this is mainly due to this richness of representation , since for each rule we must provide a set of conditions on the agents being influenced , conditions on their neighbors and their ties to their neighbors , and how capable these neighbors are of influencing them . a domain expert is likely able to provide important insights into these components , but the best way to obtain these rules is undoubtedly to leverage the presence of large amounts of data in domains like twitter ( with about 340 m messages sent per day , available through public apis ) , facebook ( over 950 m users with more complex information ; not publicly available , but data can be requested through apps ) , and blogging and photo hosting sites such as blogger and flickr ( which have millions of users as well ) . concretely , we have begun working towards this goal by extracting several time - series , multi - attribute network data sets on which to apply @xmath0 . for instance , to study the proliferation of research on different topics , we looked at research on `` niacin '' indexed by thomson reuters web of knowledge ( http://wokinfo.com ) . this topic was chosen due to its interest to a variety of disciplines , such as medicine , biology , and chemistry ; this gives the data more variety compared with more discipline - specific topics . we extracted an author - paper bipartite network consisting of @xmath132 papers with @xmath133 authors and @xmath134 edges ( cf . figure [ fig : papers12 ] ) ; from this data we can easily focus on various kinds of networks ( co - author , citation , etc . ) . we have also collected attribute and time - series data for this network , as well as the subjects of the papers ; the propagation of these subjects is a good starting point to test methods for the acquisition of @xmath0 rules . we are harvesting larger datasets from various online social networks . further details can be found in the full version of the paper . * a proposed learning architecture . * we are currently developing a learning architecture ( depicted in fig . [ fig : architecture ] ) based on the use of state - of - the - art data analysis , clustering , and influence learning techniques as building blocks for the acquisition of rules from data sets . the key question is not just the identification of the best techniques to adopt , but how to adapt them and combine them in such a way as to produce meaningful and useful outputs . consider the diagram in fig . [ fig : architecture ] : the data first flows from raw data sources to the _ cluster identification _ component , which has the goal of identifying sets of agents behaving as groups ( for instance , teens influencing other teens of the same sex in the consumption of music , or scientists of a certain field influencing the research topics of others in a related field ) @xcite ; the main output here is a set of conditions on nodes and edges that characterize groups of nodes . once clusters are identified , the _ influence recognition _ component will make use of both the clusters and the data sources to recognize what kind of influence is present in the system @xcite ; the main output of this component is the influence function to be used in the rules . the _ rule generation _ component then takes the output of the cluster identification and influence recognition components , along with the raw data ( _ e.g. _ , to analyze time stamps ) and produces rules ; the output of this component is involved in a refinement cycle with experts who can provide feedback on the rules being produced ( such as possible combinations of rules , identification of cases of overfitting , etc . ) . in this paper , we presented the @xmath0 language for modeling cascades in multi - agent systems organized in the form of complex networks . we started by establishing seven criteria in the form of desiderata for such a formalism , and proved that @xmath0 meets all of them ; to the best of our knowledge , this has not been accomplished by any previous model in the literature . we also note that @xmath0 is the first language of its kind to consider network structure in the semantics , potentially opening the door for algorithms that leverage features of network topology in more efficiently answering queries . our current work involves implementing the algorithms described in this paper , as well as the real - world applications described in section [ sec : learn ] ; though our algorithms have polynomial time complexity , it is likely that further optimizations will be needed in practice to ensure scalability for very large data sets . in the near future , we shall also explore various types of queries that have been studied in the literature , such as finding agents of maximum influence , identifying agents that cause a cascade to spread more quickly , and identifying agents that can be influenced in order to halt a cascade . p.s . is supported by the army research office ( project 2gdatxr042 ) . is supported under ( uk ) epsrc grant ep / j008346/1 proqaw . the opinions in this paper are those of the authors and do not necessarily reflect the opinions of the funders , the u.s . military academy , or the u.s . 10 s. aral and d. walker . . , 337(6092):337341 , 2012 . m. broecheler , p. shakarian , and v. subrahmanian . a scalable framework for modeling competitive diffusion in social networks . in _ proc . of socialcom_. ieee , 2010 . w. chen , c. wang , and y. wang . scalable influence maximization for prevalent viral marketing in large - scale social networks . in _ proc . of kdd 10 _ , pages 10291038 . acm , 2010 . a. goyal , f. bonchi , and l. lakshmanan . discovering leaders from community actions . in _ proc . of cikm _ , pages 499508 . acm , 2008 . a. goyal , f. bonchi , and l. lakshmanan . learning influence probabilities in social networks . in _ proc . of wsdm _ , pages 241250 . acm , 2010 . a. goyal , f. bonchi , and l. lakshmanan . a data - based approach to social influence maximization . , 5(1):7384 , 2011 . m. granovetter . . , 78(6):13601380 , 1973 . m. granovetter . threshold models of collective behavior . , 83(6):14201443 , 1978 . a. jain . data clustering : 50 years beyond k - means . , 31(8):651666 , 2010 . d. kempe , j. kleinberg , and e. tardos . maximizing the spread of influence through a social network . in _ proc . of kdd 03 _ , pages 137146 . acm , 2003 . e. lieberman , c. hauert , and m. a. nowak . evolutionary dynamics on graphs . , 433(7023):312316 , 2005 . t. c. schelling . . w.w . norton and co. , 1978 . p. shakarian , a. parker , g. i. simari , and v. s. subrahmanian . annotated probabilstic temporal logic . , 12(2 ) , 2011 . p. shakarian , v. subrahmanian , and m. l. sapino . . in _ proc . of iclp ( tech . comm . ) _ , 2010 . v. sood , t. antal , and s. redner . voter models on heterogeneous networks . , 77(4):041121 , 2008 . t. warren liao . clustering of time series data a survey . , 38(11):18571874 , 2005 . let @xmath135 be a subset of @xmath5 . we can create @xmath136 as follows . we build interpretation @xmath85 . for each @xmath137 , let @xmath138 be the least of the set @xmath139 \ > \in i(t)(c),\<l , [ \ell , u ) \ > \in i(t)(c ) \}$ ] and @xmath140 be the least of the set @xmath141 \ i(t)(c),\<l , ( \ell , u ) \ > \in i(t)(c ) \}$ ] . then , for each @xmath137 let @xmath142 be the greatest element of the set @xmath143 \ > \in i(t)(c),\<l , ( \ell , u ] \ > \in i(t)(c ) \}$ ] + and @xmath144 be the greatest of the set + @xmath145 . if there is any interpretation @xmath2 in @xmath5 where there is not some @xmath108 s.t . @xmath146 then add @xmath147 \>$ ] to @xmath148 . if @xmath149 and @xmath150 then add @xmath151 \>$ ] to @xmath148 . if @xmath149 and @xmath152 then add @xmath153 to @xmath148 . if @xmath154 and @xmath152 then add @xmath155 to @xmath148 . finally , if @xmath154 and @xmath150 then add @xmath156 \>$ ] to @xmath148 clearly , @xmath157 . in the next part of the proof , we show we can create @xmath158 as follows . we build interpretation @xmath85 . for each @xmath137 let @xmath138 be the greatest of the set + @xmath139 \ \in i(t)(c),\<l , [ \ell , u ) \ > \in i(t)(c ) \}$ ] and @xmath140 be the greatest of the set + @xmath141 \ > \in i(t)(c),\<l , ( \ell , u ) \ > \in i(t)(c ) \}$ ] . then , for each @xmath137 let @xmath142 be the least element of the set @xmath143 \ > \in i(t)(c),\<l , ( \ell , u ] \ > \in i(t)(c ) \}$ ] and @xmath144 be the least of the set @xmath145 . if @xmath159 or @xmath160 then add @xmath161 to @xmath148 . if @xmath154 and @xmath162 then add @xmath151 \>$ ] to @xmath148 . if @xmath154 and @xmath163 then add @xmath153 to @xmath148 . if @xmath149 and @xmath163 then add @xmath155 to @xmath148 . finally , if @xmath149 and @xmath162 then add @xmath156 \>$ ] to @xmath148 clearly , @xmath164 . + as both @xmath136 and @xmath158 exist and are clearly in @xmath5 then the statement follows . [ [ a - single - application - of - gamma - can - be - computed - in - polynomial - time ] ] a single application of @xmath111 can be computed in polynomial time ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ for interpretation @xmath2 , @xmath165 can be computed by conducting @xmath166 satisfaction checks where @xmath118 is the maximum in - degree of a node in the network . ( this combined with the assumption that the influence function is computed in constant time results in polynomial time computation for a single application of @xmath111 . ) we note that a given rule will require the most satisfaction checks , as a rule will potentially affect a network atom of a certain label for each vertex - time point pair . by the definition of @xmath167 , a given rule clearly causes no more than @xmath168 satisfaction checks . as the number of rules is no more than @xmath169 , the statement follows . for a given vertex @xmath119 , we will use the notation @xmath120 to denote the number of incoming neighbors ( of any edge type ) and @xmath170 . first we show that for a given @xmath121 and @xmath34 , a given rule @xmath122 can tighten the bound on a network atom formed with @xmath123 no more than @xmath124 times . this is because a given rule adjusts the bound on a network atom based on the number of eligible and qualifying neighbors , which are bounded by @xmath171 respectively . at each application of @xmath111 , at least one network atom must tighten . hence , as there are only @xmath172@xmath125 tightenings possible , this is also the bound on the number of applications of @xmath111 . suppose , bwoc , that @xmath173 . then , there exists some @xmath34 , @xmath97 and @xmath20 s.t . @xmath174 , @xmath175 , and @xmath176 s.t . @xmath177 and @xmath95 . there are four things that affect @xmath178 : facts , rules , integrity constraints and @xmath179 . clearly , we need not consider the effect that either facts or @xmath179 have on @xmath178 , as @xmath2 satisfies all facts and @xmath128 . we also note that a given integrity constraint imposed by definition [ ibound ] can tighten @xmath178 no more than the associated bound in any model . hence , there must be some rule @xmath180 that causes @xmath178 to become less than @xmath108 . as @xmath181 , we know that @xmath182 . as a result , we have @xmath183 and @xmath184 . further , as @xmath185 and no rule can modify a non - fluent atom , we have claim : if program @xmath33 is consistent then @xmath126 is a model of @xmath33 . + suppose , bwoc , that there is a fact in @xmath33 that @xmath126 does not satisfy . however , by the definition of @xmath111 and the definition of a fact , @xmath126 must satisfy all facts as the bound on the weight associated with each fact is included in the intersection . further , we can also see by the definition of @xmath111 that @xmath126 strictly satisfies all non - fluent facts in @xmath33 . we also note that the final application of the @xmath111 operator ensures that all integrity constraints are satisfied by @xmath126 . now , suppose , bwoc , that there is a rule in @xmath33 that @xmath126 does not satisfy . however , with each application of @xmath111 , for each rule , we include the bound on the weight returned by the @xmath193 function for each time step in the target time step associated with that rule . as @xmath111 is applied to convergence , and new bounds are intersected with each application , then we know that all time points in any associated target time set are considered in the intersection . + proof of theorem : the above claim tells us that @xmath194 . now consider interpretation @xmath2 s.t . as @xmath195 , multiple applications of lemma [ boundlemma ] tell us that @xmath196 . hence , the statement follows . claim 1 : if @xmath33 is consistent , then @xmath131 is a canonical model of @xmath33 . + clearly , @xmath197 satisfies all facts and integrity constraints in @xmath33 . hence , we shall consider programs that only consist of rules in this proof . we say @xmath2 @xmath123-canonically satisfies @xmath33 iff @xmath2 canonically satisfies @xmath198 . clearly , @xmath2 canonically satisfies @xmath33 if for all @xmath199 , @xmath33 @xmath123-canonically satisfies by @xmath2 . we say that @xmath2 is an @xmath200-canonically consistent interpretation if for @xmath20 , for the first @xmath201 , @xmath202 where @xmath203 . consider some @xmath204 and @xmath20 . clearly , @xmath2 is an @xmath205-model for @xmath33 . let us assume , for some value @xmath206 , that @xmath2 is an @xmath207 model for @xmath33 . let time point @xmath208 be the @xmath206-th element of @xmath209 . consider the time step before time @xmath208 is considered in the for - loop at line [ bigfor ] of , which causes the condition at line [ condline ] to be true . by line [ newcurfree ] , @xmath210[l]$ ] . this means that @xmath208 is a member of both . hence , when @xmath208 is considered at line [ bigfor ] , the condition at line [ condline ] is true , causing @xmath211 and as the element @xmath53 is not changed here , we have shown the @xmath2 is an @xmath200-model for @xmath33 . by the for - loop at line [ vlloop ] , for all @xmath199 and @xmath212 , @xmath2 is an @xmath200-model for @xmath33 . hence , at the for - loop at line [ bigfor ] , we can be assured that for @xmath199 and @xmath212 that @xmath2 @xmath213 satisfies @xmath33 which means that @xmath2 canonically satisfies @xmath33 + claim 2 : if @xmath2 is a canonical model for @xmath33 , + @xmath214 is an interpretation that also strictly satisfies all non - fluent facts in @xmath33 , and @xmath215 is @xmath216 after being manipulated in lines [ insideforbegin]-[insideforend ] of , then @xmath217 . we note that by the definition of satisfaction of a non - fluent fact , and the fact that both @xmath216 and @xmath2 must strictly satisfy all non - fluent facts in @xmath33 , we know that for all @xmath20 and @xmath34 that : @xmath218 let us assume that lines [ insideforbegin]-[insideforend ] of the algorithm are changing @xmath216 when the outer loop is considering time @xmath208 and that the condition at line [ condline ] is true . clearly , @xmath219 as a result , for any @xmath220 pair considered at this point by the algorithm , if @xmath221 and @xmath222 then we have @xmath223 . by the algorithm , if we have @xmath224 and @xmath225 we have that @xmath226 . as + @xmath227 , we know that @xmath228 . as a result , we have @xmath217 , completing the claim . proof of theorem : as initially @xmath229 and @xmath196 by theorem [ minmodelgamma ] , we note that the algorithm changes + @xmath216 either by applying @xmath111 or manipulating it in lines [ insideforbegin]-[insideforend ] , which tells us ( by claim 2 ) that for all models @xmath2 of @xmath33 that + @xmath230 . since by claim 1 we know that + @xmath231 , the statement of the theorem follows . one way in which can be used is looking at proliferation of research on different topics . we look at research conducted on niacin , an organic compound commonly used for increasing levels of high density lipoproteins ( hdl ) . using thomson reuters web of knowledge ( http://wokinfo.com ) we were able to extract information on @xmath232 articles about niacin . this information was then processed using the science of science ( sci@xmath233 ) tool ( http://sci2.cns.iu.edu ) to extract numerous different networks such as author by paper networks , citation networks , and paper by subject networks . each paper has attributes about when it was published , what journal it was published in , and what subjects the paper was about . during the first time period there is a total of @xmath234 papers with @xmath235 different authors and @xmath236 connections based on an author being cited as an author of a given paper . during the second time period , there is a total of @xmath132 papers with @xmath133 different authors and @xmath237 connections .

the modeling of cascade processes in multi - agent systems in the form of complex networks has in recent years become an important topic of study due to its many applications : the adoption of commercial products , spread of disease , the diffusion of an idea , etc . in this paper , we begin by identifying a desiderata of seven properties that a framework for modeling such processes should satisfy : the ability to represent attributes of both nodes and edges , an explicit representation of time , the ability to represent non - markovian temporal relationships , representation of uncertain information , the ability to represent competing cascades , allowance of non - monotonic diffusion , and computational tractability . we then present the @xmath0 language , a formalism based on logic programming that satisfies all these desiderata , and focus on algorithms for finding minimal models ( from which the outcome of cascades can be obtained ) as well as how this formalism can be applied in real world scenarios . we are not aware of any other formalism in the literature that meets all of the above requirements . [ representation languages ]

You are an expert at summarizing long articles. Proceed to summarize the following text: the ads / cft correspondence @xcite has revealed the deep relations between gauge theories and string theories and has provided powerful tools for understanding the dynamics of strongly coupled field theories in the dual gravity side . in recent years , this paradigm has been applied to investigate the properties of certain condensed matter systems @xcite . the correspondence between gravity theories and condensed matter physics(sometimes is also named as ads / cmt correspondence ) has shed light on studying physics in the real world in the context of holography . it is well known that in realistic condensed matter systems , the presence of a finite density of charge carriers is of great importance . according to the ads / cft correspondence , the dual bulk gravitational background should be charged black holes in asymptotically ads spacetimes . the simplest example of such charged ads black holes is reissner - nordstrm - ads(rn - ads ) black hole , which has proven to be an efficient laboratory for studying the ads / cmt correspondence . for instance , investigations of the fermionic two - point functions in this background indicated the existence of fermionic quasi - particles with non - fermi liquid behavior @xcite , while the @xmath5 symmetry of the extremal rn - ads black hole is crucial to the emergent scaling symmetry at zero temperature @xcite . moreover , adding a charged scalar in such background leads to superconductivity @xcite . a further step towards a holographic model - building of strongly - coupled systems at finite charge density is to consider the leading relevant ( scalar ) operator in the field theory side , whose bulk gravity theory is an einstein - maxwell - dilaton system with a scalar potential . such theories at zero charge density were analyzed in detail in recent years as they mimic certain essential properties of qcd @xcite . solutions at finite charge density have been considered in @xcite in the context of ads / cmt correspondence . recently a general framework for the discussion of the holographic dynamics of einstein - maxwell - dilaton systems with a scalar potential was proposed in @xcite , which was a phenomenological approach based on the concept of effective holographic theory ( eht ) . the minimal set of bulk fields contains the metric @xmath6 , the gauge field @xmath7 and the scalar @xmath8 ( dual to the relevant operator ) . @xmath8 appears in two scalar functions that enter the effective action : the scalar potential and the non - minimal maxwell coupling . they studied thermodynamics of certain exact solutions and computed the dc and ac conductivity . the main advantage of this eht approach is that it permits a parametrization of large classes of ir dynamics and allows investigations on important observarables . however , it is not clear whether concrete ehts can be embedded into string theories . for subsequent generalizations see @xcite . on the other hand , strongly coupled quantum liquids play an important role in condensed matter physics , where quantum liquids mean translationally invariant systems at zero ( or low ) temperature and at finite density . by now there are two successful phenomenological theories of quantum liquids : landau s fermi - liquid theory and the theory of quantum bose liquids , describing two different behaviors of a quantum liquid at low momenta and temperatures . in particular , the specific heat of a bose liquid at low temperature is proportional to @xmath9 in @xmath10 spatial dimensions , while the specific heat of a fermi liquid scales as @xmath11 at low @xmath11 , irrespective of the spatial dimensions . one may wonder if the newly developed techniques in ads / cft correspondence can help us understand the behavior of quantum liquids . in @xcite the authors considered a class of gauge theories with fundamental fields whose holographic dual in the appropriate limit was given in terms of the dirac - born - infeld ( dbi ) action in ads space . they found that the specific heat@xmath12 in @xmath13 spatial dimensions at low temperature and the system supported a sound mode at zero temperature , which was called `` zero - temperature sound '' . one interesting feature was that the `` holographic zero sound '' mode was almost identical to the zero sound in fermi liquids : the real part of the dispersion relation was linear in momentum ( @xmath14 ) and the imaginary part had the same @xmath15 dependence predicted by landau . the crucial difference was that the zero - temperature sound velocity coincided with the first - sound velocity , while generically the two velocities are not equal for a fermi liquid . such analysis was performed in the case of massive charge carriers in @xcite and in the case of sakai - sugimoto model in @xcite . the specific heat of general @xmath16 systems was calculated in @xcite and the specific heat of lifshitz black holes was discussed in @xcite and @xcite , while the zero sound was also investigated in @xcite . in this paper we will study the low - temperature specific heat and the holographic zero sound in effective holographic theories . here the bulk effective theory is @xmath0-dimensional einstein gravity coupled to a maxwell term with non - minimal coupling and a scalar . it was found in @xcite that the theory admitted both extremal and near - extremal solutions with anisotropic scaling symmetry . we consider dynamics of probe d - branes in the above mentioned backgrounds and find that by appropriately fixing the parameters in the effective theory , the specific heat can be proportional to @xmath11 , resembling a fermi liquid . we also compute the current - current retarded green functions at low frequency and low momentum , and clarify the conditions when a quasi - particle excitation exists . moreover , we also explore the possibility of observing the existence of fermi surfaces in such a system by numerical methods . we find that although the system possesses some features of fermi liquids , such as linear specific heat and zero sound excitation , we do not observe any characteristic structure in the wide range of @xmath17 . in addition , the ac conductivity is also obtained as a by - product . the rest of the paper is organized as follows : the exact solutions of the effective bulk theory will be reviewed in section 2 and the thermodynamics of massless charge carriers will be discussed in section 3 . we shall calculate the correlation functions in section 4 and identify the quasi - particle behavior , while the existence of fermi surfaces will be explored in section 5 via numerics . we will calculate the ac conductivity in section 6 , including the zero density limit . finally we will give a summary and discuss future directions . in this section we will review the solutions obtained in @xcite , which can be seen as generalizations of the four - dimensional near - extremal scaling solution discussed in @xcite . in the beginning we consider the following action in @xmath0-dimensions , without any reference to string theory or m / theory origin nor specifying the forms of the gauge coupling @xmath18 and the scalar potential @xmath19 explicitly , @xmath20.\ ] ] the resulting solutions are charged dilaton black holes , which have been investigated in the literature for a long period @xcite . let us focus on solutions carrying electric charge only . the general configuration with planar symmetry can be written as follows @xmath21 after plugging in the scaling ansatz @xmath22 into the equations of motion , we can arrive at several constraints on the parameters and the scalar functions : * we require that @xmath23 , so that the extremal solutions have smooth connections to the finite temperature solutions ; * the field equations indicate that @xmath24 and @xmath25 and @xmath26 when the scale invariance is restored . * the equations of motion determine that the scalar field must take the form @xmath27 and both @xmath18 and @xmath19 are constrained to be exponential in @xmath8 , power law in @xmath28 . * once we have fixed @xmath23 , the metric must have @xmath29 , where saturation occurs for vanishing flux , e.g. in @xmath30 with @xmath31 . subsequently , according to the constraints discussed above , we take the following forms of @xmath18 and @xmath19 . @xmath32 then we will consider charged dilaton black holes with a liouville potential @xcite . now the scaling ansatz turns out to be @xmath33 since @xmath34 and @xmath35 can be eliminated by rescaling @xmath28 and @xmath36 , we shall set @xmath37 and @xmath38 . the remaining parameters can be explicitly given in terms of @xmath39 , @xmath40^{2}}{(d-2)(2+\alpha^{2}+\alpha\eta ) [ 2(d-2)+(d-1)\alpha^{2}-\eta^{2}(d-3)+2\alpha\eta]}.\end{aligned}\ ] ] it can be easily seen that there is no scale invariance in such backgrounds . we will restrict to @xmath41 and @xmath42 without loss of generality , which implies that @xmath8 must diverge to positive infinity at the horizon . in the specific limit @xmath43 , the scalar potential becomes constant and the scale invariance of the solutions can be restored : * when we set @xmath44 with vanishing flux , the scalar @xmath8 becomes constant and the resulting solution is @xmath30 ; * when we set @xmath45 with flux through @xmath46 , @xmath8 becomes constant and the resulting solution is @xmath47 ; * when we set @xmath48 arbitrary , @xmath25 with flux through @xmath46 , the scalar @xmath49 and the resulting solution is the modified lifshitz solution , whose dynamical exponent @xmath50 before coming to practical calculations we should determine the range of parameters . firstly , by requiring that @xmath51 for small @xmath28 and the flux to be real , one can impose the bound on @xmath48 in terms of fixed @xmath52 , @xmath53 secondly , when the flux is zero , @xmath54 we should require @xmath55 to ensure a well - defined boundary in the sense of ads / cft , @xmath56 combining constraint derived in the general background in the beginning of this section , we can obtain the following complete restrictions @xmath57 we will impose such constraints in the subsequent calculations . the near extremal solution can be obtained in a similar way , @xmath58 where @xmath59 and the other parameters and fields remain the same as the extremal solutions . one can easily get the temperature @xmath60 and the entropy density @xmath61 in this section we will investigate thermodynamics of massless charge carriers in the backgrounds reviewed in previous section . according to ads / cft , @xmath62 probe d - branes correspond to @xmath62 fields in the fundamental representation of the gauge group in the probe limit @xmath63 @xcite . an efficient method for evaluating the dc conductivity and dc hall conductivity of probe d - branes was proposed in @xcite and @xcite . moreover , a holographic model building approach to `` strange metallic '' phenomenology was initiated in @xcite , where the bulk spacetime was a lifshitz black hole and the charge carriers were described by d - branes . here we will consider probe d - branes as massless charge carriers and explore the thermodynamics in the near - extremal background . the dynamics of probe d - branes is described by the dirac - born - infeld ( dbi ) action @xmath64 where @xmath65 denotes the tension of d - branes , @xmath66 is the induced metric and @xmath67 is the @xmath68 field strength on the worldvolume . in the second line we set @xmath69 , where @xmath70 denotes volume of the internal space that the d - branes may be wrapping . furthermore , we assume that the d - branes are extended along @xmath71 spatial dimensions of the black hole solution . if @xmath72 , the fundamental fields are propagating along certain @xmath10-dimensional defect . we will introduce a nontrivial worldvolume gauge field @xmath73 and absorb the factor @xmath74 into @xmath67 . since we are not studying realistic string theories , the wess - zumino terms will be omitted in the following discussions . before proceeding we should make sure that the backreaction of the probe branes onto the background can be neglected . our discussion is along the line of @xcite . expanding the dbi action to quadric order of @xmath75 in the background , we can obtain @xmath76 to avoid backreaction of the probes on the background , the stress energy of the probes must be smaller than that generating the bulk spacetime . it can be easily seen that the stress energy of the original background@xmath77 , where @xmath78 denotes the planck length in @xmath0-dimensional spacetime and @xmath79 is the corresponding cosmological constant . therefore by varying the quadric action of the probes with respect to @xmath80 , we can arrive at the following condition @xmath81 one can see that as long as the effective tension @xmath82 is sufficiently small , the backreaction can be neglected . in this background configuration , after performing the trivial integrations on @xmath83 and dividing out the infinite volume of @xmath84 , we can obtain the action density . @xmath85 where @xmath86 and the prime denotes derivative with respect to @xmath28 . then the charge density is given by @xmath87 we can also solve for @xmath88 @xmath89 by plugging in the solution for @xmath88 , we can find the on - shell action density @xmath90 following the methods used in @xcite , some interesting physical quantities like the chemical potential and the free energy , can be evaluated analytically . we will see that this is still the case for our background . the chemical potential is given by @xmath91 where @xmath92 notice that in order to obtain the above results , we have made use of the following useful formulae for beta function and incomplete beta function @xmath93 as well as for hypergeometric function @xmath94 after choosing the grand - canonical ensemble , the free energy density is given by @xmath95 where @xmath96 moreover , other thermodynamic quantities can also be calculated from the thermodynamic relations . the charge density can be written as @xmath97 which is consistent with previous result . the entropy density is given by @xmath98 notice that when @xmath26 , there exists a nontrivial contribution to the entropy density at @xmath99 like those observed in @xcite and @xcite . on the other hand , the entropy density is vanishing at extremality as long as @xmath100 . the specific heat is @xmath101 it is well known that for a gas of free bosons in @xmath10 spatial dimensions , the specific heat at low temperature is proportional to @xmath102 , while for a gas of fermions the low temperature specific heat is proportional to @xmath11 , irrespective of @xmath13 . when @xmath100 and @xmath82 is sufficiently small , the first term dominates . one can easily obtain @xmath103 when the specific heat is proportional to @xmath11 . then the parameter @xmath48 can be expressed in terms of @xmath52 @xmath104},\ ] ] combining with ( [ res ] ) , we can arrive at the following conclusions : * when @xmath105 both @xmath106 and @xmath107 are permitted solutions ; * when @xmath108 only @xmath107 is a permitted solution ; * when @xmath109 there is no solution , which means that we can not realize @xmath110 in this regime . when @xmath111 , the second term provides the only contribution . the linear dependence on @xmath11 fixes @xmath112 . note that in this limit , @xmath113 . so @xmath114 by taking into account of ( [ res ] ) , we can find that only @xmath115 is permitted . we can also formally evaluate the `` speed of sound '' . in grand - canonical ensemble , the pressure is given by @xmath116 while the energy density is @xmath117 therefore , @xmath118 however , it was emphasized in @xcite that this quantity is only the speed of normal / first sound in the relativistic case @xmath119 . actually the speed of normal sound is dimensionful in a system with @xmath120 . we will calculate the holographic zero sound in the next section . in this section we will calculate the holographic zero sound in the anisotropic background at extremality . the basic strategy is to consider fluctuations of the worldvolume gauge field on the probe d - branes in the background with nontrivial @xmath121 . such analysis was performed for @xmath122 background in @xcite and for lifshitz background in @xcite . we will calculate the holographic zero sound in a similar way and classify the behavior of the zero sound in different parameter ranges . zero sound should appear as a pole in the density - density retarded two - point function @xmath123 at extremality @xcite . in @xcite the authors provided a general framework for evaluating the corresponding retarded green s functions with background metric @xmath124 here we will take the nontrivial dilaton into account . the symmetries in the spatial directions allow us to consider fluctuations of the gauge fields with the following form @xmath125 where @xmath126 denotes one of the spatial directions . the quadratic action for the fluctuations is given by @xmath127,\ ] ] where @xmath128 . note that we are working in the gauge of @xmath129 . after performing the fourier transform @xmath130 the linearized equations of motion can be written as @xmath131-\frac{e^{-\phi}g^{q/2 - 1}_{xx}g_{rr } } { \sqrt{|g_{tt}|g_{rr}-a^{\prime2}_{t}}}(k^{2}a_{t}+\omega ka_{x})=0,\ ] ] @xmath132+\frac{e^{-\phi}g^{q/2 - 1}_{xx}g_{rr } } { \sqrt{|g_{tt}|g_{rr}-a^{\prime2}_{t}}}(\omega^{2}a_{x}+\omega ka_{t})=0.\ ] ] in addition , the following constraint can be obtained by writing @xmath133 s equation of motion in @xmath129 gauge @xmath134 the above equations are not independent , as we can obtain the equation of motion for @xmath135 by combining the constraint equation and the equation of motion for @xmath136 . therefore it is sufficient to solve the constraint and the equation for @xmath135 only . by introducing the gauge - invariant electric field @xmath137 we can obtain the equation of motion for @xmath138 @xmath139e^{\prime}-\frac{g_{rr}}{|g_{tt}|}(u^{2}k^{2}-\omega^{2})e=0,\ ] ] where @xmath140 moreover , the quadratic action can also be expressed in terms of @xmath138 , @xmath141,\ ] ] for our specific background , the metric can be rewritten in terms of the new radial coordinate @xmath142 as follows @xmath143 in the new coordinate system , the solution of the worldvolume gauge field and the function @xmath144 are given by @xmath145 where dot denotes derivative with respect to @xmath146 . integrating the quadratic action by parts @xmath147 introducing a cutoff at @xmath148 and taking the limit @xmath149 , the quadratic action turns out to be @xmath150 after imposing the incoming boundary condition at the `` horizon '' @xmath151 and plugging in the solutions of @xmath152 , the retarded correlation function reads @xcite @xmath153 by defining @xmath154 the retarded correlation functions can be written in terms of @xmath155 @xmath156 in order to evaluate the retarded correlation functions , we should try to solve ( [ 4eq6 ] ) , whose analytic solutions are always difficult to find . we will leave the numerical work to section 5 , while here we will obtain the low - frequency behavior of @xmath157 by solving ( [ 4eq6 ] ) in different limits and matching the two solutions in an overlapped regime , following the spirit of @xcite and @xcite . to be concrete , we will solve ( [ 4eq6 ] ) in the limit of large @xmath146 and then expand the solution in the small frequency and momentum limit . next we will take the small frequency and momentum limit first and then perform the large @xmath146 expansion . the integration constants can be fixed by matching the two solutions . first let us take @xmath158 , which leads to the following equation for @xmath138 @xmath159 the solution can be given in terms of a hankel function of the first kind , @xmath160 in the limit of small frequency with @xmath161 , the asymptotic expansion reads @xmath162 it should be pointed out that the case of @xmath163 must be treated separately . in this case the corresponding parameter is given by @xmath164 now the expansion contains a logarithmic term @xmath165 where @xmath166 is the euler constant . next we take @xmath167 with @xmath168 being fixed . then the last term in ( [ 4eq6 ] ) can be neglected and the equation of @xmath138 becomes @xmath169\dot{e}=0.\ ] ] when @xmath170 and @xmath171 , the solution is given by @xmath172,\end{aligned}\ ] ] where @xmath173 . for either @xmath170 or @xmath171 , the powers of @xmath146 do not match , which will be displayed in appendix a. we will make use of the following useful formulae for the asymptotic expansion @xmath174 @xmath175 therefore the large @xmath146 limit is given as follows when @xmath176 , @xmath177.\ ] ] when @xmath178 the expansion also contains a logarithmic term @xmath179\nonumber\\ & \simeq&d_{1}+d_{2}[\frac{c_{1}\mu_{0}k^{2}}{md}-\frac{\omega^{2}}{md}\log 2d + \frac{\omega^{2}}{(\beta-1)d}\log\omega-\frac{\omega^{2}}{(\beta-1)d}\log(\omega z^{\beta-1})].\end{aligned}\ ] ] to evaluate the retarded correlation functions , we need small @xmath146 expansion of the solution . it can be seen that the second term in the expansion always tends to zero more rapidly , being irrespective of @xmath178 or not . therefore we have @xmath180 assuming that @xmath181 can be dealt with in a similar fashion , see footnote 7 of @xcite . ] , the leading order behavior of @xmath138 reads @xmath182 , so the quadratic action turns out to be @xmath183 thus @xmath184 the relation between the integration constants @xmath185 and @xmath186 can be obtained by matching the expansions of the solutions in different limits and eliminating the other integration constant @xmath187 . finally we summarize our result for @xmath155 @xmath188 when @xmath176 the parameters are given by @xmath189 when @xmath178 the parameters are given by @xmath190 the dispersion relation of the holographic sound mode is given by setting the denominator of @xmath155 to vanish . similar to the situations discussed in @xcite , the value of @xmath191 determines which term in the denominator dominates . it can be seen that when @xmath192 , the @xmath193 term dominates . then we can expand @xmath194 @xmath195,\ ] ] and then invert to find @xmath196 notice that when @xmath192 , @xmath197 , so at low momentum @xmath198 , that is , the real part is bigger than the imaginary part . therefore this mode describes a quasi - particle excitation . as a check of consistency , we can take the specific limit @xmath199 , which just gives the @xmath30 background . one can obtain @xmath200 which agrees with @xcite . the speed of the holographic zero sound is given by @xmath201 it can be seen that in the relativistic case @xmath31 , the speed of zero sound coincides with the speed of normal / first sound . one special case is @xmath202 , where all the @xmath203 functions cancel and the speed of zero sound reads @xmath204 which turns out to be finite as @xmath205 . when @xmath206 as well as @xmath207 has a pole as @xmath205 , so @xmath208 goes to zero from above . when @xmath209 dominates , then @xmath210 inverting the relation above , @xmath211 notice that since @xmath212 is real and @xmath213 is complex , the leading term has a complex coefficient . furthermore , the real and imaginary parts are of the same order , hence this mode is not a quasi - particle . finally when @xmath178 , @xmath214 expanding for small @xmath215 , @xmath216 it can be seen that the dispersion relation differs from the holographic zero sound mode by logarithmic terms . in previous section we observed a sound - like excitation in the regime of @xmath192 . it is known that such zero sound mode is associated with the deformation of the fermi surface away from the spherical shape . the theory of normal fermi liquids tells us that the jump in the distribution function can be observed as a singularity in the retarded current - current green s function in the @xmath217 limit . to see if we can observe such fermi surface we need the complete solution to the equation for the gauge fluctuations ( [ 4eq6 ] ) with @xmath217 , at least numerically . we investigated thermodynamics of probe d - branes in section 3 , where we found that when @xmath100 , the specific heat was proportional to the temperature @xmath11 under certain conditions , which is just the behavior of fermi liquids . as we have many groups of parameters @xmath218 which lead to specific heat linear in @xmath11 , we choose the following parameters @xmath219,@xmath220 in @xmath1 dimensional spacetime as an example . for simplicity we also fix @xmath221 and @xmath222 . then the equation for the gauge fluctuations ( [ 4eq6 ] ) in the @xmath217 limit is reduced to @xmath223 in the near horizon region @xmath224 , the perturbative solution of the above equation is given by @xmath225 here , we choose the boundary condition @xmath226 at the horizon , which is compatible to impose the incoming boundary condition for @xmath227 . so we set @xmath228 . at the boundary @xmath229 , the asymptotic solution of ( [ eq : red ] ) becomes @xmath230 notice that since the first term is finite at the boundary , it corresponds to the source term and the coefficient of the second term implies the vev of the dual gauge operator . then , the green function is proportional to @xmath231 . in figure 1 , we plot the dependence of the green s function on the momentum @xmath17 for @xmath232 . .,scaledwidth=50.0% ] from the figure it can be easily seen that the characteristic structure does not appear in a wide range of @xmath17 , while the specific heat and zero sound exhibit typical features of fermi liquids . such phenomenon was also observed in @xcite . it was pointed out in @xcite that one difficulty in applying landau s theory of fermi liquids was the assumption that the particle number should be conserved as the strength of the interaction was varied , which was not obvious for the case they discussed . their conclusion seems to be still applicable to the present case . in this section we calculate the ac conductivity by making use of the correlation functions , which can be seen as a by - product of section 4 . furthermore , we will take the limit of zero density and compare the results with those appeared in previous literatures . it can be easily seen that the current - current correlation function is given by @xmath233 therefore we can obtain the following expressions in the small frequency limit , @xmath234 @xmath235 @xmath236 recalling that the definition of ac conductivity is given by @xmath237 we can arrive at the following results @xmath238 @xmath239 @xmath240 as a check of consistency , we consider the specific limit , @xmath241 . when @xmath242 , @xmath243 , which agrees with the result obtained in @xcite . the behavior of the ac conductivity is qualitatively similar to that investigated in @xcite . to be concrete , when @xmath176 @xmath244 is real while @xmath213 is complex . the conductivity is purely imaginary when @xmath192 and has a simple pole at zero frequency . then according to kramers - kronig relation , one can conclude that the real part of the conductivity , and hence the spectral function , consists of a delta function at zero frequency . when @xmath245 the conductivity , and hence the spectral function has a power - law dependence , and no explicit quasi - particle excitation exists . on the other hand , it was observed in @xcite that after transforming the equation of @xmath136 into a schrdinger - like form , the ac conductivity was directly related to the reflection amplitude for scattering off the potential . for our system , it can be observed that the equation for @xmath136 at @xmath246 becomes @xmath132+\frac{e^{-\phi}g^{q/2 - 1}_{xx}g_{rr } } { \sqrt{|g_{tt}|g_{rr}-a^{\prime2}_{t}}}\omega^{2}a_{x}=0\ ] ] in the original coordinate , @xmath247 the asymptotic behavior of @xmath136 is given by @xmath248 where we have introduced a background electric field @xmath249 , the above equation can be put in a schrdinger - like form , @xmath250 where @xmath251 and @xmath252.\ ] ] generally speaking , the ac conductivity can be obtained by numerical methods . however , in the zero density limit @xmath253 one can find @xmath254 so the potential possesses the same form as those investigated in @xcite and @xcite . assuming that the solution asymptotes to @xmath30 , we can obtain @xmath255 following their approach . in particular , when @xmath256 , the potential vanishes and the conductivity is constant at all temperatures @xmath257 which agrees with the analysis performed in @xcite . in this paper we explore the zero sound in @xmath0-dimensional effective holographic theories , whose bulk fields include the graviton @xmath6 , the @xmath68 gauge field @xmath7 and the scalar field @xmath8 . the solutions possess anisotropic scaling symmetry and they reduce to previously known examples , such as @xmath30 , @xmath47 and `` modified '' lifshitz solutions under certain conditions . we consider thermodynamics of massless probe d - branes in the near - extremal background and clarify the conditions under which the specific heat is linear in the temperature , which is a characteristic feature of fermi liquids . subsequently we study the zero sound mode by considering the fluctuations of the worldvolume gauge fields on the probe d - branes . rather than analytically solving the equations of motion , we obtain the low - frequency behavior by solving the equation in two different limits and then matching the two solutions in a regime where the limits overlap , following @xcite and @xcite . the resulting behavior of the zero sound looks similar to that investigated in @xcite , that is , when the parameter @xmath258 , the dispersion relation reveals a quasi - particle excitation ; while the zero sound is not a well - defined quasi - particle when @xmath259 . furthermore , we plot the correlation function in @xmath1 at @xmath217 with fixed parameters which lead to linear specific heat . the result is that we can not observe any characteristic structure of fermi liquids in a wide range of @xmath17 , which is similar to what was found in @xcite . as a by - product , we also evaluate the ac conductivity via the current - current correlation function , which reduces to previously known results at specific limits . by now there are mainly two approaches for studying condensed matter physics in the context of holography . one approach can be thought of as `` top - down '' , that is , we consider certain exact solutions or brane configurations in string / m theory which possess the desired properties of condensed matter systems . the main advantage is that we have clear understanding about the dual field theories , while it is difficult to find such solutions in string / m theory . a complementary approach can be seen as `` bottom - up '' , which means that we consider certain toy models of gravity which possess solutions with the desired properties . it allows a parametrization of large classes of ir dynamics and provides useful information in the dual field theory side . however , the main disadvantage is that the embeddings of such toy models into string / m theory are not obvious , thus many things in the field theory side remain unknown . however , the `` bottom - up '' approach is an efficient tool for investigating the ads / cmt correspondence . the background solutions we studied in this paper are exact solutions of theories with domain wall vacua . in @xcite the author also constructed interpolates between two exact solutions of the single- exponential , domain wall gravity theory , which lead to the argument that the domain wall / qft correspondence @xcite can be considered as an effective holographic tool which is applicable in settings beyond the regime of domain wall supergravities . therefore the domain wall / qft correspondence can also be taken as one specific class of effective holographic theories . moreover , the author of @xcite argued that even when the uv completion of some bulk theory was unknown , if the theory admitted an approximate domain wall solution at some intermediate value of @xmath28 then one could use domain wall / qft correspondence to develop a holographic map . thus it is interesting to develop the ads / cmt holography by making use of this domain wall / qft correspondence . in particular , we can establish precision holography in this anisotropic background along the line of @xcite and study the fermionic correlation functions following @xcite . we leave such fascinating projects in the future . * acknowledgments * this work was supported by the national research foundation of korea(nrf ) grant funded by the korea government(mest ) through the center for quantum spacetime(cquest ) of sogang university with grant number 2005 - 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we investigate zero sound in @xmath0-dimensional effective holographic theories , whose action is given by einstein - maxwell - dilaton terms . the bulk spacetimes include both zero temperature backgrounds with anisotropic scaling symmetry and their near - extremal counterparts obtained in 1006.2124 [ hep - th ] , while the massless charge carriers are described by probe d - branes . we discuss thermodynamics of the probe d - branes analytically . in particular , we clarify the conditions under which the specific heat is linear in the temperature , which is a characteristic feature of fermi liquids . we also compute the retarded green s functions in the limit of low frequency and low momentum and find quasi - particle excitations in certain regime of the parameters . the retarded green s functions are plotted at specific values of parameters in @xmath1 , where the specific heat is linear in the temperature and the quasi - particle excitation exists . we also calculate the ac conductivity in @xmath0-dimensions as a by - product . cquest-2010 - 0395 * zero sound in effective holographic theories * bum - hoon lee@xmath2 , da - wei pang@xmath3 and chanyong park@xmath3 + _ seoul 121 - 742 , korea + _ _ @xmath4 center for quantum spacetime , sogang university _ + _ seoul 121 - 742 , korea + _ bhl@sogang.ac.kr , pangdw@sogang.ac.kr , cyong21@sogang.ac.kr

You are an expert at summarizing long articles. Proceed to summarize the following text: in adaptive control and recursive parameter estimation one often needs to adjust recursively an estimate @xmath0 of a vector @xmath1 , which comprises @xmath2 constant but unknown parameters , using measurements of a quantity @xmath3 here @xmath4 is a vector of known data , often called the regressor , and @xmath5 is a measurement error signal . the goal of tuning is to keep both the estimation error @xmath6 and the parameter error @xmath7 as small as possible . there are several popular methods for dealing with the problem above , for instance least - squares . maybe the most straightforward involve minimizing the prediction error via gradient - type algorithms of the form : @xmath8 where @xmath9 is a constant , symmetric , positive - definite gain matrix . let us define @xmath10 and analyze differential equations and , which under the assumption that @xmath11 is identically zero read : @xmath12 the nonnegative function @xmath13 has time derivative @xmath14 hence @xmath15 inspection of the equation above reveals that @xmath16 is limited in time , thus @xmath17 , and also that the error @xmath18 ( norms are taken on the interval @xmath19 where all signals are defined ) . these are the main properties an algorithm needs in order to be considered a suitable candidate for the role of a tuner in an adaptive control system . often @xmath20 or something similar is also a desirable property . to obtain the latter , normalized algorithms can be used ; however , the relative merits of normalized versus unnormalized tuners are still somewhat controversial . another alternative is to use a time - varying @xmath9 , as is done in least - squares tuning . in [ sec : acceleration ] we present a tuner that sets the second derivative of @xmath0 , and in [ sec : covariance ] the effects of a white noise @xmath5 on the performance of the two algorithms are compared . then we show some simulations and make concluding remarks . classical tuners are such that the _ velocity _ of adaptation ( the first derivative of the parameters ) is set proportional to the regressor and to the prediction error @xmath21 . we propose to set the _ acceleration _ of the parameters : @xmath22 notice that the the formula above is implementable ( using @xmath23 integrators ) if measurement error is absent , because the unknown @xmath24 appears only in scalar product with @xmath25 . choose another function of lyapunovian inspiration : @xmath26 taking derivatives along the trajectories of gives @xmath27 integrating @xmath28 we obtain @xmath29 which leads immediately to the desired properties : @xmath30 the slow variation property @xmath31 follows without the need for normalization , and now we obtain @xmath32 instead of @xmath33 as before . we might regard @xmath34 as a modified error , which can be used in the stability analysis of a detectable or `` tunable '' adaptive system via an output - injection argument ; see @xcite . a generalization of is @xmath35 with @xmath36 and @xmath37 constant , symmetric , positive - definite @xmath38 matrices such that @xmath39 and @xmath40 . the properties of tuner , which can be obtained using the positive - definite function @xmath41 in the same manner as before , are @xmath42 we now consider the effects on the expected value and covariance of @xmath43 of the presence of a measurement error . the assumptions are that @xmath11 is a white noise with zero average and covariance @xmath44 and that @xmath45 are given , deterministic data . for comparison purposes , first consider what happens when the conventional tuner is applied to in the presence of measurement error @xmath5 : @xmath46 the solution to the equation above can be written in terms of @xmath47 s state transition matrix @xmath48 as follows @xmath49 hence @xmath50 because @xmath51 by assumption . here the notation @xmath52 , denoting the expectation with respect to the random variable @xmath5 , is used to emphasize that the stochastic properties of @xmath25 are not under consideration . the conclusion is that @xmath43 will converge to zero in average as fast as @xmath53 does . the well - known persistency of excitation conditions on @xmath54 are sufficient for the latter to happen . to study the second moment of the parameter error , write @xmath55 the covariance of @xmath43 can be written as the sum of four terms . the first is deterministic . the second term @xmath56 because @xmath11 has zero mean , and the third term is likewise zero . the fourth term @xmath57 where fubini s theorem and the fact @xmath58 were used . performing the integration and adding the first and fourth terms results in @xmath59 this equation can be given the following interpretation : for small @xmath60 , when @xmath53 is close to the identity , the covariance of @xmath43 remains close to @xmath61 , the outer product of the error in the initial guess of the parameters with itself . as @xmath62 , which will happen if @xmath54 is persistently exciting , @xmath63 tends to @xmath64 . this points to a compromise between higher convergence speeds and lower steady - state parameter error , which require respectively larger and smaller values of the gain @xmath9 . algorithms that try for the best of both worlds parameter convergence in the mean - square sense often utilize time - varying , decreasing gains ; an example is the least - squares algorithm . we shall now attempt a similar analysis for the acceleration tuner applied to , which results in the differential equation @xmath65 let @xmath66 where @xmath67 , @xmath68 , each @xmath69 is a function of @xmath70 unless otherwise noted , and the dot signifies derivative with respect to the first argument . if @xmath71 , @xmath72 following the same reasoning used for the velocity tuner , one concludes that @xmath73 and that @xmath74 however the properties of the acceleration and velocity tuners are not yet directly comparable because the right - hand side of does not lend itself to immediate integration . to obtain comparable results , we employ the ungainly but easily verifiable formula , @xmath75 ' '' '' valid for arbitrary scalars @xmath76 and @xmath77 , and make the [ [ simplifying - assumption ] ] simplifying assumption : + + + + + + + + + + + + + + + + + + + + + + + + for @xmath78 , and 3 , @xmath79 , where @xmath80 are scalars and @xmath81 is the @xmath82 identity matrix . premultiplying by @xmath83 $ ] , postmultiplying by @xmath83^\top$ ] , integrating from 0 to @xmath60 , and using the simplifying assumption gives formula . @xmath84 ' '' '' taking @xmath85 in , @xmath86 results positive - semidefinite , therefore @xmath87 the combination of and shows that @xmath88 can be increased without affecting @xmath24 s steady - state covariance . on the other hand , to decrease the covariance we need to increase @xmath89 , which roughly speaking means increasing damping in . since @xmath88 and @xmath89 can be increased without affecting the stability properties shown in [ sec : acceleration ] , a better transient @xmath90 steady - state performance compromise might be achievable with the acceleration tuner than with the velocity tuner , at least in the case when @xmath91 , @xmath92 , and @xmath37 are `` scalars . '' notice that @xmath93 by construction . [ [ approximate - analysis ] ] approximate analysis : + + + + + + + + + + + + + + + + + + + + + + the derivation of inequality does not involve any approximations , and therefore provides an upper bound on @xmath94 , valid independently of @xmath54 . a less conservative estimate of the integral in can be obtained by replacing @xmath95 by its average value @xmath96 in the definition of @xmath86 in . this approximation seems reasonable because @xmath86 appears inside an integral , but calls for more extensive simulation studies . to obtain a useful inequality , we require @xmath97 ; namely , using the schur complement @xmath98 or , using the simplifying assumption and substituting @xmath95 by its approximation @xmath96 @xmath99 suppose further that @xmath100 . looking for the least conservative estimate , we pick @xmath101 , the least value of @xmath76 that keeps @xmath97 . thus @xmath102 with @xmath103 \bar{m}_1 \left[\begin{smallmatrix}{\phi}^\top_{11}(t,0 ) \\ { \phi}^\top_{12}(t,0 ) \end{smallmatrix}\right]}{4m_1 ^ 2 m_2m_3r(1+\mu_2 ) -r}.$ ] taking @xmath104 we repeat the previous , exact result . for large positive values of @xmath77 the first term of the right - hand side of tends to @xmath105 , which indicates that the steady - state covariance of the parameter error decreases when the signal @xmath25 increases in magnitude , and that it can be made smaller via appropriate choices of the gains @xmath88 and @xmath106 . the situation for the accelerating tuner is hence much more favorable than for the conventional one . the simulations in this section compare the behavior of the accelerating tuner with those of the gradient tuner and of a normalized gradient one . all simulations were done in open - loop , with the regressor a two - dimensional signal , and without measurement noise . figure [ fig : step ] shows the values of @xmath107 and @xmath108 respectively when @xmath25 is a two - dimensional step signal . in figure [ fig : sin ] the regressor is a sinusoid , in figure [ fig : sia ] an exponentially increasing sinusoid , and in figure [ fig : prb ] a pseudorandom signal generated using matlab . no effort was made to optimize the choice of gain matrices ( @xmath91 , @xmath92 , and @xmath37 were all chosen equal to the identity ) , and the effect of measurement noise was not considered . the performance of the accelerating tuner is comparable , and sometimes superior , to that of the other tuners . = 2.5 in = 2.5 in = 2.5 in = 2.5 in = 2.5 in = 2.5 in = 2.5 in = 2.5 in other ideas related to the present one are replacing the integrator in with a positive - real transfer function @xcite , and using high - order tuning ( @xcite ) . high - order tuning generates as outputs @xmath0 as well as its derivatives up to a given order ( in this sense we might consider the present algorithm a second - order tuner ) , but unlike the accelerating tuner requires derivatives of @xmath25 up to that same order . we expect that accelerating tuners will find application in adaptive control of nonlinear systems and maybe in dealing with the topological incompatibility known as the `` loss of stabilizability problem '' in the adaptive control literature . the stochastic analysis in [ sec : covariance ] indicates that the performance and convergence properties of the accelerating tuner , together with its moderate computational complexity , may indeed make it a desirable tool for adaptive filtering applications . it seems that a better transient @xmath90 steady - state performance compromise is achievable with the accelerating tuner than with the velocity tuner . to verify this conjecture , a study of convergence properties of the accelerating tuner and their relation with the persistence of excitation conditions is in order , as well as more extensive simulations in the presence of measurement noise .

we propose a tuner , suitable for adaptive control and ( in its discrete - time version ) adaptive filtering applications , that sets the second derivative of the parameter estimates rather than the first derivative as is done in the overwhelming majority of the literature . comparative stability and performance analyses are presented . * key words : * adaptive control ; parameter estimation ; adaptive filtering ; covariance analysis .

You are an expert at summarizing long articles. Proceed to summarize the following text: the focus of this paper is three - fold : 1 . to review an historically significant problem in analytic number theory , 2 . to describe a particular branch of @xmath2 , 3 . using ( 2 ) , solve ( 1 ) in a more elementary way . in 1859 , bernhard riemann published `` on the number of primes less than a given magnitude''@xmath4 , and introduced an analytic continuation of the euler product he called the zeta function : @xmath5 . using @xmath5 , infinite product representations of the @xmath6 and @xmath7 functions and relying on advanced principles of analysis , riemann went on to derive an analytic expression for a function he called @xmath0 equal to the number of prime numbers under @xmath1 . however , if a branch of @xmath2 is carefully chosen , riemann s expression for @xmath0 can be derived via the residue theorem using simpler techniques of analysis . this approach is described below . throughout this paper , a complex variable is represented as @xmath8 , and the non - trivial zeros of the zeta function with positive imaginary component as @xmath9 ( or just @xmath10 in some cases ) , such that @xmath11 . the principal branch of the complex logarithm is denoted by log and other branches are given by log . common in complex analysis , log represents logarithm to base e , that is log(e)=1 . that convention is used here . in all cases , the variable x is taken to be a real number greater than one . riemann derives his prime counting function by beginning with the euler product : @xmath12 taking logarithms of the above gives @xmath13 and remembering @xmath14 for @xmath15z@xmath161 , we obtain @xmath17 replacing each prime power with @xmath18 we can express equation as @xmath19 with @xmath20 @xmath0 is equal to the number of primes strictly less than @xmath1 and as is common in fourier theory , at points of discontinuity , namely at prime numbers , it is defined to be one - half the difference between it s old and new value . with this definition of @xmath0 , we see @xmath21 begins at 0 for x=0 and increases by a jump of 1 at primes , by a jump of 1/2 at prime squares @xmath22 , by a jump of @xmath23 at prime cubes @xmath24 , etc . that is , @xmath21 is zero for @xmath25 , is @xmath26 for @xmath27 , is 1 for @xmath28 , is @xmath29 for @xmath30 , is @xmath31 for @xmath32 , is @xmath33 for @xmath34 , is @xmath35 for @xmath36 , is @xmath37 for @xmath38 and so on . from @xmath21 , one can find @xmath0 via mbius inversion to obtain @xmath39 with @xmath40 a positive integer such that @xmath41 . the function f(x ) is riemann s prime counting function and riemann s derivation of an analytic expression for @xmath21 and thus @xmath0 , is the principal result of his paper . he accomplished this by inverting via fourier inversion giving @xmath42 in order to evaluate , riemann expressed @xmath5 in terms of the @xmath43 function : @xmath44 and solved for @xmath45 . he then expressed both @xmath7 and @xmath46 in terms of their infinite product representations and proceeded to evaluate the resulting integrals directly to obtained . @xmath47 is the corrected version . ] @xmath48+\int_x^{\infty } \frac{1}{t\left(t^2 - 1\right ) \text{log}(t ) } \ dt-\text{log}\,(2 ) \label{eqn003}\ ] ] in which @xmath49 is the logarithmic integral and @xmath50 with @xmath51 the exponential expressions are obtained by first considering @xmath52 where the contour @xmath53 is along the real axis from zero to @xmath1 with an indentation above the ( simple ) pole at @xmath54 for @xmath55 giving rise to the @xmath56 term , or below it for @xmath57 and giving rise to the @xmath58 term . letting @xmath59 gives @xmath60 where the path of integration for the integral on the right is now along the real axis with an indentation above or below the origin . making the substitution @xmath61 , then @xmath62 in which @xmath63 the center integral of the last expression tends to zero leaving the desired result . for a further explanation of , see edwards and ingham @xcite . although riemann s work represents a beautiful case study in analysis , his approach to evaluating was based on advanced methods of analysis . in this paper , is evaluated more simply using a holomorphic branch of @xmath2 . the geometry of this branch is investigated and the results used to derive an integral expression for @xmath21 which is then evaluated via the residue theorem . the function @xmath64 is holomorphic in the half - plane @xmath65 since @xmath5 is non - zero there . however , there exists a holomorphic branch that extends to the left half - plane . proof of the following theorem can be found in @xcite . if @xmath66 is a nowhere vanishing holomorphic function in a simply connected region @xmath67 , then there exists a holomorphic function @xmath68 on @xmath67 such that @xmath69 the function @xmath70 is given by @xmath71 , and determines a `` branch '' of that logarithm . also : @xmath72 [ theorem001 ] the trivial zeros of @xmath5 are located along the negative real axis at the points @xmath73 . the non - trivial zeros are symmetric in the critical strip @xmath74 to the line @xmath75 , and @xmath76 has one simple pole at @xmath54 . therefore , we can define @xmath67 to be the complex plane with branch cuts at the real axis extending from @xmath77 to one , and branch cuts of the form @xmath78 for each zero on the critical line and branch cuts between zeros for those off the critical line . we then construct a contour @xmath79 around @xmath67 . this contour is shown in figure ( [ fig : file1 ] ) and explained below . then @xmath80 is holomorphic in @xmath67 with @xmath81 where the set @xmath82 represents the branch cuts , and the function @xmath83 is now a continuous and analytic function of @xmath84 throughout @xmath85 . therefore , @xmath86 with the path of integration remaining in @xmath67 . in order use @xmath2 in this analysis , we must first understand the geometry of @xmath87 which we do now . @xmath88 is continuous except at the zeros of the zeta function and the point @xmath54 . the imaginary sheet , @xmath87 , is however more complicated , and a description of this sheet in the neighborhood of the branch cuts is needed in order to use the residue theorem to derive riemann s expression for @xmath21 . @xmath87 along a section of the black contours of @xmath79 is shown in figure ( [ fig : realaxiscloseup ] ) . note the stair - step geometry . the dark black trace is the contour below the real axis and the gray trace , above the real axis . empirical results suggest the difference in argument changes by @xmath89 at each step . we can prove this difference using the following theorem @xcite : let @xmath90 be analytic with a simple pole at @xmath91 and @xmath92 be an arc of a circle of radius @xmath93 and angle @xmath94 centered at @xmath91 . then : @xmath95 [ theorem005 ] consider now @xmath96 and the points @xmath97 and @xmath98 in figure ( [ fig : closeupat2 ] ) where @xmath99 and @xmath100 with @xmath101 ( note the indentation around the origin applies only to @xmath102 ) . we then have in the limit as the black contours approach the real axis , along black contours of b(m ) ] @xmath103 now , the first and third integrals are analytic outside some deleted neighborhood @xmath104 of @xmath54 and therefore the integrals in this region cancel as @xmath105 . inside @xmath104 , @xmath106 with @xmath107 for @xmath108 . clearly , the residue of @xmath109 at @xmath54 is @xmath110 . ] now , because of the symmetric form of the limit , i.e. , the terms @xmath111 , the angle @xmath94 of @xmath92 will always extend from @xmath112 to @xmath113 even in the limit as @xmath105 . therefore @xmath114 consider now the remaining two integrals inside @xmath104 . substituting into the integrals , the @xmath115 terms will again because they are analytic , cancel as @xmath105 leaving @xmath116 thus as the black contours in the range @xmath117 approach the real axis , @xmath118 and therefore @xmath119 now , the real part of @xmath5 in the range @xmath117 is negative and the imaginary part of @xmath5 is negative in the range @xmath120 along the upper black contour ( gray contour in figure [ fig : realaxiscloseup ] ) . therefore @xmath87 along the upper black contour in the range @xmath101 must approach @xmath121 since this branch of @xmath83 is continuous and therefore by the expression above , @xmath87 along the lower contour is @xmath122 over the same range . now , in the vicinity of each trivial zero , @xmath123 with the residue of this expression clearly being one . referring to figure ( [ fig : closeupat2 ] ) , consider the points @xmath124 and @xmath125 . then @xmath126 and again because of the symmetric form of the limit , @xmath127 leaving for the integrals : @xmath128 substituting into , we obtain after noting the analytic parts cancel : @xmath129 since @xmath130 . then @xmath131 and @xmath132 therefore the imaginary component of @xmath2 along the upper black contour as it approaches the real axis in the range @xmath133 has the limit zero since we have shown above that @xmath134 for @xmath101 along the upper black contour has a limit of @xmath121 . this same approach can then be applied to the remaining components of the black contour to arrive at the geometry in figure ( [ fig : realaxiscloseup ] ) . @xmath87 in a region around the branch cut at @xmath135 is shown in figure ( [ fig110:subfig : a ] ) . empirical results indicate the red contour ( over the imaginary sheet of @xmath2 ) is @xmath136 that of the blue contour . a similar argument to that in section [ section31 ] can be used to explain this geometry : for a point @xmath137 on the blue contour and @xmath138 on the red contour , @xmath139 and because of symmetry , @xmath140 for a deleted neighborhood surrounding each trivial zero on the critical line , called a `` common zero '' in this paper , we have @xmath141 making the same type of substitution as above gives @xmath142 this leaves @xmath143 and thus @xmath144 which is the value observed empirically . we must also consider the possibility of zeros off the critical line . these are called `` rogue zeros '' in this paper . how would these effect the branch cuts ? we can artificially introduce such zeros by considering the function @xmath145 and numerically solve ( [ inteqn1 ] ) using this function . the plot in figure ( [ fig110:subfig : b ] ) shows @xmath146 $ ] at this branch cut with @xmath147 and @xmath148 . numerical results suggest the difference in argument from @xmath149 to the first zero is @xmath150 and between the zeros is @xmath89 . this same type of geometry would occur with actual rogue zeros and using the same argument above , this @xmath150 and @xmath89 difference is easily explained . we now consider @xmath151 where @xmath79 includes the possibility of zeros off the critical line with the horizontal gray legs of the contour passing over certain lines with imaginary components equal to @xmath152 . since @xmath79 is closed over a function analytic throughout the region of integration , @xmath153 and therefore @xmath154 now @xmath155 over the purple contour and therefore we can replace @xmath2 with @xmath64 in that integral . multiplying by @xmath156 , and taking the limit as @xmath157 , @xmath158 where in , the appropriate limits are taken for each integral . we now go on to evaluate the various color - coded components of this expression . the green contours encircle each non - trivial zero of the zeta function . we allow the diameter of these contours to approach zero and take the following limit : @xmath159 where @xmath160 represents the set of non - trivial zeros of the zeta function . writing this in terms of s=@xmath161 , we have for the common zeros : @xmath162 and @xmath163 for rogue zeros with @xmath164 . the @xmath165 term is bounded by @xmath166 and note as @xmath167 , the denominator tends to @xmath168 with the numerator tending to @xmath169 with @xmath170 being an upper bound on the remaining terms . the zeta function has simple zeros and therefore @xmath171 which tends to zero . this gives @xmath172 independent of the placement of non - trivial zeros . the cyan contours encircle the trivial zeros and yield the following integrals : @xmath173 similar to the integrals over the green indentations , we express these integrals in terms of s=@xmath174 and since these zeros are simple , we obtain a similar limiting process : @xmath175 and therefore @xmath176 the integral around the orange contour at s=1 yields @xmath177 similar to the integrals over the green indentations , we express the integral in terms of s=@xmath178 and arrive at a similar limiting process : @xmath179 the pole at s=1 is a simple pole so that this limit approaches the form @xmath180 therefore @xmath181 we wish to consider the following limit : @xmath182 referring to figure ( [ fig : file1 ] ) , we can write this in terms of two integrals : @xmath183 note each integral traverses surfaces separated by a branch cut with @xmath184 as described in section [ section31 ] . as discussed in that section , @xmath185 over the upper contour is @xmath121 and over the lower contour is @xmath122 as shown in figure ( [ fig : realaxiscloseup ] ) . we then have for the upper contour : @xmath186 with @xmath187 . as @xmath188 , the first term of the last integral tends to @xmath189 and the imaginary component of @xmath190(s ) over the contour near zero tends to @xmath191 . that is , @xmath192 using these values we have @xmath193 in the same way , @xmath87 over the lower contour approaches @xmath194 according to section [ section31 ] , and we obtain for that result : @xmath195 therefore the sum of the integrals over the central brown contours as the diameters tend to zero is @xmath196 we consider two cases : case i takes the zeros on the critical line and case ii considers zeros off the critical line . in the case of zeros on the critical line , the value of log @xmath197(s ) on the blue contour is @xmath198 and the value on the red contour is @xmath199 as discussed in section [ subsection32 ] . in the limit as @xmath200 , the integrals over these contours as @xmath79 extends to infinity are @xmath201 letting @xmath202 , then @xmath203 and with @xmath204 gives @xmath205.\ ] ] for the conjugate contours at @xmath206 , @xmath207\ ] ] with the last integrals expressed in terms of the logarithmic integral as per . in this case we have sets of zeros of the form @xmath208 the presence of these zeros will cause the values between the red and blue contours to differ by @xmath150 between the range @xmath77 to @xmath209 and by @xmath210 between the range @xmath211 as was discussed in section [ subsection32 ] . we obtain for each set of zeros @xmath212 and with @xmath213 , @xmath214 this then becomes @xmath215 with a similar case for the pair with negative imaginary parts . these expressions are again equal to @xmath216 . that is , @xmath217 the set of conjugate zeros gives a similar expression in terms of @xmath218 . therefore we can write for the set of common and rogue zeros , @xmath219 so that for the set of all non - trivial zeros @xmath220 : @xmath221.\ ] ] the integrals over the black contours are determined by the change in argument over these contours described in section [ section31 ] . those over the interval ( -6,-4 ) are @xmath222 , \end{aligned}\ ] ] with @xmath223 being the incomplete gamma function . over the interval @xmath224 , these integrals are @xmath225 . \end{aligned}\ ] ] and for the interval @xmath226 , @xmath227 . \end{aligned}\ ] ] and in general , @xmath228 . \end{aligned}\ ] ] adding the first few terms of this set gives @xmath229 summing all the terms , we have for the black contours in the interval @xmath230 : @xmath231,\ ] ] with the sum converging due to the exponential decay of the gamma integrals . as discussed in section [ section31 ] , in the range ( -2,1 ) , the imaginary surface over the lower contour is @xmath122 and over the upper contour , @xmath121 . therefore , log @xmath197(s ) differs by 2@xmath194i over these two contours . however , the integrand is singular at the origin and thus we take the principal value for these integrals . we then have @xmath232 we wish to prove the following : @xmath233 where the gray boundary is incremented through certain lines crossing the critical strip of the zeta function . proofs of the following theorems can be found in ingham @xcite . there exist a sequence of numbers , @xmath234 , such that : @xmath235 [ thm001 ] for some positive constant @xmath236 . these are the lines at @xmath152 of figure ( [ fig : file1 ] ) . in the region obtained by removing from the half - plane @xmath237 , the interiors of a set of circles surrounding each trivial zero with radius @xmath26 , we have : @xmath238 [ thm002 ] and for @xmath239 : @xmath240 theorem @xmath241 implies that between each integer @xmath242 and @xmath243 , there exists a horizontal contour passing between the zeta zeros over which the order of @xmath109 does not exceed @xmath244 . theorem @xmath245 states this order does not exceed @xmath246 when @xmath247 . contour ( for @xmath248 sufficiently large ) ] figure([fig : tmcontour ] ) shows the top gray contour , @xmath249 , over a finite path of length @xmath250 . then @xmath251 where @xmath252 . that is , for @xmath253 sufficiently negative and @xmath248 sufficiently large , we have the inequality @xmath254 the value of @xmath2 over this contour given by is @xmath255 and therefore @xmath256 where @xmath236 is a positive constant and @xmath257 for @xmath258 . substituting into , we can write for the first integral : @xmath259 since @xmath260 . now consider the second integral and the expression @xmath261 therefore the bound on @xmath2 over this segment becomes @xmath262 by . now , for @xmath263 we have @xmath264 , then @xmath265 and therefore @xmath266 this is because for @xmath267 , @xmath268 . combining and gives @xmath269 and as @xmath270 , the quantity on the left side tends to zero . the same argument then applies to the lower contour at @xmath271 . we wish to first place upper bounds on the quantity @xmath2 over the contours @xmath272 shown at the left in figure ( [ fig : file1 ] ) . for @xmath273 we can write @xmath274 with @xmath275 . using then : @xmath276 for the contour @xmath277 over the vertical interval @xmath278 we have @xmath279 now , @xmath273 is separated from @xmath277 by a branch cut with @xmath280 we can then write @xmath281 and using we obtain @xmath282 and in general : @xmath283 now denote the number of non - trivial zeros of @xmath5 in the range @xmath284 by @xmath285 . a proof of the following theorem , first proposed by riemann , can be found in ingham @xcite : when @xmath286 , @xmath287 [ zerolimit ] thus in the limit as @xmath270 , the number of zeros in the range @xmath288 and thus the number of contours @xmath272 does not have an order which exceeds some constant times the factor of @xmath289 . writing as @xmath290 we have @xmath291 since : @xmath292 and @xmath293 this gives : @xmath294 and as @xmath270 , this limit approaches zero . summing over all the contours @xmath272 as the boundary extends to infinity we obtain : @xmath295 a similar argument then applies to the lower set of vertical contours . we now have all the contours of b(m ) and can write as @xmath296+\sum_{n=2}^{\infty } \gamma [ 0,2n \log ( x)]-\text{log } ( 2 ) . \label{eqn62}\ ] ] riemann obtained @xmath297+\int_x^{\infty } \frac{1}{t\left(t^2 - 1\right ) \text{log}(t ) } \ , dt-\text{log}(2 ) . \label{eqn003b}\ ] ] therefore , if we can show : @xmath298,\ ] ] then the results calculated via the residue theorem are equal to that obtained by riemann . note first if we let @xmath299 in the integral @xmath300 we obtain @xmath301 this leaves us to verify : @xmath302-\int_0^{1\left / x^2\right . } \frac{1}{{\text{log}\,}u } \ , du.\ ] ] now @xmath303 and @xmath304 we therefore wish to show @xmath305 we can write @xmath306 we are left then with showing @xmath307 which we can show by making two change of variables . the first let @xmath308 to obtain @xmath309 now let v =- r : @xmath310 which was to be shown . using a particular holomorphic branch of @xmath2 and the residue theorem , we obtain the same expression for @xmath21 as riemann .

in his paper `` on the number of primes less than a given magnitude''@xcite , bernhard riemann introduced a prime counting function @xmath0 which counts the number of primes under @xmath1 . riemann obtained an analytic expression for @xmath0 by evaluating an inverse laplace transform . his method involved advanced techniques of analysis . however , this transform can be evaluated using the residue theorem when an appropriate branch of @xmath2 is defined . in this paper , a method for constructing a holomorphic branch of @xmath2 extending to the left half - plane is described along with it s geometry surrounding the logarithmic branch points . using this information , an integral representation of @xmath0 is formulated in terms of this branch of @xmath3 which is then evaluated . the results are shown equal to riemann s expression .

You are an expert at summarizing long articles. Proceed to summarize the following text: there are two general approaches to the study of the complex effects of galaxy collisions . the first is statistical , such as the study of particular properties of a reasonable sample of interacting galaxy systems , or a grid of numerical models covering some range of initial conditions . the second approach is the case study , the detailed investigation of a particular collisional system . the first approach is the path more frequently followed , in the literature . a great deal can be learned from observations of the global properties of many systems , which are easier to acquire ( individually ) , than the high sensitivity , high resolution , multi - wavelength data needed for a good case study . in the realm of simulations , the statistical approach also has many attractions . for example , in many regions of parameter space the model outcomes are quite sensitive to collision parameters , so converging on the correct parameters can be a prolonged task . ( at the same time , such sensitivities can go a long way toward guaranteeing the uniqueness of a successful model . ) one of the greatest successes of the statistical approach ( spurred by iras results ) in the last two decades is an understanding of how gas is redistributed in major mergers , and how ultra - luminous , super - starbursts result . insights obtained from numerical models ( e.g. , barnes and hernquist 1972 ) also played a crucial role . statistical studies also provide important inputs to the topics of mass transfer and induced star formation ( sf ) in galaxy collisions . even aside from studies of luminous major merger remnants , there has been much work on the questions of whether and by how much sf is enhanced by interactions . the early color analysis of larson and tinsley ( 1978 ) , suggested color dispersion enhancement , rather than bluer colors . the nuclear spectrophotometry by keel et al . ( 1985 , also kennicutt et al . 1987 ) suggested enhancement of nuclear sfrs . more recently , bergvall et al . ( 2003 and earlier work cited therein ) find no difference in the broadband colors of sample of 59 interacting systems relative to a control sample of 38 galaxies . they do find a moderate increase in central sf and far - infrared emission in the interacting sample . they argue that earlier work , claiming greater enhancements was biased towards ir luminous merger remnants , in contrast to their sample . barton , geller & kenyon ( 2003a , and barton gillespie , geller & kenyon 2003b ) obtained b and r band photometry and optical spectroscopy of 190 galaxies in pairs and compact groups . they also found enhanced core sf , and a number of post - starburst , as well as starburst cores . in addition they found an anti - correlation between galaxy separation and sf , and suggested starbursts were preferentially triggered at closest approach , decaying thereafter . these statistical studies shed the most light on nuclear sf , suggesting that it has a short duty cycle , and perhaps requires a relatively strong disturbance to funnel the gas fuel inward ( keel 1993 ) . in this symposium we have seen some beautiful observations of sf in tidal structures . the statistical studies suggest that such sf does not add up to a very large amount . this echos the result of the statistical study of schombert , wallin & struck - marcell ( 1990 ) on the colors of tidal bridges and tails . yet , in tails it is the nature , rather than the quantity of sf that is of interest . case studies with more details on the sf history , as well as instantaneous rates in individual sf regions are needed to address the many unanswered questions . the processes of tidal or splash mass transfer are very hard to study either individually , or statistically . we can observe the amount of gas between or outside the interacting galaxies , but how much has been transferred already , and how much will yet be pulled out ? the statistical question can be addressed with models , but there have been few such studies . wallin & stuart ( 1992 ) gave us a survey and analysis of 1000 restricted 3-body encounters with mass transfer from a particle disk around the primary . these models evenly sampled a number of collision parameters , and quantified dependences , such as inclination , some of which had been known since toomre and toomre 1972 ( see struck 1999 , sec . howard et al . ( 1993 ) published a nice atlas of 86 n - body simulations ( with rigid halos ) of the effects of encounters on ( 2-d ) disks . i produced a small model grid to study the hydrodynamics of colliding gas disks ( struck 1997 ) . statistical studies show us the big picture , and give the average answer to big picture questions , but often leave us wondering about the specific mechanisms . case studies provide detailed answers to some of those questions , but they require a large amount of high quality observational data , and a substantial modeling effort to interpret it . as yet , not many have been published . however , we can now optically resolve individual star clusters in nearby interacting systems , map tenuous gas distributions , and we have the computer power to do the modeling , so it is a great time for case studies . i will justify that statement with two examples . bev smith and i have been working for some years on this beautiful system ( see figure 1 , and smith & wallin 1992 , smith et al . 1997 , struck & smith 2003 ) . it is an asymmetric collisional ring galaxy , with a substantial bridge connecting it to a nearly edge - on companion , and it also contains some distinctive tidal tails . it has a variety of features that make it an especially interesting case for the study of induced sf and mass transfer . the primary has a prototypical starburst nucleus , but their is little evidence for recent sf in the ring wave , in contrast to most gas - rich collisional ring systems ( see appleton and struck - marcell 1996 ) . the companion has a post - starburst spectrum . there are intriguing knots of recent sf in one of the countertails , and roughly in a line on the north side of the bridge . more than half of the gas mass is located outside the main disks , in the bridge and tails ( smith et al . 1997 ) . in the bridge , there is an offset between the center line of the old stars , and that of the gas . lanon et al . ( 2001 ) have published detailed spectral evolutionary models of hst observations , so we have a good picture of the sf history of the nucleus of ngc 7714 . in a word , this history seems to be characterized by repetitive bursts . unfortunately , high resolution hst spectra are not available for extra - nuclear regions , or for the companion . ( a. lanon is working to obtain vlt data , however . ) we have recently published n - body hydrodynamic ( sph ) models of this system that can account for nearly all of the observed morphological and kinematic features ( smith & struck 2003 ) . successful models require a high inclination collision , which is somewhat prograde for galaxies . such an orbit allows the simultaneous production of a complete ring , and the tails . in terms of mass transfer we found that we could indeed fling a large fraction of the gas out of the two disks in such model collisions , and without unusual initial conditions , like especially extended gas disks . specifically , our models put a somewhat large gas fraction than observed into the great hi loop to the north , somewhat less than observed in the bridge , and a good deal in the companion countertail . this last feature is not isolated in the observations , except as a slight eastern extension of the companion disk . the models suggest that it lies behind both the companion disk and the bridge , and can not be easily distinguished . interestingly , we found that the bridge consists of several superposed , but dynamically distinct components . in addition to the two usual tidal components stretching from the near side of each galaxy towards its companion , there is also the superposed countertail of ngc 7715 , and the remnants of an old tail winding around from the far - side of ngc 7715 to the primary ( ngc 7714 ) . mass transfer onto the primary from this old bridge started before closest approach . some of the early transfer material was incorporated into the primary disk , or in a plane very close to it , and subsequently flung out ( again ! ) as the inner sw countertail . this feature , and the fact that the location and mass of some of the bridge components are very sensitive to initial collision parameters , illustrate the intricacies of mass transfer that can be found in detailed case studies . our models include simple feedback prescriptions that can tell us something about induced sf . the gas compression history can also suggest sf phenomena even when we do nt really have enough particles to fully resolve them . examples of the latter include the result that the ( star - forming ) inner sw countertail of ngc 7714 may consist of mass transferred material that has been compressed in the primary disk . similarly , the models show that the old bridge component is overrun and shocked by a newer bridge component . this might account for the line of young star clusters on the north side . the feedback model further suggests that the companion experienced a starburst at closest approach , consistent with its post - starburst spectrum . it also suggests multiple bursts have occurred in the core of the primary , in agreement with the spectral synthesis results . however , the exact timing and number of these bursts are not completely predictable . the models suggest that they could be driven by compressions from the ( m=0 ) ring wave component of the disturbance , and fed by inward mass transfer resulting from the spiral component . on the other hand , repetitive starbursts in the gas - rich core are driven by the feedback terms even without an external disturbance . ( this result derives from control runs and other unpublished modeling . ) it is likely that the gas density in the core or its susceptibility to sf was less than in the models . i have been working on this galaxy with the ocular galaxy collaboration ( see elmegreen et al . 1991 , 1995a , b , 2000 , 2001 , kaufman et al . the hubble heritage image of this graceful pair has been often reproduced . the models suggest that it has been one of the gentlest close interactions known . they further suggest that the companion approached from slightly above the primary ( in the west ) , interacting with its outer edge , and then moving slightly below and to the north and then east of the primary . the encounter is retrograde relative to the primary , so its optical disk is not greatly perturbed , though its larger hi disk is . the encounter is prograde and nearly in - plane relative to the companion , which is estimated to be about 60 - 80% of the mass of the primary . encounters that strongly torque the disk produce long tidal tails , and also the transient ocular ( eye - shaped ) morphology from material that loses angular momentum ( elmegreen et al . this system is a prototypical example of that process of disk rearrangement . in addition , the models suggest that there is some moderate mass transfer from the companion to the primary , both presently and earlier in a glancing interaction on the west side . they also suggest that the collisional perturbation did not produce the spirals in the primary ; they were almost certainly pre - existing . observationally , the recent star formation in this system is mostly extended ( and beautiful ! ) with no excess in the galaxy cores . specifically , the young star clusters are concentrated in the long spiral arms . in the companion , the young clusters are concentrated on the rim of the ocular . the tidal tail is younger and shorter than many we have seen at this symposium and is not presently the site of active sf . the models , projected into the future , indicate that it will grow ! the feedback models yield a bit more sf in the core of the primary than observed , but otherwise with the same widely distributed sf in the primary , and concentration of sf in the ocular and tidal tail of the companion . the models suggest some interesting patterns in the sf in the primary over the course of the interaction , but these must be further analyzed . what have we learned from these and other published case studies , and how do they complement the statistical studies ? in the area of mass transfer , the generally good agreement between model and observational gas distributions in two very different cases - arp 284 ( extreme gas removal from the disks ) and ngc 2207 ( little perturbation of the primary disk ) - is very encouraging . these general results also agree with expectations derived from exploratory model grids . the more detailed comparisons to observation in these two systems give us confidence that hydrodynamic models can quite accurately reproduce details of collisional morphology and kinematics on scales of about a few kpc . this could motivate further checks of model predictions , e.g. , a metallicity study of the countertails of ngc 7714 to see if there is evidence of differences that might be expected if the inner tail gas came from the companion . such specific predictions are not possible without detailed modeling of individual systems . in terms of sf , case studies are required to model modes of sf that are either unique to a specific system , or nearly so . possible examples from the cases above include the inner sw tail of ngc 7714 and , the line of sf regions on the northern edge of the arp 284 bridge . less unique examples , but still with system specific characteristics include the absence of sf in the ngc 7714 ring , sf in the rim of the ic 2163 ocular , and the scattered sf in the ngc 2207 disk . the bulk of interaction induced sf in the universe occurs in merger remnants and the cores of unmerged collision partners . statistical studies are ideal for studying the mean characteristics of this type of sf , and case studies would contribute little if this sf has a large stochastic component . however , understanding anomalous sf , like the examples of the previous paragraph , may be essential to understanding the formation of globular clusters and tidal dwarfs . these are minority populations , but still very interesting . in fact , answering the questions of how tidal dwarfs and globulars form , and also understanding the sytematics of wave induced sf in disks will require many detailed case studies . appleton , p. n. , & struck , c. 1996 , fund . phys . , 16 , 111 barnes , j. e. , & hernquist , l. 1992 , , 30 , 705 barton , e. j. , geller , m. j. , & kenyon , s. j. 2000 , , 530 , 660 barton gillespie , e. , geller , m. j. , & kenyon , s. j. 2003 , , 582 , 668 bervall , n. , laurikainen , e. , & aalto , s. 2003 , astro - ph 0304384 ( a&a in press ) elmegreen , b. g. , et al . 1995a , , 453 , 139 elmegreen , b. g. , et al . 2000 , , 120 , 630 elmegreen , d. m. , et al . 1995b , , 453 , 100 elmegreen , d. m. , et al . 2001 , , 121 , 182 elmegreen , d. m. , et al . 1991 , , 244 , 52 howard , s. , keel , w. c. , byrd , g. , & burkey , j. 1993 , , 417 , 502 kaufman , m. , et al . 1997 , , 114 , 2323 keel , w. c. 1993 , , 106 , 1771 keel , w. c. , et al . 1985 , , 90 , 708 kennicutt , jr . , r. c. , et al . 1987 , , 93 , 1011 lanon , a. , et al . 2001 , , 552 , 150 larson , r. b. , & tinsley , b. m. 1978 , , 219 , 46 schombert , j. m. , wallin , j. f. , & struck - marcell , c. 1990 , , 99 , 497 smith , b. j. , struck , c. , & pogge , r. w. 1997 , , 483 , 754 smith , b. j. , & wallin , j. f. 1992 , , 393 , 544 struck , c. 1997 , , 113 , 269 struck , c. 1999 , physics reports , 321 , 1 struck , c. , & smith , b. j. 2003 , , 589 , 157 toomre , a. , & toomre , j. 1972 , , 178 , 623 wallin , j. f. , & stuart , b. v. 1992 , , 399 , 29

the amount , timing and ultimate location of mass transfer and induced star formation in galaxy collisions are sensitive functions of orbital and galaxy structural parameters . i discuss the role of detailed case studies and describe the results for two systems , arp 284 and ngc 2207/ic 2163 , that have been studied with both multiwaveband observations , and detailed dynamical models . the models yield the mass transfer and compressional histories of the encounters and the `` probable causes '' or triggers of individual star - forming regions . # 1_#1 _ # 1_#1 _ = # 1 1.25 in .125 in .25 in

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Proceed to summarize the following text: adaptive network consists of a collection of agents that are interconnected to each other and solve distributed estimation and inference problems in a collaborative manner . two useful strategies that enable adaptation and learning over such networks in real - time are the incremental strategy @xcite and the diffusion strategy @xcite . incremental strategies rely on the use of a hamiltonian cycle , i.e. , a cyclic path that covers all nodes in the network , which is generally difficult to enforce since determining a hamiltonian cycle is an np - hard problem . in addition , cyclic trajectories are not robust to node or link failure . in comparison , diffusion strategies are scalable , robust , and able to match well the performance of incremental networks . in adaptive diffusion implementations , information is processed locally at the nodes and then diffused in real - time across the network . diffusion strategies were originally proposed in @xcite and further extended and studied in @xcite . they have been applied to model self - organized and complex behavior encountered in biological networks , such as fish schooling @xcite , bird flight formations @xcite , and bee swarming @xcite . diffusion strategies have also been applied to online learning of gaussian mixture models @xcite and to general distributed optimization problems @xcite . there have also been several useful works in the literature on distributed consensus - type strategies , with application to multi - agent formations and distributed processing @xcite . the main difference between these works and the diffusion approach of @xcite is the latter s emphasis on the role of adaptation and learning over networks . in the original diffusion least - mean - squares ( lms ) strategy @xcite , the weight estimates that are exchanged among the nodes can be subject to quantization errors and additive noise over the communication links . studying the degradation in mean - square performance that results from these particular perturbations can be pursued , for both incremental and diffusion strategies , by extending the mean - square analysis already presented in @xcite , in the same manner that the tracking analysis of conventional stand - alone adaptive filters was obtained from the counterpart results in the stationary case ( as explained in ( * ? ? ? * ch . 21 ) ) . useful results along these lines , which study the effect of link noise during the exchange of the weight estimates , already appear for the traditional diffusion algorithm in the works @xcite and for consensus - based algorithms in @xcite . in this paper , our objective is to go beyond these earlier studies by taking into account additional effects , and by considering a more general algorithmic structure . the reason for this level of generality is because the analytical results will help reveal which noise sources influence the network performance more seriously , in what manner , and at what stage of the adaptation process . the results will suggest important remedies and mechanisms to adapt the combination weights in real - time . some of these insights are hard to get if one focuses solely on noise during the exchange of the weight estimates . the analysis will further show that noise during the exchange of the regression data plays a more critical role than other sources of imperfection : this particular noise alters the learning dynamics and modes of the network , and biases the weight estimates . noises related to the exchange of other pieces of information do not alter the dynamics of the network but contribute to the deterioration of the network performance . to arrive at these results , in this paper , we first consider a generalized analysis that applies to a broad class of diffusion adaptation strategies ( see further ahead ; this class includes the original diffusion strategies and as two special cases ) . the analysis allows us to account for various sources of information noise over the communication links . we allow for noisy exchanges during _ each _ of the three processing steps of the adaptive diffusion algorithm ( the two combination steps and and the adaptation step ) . in this way , we are able to examine how the three sets of combination coefficients @xmath0 in influence the propagation of the noise signals through the network dynamics . our results further reveal how the network mean - square - error performance is dependent on these combination weights . following this line of reasoning , the analysis leads to algorithms and further ahead for choosing the combination coefficients to improve the steady - state network performance . it should be noted that several combination rules , such as the metropolis rule @xcite and the maximum degree rule @xcite , were proposed previously in the literature especially in the context of consensus - based iterations @xcite . these schemes , however , usually suffer performance degradation in the presence of noisy information exchange since they ignore the network noise profile @xcite . when the noise variance differs across the nodes , it becomes necessary to design combination rules that are aware of this variation as outlined further ahead in section vi - b . moreover , in a mobile network @xcite where nodes are on the move and where neighborhoods evolve over time , it is even more critical to employ adaptive combination strategies that are able to track the variations in the noise profile in order to cope with such dynamic environments . this issue is taken up in section vi - c . we use lowercase letters to denote vectors , uppercase letters for matrices , plain letters for deterministic variables , and boldface letters for random variables . we also use @xmath1 to denote conjugate transposition , @xmath2 for the trace of its matrix argument , @xmath3 for the spectral radius of its matrix argument , @xmath4 for the kronecker product , and @xmath5 for a vector formed by stacking the columns of its matrix argument . we further use @xmath6 to denote a ( block ) diagonal matrix formed from its arguments , and @xmath7 to denote a column vector formed by stacking its arguments on top of each other . all vectors in our treatment are column vectors , with the exception of the regression vectors , @xmath8 , and the associated noise signals , @xmath9 , which are taken to be row vectors for convenience of presentation . we consider a connected network consisting of @xmath10 nodes . each node @xmath11 collects scalar measurements @xmath12 and @xmath13 regression data vectors @xmath8 over successive time instants @xmath14 . note that we use parenthesis to refer to the time - dependence of scalar variables , as in @xmath12 , and subscripts to refer to the time - dependence of vector variables , as in @xmath8 . the measurements across all nodes are assumed to be related to an unknown @xmath15 vector @xmath16 via a linear regression model of the form @xcite : @xmath17 where @xmath18 denotes the measurement or model noise with zero mean and variance @xmath19 . the vector @xmath16 in denotes the parameter of interest , such as the parameters of some underlying physical phenomenon , the taps of a communication channel , or the location of food sources or predators . such data models are also useful in studies on hybrid combinations of adaptive filters @xcite . the nodes in the network would like to estimate @xmath16 by solving the following minimization problem : @xmath20 in previous works @xcite , we introduced and studied several distributed strategies of the diffusion type that allow nodes to cooperate with each other in order to solve problems of the form in an adaptive manner . these diffusion strategies endow networks with adaptation and learning abilities , and enable information to diffuse through the network in real - time . we review the adaptive diffusion strategies below . in @xcite , two classes of diffusion algorithms were proposed . one class is the so - called combine - then - adapt ( cta ) strategy : @xmath21\\ \end{aligned}\right.\end{aligned}\ ] ] and the second class is the so - called adapt - then - combine ( atc ) strategy : @xmath22\\ { \bs{w}}_{k , i}&=\sum_{l\in{\mc{n}}_k}a_{2,lk}{\bs{\psi}}_{l , i}\\ \end{aligned}\right.\end{aligned}\ ] ] where the @xmath23 are small positive step - size parameters and the @xmath0 are nonnegative entries of the @xmath24 matrices @xmath25 , respectively . the coefficients @xmath0 are zero whenever node @xmath26 is not connected to node @xmath11 , i.e. , @xmath27 , where @xmath28 denotes the neighborhood of node @xmath11 . the two strategies and can be integrated into one broad class of diffusion adaptation @xcite : @xmath29\\ \label{eqn : idealdiffusionpostdiff } { \bs{w}}_{k , i}&=\sum_{l\in{\mc{n}}_k}a_{2,lk}{\bs{\psi}}_{l , i}\end{aligned}\ ] ] several diffusion strategies can be obtained as special cases of through proper selection of the coefficients @xmath0 . for example , to recover the cta strategy , we set @xmath30 , and to recover the atc strategy , we set @xmath31 , where @xmath32 denotes the @xmath24 identity matrix . in the general diffusion strategy , each node @xmath11 evaluates its estimate @xmath33 at time @xmath34 by relying solely on the data collected from its neighbors through steps and and on its local measurements through step . the matrices @xmath35 , @xmath36 , and @xmath37 are required to be left or right - stochastic , i.e. , @xmath38 where @xmath39 denotes the @xmath40 vector whose entries are all one . this means that each node performs a convex combination of the estimates received from its neighbors at every iteration @xmath34 . the mean - square performance and convergence properties of the diffusion algorithm have already been studied in detail in @xcite . for the benefit of the analysis in the subsequent sections , we present below in the recursion describing the evolution of the weight error vectors across the network to do so , we introduce the error vectors : @xmath41 and substitute the linear model into the adaptation step to find that @xmath42 where the @xmath43 matrix @xmath44 and the @xmath15 vector @xmath45 are defined as : @xmath46 we further collect the various quantities across all nodes in the network into the following block vectors and matrices : @xmath47 then , from , , and , the recursion for the network error vector @xmath48 is given by @xmath49 where @xmath50 each of the steps in involves the sharing of information between node @xmath11 and its neighbors . for example , in the first step , all neighbors of node @xmath11 send their estimates @xmath51 to node @xmath11 . this transmission is generally subject to additive noise and possibly quantization errors . likewise , steps and involve the sharing of other pieces of information with node @xmath11 . these exchange steps can all be subject to perturbations ( such as additive noise and quantization errors ) . one of the objectives of this work is to analyze the _ aggregate _ effect of these perturbations on general diffusion strategies of the type and to propose choices for the combination weights in order to enhance the mean - square performance of the network in the presence of these disturbances . to node @xmath11 . ] so let us examine what happens when information is exchanged over links with additive noise . we model the data received by node @xmath11 from its neighbor @xmath26 as @xmath52 where @xmath53 and @xmath54 are @xmath15 noise signals , @xmath9 is a @xmath13 noise signal , and @xmath55 is a scalar noise signal ( see fig . [ fig : noise ] ) . observe further that in , we are including several sources of information exchange noise . in comparison , references @xcite only considered the noise source @xmath53 in and one set of combination coefficients @xmath56 ; the other coefficients were set to @xmath57 for @xmath58 and @xmath59 . in other words , these references only considered and the following traditional cta strategy without exchange of the data @xmath60 compare with ; note that the second step in only uses @xmath61 : @xmath62\\ \end{aligned}\right.\end{aligned}\ ] ] the analysis that follows examines the aggregate effect of all four noise sources appearing in , in addition to the three sets of combination coefficients appearing in . we introduce the following assumption on the statistical properties of the measurement data and noise signals . [ asm : all ] 1 . the regression data @xmath8 are temporally white and spatially independent random variables with zero mean and covariance matrix @xmath63 . 2 . the noise signals @xmath64 , @xmath53 , @xmath55 , @xmath9 , and @xmath54 are temporally white and spatially independent random variables with zero mean and ( co)variances @xmath19 , @xmath65 , @xmath66 , @xmath67 , and @xmath68 , respectively . in addition , @xmath65 , @xmath66 , @xmath67 , and @xmath68 are all zero if @xmath27 or @xmath69 . 3 . the regression data @xmath70 , the model noise signals @xmath71 , and the link noise signals @xmath72 , @xmath73 , @xmath74 , and @xmath75 are mutually - independent random variables for all @xmath76 and @xmath77 . using the perturbed data , the diffusion algorithm becomes @xmath78\\ \label{eqn : noisydiffusionpostdiffold } { \bs{w}}_{k , i}&\!=\!\sum_{l\in{\mc{n}}_k}a_{2,lk}{\bs{\psi}}_{lk , i}\end{aligned}\ ] ] where we continue to use the symbols @xmath79 to avoid an explosion of notation . from and , expressions can be rewritten as @xmath80\\ \label{eqn : noisydiffusionpostdiff } { \bs{w}}_{k , i}&\!=\!\sum_{l\in{\mc{n}}_k}a_{2,lk}{\bs{\psi}}_{l , i}\!+\!{\bs{v}}_{k , i}^{(\psi)}\end{aligned}\ ] ] where we are introducing the symbols @xmath81 and @xmath82 to denote the aggregate @xmath15 zero - mean noise signals defined over the neighborhood of node @xmath11 : @xmath83 with covariance matrices @xmath84 it is worth noting that @xmath85 and @xmath86 depend on the combination coefficients @xmath56 and @xmath87 , respectively . this property will be taken into account when optimizing over @xmath56 and @xmath87 in a later section . we further introduce the following scalar zero - mean noise signal : @xmath88 for @xmath89 , whose variance is @xmath90 to unify the notation , we define @xmath91 . then , from , , and , it is easy to verify that the noisy data @xmath92 are related via @xmath93 for @xmath94 . continuing with the adaptation step and substituting , we get @xmath95\end{aligned}\ ] ] then , we can derive the following error recursion for node @xmath11 ( compare with ): @xmath96 where the @xmath43 matrix @xmath97 and the @xmath98 vector @xmath99 are defined as ( compare with and ): @xmath100 we further introduce the block vectors and matrices : @xmath101 and the corresponding covariance matrices for @xmath102 and @xmath103 : @xmath104 then , from , , and , we arrive at the following recursion for the network weight error vector in the presence of noisy information exchange : @xmath105\!-\!{\bs{v}}_i^{(\psi)}\nonumber\\ { } & = { \mc{a}}_2^{\t}\left[(i_{nm}\!-\!{\mc{m}}{\bs{\mc{r}}}_i')({\mc{a}}_1^{\t}{\wt{\bs{w}}}_{i-1}\!-\!{\bs{v}}_{i-1}^{(w ) } ) \!-\!{\mc{m}}{\bs{z}}_i\right]\!-\!{\bs{v}}_i^{(\psi)}\end{aligned}\ ] ] that is , @xmath106 compared to the previous error recursion , the noise terms in consist of three parts : * @xmath107 is contributed by the noise introduced at the information - exchange step _ before _ adaptation . * @xmath108 is contributed by the noise introduced at the adaptation step . * @xmath103 is contributed by the noise introduced at the information - exchange step _ after _ adaptation . given the weight error recursion , we are now ready to study the mean convergence condition for the diffusion strategy in the presence of disturbances during information exchange under assumption [ asm : all ] . taking expectations of both sides of , we get @xmath109 where @xmath110 from , , , and , it can be verified that @xmath111 whereas , from and assumption [ asm : all ] , we get @xmath112\nonumber\\ { } & = -\left(\sum_{l\in{\mc{n}}_k}c_{lk}r_{v , lk}^{(u)}\right)w^o\end{aligned}\ ] ] let us define an @xmath113 matrix @xmath114 that collects all covariance matrices @xmath115 , @xmath116 , weighted by the corresponding combination coefficients @xmath117 , such that its @xmath118th @xmath43 submatrix is @xmath119 . note that @xmath114 itself is _ not _ a covariance matrix because @xmath120 for all @xmath11 . then , from and , we arrive at @xmath121 therefore , using and , expression becomes @xmath122 with a driving term due to the presence of @xmath123 . this driving term would disappear from if there were no noise during the exchange of the regression data . to guarantee convergence of , the coefficient matrix @xmath124 must be stable , i.e. , @xmath125 . since @xmath126 and @xmath127 are right - stochastic matrices , it can be shown that the matrix @xmath124 is stable whenever @xmath128 itself is stable ( see appendix [ app : meanconvergence ] ) . this fact leads to an upper bound on the step - sizes @xmath23 to guarantee the convergence of @xmath129 to a steady - state value , namely , we must have @xmath130 for @xmath131 , where @xmath132 denotes the largest eigenvalue of its matrix argument . note that the neighborhood covariance matrix @xmath133 in is related to the combination weights @xmath117 . if we further assume that @xmath37 is doubly - stochastic , i.e. , @xmath134 then , by jensen s inequality @xcite , @xmath135 since ( i ) @xmath132 coincides with the induced @xmath136-norm for any positive semi - definite hermitian matrix ; ( ii ) matrix norms are convex functions of their arguments @xcite ; and ( iii ) by , @xmath117 are convex combination coefficients . thus , we obtain a sufficient condition for the convergence of in lieu of : @xmath137 } } \ ] ] for @xmath131 , where the upper bound for the step - size @xmath138 becomes independent of the combination weights @xmath117 . this bound can be determined solely from knowledge of the covariances of the regression data and the associated noise signals that are accessible to node @xmath11 . it is worth noting that for traditional diffusion algorithms where information is perfectly exchanged , condition reduces to @xmath139}\end{aligned}\ ] ] for @xmath131 . comparing with , we see that the link noise @xmath9 over regression data reduces the dynamic range of the step - sizes for mean stability . now , under , and taking the limit of as @xmath140 , we find that the mean error vector will converge to a steady - state value @xmath141 : @xmath142 it is well - known that studying the mean - square convergence of a single adaptive filter is a challenging task , since adaptive filters are nonlinear , time - variant , and stochastic systems . when a network of adaptive nodes is considered , the complexity of the analysis is compounded because the nodes now influence each other s behavior . in order to make the performance analysis more tractable , we rely on the energy conservation approach @xcite , which was used successfully in @xcite to study the mean - square performance of diffusion strategies under perfect information exchange conditions . that argument allows us to derive expressions for the mean - square - deviation ( msd ) and the excess - mean - square - error ( emse ) of the network by analyzing how energy ( measured in terms of error variances ) flows through the nodes . from recursion and under assumption [ asm : all ] , we can obtain the following weighted variance relation for the global error vector @xmath48 : @xmath143\}\\ { } & \quad+\e\|{\mc{a}}_2^{\t}(i_{nm}\!-\!{\mc{m}}{\bs{\mc{r}}}_i'){\bs{v}}_{i-1}^{(w)}\|_{\sigma}^2+\e\|{\bs{v}}_i^{(\psi)}\|_{\sigma}^2 \end{aligned } } \ ] ] where @xmath144 is an arbitrary @xmath113 positive semi - definite hermitian matrix that we are free to choose . moreover , the notation @xmath145 stands for the quadratic term @xmath146 . the weighting matrix @xmath147 in can be expressed as @xmath148 where @xmath124 is given by and @xmath149 denotes a term on the order of @xmath150 . evaluating the term @xmath149 requires knowledge of higher - order statistics of the regression data and link noises , which are not available under current assumptions . however , this term becomes negligible if we introduce a small step - size assumption . [ asm : smallstepsize ] the step - sizes are sufficiently small , i.e. , @xmath151 , such that terms depending on higher - order powers of the step - sizes can be ignored . hence , in the sequel we use the approximation : @xmath152 observe that on the right - hand side ( rhs ) of relation , only the first and third terms relate to the error vector @xmath153 . by assumption [ asm : all ] , the error vector @xmath153 is independent of @xmath154 and @xmath155 . thus , from , the third term on rhs of can be expressed as @xmath156\cdot\e{\wt{\bs{w}}}_{i-1}\}\nonumber\\ { } & = -2\,{\mathfrak{re}}(z^*{\mc{m}}{\mc{a}}_2\sigma{\mc{a}}_2^{\t}{\mc{a}}_1^{\t}\cdot\e{\wt{\bs{w}}}_{i-1})+o({\mc{m}}^2)\end{aligned}\ ] ] since we already showed in the previous section that @xmath157 converges to a fixed bias @xmath141 , quantity will converge to a fixed value as well when @xmath140 . moreover , under assumption [ asm : all ] , the second , fourth , and fifth terms on rhs of relation are all fixed values . therefore , the convergence of relation depends on the behavior of the first term @xmath158 . although the weighting matrix @xmath147 of @xmath153 is different from the weighting matrix @xmath144 of @xmath159 , it turns out that the entries of these two matrices are approximately related by a linear equation shown ahead in . introduce the vector notation @xcite : @xmath160 then , by using the identity @xmath161 , it can be verified from that @xmath162 where the @xmath163 matrix @xmath164 is given by @xmath165 to guarantee mean - square convergence of the algorithm , the step - sizes should be sufficiently small and selected to ensure that the matrix @xmath164 is stable @xcite , i.e. , @xmath166 , which is equivalent to the earlier condition @xmath125 . although more specific conditions for mean - square stability can be determined without assumption [ asm : smallstepsize ] @xcite , it is sufficient for our purposes here to conclude that the diffusion strategy is stable in the mean and mean - square senses if the step - sizes @xmath23 satisfy or and are sufficiently small . the conclusion so far is that sufficiently small step - sizes ensure convergence of the diffusion strategy in the mean and mean - square senses , even in the presence of exchange noises over the communication links . let us now determine expressions for the error variances in steady - state . we start from the weighted variance relation . in view of , it shows that the error variance @xmath167 depends on the mean error @xmath129 . we already determined the value of @xmath168 in . we continue to use the vector notation and proceed to evaluate all the terms , except the first one , on rhs of in the following . for the _ second _ term , it can be expressed as @xmath169^*\sigma\end{aligned}\ ] ] where we used the identity @xmath170^*\sigma$ ] for any hermitian matrix @xmath171 , and @xmath172 denotes the autocorrelation matrix of @xmath173 . it is shown in appendix [ app : rz ] that @xmath172 is given by @xmath174 where @xmath175 is defined in , @xmath123 is in , and @xmath176 are two @xmath113 positive semi - definite block diagonal matrices : @xmath177\end{aligned}\ ] ] from expression and assumption [ asm : smallstepsize ] , the _ third _ term on rhs of is given by @xmath178\sigma\}\nonumber\\ { } & = -\!\left[\vec\left({\mc{a}}_2^{\t}{\mc{a}}_1^{\t}(\e{\wt{\bs{w}}}_{i-1})z^*{\mc{m}}{\mc{a}}_2 + { \mc{a}}_2^{\t}{\mc{m}}z(\e{\wt{\bs{w}}}_{i-1})^*{\mc{a}}_1{\mc{a}}_2\right)\right]^*\sigma\end{aligned}\ ] ] likewise , the _ fourth _ term on rhs of is approximated by @xmath179\right)\right]^*\sigma\nonumber\\ { } & \approx\left[\vec({\mc{a}}_2^{\t}{\mc{r}}_v^{(w)}{\mc{a}}_2)\right]^*\sigma\end{aligned}\ ] ] where we are now ignoring terms on the order of @xmath180 and @xmath150 . the _ fifth _ term on rhs of is given by @xmath181^*\sigma\end{aligned}\ ] ] let us introduce @xmath182 at steady - state , as @xmath140 , by and , the weighted variance relation becomes @xmath183^*\sigma\end{aligned}\ ] ] where we are using the compact notation @xmath184 to refer to @xmath185 doing so allows us to represent @xmath147 by the more compact relation @xmath186 on rhs of ; we shall be using the weighting matrix @xmath144 and its vector representation @xmath187 interchangeably for ease of notation ( likewise , for @xmath147 and @xmath188 ) . the steady - state weighted variance relation can be rewritten as @xmath189^*\sigma\end{aligned}\ ] ] where the term @xmath190 is contributed by the model noise @xmath191 while the remaining terms @xmath192 are contributed by the link noises @xmath193 . recall that we are free to choose @xmath144 and , hence , @xmath187 . let @xmath194 , where @xmath195 is another arbitrary positive semi - definite hermitian matrix . then , we arrive at the following theorem . [ lemma : steadystatevariancerelation ] under assumptions [ asm : all ] and [ asm : smallstepsize ] , for any positive semi - definite hermitian matrix @xmath195 , the steady - state weighted error variance relation of the diffusion strategy is approximately given by @xmath196^*(i_{n^2m^2}-{\mc{f}})^{-1}\vec(\omega ) \end{aligned } } \ ] ] where @xmath197 is given in , @xmath198 in , @xmath199 in , and @xmath164 in . each subvector of @xmath200 corresponds to the estimation error at a particular node , say , @xmath201 for node @xmath11 . the network msd is defined as @xcite : @xmath202 since we are free to choose @xmath195 , we select it as @xmath203 . then , expression gives @xmath204^*(i_{n^2m^2}-{\mc{f}})^{-1}\vec(i_{nm } ) \end{aligned } } \ ] ] similarly , if we instead select @xmath205 , where @xmath206 then expression would allow us to evaluate the network emse as : @xmath207^*(i_{n^2m^2}-{\mc{f}})^{-1}\vec({\mc{r}}_u ) \end{aligned } } \ ] ] where , under assumption [ asm : all ] , the network emse is given by @xmath208 we showed in the earlier sections that the link noise over regression data biases the weight estimators . in this section we examine how the results simplify when there is no sharing of regression data among the nodes . [ asm : nodatasharing ] nodes do not share regression data within neighborhoods , i.e. , assume @xmath209 . by assumptions [ asm : smallstepsize ] and [ asm : nodatasharing ] , matrices @xmath210 in , , and become @xmath211 where @xmath212 is given in . then , the network msd and emse expressions and simplify to : @xmath213^ * ( i_{n^2m^2}-{\mc{f}})^{-1}\vec(i_{nm } ) \end{aligned } } \ ] ] and @xmath214^ * ( i_{n^2m^2}-{\mc{f}})^{-1}\vec({\mc{r}}_u ) \end{aligned } } \ ] ] recalling that @xmath198 and @xmath164 are related to the combination matrices @xmath215 , or , equivalently , @xmath216 , results and express the network msd and emse in terms of @xmath216 . however , it is generally difficult to use these expressions to optimize over @xmath216 to reduce the impact of link noise . instead , by substituting into and using the fact that @xmath164 is stable , we can arrive at another useful expression for the network msd : @xmath217^ * \sum_{j=0}^{\infty}{\mc{f}}^j\vec(i_{nm})\nonumber\\ { } & = \frac{1}{n}\!\left[\vec({\mc{a}}_2^{\t}{\mc{m}}{\mc{s}}{\mc{m}}{\mc{a}}_2\!+\!{\mc{r}}_v)\right]^ * \sum_{j=0}^{\infty}({\mc{b}}^{\t}\otimes{\mc{b}}^*)^j\vec(i_{nm})\nonumber\\ { } & = \frac{1}{n}\!\left[\vec({\mc{a}}_2^{\t}{\mc{m}}{\mc{s}}{\mc{m}}{\mc{a}}_2\!+\!{\mc{r}}_v)\right]^ * \sum_{j=0}^{\infty}\vec({\mc{b}}^{*j}{\mc{b}}^j)\end{aligned}\ ] ] that is , @xmath218 } \ ] ] where @xmath124 is given in . similarly , the network emse can be expressed as @xmath219 } \ ] ] expressions and reveal in an interesting way how the noise sources originating from any particular node end up influencing the overall network performance . let us denote @xmath220 the error recursion can be rewritten as @xmath221 where @xmath222 then , @xmath223 under assumption [ asm : nodatasharing ] , @xmath224 in and can be simplified as @xmath225 where @xmath226 are given in and . by assumption [ asm : all ] , @xmath224 are temporally independent for different @xmath34 and @xmath227 where @xmath124 is given by . as @xmath140 , the first term on rhs of becomes @xmath228\right\}\nonumber\\ { } & \stackrel{(a)}{\approx}\lim_{i\rightarrow\infty}\tr\left[\left(\e{\bm{\phi}}_{0,i}\right ) ( \e{\wt{\bs{w}}}_{-1}{\wt{\bs{w}}}_{-1}^*)\left(\e{\bm{\phi}}_{0,i}\right)^*\right]\nonumber\\ { } & = \lim_{i\rightarrow\infty}\tr\left[{\mc{b}}^{i+1}(\e\,{\wt{\bs{w}}}_{-1}{\wt{\bs{w}}}_{-1}^*){\mc{b}}^{(i+1)*}\right]\nonumber\\ { } & \stackrel{(b)}{=}0\end{aligned}\ ] ] where ( a ) is obtained by approximating the expectation of the product by the product of expectations and ( b ) is due to the stability of @xmath124 . therefore , the steady - state value of gives @xmath229\nonumber\\ { } & \stackrel{(a)}{\approx}\lim_{i\rightarrow\infty}\sum_{m=0}^{i}\tr\left[{\mc{b}}^{i - m}({\mc{a}}_2^{\t}{\mc{m}}{\mc{s } } { \mc{m}}{\mc{a}}_2+{\mc{r}}_v){\mc{b}}^{(i - m)*}\right]\nonumber\\ { } & \stackrel{(b)}{=}\lim_{i\rightarrow\infty}\sum_{j=0}^{i}\tr\left[{\mc{b}}^{j}({\mc{a}}_2^{\t}{\mc{m}}{\mc{s } } { \mc{m}}{\mc{a}}_2+{\mc{r}}_v){\mc{b}}^{j*}\right]\nonumber\\ { } & = \sum_{j=0}^{\infty}\tr\left[{\mc{b}}^{j}({\mc{a}}_2^{\t}{\mc{m}}{\mc{s}}{\mc{m}}{\mc{a}}_2+{\mc{r}}_v){\mc{b}}^{*j}\right]\end{aligned}\ ] ] where , by and , ( a ) is due to @xmath230 and ( b ) is simply a change of variable : @xmath231 . since the @xmath232th term of the summation in or is contributed by the term @xmath233 , which consists of all the noise sources at time @xmath234 , expression shows how various sources of noises are involved and how they contribute to the network msd . before we optimize the combination matrices @xmath216 , we first specialize the msd expression and the emse expression for the atc and cta algorithms . for the atc algorithm , we set @xmath31 and @xmath235 , and for the cta algorithm , we set @xmath236 and @xmath30 . let us denote @xmath237 then , we get @xmath238\!\!\\ \label{eqn : noisyatcemse } \!\!{\textrm{emse}}_{\textrm{atc}}&\!\approx\!\frac{1}{n}\!\sum_{j=0}^{\infty}{\mathrm{tr}}\!\left[{\mc{b}}_{\textrm{atc}}^j ( { \mc{a}}^{\t}{\mc{m}}{\mc{s}}{\mc{m}}{\mc{a}}\!+\!{\mc{r}}_v^{(\psi ) } ) { \mc{b}}_{\textrm{atc}}^{*j}{\mc{r}}_u\right]\!\!\end{aligned}\ ] ] and @xmath239\\ \label{eqn : noisyctaemse } { \textrm{emse}}_{\textrm{cta}}&\approx\frac{1}{n}\sum_{j=0}^{\infty}{\mathrm{tr}}\left[{\mc{b}}_{\textrm{cta}}^j ( { \mc{m}}{\mc{s}}{\mc{m}}+{\mc{r}}_v^{(w)}){\mc{b}}_{\textrm{cta}}^{*j}{\mc{r}}_u\right]\end{aligned}\ ] ] minimizing the msd expression or the emse expression for the atc algorithm over left - stochastic matrices @xmath240 is generally nontrivial . we pursue an approximate solution that relies on optimizing an upper bound and performs well in practice . let us use @xmath241 to denote the nuclear norm ( also known as the trace norm , or the ky fan @xmath242-norm ) of matrix @xmath243 @xcite , which is defined as the sum of the singular values of @xmath243 . therefore , @xmath244 for any @xmath243 and @xmath245 when @xmath243 is hermitian and positive semi - definite . let us also denote @xmath246 as the block maximum norm of matrix @xmath243 ( see appendix [ app : meanconvergence ] ) . then , @xmath247 & = \big\|{\mc{b}}_{\textrm{atc}}^j({\mc{a}}^{\t}{\mc{m}}{\mc{s}}{\mc{m}}{\mc{a } } + { \mc{r}}_v^{(\psi)}){\mc{b}}_{\textrm{atc}}^{*j}\big\|_*\nonumber\\ { } & \le\|{\mc{b}}_{\textrm{atc}}^j\|_*\cdot\|{\mc{a}}^{\t}{\mc{m}}{\mc{s } } { \mc{m}}{\mc{a}}+{\mc{r}}_v^{(\psi)}\|_*\cdot\|{\mc{b}}_{\textrm{atc}}^{*j}\|_*\nonumber\\ { } & \le c^2\cdot\|{\mc{b}}_{\textrm{atc}}^j\|_{b,\infty}^2\cdot { \mathrm{tr}}({\mc{a}}^{\t}{\mc{m}}{\mc{s}}{\mc{m}}{\mc{a}}+{\mc{r}}_v^{(\psi)})\nonumber\\ { } & \le c^2\cdot\|{\mc{b}}_{\textrm{atc}}\|_{b,\infty}^{2j}\cdot { \mathrm{tr}}({\mc{a}}^{\t}{\mc{m}}{\mc{s}}{\mc{m}}{\mc{a}}+{\mc{r}}_v^{(\psi)})\nonumber\\ { } & \le c^2\cdot(\|{\mc{a}}\|_{b,\infty}\cdot\|i_{nm}-{\mc{m}}{\mc{r}}_u\|_{b,\infty})^{2j } { \mathrm{tr}}({\mc{a}}^{\t}{\mc{m}}{\mc{s}}{\mc{m}}{\mc{a}}+{\mc{r}}_v^{(\psi)})\nonumber\\ { } & = c^2\cdot\rho(i_{nm}-{\mc{m}}{\mc{r}}_u)^{2j}\cdot{\mathrm{tr } } ( { \mc{a}}^{\t}{\mc{m}}{\mc{s}}{\mc{m}}{\mc{a}}+{\mc{r}}_v^{(\psi)})\end{aligned}\ ] ] where @xmath248 is some positive scalar such that @xmath249 because @xmath241 and @xmath246 are submultiplicative norms and all such norms are equivalent @xcite . in the last step of we used lemmas [ lemma : rghtstocmat ] and [ lemma : blockdiagonal ] from appendix [ app : meanconvergence ] . thus , we can upper bound the network msd by @xmath250 ^ 2}\end{aligned}\ ] ] where the combination matrix @xmath251 appears only in the numerator . the result motivates us to consider instead the problem of minimizing the upper bound , namely , @xmath252 using and , the cost function in can be expressed as @xmath253\end{aligned}\ ] ] problem can therefore be decoupled into @xmath10 separate optimization problems of the form : @xmath254\\ \st & \;\ ; \sum_{l\in{\mc{n}}_k}a_{lk}=1,\;\;a_{lk}\ge0,\;\;a_{lk}=0\;\mbox{if}\;l\notin{\mc{n}}_k \\ \end{aligned } } \ ] ] for @xmath255 . with each node @xmath94 , we associate the following nonnegative _ variance product _ measure : @xmath256 this measure incorporates information about the link noise covariances @xmath257 . the solution of is then given by @xmath258 we refer to this combination rule as the relative variance combination rule ; it is an extension of the rule devised in @xcite to the case of noisy information exchanges . in particular , the definition of the scalars @xmath259 in is different and now depends on both subscripts @xmath26 and @xmath11 . minimizing the emse expression for the atc algorithm over left - stochastic matrices @xmath240 can be pursued in a similar manner by noting that @xmath260\le c^2[\rho(i_{nm}\!-\!{\mc{m}}{\mc{r}}_u)]^{2j}\ , { \mathrm{tr}}({\mc{a}}^{\t}{\mc{m}}{\mc{s}}{\mc{m}}{\mc{a}}\!+\!{\mc{r}}_v^{(\psi)})\,{\mathrm{tr}}({\mc{r}}_u)\end{aligned}\ ] ] thus , minimizing the upper bound of the network emse leads to the same solution . using the same argument , we can also show that the same result minimizes the upper bound of the network msd or emse for the cta algorithm . to apply the relative variance combination rule , each node @xmath11 needs to know the variance products , @xmath259 , of their neighbors , which in general are not available since they require knowledge of the quantities @xmath261 . therefore , we now propose an adaptive combination rule by using data that are available to the individual nodes . for the atc algorithm , we first note from and that @xmath262 for @xmath89 . since the algorithm converges in the mean and mean - square senses under assumption [ asm : smallstepsize ] , all the estimates @xmath263 tend close to @xmath16 as @xmath140 . this allows us to estimate @xmath264 for node @xmath11 by using instantaneous realizations of @xmath265 , where we replace @xmath51 by @xmath266 . similarly , for node @xmath11 itself , we can use realizations of @xmath267 to estimate @xmath268 . to unify the notation , we define @xmath269 . let @xmath270 denote an estimator for @xmath264 that is computed by node @xmath11 at time @xmath34 . then , one way to evaluate @xmath270 is through the recursion : @xmath271 for @xmath272 , where @xmath273 is a forgetting factor that is usually close to one . in this way , we arrive at the adaptive combination rule : @xmath274^{-1}}{\sum_{m\in{\mc{n}}_k}[{\widehat{\bm{\gamma}}}_{mk}^{2}(i)]^{-1 } } , & { \textrm{if $ l\in{\mc{n}}_k$ } } \\ 0 , & { \textrm{otherwise } } \\ \end{cases } } \ ] ] the diffusion strategy is adaptive in nature . one of the main benefits of adaptation ( by using constant step - sizes ) is that it endows networks with tracking abilities when the underlying weight vector @xmath16 varies with time . in this section we analyze how well an adaptive network is able to track variations in @xmath16 . to do so , we adopt a random - walk model for @xmath16 that is commonly used in the literature to describe the non - stationarity of the weight vector @xcite . [ asm : randomwalk ] the weight vector @xmath16 changes according to the model : @xmath275 where @xmath276 has a constant mean @xmath16 for all @xmath34 , @xmath277 is an i.i.d . random sequence with zero mean and covariance matrix @xmath278 ; the sequence @xmath277 is independent of the initial conditions @xmath279 and of all regression data and noise signals across the network for all time instants . we now define the error vector at node @xmath11 as @xmath280 so that the global error recursion ( [ eqn : noisyerrorrecursion1 ] ) for the network is replaced by @xmath281 where the @xmath282 vector @xmath283 is defined as @xmath284 by assumptions [ asm : all ] and [ asm : randomwalk ] , it can be verified that the condition for mean convergence continues to be @xmath285 , where @xmath124 is defined in . in addition , it can also be verified that the error recursion converges in the mean sense to the same non - zero bias vector @xmath141 as in . from and under assumption [ asm : smallstepsize ] , we can derive the weighted variance relation : @xmath286\}\nonumber\\ { } & \qquad+{\mathbb{e}}\|{\mc{a}}_2^{\t}(i_{nm}-{\mc{m}}{\bs{\mc{r}}}_i'){\bs{v}}_{i-1}^{(w)}\|_{\sigma}^2\nonumber\\ { } & \qquad+{\mathbb{e}}\|{\mc{a}}_2^{\t}{\mc{m}}{\bs{z}}_i\|_{\sigma}^2+{\mathbb{e}}\|{\bs{v}}_i^{(\psi)}\|_{\sigma}^2\end{aligned}\ ] ] where @xmath164 is given in . if the step - sizes are sufficiently small , then we can assume that the network continues to be mean - square stable . the steady - state performance is affected by the non - stationarity of @xmath16 . from assumption [ asm : smallstepsize ] , at steady - state , expression becomes @xmath287^*(i_{n^2m^2}\!-\!{\mc{f}})^{-1}{\mathrm{vec}}(\omega)\end{aligned}\ ] ] where @xmath197 is given in , @xmath288 in , @xmath199 in , @xmath164 in , and @xmath289 is the covariance matrix of @xmath283 : @xmath290 by , , and , we get @xmath291 then , following the same argument that led to , we find that the network msd is now given by : @xmath292^*(i_{n^2m^2}-{\mc{f}})^{-1}\vec(i_{nm } ) \end{aligned } } \ ] ] similarly , the network emse is given by : @xmath293^*(i_{n^2m^2}-{\mc{f}})^{-1}\vec({\mc{r}}_u ) \end{aligned } } \ ] ] where @xmath212 is defined in . observe that the main difference relative to and is the addition of the term @xmath289 . therefore , all the results that were derived in the earlier section , such as and , continue to hold by adding @xmath289 . in particular , if assumptions [ asm : smallstepsize ] and [ asm : nodatasharing ] are adopted , expressions and can be approximated as @xmath294^*(i_{n^2m^2}-{\mc{f}})^{-1}\vec(i_{nm } ) \end{aligned } } \ ] ] and @xmath295^*(i_{n^2m^2}-{\mc{f}})^{-1}\vec({\mc{r}}_u ) \end{aligned } } \ ] ] where @xmath198 is now given in . we simulate two scenarios : noisy information exchanges and non - stationary environments . we consider a connected network with @xmath296 nodes . the network topology is shown in fig . [ fig : topology ] . the unknown complex parameter @xmath16 of length @xmath297 is randomly generated ; its value is @xmath298 $ ] . we adopt uniform step - sizes , @xmath299 , and uniformly white gaussian regression data with covariance matrices @xmath300 , where @xmath301 are shown in fig . [ fig : regressor ] . the variances of the model noises , @xmath302 , are randomly generated and shown in fig . [ fig : modelnoise ] . we also use white gaussian link noise signals such that @xmath303 , @xmath304 , and @xmath305 . all link noise variances , @xmath306 , are randomly generated and illustrated in fig . [ fig : linknoise ] from top to bottom . we assign the link number by the following procedure . we denote the link from node @xmath26 to node @xmath11 as @xmath307 , where @xmath308 . then , we collect the links @xmath309 in an ascending order of @xmath26 in the list @xmath310 ( which is a set with _ ordered _ elements ) for each node @xmath11 . for example , for node @xmath311 in fig . [ fig : topology ] , it has @xmath312 links ; the ordered links are then collected in @xmath313 . we concatenate @xmath314 in an ascending order of @xmath11 to get the overall list @xmath315 . eventually , the @xmath316th link in the network is given by the @xmath316th element in the list @xmath317 . nodes.,height=172 ] we examine the simplified cta and atc algorithms in and , namely , no sharing of data among nodes ( i.e. , @xmath209 ) , under various combination rules : ( i ) the relative variance rule in , ( ii ) the metropolis rule in @xcite : @xmath318 where @xmath319 denotes the degree of node @xmath11 ( including the node itself ) , ( iii ) the uniform weighting rule : @xmath320 and ( iv ) the adaptive rule in with @xmath321 . we plot the network msd and emse learning curves for atc algorithms in figs . [ fig : msd_atc ] and [ fig : emse_atc ] by averaging over 50 experiments . for cta algorithms , we plot their network msd and emse learning curves in figs . [ fig : msd_cta ] and [ fig : emse_cta ] also by averaging over 50 experiments . moreover , we also plot their theoretical results and in the same figures . from fig . [ fig : sim2 ] we see that the relative variance rule makes diffusion algorithms achieve the lowest msd and emse levels at steady - state , compared to the metropolis and uniform rules as well as the algorithm from @xcite ( which also requires knowledge of the noise variances ) . in addition , the adaptive rule attains msd and emse levels that are only slightly larger than those of the relative variance rule , although , as expected , it converges slower due to the additional learning step . the value for each entry of the complex parameter @xmath322 is assumed to be changing over time along a circular trajectory in the complex plane , as shown in fig . [ fig : track ] . the dynamic model for @xmath323 is expressed as @xmath324 , where @xmath325 , @xmath326 , and @xmath327 . the covariance matrices @xmath328 are randomly generated such that @xmath329 when @xmath330 , but their traces are normalized to be one , i.e. , @xmath331 , for all nodes . the variances for the model noises , @xmath332 , are also randomly generated . we examine two different scenarios : the low noise - level case where the average noise variance across the network is @xmath333 db and the noise variances are shown in fig . [ fig : hsnr ] ; and the high noise - level case where the average variance is @xmath334 db and the variances are shown in fig . [ fig : lsnr ] . we simulate 3000 iterations and average over 20 experiments in figs . [ fig : track_hsnr ] and [ fig : track_lsnr ] for each case . the step - size is 0.01 and uniform across the network . for simplicity , we adopt the simplified atc algorithm where @xmath209 , and only use the uniform weighting rule . the tracking behavior of the network , denoted as @xmath335 , is obtained by averaging over all the estimates , @xmath336 , across the network . figs . [ fig : track_hsnr ] and [ fig : track_lsnr ] depict the complex plane ; the horizontal axis is the real axis and the vertical axis is the imaginary axis . therefore , for every time @xmath34 , each entry of @xmath323 or @xmath337 represents a point in the plane . when @xmath34 is increasing , @xmath338 moves along the red trajectory ( in @xmath339 ) , @xmath340 along the blue trajectory ( in @xmath341 ) , @xmath342 along the green trajectory ( in @xmath343 ) , and @xmath344 along the magenta trajectory ( in @xmath345 ) . from fig . [ fig : track ] , it can be seen that diffusion algorithms exhibit the tracking ability in both high and low noise - level environments . in this work we investigated the performance of diffusion algorithms under several sources of noise during information exchange and under non - stationary environments . we first showed that , on one hand , the link noise over the regression data biases the estimators and deteriorates the conditions for mean and mean - square convergence . on the other hand , diffusion strategies can still stabilize the mean and mean - square convergence of the network with noisy information exchange . we derived analytical expressions for the network msd and emse and used these expressions to motivate the choice of combination weights that help ameliorate the effect of information - exchange noise and improve network performance . we also extended the results to the non - stationary scenario where the unknown parameter @xmath16 is changing over time . simulation results illustrate the theoretical findings and how well they match with theory . following @xcite , we first define the block maximum norm of a vector . [ def : vecblkmaxnorm ] given a vector @xmath347 consisting of @xmath10 blocks @xmath348 , the block maximum norm is the real function @xmath349 , defined as @xmath350 where @xmath351 denotes the standard @xmath136-norm on @xmath352 . similarly , we define the matrix norm that is induced by the block maximum norm as follows : [ def : blkmaxnorm ] given a block matrix @xmath353 with block size @xmath43 , then @xmath354 denotes the induced block maximum ( matrix ) norm on @xmath355 . [ lemma : blkunitaryinvariant ] the block maximum matrix norm is block unitary invariant , i.e. , given a block diagonal unitary matrix @xmath356 consisting of @xmath10 unitary blocks @xmath357 , where @xmath358 , for any matrix @xmath353 , then @xmath359 where @xmath360 denotes the block maximum matrix norm on @xmath355 with block size @xmath43 . [ lemma : rghtstocmat ] let @xmath361 be a right - stochastic matrix . then , for block size @xmath43 , @xmath362 from definition [ def : blkmaxnorm ] , we get @xmath363_{lk}x_k\|_2}{\max_{k}\|x_k\|_2}\nonumber\\ { } & \le\max_{x\in{\mathbb{c}}^{mn}\backslash\{0\ } } \frac{\max_{l}\sum_{k=1}^{n}[a]_{lk}\|x_k\|_2}{\max_{k}\|x_k\|_2}\nonumber\\ { } & \le\max_{x\in{\mathbb{c}}^{mn}\backslash\{0\ } } \frac{\max_{l}(\sum_{k=1}^{n}[a]_{lk})\cdot\max_k\|x_k\|_2}{\max_{k}\|x_k\|_2}\nonumber\\ { } & \le\max_{x\in{\mathbb{c}}^{mn}\backslash\{0\}}\frac{\max_{l}1\cdot\max_{k}\|x_k\|_2}{\max_{k}\|x_k\|_2}\nonumber\\ { } & = 1\end{aligned}\ ] ] where @xmath364 consists of @xmath10 blocks @xmath348 , and @xmath365_{lk}$ ] denotes the @xmath366th entry of @xmath240 . on the other hand , for any induced matrix norm , say , the block maximum norm , it is always lower bounded by the spectral radius of the matrix @xcite : @xmath367 combining and completes the proof . [ lemma : blockdiagonal ] let @xmath368 be a block diagonal hermitian matrix with block size @xmath43 . then the block maximum norm of the matrix @xmath251 is equal to its spectral radius , i.e. , @xmath369 denote the @xmath11th @xmath43 submatrix on the diagonal of @xmath251 by @xmath370 . let @xmath371 be the eigen - decomposition of @xmath370 , where @xmath372 is unitary and @xmath373 is diagonal . define the block unitary matrix @xmath374 and the diagonal matrix @xmath375 . then , @xmath376 . by lemma [ lemma : blkunitaryinvariant ] , the block maximum norm of @xmath251 with block size @xmath43 is @xmath377 where we used the fact that the induced @xmath136-norm is identical to the spectral radius for hermitian matrices @xcite . on the other hand , any matrix norm is lower bounded by the spectral radius @xcite , i.e. , @xmath378 combining and completes the proof . now we show that the matrix @xmath379 is stable if @xmath128 is stable . for any induced matrix norm , say , the block maximum norm with block size @xmath43 , we have @xcite @xmath380 where , from and , @xmath126 and @xmath127 satisfy lemma [ lemma : rghtstocmat ] . by and , it is straightforward to see that @xmath128 is block diagonal with block size @xmath43 . then , by lemma [ lemma : blockdiagonal ] , expression can be further expressed as @xmath381 which completes the proof . in appendix i of @xcite and the matrix @xmath180 in lemma 2 of @xcite are block diagonal , the @xmath382 norm used in these references should simply be replaced by the @xmath360 norm used here and as already done in @xcite . ] let us denote the @xmath366th submatrix of @xmath383 by @xmath384 . by assumptions [ asm : all ] and expression , @xmath385 can be evaluated as @xmath386 where , by expressions and , @xmath387 when @xmath388 , expression reduces to @xmath389 when @xmath390 , expression becomes @xmath391 where @xmath392 denotes the kronecker delta function . evaluating the last term on rhs of requires knowledge of the excess kurtosis of @xmath393 , which is generally not available . in order to proceed , we invoke a separation principle to approximate it as @xmath394 substituting into leads to @xmath395\end{aligned}\ ] ] from and , we get @xmath396\end{aligned}\ ] ] substituting into , we obtain @xmath397\end{aligned}\ ] ] from and , we arrive at expression . l. li and j. a. chambers , `` distributed adaptive estimation based on the apa algorithm over diffusion netowrks with changing topology , '' in _ proc . ieee workshop stat . signal process . 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adaptive networks rely on in - network and collaborative processing among distributed agents to deliver enhanced performance in estimation and inference tasks . information is exchanged among the nodes , usually over noisy links . the combination weights that are used by the nodes to fuse information from their neighbors play a critical role in influencing the adaptation and tracking abilities of the network . this paper first investigates the mean - square performance of general adaptive diffusion algorithms in the presence of various sources of imperfect information exchanges , quantization errors , and model non - stationarities . among other results , the analysis reveals that link noise over the regression data modifies the dynamics of the network evolution in a distinct way , and leads to biased estimates in steady - state . the analysis also reveals how the network mean - square performance is dependent on the combination weights . we use these observations to show how the combination weights can be optimized and adapted . simulation results illustrate the theoretical findings and match well with theory . diffusion adaptation , adaptive networks , imperfect information exchange , tracking behavior , diffusion lms , combination weights , energy conservation .

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Proceed to summarize the following text: recent advances in studying the world - volume theory of m2-branes @xcite have allowed new perspectives on the ads / cft correspondence . indeed , the three - dimensional theory on the m2-branes is now understood as an ordinary gauge theory of chern - simons ( cs ) type @xcite . the investigation of ads@xmath3/cft@xmath4 subsequently took flight , ever augmenting our arsenal of dual pairs of theories . one strand of development has been the extension of the elaborate structure of the ads@xmath5/cft@xmath3 situation wherein the d3-brane world - volume gauge theory and the corresponding calabi - yau cone over sasaki - einstein 5-folds have been probed in great detail over the past decade . of particular interest had been the cases where the calabi - yau threefold admits toric description . there , a rich tapestry , enabled by the abundant techniques of toric geometry , had been woven over such themes as toric duality @xcite , dimer models and brane tilings @xcite . along this vein , the parallel story for m2-branes probing toric calabi - yau 4-fold singularities met with rapid progress @xcite , addressing such issues as tiling or its 3-dimensional counter - part of crystal models , toric duality , as well as partition functions in the light of the plethystic programme @xcite etc . thus inspired , especially by the fortitude to attack the general singularity @xcite , it is expedient for us to take a synthetic approach . the space of toric singularities of calabi - yau 4-folds is largely unchartered ; nevertheless , we can gradually and systematically list , starting from the simplest imaginable , the possible @xmath6-dimensional quiver chern - simons theories . these theories are , of course , infinite in number ; however , we will see how the number of nodes in the quiver and the number of terms in the superpotential can be used as order parametres to begin a classification . even at low numbers , we will encounter many highly non - trivial theories . furthermore , we will establish generating functions which counts the number of inequivalent theories at each level of our enumeration . remarkably , we can geometrically interpret these functions . thus , let us we embark on a journey through the complex _ terra incognita _ of new graphs and associated superpotentials , guided by our experience with the techniques of @xmath7-dimensional @xmath8 gauge theory , _ mutatis mutandis _ , and slowly probe the dual pairs of toric geometries and cs theories . a wealth of interesting properties will be encountered . our taxonomic study is organised as follows . we begin , in section [ s : mcs ] , with a brief review of the computation of the moduli space of quiver chern - simons theories in @xmath6-dimensions , culminating in an algorithmic flow - chart which generalises ( and encompasses ) the `` forward - algorithm '' of @xcite to the present case . next , in section [ s : tax ] , we begin our systematic investigation of what toric chern - simons quiver theories can exist . we start with two terms in the superpotential satisfying the toric condition and exhaustively present what possibilities could arise , using the number of nodes @xmath9 and the number of fields @xmath10 as order parametres . we then explicitly show how the moduli spaces of these theories are families of toric calabi - yau 4-folds , indexed by the chern - simons levels . we find a new manifestation of toric duality . in section [ s : dimer ] , we show how all these theories admit a dimer - model , or periodic - planar - tiling description , contrary to what one would naively expect . finally , stepping back to have a bird s eye view , we find a generating function in section [ s : count ] which counts all the theories encountered . we conclude with prospect in section [ s : conc ] . in this section , we briefly outline the computation of moduli space of the theories of our interest : @xmath6-dimensional quiver chern - simons theories with @xmath11 supersymmetry . we point out that henceforth we are interested in the classical mesonic moduli space , which we will denote as @xmath12 . the theories are with four supersymmetries and the gauge groups have no kinetic terms but instead have cs terms , moreover , they have product gauge groups together with bifundamental and adjoint matter . the computation of @xmath12 is a direct generalisation of the so - called `` * forward algorithm * '' for @xmath7-dimensional @xmath8 gauge theories @xcite . let the quiver cs theory have gauge group consisting of @xmath9 factors , and a total of @xmath10 fields , we then have the action , written in @xmath11 superspace notation : @xmath13 where @xmath14 indexes the factors in the gauge group , @xmath15 are the superfields accordingly charged , @xmath16 are the gauge multiplets , @xmath17 is the superspace derivative , @xmath18 is the superpotential and @xmath19 are the chern - simons levels which are integers ; an overall trace is implicit since all the fields are matrix - valued . we take the following two constraints on the cs levels , the reader is referred to @xcite for details : @xmath20 the classical mesonic moduli space @xmath12 is determined by the following equations @xmath21 & = & 4k_a\sigma_a \\ \label{df } \sigma_a x_{ab } - x_{ab } \sigma_b & = & 0 \ , \end{aligned}\ ] ] where @xmath22 is the scalar component of @xmath16 . indeed , this is in analogy to the f - term and d - term equations of @xmath8 gauge theories in @xmath7-dimensions , with the last equation being a new addition . we are particularly interested in the case when @xmath12 is a toric variety where the forward algorithm conveniently uses the combinatorial power of lattice geometry and toric cones @xcite . this is the abelian case where the gauge group is simply @xmath23 . physically , @xmath12 is a toric calabi - yau 4-fold transverse to an m2-brane on whose world - volume lives the @xmath6-dimensional cs theory . for a stack of @xmath24 parallel , coincident m - branes , the moduli space is the @xmath24-th symmetrised product of ( or more precisely the @xmath24-th hilbert scheme of points on ) the 4-fold @xcite . in this abelian , toric case then , the third equation of ( [ df ] ) sets all @xmath22 to a single field , say @xmath25 , on the coherent component of the moduli space . the second equation causes the d - terms to have fi - parametres @xmath26 . the moduli space @xmath12 is a symplectic quotient of the space of solutions to the f - terms prescribed by the first equation modulo the gauge conditions prescribed by the d - terms . because of the condition that all @xmath19 sum to 0 imposed in ( [ k - con ] ) there is an overall @xmath27 ( corresponding to the center of mass motion of the m2-brane ) which can be factored out . furthermore , there is another @xmath27 which can be factored . this is because the presence of cs couplings induces fayet - iliopoulos ( fi ) parameters on the space of d - terms . in a generic @xmath7-dimensional theory all these fi parameters are arbitrary , but here they are all aligned along a line which is parameterized by @xmath25 and has a direction set by the cs integers . this picks out a very specific baryonic direction out of all possible directions which are present in @xmath7-dimensions . this direction becomes mesonic in @xmath6-dimensions and fibers over the calabi - yau 3-fold to give a total space as a calabi - yau 4-fold . thus , in all , there is a net of @xmath28 d - terms . the space of solutions of the f - terms is itself a toric variety , of dimension @xmath29 , this is the so - called `` * master space * '' @xmath30 , studied in detail in @xcite . indeed , the @xmath31 d - terms are all the directions which remain baryonic in @xmath6-dimensions and give rise to the master space baryonic directions for a given lagrangian with @xmath9 gauge groups . this does not imply that all possible baryonic directions of the particular cy4 are given by these @xmath31 directions . it only provides a lower bound . there are at least @xmath31 such baryonic directions and a different formulation may give more than this number . such a situation is evident from the study of models presented in @xcite . ] . since we are interested in the mesonic moduli space , we impose gauge invariance with respect to all of them . in summary , @xmath32 where we have marked the complex dimensions explicitly as subscripts . the @xmath31 fi - parametres are shown in @xcite to be in the integer kernel of the matrix @xmath33 to proceed we now recall the essential features in the computation of the 3 + 1 dimensional mesonic moduli space of complex dimension 3 . computationally , the charges of the fields are given by the so - called * incidence matrix * of the quiver ; this is a @xmath34 matrix . each column of this matrix representation of the quiver consists of two possible choices ( 1 ) a single pair @xmath35 and @xmath36 denoting an arrow beginning and ending at the appropriate node and zero elsewhere , or ( 2 ) an entire column of zeros , denoting an adjoint field charged only under one single node . the incidence matrix is customarily denoted as @xmath37 and specifies the d - terms , i.e. , the @xmath27 toric actions . the crucial property of @xmath37 is that each of its columns sums to 0 ( since each column corresponds to one arrow that necessarily has one head and one tail . ) and that each of its rows also sums to 0 ( this ultimately ensures that the moduli space be calabi - yau ) . we emphasise the ambiguity for an adjoint : we do not know under which precise node it is charged ; from the point of view of the incidence matrix , there is no way to distinguish . we will see later how using information from the superpotential one may overcome this ambiguity . of course , one row of @xmath37 is redundant because of the summation rule , we delete it and customarily call the result @xmath38 . on the other hand , the f - terms can be solved in terms of a matrix @xmath39 , whose dual cone we call @xmath40 . from these we can extract so - called @xmath41 and @xmath42 matrices . we refer the reader to section 2 of @xcite for the details . schematically we can summarise the procedure of the `` forward algorithm '' , taken from _ cit . as follows : @xmath43^t & & \longrightarrow & & ( q_t)_{(c-3 ) \times c } = \left ( \begin{array}{c } ( vu)_{(g-1)\times c } \\ q_{(c - g-2 ) \times c } \\ \end{array } \right ) & \end{array}\ ] ] we have marked the dimensions of each matrix for clarity . note that @xmath44 is the number of perfect matching in the dimer model @xcite description of the theory to which we shall later turn . the key point is that we can combine the matter content ( specified by the incidence matrix ) and the superpotential ( specified by the @xmath39-matrix ) into a single charge matrix @xmath45 ; its kernel , @xmath46 , of dimension @xmath47 , encodes the toric diagram of the calabi - yau threefold moduli space . refreshed with this recollection let us make one more step before getting back to the main case of interest . having outlined the computational procedure , let us now proceed with a systematic study of examples . ideally , one would wish for a classification of all possible theories , their quiver diagrams , interactions and subsequent geometries . this is a daunting task of organising an infinite number of models . nevertheless , we can proceed cautiously and modestly : let us start with theories with a single pair of black - white nodes in the dimer picture . these correspond to cases where the superpotential vanishes for a single brane ; this is because the so - called * toric condition * @xcite requires that each field appears exactly twice with opposite signs . in terms of the dimer model @xcite , this condition is what gives rise to the bi - partite nature of the tiling . of course , for multiple branes , because the fields become matrix - valued and do not commute necessarily , the superpotential may no longer vanish . this is familiar to us . for example , in the 3 + 1 dimensional gauge theory for the conifold 3-fold singularity there is a quartic superpotential which vanishes for a single brane . we will encounter this situation in detail below . hence , with a single m2 brane , and with only two terms therein , the superpotential actually vanishes , subsequently , the master space is freely generated by the @xmath10 fields and is simply @xmath57 . thus , the matter content completely specifies the 4-fold singularity . the symplectic quotient ( [ symp ] ) and the traditional approach both become relatively simple in this case . in fact , in ( [ gt ] ) , @xmath58 and the perfect matching matrix @xmath54 is just the identity matrix . the @xmath53 matrix for the f - terms are not present and we have that : @xmath59 where @xmath60 means @xmath12 has the toric diagram given by @xmath46 of dimensions @xmath61 . indeed , from ( [ symp ] ) , we see that @xmath62 ; let us hence use @xmath10 as a single order parametre and proceed gradually . we initiate with the case of two nodes , i.e. , @xmath63 . for each incremental value of @xmath10 , we classify all @xmath37-matrices satisfying the constraints which define the incidence matrix . indeed , in ( [ symp2 ] ) , @xmath64 has dimensions @xmath65 and its kernel @xmath46 will have columns of length 4 , as is required for a toric diagram of a 4-fold . in summary , our classification scheme for 2 terms in the superpotential proceeds as follows : 1 . fix @xmath9 , the number of nodes in the quiver . this also fixes the number of arrows as @xmath62 ; 2 . find all @xmath66 matrices which are incidence matrices , i.e. , ( a ) each column is one of only zeros ( adjoint ) or consists of a single pair of @xmath35 and @xmath36 and zero otherwise ( bi - fundamental ) , ( b ) each row sums to 0 ( calabi - yau condition ) ; 3 . identify all loops ( gauge invariant operators ) in the quiver drawn from each incidence matrix and construct possible 2-term superpotentials satisfying ( a ) the toric condition , i.e. , each bi - fundamental field occurs exactly twice and with opposite sign and ( b ) for a single brane when all the fields reduce from matrix - valued to complex numbers , the 2 terms conspire to cancel . how do we attack step 2 , the most computationally intense one ? luckily , the algorithms for such a matrix - partitioning problem were implemented in @xcite and we can thus happily proceed with presenting the solutions . we adhere to the standard notation that _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ for bi - fundamentals @xmath67 denotes the @xmath68-th arrow from node @xmath14 to @xmath69 and similarly that @xmath70 denotes the @xmath68-th adjoint on node @xmath14 ( when there is only a single arrow the @xmath68-index is dropped ) . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ already , there are many highly non - trivial theories . here , there are 2 nodes and we find only two non - isomorphic solutions . these are presented in figure [ f : e=4 ] in the appendix . there are two possibilities . in model ( 1 ) we see that this is a non - chiral theory with 4 bi - fundamentals . we recognise the quiver as that of the conifold quiver . indeed , the full superpotential for arbitrary @xmath24 of that theory also has 2 terms : calling the bi - fundamentals @xmath71 and @xmath72 , ( @xmath73 and @xmath74 matrices ) we have that @xmath75 , which indeed satisfies the toric condition and vanishes for @xmath76 . the fact that we have cs constraints , of course , modifies the moduli space from the 3-dimensional conifold to a cy 4-fold . in model ( 2 ) , there are 2 adjoint fields and 2 bi - fundamentals . where shall we place the 2 non - bi - fundamental fields in model ( 2 ) ? we can be guided by promoting to @xmath77 number of branes . there could be two possible placements : both as adjoints on 1 node or one on each node . calling the two bi - fundamentals @xmath78 and @xmath79 , for both adjoints on the same node , say node 2 without loss of generality , we have 2 gauge invariant terms : @xmath80 and @xmath81 . on the other hand , for one adjoint on each node we have only one possible invariant : @xmath82 ; this is because we need to be careful in matrix composition when we go about the loops in the quiver to construct the gauge invariants . therefore , model ( 2 ) has a natural candidate for a 2-term superpotential satisfying the toric condition and having two terms , namely @xmath83x_{21})$ ] . in summary , the two full models are drawn in figure [ f : e=4full ] . = 10 cm @xmath84 ; @xmath85 @xmath86x_{21})$ ] of course , the actual moduli space @xmath12 for both models are easily determined using ( [ symp2 ] ) . here @xmath87 and there is no symplectic action . thus the moduli space is simply the master space , viz . , @xmath88 . the toric diagram is merely 4 corners of a tetrahedron . we are assured by the fact that these two models have appeared in the literature , model ( 1 ) , in @xcite and model ( 2 ) , in @xcite , in each case from different lines of reasoning . here we have arrived at these from yet another viewpoint a systematic scan of theories . it is perhaps important to emphasize that model ( 2 ) is not considered to be a consistent model in 3 + 1 dimensions even though it does admit a two dimensional tiling @xcite ( here consistent is taken to mean that it leads to a 3 + 1 dimensional scft under the rg flow to the ir , with a known ads dual ) . the essential feature of model ( 2 ) is node 2 which has 1 flavor ( or taking the sqcd conventions of the number of flavors @xmath89 equal to the number of colors ) . henceforth we shall call such nodes as `` * one - flavored nodes * '' . indeed it is a standard lore that such theories develop a scale and confine in 3 + 1 dimensions , thus not leading to scft . on the other hand , such theories in 2 + 1 dimensions do not necessarily develop a scale @xcite and can perfectly lead to a non - trivial scft under the rg flow . once we allow for theories with one - flavored nodes , a whole space of opportunities reveals itself and a zoo of new scft s in 2 + 1 dimensions becomes available . we encounter more examples of this type in the following sections . here , there are 3 nodes and we find 5 distinct solutions . these are shown in figure [ f : e=5 ] in the appendix . model ( 1 ) has 5 bi - fundamentals . models ( 2 ) and ( 3 ) have 4 bi - fundamentals and 1 adjoint field . note that model ( 2 ) has a disconnected node labelled 2 . model ( 4 ) has 3 bi - fundamentals and 2 adjoint fields and finally model ( 5 ) , also with a disconnected node , has only 2 bi - fundamentals and 3 adjoint fields . let us ignore models ( 2 ) and ( 4 ) since they inevitably have reducible moduli spaces due to disconnected nodes . for model ( 1 ) , using the standard notation , we clearly have the 2-term superpotential : @xmath90 , which satisfies the toric condition and vanishes for @xmath76 when all fields are simply complex numbers . for model ( 3 ) , clearly there are two invariants @xmath91 and @xmath92 . where shall we place the missing adjoint @xmath93 ? it is easy to see that placing it on node 2 gives two invariants once we promote to @xmath77 branes , i.e. , to matrix - valued fields : @xmath94 and @xmath95 . this indeed gives a two - term superpotential satisfying the toric condition of each field appearing exactly twice with opposite signs : @xmath96 ; moreover , @xmath18 indeed vanishes at @xmath76 when all field are merely complex numbers . finally , for model ( 4 ) , there are 2 ways of placing the 2 adjoints : both on the same node , say node 1 , or one each on 2 of the nodes . the latter possibility is ruled out since it would be impossible to have 2 terms and satisfying the toric condition . however , putting both @xmath97 on node 1 , gives us a superpotential satisfying all requisites : @xmath98x_{13}x_{32})$ ] . we summarise all these good models in figure [ f : e=5full ] . it is important to note that none of these 3 models appeared previously in the literature . indeed , our systematic study revealed the existence of these 3 prototypical models . we will proceed later by analyzing their properties and further introduce them from yet another taxonomic viewpoint . it is further important to notice that all these models have a one - flavored node and therefore do not correspond to consistent models in 3 + 1 dimensions even though they all admit a brane tiling . nevertheless they present a rich structure of non - trivial scft s in 2 + 1 dimensions with a highly intricate spectrum of scaling dimensions . = 14 cm @xmath99x_{13}x_{32 } ) . \ \\ \end{array}$ ] here , there are 4 nodes and there is a total of 18 distinct solutions . these are shown in figure [ f : e=6 ] in the appendix . models ( 1 ) to ( 10 ) have 6 bi - fundamentals . note that models ( 1 ) , ( 2 ) , ( 5 ) and ( 9 ) all have disconnected nodes . model ( 8) , though apparently acceptable , has a superpotential in terms of its bi - fundamentals which looks like @xmath100 . this vanishes by cyclicity of the trace and hence it is really just a theory without superpotential at all . models ( 11 ) and ( 12 ) have 5 bi - fundamentals and 1 adjoint field . models ( 13 ) to ( 16 ) have 4 bi - fundamentals and 2 adjoint fields . note that models ( 13 ) and ( 14 ) both have disconnected nodes . model ( 17 ) has 3 bi - fundamentals and 3 adjoint fields , as well as a disconnected node . finally , model ( 18 ) has 2 bi - fundamentals , 4 adjoint fields and 2 disconnected nodes . once again , we select the diagrams without detached nodes , insert the appropriate adjoint fields , and also write down the possible 2-term superpotentials . we find a total of 6 possible models . these are presented in figure [ f : e=6full ] . = 14 cm @xmath101 ) \end{array } $ ] happily , we have again recovered and extended some of the known models in the literature . model ( 6 ) was proposed in figure 3 of @xcite for the @xmath102 geometry while model ( 10 ) was proposed in figure 7 of @xcite for the so - called @xmath103 theory . the other models appear for the first time in the literature . all models have a one - flavored node and so none correspond to consistent models in 3 + 1 dimensions even though they all admit a brane tiling description . for completeness we also include the two disconnected quivers , models ( 3 ) and ( 15 ) . these engender multi - trace superpotentials because of the quivers factorise . we present these in figure [ f : e=6disconnect ] . we see that they are essentially products of models ( 1 ) and ( 2 ) of the 4-edged case , together with a 2-edged bi - fundamental non - chiral quiver . also , we include model ( 8) , which has a completely vanishing superpotential if it were to be 2-termed . we will exclude these models from discussion in the following . = 14 cm @xmath104x_{41 } ) \ ; \\ w_{(8 ) } = { \mathop{\rm tr}}(x_{31}x_{14}x_{42}x_{24}x_{41}x_{13 } - x_{42}x_{24}x_{41}x_{13}x_{31}x_{14 } ) = 0 \ ; \\ \end{array } $ ] let us take , for concreteness , model ( 1 ) of the @xmath105 quivers , given in figure [ f : e=5full ] . there are 3 nodes and hence 2 independent cs levels , @xmath106 and @xmath107 . the toric diagram , according to ( [ gt ] ) , is given by @xmath108 in the penultimate step , we have been mindful of the fact we need to find the integer kernel of the charge matrix and not merely the nullspace and hence we wrote gen ( ) therein to denote that we should reduce to a basis over the integers for whichever choice of @xmath106 and @xmath107 . this means that each row of @xmath46 must have gcd being 1 . guaranteed by the condition @xmath109 , rows 2 and 3 are acceptable but rows 1 and 4 need to divide out the common factor of @xmath107 ; this gives the last step . indeed , each column of @xmath46 is of length 4 , signifying that the resulting moduli space corresponding to this theory is a toric 4-fold . furthermore , we see that the vector @xmath110 is perpendicular to every pair - wise linear combination between the 5 column vectors , this means that the columns are actually co - spatial , i.e. , live on a dimension 3 hypersurface in @xmath111 ; this , of course , guarantees that our 4-fold is in fact calabi - yau . thus re - assured , we can , without loss of generality , delete any row of @xmath46 ( call it @xmath112 ) and represent the toric 4-fold by an integer polytope in 3-dimensions . thus , we can write , for the toric diagram , @xmath113 where for convenience we have deleted the second row . now , depending on the choice of @xmath106 and @xmath107 obeying the coprimarity condition , we have an infinite family of toric cy4s . as two illustrious examples we have that @xmath114 we see that the two are not related by any @xmath115 transformations and are thus inequivalent toric varieties . we draw these diagrams explicitly in figure [ f : torice=5 ] . = 12 cm the other two models of the @xmath105 quiver theories can be similarly treated . for model ( 3 ) , we have that @xmath116 this , as above , gives an infinite family , parametrised by choices of @xmath106 and @xmath107 , of possibilities for the moduli space @xmath12 and hence inequivalent theories . finally , for model ( 4 ) , we have @xmath117 [ [ an - interesting - family ] ] an interesting family : + + + + + + + + + + + + + + + + + + + + + + + examining model ( 1 ) of figure [ f : e=5full ] and model ( 4 ) of figure [ f : e=6full ] , it is clear that for the general case of @xmath9 nodes and @xmath118 fields , there will always be a cyclically - directed @xmath9-gon graph with 1 edge having an extra pair of arrows in opposite directions . thus all @xmath10 fields are bi - fundamentals and let us , without loss of generality , place the extra pair of bi - fundamentals between nodes 1 and @xmath9 , and let the direction of the arrows be @xmath119 , @xmath120 , , @xmath121 and @xmath122 . the quiver and adjacency matrices are @xmath123 the 2-term superpotential is also straight - forward to write down : @xmath124 we can also apply ( [ gt ] ) to the incidence matrix to find the toric diagram . the result has @xmath118 lattice points : @xmath125 [ [ six - field - models ] ] six field models : + + + + + + + + + + + + + + + + + + for completeness , let us present the matrices @xmath46 encoding the toric diagrams for the 6 models at @xmath126 , drawn in figure [ f : e=6full ] . they are , respectively , @xmath127 in its original guise , toric duality @xcite referred to the phenomenon of 4-dimensional , @xmath8 gauge theories having the same vacuum moduli space as toric varieties , some classes of these were later realised to be seiberg dualities . in our host of examples above , we have again encountered this duality , now in a more general setting . take the two models of @xmath128 , they , even though having quite different quivers , share the same infrared moduli space as @xmath88 . perhaps more dramatic is the following pair : take model ( 4 ) of @xmath105 at @xmath129 and inspect ( [ gt5 - 4 ] ) , then take model ( 16 ) of @xmath126 at @xmath130 and inspect ( [ gt6 ] ) , we see that they are @xmath131 we see that if we removed a repeated column in the latter , which does not influence the toric description , the moduli spaces of the two theories , which have different number of gauge group factors , different matter content and interactions and different chern - simons levels , are identical as toric varieties . clearly there are infinitely many such cases and it is interesting to study the systematics of this phenomenon . it is now well - established that the most convenient and elegant way of encoding the @xmath7-dimensional quiver gauge theory of d3-branes probing toric calabi - yau threefold singularities is through the formalism of dimer models , or , equivalently , brane - tilings @xcite . the afore - mentioned `` toric condition '' @xcite of the superpotential is naturally interpreted as the bi - partite ( 2-colour ) property of the tiling while the calabi - yau condition of the threefold , which compels the toric diagram to be planar , gives the inherent structure of periodic tiling of the 2-dimensional plane . one question which has emerged is how this would generalise to toric varieties of higher dimension . indeed , proposals have been made which suggest that a 3-dimensional analogue of the dimer , a so - named `` crystal model '' , should encode the toric calabi - yau 4-fold case @xcite . is this so for the 4-folds we have encountered in our investigation ? surprisingly , we find all our above cs theories to afford 2-dimensional tilings , rather than the naively expected crystals which tile 3-dimensions . let us recall that for the @xmath7-dimensional gauge theories , an important relation exists @xcite : @xmath132 where @xmath133 is the number of terms in the superpotential , while @xmath10 and @xmath9 , as above , are the number of fields and gauge group factors . the recognition of ( [ euler ] ) as the topological euler equation for the simplex decomposition of a genus 1 riemann surface @xmath134 , with a number @xmath133 of vertices , a number @xmath10 of edges and a number @xmath9 of faces was key to the birth of the dimer model . indeed , the graph dual of this simplex is a periodic version of writing the quiver diagram together with the superpotential , the periodicity further supporting the existence of @xmath134 as a torus . how does this crucial relation read for our @xmath6-dimensional cs theories ? here , since we are only considering 2-term superpotentials , @xmath135 . moreover , recall that @xmath62 from ( [ symp ] ) . therefore @xmath136 and ( [ euler ] ) is still satisfied ! this is not what one would expect from a crystal model which is not a periodic tiling of the plane but which is perhaps at first expected for all toric 4-fold theories . in summary , we have that _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ all quiver chern - simons theories corresponding to a m2-brane probing a toric calabi - yau 4-fold , such that the superpotential has two terms , admit a dimer model ( 2d - tiling ) description . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ we could have continued the process of section [ s : tax ] _ ad infinitum _ , listing more and more graphs and then for each , construct possible superpotentials with 2 terms or fix various values of chern - simons levels to obtain infinite families of toric moduli spaces . this , though explicit , is perhaps not so illustrative , let alone computationally prohibitive . it would , however , be most enlightening if we could count , say , the number of possible quivers for a given number of nodes . in this section , let us give a generating function to perform this count ; we will find an elegant result very much in the spirit of the plethystic programme @xcite . to this end , we shall introduce a systematic enumeration and construction of the quivers . let us do so by the concept of _ base node _ which we now introduce . examining model ( 2 ) of figure [ f : e=4full ] , models ( 3 ) and ( 4 ) of figure [ f : e=5full ] , as well as models ( 7 ) , ( 11 ) and ( 16 ) of figure [ f : e=6full ] , we see that they , perhaps not immediately obviously , fall into a family . these are all models which have a single base node , viz . , a single reference node whence loops depart and thence return . let us consider a chain of closed paths beginning and ending on this same node ( corresponding to a possibly multi - trace gauge invariant operator ) and denote it by a sequence of non - negative integers each entry of which encodes the length of one loop . clearly , this sequence is unordered . for example , for the simple quiver which has a single node with 3 self - loops attached ( incidentally , this is the quiver of the @xmath137 super - yang - mills theory for d3-branes in flat @xmath138 paradigmatic in the first ads / cft pair ) , we would denote it as 000 , drawn in figure [ f : c3 ] . = 3 cm next , what would 001 ( and of course any of its permutations ) denote ? this is model ( 2 ) of figure [ f : e=4full ] , which has 2 loops of length 0 and 1 loop of length 1 . note that by length here we mean the number of different nodes traversed before returning . similarly , 011 and 002 correspond to models ( 3 ) and ( 4 ) of figure [ f : e=5full ] respectively . likewise , we retrieve the 3 aforementioned models for @xmath126 fields corresponding to 111 , 012 and 003 . the systematics is thus clear . for @xmath10 fields and @xmath139 nodes , we are counting graphs which are in correspondence with unordered partitioning of @xmath140 into 3 parts of non - negative integers . this is a standard problem , whose solution is simply @xmath141 this function was indeed encountered in @xcite as the generating function for 1 adjoint field with @xmath142 d3 branes . the above is the generating function such that the coefficient in front of @xmath143 gives the number of quivers of 1 base node with @xmath9 nodes . these are all quivers with 3 primitive loops . having addressed one family of quivers transcending across @xmath10 , the family with a single base node , let us move onto 2 base nodes , say 1 and 2 . now , it is more convenient to count by open paths : we let @xmath144 denote a configuration which has open paths of length @xmath145 , @xmath146 , @xmath147 , @xmath148 respectively starting at node 1 , ending at node 2 , starting at node 2 and ending at node 1 . again , by length we mean distance away from the base pair . these are all quivers with 4 primitive paths between the 2 base nodes . specifically , 0000 would correspond to model ( 1 ) of figure [ f : e=4full ] . next , taking the base pair to be nodes 3 and 1 , 0001 would denote model ( 1 ) of figure [ f : e=5full ] . moving on to 4 nodes and 6 fields , we see that 0002 , 0011 and 0101 denote respectively models ( 4 ) , ( 6 ) and ( 10 ) of figure [ f : e=6full ] when taking nodes 4 and 1 and the base pair . this is analogous to the above , but is the unordered partitioning of @xmath31 into 4 non - negative integers with the extra complication that we must respect the di - hedral symmetry . specifically , the transpositions @xmath149 , @xmath150 , @xmath151 generate a dihedral group of order 8 . orbits under these 8 elements must be quotiented out . the generating function for this also admits a standard solution by method of molien series @xcite for the dihedral group , in a 4-dimensional representation acting on our 4-vector : @xmath152 here , the coefficient of @xmath153 is the number of models with @xmath154 nodes . what about triplets or more of base nodes ? note that our above two have already exhausted all the quivers we have so far explicitly constructed . could there be more families which one might encounter at higher @xmath10 ? we now argue that we shall , in fact , not . first , let us recall that our counting procedure above of course does not include any disconnected graphs and moreover excludes shapes such as model ( 8) of figure [ f : e=6disconnect ] . this is a 3-base - node example ; however , we have explicitly shown that the superpotential vanished . indeed , the cases of base nodes being one or two precisely permitted us to write a commutation relation allowing for the two terms in the superpotential : the case of the 1-base - node achieved with the adjoint and the case of the 2-base - node , with the pair of bi - fundamentals between them . any other case must either be reducible to these two situations or have vanishing superpotential . indeed , we can use an additional symmetry to restrict the possible models . consider the number of flavours @xmath155 for a given node @xmath68 : this is either an incoming - outgoing pair of arrows or an adjoint . the vector @xmath155 for @xmath156 is an additional order parameter . from our examples , we see that it is 1 for most nodes . this is the one - flavoured node we discussed earlier . however , it must be at least 1 for every node since we are not counting disconnected quivers and it must never exceed 3 since there are only @xmath118 number of arrows in total . clearly , the sum of @xmath155 over @xmath68 is equal to @xmath118 . therefore , there are only 2 possibilities for the components of the vector : ( a ) @xmath140 of them being 1 and a single 3 or ( b ) @xmath31 of them being 1 and 2 of them being 2 . subsequently , this places a significant restriction on the total number of adjoints . since each node must have at least one bi - fundamental , case ( a ) could have up to 2 adjoints on the last node and case ( b ) could have up to a single adjoint on the last two nodes . hence , in total there could really only be 0 , 1 or 2 adjoints . now , we see that any node which has no adjoint and only an incoming / outgoing pair of arrows is a descendent of a simpler configuration : namely replace this node , together with its 2 attached arrows , with a single arrow between the 2 nodes from which the said node emanates . therefore , we need only consider the parent quivers of ( a ) and ( b ) , viz . , a single node with flavour number 3 or two nodes each with flavour number 2 , respectively . the former is 000 . the latter has two possibilities : 0000 and a quiver which looks like @xmath157 ; we can easily show that the latter admits no non - vanishing 2-term superpotential . indeed , model ( 8) of figure [ f : e=6disconnect ] discussed above is a descendent thereof and also needs to be eliminated . in summary , we only need to consider descendents of the 3-vector 000 and the 4-vector 0000 and no quivers with more than 2 base nodes survive . hence , @xmath158 and @xmath159 together exhaust all counting . furthermore , noticing the shift in power of @xmath159 with respect to @xmath158 , we conclude that : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ the number of theories of our interest , i.e. , all non - trivial , connected models with 2 terms in a non - vanishing superpotential and @xmath154 nodes , is counted by the coefficient of @xmath160 in the expansion of the generating function _ @xmath161 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ the forms of @xmath158 and @xmath159 are perhaps familiar to the astute reader . the first , is the hilbert series ( cf . @xcite ) for the weighted projective plane @xmath162}$ ] ( or , alternatively the quotient @xmath163 of the symmetric group on 3 objects ) , and the second , that of the 4-fold quotient singularity @xmath164 , i.e. , the orbifold of @xmath88 by the dihedral group of order 8 ( cf . @xcite for discrete subgroups of @xmath165 ) . it is quite elegant that these two geometrical spaces should encode the entire space of quivers of our type . in fact , quivers of the first family , the single - base - node type , obey a simple rule of composition , reflecting the fact that there is an underlying commutative algebra which is freely generated : the zero element is the self - adjoining loop , 1 is the quiver with 2 nodes and a pair of bi - fundamentals in opposite directions between them , and thus generalising to @xmath166 , which is the quiver that is an @xmath166-gon with bi - fundamentals cyclically going around once . we can create a quiver by selecting any 3 from this list and composition is by pasting the three at a chosen common node : @xmath167 in terms of the 3-vector of integers , we can also see this composition . for example , 001 + 011 = 012 refers to the following pasting of quivers @xmath168 indeed , we have drawn the result in a suggestive form , so as to indicate that such a composition corresponds to the insertion of nodes . specifically , adding @xmath169 signifies inserting , after choosing an absolute ordering of the loops , @xmath145 , @xmath148 and @xmath170 nodes into the loops . in this fashion , we readily see that the quivers @xmath171 , @xmath172 and @xmath173 generate , by this insertion procedure , all quivers with one base node . thus , we have an algebra freely generated by the three elements : @xmath174 the second family is a little more involved . we can define the 0 element as a node with an out - going arrow , 1 as a node with an incoming and an out - going arrow , etc . : @xmath175 composition is by again pasting , though this time we must first paste to create a pair of base nodes , unto which we may then attach the open arrows of the element @xmath166 . finally , we must eliminate the redundancies of quivers which obey dihedral symmetry . in terms of the 4-vector , we now have an algebra generated by 5 elements , 0001 , 0011 , 0101 , 0111 and 1111 : @xmath176 obeying a single relation . we have thus uncovered an algebraic structure on the space of cs quivers with 2-term superpotentials . this is expected to persist to higher number of terms and indeed to the space of all quivers ; we have thus found an underlying algebraic variety for the set of chern - simons quivers . we remark also that taking the plethystic logarithm @xcite of the total @xmath177 reveals that it is a non - terminating series suggesting that the algebraic variety is not a complete intersection . it is in the form of a set union of the two orbifolds , though there is a degree shift on the second . in this paper we have started a taxonomic study of quiver chern - simons theories in @xmath6-dimensions . in particular , we study the theories arising from m2-branes probing toric calabi - yau 4-fold singularities . our purpose is not to merely provide a catalogue of such theories since there are clearly infinite families thereof , but to study the space of these theories from both a synthetic and an analytic approach , and to uncover interesting physical phenomena as well as mathematical structure . some of the techniques and concepts can indeed be generalised to quiver theories in arbitrary dimension and with any supersymmetry . we have first presented , in flow - chart ( [ gt ] ) , a succinct way of a `` forward algorithm '' which takes the quiver and superpotential data as input and the moduli space as an output : the former as a pair of matrices , respectively the incidence and the perfect matching matrices , and the latter , as a matrix representing the integer coordinates of the toric diagram of the calabi - yau 4-fold . the intermediate computations involve nothing more than matrix multiplications and finding kernels . indeed , this generalises and significantly improves on the forward - algorithm of the case of @xmath7-dimensional gauge theories of @xcite : we only need to insert a @xmath48-matrix of the chern - simons levels and have obviated the need for finding dual cones . thus armed , we have commenced the classification of all quiver chern - simons theories . listing the quivers is matter of combinatorics and for simplicity we have taken the superpotential to have two terms . it is clearly a pressing problem what happens at an arbitrary number @xmath133 of terms , this we shall leave to forth - coming work . subsequently our order parametre becomes @xmath9 , the number of nodes and @xmath10 , the number of arrows , is equal to @xmath178 since the master space here is @xmath179 . in general , we should have 3 order parametres @xmath180 . we exhaustively list the first members at @xmath181 and find many non - trivial theories , including and extending beyond what has so far emerged in the literature . because of our forward algorithm , we can readily determine the toric calabi - yau 4-fold moduli spaces for all the theories presented . these exhibit as many toric varieties , each of which is an infinite family , indexed by the integer chern - simons levels . remarkably we have once more encountered the phenomenon of `` toric duality '' first noted in @xcite for @xmath7-dimensional gauge theories . now , it has manifested in perhaps some more striking guises : we find not only theories with the same number of nodes but also those with different number of nodes , fields , as well as chern - simons levels , to flow to the same ir moduli space as toric calabi - yau 4-folds . it is certainly of importance to study what is precisely happening in the field theory of such pairs . building upon our host of examples , we have stepped back for a panoramic view of all our theories : connected quiver chern - simons with 2-term toric superpotentials . though the graphs seem unwieldy and growth rapidly in number as we increase the number of nodes , we have found a generating function ( [ gen ] ) which counts the number of inequivalent theories given the number of nodes . remarkably , this generating function splits conveniently into 2 pieces , each of which is the hilbert series of a precise algebraic variety : the non - abelian quotient space @xmath163 by the symmetric group on 3 objects of order 6 and the abelian quotient @xmath164 by the ordinary dihedral group of order 8 ( i.e. , the symmetry group of the square ) . could we package the case with arbitrary @xmath182 into a convenient tri - variate generating function and would the result have an interpretation as a hilbert series , whereby suggesting that the space of quiver theories is itself some algebraic variety ? in answering this and many questions raised we are presently engaged . indeed , we have only tread upon the fringe of a vast and fertile land , into her bountiful bosom we must onwardly march . _ scientiae et technologiae concilio y .- h . h. hoc opusculum dedicat cum gratia ob honorem officiumque socii progressi collatum . et ricardo fitzjames , episcopo londiniensis , ceterisque omnibus benefactoribus collegii mertonensis oxoniensis quorum beneficiis pie , studiose , iucunde vivere licet , pro amore catharinae sanctae alexandriae et ad maiorem dei gloriam . _ in this appendix , we present the results for a preliminary classification of the case of 2 terms in the superpotential . this will include cases of disjoint quivers and unmarked adjoint fields , we present them here for completeness . the cases of 4 , 5 and 6 fields are respectively shown in figures [ f : e=4 ] , [ f : e=5 ] and [ f : e=6 ] . in the text we will sift through these graphs carefully , adding adjoints wherever consistent . = 10 cm = 14 cm = 16 cm in this appendix , we extract the matrix @xmath54 of perfect matchings directly from the superpotential @xmath183 of the fields @xmath184 . in the archaic days of the forward algorithm , @xmath54 is found to be the product of the matrix @xmath39 ( which encodes how a total of @xmath10 f - terms can be solved in terms of a fewer number ) with its dual cone matrix @xmath40 . now we can follow the ensuing prescription to directly obtain @xmath54 without the rather computationally intensive procedure of finding dual cones : 1 . group all the monomials appearing in @xmath18 into those with positive coefficients and those with negative , giving us two sets of equal length ( since each field must appear exactly twice with opposite signs by the toric condition ) , on each of which we choose an order ; 2 . find the position of field @xmath185 in each of the two sets ; say it is @xmath68-th in the positive set and @xmath186-th in the negative set . construct a matrix whose @xmath187-th entry is @xmath185 ( should the index ever repeat , simply add , to the entry , the new field ) . do so for all @xmath10 fields and patch in zeros otherwise . this is the kasteleyn matrix . compute its determinant @xmath17 . 3 . @xmath17 is a new polynomial in the @xmath10 fields with @xmath44 ( the number of perfect matchings ) terms . fix an order for these terms , and construct and @xmath188 matrix . for each of the @xmath10 fields find the location in the monomial in @xmath17 . we place a 1 at this location in the @xmath10-th row correspondingly and 0 otherwise . the resulting matrix is the desired @xmath54 . as an illustrative example , let is consider the famous cone over the zeroth del pezzo surface , i.e. , the abelian orbifold @xmath189 . its quiver is a directed loop over 3 nodes , each with multiplicity 3 and its superpotential is a six term cubic in 9 fields : @xmath190 we recall that ( cf . e.g. , eq . ( 2.12 ) - ( 2.17 ) of @xcite ) the explicit solutions to the f - terms are @xmath191 and hence @xmath192 where dual refers to finding the dual cone . the product @xmath193 is the matrix @xmath54 . using our new algorithm , we find that there are 3 positive terms and 3 negative terms , giving us a kasteleyn matrix and its determinant @xmath17 as : @xmath194 upon comparison , we find that both procedures give ( up to trivial permutation of columns , note that we have fixed the row by a canonical ordering of the fields ) @xmath195 @xmath196 . we must point out that on a simple timing contrast , our new algorithm for this simple example is about a factor of 10 faster ! this is significant since the algorithm for finding dual cones is exponential running time so for bigger examples , we expect that our new method to be a substantial improvement . for example , for the cone over the third del pezzo surface , with 14 fields and 8 terms in the superpotential , our present algorithm constitutes a time reduction of more than 1000-fold over the old dual - cone method ! it would be interesting to see whether this technique could be generalised to rapidly find the dual cone of arbitrary integer cones ; this would of tremendous use to toric geometry . 99 j. bagger and n. lambert , `` modeling multiple m2 s , '' phys . d * 75 * , 045020 ( 2007 ) [ arxiv : hep - th/0611108 ] . `` gauge symmetry and supersymmetry of multiple m2-branes , '' phys . d * 77 * , 065008 ( 2008 ) [ arxiv:0711.0955 [ hep - th ] ] . `` comments on multiple m2-branes , '' jhep * 0802 * , 105 ( 2008 ) [ arxiv:0712.3738 [ hep - th ] ] . s. benvenuti , b. feng , a. hanany and y. h. he , `` counting bps operators in gauge theories : quivers , syzygies and plethystics , '' jhep * 0711 * , 050 ( 2007 ) [ arxiv : hep - th/0608050 ] . b. feng , a. hanany and y. h. he , `` counting gauge invariants : the plethystic program , '' jhep * 0703 * , 090 ( 2007 ) [ arxiv : hep - th/0701063 ] . c. beasley , b. r. greene , c. i. lazaroiu and m. r. plesser , `` d3-branes on partial resolutions of abelian quotient singularities of calabi - yau threefolds , '' nucl . b * 566 * , 599 ( 2000 ) [ arxiv : hep - th/9907186 ] . j. gray , y. h. he , v. jejjala and b. d. nelson , `` exploring the vacuum geometry of n = 1 gauge theories , '' nucl . b * 750 * , 1 ( 2006 ) [ arxiv : hep - th/0604208 ] ; y. h. he , `` vacuum geometry and the search for new physics , '' in the proceedings of 9th workshop on non - perturbative qcd , paris , france , 4 - 8 jun 2007 , pp 04 . d. forcella , a. hanany , y. h. he and a. zaffaroni , `` the master space of n=1 gauge theories , '' jhep * 0808 * , 012 ( 2008 ) [ arxiv:0801.1585 [ hep - th ] ] ; ` mastering the master space,' lett . phys . * 85 * , 163 ( 2008 ) [ arxiv:0801.3477 [ hep - th ] ] . b. feng , a. hanany , y. h. he and a. m. uranga , `` toric duality as seiberg duality and brane diamonds , '' jhep * 0112 * , 035 ( 2001 ) [ arxiv : hep - th/0109063 ] . + c. e. beasley and m. r. plesser , `` toric duality is seiberg duality , '' jhep * 0112 * , 001 ( 2001 ) [ arxiv : hep - th/0109053 ] . o. aharony , a. hanany , k. a. intriligator , n. seiberg and m. j. strassler , `` aspects of n = 2 supersymmetric gauge theories in three dimensions , '' nucl . b * 499 * , 67 ( 1997 ) [ arxiv : hep - th/9703110 ] . a. hanany and y. h. he , `` a monograph on the classification of the discrete subgroups of su(4 ) , '' jhep * 0102 * , 027 ( 2001 ) [ arxiv : hep - th/9905212 ] .

we initiate a systematic investigation of the space of 2 + 1 dimensional quiver gauge theories , emphasising a succinct `` forward algorithm '' . few `` order parametres '' are introduced such as the number of terms in the superpotential and the number of gauge groups . starting with two terms in the superpotential , we find a generating function , with interesting geometric interpretation , which counts the number of inequivalent theories for a given number of gauge groups and fields . we demonstratively list these theories for some low numbers thereof . furthermore , we show how these theories arise from m2-branes probing toric calabi - yau 4-folds by explicitly obtaining the toric data of the vacuum moduli space . by observing equivalences of the vacua between markedly different theories , we see a new emergence of `` toric duality '' . imperial / tp/08/ah/10 + 0.25 in m2-branes and quiver chern - simons : @xmath0 a taxonomic study + amihay hanany@xmath1 and yang - hui he@xmath2 + + [ cols= " < , < " , ]

You are an expert at summarizing long articles. Proceed to summarize the following text: following einstein s insight that gravity is encoded into the geometry of spacetime , quantum gravity aims at providing realizations of quantum geometry . in this task , one key technical and conceptual challenge is to reconcile the regularizations required in quantum field theory with the diffeomorphism symmetry which underlies general relativity . indeed , a number of approaches employ discretizations as regulators , which is the case for instance of regge calculus @xcite , and in this class of theories , where one attempts to represent geometrical data on a triangulation , diffeomorphism symmetry is generically broken @xcite . in light of this , a very important result is therefore the construction of a continuum notion of quantum geometry , which was achieved in the context of loop quantum gravity @xcite ( lqg hereafter ) by ashtekar , isham , and lewandowski @xcite . the so - called ashtekar lewandowski ( al ) representation provides a hilbert space representation of the kinematical algebra of observable of the full _ continuum _ theory . this kinematical algebra of observable encodes the intrinsic and extrinsic geometry of a spatial hypersurface into holonomies measuring curvature ( of the ashtekar barbero connection @xcite ) and fluxes measuring the spatial geometry . most importantly , this kinematical setup allows to deal successfully with spatial diffeomorphisms . indeed , there exists a fully kinematical hilbert space describing geometry in a coordinate - dependent way , and on which the diffeomorphisms act unitarily . the fact that the diffeomorphisms act unitarily also allows to define a hilbert space of spatially diffeomorphism - invariant states , and , quite noticeably , this task has so far only been achieved in the ( ashtekar ) connection formulation of general relativity @xcite . the kinematical hilbert space allows for a rigorous definition of the quantum dynamics in the form of hamiltonian constraints @xcite , and one can hope to construct the physical hilbert space , which would incorporate spacetime diffeomorphism - invariant states , from the solutions to these constraints . in the present article , building upon the earlier work @xcite , we construct an alternative realization of a continuum quantum geometry , which is unitarily inequivalent to the ashtekar lewandowski representation . we hope that this new framework will make the description of states describing configurations with macroscopic geometry much easier . the reason for this expectation is that the new representation which we are constructing here supports states which are peaked on an almost everywhere flat connection ( we therefore call this representation the bf representation , since the bf vacuum is peaked on a flat connection ) . curvature has only distributional support on defects , which brings us much nearer to regge s proposal @xcite of approximating general relativity by a very dense gas of defects in an otherwise flat geometry . as explained in more detail in the overview given in section [ overview ] , our construction leads to a continuum hilbert space which supports arbitrarily many excitations in the form of defects . in fact , both the al representation and the bf representation can be interpreted in this way . for the al representation , it is the defects themselves which generate ( non - degenerate ) geometry , and therefore a macroscopic geometry corresponds to a highly excited state . for the bf representation however , the states have ( almost everywhere ) maximal uncertainty in spatial geometry since they are peaked on the conjugated variable , namely on flat connections . there are numerous proposals to approximate the dynamics of general relativity by mostly flat geometries with defects , either in the classical theory @xcite , or in the context of lqg @xcite . another related class of approaches is the combinatorial quantization of flat connections in three spacetime dimensions @xcite . in this work , we construct for the first time a hilbert space carrying a representation of a continuum observable algebra and supporting states which are peaked on almost everywhere flat connections . this continuum construction has important consequences . in particular , it requires the compactification ( or the exponentiation ) of the fluxes , and with it the introduction of a discrete topology for the holonomy group parameters . this in turn changes the properties of , e.g. , the spectra of observables like the area operator . our construction of the continuum hilbert space is done via a so - called inductive limit . as is explained in more detail in section [ overview ] , such a constructions allows one to keep the cake and to eat it too . for most purposes , it is sufficient to deal with a discretization . however , all the hilbert spaces describing states restricted to such discretizations are embedded into a continuum hilbert space . the properties of observables are therefore changed further due to redefining ( as compared to the al representation ) the way in which the observables on hilbert spaces associated to discretizations are related to the observables on the continuum hilbert space . for the bf representation , this relation is based on a geometric coarse - graining of the fluxes , which allows to address for example the staircase problem appearing for geometric operators in the al representation . this work is divided in two main parts . we start in section [ sec : configuration space ] by providing a characterization of the ( cotangent ) space of flat connections on manifolds with a fixed number of defects , where the positioning of the defects is prescribed by a choice of triangulation . we then define in section [ sec : inner product ] an inner product ( compatible with the inductive limit construction which comes afterwards ) which leads to a discrete topology on the group . this discrete topology on the group implies that dealing with gauge invariance is far less trivial than in the al case . we bypass this difficulty by working with the space of almost gauge - invariant wave functions , and employ a group averaging for the remaining global adjoint action . in section [ sec : quantization algebra ] , we construct a representation of holonomies and ( exponentiated ) fluxes which are supported by the triangulation . this already allows us to discuss the spectra of these operators , and also that of the area operator , which we do in section [ sec : area ] . in the second main part of this work , we construct the continuum hilbert space via an inductive limit . this first requires the definition of embeddings that map hilbert spaces associated to coarser triangulations into hilbert spaces associated to finer triangulations . we therefore develop a number of technical results in section [ sec : refinement ] , which allow in particular to obtain a convenient splitting of the observables into coarser observables and finer observables . in section [ sec : inductive limit ] , we describe the construction of the inductive limit hilbert space . we then discuss how to relate the operators on the various hilbert spaces in order to define a notion of continuum operators in sections [ sec : extop ] and [ contop ] . finally , we close with a discussion of our results in section [ sec : discussion ] . the appendices include a more detailed elaboration on the configuration space of flat connections , a number of proofs of theorems needed in the main text , as well as the construction of the ( intertwiner- ) spin representation for the wave functions , which replaces the usual spin network states of the al representation . in order to construct the _ continuum _ hilbert space , we proceed in the same way as for the al representation , namely by considering an inductive limit of hilbert spaces . let us explain why this construction is very natural in a background - independent context . this will also allow us to explain how the new realization of quantum geometry presented here can be seen as one member of a potentially much bigger class of quantum geometries @xcite . we usually organize hilbert space representations into a vacuum and excitations , where the excitations are generated by applying the observable algebra to the vacuum state . the vacuum states we are dealing with for both the al and the new representation are states which are sharply peaked on a certain configuration , as opposed to the gaussian states of free field theory for which one needs to specify a background metric . for the al representation , one has a vacuum which is sharply peaked on a totally degenerate spatial geometry , i.e. in which all flux observables annihilate the vacuum . for the representation constructed in this work , the vacuum state is very different : it is sharply peaked on a ( globally and locally ) flat connection . in fact , such states which are sharply peaked on a flat connection can be described by ( first class ) constraints arising from a topological field theory ( these constraints play also an important role in the classical theory @xcite ) . for the new representation presented here , this topological field theory is given by bf theory @xcite , which is also the theory underlying the construction of spin foam models @xcite . we will therefore refer to the new representation as the bf representation . the topological field theory for the al representation is a trivial one , and coincides with the strong coupling limit of yang mills theory , whereas bf theory arises from a zero coupling limit @xcite . the topological field theory in itself does however not provide room for local excitations , and therefore does not give a representation of the observable algebra which is supposed to be generating such excitations . in order to add excitations to the theory , one needs to allow for the presence of defects , e.g. in the form of a violation of the flatness constraints in the bf case , or of a violation of the flux constraints in the al case . therefore , the notion of what constitutes a `` defect '' strongly depends on the choice of vacuum . in the case of the al representation , a `` defect '' actually means an excitation which allows to have at least some of the spatial geometrical operators with non - vanishing values . this would rather be an `` anti - defect '' of geometry if one understands a degenerate geometry as being highly defected . defects play also an important role in condensed matter applications @xcite . in this case , one wishes to describe localized excitations which are separated by very large distances so that they do not interact with each other . therefore , the precise values of the distances , i.e. the metric geometry , does not actually matter . this justifies a posteriori the use of the vacuum ( or the physical state ) of a topological field theory : it ensures that only relations of topological nature between the defects actually matter ( for instance how two line defects wind around each other ) . furthermore , it explains why frameworks for describing defects are applicable to quantum gravity , since the metric properties are then not encoded in a background , but in the defect configurations themselves . a second property which one usually assigns to defects is a discreteness for their `` charge '' labels . in condensed matter , this arises since one needs finite energy gaps between the vacuum and the excitations described by the defects , so that there is a notion of adiabatically moving the defects around . interestingly , there is a similar discreteness of the charges describing the defects in the framework of quantum geometry . this arises however due to a different reason , namely in the way in which the continuum limit is constructed . to construct the continuum limit , one starts with a family of hilbert spaces where each hilbert space describes a fixed number of defects . for a diffeomorphism - covariant description , one does not only fix the number of defects , but also their positions . this is achieved by embedding a discrete structure , e.g. a graph or a triangulation , into the spatial manifold . defects are then confined to this discrete structure , for instance line defects to the links of the graph or to the edges of the triangulation . the degrees of freedom are given by the values of the charges for the defects . the values can also describe a vanishing charge , which is then equivalent to having no defect . this is an important point since it allows to embed states realized in a hilbert space based on e.g. a coarse triangulation into a hilbert space based on a refinement of this triangulation . for the continuum theory , one however wishes to allow for an arbitrary number of defects ( at arbitrary positions ) . one can construct such a continuum hilbert space as an inductive limit of hilbert spaces . a key point in this construction is the notion of refinement for the discrete structures carrying the defects on the one hand , and for the states on the other hand . let us consider the example of line defects confined to a graph embedded into the manifold . one possible refinement of a graph is to add links to it . the corresponding state for the refined graph is then defined by transposing over all the charge labels from the coarse state , and by assigning the charge labels describing a vanishing defect for the new links . in other words , one puts the new additional degrees of freedom into a vacuum state @xcite . in order to define the inductive hilbert space structure , the refinement operator acting on the states needs to be isometric ( with respect to the inner product defined on the hilbert spaces with a fixed discrete structure ) . in particular , the vacuum state where all the charges describe vanishing defects should have a finite norm . moreover , what happens ( in both the bf and the al representation ) is that the charges label an orthonormal basis for the hilbert spaces associated to a fixed discrete structure . therefore , even if the charge labels are continuous ( as it is the case with the bf representation , where the labels are elements of the ( lie ) group ) , we will have to equip the space of charge values with a discrete topology . if we see the charge labels as momentum space , then a discrete topology on momentum space corresponds to a compactification of the configuration space . this compactification can be understood in the following way for the two representations : * in the al representation , the configuration space associated to a fixed discrete structure is given by group holonomies along the links of embedded graphs , and therefore corresponds to @xmath0 , where @xmath1 is a compact lie group ( usually @xmath2 ) , and @xmath3 denotes the number of allowed defects , which in this case coincides with the number of links of the graph . the inductive limit of hilbert spaces amounts to a projective limit for the configuration space , in which compactness is preserved ( see for instance @xcite ) . the space of charge labels is constructed through the group fourier transform and given by representation labels ( and magnetic indices ) . this space does indeed naturally carry a discrete topology ( for a compact group @xmath1 ) . * in the bf representation , the configuration variables are actually the fluxes and [ intertwinerspin ] . ] , which correspond to the dual of the lie group . we therefore need to consider a compactification of this space . in fact , for the group @xmath4 the ( pontryagin ) dual is given by the abelian group @xmath5 , which can be bohr - compactified . this in turn leads to a discrete topology on @xmath4 . for @xmath6 , it is however easier to start with a discrete topology , and to then derive the compactification of the dual space via a ( generalized ) group fourier transform . this is described in appendix [ spinrepsu2 ] . therefore , we see that the inductive limit construction leads to a compactification of the underlying configuration spaces . this also explains the difficulty to define a projective limit hilbert space ( in the al representation ) for a non - compact group like @xmath7 , since this latter can not easily be compactified , one can use the bohr compactification @xcite . it might actually be easier to compactify the dual of @xmath7 , which would then allow for a bf representation . ] . the compactification of the fluxes has important repercussions for the properties , and in particular the spectrum , of the observables . due to the discrete topology on the ( lie ) group , the fluxes can not act anymore as ( lie ) derivatives like in the al representation . instead , one has to replace the fluxes with translation operators labelled by a translation group element @xmath8 . due to the compactification , the spectrum of these translation operators is bounded . however , depending on the choice of @xmath8 , the spectrum can be either discrete or continuous . the continuous spectrum arises if the action of the translation operator on the group is ergodic . we will provide a more detailed discussion of these features in section [ sec : area ] . a ( bohr ) compactification is also employed in another context in loop quantum cosmology ( lqc ) @xcite . in this case , one compactifies @xmath9 , which describes the momentum conjugated to the scale factor ( this momentum @xmath10 is proportional to the connection degree of freedom ) . this then leads to a discrete topology for the scale factor . note that in lqc ( point ) holonomies are compact while the conjugated momentum discrete , which is the reversed situation as compared to the bf representation . it is also well - known that compactifications lead to non - separable hilbert spaces . let us comment more on this issue , and in particular on the fact that there are now two different sources of non - separability . * the first source of non - separability arises from the inductive limit hilbert space construction based on _ embedded _ discrete structures . this can be interpreted as introducing a discrete topology for the underlying spatial manifold . this is also mirrored in the interpretation of defects in condensed matter as charges which lie at large distance from one another . here , we declare for instance any point ( in the case in which the defects are point - like ) of the underlying manifold to be separated from any other point ( mathematically , these points define open sets ) , which gives rise to the discrete topology . + one can however expect that this kind of non - separability will be lifted ( to a large extent ) if one constructs the spatial diffeomorphism - invariant hilbert space . indeed , diffeomorphisms act by changing the embeddings of the discrete structures , and therefore the positioning of the defects . for a diffeomorphism - invariant description only the relative ( topological ) positions of the defects matter . * in the bf representation , we encounter a second kind of non - separability , which results from the discrete topology on the space of charge labels . this non - separability will not be lifted by implementing spatial diffeomorphism symmetry . there is however the possibility that it can be lifted for the physical hilbert space if one includes a positive cosmological constant ( at least in the euclidean theory ) . in this case , the large spin @xmath11 labels in the spin representation might become irrelevant : for instance , the area eigenvalues first grow with growing spin , and then start ( depending on the value of @xmath8 ) to oscillate . therefore , very large spins are in a sense not needed since they lead to similar area eigenvalues as the smaller spins . + one can also introduce a quantum deformation of the group at root of unity , i.e. work with @xmath12 . in this case , the cutoff on the spin is implemented explicitly . the charge labels are then described by the drinfeld center of the @xmath12 module category , which has only finitely many objects @xcite . in three spacetime dimensions , the corresponding vacuum is given by the turaev viro state sum model @xcite and describes ( euclidean ) gravity with a positive cosmological constant . future work @xcite will be concerned with constructing a continuum hilbert space representation based on quantum groups . in this section we briefly summarize the setup of our construction , and present the notations and notions which are used throughout the rest of the article . we refer the reader to the companion paper @xcite for more details . let @xmath13 be a @xmath14-dimensional closed ( i.e. compact and without boundary ) manifold , which we choose to be real , analytic , orientable , and path - connected . we will be interested in particular in the cases @xmath15 and @xmath16 , which we treat simultaneously . the manifold @xmath13 can be thought of as a spatial boundary , and our aim is to construct states corresponding to @xmath14-dimensional boundary quantum geometries on it . for this , we consider a principal @xmath1-bundle @xmath17 , where the gauge group @xmath1 is an arbitrary compact lie group whose lie algebra will be denoted by @xmath18 . if @xmath1 is simply - connected ( which is the case for @xmath2 , the gauge group of interest in this work ) , since we consider the case @xmath19 it is always possible to pick a global trivialization of @xmath20 , which we do for simplicity . the space @xmath21 of smooth @xmath1 connections is then affine over @xmath22 , the space of @xmath18-valued one - form fields over @xmath13 , and the group of local gauge transformations is given by @xmath23 . on the phase space of lqg , a connection @xmath24 is canonically conjugated to a @xmath25-valued vector density @xmath26 ( the electric field ) of weight + 1 . if we write @xmath27 and @xmath28 , where @xmath29 are coordinates on @xmath13 and @xmath30 forms a basis of @xmath18 , the non - vanishing poisson brackets are given by [ firstequ ] a^i_a(x),e^b_j(y)=^b_a^i_j^d(x , y ) . following the logic of lqg , we are going to consider holonomies of the connection and fluxes of the electric field as our elementary variables . however , instead of assigning this data to arbitrary oriented embedded ( dual ) graphs and @xmath31-dimensional submanifolds , as is done in the usual construction of the holonomy - flux @xmath32-algebra , here we adopt a simplicial point of view and focus rather on the triangulation itself . we now explain the type of discrete data which is relevant for our construction . we consider a fiducial riemannian metric on the manifold @xmath13 , and the infinite set of embedded ( or geometric ) triangulations @xmath33 defined by this metric . in both @xmath15 and @xmath16 dimensions , these embeddings refer to the vertices of the triangulations . the edges and ( in @xmath16 ) the triangles of a given triangulation are defined respectively as geodesics and minimal surfaces with respect to the auxiliary metric . we furthermore assume that the triangulations are fine enough so that these geodesics and minimal surfaces are unique . to each triangulation @xmath33 we assign a dual graph @xmath34 , which is the one - skeleton of the cellular complex dual to @xmath33 . this dual graph is such that its nodes @xmath35 are dual to the @xmath14-dimensional simplices @xmath36 of @xmath33 , while its links @xmath37 are dual to the @xmath31-dimensional simplices @xmath38 of @xmath33 . for each link we choose a fiducial orientation , and denote by @xmath39 and @xmath40 the source and target nodes . to each pair given by a triangulation and its dual graph we then assign a root , which is a choice of a preferred @xmath14-dimensional root simplex @xmath41 , or equivalently ( by duality ) a choice of a preferred root node @xmath42 . this in turn defines a rooted geometric triangulation . in order to describe the behavior of this root under the refining operations , we further equip the root simplex with a flag structure , which consists simply in a choice of nondisjoint lower - dimensional simplices @xmath43 . with these structures , we can finally consider the infinite set of flagged rooted geometric triangulations of a given manifold @xmath13 . on this set , we now choose the refinement operations to be the alexander moves . these alexander moves refine a given triangulation by subdividing a @xmath44-dimensional simplex , where @xmath45 , with the addition of a new ( embedded ) vertex . as explained in @xcite , if the root of a triangulation is subdivided by a move , the flag structure defines in a canonical manner a new root for the refined triangulation . a geometric triangulation @xmath46 is said to be finer than a geometric triangulation @xmath33 if it can be obtained form @xmath33 by a ( finite ) sequence of refining alexander moves , and if its vertices after the moves have the same coordinates as those of @xmath33 . we will denote this relation by @xmath47 . in oder to unambiguously specify paths @xmath48 in a graph @xmath34 and to obtain a characterization of the fundamental cycles of this graph ( and to perform a gauge - fixing ) , it is convenient to introduce ( maximal spanning ) trees . a maximal or spanning tree @xmath49 in a graph @xmath34 is a connected subgraph of @xmath34 which does not contain any closed cycles and includes all the nodes of @xmath34 . in the rest of the paper we use ` tree ' as short form for maximal connected tree . the links composing a tree are called branches and denoted by @xmath50 , while the links of the graph that are not in the tree are called leaves and denoted by @xmath51 . although the leaves correspond to the links in @xmath52 , by abuse of language we will refer to the leaves of the tree . the set of leaves in a tree will be denoted by @xmath53 . a rooted tree with root @xmath42 is a tree where a preferred node is identified and called the root . there is a one - to - one correspondence between the leaves of a tree and the fundamental cycles of the underlying graph . for a rooted tree , one way to characterize these cycles is to associate them with paths @xmath54 labeled by the leaves in @xmath53 . a path @xmath54 starts at the root , goes along the branches of the tree until the source node of the leaf @xmath51 , then goes along @xmath51 , and back to the root along branches . the cardinality of the set of fundamental cycles is independent of the choice of tree . explicitly , if we denote by @xmath55 the number of nodes in the graph @xmath34 and by @xmath56 the number of links , then any given tree has @xmath57 branches and @xmath58 leaves ( and fundamental cycles ) . as explained in the introduction , to construct the full continuum quantum theory we will proceed in two steps . in this first part , we are going to define the quantum theory on a fixed triangulation . for this , we will first focus on the classical connection configuration space , then on the inner product and hilbert space structure built upon it , and finally on the quantization of the algebra of holonomies and fluxes . this will enable us in particular to discuss the properties of the area operator . in the second part we will then show the existence of a continuum ( inductive limit ) hilbert space . in this section , we focus on the connection degrees of freedom and describe the configuration space of smooth @xmath1 connections on the spatial manifold @xmath13 . in the al framework , the quantum theory is built over a suitable generalization of @xmath21 which is the space @xmath59 of generalized connections . in the present context however , as we shall see , by configuration space we mean the space of classical states which reflect the geometrical properties of the states of our ( yet - to - be built ) quantum theory , which turns out to be the moduli space of flat connections on the spatial manifold @xmath13 with certain defects . ] over which we are going to build the quantum theory . the bf vacuum is a state locally and globally peaked on flat connections . this property can be encoded in the triviality of the holonomies of the connection around the faces of the dual graph @xmath34 ( more precisely , along the fundamental cycles of the graph ) . excitations over this vacuum locally shift the holonomies away from the identity , thereby creating curvature defects which can be thought of as being carried by the @xmath60-dimensional simplices ( the hinges ) of the triangulation . for this reason , we are interested in connections which are flat appart from possible conical singularities along the hinges . if we denote by @xmath61 the @xmath60-skeleton of the triangulation @xmath33 , we are therefore led to considering the space of flat connections on @xmath62 . as is well - known @xcite , the moduli space of flat connections modulo gauge transformations on a compact and connected manifold @xmath13 is a finite - dimensional orbifold which admits several different ( although equivalent ) mathematical descriptions . in particular , it is common to describe the quotient _ 0 a|f(a)=0/ in terms of holonomies ( i.e. group homomorphisms ) as [ without defects ] _ 0=(_1(),g)/g , where @xmath1 acts on the space @xmath63 of homomorphisms by diagonal adjoint action . this defines the space of locally flat connections for a compact and connected manifold @xmath13 without defects . details about the construction of the first fundamental group @xmath64 can be found in appendix [ appendix : pi1 ] . in the present work , since we are interested in flat connections on the defected manifold @xmath62 , we first have to worry about the fact that this latter is not compact . in order to avoid the inconvenience of working with infinite triangulations , one can resort to the following construction : first , choose a flat metric on the simplices of @xmath33 ( which can be the auxiliary metric on @xmath13 ) and let , for some small @xmath65 , @xmath66 be the open ball of radius @xmath67 . then , the set _ ( d-2)^_x_(d-2)b_(x ) , which represents a `` blown - up '' version of @xmath61 , is open in @xmath13 , and if @xmath67 is small enough one has that @xmath62 is homotopy - equivalent to the new manifold @xmath68 . in fact , one has that @xmath69 is a deformation - retract of @xmath62 . as such , these two spaces have isomorphic fundamental groups , and we can define our configuration space to be _ ( _ 1(),g)/g . the question is now how to characterize @xmath70 . to this end we note that the graph @xmath34 , which is the one - skeleton of the simplicial complex dual to the triangulation @xmath33 of @xmath13 , is a deformation - retract of @xmath71 . as a consequence , their first fundamental groups are isomorphic , i.e. @xmath72 . we are therefore led to considering the first fundamental group @xmath73 of the graph @xmath34 dual to the triangulation @xmath33 . the first fundamental group @xmath73 of a graph @xmath34 is a free group , and is generated by a set of fundamental cycles of the graph @xcite ( we refer the reader to appendix [ appendix : pi1defects ] for more details ) . as noted in section [ sec : setup ] , a set of fundamental cycles can be obtained by choosing a maximal spanning tree @xmath49 ( which here in addition will be rooted ) in @xmath34 . the links of @xmath34 which are not part of the tree are called leaves and denoted by @xmath51 . the set @xmath53 of leaves with respect to a given choice of tree @xmath49 defines a set of fundamental cycles @xmath54 . such a fundamental cycle @xmath54 starts at the root , goes along the branches of the tree until the source node of the leaf @xmath51 , then goes along @xmath51 , and back to the root along branches . by definition , a set of fundamental cycles obtained by a choice of tree does freely generate the first fundamental group @xmath73 . in other words , if we denote by @xmath54 the cycle associated to the leaf @xmath51 and choose a numbering @xmath74 of the leaves , we have that _ 1 ( ) = _ 1, ,_|| . therefore , the space @xmath75 is parametrized by the set of group - valued maps @xmath76 , and by associating to each cycle @xmath54 a group element @xmath77 we can write that ( _ 1(),g ) = \{g_1, ,g_|||g_ig}. from this space , we then obtain the space of locally flat connection by dividing out by the diagonal adjoint action of the group on @xmath78 , i.e. _ \{g_1, ,g_|||g__ig}/g . as we will now explain , this space in turn agrees with the space of gauge - invariant ( or gauge - fixed ) holonomies on the graph @xmath34 . since the @xmath79-tuple @xmath80 determines the holonomies along the fundamental cycles of the graph , it can be used to reconstruct the holonomies along all the cycles ( starting and ending at the root ) in @xmath34 . a ( suitable ) space of functions of these cycle holonomies therefore gives the space of functions of the graph connection which are invariant under gauge transformations at all the nodes except the root . let us explain how this comes about . a graph connection is given by an assignment of group elements @xmath81 to the links @xmath82 of the graph , and the action of a gauge transformation with parameter @xmath83 on @xmath81 factorizes at the nodes and gives u_nh_l = u^-1_l(1)h_lu_l(0 ) . given a closed cycle @xmath84 with @xmath85 starting at the root and @xmath86 ending at the root , the cycle holonomy @xmath87 is therefore invariant under gauge transformations at all the nodes except at the root . instead , under gauge transformations at the root , it transforms with the adjoint action [ rootgaugetrafo1 ] g^|n|\{u_n}g_=u^-1_r g_u_r . by choosing a tree , the space of holonomies which are invariant under non - root gauge transformations can again be characterized by the association of group elements to the leaves of the graph . as explained above , for ever leaf @xmath51 we can consider paths @xmath54 describing the fundamental cycles by starting at the root , going along the tree until the source of @xmath51 , then going along @xmath51 , and back to the root along the tree . to every leaf we can therefore associate a group element [ hcycle ] g_h_e^1 = g_r(1)^-1h_g_r(0 ) . any other arbitrary cycle holonomy can be obtained by composing the holonomies @xmath77 of the fundamental cycles . the fundamental cycle holonomies therefore parametrize the space of graph connections which are invariant under non - root gauge transformations . furthermore we can use the tree to perform a ( partial ) gauge - fixing . given any graph connection @xmath88 there exist a unique gauge transformation @xmath89 , acting on all nodes except the root , such that @xmath90 for all the links @xmath50 which are branches of the tree . here we use @xmath91 to denote an equality which holds with a particular gauge - fixing . in this gauge , the holonomy along a fundamental cycle reduces to the holonomy associated to the leaf @xmath51 , i.e. @xmath92 . thus @xmath93 parametrizes the space of almost gauge - invariant graph connections , that is connections invariant under all gauge transformations except the ones at the root . we can therefore conclude that @xmath78 describes the space of almost gauge - invariant connections on @xmath34 , as well as the space of almost gauge - invariant locally flat connections on the manifolds with defects @xmath71 , which we will denote by @xmath94 . in this section we introduce the inner product and hilbert space structure on a fixed triangulation . for this , we first focus on states which are gauge - invariant everywhere appart from the root , and then use the technique of group averaging in order to account for these residual gauge transformations . our goal is to construct a hilbert space @xmath95 for which the space of almost gauge - invariant graph connections @xmath94 serves as the configuration space , as it would be the case for example in ( at least one approach to ) the quantization of chern simons theory . ] . the reason for working with @xmath94 instead of the full gauge - invariant space @xmath96 is that the configuration space @xmath78 is much simpler to handle than the orbifold @xmath97 . we will therefore first consider @xmath94 , and then impose invariance under the residual gauge transformations at the root via a group averaging procedure . as discussed in section [ sec : configuration space ] , the configuration space @xmath94 is parametrized by a set of group elements in @xmath78 , and the identification of the leaves @xmath51 implicitly means that one has made a choice of tree @xmath49 . we therefore consider some space @xmath98 of functions as a candidate for our hilbert space @xmath95 , and consider states of the form |\{g _ } ( g_1, ,g_|| ) . here we can interpret the group elements @xmath77 as giving the holonomies associated to the fundamental cycles @xmath54 of the graph @xmath34 . alternatively , if we use the tree @xmath49 to gauge - fix the branch holonomies to the identity , since in this gauge we have @xmath92 , we can interpret the group elements @xmath77 as the ( edge ) holonomies associated to the leaves . let us now propose an inner product on the space @xmath98 . our choice for this inner product is motivated by the inductive limit hilbert space construction which will be carried out in the second main part of this work . as we will see , this construction requires to isometrically embed hilbert spaces based on coarser triangulations @xmath33 into hilbert spaces based on finer triangulations @xmath46 . recall that for a finer triangulation @xmath46 with @xmath99 , the associated manifold @xmath100 has more defects than the manifold @xmath62 , and the dual graph @xmath101 has more independent cycles than the dual graph @xmath34 . this means that the space @xmath102 of almost gauge - invariant holonomies is isomorphic to @xmath103 with @xmath104 . now , the coarser configuration space @xmath94 can be regained from the finer configuration space @xmath102 by imposing constraints @xmath105 ( where the index @xmath106 has range @xmath107 ) . these constraints impose that the `` finer '' holonomies , i.e. holonomies of cycles around the additional defects in @xmath100 , are trivial ( this will be explained in much more detail in section [ sec : refinement ] ) . the presence of these constraints means that states in the finer hilbert space @xmath108 which arise as the embedding of states in the coarser hilbert space @xmath109 are sharply peaked on vanishing `` finer '' holonomies . these states need to have a finite norm as otherwise the embedding would not be isometric ( and the inner product would not be cylindrically consistent , which would prevent the inductive limit construction of the continuum hilbert space from existing ) . furthermore , the motivation for this work is to construct a representation based on a vacuum state which is peaked on locally and globally flat connections , and this vacuum state should of course also have a finite norm . if the gauge group @xmath1 is a finite group , the inner product provided by the discrete measure ( which agrees in this case with the haar measure ) leads to the required finiteness . however , if @xmath1 is a lie group ( even a compact one ) , the states of interest will have infinite norm . we therefore need to consider an appropriate regularization procedure . our regularization procedure goes as follows . first , we start with an auxiliary inner product @xmath110 , together with a family @xmath111 of regulator states having a finite norm with respect to this auxiliary inner product for finite @xmath112 . second , we need a family @xmath113 of regulator states approaching the ( locally and globally ) flat vacuum state @xmath114 when @xmath115 . with this , we can then define the inner product _ 1|_2=_0 . the use of such an inner product , which employs the vacuum state as a reference vector , was suggested already in @xcite and @xcite . as we are going to show , it leads to a discrete mesure on the space of locally flat connections , which in turn will make the proof of cylindrical - consistency of the inner product rather simple to obtain . it is interesting to notice that other proposals using a non - discrete measure on the space of flat connections have appeared earlier in the literature @xcite , but so far do not allow for a clear understanding of the cylindrical - consistency properties of the inner product ( as opposed to the present framework ) @xcite . to simplify the discussion , let us now consider that the configuration space is given by a single copy of the ( compact and semi - simple ) gauge group @xmath1 , and choose the auxiliary inner product to be the one induced by the haar measure on @xmath1 . furthermore , let us use a heat kernel regularization in order to regularize delta - peaked states . for @xmath116 , we can then define @xmath112-regulated families of states converging to the delta function peaked on @xmath117 as _ d_(-())_(g^-1 ) . here we have chosen representatives @xmath118 , given by representation matrices @xmath119 and with dimension @xmath120 , for the equivalence classes of unitary irreducible representations of @xmath1 . @xmath121 denotes the quadratic casimir ( the eigenvalue of the laplacian on the group manifold ) associated to the representation @xmath118 , and @xmath122 is the character of @xmath118 . since the representation matrices satisfy @xmath123 , we can readily compute = = . using the asymptotic expansion of the heat kernel on compact lie groups ( see for instance @xcite ) , which takes the form ^_(g ) ( - ) , where @xmath124 is the geodesic distance in @xmath1 between @xmath125 and the unit element @xmath126 , we finally get the following limit of vanishing regulator for the inner product : [ limitip ] ( , ) , here @xmath127 if and only if @xmath128 and is vanishing otherwise . this delta symbol should therefore be treated as a kronecker delta with continuous parameters . this is similar to what happens in the case of the bohr compactification of abelian groups , in particular for example in loop quantum cosmology ( lqc ) @xcite . however , in lqc the bohr compactification typically corresponds to the configuration space ( i.e. to the holonomies , which are proportional to the momentum @xmath129 conjugate to the scale factor @xmath130 ) , whereas here , in the case of the abelian group @xmath131 , we are compactifying the dual group @xmath132 ( i.e. the space of fluxes ) , which in turn leads to a discrete topology on @xmath131 itself . let us now turn to the basis states for the hilbert space @xmath95 . introducing the collection @xmath133 of group elements , the basis states @xmath134 are of the form [ basisvectors ] _ \{_}\{g _ } = _ ( _ , g _ ) , and are orthonormal in the above - introduced inner product : [ eq : hsinnerproduct ] _ \{_}|_\{_}= _ ( _ , _ ) . note that the hilbert space @xmath95 , although being associated to a finite triangulation , is therefore non - separable . more precisely , the hilbert space is the closure of states of the form \{g _ } = _ \{_}f\{_}_\{_}\{g _ } , where the sum runs over a _ finite _ set of values @xmath135 and @xmath136 is a function on this finite set . although we are here developing a flux formulation of lqg , it is worth noticing that we work still in the holonomy representation for the arguments of the wave functions . in appendix [ appendix : spin ] , we introduce a dual spin representation based on a suitable discrete generalization of the ( group ) fourier transform . this spin representation can also be used to express the measure functional , as explained in appendix [ appendix : spin ] . the construction which we have carried out so far depends on a specific choice of maximal tree @xmath49 with ( fixed ) root node @xmath42 . however , it is not difficult to see that the inner product is independent of the choice of tree , which means that the construction of @xmath95 is also independent of this choice . in order to see this explicitly , let us choose another tree @xmath137 in @xmath34 with the same root @xmath42 . the trees @xmath49 and @xmath137 define two different isomorphisms between @xmath94 and @xmath78 , which means that the same connection in @xmath94 corresponds to two different sets @xmath138 and @xmath139 of group elements . the explicit isomorphism between the two sets of group elements can be constructed as follows . for both sets @xmath53 and @xmath140 of leaves , the corresponding sets @xmath141 and @xmath142 of cycles are fundamental . therefore , a given cycle @xmath54 can be expressed as a word in the cycles @xmath142 ( and their inverses ) , and the other way around , where the orientation ( and ordering ) of @xmath54 in the product is according to the orientation ( and ordering ) in which the leaves @xmath51 appear in @xmath143 . ] . we denote these relations by _ = w_\ { _ } , _ = w_\{_}. because of this , the corresponding group holonomies for the same flat connection in @xmath94 can therefore be expressed as [ eq : isomorphism ] g_=w_\{g_},g _ = w_\{g_}. let us now consider a state @xmath144 in the representation based on the choice of tree @xmath49 . a change of tree @xmath145 induces a gauge transformation @xmath146 of the function @xmath147 on @xmath78 , which we denote by @xmath148 . the gauge - transformed state is then given by \{g _ } = ( ) \{g _ } = \{w_\{g_}}. in particular , for the basis states @xmath134 we have that ( _ \{_})\{g _ } = _ \{_}\{w_\ { g _ } } = _ ( _ , w_\{g _ } ) = _ ( w_\{_},g _ ) = _ \{w_\{_}}\{g_}. here we have used the fact that the words @xmath149 ( or their inverses @xmath150 ) define a bijective map @xmath151 , which means that @xmath152 if and only if @xmath153 . with these ingredients , we can finally compute the inner product using the representation based on the choice of tree @xmath137 , which leads to ( _ \{_})|(_\{_})=_\{w_\{_}}|_\{w_\{_}}= _ ( _ , _ ) . this is identical to the inner product between states expressed with respect to the choice of tree @xmath49 , which shows that the inner product is independent from the choice of tree , or in other words invariant under gauge transformations not acting at the root . we are now going to deal with these residual gauge transformations . the states in @xmath154 which we have constructed correspond to states on almost gauge - invariant connections @xmath94 . under the gauge transformation at the root @xmath42 , the group elements in @xmath155 become conjugated as [ eq : gaugetransformationatroot ] \{_}^u = \{_1 ,_||}^u \{u_1u^-1, ,u_||u^-1 } , where the group element parametrizing the gauge transformation is @xmath156 . due to the form of the inner product , it is clear that the gauge - averaged states which result from integrating the action over the gauge group @xmath1 have infinite norm . therefore , the hilbert space @xmath157 of gauge - invariant states has to be constructed differently , and one way to do so is by using the technique of refined algebraic quantization ( raq ) @xcite . for this , let us denote by @xmath158 the dense subspace of finite linear combinations of @xmath134 with in @xmath34 , it is easy to see that @xmath158 is actually invariant under a change of tree . ] @xmath135 . instead of being the image of a projector on @xmath154 , the hilbert space @xmath157 of fully gauge - invariant functions is going to be obtained as a subspace of the algebraic dual @xmath159 of @xmath158 . these various spaces are organized in a so - called gelfand triple _ _ _ , where the last inclusion is given by the riesz representation theorem , using the inner product on @xmath95 . the elements in @xmath159 can be thought of as distributional states . instead of a projector onto gauge - invariant states , raq employs a _ rigging map _ , i.e. a linear function @xmath160 which maps to the gauge - invariant distributional states in the sense that _ = _ for all @xmath161 . we choose this rigging map to be [ eq : riggingmap ] _ _ g/__(u)_\{_}^u|=_ug_\{_}^u| , where @xmath162 is the stabilizer of @xmath163 under the adjoint action of @xmath1 , and where we use the discrete measure @xmath164 on @xmath1 . because an element @xmath165 is a finite linear combination of basis elements @xmath134 , the integral in is finite since only finitely many elements in the sum are non - vanishing . on the image of @xmath166 , we can now define the pre - inner product form [ eq : preinnerproduct ] _ |___\ { _ } = \ { l , + . since is sesquilinear and non - negative , the image of non - zero vectors which have vanishing norm in @xmath166 can be completed with respect to it and we can construct the hilbert space as the closure ^u_. because the discrete measure @xmath164 is translation - invariant on @xmath1 , the kernel of @xmath166 is generated by elements of the form @xmath167 . in other words , gauge - equivalent states are mapped to the same distributional state . as a result , an orthonormal basis of states in @xmath157 is given by states which are labelled by gauge - orbits @xmath168 $ ] in @xmath78 . the inner product in @xmath157 is such that two such states have zero norm precisely if two of these orbits do not coincide , i.e. _ [ \{_}]|_[\{_}]=([\{_}],[\ { _ } ] ) . it should be noted that the isomorphisms commute with the gauge transformations at the root , which implies that the resulting hilbert space @xmath157 is of course independent of the choice of tree . finally , note that there exist in principle alternative versions of the rigging map which include an @xmath168$]-dependent factor . with these other choices , the resulting inner product will simply change by state normalization . now that we have introduced the hilbert space , the inner product , and a basis of states on a fixed triangulation , we are ready to discuss the quantization of the algebra of holonomy - flux observables . this will then enable us to discuss , in the next section , the construction of the area operator and the associated notion of quantum geometry . in the previous section , we have focused on the configuration space of connections associated to a triangulation . in order to obtain the phase space of lqg , we need to supplement the holonomies encoding the connection degrees of freedom with their conjugated fluxes of the electric field . this will allow us to discuss in which sense the quantum operators which we are going to define provide a representation of the holonomy - flux algebra . as discussed in @xcite , the finite - dimensional ( almost ) gauge - invariant phase space @xmath169 associated to a triangulation @xmath33 is spanned by cycle holonomies @xmath77 and rooted fluxes @xmath170 associated to the leaves . these variables , which form the rooted holonomy - flux algebra @xmath171 , are defined as @xmath172 and @xmath173 , where @xmath174 is the holonomy along the tree going from the root to the source node of the leaf @xmath51 . the flux @xmath175 is an element of the lie algebra of @xmath1 . in the case where @xmath176 for example , the poisson brackets between these variables are given by ^i_,^j_=_,^ij_k^k_,^k_,g_=_,g_^k-_^-1,^kg_,g_,g_=0 , where the @xmath177 denote the generators of @xmath178 . this is nothing but the poisson structure on @xmath179 , where @xmath180 . note that the same poisson algebra ( however with dirac brackets ) , between the holonomies @xmath181 and fluxes @xmath175 associated to the leaves , can be obtained by starting from the gauge - covariant phase space associated to the graph @xmath34 and performing a gauge - fixing holonomies associated to paths along several links , and by @xmath182 holonomies associated to single links . therefore , the gauge - fixed data is given by elements @xmath183 , while at the almost gauge - invariant level we have @xmath184 . ] . the gauge - covariant phase space is parametrized by holonomies @xmath81 and fluxes @xmath185 associated to all the links of the graph , and the poisson structure is that of @xmath186 . by choosing a tree , one can gauge - fix all the branch holonomies to be @xmath187 . the associated dirac brackets , involving @xmath181 and @xmath175 only , do then agree with the poisson brackets and lead again to the poisson structure on @xmath179 . since we are working in the connection representation , we can expect that the holonomies will be quantized as multiplication operators and the conjugated fluxes as derivative operators . while it is true that the holonomies will act by multiplication , the fluxes can however not be realized as derivative operators . this is due to the peculiar hilbert space topology which we have chosen in order to accommodate the flat state as a normalizable state an to allow for the construction of an inductive limit hilbert space . states in this hilbert space , seen as functions on @xmath78 , are generically not continuous . this effect is analogous to what happens in lqc due to the bohr compactification of the configuration space ( remember however , from the discussion of section [ overview ] , the difference between lqc and the construction based on the bf vacuum presented here ) . the way around this difficulty is to introduce exponentiated derivative operators , or , in other words , translations . indeed , we are going to see that the quantized exponentiated fluxes act as left or right translations on the appropriate copy of the group . in addition to considering the exponentiated version of an elementary flux variable , we will also have to discuss the action of exponentiated parallel - transported fluxes . it turns out that the parallel transport will appear as an adjoint action of certain holonomies on the group element parametrizing the translation . to be more precise , the exponentiation of the fluxes is to be understood as exponentiating the symplectic flow generated by the fluxes . indeed , if one sees the fluxes as representing vector fields ( acting as derivatives ) on the phase space ( see for instance @xcite ) , the exponentiated fluxes correspond to integrating these vector fields to a finite flow . let us illustrate this for one copy of the group . the symplectic flow induced by an elementary flux @xmath185 associated to a link @xmath82 is given by [ flux1 ] ( _ ix^i_l,)f(h_l ) _ k\{_ix^i_l , f(h_l)}_k = r_l^(_i^i)f(h_l ) f(h_l(_i^i ) ) , where the iterated poisson brackets are defined by @xmath188 and @xmath189 . for the link with a reversed orientation , we have that ( _ ix^i_l^-1,)f(h_l ) r_l^-1^(_i^i)f(h_l ) = l_l^(_i^i)f(h_l ) = f((-_i^i)h_l ) . in an abuse of notation , we will denote the group element parametrizing this flow by @xmath190 . so far we have considered the flow induced by an elementary flux which is not parallel - transported . note that if we understand this description as arising from a gauge - fixing , then the flux @xmath175 in the gauge - fixed version does indeed correspond to the rooted ( parallel - transported along the tree ) flux @xmath170 . hence , the ( exponentiated ) action of @xmath175 on @xmath181 agrees with the ( exponentiated ) action of @xmath170 on the cycle holonomy @xmath77 . we might however encounter further parallel transport for the fluxes , not necessary along the tree . in the almost gauge - invariant description , such a parallel transport will be along a closed loop @xmath48 , and will appear through the adjoint action of a product @xmath191 of cycle holonomies @xmath192 , which in the gauge - fixed description is just given by the corresponding product @xmath193 of leaf holonomies @xmath194 . let us assume for now that @xmath181 is not appearing in this product . we then have for the symplectic flow generated by @xmath195 the following action : ( _ i\{h^-1_x^i_h_,})f(h _ ) = f(h_h_h_^-1 ) . if @xmath193 includes a leaf holonomy @xmath181 ( such that it can not be reduced by using the inversion formula @xmath196 ) the evaluation of the flow does become very involved ( since the flux @xmath175 is meeting in the iterated poisson brackets the element @xmath181 in the parallel transport ) . we will exclude this case and not allow such a parallel transports to happen . in summary , we are going to quantize the parallel - transported fluxes @xmath197 in their exponentiated form as parallel - transported ( right ) translations , i.e. g_^-1_g _ r_^g_g_^-1 , and the parallel - transported fluxes associated to inverted links as left translations , i.e. g_^-1_^-1 g _ l_^g_g_^-1 . in all cases , we assume that the parallel transport @xmath191 does not contain the holonomy @xmath181 ( except in the case that the right translation can be rewritten into a left one and vice versa ) . note that although a representation of the fluxes as derivative operators is not available , it is still possible to approximate derivative operators in terms of translations generated by the exponentiated fluxes . this is what we will use in the next section in order to discuss the area operator . we have here considered only the fluxes associated to the leaves . it is however possible to reconstruct the fluxes associated to the branches starting from the ones associated to the leaves by solving the gau constraints . the corresponding translation operators are then products of translation operators associated to the leaves . this construction is explained in appendix [ appendix : branchx ] . let us consider the space @xmath198 of continuous functions @xmath199 over @xmath79 copies of the gauge group . being continuous functions on a compact space , these functions are bounded , and in the holonomy representation the holonomy operators acting as multiplication of states by elements of @xmath198 are therefore bounded operators . examples of such functions are matrix elements @xmath200 for a unitary irreducible representation @xmath118 . although states of the form @xmath201 are not normalizable in our inner product ( if @xmath1 is not a finite group ) , such functions are allowed to be used as multiplication operators , i.e. lead to densely defined operators . another example is given by functions of the form f\{g _ } = _ if_i_(g_,^(i ) _ ) , where the index @xmath202 runs over a finite set , and @xmath203 is the @xmath51-th entry in the @xmath202-th element @xmath204 of a finite subset of @xmath78 . again , we denote by @xmath205 the kronecker delta as defined below equation . the action of an holonomy operator on a state in @xmath95 is given by [ holop1 ] \{g _ } = f\{g_}\{g_}. the holonomy operators are straightforward to diagonalize , and the eigenvectors agree with the basis vectors @xmath134 . therefore , such an operator has a discrete spectrum given by the image of @xmath206 in @xmath207 ( or in @xmath9 if one takes the real or imaginary parts of @xmath206 ) . note that the spectrum can appear as continuous , but always comes equipped with a discrete spectral measure . as discussed in the beginning of this section , we do not quantize the fluxes themselves , but rather their exponentiated version , which leads to an action as right translation operators . to the flux associated to the @xmath202-th leaf @xmath208 we associate the operator r__i^\{g _ } = ( g_1, ,g_i, ,g_|| ) . a ( first order ) difference operator can then be defined as d^__i ( r__i^-r__i^^-1 ) , with @xmath209 denoting the geodesic distance in @xmath1 between @xmath210 and @xmath126 . the translation operators are isometric since the discrete measure employed for the inner product on @xmath95 is invariant under translations . they are also invertible , i.e. ( r_^)^-1 = ( r_^)^=r_^^-1 , and therefore unitary . it is quite instructive to discuss the spectrum of such a translation operator , and to compare it to the spectrum of a flux operator in the al representation . this will be the focus of the next subsection . let us now also discuss the action of parallel - transported exponentiated fluxes , which , as discussed above , are quantized as operators acting in the following way : r__i^g_g_^-1\{g _ } = ( g_1, ,g_ig_g_^-1, ,g_|| ) . these operators are also isometric ( and unitary ) because of the invariance of the discrete measure under translations . the action of a parallel - transported exponentiated flux on a basis state is given by r__i^g_g_^-1_\{_}\{g _ } & = _ \{_}(g_1, ,g_ig_g_^-1, ,g_|| ) + & = ( _ i_^-1_,g_i)_i(_,g _ ) + & = _ \{\{_}_i;_i_^-1_}\{g _ } , where @xmath211 denotes the same product in terms of @xmath163 as @xmath191 in terms of the fundamental cycle holonomies @xmath77 ( remember that we do not allow the link @xmath208 to appear in the parallel transport , as otherwise the last equality would not hold ) . we are now going to discuss the spectrum of the translation operator . for simplicity , let us focus on one single copy of the group , and consider the hilbert space @xmath212 , where @xmath164 denotes the discrete measure . this hilbert space being non - separable , the spectral analysis is a priori not straightforward to carry out . fortunately , since the translation operator on @xmath2 and @xmath131 is invertible , it leaves separable subspaces of @xmath213 invariant , and its spectral analysis can therefore be done on each separable subspace separately . for a fixed @xmath116 , consider the invariant subspace _ ^= , which is the closure of the space of states obtained by applying an arbitrary integer power of the translation operator to a basis state @xmath214 . we can transform the matrix representation of the group with the adjoint transformations @xmath215 and @xmath216 , in such a way that @xmath217 is diagonal with @xmath218 , and restrict ourselves to considering the translation operator for @xmath131 . let us therefore turn to the spectral analysis of the translation operator @xmath219 on @xmath131 equipped with the discrete topology . we consider basis functions @xmath220 given by @xmath221 , which we declare to be orthonormal in the kronecker delta inner product @xmath222 , where the arguments are understood modulo @xmath223 . these functions form a basis of the non - separable hilbert space = , a typical element of which is given by a function of the form ( ) = _ if(_i)__i ( ) , where @xmath202 takes values in some finite subset of @xmath224 , and @xmath225 is the @xmath131 angle . the inner product between two such functions is simply given by _ 1|_2=_()_2 ( ) = _ if_2(_i ) . the action of the translation operator on elements of @xmath213 is given by r^ ( ) = ( + ) , and its action on basis functions by . on basis vectors @xmath134 , the group elements @xmath226 evaluate to @xmath227 ( where @xmath228 does not appear ) . the discussion is therefore exactly the same , since the class angle of @xmath210 is invariant under conjugation . ] r^_=_- . as mentioned above , since this operator is invertible it leaves separable subspaces of @xmath213 invariant . the spectral analysis can therefore be done separately on each separable subspace . let us consider the sub - hilbert space which is the closure of the space of states generated by applying an arbitrary integer power of the translation operator to the basis @xmath214 , i.e. _ ^ , where the label @xmath117 in @xmath214 is to be understood in @xmath9 modulo @xmath223 . the properties of this subspace and of the translation operator acting on it will depend on whether the angle @xmath229 parametrizing the translation is rational or not . we now discuss these two cases separately . 1 . + this case corresponds to @xmath230 , which means that @xmath231 with @xmath232 such that @xmath233 . then we have that @xmath234 , and the repeated action of @xmath219 on a basis state @xmath214 generates finite - dimensional ( in fact @xmath235-dimensional ) subspaces _ ^\{_-n|}. in terms of these finite - dimensional spaces , the whole hilbert space decomposes as = _ [ 0,2/q)^_. it is straightforward to diagonalize the translation operator on the finite - dimensional spaces @xmath236 . in particular , the normalized eigenvectors are of the form [ eigen1 ] v _ , = ^q-1_n=0e^in_+n , and the associated eigenvalues are ( r^ ) = \{e^i|=0, ,q-1}. let us now fourier transform these eigenvectors using the ( group ) fourier transform introduced in appendix [ appendix : spin ] ( this will also help illustrate the case of an irrational angle @xmath229 ) . in the case of @xmath131 , the momentum representation is defined with the discrete measure by ( k ) = _ ( ) e^-ik ( ) , where @xmath237 . the inner product expressed in @xmath44-space ( and which actually defines the bohr compactification on @xmath132 ) is [ inp ] _ 1|_2=_t_|k|t_2(k ) . one would expect the functions @xmath238 to be eigenvectors of the translation operator . however , these functions have vanishing norm in the inner product . let us therefore consider the @xmath44-representation of the eigenvectors . with @xmath239 , we have v_,(k ) = _ n=0^q-1e^in_+n(k ) = e^-ik_n=0^q-1e^in(-k ) = e^-ik_(q)(,k ) , where we have used @xmath231 , the summation representation of the periodic kronecker delta @xmath240 ( with arguments modulo @xmath235 ) , and the fact that @xmath241 since @xmath235 does not divide @xmath242 . therefore , instead of vectors @xmath243 , which have vanishing norm , we have eigenvectors @xmath244 with finite norm . letting @xmath245 illustrates the fact that we have to expect generalized eigenvectors in the case of an irrational angle . this case corresponds to @xmath246 . the angle @xmath117 then labels an equivalence class @xmath247 $ ] of angles defined by the equivalence relation [ eclass ] . let us introduce the set @xmath248 of all such equivalence classes , as well as the ( countably ) infinite - dimensional hilbert spaces _ ^. the whole hilbert space @xmath249 then decomposes as = _ [ ] i_^. we prove in appendix [ appendix : spectrum ] that the spectrum of the translation operator @xmath219 on each subspace @xmath236 consists of the whole circle , i.e. ( r^ ) = ( 1 ) , and is continuous ( in the sense that the spectral measure is continuous ) . the generalized eigenvectors are given by [ irreigen ] w _ , = _ ne^in_+n , with generalized eigenvalues @xmath250 where @xmath251 . choosing a different representative @xmath252 $ ] in the equivalence class @xmath247 $ ] defined in just leads to a phase factor for @xmath253 . in @xmath44-space , where again @xmath239 , the generalized eigenvectors are given by w_,(k ) = _ ne^in_+n(k ) = e^-ik_ne^in(-k ) = 2e^-ik(-k ) , where @xmath254 is now the delta _ function _ with respect to the continuous measure @xmath255 on @xmath131 . we can in fact understand these generalized eigenvectors via a regularization procedure which uses the variable @xmath256 in the inner product as a cutoff . one considers the inner product with fixed @xmath256 and eigenvectors [ cutoffvec ] w^t_,(k ) = c(t)e^-ik(,k ) , where @xmath257 $ ] , and @xmath258 is an appropriate normalization factor . to keep the norm of @xmath259 finite when @xmath260 , the normalization factor has to grow to infinity . let us finish this subsection by comparing the spectra of the translation operator for the different cases discussed above , with the spectra of the translation operator defined on the hilbert space @xmath261 . these can be obtained by evaluating the function @xmath262 on the spectrum @xmath132 of the momentum operator @xmath263 . we then have to discuss separately the two cases for the angle @xmath229 . 1 . + if @xmath231 with @xmath232 such that @xmath233 , then we have a discrete spectrum @xmath264 , and eigenvectors @xmath238 in the @xmath44-space representation which now have finite norm . if @xmath229 is irrational , then we have a `` discrete '' spectrum @xmath265 , in the sense that the spectral measure is discrete since it derives from that of the momentum operator with discrete spectrum . furthermore , the eigenvectors are again given by @xmath238 . we therefore see that the difference between the two quantizations expresses itself in a rather subtle way , namely in the different nature of the spectral measure for the translation operator with irrational angle . now that we have an understanding of these subtleties at the level of the translation operator , we can turn to the more interesting case of the area operator . in this section we are going to introduce the area operator and discuss some of its properties . for this , we focus on the case with @xmath16 spatial dimensions . the length operator in @xmath15 spatial dimensions exhibits similar features , but it is more instructive to study the three - dimensional case in order to grasp all of the subtleties and to understand the role of the barbero immirzi parameter . recall that the fluxes arise as the smearing of the densitized triad fields , which themselves encode the information about the intrinsic geometry , and in particular the spatial metric tensor . the area of a surface can therefore naturally expressed in terms of the fluxes . let us fix a surface @xmath266 , which we can think of as being built from minimal triangular surfaces spanned by piecewise geodesic arcs . it is therefore naturally a triangulated surface @xmath267 , and we denote its elementary triangles by @xmath268 . the area of this triangulated surface is given by [ classical - area ] _ _ t_s|_i=1,2,3^i_t^i_t|^1/2 _ t_s_t , where the flux @xmath269 associated to a triangle @xmath270 is going to be defined below . here we have introduced the barbero immirzi ( bi ) parameter and its conjugated electrical field variable @xmath271 . the barbero immirzi parameter appears in the ashtekar barbero connection through @xmath272 , where @xmath273 is the spin connection and @xmath274 the extrinsic curvature ( contracted with a triad ) . the electric field result from a rescaling by @xmath275 of the densitized triad fields . therefore , the geometric fluxes are given by the fluxes @xmath276 multiplied by @xmath277 . ] @xmath277 , and set @xmath278 . before quantizing this expression , let us collect a couple of remarks which already highlight at the classical level some key differences between the area operator in the bf representation and in the al one . * the subdivision of the surface @xmath266 into a set @xmath267 of triangles @xmath268 is an essential ingredient of the definition . the choice of @xmath267 provides a `` scale '' on which the area will be measured , which implies in particular that two different triangulations of @xmath266 may a priori lead to different results . this should not be seen as a disadvantage of the framework . on the contrary , it addresses the so - called staircase ( or `` coast of britain '' ) problem for fractal geometries , which is that since the measurement of a given length depends on a choice of measuring scale , for fractal - like geometries one can in principle get infinite results in the limit where the measuring scale approaches zero . in the present framework , the triangles @xmath268 play the role of the smallest available rulers with which one can measure the geometry . as usual in loop quantum gravity , `` smallest '' should here be understood in relative terms , since the metric properties are encoded in the states themselves . what actually matters is rather the fineness of the triangulation @xmath267 as compared to the fineness of the states onto which the area operator is applied . + since one can choose @xmath267 to be much coarser than the triangulations underlying the states , the bf - based representation provides a natural way of defining coarse - grained geometric quantities . this possibility does a priori not exist in the al representation , which leads to problems when trying to construct suitable semi - classical states approximating smooth geometries ( see for instance the discussion in @xcite ) . + because of its behavior under refinement operations , the area operator introduced here is similar to the one proposed by livine and terno in @xcite . this latter is an alternative to the usual area operator in the al hilbert space @xcite . the usual area operator @xcite involves a sum over the casimir operators for each spin network edge piercing the surface . under refinement of the states , the sum is extended in order to include the casimir operators associated to the additional edges . the alternative suggestion made in @xcite is to first couple the representation spaces associated to the spin network edges going through a piece of surface ( which here is defined to be a triangle in @xmath267 ) to a total angular momentum vector space . one should then considers the @xmath2 casimir with respect to this total angular momentum . therefore , under refinement one still considers just one casimir operator , but now for a different total angular momentum vector space arising from more recoupling steps . + the difference with our new proposal is that here we have to replace the angular momentum operators with exponentiated operators ( due to the compactification of the lie algebra of @xmath2 ) , while the behavior under refinement of states is similar . indeed the cylindrical - consistency of the area operator ( i.e. its consistent behavior under refinement ) in the bf - based representation is ensured by the recoupling prescription . this follows from the discussion in section [ extflux ] , where we define the action of exponentiated flux operators on refined states , which we summarize in the next two points . * the area @xmath279 given by can be defined on all the triangulations @xmath33 which support @xmath267 , i.e. in which the triangles @xmath268 of @xmath267 arise as the possible coarse - graining of ( unions of ) triangles @xmath280 . because of this , the fluxes @xmath269 have to be thought of as composed objects . one possibility is to define these as `` integrated fluxes '' associated to the triangles @xmath268 following the definition of @xcite ( the triangle @xmath268 is then to be understood as a co - path @xmath281 in @xcite ) , i.e. [ xt ] _ t g^-1_rl_1(0)(_l_itg^-1_l_1(0)l_i(0)x_l_ig_l_1(0)l_i(0))g_rl_1(0 ) . here @xmath282 labels all the links which are dual to the triangles of @xmath33 composing @xmath268 , the source node @xmath283 of the link @xmath85 serves as a root for the surface @xmath268 , and @xmath284 is the parallel transport from this source node to the source node of the link @xmath282 . this parallel transport takes place along a surface tree ( defined in detail in @xcite ) , which is as close to the surface as possible ( and below the oriented surface ) . this means that @xmath285 ( and therefore the area operator itself ) depends on this choice of surface tree ( for every triangulation @xmath33 on which one wishes to define @xmath285 ) . note that in we also have to consider in general fluxes associated to links that are not necessarily leaves . such branch fluxes can however be expressed in terms of fluxes associated to the leaves , as explained in appendix [ appendix : branchx ] . + one can come up with alternative definitions for @xmath285 by changing the prescription for the parallel transports . for instance , one can extend the triangulation @xmath267 to a @xmath14-dimensional triangulation @xmath286 in such a way that each @xmath268 is a boundary triangle ( with outward orientation ) of a @xmath14-simplex in @xmath286 . we can then allow for a parallel transport which is not necessarily as close to the surface as possible , but rather confined to be inside the coarse @xmath14-simplex in @xmath286 which bounds @xmath268 . this definition corresponds to the notion of coarse flux observables which we will define in section [ splitting algebra ] . the area operator will then depend on this additional data @xmath286 , and on a definition of parallel transport in each of its simplices . * with the definition(s ) of @xmath285 discussed above , we can write down a consistent area operator which can be applied to states based on different triangulations . consistency means that the area operator will give the same result for states connected by the refinement operations which we define in section [ sec : refinement ] . this allows for the area operator to be defined on a continuum ( inductive limit ) hilbert space . + given a triangulation @xmath33 supporting @xmath267 or @xmath286 , we need to specify the parallel transport for the definition of the fluxes @xmath285 . this will then give a full definition of the area operator @xmath279 or @xmath287 for the triangulation @xmath33 . the area operator can also be defined for all states based on a triangulation which are coarser than @xmath33 . to this end , one just needs to refine the coarser states to states defined on @xmath33 ( in a manner defined in the second part of this work ) . however , the extension to states defined on triangulations @xmath46 finer than @xmath33 is less trivial . the basic problem is to define a notion of parallel transport which leads to consistent results for arbitrary fine triangulations . this prescription corresponds therefore to a continuum prescription , and what one needs to specify is a method for projecting this prescription to a given triangulation @xmath46 . we outline in section [ contop ] a prescription for parallel transport in the continuum which can be projected in a consistent manner to triangulations . using this method , we can define the area operator as a continuum operator . * since only the exponentiated fluxes can be quantized , we have to approximate the differential operators by difference operators . such approximations to the gauge - invariant expressions of the form @xmath288 are however not completely gauge - invariant anymore ( this concerns only the gauge transformations at the root ) . one could therefore introduce a further change of parallel transport for the ( exponentiation of the fluxes ) @xmath285 , so that the fluxes associated to the triangles @xmath268 are first transported to some node in @xmath286 , and only then to the root . this makes the definition of the area operator more local in its dependence on data near or far away from the surface . by first transporting the fluxes along the surface , we will have a more local dependence . data off of the surface appear only via the global parallel transport from the chosen node in @xmath286 to the root . let us now discuss the quantization of the area operators . as mentioned above , we have to approximate the fluxes with difference operators . to this end , we choose a parameter @xmath289 , along with a basis @xmath290 , and define @xmath291 as ^_i ( ^i ) . we then define a quantization of the quantity @xmath292 as [ areaelement ] _ i^i_l^i_l _ i(r^^+_i_l+r^^-_i_l-2 ) this quantization covers the case in which @xmath267 coincides with a triangulated surface of the triangulation of @xmath33 . in this case , the fluxes @xmath285 appearing in the expression for the area are given by the fluxes @xmath293 . the quantized area operator is then given by [ areaoperator ] ^__s = _ l_s|-_i(r^^+_i_l+r^^-_i_l-2)|^1/2 . before discussing the spectrum of this operator , let us gather a few remarks concerning its definition . * in section [ contop ] , we discuss the way in which the translation operators can be defined on the continuum hilbert space . this can be applied in order to obtain a consistent continuum definition of the area operator , as discussed above for the classical expression . * as mentioned above , the quantized area operators are not invariant under the frame rotations at the root anymore . we therefore discuss later on in this section strategies for defining a fully gauge - invariant version of the area operator . * interpretation of the parameter @xmath8 : the approximation of the flux derivative operator by difference operators is similar to the approximation of the extrinsic curvature ( or the momentum conjugated to the scale factor ) by ( point ) holonomies in loop quantum cosmology ( lqc ) @xcite . in fact , in lqc the configuration space of the scale factor is also quantized with a discrete topology , which leads to an evolution equation which is discrete in @xmath8 steps . in the case of lqc , it is necessary to adjust the step parameter @xmath8 to the value of the scale factor @xcite . in our case , @xmath8 parametrizes a translation on the group , which encodes the values of the curvature defects . one could attempt to introduce a dependence on the class angle of the group element which is being translated , which would lead to a curvature - dependent discretization step . this would in turn change the properties ( and the spectrum ) of the area operator . we will discuss such state - dependent @xmath8 parameters for the fully gauge - invariant area operator . * the barbero immirzi parameter : in , we have introduced the bi parameter by treating it as a factor which multiplies the difference operator . there exists however an alternative ( quantization ) ambiguity , which is to define the group elements by which one translates as [ altversion ] ^_i ( _ ^i ) , and to omit the prefactor @xmath277 in . in this case ( at least for @xmath131 ) the bi parameter can redefine the values of the parameter @xmath8 which lead to discrete and continuous spectra respectively . this alternative changes also the bound on the spectrum of the area operator : whereas in the version it multiplies the bound , version has a bound which is independent of the bi parameter . also , at least in the case of @xmath131 , if the barbero immirzi parameter is included using , the spectrum of the area operator does then not depend on this parameter as long as @xmath294 is irrational . let us consider the area operator associated to one triangle @xmath268 , and focus on a triangulation @xmath33 where this triangle @xmath268 appears and is not further refined . with this choice , the triangle @xmath268 is dual to a link @xmath82 , and we have that [ arqu ] ^_l = |-_i(r^^+_i_l+r^^-_i_l-2)|^1/2 . note that the ( squared ) area operator is a linear combination of bounded operators , and therefore is itself bounded . the bound does however grow with @xmath295 . this boundedness results from the compactification of the dual of the lie group , and is similar to the bound on curvature which appears in lqc due to the compactification of the momentum variable conjugated to the scale factor . this bound on the area is also similar to the quantum group deformation ( at root of unity ) , where only a finite range of representations @xmath11 are admissible . on the other hand , it is not so clear that the spectrum of the area operator is still discrete , as it is in the al representation . let us now discuss the spectra for the two gauge groups of interest . we start by discussing the case @xmath296 , where the issue with gauge transformations at the root does not arise . furthermore , in this case the translation operators commute , so we can use directly the results of the spectral analysis of the translation operators themselves . again , the properties of the area operator will depend on whether the parameter @xmath8 is rational or irrational . 1 . + this case corresponds to @xmath297 , which means that @xmath298 with @xmath232 such that @xmath233 . then we have eigenvectors of the form v__1,_1v__2,_2v__3,_3 , where @xmath299 is given by . this leads to a discrete spectrum \{|_i(2 - 2(_i))|^1/2|(_1,_2,_3)(z_q)^3 } for the @xmath296 area operator . this case corresponds to @xmath300 . we can then apply the spectral theorem on the separable invariant subspaces discussed in section [ spectrans ] . thus , we have generalized eigenvectors w__1,_1w__2,_2w__3,_3 , where @xmath301 is given by , and a @xmath302 labels an eigenvalue @xmath303 of the translation operator @xmath304 . in order to make the similarity with the discrete case more transparent , we can define implicitly @xmath305 as the integer for which @xmath306 mod @xmath223 holds ( remember that @xmath307 can indeed be understood as being concentrated on @xmath308 in the @xmath44-representation ) . in this sense , the spectrum is given by the closure of the set \{|_i(2 - 2(_i))|^1/2|(_1,_2,_3)(z_q)^3}. if we choose @xmath8 as well as the @xmath309 to be small , one finds a very good approximation to the usual case , i.e. spectral values @xmath310 . for growing @xmath309 , the bound sets in and one has an oscillatory behavior which fills densely all the allowed range , which is given by . let us now turn to the more complicated case of the gauge group @xmath2 . here , the translation operators @xmath304 do not commute , so we can not apply the results from the spectral analysis of these operators straightforwardly . however , we can define as before subspaces which are left invariant by the area operator , by taking the closure of the space of states generated by applying arbitrary integer powers of @xmath304 to a given basis state @xmath214 . if the parameters @xmath311 are elements of a finite subgroup of @xmath2 , this results in a finite - dimensional subspace with discrete spectrum . these are however only the subgroups corresponding to the platonic solids , and thus no regular family of finite groups that generates rotations in all three directions . therefore , we can rather expect to have a continuous spectrum , at least in the sense of having generalized eigenvectors ( it still could happen that there are normalizable eigenvectors , as explained below ) . we can nevertheless diagonalize the area operator by employing the spin representation laid out in appendix [ spinrepsu2 ] . this uses a ( discrete generalization of the ) group fourier transform , so that the states are wave functions of the form @xmath312 , where @xmath11 is the spin representation label and @xmath313 are magnetic indices . the translation operators act in this representation as ( r^)(j ) = d^j ( ) ( j ) , and therefore leaves the @xmath11 label and one of the magnetic indices @xmath314 invariant . the difference with the usual case is that @xmath315 is a state with vanishing norm , since the inner product is given by [ inpl ] _ 1|_2=__j=0^((_1^(j)_2(j ) ) , with a normalization factor @xmath316 ( see ) . in order to consider the ( so far non - existant ) subspaces associated to one fixed spin @xmath11 , we introduce a fixed cutoff @xmath317 on the spins @xmath11 , and define an inner product on the space of functions @xmath318 as in but without taking the limit . the associated hilbert spaces @xmath319 are then finite dimensional ( and just correspond to ( rescalings of ) @xmath320 ) . this enables us to consider vectors of the form [ jn ] ^jn_mn(j ) = c()a_m(j , n)_j , j_n , n , where @xmath321 is a normalization factor which grows to infinity if the regulator is taken away , i.e. in the limit @xmath322 . the action of the square of the area operator on a subspace spanned by the vectors , for fixed @xmath323 and any fixed @xmath324 , is given by the matrix a_mn^j ( ) = -_i(d^j_mn(_i)+d^j_mn(^-1 _ i)-2_m , n ) . we now have to diagonalize these matrices and examine the eigenvalues of the area operator on the cutoff hilbert space @xmath319 . the general behavior of the eigenvalues is similar to the case of the gauge group @xmath296 treated above , but with an important difference : for small @xmath8 and @xmath11 , the matrix @xmath325 is nearly diagonal with almost equal eigenvalues , approximating well the casimir eigenvalues @xmath326 . for growing spin , the approximate degeneracy of the eigenvalues is lifted , and the eigenvalues spread more and more . moreover , the eigenvalues are bounded , and an oscillatory behavior sets in , as can be seen on figure [ fig : area2 ] . however , it is not clear whether the oscillatory behavior will cover the entire allowed interval . axis ) of the squared area operator as a function of the spin @xmath11 ( @xmath327 axis ) , for @xmath328 , @xmath329 , and @xmath330 . ] axis ) of the squared area operator as a function of the spin @xmath11 ( @xmath327 axis ) , for @xmath331 , @xmath329 , and @xmath330 . ] so far , we have discussed the area operator associated to one single link . under refinement , this link can be subdivided into several links , and in this case the translation operators @xmath332 have to be replaced by a product of ( possibly parallel - transported ) translation operators associated to the finer links @xmath333 . however the essential features do not change : using the intertwiner - spin representation laid out in appendix [ intertwinerspin ] , one can couple the representation spaces associated to the finer links into a total spin @xmath11 . the discussion then proceeds as before , with the difference that one now has higher multiplicities for the eigenvalues caused by the recoupling procedure . as before , the spectrum is bounded by @xmath334 . moreover , we have only considered so far the case of a basic triangle appearing in the triangulation of the surface @xmath266 underlying the definition of the area operator for this surface . these basic area operators will usually commute with each other . in order to obtain the spectrum of the full area operator , one needs to add up the ( square root ) of each of these contributions of the basic triangle ( squared ) area operators . the bound on the spectrum , which is given by @xmath334 , is therefore multiplied by the number of triangles appearing in the surface @xmath266 . although the basic area operators are bounded , we can achieve a large area by subdividing a given area in many pieces . this exemplifies even more the coast of britain " paradox : the measurement scale , which here is the subdivision of the surface into triangles , determines the maximal area of this surface . to summarize our discussion on the spectrum of the area operator , let us emphasize that the compactification of the flux space , i.e. of the dual to the lie group , naturally leads to a compactification of the area spectrum for the area operator associated to a basic triangle . note that the subdivision of the surface @xmath266 is inherent to the definition of the are operator associated to this ( triangulated ) surface , and not to the triangulation on which the states are defined . therefore , by adding up the area operators associated to many triangles , we can increase the bound on the spectrum of the composed operator . furthermore , we have seen that it is possible to account for the barbero immirzi parameter @xmath277 in two different ways , which does also influence the bound on the spectrum . since the spectrum associated to a given basic area operator is bounded , it will wind around in the compact range of values specified by the bound @xmath335 $ ] . for @xmath131 , with a rational angle @xmath8 this winding happens in a periodic way and results in a discrete spectrum , while for @xmath8 irrational the winding is ergodic and gives a continuous spectrum . for @xmath2 , we can also diagonalize the area operator in the spin representation . these are indeed proper eigenvalues if one introduces a cutoff hilbert space which only admits spins smaller than or equal to @xmath317 . however , it is not clear how the spectrum changes if the cutoff is removed . the spectrum associated to small spins @xmath11 ( and sufficiently small @xmath8 ) approximates the usual area spectrum @xmath336 quite well . for larger spin values @xmath11 ( or larger @xmath8 ) , an oscillatory behavior appears . one can expect that this leads to a dense filling of at least a part of the space of allowed values . it will therefore be interesting to see what happens for the fully gauge - invariant area operator , which we now discuss . the lifting of the degeneracy of the eigenvalues in figure [ fig : area1 ] and figure [ fig : area2 ] is due to the non - invariance of the area operator under frame rotations ( i.e. gauge transformations ) at the root . let us therefore discuss possibilities for defining a fully gauge - invariant area operator . one possible starting point is to take the action of the area operator in the spin representation , and to consider the following ( normalized ) trace of the area operator : [ areag ] a^j _ ( ) = _ ma_mm^j ( ) = . if would give a well - defined operator ( on either the kinematical hilbert space or on the hilbert space of fully gauge - invariant wave functions ) , and also if we took the limit @xmath337 , we could read off the spectrum from the representation in . there , the oscillatory behavior of the sine function is suppressed by a factor of @xmath338 , which leads to a discrete spectrum for sufficiently small spins @xmath11 . to answer the question whether can give an operator on the kinematical hilbert space @xmath95 , we can consider the ( kernel of the ) operator in the holonomy representation , namely ( a_^)(h ) = _ ( h)a^_(h,h)(h ) , with [ akernel ] a^_(h,h ) & = _ _ jd_j_j(hh^-1)a^j _ ( ) + & = ( h,h)-__j_j((hh^-1))_j ( ( ) ) , where @xmath339 , and @xmath340 $ ] is the class angle of the group element @xmath117 . the sum in the second term gives the ( dirac ) delta function ( without the factor of @xmath341 ) on the equivalence classes of the adjoint action , i.e. [ suml ] _ j/2_j()_j ( ) = _ ( g)(g^-1g^-1 ) , and one could in principle interpret the area operator as a translation operator acting on these equivalence classes . however , the sum with a cutoff @xmath342 scales rather as _ j_j ( ) _ j ( ) ~\ { cl & , + ^3 & . because of this , the kernel would be equal to the kronecker delta for @xmath343 ( since the second term is suppressed in the limit @xmath322 ) , but would be vanishing for @xmath344 . the expression does therefore not define a useful area operator on the kinematical hilbert space . nevertheless , it could be possible to define matrix elements in the inner product of the fully gauge - invariant hilbert space , as defined in section [ sec : raq ] , via the raq procedure . this would lead to an averaging over the orbits of the adjoint action ( acting on the entry @xmath345 in ) with respect to the discrete measure on the group . the difficulty is however to exchange the sum and the limit in @xmath342 in with this discrete measure integral in a well - defined way . in summary , the derivation of a fully gauge - invariant area operator from the kinematical version is rather involved ( this difficulty is however lifted if one changes to a quantum group at root of unity , as will be discussed in section [ sec : discussion ] ) . an alternative way to proceed is offered by the interpretation of the kernel with respect to the continuous topology : it would then define a translation operator for the class angle . indeed , the usual area operator is given as a laplace operator on the group if we work with @xmath346 . we could therefore consider the laplace operator restricted to class functions , which is [ lap ] = _ ( ^2 _ ) , and approximate this differential operator by a difference operator . this ( discretization ) procedure does however involve lots of ambiguities , and one would rather wish for a more natural connection with the area operator defined on the kinematical hilbert space . let us therefore study how the translation operators @xmath347 , which are instrumental for the area operator on the kinematical hilbert space , act on the class angle of the argument @xmath125 being translated in the wave function . for this , let us use the parametrization g=(g)-i(g)(g),_i^=i_i , where @xmath348 are the pauli sigma matrices , @xmath349 is a unit vector , and @xmath350 . with this , the class angle of the product @xmath351 is given by [ classa ] ( g_i^ ) = ( ( g)((g))_i(g ) ) . the problem is now that this class angle depends through @xmath349 on the full group element @xmath125 , and not only on its class angle . to get rid of this dependency , we can integrate over @xmath161 the adjoint action @xmath352 , which happens to be equal to a rotation of the vector @xmath349 over the sphere @xmath353 . this will lead to the same averaged class angles for the product of @xmath125 with @xmath354 and @xmath355 , and for all @xmath356 . with the parametrization n_1=,n_2=,n_3= , where @xmath218 and @xmath357 $ ] , the integration is easiest to perform for @xmath358 . choosing the normalized invariant measure on the sphere , we obtain r^((g ) ) & _ 0 ^ 2_0 ^ 2(g_2^+ ) + & = _ -1 ^ 1((g)-x(g))x + & = ( -((g)+)+ + & + ( ( g)-)-).[theta1 ] here one can choose the roots such that @xmath359 , and furthermore ( ( x ) ) = \ { cl -x & + x & + -x & . this ensures that the result is finite for @xmath360 ( and equal to @xmath361 ) and for @xmath362 ( and equal to @xmath363 ) , as can be seen on figure [ fig : theta1 ] . between the translated and untranslated class angles , with translation parameter @xmath364 , as a function of the untranslated class angle @xmath365 . ] consider now a fully gauge - invariant hilbert space @xmath157 as defined in section [ sec : raq ] , with a basis of wave functions @xmath366}$ ] , and an area operator acting on a leaf @xmath51 . we parametrize the orbits @xmath168 $ ] under the adjoint action in such a way that the leaf @xmath51 on which the area operator is acting appears only with a class angle , i.e. @xmath168=\{\theta_\ell,\dots\}$ ] . we can then define the square of the area operator as [ areainv1 ] _ ^_\ { _ , } = -(_\{r^(_), }-_\ { _ , } ) . we therefore have a translation operator acting on the class angle with a configuration - dependent translation parameter . this makes the investigation of the spectrum more complicated , and we leave this rather important question for future work . it should be emphasized that is one particular proposal for a fully gauge - invariant area operator , obtained by averaging the shift in the class angle over the orbits of the adjoint action . it is certainly possibly to define alternative versions , for instance by discretizing the action of the laplacian on class functions . we now turn to the second main part of this work , which concerns the construction of the continuum inductive limit hilbert space . for this , we are going to introduce in section [ sec : refinement ] a few technical results concerning the refinement operations . this will then enable us to establish the cylindrical - consistency of the inner product , which is required in order for the inductive limit to be constructed . we will then discuss the relationship between continuous operators and operators defined on a fixed triangulation . in this section , we are going to construct refining maps which embed hilbert spaces @xmath95 into hilbert spaces @xmath108 associated to finer triangulations . these refinement maps are needed in order to obtain the continuum hilbert space @xmath367 as an inductive limit from the family of hilbert spaces @xmath95 . since our description of the hilbert spaces @xmath95 involves choices of trees ( for describing bases for these hilbert spaces ) , we are first going to discuss the notion of refined trees . this will then allow us to express the refinement maps in a compact way . it will also facilitate the expression of data on a finer phase space in terms of data on a coarser phase space , which will be used later on in order to provide extensions of the holonomy - flux observables associated to a given triangulation to a finer triangulation . in order to help the reader , we provide here a short summary of the various notations which are used throughout the rest of this work to describe structures related to graphs , trees , and their refinements . 1 . ll & + ^-1(l ) & + ^-1_(l ) & + & + & + _ & + _ & + _i & + _i & let us consider two triangulations @xmath33 and @xmath46 such that @xmath47 , along with their respective dual graphs @xmath34 and @xmath101 . in @xcite , we have described the projection maps @xmath368 which send paths @xmath369 in @xmath101 ( which can be viewed as paths in @xmath46 ) to paths @xmath48 in @xmath34 ( viewed as paths in the coarser triangulation @xmath33 ) . let us recall how simplices and paths in @xmath33 and @xmath46 ( and their dual graphs ) are related by @xmath368 . since @xmath46 is a refinement of the triangulation @xmath33 , we can identify each simplex @xmath370 of @xmath33 which is involved in the refinement with a union @xmath371 of simplices ( of the same dimension as the simplices @xmath370 ) in the finer triangulation @xmath46 . in other words , @xmath371 gets coarse - grained to @xmath370 under the inverse of the refining map . we denote this relationship by to denote a map from the space of path in @xmath101 to the space of path in @xmath34 , and for denoting the relationship between simplices @xmath370 in @xmath33 and the set of simplices @xmath372 in @xmath46 into which the simplices @xmath370 are refined . ] @xmath373 . consider now a path @xmath369 in @xmath101 . when seen as a path in @xmath46 , this @xmath369 goes inside @xmath14-dimensional simplices and intersects some of their boundary @xmath31-dimensional simplices . we define the path @xmath374 in @xmath33 in such a way that it enters and leaves a @xmath14-dimensional simplex @xmath375 in the same order and through the same boundary simplices @xmath376 as @xmath369 enters and leaves the complex @xmath377 with respect to the boundary complexes @xmath378 . an important property which follows from this definition of the projection maps @xmath368 is their transitivity . this can be summarized in the following result ( here we denote by @xmath379 the projection map from @xmath46 to @xmath33 ) : [ lemma : ptransitivity ] given three triangulations @xmath380 and the corresponding projection maps @xmath379 , @xmath381 , and @xmath382 , then we have that [ ptransitivity ] _ ,_, = _ , . every link @xmath37 , whether it is a leaf or a branch of a tree , has a pre - image @xmath383 in the set of paths in @xmath101 whose nature depends on the @xmath14-dimensional simplices dual to the end nodes @xmath39 and @xmath40 . if the @xmath14-dimensional simplices dual to the end nodes of @xmath82 are not involved in the refinement of the triangulation , then the pre - image @xmath383 consists of a single link @xmath333 . if at least one of the @xmath14-dimensional simplices dual to the end nodes of @xmath82 is being refined , then the pre - image @xmath383 consists of a set of paths ( which can a priori differ in their source and/or target nodes ) and therefore contains several links . in general , @xmath383 will include paths composed out of several links of the finer triangulation . it will turn out to be useful to isolate the paths which include only one link . for this , we therefore introduce the notation @xmath384 for the `` link pre - image '' , i.e. the subset of @xmath383 containing all the links @xmath333 such that @xmath385 . this link pre - image will be essential for discussing the refinement of flux observables , which are based on so - called co - paths ( a set of connected edges in @xmath15 and a set of connected triangles in @xmath16 ) . basically , @xmath386 contains all the links which are dual to the @xmath31-dimensional simplices appearing in the refinement of the @xmath31-dimensional simplex dual to the link @xmath82 . we are now going to use these projections of paths to characterize refined trees as , roughly speaking , trees whose fundamental cycles can be projected to the fundamental cycles determined by the coarser tree . consider a triangulation @xmath33 , its dual graph @xmath34 , and a finer triangulation @xmath46 obtained from @xmath33 by alexander moves . given a tree @xmath49 in @xmath34 , it turns out that not all the trees @xmath387 will be useful in our construction , not even if they satisfy the condition @xmath388 . this is illustrated with some examples in appendix [ appendix : trees ] . in what follows , we will therefore characterize the trees @xmath387 which arise as `` natural extensions '' of @xmath49 , and call them refined trees . [ def : refine - tree ] consider two triangulations such that @xmath47 , and a tree @xmath49 in the graph dual to the coarser triangulation @xmath33 . we say that a tree @xmath389 is a refinement of a coarser tree @xmath49 if the following two conditions are satisfied : * all the paths along @xmath389 are mapped under @xmath368 to paths along @xmath49 , which we write as @xmath388 . * for every @xmath14-dimensional simplex @xmath390 being refined when going to the finer triangulation @xmath46 , the nodes dual to any two simplices in the union @xmath391 can be connected by a path in @xmath389 without ever crossing a @xmath31-dimensional simplex in the boundary @xmath378 . from these two conditions , it follows in fact that * _ there exists a subset @xmath392 of the set of leaves of @xmath389 such that all the fundamental cycles determined by @xmath393 can be mapped bijectively to the fundamental cycles determined by @xmath53 . _ one can select such a set @xmath393 in the following way : for a given leaf @xmath51 in @xmath53 , consider the corresponding dual simplex @xmath394 and the subdivision of this simplex under refinement , i.e. the set @xmath395 of simplices . because of condition @xmath396 , all the simplices in this set are dual to leaves with respect to the refined tree @xmath389 . one can then choose one of the leaves @xmath397 dual to one of these simplices to be in @xmath393 and to `` represent '' the leave @xmath51 . by making this choice for all the leaves @xmath398 , we determine the set @xmath393 . note that this choice specifies additional information on top of the refined tree , and in the following we implicitly assume that with the refined tree we have also chosen such a subset . we also adopt the convention according to which the orientations of the leaves @xmath399 coincide with the orientations of the corresponding leaves in @xmath53 . therefore , if necessary , we need to invert the orientations of some of the leaves @xmath397 . one can replace condition @xmath396 with @xmath400 , which results in a weaker notion of refined tree , as illustrated with the examples of appendix [ appendix : trees ] . this weaker notion still allows for the identification of constraints in section [ constraints ] . however , condition @xmath396 makes the splitting of ( in particular the flux ) observables , into coarse and fine ones ( detailed in section [ splitting algebra ] ) much easier . there is a way to explicitly construct refined trees , which we will now explain . the construction follows two steps , and requires some choices along the way . 1 . for each @xmath14-dimensional simplex @xmath401 in @xmath33 , consider the collection of @xmath14-dimensional simplices which build up @xmath401 , i.e. the set @xmath402 . choose a maximal tree dual to this piece of the triangulation , i.e. a set of edges connecting the simplices @xmath403 , which meets every vertex , but which has no closed loops . by doing this for all the @xmath14-dimensional simplices in @xmath33 , one gets a ( non - necessarily maximal ) tree in @xmath101 . 2 . for each branch in @xmath49 , take the dual @xmath31-dimensional simplex @xmath404 in @xmath33 , and consider the collection of @xmath31-dimensional simplices in @xmath46 which build up this simplex , i.e. @xmath405 . choose one of these simplices , and add its dual link to the list of branches in @xmath389 . by doing this for all the branches in @xmath49 , one ends up with a maximal tree @xmath389 in @xmath46 . it is easy to check that the properties @xmath406 and @xmath396 of definition [ def : refine - tree ] are satisfied for the resulting maximal tree @xmath389 . in fact , it can be shown that every refined tree is the result of such a construction process . a proof of this fact is given in appendix [ app : construction ] . in the following section , we are going to use different subdivisions of the set of leaves @xmath407 of the refined tree . we now summarize what these various leaves are . * one subdivision of @xmath407 is into the representatives @xmath408 for the coarse leaves and the remaining leaves @xmath409 . this corresponds to a subdivision of the holonomy degrees of freedom into coarse holonomies and finer holonomies . the latter are constrained for configurations which arise from a refining procedure . * furthermore , for each coarse leave @xmath51 , we have a subset @xmath410 of leaves . the leaves in a given set @xmath411 give all the links which are dual to simplices @xmath412 which compose the coarse simplex @xmath413 dual to @xmath51 . for a leave @xmath414 we have @xmath415 . * the remaining leaves , i.e. the ones in @xmath416 , are either links `` inside '' the coarser simplices @xmath375 , or are part of @xmath417 ( which is not a subset of @xmath418 since it contains exactly one branch @xmath419 ) . the links in @xmath411 cross a coarser @xmath31-dimensional simplex @xmath420 which is dual to a branch @xmath50 of the tree @xmath49 . for all the leaves @xmath421 we have that @xmath422 , i.e. the corresponding cycles are mapped to the trivial cycles . finally , let us conclude this subsection by mentioning an additional important property satisfied by the notion of refined trees , which is that of transitivity . this can be summarized in the following lemma , which we prove in appendix [ proof : ttransitivity ] : [ lemma : ttransitivity ] let us consider three triangulations @xmath380 . let @xmath389 be a refined tree for @xmath49 , and @xmath423 be a refined tree for @xmath389 . then we have that @xmath423 is also a refined tree for @xmath49 . now that we have obtained criteria for characterizing refined trees and shown that such trees can always be constructed , we are going to use this to derive the constraints satisfied by the finer cycles of the refined graph . these constraints will in turn enable us to define the refinement maps , and then to separate the holonomy - flux algebra on the finer phase space into two mutually - commuting subalgebras . with our definition of refined trees , we had to choose a subset @xmath393 of `` coarse '' leaves in @xmath101 which are representatives for the leaves in @xmath34 . this means that if we index the leaves of @xmath53 with small latin letters @xmath424 , then for each @xmath425 there exists a unique leaf @xmath399 such that [ pell ] _ i = ( _ ) . we will denote this unique leave @xmath399 which maps under @xmath368 to @xmath208 as @xmath426 . therefore , @xmath427 restricted to @xmath393 is a bijection . we will use the symbol @xmath428 for the corresponding map , so that @xmath429 . for the remaining leaves in @xmath430 , there can _ a priori _ be three possibilities . the corresponding cycles can be mapped under @xmath368 to @xmath431 the trivial cycle of @xmath34 ( i.e. to the root ) , @xmath432 to a fundamental cycle @xmath433 or @xmath434 to the inverse @xmath435 of a fundamental cycle , or @xmath436 to an arbitrary non - fundamental cycle ( which can in turn be described as a combination of fundamental cycles ) . explicitly , if we index the leaves of @xmath430 with capital latin letters @xmath437 , for each @xmath438 we can therefore define , where @xmath42 and @xmath439 stand for the trivial cycles ( i.e. the roots ) in @xmath34 and @xmath101 respectivly . ] [ well ] w_i\{__i } ^-1_((__i ) ) as a word in the set @xmath440 of fundamental cycles determined by the leaves @xmath426 . however , one can show that with the definition [ def : refine - tree ] of a refined tree option @xmath436 does not occur . this is summarized in the following result : [ lemma : words ] a refined tree , together with a separation of its leaves into leaves @xmath441 and @xmath438 , is such that the words @xmath442 are either trivial or one - letter words in the cycles @xmath443 determined by the leaves in @xmath393 ( or their inverses ) . this result is tight to condition @xmath396 in the definition of the refined trees . without this condition , one can obtain in words of several letters in the cycles @xmath443 . the proof of this lemma follows from the discussion in the previous subsection , about the partition of the refined leaves into different subsets . if the leaf @xmath444 leads to the case @xmath431 , we have @xmath445 . in this case , the leave @xmath444 will not appear in any of the sets @xmath446 or @xmath447 . such a leave @xmath444 does not cross any of the @xmath60-dimensional simplices in @xmath46 which coarse grain to the @xmath60-dimensional simplices in @xmath33 . if the leave @xmath444 leads to the cases @xmath432 or @xmath434 , then there exists a unique @xmath208 such that either @xmath432 @xmath448 or @xmath434 @xmath449 . accordingly , we have @xmath450 or @xmath451 . let us now discuss how to obtain the constraints on the finer cycles of the refined tree . from equation , it follows that we have the composition of paths @xmath452 , where @xmath42 is the trivial path ( i.e. the root ) . in terms of holonomies @xmath453 along the paths @xmath454 , this means that we have found a set of independent constraints given by [ constraints1 ] _ iw_i\{g__i}g^-1__i . let us now show that these constraints are sufficient in order to assign trivial holonomies to all cycles of @xmath101 which are mapped by @xmath368 to trivial cycles . consider a non - fundamental cycle in @xmath101 which is mapped by @xmath368 to the trivial cycle ( i.e. to the root @xmath42 ) . any arbitrary cycle in @xmath101 can be described by a word @xmath455 in the fundamental cycles , which are given by the union set @xmath456 . we can therefore write that [ step1 ] ( w\{__i;__i } ) = r. we therefore have to show that the constraints imply that @xmath457 . since @xmath368 is a homeomorphism and @xmath458 is a bijection , using @xmath459 , equation is equivalent to w\{(__i);(__i ) } = w\{__i;_(_^-1((__i ) ) ) } = r. using @xmath460 and again @xmath459 we can now rewrite this equation into [ trivialw ] w\{__i;_(w_i\{__i } ) } = w\{__i;w_i\{_(__i ) } } = w\{__i;w_i\{__i } } = r. the last relation is written in terms of the fundamental cycles of @xmath34 , which generate a free group . the word @xmath461 on the right - hand side of can therefore be reduced to the identity by applying successively the concatenation of paths @xmath462 . in terms of holonomies , the constraints imply that @xmath463 . by using these relations in the word @xmath464 and by following the same steps of reductions ( using @xmath465 ) as for the cycles , we can show that the word @xmath464 indeed reduces to the unit element of the group . the notion of refined tree allows us to define a splitting of the algebra of observables on a finer triangulation , into a set of coarser observables which can be mapped to the observable algebra on a coarser triangulation , and into a set of finer observables . this split will preserve the symplectic structure and can therefore be used to define the coarse - graining of spin network states ( see for instance @xcite ) . this circumvents a problem noted in @xcite and coined `` curvature - induced torsion '' , namely the fact that if one coarse - grains an entire region into a ( spin network ) node , the fluxes associated to the links adjacent to this note do not necessarily need to satisfy the closure ( or gau ) constraint anymore . this problem is avoided here by making the splitting of the algebra at the ( almost ) gauge - invariant level . here we use a tree to determine the ( almost ) gauge - invariant algebra of observables . therefore , every region which is coarse - grained into a node is entered by at least one tree . the flux associated to this tree is however not part of the observable algebra , so one can thus not keep track of the closure constraint after coarse - graining . a splitting of the algebra of observables underlies also the framework of @xcite aimed at enlarging the lqg state space . however , @xcite leaves open the treatment of the gau constraints . it was noted there that this requires indeed to work with parallel - transported fluxes . here we do provide a full description of the corresponding algebra and a splitting of this algebra into finer and coarser observables . let us recall the rooted holonomy - flux algebra @xmath171 , given by holonomies @xmath172 and fluxes @xmath173 . here @xmath174 is the holonomy along the tree going from the root to the source node of the leaf @xmath51 . the poisson brackets between these variables are given by ( for @xmath176 ) ^i_,^j_=_,^ij_k^k_,^k_,g_=_,g_^k-_^-1,^kg_,g_,g_=0 , where the @xmath177 denote the generators of @xmath178 . we are going to show that , when considering a refined triangulation @xmath99 , the algebra of ( almost ) gauge - invariant observables can be split into two mutually commuting ( with respect to the symplectic structure ) algebras @xmath466 and @xmath467 , which correspond respectively to coarse and fine observables . let us choose a refined tree in the graph dual to the refined triangulation and , following the notation introduced above , denote by @xmath468 the set of oriented leaves @xmath469 which satisfy the condition @xmath470 . furthermore , let us denote by @xmath471 the set of oriented leaves in @xmath407 which satisfy @xmath472 . the set @xmath473 therefore contains all the links dual to the fine @xmath31-dimensional simplices of @xmath46 which coarse - grain to the coarser @xmath31-dimensional simplex of @xmath33 dual to @xmath51 . with this notation , we can now define on the phase space @xmath474 the refined extension is an extension of @xmath170 , which is different from @xmath475 . ] @xmath476 of a flux observable @xmath477 as [ fluxrefined ] __i _ _^-1(_i)_+__^-1(_i^-1)_()^-1 . this definition is a priori not the same as that of the integrated fluxes @xmath478 introduced in @xcite for co - paths @xmath281 of @xmath31-dimensional simplices in @xmath33 . for the fluxes @xmath478 , the parallel transport of the elementary fluxes ( associated to the @xmath31-dimensional simplices building up @xmath281 ) is defined through the notion of a shadow tree in the shadow graph of the co - path @xmath281 , and therefore stays as close as possible to @xmath281 . here however , the parallel transport of the fluxes used to define is along the sub - tree which remains inside a set @xmath391 of @xmath14-dimensional simplices . fortunately , the constraints impose the flatness of the parallel transport inside the coarse simplices . one can thus replace the parallel transport for the integrated flux by a parallel transport along a surface tree without changing its action on refined states ( classically and on the constraint hypersurface ) . for the sake of obtaining a splitting into mutually commuting subalgebras , we will stick with the definition . the changes in the parallel transport described above might lead to commutators that vanish ( only ) on the constraint hypersurface . we are now ready to split the holonomies and fluxes on @xmath474 into coarse and fine observables . for this , remember that with our definition of refined trees there is a natural separation of the set of leaves of @xmath389 into two subsets @xmath393 and @xmath479 . the decomposition of the holonomy - flux algebra on @xmath474 consists in assigning different observables for the leaves in these subsets . explicitly , the coarse observables in @xmath466 are given by [ hfcoarse ] ( g__i,__i ) , _i _ , with the fluxes @xmath476 defined as in . depending on the type of leaf @xmath438 , the fine observables in @xmath467 are given by [ hffine ] @xmath480 this separation of the holonomies follows from the constraints and the result of lemma [ lemma : words ] . with this separation , we now have the following result , which we prove in appendix [ proof : separation ] : [ lemma : separation ] the coarse observables and the fine observables form two mutually - commuting subalgebras of the rooted holonomy - flux algebra @xmath171 , which we denote respectively by @xmath481 and @xmath482 . we can now introduce the refinement maps @xmath483 acting on the hilbert spaces @xmath95 . we have seen that given a tree @xmath484 and a refined tree @xmath387 , there is a subdivision of the set @xmath407 of leaves of @xmath389 into the two disjoint sets @xmath393 and @xmath430 . the leaves in the first set are labelled with small latin indices , while that of the second set carry capital latin indices . for every leave @xmath438 , we have a constraint [ constraints2 ] _ i w_i\{g__i}g^-1__i , as introduced in . this constraint can be used to to define the refinement map in the following way : [ def : refinement ] for a state @xmath485 , we define the action of the refinement map as [ refined ] _ , ()\{g__i;g__i } \{g__i;g__i } = \{g__i}_i(w_i\{g__i}g^-1__i , ) . this definition means that in order to obtain the refined wave function @xmath486 , we first have to replace the variables @xmath487 with the variables @xmath488 associated to the leaves @xmath489 in the set @xmath490 ( this set resulted from a choice of one representative leave in the finer triangulation @xmath46 per leave @xmath208 of the coarser triangulation @xmath33 ) , and then multiply the state with the ( discrete measure ) delta functions peaked on the constraints @xmath491 . one can equivalently replace any of the constraints @xmath492 in the argument of the delta functions in with constraints @xmath493 . in this way , we match the splitting of the observable algebra associated to @xmath46 into coarser and finer observables as described in section [ splitting algebra ] . we are now going to address an important point , which is the gauge - independence of the refinement maps . in the definition , the state and its refinement are expressed with respect to a particular gauge - fixing , i.e. a choice of tree , a choice of refined tree , and a choice @xmath393 for the set of coarse leave representatives . we are going to prove that our construction is independent from this gauge information , i.e. that gauge transformations commute with the refinement operations . let us consider a change @xmath145 of tree , a change @xmath494 of refined tree , and with it a change @xmath495 of the set of coarse leave representatives . this leads to a different basis for the fundamental cycles . if we denote the two different basis and their splitting into coarse and fine cycles by @xmath496 and @xmath497 , the change of basis can be written as _ _i = w__i\{__j;__j},__i = w__i\{__j;__j}. therefore as discussed in section [ change of tree ] , the associated gauge transformation of the refined state is given by [ refrans ] ( )\{g__j;g__j } = \{w__i\{g__j;g__j}}_i(w_i\{w__i\{g__j;g__j}}w^-1__i\{g__j;g__j } , ) . our task is to show that this state is equal to the state which we would obtain by applying first a gauge transformation @xmath145 and then the refinement operation . we are going to study the two terms of this expression separately . let us first consider the delta function factors . these give only a non - vanishing value equal to @xmath126 to holonomy configurations which satisfy the constraints . these constraints can be expressed with respect to different choices of trees and sets @xmath393 . let us denote the constraints in the cycle basis @xmath143 by _ j = _ j\{g__j}g^-1__j . as shown below , these constraints are independent and sufficient to determine the constraint hypersurface on which every `` finer '' cycle carries a trivial holonomy . this property does not depend on the choice of cycle basis being used , and therefore we can write that _ i(w_i\{w__i\{g__j;g__j}}w^-1__i\{g__j;g__j } , ) = _ j(_j\{g__j}g^-1__j , ) . let us now focus on the first factor on the right - hand side of . first , notice that due to the delta functions we can proceed to the following replacement : \{w__i\{g__j;g__j } } \{w__i\{g__j;_j\{g__j}}}. now , the argument of the state on the right - hand side also gives the transformation law for the fundamental cycles in @xmath34 , which can be obtained by applying @xmath368 to the fundamental cycles in @xmath34. this means that _ _ i = w__i\{__j;_j\{__j } } implies for the cycles @xmath433 and @xmath498 with respect to the trees @xmath49 and @xmath137 respectively the relation _ _ i = w__i\{__j;_j\{__j } } = w__i\{__j}. therefore , we can finally write that ( )\{g__j;g__j } = \{w__i\{g__j}}_j(_j\{g__j}g^-1__j , ) , and the right - hand side corresponds to the state which would have been obtained by applying first a gauge transformation @xmath145 in the coarse triangulation and then refining using a refined tree @xmath499 and a choice of set @xmath500 . this shows that the refinement commutes with gauge transformations , and that the refinement operation is independent , the delta function factor in leads to the same set of configurations on which the refined state does not vanish . second , a different choice for the set of coarse leave representatives will also replace some leaves @xmath426 by @xmath444 in the argument of @xmath147 in . however , any @xmath444 that replaces a @xmath426 needs to satisfy @xmath501 . due to the delta function factors we can therefore replace back @xmath502 with @xmath488 . ] from the particular choice of coarse leave representatives . we now prove another very important property of the refinement maps , which is that they satisfy a transitivity condition . this is needed in order for the inductive limit construction of the continuum hilbert space to be possible . we have the following result : [ lemma : transitivity ] the embedding maps @xmath503 defined in satisfy _ ,_, = _ , . in order to prove this result , it is sufficient to show that it holds for the basis vectors , and for a choice of tree in @xmath34 and of refined tree in @xmath101 . the trees define an orthonormal basis for the two hilbert spaces @xmath95 and @xmath108 respectively . in terms of these bases , the refinement operation is given by [ eq : transitivityofrefinements ] _ , _\{_}=_\{_ } , where @xmath504 is determined as follows : [ labelref ] @xmath505 \alpha_{\ell'}'&=\alpha_\ell\q&&\text{if $ \mathbf{p}_{\delta',\delta}(\gamma_{\ell'})=\gamma_\ell$},\\[5pt ] \alpha_{\ell'}'&=\alpha^{-1}_\ell\q&&\text{if $ \mathbf{p}_{\delta',\delta}(\gamma_{\ell'})=\gamma_\ell^{-1}$.}\end{aligned}\ ] ] now , let us consider three triangulations @xmath506 , a tree @xmath49 , a refined tree @xmath389 for @xmath49 , and a refined tree @xmath423 for @xmath389 . by virtue of lemma [ lemma : ttransitivity ] , @xmath423 is then also a refined tree for @xmath49 . therefore , we have that @xmath507 with @xmath508 \alpha_{\ell''}''&=\alpha_\ell\q&&\text{if $ \mathbf{p}_{\delta'',\delta}(\gamma_{\ell''})=\gamma_\ell$},\\[5pt ] \alpha_{\ell''}''&=\alpha^{-1}_\ell\q&&\text{if $ \mathbf{p}_{\delta'',\delta}(\gamma_{\ell''})=\gamma_\ell^{-1}$.}\end{aligned}\ ] ] a similar statement holds for @xmath509 . by employing the transitivity of the projection maps @xmath368 described in lemma [ lemma : ptransitivity ] , this shows the transitivity of the refinement maps @xmath510 . @xmath511 with the refinement maps at hand , we can finally define the inductive limit hilbert space @xmath367 . such an inductive limit construction is based on a partially - ordered directed set , which is here provided by the set of triangulations @xmath512 . the refinement maps ) satisfy @xmath513 the inductive limit is defined as the following disjoint union on which an equivalence relation is imposed : _ = , where @xmath514 and @xmath515 are equivalent , i.e. @xmath516 , if there exists a triangulation @xmath517 such that @xmath518 . in words , the two states are equivalent if they become eventually equal under refinement . the inductive limit carries a linear vector space structure which is inherited from the vector space structure on @xmath95 together with the linearity of the refinement maps . the inductive limit can also inherit the inner product from the hilbert spaces @xmath95 . for this , we need the refinement maps to be isometric , i.e. to satisfy the condition [ isometry ] _ , (_1)|_,(_2)_ = _1|_2_=_1|_2 _ , which is also known as cylindrical - consistency for the inner product . this condition makes the inner product well - defined on the equivalence classes @xmath519 $ ] of states defined above . in order to compute the inner product between two states @xmath514 and @xmath515 , one chooses a common refining triangulation @xmath517 and takes the inner product between the states @xmath520 and @xmath521 in @xmath522 . the condition ensures that the result does not depend on the choice of the common refining triangulation @xmath517 . let us therefore show that the refinement maps are indeed isometric . it is sufficient to show this for a set of basis states . in , we have considered the following refinement of the basis states : [ eq : transitivityofrefinements2 ] _ , _\{_}=_\{_ } , where @xmath504 is determined by [ labelref ] @xmath505 \alpha_{\ell'}'&=\alpha_\ell\q&&\text{if $ \mathbf{p}_{\delta',\delta}(\gamma_{\ell'})=\gamma_\ell$},\\[5pt ] \alpha_{\ell'}'&=\alpha^{-1}_\ell\q&&\text{if $ \mathbf{p}_{\delta',\delta}(\gamma_{\ell'})=\gamma_\ell^{-1}$.}\end{aligned}\ ] ] this refinement operation does indeed leave the inner product between basis states invariant in the sense that [ innerproductc ] _ \{_}|_\{_}_= _ ( _ , _ ) = _ , _\{_}|_,_\{_}_ , where we have made explicit the triangulation on which the inner product is computed . this gives an inner product which is well - defined ( with respect to the equivalence relation ) on the inductive limit hilbert space @xmath367 . in equation of appendix [ spinrepsu2 ] , we have defined a measure functional @xmath523 which reproduces the inner product . this measure is also well - defined on the continuum hilbert space @xmath367 . it can be expressed via its action on basis states as _ ( _ \ { _ } ) = _ ( _ , ) . finally , let us note that the construction of the fully gauge - invariant states via the raq procedure in section [ raqpro ] can also be extended to the continuum hilbert space . these fully gauge - invariant states are then linear functionals on ( a dense subspace of ) the kinematical hilbert space @xmath367 . the basic argument why this construction is cylindrical consistent is that the stabilizer group associated to a label @xmath135 for a basis state @xmath134 does not change if this state is refined . this follows from the action of the refinement on the labels which is detailed in : an @xmath79-tuple @xmath135 gets extended to an @xmath524-tuple @xmath525 by @xmath406 including entries equal to the identity group element , @xmath396 copying an entry @xmath117 a number of times , and @xmath526 copying the inverse of an entry @xmath527 a number of times . therefore , any @xmath161 which leaves the label @xmath155 invariant under the adjoint action does also leave @xmath504 invariant . in addition to having a well - defined inner product on the inductive limit hilbert space @xmath367 , we would also like to have well - defined operators @xmath528 acting on @xmath367 ( here and in what follows we will denote operators without using hats on top of the symbols ) . assume that we have at our disposal a continuum operator @xmath528 , i.e. a prescription for how @xmath528 acts on states @xmath529 for any triangulation @xmath33 ( possibly by mapping these states to finer hilbert spaces ) . the condition of cylindrical - consistency for such an operator @xmath528 is then [ cco ] _ , = _ , . this condition guarantees that applying the operator to states in the same equivalence class @xmath519\in\mathcal{h}_\infty$ ] gives the same result . we will comment more in section [ contop ] on how to obtain such continuum operators . here , we are rather going to focus on the following question : given an operator @xmath530 and a finer triangulation @xmath46 , we would like to define an extension @xmath531 which acts on the finer hilbert space @xmath108 and satisfies [ consistency o ] _ , (_)_, = _ , _. one could hope that starting with a given @xmath532 and extending it to all possible finer triangulations , one will end up with a description of a continuum operator @xmath528 . however , the difficulty is that the extension maps which we are going to define are in general not transitive . transitivity can nevertheless be obtained if one has a consistent way to project a continuum operator onto the different @xmath95 . we will discuss this option in section [ contop ] . for this discussion it will be helpful to first understand how an operator on a fixed triangulation can be extended to a finer triangulation . this is going to be the topic of the present section . an operator @xmath532 determines the properties of any extension @xmath533 on the subspace of states which are cylindrical consistent over @xmath33 ( i.e. states which can be obtained via refinements from @xmath95 ) . for instance , the spectrum of @xmath532 coincides with the spectrum of @xmath533 restricted to this subspace . for a discussion of the notion of ( extended ) observables in the classical theory , and the relation to symplectic reductions , we refer the reader to @xcite . let us recall how the refinement maps are defined . for a state @xmath534 , the refinement map acts as [ refa ] _ , ()\{g__i;g__i } \{g__i;g__i } = \{g__i}_i(w_i\{g__i}g^-1__i , ) . in words , this means that we first replace the variables @xmath487 associated to the leaves @xmath425 with the variables @xmath488 associated to the leaves @xmath426 in the set @xmath490 , and then multipliy the state with the ( discrete measure ) delta functions peaked on the constraints @xmath535 . we are now going to discuss separately the extension of the holonomies , of the fluxes , and of the parallel - transported fluxes . let us start the discussion with a holonomy operator @xmath536 . we define ( one version of ) an extension of this holonomy operator as [ exthol ] _ , (f\{g__i } ) f\ { g__i}. this definition means that we just replace the arguments @xmath487 in @xmath206 with @xmath488 ( with @xmath537 ) , and then see the resulting @xmath206 as a function on @xmath103 which is constant in all arguments @xmath502 ( with @xmath538 ) which parametrize the finer holonomies . it is straightforward to see that this refinement map satisfies the extension property . indeed , we have that [ refinedhol ] _ , (f)\{g__i;g__i } & = f\{g__i}\{g__i}_i(_i^1\{g_ } , ) + & = _ , (f)_,()\{g__i;g__i}. the prescription for extending holonomy operators is not the only possible one . for example , the following changes do not affect the action of @xmath539 on refined states , and therefore do not affect the extension property : 1 . one can multiply @xmath539 with any other function on @xmath103 which evaluates to the identity on the constraint hypersurface . any of the arguments @xmath488 in the extended holonomy operator @xmath540 can be replaced by @xmath488 multiplied with any word in the constraints @xmath535 and @xmath541 ( or their inverses ) , i.e. one can proceed to the replacement g__i g__iw\{_i;_i^-1}. this second modification amounts to deforming the path underlying the holonomies . the original and deformed path must however be in the same homotopy class of paths on @xmath62 . this freedom in changing paths is essential for the construction of continuum observables as discussed in section [ contop ] . as explained in section [ sec : qfluxes ] , we obtain right translation operators @xmath542 as quantizations of the exponentiated ( flow of the ) fluxes @xmath170 , and left translations as quantizations of the exponentiated ( flow of the ) fluxes @xmath543 . in section [ splitting algebra ] , we have defined one possible extension for the fluxes , when seen as classical phase space functions . based on the choice of refined tree @xmath389 in @xmath46 , the extended flux is given by [ fluxrefined2 ] __i _ _^-1(_i)_+__^-1(_i^-1)_()^-1 . accordingly , we can define an extension for the operators @xmath542 as _ , (r^ _ ) _ _^-1(_i)r^___^-1(_i^-1)l^_. in order to see that this definition satisfies the extension property , first note that @xmath544 leaves the ( kronecker ) delta functions in the constraints invariant , i.e. [ cflux1 ] _ , (r^_)(_i , ) = ( _ i , ) . this relation follows at the classical level from the fact that the fluxes @xmath476 poisson - commute with either @xmath535 or @xmath541 ( they commute with both versions weakly ) . to see this in the quantum theory , consider for instance the case described by equation . there , we have a constraint @xmath545 associated to a leave @xmath444 which is in the link pre - image @xmath546 of some leave @xmath425 . it is therefore not included in any other link pre - image @xmath547 with @xmath548 . for this reason , @xmath549 will not affect @xmath545 for any @xmath548 . for the case @xmath550 , we have that _ , (r^__i)f(g__ig^-1__i ) = ( r^__ir^__i)f(g__ig^-1__i ) = f(g__ig^-1__i ) , for any function @xmath551 . therefore , the refined fluxes leave in particular @xmath552 ( and also @xmath553 since the argument is only changed by a conjugation ) invariant . similarly , one can go through the remaining cases . furthermore , it is easy to see that the action of @xmath554 on @xmath555 equals the action of @xmath556 on @xmath557 . since the refinement maps multiply the states with delta functions in the constraints , and then replace the arguments @xmath557 in the coarse wave function with @xmath555 in the finer wave function , we can now conclude that the extension map for the fluxes satisfies the condition . again , we can change a given extension of a translation operator without violating the extension property . a particularly important class of changes is to include a parallel transport for the fluxes , in the form ^_,(r^ _ ) _ _^-1(_i)r^g[](g[])^-1__^-1(_i^-1)l^g[](g[])^-1_. here @xmath558 $ ] is an association to @xmath397 of an holonomy along a cycle ( starting and ending at the root ) which is homotopy - equivalent to the trivial cycle in @xmath62 . this means that @xmath558=\openone$ ] if the ( coarse - graining ) constraints @xmath559 are satisfied . in order to avoid ordering issues , one can demand that for a fixed coarse leave @xmath282 , none of the parallel transports @xmath558 $ ] associated to leaves @xmath397 in the set @xmath560 ( i.e. the set of leaves that appear in the refined translation ) include any of the leaves in @xmath561 . indeed , the parallel transport along a surface tree , as defined in @xcite , does not include any links which cross this surface . one can therefore define an extension which matches the definition for an integrated flux ( where the finer fluxes are parallel - transported along a surface tree ) given in @xcite . the restrictions on the set of leaves which can appear in the parallel transport can however be weakened . for this , one can choose an ordering of the elementary translation operators by numbering the leaves in @xmath562 , so that @xmath563 , and then order the product of translation operators from @xmath564 ( appearing on the right ) to @xmath565 ( appearing on the left end ) . the condition on a parallel transport @xmath566 $ ] is then that it should @xmath431 be homotopy - equivalent to the trivial cycle in @xmath62 , @xmath432 not include the link @xmath567 itself , and @xmath436 be such that @xmath566 $ ] is left invariant by the product of translation operators associated to the leaves @xmath568 . let us now close this section by discussing the refinement of the parallel - transported translation operator @xmath569 , where @xmath191 is a holonomy associated to a ( rooted ) cycle @xmath48 which does not include the link @xmath208 itself . the parallel transport @xmath191 is a word in the holonomies @xmath570 and their inverses for @xmath548 , i.e. @xmath571 . we define @xmath572 as the same word in the holonomies @xmath573 , i.e. @xmath574 . we can then define the extension of a parallel - transported translation operator as [ extensionpf ] _ , (r^g_g^-1___i ) _ _^-1(_i)r^g_(g_)^-1___^-1(_i^-1)l^g_(g_)^-1_. note that @xmath572 does not include any leaves in @xmath560 , and that therefore there are no ordering issues . let us now discuss the interpretation of the parallel transports @xmath572 in the finer triangulation . the holonomy @xmath191 describes a priori a rooted cycle in the coarse triangulation . but it can also be interpreted as giving a parallel transport , alternative to the one provided by the tree , for the flux based at the source node @xmath575 of @xmath282 . let us write @xmath191 as @xmath571 . then , this alternative path is going from the root @xmath42 to the source vertex @xmath576 of @xmath208 , and is given by t_r_i(0)w_\{t_r_j(1)^-1l_jt_r_j(0)}. here @xmath577 is the path along the tree from the root @xmath42 to the source node of @xmath578 , and @xmath579 is the path along the tree from the root @xmath42 to the target node of @xmath578 . in the refined triangulation , @xmath572 provides in the same way alternative paths for the parallel transports of the fluxes @xmath580 . for a fixed @xmath581 , the path starts at the root @xmath439 and ends at @xmath582 . it is given by t_r(0)w_\{t_r_j(1)^-1l_jt_r_j(0)}. likewise , if @xmath583 , the path ends at @xmath584 , and is given by t_r(1)w_\{t_r_j(1)^-1l_jt_r_j(0)}. again , one can also choose alternative extensions for the parallel - transported translation operators @xmath585 , in particular by changing the parallel transports @xmath572 . for each @xmath580 , one can replace @xmath572 with a ( @xmath397-dependent ) transport along an homotopy - equivalent ( in @xmath62 ) path to @xmath48 . as before , in order to avoid ordering issues , one can demand that these new paths should not include any @xmath580 . this condition can be weakened in a similar way as discussed previously for the translation operators . in the previous section , we have discussed extensions of operators on a given triangulation @xmath33 to a refined triangulation @xmath46 . this allows to define a consistent family of operators given on a sequence @xmath586 of finer and finer triangulations . by taking the operator on a very fine triangulation @xmath587 in this sequence , it is also possible to extend this consistent family to all triangulations coarser than @xmath587 . for this , we just have to first refine states from the coarser triangulation to @xmath587 , and then apply the operator @xmath588 to such states . this can be repeated with an even finer triangulation @xmath589 , which then gives a consistent definition on all triangulations which are coarser than @xmath590 ( agreeing with the previous definition on triangulations where it has already been defined ) . however , there is no guarantee that starting with an operator @xmath532 and then extending this operator to @xmath46 and @xmath517 independently will give a definition which agrees on a triangulation that can arise as coarse - graining of @xmath46 and @xmath517 respectively . the challenge is therefore to extend a given observable @xmath532 to all triangulations at once in a consistent way . let us first discuss one possibility which does lead to a consistent family of observables , but however _ not _ to a continuum observable in the usual sense ( although it can be defined on the inductive limit hilbert space @xmath367 ) . given @xmath591 on @xmath592 we define extensions to any finer @xmath108 by [ nullextension ] ( _ _ 0)_ ( _ _ 0)_i(_i , ) , where @xmath535 are the ( coarse - graining ) constraints describing states in @xmath108 which arise as refinements from @xmath592 . here @xmath593 is any of the allowed extensions of @xmath591 to @xmath46 . it does not matter which extension one chooses , since the operator @xmath594 is only non - vanishing on states which arise as refinements of @xmath595 . the actions of different extensions @xmath593 do agree on such states . we can also define @xmath596 for any @xmath33 which is not a refinement of @xmath595 , by first refining a state in @xmath95 to a common refinement of @xmath595 and @xmath33 , and then applying the prescription . the family @xmath594 of observables constructed in this way satisfies the cylindrical - consistency conditions . it therefore defines an operator on the inductive limit hilbert space @xmath367 . the observables do however vanish on any state which is not cylindrical over @xmath595 ( i.e. which is not a refinement of a state in @xmath592 ) . such operators do therefore not correspond to quantizations of `` continuum '' flux and holonomy functions . the classical version they correspond to are rather such functions multiplied with a ( infinite ) product of delta functions imposing that curvature is vanishing almost everywhere . let us emphasize again that , nevertheless , the observables are well defined . likewise , their spectrum has a well defined meaning . this holds in particular with respect to the interpretation of the inductive hilbert space construction as corresponding classically to a symplectic reduction @xcite . that is , the spectrum of @xmath532 coincides with the spectrum of @xmath597 restricted to the subspace of states which are cylindrical over @xmath33 . the spectrum of @xmath594 is given by @xmath598 . this leaves us with the question of whether one can construct quantum observables on @xmath367 which would actually correspond to quantizations of continuum versions of flux and holonomy observables . let us sketch such a construction here , starting with an holonomy operator . to discuss the holonomy operator , we will assume that the path @xmath48 underlying the holonomy is piecewise - geodesic in the auxiliary metric on @xmath13 . the holonomy operator is unambiguously defined on a triangulation @xmath33 if the underlying path is a path in the inside of the manifold @xmath62 , i.e. if it avoids the defects of this manifold . we call the set of triangulations @xmath33 on which this is the case @xmath599 . this is itself a partially - ordered directed set , and the corresponding inductive limit hilbert space gives a subspace @xmath600 of the full inductive limit hilbert space @xmath367 . the holonomy operator can unambiguously be defined on this subspace . in the case in which the path @xmath48 does hit the boundary of @xmath62 one needs to choose a regularization of the operator ( or rather of the underlying path ) . any such choice of regularization corresponds to an extension of an operator from @xmath601 to a bigger subspace . such extensions of a ( bounded ) operator from @xmath600 to a ( bounded ) operator @xmath367 always exist , but they are in general not unique . for the translation operator ( or the parallel - transported translation operator ) , the main issue is again to unambiguously define the paths underlying the parallel transports of the translation operators associated to the finer links . in particular , for the notion of integrated flux over a surface @xmath266 ( in @xmath16 spatial dimensions ) , we have to specify a parallel transport for all points on this surface to , e.g. , a surface root . the complicated case is here @xmath16 . indeed , for @xmath15 there is a canonical construction for the parallel transport of the fluxes , which are integrated along a one - dimensional `` shadow graph '' as explained in @xcite . for the following discussion , we therefore restrict ourselves to @xmath16 , and furthermore assume that the auxiliary metric is flat , so that the notion of piecewise - geodesic reduces to that of piecewise - linear . let us consider the integrated flux observables associated to a surface @xmath266 , which is itself piecewise - linear and more precisely comes from the gluing of ( flat ) triangular surfaces @xmath268 . the continuum parallel transport for this surface is specified as follows . first , this parallel transport is required to take place in the surface @xmath602 which is infinitesimally shifted below @xmath266 ( with respect to the orientation of @xmath266 ) . second , we choose in each triangle @xmath268 of @xmath266 an inner reference point @xmath603 , referred - to as a @xmath268-root , to which all the inside points of @xmath268 are parallel - transported along geodesics ( which are here straight lines ) . finally , we choose a tree , referred - to as an @xmath266-tree , in the graph dual to the triangulation of @xmath266 . the root of this @xmath266-tree should agree with that of the @xmath268-roots @xmath604 . furthermore , the tree should connect the remaining @xmath268-roots @xmath603 to @xmath604 as follows : the nodes @xmath603 of two neighboring triangles are connected by a geodesic restricted to @xmath602 , i.e. a possibly kinked line in the @xmath16 flat space , and a straight line if considered from the induced ( flat ) metric of the two triangles . this description defines the parallel transport for the continuum observable . we now have to construct a projection of this parallel transport to a given triangulation @xmath33 . this triangulation should be sufficiently fine in order to support the surface @xmath266 , i.e. the triangular surfaces @xmath268 are unions of two - dimensional simplices @xmath605 of @xmath33 . consider the cut of the surface @xmath602 with the triangulation @xmath33 . by shifting @xmath266 infinitesimally to @xmath602 , we avoid the potential curvature singularities ( which in @xmath16 are along line defects ) along the edges of the triangulation @xmath33 ( remember that we also assume that the edges of the triangulation are geodesics , i.e. straight lines , and that the triangles of the triangulation are minimal surfaces , i.e. flat ) . the edges of @xmath602 might now be `` thickened '' , since it could happen that more than two ( downside ) tetrahedra hinge on a given edge of @xmath266 . we can however ignore this thickening since the path of the parallel transport across these additional tetrahedra is unambiguous . what needs to be clarified is the path for the parallel transport across the two - dimensional triangulation of @xmath266 which is induced by @xmath33 . this induced triangulation @xmath606 of @xmath266 will be a refinement of the original triangulation of @xmath266 into triangles @xmath268 , as shown on figure [ fig : subdivision ] . we have to specify the parallel transport paths on @xmath607 , where @xmath608 denotes the vertices of @xmath606 . we assume that the triangulation @xmath33 is such that none of the @xmath268-roots @xmath603 coincide with a vertex of the induced triangulation or lies on an edge of this induced triangulation . we furthermore assume that the vertices of @xmath608 do not touch any of the paths along the @xmath266-tree . therefore , we are just left with the task of specifying the tree for the parallel transport inside a given triangle @xmath268 . we construct the parallel transport on @xmath609 as follows . since the @xmath268-root @xmath603 lies in the inside of one of the induced triangles @xmath605 , the parallel transport for a point @xmath242 of this triangle @xmath610 to @xmath603 is therefore uniquely specified along the geodesic from @xmath242 to @xmath603 . let us consider some other induced triangle @xmath611 of @xmath268 . then we can have the following two cases : 1 . the parallel transports along straight lines of all the points in @xmath611 are homotopy - equivalent in @xmath607 . this means that for any two points @xmath612 and @xmath613 lying inside this induced triangle , the piecewise - geodesic path from @xmath603 to @xmath612 , then to @xmath613 , and back to @xmath603 , is homotopy - equivalent to the trivial cycle . we can then choose any of the inside points @xmath242 , and adopt as a parallel transport for all points of this triangle a representative path which crosses through all the triangles which are crossed by the path from @xmath242 to @xmath603 . there are points @xmath612 and @xmath613 of the inside of the induced triangle for which the piecewise geodesic path from @xmath603 to @xmath612 , then to @xmath613 , and back to @xmath603 , is not homotopy - equivalent ( in @xmath609 ) to the trivial cycle . in this case , we have to subdivide the triangulation @xmath33 further , until this situation eventually does not appear ( now with respect to @xmath614 ) . to see that this procedure does indeed stabilize after a finite number of steps ( and does not produce further and further subdivisions ) , consider the example of figure [ fig : subdivision ] . there , we construct the subdivision of @xmath268 into regions which are homotopy - equivalently transported to @xmath603 . to this end , we need to draw straight lines ( appearing as dashed lines in the middle panel of figure [ fig : subdivision ] ) from the @xmath268-root @xmath603 to all the vertices @xmath608 , and continue these lines until they hit the boundary of @xmath268 . we can ignore the parts of the lines which lie in @xmath610 ( which are in dashed green in figure [ fig : subdivision ] ) . the remaining parts of the lines ( in dashed red ) will subdivide the other triangles @xmath611 into homotopy - equivalent regions . we then need to subdivide the triangulation further , until these regions are unions of triangles @xmath615 of the refined triangulation . this introduces the finely dashed red line in the middle panel of figure [ fig : subdivision ] . however , note that the lines emanating from @xmath603 do not cross themselves , and therefore only form new vertices with the edges which are already subdividing the triangle @xmath268 ( this is the vertex @xmath616 in the middle panel ) . these new vertices can in principle refine the notion of homotopy - equivalent regions , which would then require an iteration of the procedure . all the new vertices are however exactly on the boundary of the homotopy - equivalent regions which have just been defined . therefore , an iteration of this procedure is not necessary . we now have a triangulation of @xmath268 which is such that any two points in the inside of any given triangle are homotopy - equivalent as far as the transport to @xmath603 is concerned . this allows to define a sub - tree , which specifies the parallel transport for the triangle @xmath268 itself ( see the right panel of figure [ fig : subdivision ] ) . as a last step , one needs to find a refined triangulation @xmath46 of @xmath13 which induces a triangulation of @xmath268 that coincides with the one we have just constructed . and its induced subdivision by a triangulation @xmath33 . the vertices @xmath617 and @xmath618 lead to a further subdivision into homotopy - equivalent regions with respect to the transport to @xmath603 , which is shown in the middle panel . this subdivision is completed to a triangulation of @xmath268 , and therefore the triangulation @xmath33 of @xmath13 needs to be refined to a triangulation @xmath46 such that the induced triangulation for @xmath268 coincides with the one in the middle panel . the right panel shows the ( surface ) tree for the parallel transport in the triangulation @xmath46 . ] in summary , if the above assumptions are satisfied , we will eventually be able to define the parallel transport on a possibly refined triangulation @xmath46 , which is required in order to specify the translation operator ( on @xmath108 ) . we can also apply this translation operator to states in @xmath95 since we just need to refine these states to @xmath46 . the discussion on alternative extensions to @xmath46 for a given translation operator defined on a triangulation @xmath33 in section [ extflux ] included in particular the change of parallel transport . therefore , we can adjust the parallel transport of a given extension to @xmath46 so that it matches the one given by the projection of the continuum translation operator . consider the ( integrated ) flux associated to a fixed choice of a triangulated surface @xmath266 , a choice of @xmath268-roots , and a choice of @xmath266-surface tree . to define the corresponding translation operator on the continuum hilbert space , we first consider a partially - ordered subset of triangulations such that the assumptions regarding the reconstruction of an unambiguous parallel transport are satisfied for each of the triangulations in this subset . we can then argue in a similar manner as for the holonomy operators , and define an integrated translation operator on the corresponding subspace of @xmath367 . extensions of this operator always exist , but are again not unique . we therefore obtain continuum operators which are quantizing the holonomies and ( the exponentiated flow of the ) fluxes . this allows to consider the ( weyl form of ) the holonomy - flux algebra , or more precisely its commutation relations . due to the cylindrical - consistency of the operators , one can evaluate the commutators between two operators on a triangulation @xmath33 on which both operators can be unambiguously defined . due to our quantization prescription , one then obtains the expected result for the commutators between holonomies and translation operators . in @xcite , we have also evaluated the poisson algebra of the fluxes explicitly . note that this can not be performed in the al framework , since there the poisson bracket of two fluxes might lead to a more singular object , e.g. a flux only smeared over a one - dimensional path in @xmath16 spatial dimensions . in contrast , in @xcite one obtains again ( parallel - transported ) fluxes smeared over @xmath31-dimensional surfaces as a result of the commutator between two fluxes . the reason is that the main contribution to the commutators arises from the parallel transport and the basic fluxes . therefore , all the parallel transport holonomies for a certain ( @xmath31-dimensional ) part of the flux surface contribute to the commutator . the result of the commutator is therefore a flux which is smeared over this part of the surface ( see @xcite for details ) . the complete consideration of these commutators requires to go through many separate cases , and we therefore leave the consideration of the quantum commutators for future work . we have constructed in this paper a new realization of quantum geometry . our construction being based on the holonomy and flux variables of loop quantum gravity ( lqg ) , it shares some features with the already existing ashtekar lewandowski ( al ) representation , but , most importantly , exhibits crucial differences . first of all , both the al and the bf representations provide a concrete realization of the idea of seeing ( quantum ) gravity as a topological field theory with defects . however , while in the al representation the defects are the carriers of the geometrical excitations themselves , which leads to complications when trying to construct semi - classical smooth geometries , in the bf representation the situation is reversed . indeed , as we have seen , the bf representation is based on a vacuum peaked on locally and globally flat connections , and excitations on top of this vacuum are carried by defects encoding non - trivial local curvature ( while the conjugated intrinsic geometry is maximally uncertain ) . second , the construction of the bf representation relies on the triangulation of the spatial manifold itself , and does not consider the ( arbitrary ) dual graphs as fundamental structures , as it is the case in the al representation . because of this , the notion of refinement and embedding of dual graphs in the al representation is replaced here by the refinement of triangulations and the associated notion of embedding of coarser triangulations into finer ones . this notion of embedding is crucial for the construction of a continuum inductive limit hilbert space . as recalled in the introduction , a fundamental result of lqg , which is based on the al representation , is the existence of a continuum inductive limit hilbert space which carries a unitary representation of the holonomy - flux algebra and of the action of spatial diffeomorphisms . we have shown here that this result can be extended to the bf representation as well , and that it is possible to construct an inductive limit hilbert space in which the hilbert spaces associated to triangulations carrying a finite number of curvature excitations are embedded . the existence of this continuum hilbert space relies on several key technical points . the most important one is that we need to work with exponentiated ( i.e. compactified ) fluxes , which is in turn equivalent to considering a discrete topology on the group labelling the curvature excitations . this is the reason why the bf representation evades the f lost uniqueness theorem @xcite . indeed , this theorem states that the al representation is unique given a number of assumptions , and one of these is that the fluxes have to exist as well - defined operators . the introduction of a discrete topology on the group is one of the key features which leads to different properties for the geometric observables in the bf representation as compared to the al representation . in particular , the spectrum of the area operator , discussed in section [ sec : spectrumar ] , is bounded in the bf representation . for the structure group @xmath131 , the spectrum can be either discrete or continuous , depending on a parameter @xmath8 appearing in the definition of the area operator . we expect a similar result for the ( fully gauge - invariant ) @xmath2 area operator . furthermore , there are different possibilities to implement the presence of the barbero immirzi parameter into the area operator , and it would be very interesting to see whether this parameter will continue to appear in the black hole entropy state counting , for which the area spectrum plays a crucial role @xcite . the fact that the spectra of the operators can be different in different representations emphasizes also the point raised in @xcite , namely that the spectra of physical observables on the physical hilbert space might be quite unrelated to the spectra of ( kinematical ) observables on kinematical hilbert spaces . further key differences between the al and bf representations are the different requirements for the behavior of operators under refinement . in the case of the bf representation , we can introduce for the geometric operators a measurement scale in the form of a subdivision of the region to be measured into a triangulation . this triangulation is attached to the operator and not the the state . therefore , the area of each triangle in such a triangulation is bounded . this still allows to achieve large areas by sufficiently subdividing a region into pieces . it will be important to investigate how various results relying on the al representation will change if one works with the bf representation instead . for instance , the results of the black hole entropy calculation @xcite might change if one uses the bf representation . in this direction , the work @xcite is interesting since it uses a kind of mixture between the bf and al representations in order to accommodate the isolated horizon constraints describing a certain family of black holes @xcite . the advantages of considering this new bf representation are manifold , and there are lots of interesting and important directions to be explored , as explained in the following points : * the bf representation allows to have a nicer geometric interpretation since the states are not based anymore on degenerate spatial geometrical excitations . one should explore this interpretation in particular in @xmath619 spacetime dimensions , and its relation to simplicial geometry . since the classical phase space on a fixed triangulation agrees with that of loop quantum gravity on a fixed ( dual ) graph , one will find that the space of geometries , if interpreted in a simplicial manner , does not agree with regge geometries @xcite , but rather defines twisted geometries @xcite . however , the basis states are in fact not semi - classical , but rather squeezed states . in particular , the spatial geometrical operators have maximal uncertainties , whereas the curvature operators are maximally peaked . an interesting question is therefore to what extent the connection determines the four - dimensional geometry . note that the construction of such states with curvature line defects was also considered in @xcite . * the bf representation forces us to work with the gauge - covariant fluxes , i.e. the fluxes with parallel transport . this is intimately related to the behavior of the fluxes under refining and coarse - graining . it furthermore explains the effect of `` curvature - induced torsion '' @xcite , which can be ( easily ) understood as the fact that a curved triangle or curved tetrahedron does not satisfy the so - called closure constraints . this effect leads to a difficulty for coarse - graining , in the sense of mapping spin networks to coarser spin networks . as noted in @xcite , the coarse spin networks do not necessarily need to satisfy the coupling rules . however , this issue is largely connected to not providing a clean splitting of the observable algebra , as for instance used in @xcite for the case of the standard non - parallel - transported fluxes . here , we provided such a splitting in section [ splitting algebra ] , which allows for a clear geometric interpretation of coarse - graining in terms of which observables are kept and which are averaged over . this also answers the question raised in @xcite of how may quanta there are in a quantum spacetime : the phase space factor associated to the observables averaged over counts the number of quanta . * in @xmath620 spacetime dimensions , the bf representation constructed here shares many properties with the so - called combinatorial quantization scheme @xcite , which is based on the quantization of the space of flat connections on a manifold with defects . in this combinatorial approach , the number of defects is fixed , so there is no need to invoke a continuum limit and therefore to use a bohr compactification like we do here . nevertheless , it would be interesting to construct an explicit maps between these approaches . reference @xcite already provides a map between the observable algebras . * since the roles of the holonomies and the fluxes are inverted as compared to the al representation , one can now work with the fluxes themselves , and therefore describe the coarse - graining of geometrical quantities . this would be a completion of the proposal @xcite to use a ( non - commutative ) flux representation for loop quantum gravity . in fact , in appendix [ appendix : spin ] , we discuss the spin representation and a non - commutative product ( which is basically the matrix product ) based on this representation . one needs this product in order to express the inner product of two states as a functional applied to the ( non - commutative ) product of these states . + in this respect , it is also interesting to explore the full algebra of the exponentiated flux operators . in contrast , the full commutator algebra of the flux operators can not be described within the al representation since it would lead to fluxes smeared over singular surfaces in the case where the flux surfaces cut each other . it has been shown in @xcite that in the case of the bf representation the classical poisson algebra of the fluxes is well - defined and does not lead to such an appearance of singular surfaces . * in this work , we have left largely open the question of how to impose the spatial diffeomorphism and the hamiltonian constraints . let us first comment on the spatial diffeomorphism constraints . here , we expect that spatial diffeomorphisms will act by moving the defects around , i.e. by changing the embedding of the triangulation network which supports the defects . this is basically the same action as in the al representation , and one would therefore use , as in the al representation , a group averaging procedure in order to define the ( spatially ) diffeomorphism invariant hilbert space . + in @xcite , we have provided a generator ( on phase space ) for this action in the case of @xmath620 spacetime dimensions . as part of the task of understanding the geometries encoded in the @xmath619-dimensional case , it would be important to investigate also this case . the question there is whether and how the diffeomorphism symmetry coincides with the bf translation symmetry . this is related to a proposal by zapata @xcite to construct a theory of topological gravity as a bf theory with certain broken symmetries . * let us now turn to the hamiltonian constraints . we do believe that hamiltonian constraint operators can in principle be constructed , since the main regularization mechanism pointed out in @xcite should also hold in our case . however , we expect that the problems with the constraint algebra @xcite will persist . this is ultimately related to the problem of preserving full diffeomorphism symmetry if lattices are introduced , even if this happens only on an auxiliary level @xcite . an advantage of the bf representation is however the nicer geometric interpretation , which can facilitate the discussion of these issues . the work @xcite is also aimed at understanding the dynamics of spin foam gravity as a continuum theory , starting from bf theory . + an alternative to directly imposing the hamiltonian constraints is to use a discrete time dynamics @xcite , and then to consider the continuum limit . this would in fact fully follow the philosophy of approximating the dynamics by using defects in a regge - like manner @xcite . a framework for describing a simplicial canonical dynamics has been described in @xcite . the question of how to reconstruct the continuum limit has been considered in e.g. @xcite . in this continuum limit , one can also hope to restore diffeomorphism symmetry as exemplified in @xcite . the above points concern possible future work directions related to the bf representation . in addition , we believe that the techniques developed in this paper will also be helpful to construct further realizations of quantum geometry . any topological field theory whose defects support the holonomy - flux algebra ( or an alternative observable algebra ) can serve as a starting point for such a construction , as explained in @xcite . a main question for future research is therefore whether there exists any such topological field theories in the case of four spacetime dimensions . positive indications have been found in @xcite by studying the coarse - graining of ( simplified ) spin foam models , and work is in progress to find new topological field theories starting from spin foam models @xcite . again , there are a number of further directions to explore , including the following ones : * a long standing problem has been that of constructing a hilbert space based on the structure group @xmath7 within the al representation . the main difficulty in trying to do so is the non - compactness of the group . even if one finds a way to appropriately compactify the group , there is the problem of how to construct an invariant measure . in the case of the bf representation , we need to compactify instead the dual of the group , which leads to a discrete topology on the group itself . therefore , the advantage might be that there is an invariant discrete measure on the group which might allow to construct a bf - like representation for a non - compact gauge group . * an important question is whether there is a way to avoid the requirement of compactifying the configuration space ( which , as we have discussed , can be understood from the necessity to define defects as stable physical entities ) , possibly along the lines of @xcite . in this vein , it would be interesting to understand the general properties of inductive limit hilbert spaces and their limitations , and in particular to revisit the f lost uniqueness theorem @xcite . in particular , the question is what kind of representations are allowed if one relaxes the assumption of the existence of ( non - exponentiated ) flux operators ? * one can also consider cases in which the dual of the group is already compact . apart from finite groups , this is the case of quantum groups at root of unity . these describe ( at least in three spacetime dimensions , since the results are only conjectured for the four - dimensional case ) quantum gravity with a positive cosmological constant ( and a euclidean signature ) . there is by now a large body of work concerned with the issue of implementing the cosmological constant or quantum groups @xcite . the introduction of a quantum group changes the almost everywhere flat connection to an almost everywhere homogeneously - curved connection . a corresponding description in terms of simplicial geometry is given in @xcite . + when replacing @xmath2 with its quantum deformation at root of unity , the first question is to understand the structure of the defects . in the case of @xmath620 dimensions , the defects have been understood to be given ( in the context of higher categories ) by the drinfeld center associated to the topological theory ( here the tuare viro @xcite topological quantum field theory ) , as explained for instance in @xcite . therefore , the construction of a tuarev viro - based representation can be achieved by combining the methods of @xcite and the methods developed in this work @xcite . for the four - dimensional case , recent developments have been achieved in @xcite . + using a quantum group at root of unity will change the results for the spectrum of the area operator . the compactification of the spectrum for @xmath2 means that the eigenvalues have to wind around in the interval allowed by the bound . this can lead to a continuous spectrum in the case of lie groups . in the case of a quantum group at root of unity , the bound is on the ( admissible ) spins themselves , which means that the number of eigenvalues is itself actually bounded . this would prevent the spectrum from being continuous . in summary , the existence of this new representation should not come as a surprise , and in fact indicates the presence of unsuspected rich phase structures in discrete approaches to quantum gravity . in ordinary quantum field theory , the existence of different vacua is tight to the description of condensation mechanisms , along with phase transitions and symmetry breakings . different physical situations might be easier to describe within different representations based on different vacua . recent results @xcite indicate that spin foam models can have non - trivial phase diagrams , with phases corresponding to particular topological field theories . given such a topological field theory , one can describe excitations on top of a vacuum , and then dynamics for these excitations . in this way , it can be expected that each phase will correspond to a different realization of quantum geometry . this will not only open up a rich field of research , but also enrich our methods for constructing and understanding quantum geometries . we would like to thank abhay ashtekar , klaus fredenhagen , and aldo riello for helpful discussions . this research was supported by perimeter institute for theoretical physics . research at perimeter institute is supported by the government of canada through industry canada and by the province of ontario through the ministry of research and innovation . bb was funded by project 4966/1 - 1 of the german research foundation ( dfg ) . mg is supported by the nsf grant phy-1205388 and the eberly research funds of the pennsylvania state university . in this appendix , we briefly recall how the first fundamental group @xmath64 can be obtained in the case of two- and three - dimensional manifolds without defects . first , if @xmath13 is a closed and orientable riemann surface of genus @xmath125 , its fundamental group @xmath64 is given by the free group on @xmath621 generators @xmath622 , divided by a normal subgroup generated by the single relator @xmath623\dots[a_g , b_g]$ ] , where the commutator is @xmath624=aba^{-1}b^{-1}$ ] . this can be shown by using the seifert van kampen theorem to write @xmath625 , where @xmath626 is the bouquet obtained from the wedge sum of @xmath621 circles , and @xmath314 is the normal subgroup generated by @xmath627 . since this relator corresponds to a null - homotopic loop , we can write the presentation [ 2dpresentation ] _ 1 ( ) = a_1,b_1, ,a_g , b_g|[a_1,b_1] [a_g , b_g]=. notice that this first fundamental group , although always being finitely presented , is not free for @xmath628 . for a three - dimensional compact , connected , and orientable manifolds @xmath13 , the first fundamental group can also be finitely presented . this is most easily seen by using the fact that any such manifold admits a cw complex . indeed , if we denote by @xmath629 the two - skeleton of the cw complex of @xmath13 , then we have that @xmath630 , which can in turn be computed using the seifert van kampen theorem and the fact that @xmath629 is the union of the one - skeleton @xmath631 with all the two - cells of the cw complex . more precisely , @xmath632 is always given by a free group @xmath633 on @xmath634 generators , and the presentation of the first fundamental group of @xmath13 is given by _ 1 ( ) = a_1, ,a_n_1|r_1, ,r_n_2 , where the relator @xmath635 is the homotopy class in @xmath632 of the boundary of the @xmath202-th two - cell ( with @xmath636 ) , and therefore a word in the generators . this can equivalently be described in terms of the so - called spine of the manifold @xmath13 , which is a two - complex consisting of a single vertex , one - cells and two - cells . if the manifold @xmath13 is closed , its spine can be obtained by collapsing it after first removing an open three - ball . in this language , the generators are in one - to - one correspondence with the one - cells of the spine , the relators are in one - to - one correspondence with the two - cells , and the explicit expression for a relator @xmath635 describes how the @xmath202-th two - cell is attached to the one - skeleton . the case of the defected manifold @xmath71 is simpler to deal with than the case of a manifold without defects . this is due to the fact that the graph @xmath34 , which is the one - skeleton of the simplicial complex dual to the triangulation @xmath33 of @xmath13 , is a deformation - retract of @xmath71 . as a consequence , their first fundamental groups are isomorphic , i.e. @xmath72 , which is in turn immediate to obtain . indeed , it is well - known that the first fundamental group of a graph is free and generated by @xmath637 generators corresponding to its fundamental cycles @xcite . by using a maximal spanning tree to characterize the fundamental cycles of the graph , we can therefore say that the first fundamental group of @xmath62 is freely generated by generators associated to the leaves of the tree . for the sake of completeness , let us comment on the counting of the number of generators for the first fundamental group in the case of a punctured riemann surface of arbitrary genus . any such surface with @xmath242 ( thickened ) punctures can be decomposed as a connected sum @xmath638 , where @xmath639 has genus @xmath640 and one puncture , while @xmath641 has genus one and @xmath642 punctures . in this decomposition , the surfaces @xmath639 and @xmath641 are glued along one common puncture ( so that the total number of punctures is indeed @xmath242 ) , and @xmath643 . by the seifer van kampen theorem , we have that the first fundamental group of @xmath71 is given by the amalgamated free product [ freeproduct ] _ 1()_1(a)*__1(ab)_1(b ) = _ 1(a)*__1(^1)_1(b ) . each factor in this expression can easily be computed . first , for a surface of genus @xmath125 with a single puncture , the first fundamental group is isomorphic to that of a wedge sum of @xmath621 circles , so we can write that @xmath644 , which is the free group @xmath645 on @xmath646 generators . second , for a surface of genus one with @xmath242 punctures , the first fundamental group is isomorphic to that of a wedge sum of @xmath642 circles . this implies that @xmath647 , which is the free group on @xmath648 on @xmath649 generators . finally , we have that @xmath650 , which is the free group on a single generator . gathering these results , we get that the amalgamated free product defining @xmath70 is generated by the generators of @xmath651 and @xmath652 , with one generator removed due to @xmath653 . therefore , this leads to a free group on @xmath654 generators . this result is consistent with the usual presentation which generalizes to the case of punctured surfaces and takes the form [ punctured presentation ] _ 1 ( ) = a_1,b_1, ,a_g , b_g , g_1, ,g_p|[a_1,b_1] b_g]=g_1 g_p . this presentation is written in terms of @xmath621 generators corresponding to the non - contractible paths in @xmath71 , and the @xmath242 generators corresponding to the paths around the punctures . now , this presentation can be reduced by a so - called nielsen transformation which , since there is only one relation in the presentation , indeed leads to the free group on @xmath655 generator . now , since for triangulated surfaces the euler characteristic is given by @xmath656 , and since the punctures are `` piercing '' the faces ( which means that @xmath657 ) , we get that @xmath658 , which indeed corresponds to the number of leaves in a maximal spanning tree . we explain in this appendix how to reconstruct the fluxes ( and operators ) associated to the branches from the ones associated to the leaves . to this end we have to employ the gau constraints branches and @xmath659 leaves . as far as the counting is concerned , it is therefore possible to reconstruct all the fluxes associated with the links @xmath82 from the knowledge of the fluxes associated to the leaves @xmath51 and @xmath57 gau constraints . ] . recall that the gau constraint for a node @xmath660 takes the form [ gauss1 ] _ l|l(0)=n_l+_l|l(1)=n_l^-1 0 , where the rooted fluxes are given by _ l = g^-1_rl(0)x_lg_rl(0),_l^-1=g^-1_rl^-1(0)x_l^-1g_rl^-1(0)=-g^-1_rl^-1(0)h_lx_lh_l^-1g_rl^-1(0 ) . the last definition implies that for branches @xmath661 we simply have @xmath662 , while for leaves @xmath663 we have @xmath664 , where @xmath665 . given a choice of tree , one can successively solve for all the fluxes by starting from the end nodes of the tree , where the gau constraints each involve only one flux associated to a branch . by solving for these fluxes associated to the end nodes of the tree , we can go to the nodes which are only one step away from the end nodes along the tree . again , for every node there is only one unknown branch flux . one can then iterate this procedure and solve all the gau constraints except for the one at the root . the fluxes associated to branches are then given by linear combinations of the fluxes associated to the leaves and to the inverse leaves . to give the result of this procedure in a compact form , let us denote by @xmath666 the set of end nodes of the tree which lie after the node @xmath667 when following the orientation of the tree . for each end node @xmath668 in @xmath666 there is a unique path @xmath669 going from the target node @xmath667 of the branch to the end node @xmath668 . considering the set of all such paths for a given @xmath50 , we can then denote by @xmath670 the set of leaves which have their source node @xmath671 along the paths @xmath669 , and by @xmath672 the set of leaves which have their target node @xmath673 along the paths @xmath669 . as suggested by the notation , a choice of branch @xmath50 automatically determines the sets @xmath670 and @xmath672 ( which can have a non - empty intersection ) . the flux is then given by _ b = _ _ ( b)_+__(b)_^-1 . in this formula , each leaf @xmath51 can appear at most once with the orientation @xmath51 and at most once with the orientation @xmath674 . the right translations corresponding to a branch can therefore be defined as r_b^=__(b)r_^__(b)l_^. in most of this article we have worked in the holonomy representation for the wave functions . in this appendix we discuss a group fourier transform with respect to a discrete measure on the group , which defines a spin representation . the discrete group @xmath131 can be understood as arising from a bohr compactification of its dual group @xmath132 . consider the space of almost periodic functions @xmath675 on @xmath132 . almost periodic functions are constructed as finite combinations of functions from the set @xmath676 . since there exists an inner product , we also take the norm completion of this space . this bohr compactification inner product is given by _ 1|_2=_t_|k|t_2(k ) , and it turns the functions @xmath677 into an ( over - countable ) orthonormal basis . this basis can be used to transform states to the `` holonomy representation '' , which is here in terms of an angle @xmath225 . this transformation is given by [ trafo1 ] ( ) = _ |=_t_kte^ik(k ) , from which we recover the following kronecker delta form of the basis : _ ( ) = ( , ) , where @xmath127 if @xmath128 mod @xmath223 and is vanishing otherwise . the inverse transformation to is given by ( k ) = _ ( ) e^-ik ( ) , with a discrete measure @xmath164 . the translation operator r^ ( ) ( + ) acts on a basis vector as @xmath678 , and therefore in the @xmath44-representation as multiplication r^(k ) = e^ik(k ) . we can define a measure _ ( _ ) = ( , 0 ) for the basis functions @xmath677 , from which one can recover the inner product in the @xmath44-representation as _ 1|_2=_(_k_2 ) . this measure is the analogue ( or rather the dual ) of the haar measure ( which underlies the ashtekar lewandowski measure ) , which in the context of @xmath131 would be defined as _ ( _ k ) = ( k,0 ) for basis functions @xmath679 in the @xmath365-representation . in order to define a spin representation for @xmath2 , we proceed as in the case of @xmath131 . consider a suitable space of functions on the set \{j , m , n|j/2 , m , n =- j ,- j+1, ,j-1,j}. by analogy with the case of @xmath131 , we define as `` suitable '' functions the functions which arise as finite linear combinations of functions from the set \{d^j_mn()|(2 ) } , where @xmath680 are the spin-@xmath11 representation matrices . we write these functions into the matrix form @xmath681 , and equip the space of functions with the inner product [ c12 ] _ 1|_2=__j=0^((_1^(j)_2(j ) ) , where @xmath682 . in order to define this inner product , we have introduced the regularization factor [ nfactor ] ( ) = _ j=0^d_j^2 = ( 4 + 3)(2 + 2)(2 + 1 ) , where @xmath683 . in this inner product , the following functions are orthonormal : _ ( j)_mn = j , m , n|_=(d^j_mn())^ , and with @xmath684 constitute a ( over - countable ) basis . we can now define the transformation to the group representation as [ c15 ] ( h ) = _ h|=__j=0^(d^j(h)(j ) ) , from which we therefore obtain _ ( h ) = _ _ j=0^d_j(d^j(h^-1 ) ) = ( , h ) , with @xmath685 iff @xmath686 and vanishing otherwise . the inverse transformation to is given by ( j ) = _ ( h)(d^j(h))^(h ) , with the discrete measure @xmath164 on the group . the translation operator r^(h ) = ( h ) acts in the spin representation as ( r^)(j ) = ( d^j())(j ) . we can now also attempt to define a measure in the spin representation space which would recover the inner product however , this can only be accomplished with a non - commutative product . we therefore define a measure [ measurebf ] _ ( _ ) = ( , ) , and introduce a non - commutative ( matrix ) product for the basis states via ( _ _ ) ( j ) = _ ( j ) , which we extend by bi - linearity . we can then express the inner product as _ 1|_2=_(_1^_2 ) . the spin representation discussed in the previous section can easily be extended from functions on @xmath1 to functions on @xmath78 . for this , we consider states of the form \{j}_\{mn } = _ \{}c_\{}(_d^j_(_^-1)_m_n _ ) , where we have used the basis states _ \{j}_\{mn } = ( _ d^j_(_^-1)_m_n _ ) which are orthonormal in the inner product [ c25 ] _ 1|_2=__\{j}^(_1^\{j}_2\{j } ) . here @xmath687 is a shorthand notation for ( _ 1^_2 ) = _ m_1,n_1, ,m_||,n_||()_m_1n_1 m_||n_||(_2)_m_1n_1 m_||n_|| . we have therefore expressed the inner product as a summation over @xmath11 as well as on the magnetic indices @xmath688 . these latter transform under the adjoint action , namely in a representation v_j_1v_j_1^* v _ j_||v_j_||^*. let us now choose a recoupling scheme and a unitary intertwining map u^\{j}_,j , l;\{m , n}:v_j_1v_j_1^* v _ j_||v_j_||^ * _ , jv_j , where @xmath689 is labelling the resulting representation , @xmath3 is the associated magnetic index , and @xmath690 summarizes the intertwiner ( i.e. intermediate spin ) labels . such an intertwining map can be built from @xmath691 symbols , which couple two representations @xmath692 and @xmath693 to @xmath694 . in this case ( and with certain phase conventions ) , the matrix elements of @xmath695 are real . the condition of unitarity then means that u^\{j}_,j , l;\{m , n}u^\{j}_,j,l;\{m , n } = _ , _j , j_l , l , and [ c30 ] _ , j , lu^\{j}_,j , l;\{m , n}u^\{j}_,j , l;\{m,n } = _ \{m},\{m}_\{n},\{n}. we can then change our spin representation to ( \{j},,j)_l _ \{mn}u^\{j}_,j , l;\{m , n}\{j}_\{mn } , and also rewrite the inner product as a sum over the @xmath696 indices . to obtain the ( completely ) gauge - invariant part of a wave function @xmath697 , one would set @xmath698 ( and hence @xmath699 ) . note however that a wave function with ( kronecker ) @xmath700 behavior would have vanishing norm in the ( rewritten ) inner product . this is due to the scaling factor @xmath701 in ) . however , it is not possible to change this scaling factor such that a sufficiently large class of wave functions have finite norm . the reason for this is that the conjugacy classes @xmath702 require different scaling behaviors depending on which type of adjoint orbit they describe . we can nevertheless apply the rigging map to a basis state in the recoupled spin representation : [ c32 ] ( _ \{})(\{j},,j)_l = _ g/(u)_(u)_\{m , n}(__m_n_)u^\{j}_,j , l;\{m , n}. note that this will not give a regular function in the indices @xmath703 since the @xmath698 is given by a ( discrete measure ) integral over a constant . this integral only makes sense if one applies @xmath704 to a state in the kinematical hilbert space . therefore , the physical inner product between two basis states can also be expressed as a sum over @xmath705 and @xmath706 . the sum over @xmath705 leads , due to , to kronecker delta symbols in the indices @xmath707 , and the sum over these indices leads to the representation of the ( kronecker ) delta in the group elements , i.e. ( _ \{})_&=_g/(u)_(u)___j=0^d_j_j(u_u^-1_^-1 ) + & = _ g/(u)_(u)_(u_u^-1 , _ ) + & = ( [ \{}],[\ { } ] ) . in this appendix , we prove that the translation operator @xmath219 acting on @xmath708 has a spectrum given by @xmath131 when the angle @xmath229 is irrational . this actually holds when equipping @xmath131 with either a continuous or a discrete topology . the difference between these two cases is just that the spectral measure for @xmath708 is continuous , while for @xmath709 it is discrete . let us introduce the ket @xmath710 for @xmath711 . note that this is _ not _ the usual eigenvector of the momentum operator for eigenvalue @xmath660 . the translation operator @xmath219 acts as a right shift , i.e. @xmath712 . let @xmath713 for @xmath251 . we are now going to show that @xmath250 is in the spectrum of @xmath219 . this is equivalent to saying that the operator @xmath714 does not have a bounded inverse . first , if this operator has no inverse , we have the desired result and there is nothing to show . if it does have an inverse , then we have to show that this inverse is not bounded . for this , it is sufficient to show that there exists a sequence @xmath715 of vectors in @xmath249 such that @xmath716 but @xmath717 . we now define this sequence . it is given by @xmath718 , with c_n^(n ) = \ { cl e^in & , + 0&. . this sequence satisfies @xmath719 , as well as ( r^-e^i)_n^2 = , where the right - hand side converges to zero . this finishes the proof . in fact , the ( formal ) limit @xmath720 does not converge in @xmath213 , but @xmath721 converges ( in the dual space of a suitable dense subspace ) to the generalized eigenvector corresponding to the point @xmath250 in the spectrum . the whole spectrum is @xmath131 , and is continuous because the operator does not have any ( normalizable ) eigenvectors , but only non - normalizable ones . in order to understand the subtleties underlying the notion of refined trees , let us focus on the example depicted on figure [ fig : refine - tree-1 ] . there , we have a triangulation @xmath33 consisting of four triangles , a tree @xmath49 in its dual graph , and three leaves , @xmath722 , @xmath723 and @xmath724 , labeling the fundamental cycles @xmath725 , @xmath726 and @xmath727 . by subdividing the edge shared by the two upper triangles we get a finer triangulation @xmath46 , and we have represented on figure [ fig : refine - tree-2 ] three possible trees ( there are finitely many other possibilities ) in its dual graph @xmath101 . for each tree @xmath389 and the corresponding set @xmath407 of leaves , we can build paths @xmath454 describing the fundamental cycles by starting at the root , going along the tree until the source of @xmath397 , then going along @xmath397 , and back to the root along the tree . then , we can project these paths with @xmath368 , and compare the resulting projections with the fundamental cycles described by @xmath49 . let us do this for the three trees @xmath389 represented on figure [ fig : refine - tree-2 ] . 1 . with @xmath728 , we have that @xmath729 , @xmath730 , and @xmath731 . this means that the projections of the fundamental cycles described by the leaves of @xmath728 correspond to the fundamental cycles described by the leaves @xmath53 of @xmath49 . in addition to this , one can see that all possible paths in @xmath728 are mapped under @xmath368 to paths in @xmath49 . 2 . with @xmath732 , we have that @xmath729 , @xmath730 , and @xmath733 , while @xmath734 can be described in terms of the cycles @xmath726 and @xmath727 in @xmath34 . however , at the difference with the tree @xmath728 , one can see that the tree @xmath732 has paths which under @xmath368 are not mapped to paths in @xmath49 . 3 . with @xmath735 , there are paths which under @xmath368 are not mapped to paths in @xmath49 , and the projections of the fundamental cycles do not correspond to the fundamental cycles determined by @xmath49 . indeed , although we have that @xmath729 , there is no leaf @xmath397 in @xmath735 such that @xmath736 or @xmath737 . with a tree @xmath49 ( thick dashes ) in its dual graph , three leaves ( thin dashes ) labeling the three fundamental cycles , and a root @xmath42 ( thick node ) . we have omitted the orientation of the simplices and of the dual graph for the sake of clarity . ] with three different trees in its dual graph . ] with this example , we see that the different possible trees which can be chosen in the refined graph @xmath101 do not characterize in an equivalent manner under @xmath368 the paths and fundamental cycles described by the coarser tree @xmath49 . let @xmath47 , and @xmath49 be a maximal tree with a refined tree @xmath389 . then @xmath389 is a result of the construction procedure described in section [ refinementtrees ] . in order to prove this result , we first look at a single @xmath14-dimensional simplex @xmath738 . consider the edges of @xmath389 which start and end in this simplex , i.e. which start and end at vertices dual to @xmath14-dimensional simplices @xmath739 . this collection gives a sub - tree of @xmath389 , which is connected due to property @xmath396 of a refined tree . next , consider a @xmath31-dimensional simplex @xmath404 dual to a branch in @xmath49 . because of @xmath406 , @xmath405 contains at least one @xmath31-dimensional simplex dual to a branch in @xmath389 . if it did contain more than one , one could construct a closed loop in @xmath389 because of property @xmath406 . this means that @xmath405 contains precisely one branch in @xmath389 . all that remains to show is that the branches which we have singled out so far are the only ones in @xmath389 . but the only edges which we have not considered yet are the ones which are mapped under @xmath368 to a leaf in @xmath49 . so none of these edges can be a branch , because of property @xmath406 , and we are therefore done . * lemma [ lemma : ttransitivity ] * ( transitivity of the refined trees)*. * _ let us consider three triangulations @xmath380 . let @xmath389 be a refined tree for @xmath49 , and @xmath423 be a refined tree for @xmath389 . then we have that @xmath423 is also a refined tree for @xmath49 . _ + in order to prove this lemma , we need to show that the tree @xmath423 satisfies the properties @xmath406 and @xmath396 of definition [ def : refine - tree ] with respect to the tree @xmath49 . property @xmath406 follows from result of lemma [ lemma : ptransitivity ] , and from the fact that the trees @xmath389 and @xmath423 are refinements of @xmath49 and @xmath389 respectively . indeed , since @xmath740 and @xmath741 , by virtue of we also have that @xmath742 . for property @xmath396 , we have to show that any two nodes @xmath743 and @xmath744 , such that the dual simplices @xmath745 and @xmath746 are in @xmath747 for a fixed @xmath375 , can be connected by a path in @xmath423 without crossing the boundary @xmath748 . let @xmath749 be the simplex in @xmath46 containing the simplex @xmath745 dual to @xmath750 , i.e. such that @xmath751 , and likewise for @xmath752 . we have @xmath753 for @xmath754 . then the following two situations can occur : 1 . if @xmath755 , the tree path from @xmath743 to @xmath744 is `` inside '' @xmath756 by virtue of property @xmath396 with respect to the pair @xmath757 of trees . therefore , this path does also not cross the boundary of @xmath747 , which shows that @xmath396 holds in this case also for the pair @xmath758 . 2 . if @xmath759 , the path from @xmath743 to @xmath744 along @xmath423 projects under @xmath381 to a path along the tree @xmath389 , which is from the node dual to @xmath749 to the node dual to @xmath752 . by virtue of property @xmath396 for the pair @xmath760 of trees , the path does not cross the boundary of @xmath761 . together with the first point above , this shows that also the original path from @xmath743 to @xmath744 along @xmath423 does not cross the boundary of @xmath762 . therefore , @xmath396 holds for the pair @xmath758 in general , which finishes the proof . * lemma [ lemma : separation ] . * _ the coarse observables and the fine observables form two mutually - commuting subalgebras of the rooted holonomy - flux algebra @xmath171 , which we denote respectively by @xmath481 and @xmath482 . _ + the proof of lemma [ lemma : separation ] can be obtained by a direct calculation . first of all , it is clear that @xmath481 and @xmath482 each form a subalgebra which reproduces the poisson structure on @xmath763 . the nontrivial statement is to show that these subalgebras are mutually commuting . since holonomies are always poisson - commuting , the holonomies in @xmath481 have vanishing poisson brackets with the holonomies in @xmath482 . in addition to this , since the leaves @xmath426 and @xmath444 are always distinct , the holonomies in @xmath481 commute with all the fluxes in @xmath482 . let us now turn to the commutation relations between the fluxes in @xmath481 and the holonomies in @xmath482 . in , in addition to the leaf @xmath426 which always exists in @xmath764 , we have a leaf @xmath444 in the same pre - image . therefore , for a given @xmath208 the poisson bracket between @xmath476 and @xmath535 contains two contributions , which are in turn vanishing since ( __i)^k,_i=^k__i+^k__i , g__ig__i^-1=g__i^kg__i^-1-g__i^kg__i^-1 = 0 . similarly , for we have that ( __i)^k,_i=^k__i+^k_(_i)^-1,g__ig__i = g__i^kg__i - g__i^kg__i = 0 . for , since the leaf @xmath444 does not belong to the pre - image of any leaf @xmath208 , it does not appear in the coarse observable @xmath476 and therefore the poisson brackets @xmath765 are identically vanishing . this shows that the fluxes in @xmath481 commute with the holonomies in @xmath482 . finally , let us look at the commutation relations between the fluxes in @xmath481 and the fluxes in @xmath482 . for , since @xmath766 , the unique contribution coming from this finer leaf to will be a flux @xmath767 . however , because of the reversed orientation of the leaf in , the poisson brackets are vanishing since @xmath768 . a similar reasoning applies to . for , since @xmath769 , the flux @xmath770 never appears in and the poisson brackets are therefore identically vanishing . @xmath511 h. m. haggard , m. han , w. kamiski and a. riello , `` sl(2,c ) chern simons theory , a non - planar graph operator , and 4d loop quantum gravity with a cosmological constant : semi - classical geometry '' , ( 2014 ) , ` arxiv:1412.7546 [ hep - th ] ` . c. meusburger and b. schroers , `` the quantization of poisson structures arising in chern simons theory with gauge group @xmath771 '' , adv . * 7 * , 10031043 ( 2004 ) , ` arxiv : hep - th/0310218 ` . m. campiglia , c. di bartolo , r. gambini , j. pullin , uniform discretizations : a new approach for the quantization of totally constrained systems " , phys . rev . * d74 * ( 2006 ) 124012 , ` arxiv : gr - qc/0610023 ` . b. dittrich , `` the continuum limit of loop quantum gravity - a framework for solving the theory '' , in a. ashtekar and j. pullin , ed . , to be published in the world scientific series `` 100 years of general relativity '' , ( 2014 ) , ` arxiv:1409.1450 [ gr - qc ] ` . b. bahr and b. dittrich , `` breaking and restoring of diffeomorphism symmetry in discrete gravity '' , proceedings of the xxv max born symposium `` the planck scale '' , wroclaw , 29 june - 3 july , 2009 , ` arxiv:0909.5688 [ gr - qc ] ` .

we construct in this article a new realization of quantum geometry , which is obtained by quantizing the recently - introduced flux formulation of loop quantum gravity . in this framework , the vacuum is peaked on flat connections , and states are built upon it by creating local curvature excitations . the inner product induces a discrete topology on the gauge group , which turns out to be an essential ingredient for the construction of a continuum limit hilbert space . this leads to a representation of the full holonomy - flux algebra of loop quantum gravity which is unitarily - inequivalent to the one based on the ashtekar isham lewandowski vacuum . it therefore provides a new notion of quantum geometry . we discuss how the spectra of geometric operators , including holonomy and area operators , are affected by this new quantization . in particular , we find that the area operator is bounded , and that there are two different ways in which the barbero immirzi parameter can be taken into account . the methods introduced in this work open up new possibilities for investigating further realizations of quantum geometry based on different vacua .

You are an expert at summarizing long articles. Proceed to summarize the following text: conjunction fallacy was first empirically documented by tversky and kahneman ( 1982 , 1983 ) through a now renowned experiment in which subjects are presented with a description of someone called linda " : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ linda is 31 years old , single , outspoken , and very bright . she majored in philosophy . as a student , she was deeply concerned with issues of discrimination and social justice , and also participated in anti - nuclear demonstrations . " _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ then , subjects are shown a list of 8 possible outcomes describing her present employment and activities , and are asked to rank the propositions by representativeness or probability . two items were specifically tested : * linda is a bank teller " , * linda is a bank teller and is active in the feminist movement " . empirical results show that most people judge ( 2 ) more probable than ( 1 ) . in the framework of classical probabilities , this is a fallacy the conjunction fallacy , since a conjunction can not be more probable than one of its components . if linda being active in the feminist movement is denoted by @xmath0 and linda being a bank teller by @xmath1 , then @xmath2 should classically prevail . the conjunction fallacy has been shown to be particularly robust under various variations of the initial experimental protocol ( cf . tversky and kahneman 1982 , 1983 , gigerenzer 1996 , kahneman and tversky 1996 , hertwig 1997 , hertwig and chase 1998 , hertwig and gigerenzer 1999 , mellers et al . 2001 , stolartz - fantino et al . 2003 , bonini et al . 2004 , tentori et al . 2004 , hertwig et al . 2008 , moro 2009 , kahneman 2011 , erceg and galic 2014 ; for a review , cf . moro 2009 ) . it has been observed in other cases than the linda story , about topics like sports , politics , or natural events , and in scenarios in which the propositions to be ranked are not preceded with a description . the fallacy also persists when the experimental setting is changed , e.g. in between subjects " experiments in which ( 1 ) and ( 2 ) are presented to different subjects only . semantic and syntactic aspects have also been discussed , in relation with possible misunderstandings , like the implicit meaning of the words probability " and and " . careful experiments show that the conjunction fallacy persists . the conjunction fallacy questions the fact that classical probability theory can be used to describe human judgment and decision making , and it can also be viewed as a challenge to the definition of what a rational judgment is . thus , it is no surprise that the conjunction fallacy has been the subject of a big amount of research ( tentori and crupi 2012 give the number of a hundred papers devoted to it ) . it has interested psychologists , economists and philosophers alike . for instance , behavioral economists have looked at the consequences of the fallacy for understanding real life economic behavior , measuring the robustness of this bias in an economic context with incentives or in betting situations ( e.g. charness et al . 2010 , nilsson and anderson 2010 , erceg and galic 2014 ) . they have also investigated whether the cognitive abilities of subjects are related to behavioral biases in general ( and to the conjunction fallacy in particular , cf . oechssler et al . 2009 ) , and this has led to stimulating research with applications in finance . epistemologists have made confirmation and bayesianism enter the debate ( e.g. tentori and crupi 2008 and 2012 , hartmann and meijs 2012 , schupbach 2012 , shogenji 2012 ) . given that a conjunction fallacy occurs under robust experimental conditions , a natural question arises : how can this fallacy be explained ? several accounts have been argued for , but no one has reached an uncontroversial status today ( as noted by fisk 2004 , nilsson et al 2009 , jarvstad and hahn 2011 , tentori et al . first , tversky and kahneman originally suggested that a representativeness heuristic ( i.e. the probability that linda is a feminist is evaluated from the degree with which the instance of linda corresponds to the general category of feminists ) could account for some conjunction fallacy cases . but it has been argued that the representativeness concept involved is informal and ill - specified ( gigerenzer 1996 , birnbaum et al 1990 ) , and suggestions to specify it in the technical sense of a likelihood value ( shafir et al 1990 , massaro 1994 ) account for limited cases only ( crupi et al . according to another suggestion , agents actually evaluate the probability of the conjunction from some combination of the probabilities of the components , like averaging or adding ( fantino et al . 1997 , nilsson et al . however , such explanations do not resist empirical tests , as tentori et al . ( 2013 ) have argued . the latter propose an account of the conjunction fallacy based on the notion of inductive confirmation as defined in bayesian theory , and give experimental grounds for it it is one of the currently promising accounts . others have argued , also within a bayesian framework , that there are cases in which the conjunction fallacy is actually not a fallacy and can be accounted for rationally ( hintikka 2004 , von sydow 2011 , hartmann and meijs 2012 ) . finally , another prominent proposal to account for the conjunction fallacy , on which we focus here , makes uses of so - called quantum - like " models , which rely on the mathematics of a major contemporary physical theory , quantum mechanics ( franco 2009 , busemeyer et al . 2011 , yukalov and sornette 2011 , pothos and busemeyer 2013)note that only mathematical tools of quantum mechanics are exploited , and that the models are not justified by an application of quantum physics to the brain . the quantum - like account of the conjunction fallacy is particularly promising as it belongs to a more general theoretical framework of quantum - like modeling in cognition and decision making , which has been applied to many fallacies or human behavior considered as irrational ( for reviews , see pothos and busemeyer 2013 , ashtiani and azgomi 2015 , or bruza et al . 2015 ; textbooks include busemeyer and bruza 2012 , haven and khrennikov 2013 ) . for instance , quantum - like models of judgments have been proposed to account for order effect , i. e. when the answers given to two questions depend on the order of presentation of these questions ( atmanspacher and rmer 2012 , busemeyer and bruza 2012 , wang and busemeyer 2013 , wang et al . 2014 ) ; for the violation of the sure thing principle , which states that if an agent prefers choosing action a to b under a specific state of the world and also prefers choosing a to b in the complementary state , then she should choose a over b regardless of the state of the world ( busemeyer et al . 2006a , busemeyer et al . 2006b , busemeyer and wang 2007 , khrennikov and haven 2009 ; for ellsberg s paradox more specifically , cf . aerts et al . 2011 , aerts and sozzo 2013 , aerts et al . 2014 ; for allais paradox , cf . khrennikov and haven 2009 , yukalov and sornette 2010 , aerts et al . 2011 ) ; for asymmetry judgments in similarity , i.e. that a is like b " is not equivalent to b is like a " ( pothos and busemeyer 2011 ) ; for paradoxical strategies in game theory such as in the prisoner s dilemma ( piotrowski and sladowski 2003 , landsburg 2004 , pothos and busemeyer 2009 , brandenburger 2010 ) . more generally , new theoretical frameworks with quantum - like models have been offered in decision theory and bounded rationality ( danilov and lambert - mogiliansky 2008 and 2010 , lambert - mogiliansky et al . 2009 , yukalov and sornette 2011 ) . as the quantum - like account of the conjunction fallacy is one of the few promising accounts of the conjunction fallacy that are discussed today , we choose to focus on it in this paper . more specifically , we focus on the class of quantum - like models which are presented or defended in franco ( 2009 ) , busemeyer et al . ( 2011 ) , busemeyer and bruza ( 2012 ) , pothos and busemeyer ( 2013 ) and busemeyer et al . ( 2015 ) . in these models , an agent s belief is represented by a quantum state and not for instance by a measurement context . our aim is to assess the empirical adequacy of these quantum - like models that are used to account for the conjunction fallacy . we think that two points deserve particular scrutiny . first , it is not always clear which version of the models are supposed to account for particular cases of conjunction fallacies are the simplest ones , called non - degenerate , sufficient ? or are the more general ones , called degenerate , needed ? more recent works tend to favor degenerate models over non - degenerate ones , and non - degenerate models have received some recent criticisms ( cf . tentori and crupi 2013 and pothos and busemeyer 2013 , p. 315 - 316 ) , but a clear and definitive argument on the matter would be welcome . second , the models have not yet been much tested on other predictions than the ones they were intended to account for . it should be checked that they are not _ ad hoc _ by testing their empirical adequacy in general . it is understandable that these two points have not been tested beforehand , as a new general pattern of explanation for the conjunction fallacy is hard to come up with . but since the models have come to be seen as one of the most promising accounts , it becomes urgent to assess them empirically more thoroughly this is our goal in this paper . as for the first point discriminate between non - degenerate and degenerate models , we follow a suggestion made by boyer - kassem et al . ( 2016 ) to test so - called gr equations " , that are empirical predictions made by non - degenerate models . such a gr test requires a new kind of experiment : not the original linda experiment , in which agents have to rank propositions , but an order effect experiment , in which two yes - no questions are asked in one order or in the other , to different agents . existing data can not answer the question of whether the gr equations are verified , as was already noted in 2009 by franco : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ there are no experimental data on order effects in conjunction fallacy experiments , when the judgments are performed in different orders . such an experiment could be helpful to better understand the possible judgment strategies . " ( franco 2009 , 421 ) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ we fill this gap here by running several order effect experiments that collect the needed data . as for the second point test new empirical predictions of the models , we consider two tests that apply to any version of the quantum - like models , whether degenerate or not , that are used in the account of the conjunction fallacy . it is well - known in the literature that quantum - like models that account for the conjunction fallacy predict an order effect for the two questions associated with the conjunction ( is linda a bank teller ? " and is linda a feminist ? " ) . actually , this predicted order effect is not a side effect of the quantum - like models , but a core feature of them : they can not account for the conjunction fallacy without it . this enables a direct test of the quantum - like account of the conjunction fallacy , that we apply to our collected experimental data . also , it has been shown that any quantum - like model of the kind involved in the account of the conjunction fallacy must make an empirical prediction called the qq equality " ( wang and busemeyer 2013 , wang et al . we thus test whether the qq equality is verified . the failure of any of these last two tests will be enough to refute the current quantum - like account of the conjunction fallacy . here also , the needed data is not available in the literature , but can be conveniently obtained from the same above - mentioned new experimental configuration , with two yes - no questions in both order . note that our methodology is novel : we are not testing the quantum - like models against data produced by traditional conjunction fallacy experiments that the model were designed to explain , but we are testing them against other data , in a new experimental framework on which the models actually make some predictions , and it is why the experimental situation we shall consider is different from the usual linda experiment . our experiment instantiates the mechanism that the quantum - like account claims agents follow : to evaluate a conjunction like feminist and bank teller " , agents are supposed to evaluate one characteristic after another , answering for themselves to two yes - no questions ( is linda a feminist ? " , is linda a bank teller ? " ) . in other words , the experiment we run somehow forces agents to follow the purported quantum - like mechanism . to have more powerful tests , we have conducted several experiments , with variations of the scenario ( linda , but also others known as bill , mr . f. and k. ) , of the protocol ( questionnaires or computer - assisted experiment ) and with or without monetary incentives . the results we obtain show that current quantum - like models are not able to account for the conjunction fallacy . the outline of the paper is the following . in section 2 , a general quantum - like model is introduced . section 3 presents the three empirical tests that will be performed : the gr equations , order effect , and the qq equality . the experimental protocol is presented in section 4 , and the results in section 5 . section 6 presents the statistical analysis , and section 7 discusses the scope of the results and the future of the research on the conjunction fallacy account . as indicated in the introduction , we focus in this paper on a family of quantum - like models based on similar hypotheses that have recently been proposed to account for the conjunction fallacy . they are presented or defended in franco ( 2009 ) , busemeyer et al . ( 2011 ) , busemeyer and bruza ( 2012 ) , pothos and busemeyer ( 2013 ) and busemeyer et al . ( 2015 ) . for simplicity , we choose here to summarize them with a single model with our own notations , and the correspondence with the various models from the literature can easily be made by the reader . for illustrative purposes , we shall consider the conjunction fallacy through the linda case , but the generalization to other instances of the conjunction fallacy are straightforward . according to this literature , after reading linda s description , the subject who has to choose the more likely proposition between * linda is a bank teller " , * linda is a feminist and a bank teller " . has the following mental process . to compare the propositions , she evaluates each one in terms of a yes - no question : * is linda a bank teller ? " , * is linda a feminist and a bank teller ? " . an important hypothesis of the quantum - like models is that , when the subject considers ( @xmath3 ) , she actually answers for herself successively two simple yes - no questions : * is linda a feminist ? " , * is linda a bank teller ? " . yes " to @xmath3 amounts to answering yes " to both @xmath4 and @xmath5 . also , the hypothesis is made that the more probable outcome ( bank teller or feminist ) is evaluated first . as the description of linda makes her more likely a feminist than a bank teller , this means that @xmath3 is answered by answering first @xmath4 and then @xmath5 . _ and then _ @xmath5 ( p. 418 ) . anyway , the tests we consider in the forthcoming sections do not depend on this hypothesis . ] let us now turn to the quantum - like framework that enable the quantitative prediction of the conjunction fallacy , @xmath6 . for pedagogical purposes , the non - degenerate versions of the quantum - like models are presented first , and the degenerate versions afterwards . the belief states of agents are represented within a vector space . in the simple case where an agent have just given an answer ( respectively no " ) to question @xmath4 , her belief state is represented by the vector @xmath7 ( respectively , @xmath8 ) . in accordance with the literature , we shall say for short that these vectors represent the answers themselves . similarly with @xmath9 and @xmath10 for answers to question @xmath5 . the sets ( @xmath9 , @xmath10 ) and ( @xmath7 , @xmath8 ) respectively represent all possible answers to questions @xmath5 and @xmath4 , and thus each one is a basis of the same 2-dimension vector space . the vector space is equipped with a scalar product , thus becoming a hilbert space : for two vectors @xmath11 and @xmath12 , the scalar product @xmath13 is a complex number . the order of the vectors within a scalar product here matters : @xmath14 is the complex conjugate of @xmath13 . the above bases are supposed to be orthogonal : @xmath15 , and of unitary norm : @xmath16 . a representation of the bases in the special case of real coefficients can be found on figure [ fig_bases - b - and - f ] [ left ] . ( 100,80)(-30,-10 ) ( 0 , 0)(1,0)50 ( 52 , -2)@xmath9 ( 0 , 0)(0,1)50 ( -5 , 55)@xmath10 ( 0 , 0)(3,1)47.4 ( 50 , 15)@xmath7 ( 0 , 0)(-1,3)15.8 ( -25 , 50)@xmath8 ( 100,80)(-30,-10 ) ( 0 , 0)(1,0)50 ( 52 , -2)@xmath9 ( 0 , 0)(0,1)50 ( -5 , 55)@xmath10 ( 0 , 0)(3,1)47.4 ( 50 , 15)@xmath7 ( 0 , 0)(-1,3)15.8 ( -25 , 50)@xmath8 ( 0 , 0)(4,3)40 ( 42 , 33)@xmath17 ( 40,0)(40,15)(40,30 ) ( 30 , -7)@xmath18 ( 0,30)(20,30)(40,30 ) ( 2 , 33)@xmath19 ( 45,15)(42.5,22.5)(40,30 ) ( -5,15)(17.5,22.5)(40,30 ) an agent s state of belief is represented by a normalized vector @xmath17 within the hilbert space . this vector can be decomposed in either of the two above - mentioned bases , as indicated on figure [ fig_bases - b - and - f ] [ right ] : @xmath20 with the specific values taken in figure [ fig_bases - b - and - f ] [ right ] in a hilbert space on real numbers , this equation becomes for instance : @xmath21 the belief state @xmath17 gathers all the relevant information needed to predict the behavior of the agent , in the following way . predictions made by the quantum - like models are probabilistic . when a question @xmath22 ( @xmath23 = @xmath1 or @xmath0 ) is asked , the probability that the agent answers @xmath24 ( ) is given by the squared modulus of the scalar product between the belief state and the vector representing the answer : @xmath25 this rule is usually called the born rule , in analogy with the quantum mechanics denomination . it enables to compute the probability that the agent gives each of the 4 answers , in case questions @xmath5 or @xmath4 are asked ( as @xmath17 is normalized , @xmath26 ) . in the case of a real hilbert space like on figure [ fig_bases - b - and - f ] , a geometric interpretation of the born rule is the following : to compute the probability to answer , say , yes " to question @xmath5 , orthogonally project @xmath17 on @xmath9 this gives the length @xmath18 , and the wanted probability is just the squared of it . so , the more @xmath17 is aligned with a basis vector @xmath27 , the larger the probability is that the agent will answer @xmath28 if question @xmath22 is posed ( note the if question @xmath22 is posed " part : in quantum - like models , the probability of an answer is only defined in the context in which the corresponding question is posed ) . for instance , with the specific values in figure [ fig_bases - b - and - f ] [ right ] , @xmath29 , @xmath30 , @xmath31 and @xmath32 , which is consistent with the relative alignments of the basis vectors with @xmath17 . the last postulate of the quantum - like model has to do with the way @xmath17 changes over time . first , @xmath17 does not change unless the agent answers a question . this conveys the fact that the agent s beliefs are not externally influenced . this hypothesis is supposed to be relevant for cases in which the questions are posed to the agent relatively quickly . second , when the agent answers a question @xmath5 or @xmath4 , her state of belief changes . if her answer to question @xmath22 is @xmath24 , then her new state of belief just after giving the answer is : @xmath33 as the fraction in eq . [ eq_projection - postulate ] is a complex number , the state of belief after an answer @xmath24 is proportional to the vector @xmath27 representing this answer . in the case of a real hilbert space like on figure [ fig_bases - b - and - f ] , after answering yes " to question @xmath5 , @xmath17 becomes either @xmath9 or @xmath34 , whatever the state of belief before the question . in other words , after a question @xmath23 has been posed , the state of belief is bound to be along the basis vectors representing its answers . . [ eq_projection - postulate ] can be interpreted as follows : the @xmath35 part represents the fact that @xmath17 is projected on @xmath27 , the basis vector representing the given answer ; the @xmath36 part is then just a multiplicative factor that ensures that the new state of belief is normalized . hence , the above rule is often called the projection postulate . because of the projection postulate , the states before and after an answer are in general different . they are the same only if the state previous to the answer is proportional to one of the basis vectors representing the possible answers to the question , i. e. when @xmath37 , where @xmath38 is a complex number such that @xmath39 ( in the real case , @xmath40 ) . in such a case , the agent answers @xmath28 to question @xmath23 with probability 1 , and eq . [ eq_projection - postulate ] states that @xmath41 the fact that the state of belief changes when a question is answered is a real departure from the classical viewpoint . classically , the answer is supposed to _ reveal _ a belief , which is pre - existent to the question , and is the same before and after . however , the quantum - like models predict that once a question has been answered , the same answer will be given if the same question is posed again just after . let us now turn to the more general versions of these models , the degenerate ones . the difference lies in the fact that an answer is not represented by a vector belonging to a 1d space , but by any subspace of dimension @xmath42 , for instance a plane . then , the hilbert space is not of dimension 2 , but of a higher one . when question @xmath22 is posed , the probability that the agent answers @xmath24 is now defined as : @xmath43 where @xmath44 is the orthogonal projector onto the subspace representing answer @xmath28 to question @xmath22 . the change in the state of belief is now : @xmath45 for the rest , the model is the same . the mental process that gives rise to the conjunction fallacy that has been described at the beginning of this section is graphically illustrated on figure [ fig_bases - qp - explanation ] . the probability of considering that linda is a bank teller corresponds to the squared length of the projection of @xmath17 onto the bank teller vector @xmath9 , and @xmath46 . for instance , with the specific values used in figure [ fig_bases - qp - explanation ] with a real hilbert space , @xmath47 and @xmath48 . on the other hand , the probability of considering her to be feminist and bank teller corresponds to the squared length of the projection of @xmath17 onto two successive vectors , first @xmath7 and then @xmath9 , and @xmath49 . in the example of figure [ fig_bases - qp - explanation ] , @xmath50 and @xmath51 . ( 90,90)(-10,-30 ) ( 0 , 0)(1,0)50 ( 52 , -2)@xmath10 ( 0 , 0)(0,1)50 ( -5 , 55)@xmath9 ( 0 , 0)(3,1)47.4 ( 50 , 15)@xmath17 ( 0 , 0)(3,-4)30 ( 34 , -39)@xmath8 ( 0 , 0)(4,3)40 ( 42 , 33)@xmath7 ( -7 , 28)@xmath52 ( 47,15.5)(42,22)(37,28 ) ( 37,28)(18.5,28)(0,28 ) ( 45,16)(22.5,16)(0,16 ) ( -7 , 14)@xmath53 so , there exist some model configurations , like the one plotted on figure [ fig_bases - qp - explanation ] , in which the probability to be judged feminist and bank teller is higher than the probability to be judged bank teller , leading to @xmath54 in accordance with empirical results . a quantum - like model of the conjunction fallacy has been provided . projects her state vector onto either @xmath7 or @xmath8 , but they do not suppose that answering for herself the second question @xmath5 projects the state vector onto @xmath9 or @xmath10 , because they argue that what is needed at this time is only an evaluation of the probability , and not a firm answer ( the authors acknowledge the validity of the projection postulate as soon as the agent gives a definite answer to a question ) . so , the authors actually do not specify what the state vector is after the evaluation of the conjunction ( personal communication , 2014 ) . for simplicity , we have made as if the state vector was projected onto either @xmath9 or @xmath10 , like for other questions , but this has no consequence for our forthcoming tests . see also section [ sec_expdesign ] . ] this section presents the three empirical predictions of the above quantum - like model that we will test . the first one applies to non - degenerate models , while the others apply to non - degenerate and degenerate models . following boyer - kassem et al . ( 2016 ) , some specific empirical predictions can be derived for non - degenerate models , i.e. in which the answers are represented by subspaces of dimension 1 . it can be shown that a well - known law from quantum mechanics , the law of reciprocity , holds . consider the two questions @xmath4 and @xmath5 in one order or in the other . the law of reciprocity states that , for @xmath55 , and @xmath56 , @xmath57 this law asserts that conditional probabilities of an answer given another answer are the same whatever the order of the questions @xmath5 and @xmath4 . note that this law is typically quantum : it is not true in general for a classical model , in which @xmath58 , and thus @xmath59 as soon as @xmath60 . the law of reciprocity can be instantiated in the following ways : [ eq_pb0a0=pa0b0 ] p(b_y|f_y ) = p(f_y|b_y ) , + [ eq_pb1a0=pa0b1 ] p(b_n|f_y ) = p(f_y|b_n ) , + [ eq_pb0a1=pa1b0 ] p(b_y|f_n ) = p(f_n|b_y ) , + [ eq_pb1a1=pa1b1 ] p(b_n|f_n ) = p(f_n|b_n ) . some easy computation enable to show that the following equations , called the grand reciprocity ( gr ) equations , hold ( cf . boyer - kassem et al . 2016 , section 3.1 ) : [ eq_mix0011 ] p(b_y|f_y ) = p(f_y|b_y ) = p(b_n|f_n ) = p(f_n|b_n ) , + [ eq_mix1001 ] p(b_n|f_y ) = p(f_y|b_n ) = p(b_y|f_n ) = p(f_n|b_y ) . these equations [ eq_mix0011 ] and [ eq_mix1001 ] are equivalent to one another and to the law of reciprocity itself . they state that the conditional probabilities that exist when @xmath5 is asked before @xmath4 is asked call it situation ( @xmath61 ) and in the ( @xmath62 ) situation are actually much constrained : among the eight quantities that can be experimentally measured , there is just one free real parameter . in other words , the non - degenerate quantum - like model presented in section [ subsec_ql - models ] actually leaves very little freedom to conditional probabilities . the fact that the conditional probabilities are constrained by the gr equations had not been noticed beforehand for quantum - like models for the conjunction fallacy . note that these empirical predictions are _ consequences _ of the quantum - like models that are used to explain the conjunction fallacy in the linda experiment , and that these consequences are observable in experimental situations @xmath63 and @xmath64 situations that are _ not _ the ones of the original linda experiment . in other words , the gr equations show that a non - degenerate quantum - like model that is used to explain a linda experiment can be further tested on another kind of experiment . we shall come back on this point in section [ sec_expdesign ] . the interpretation of the conditional probabilities is clear : they have been defined as the probability of some answer to a second question given the answer to a first question . this is straightforwardly consistent with the models presented in section [ sec_ql - models ] , and in accordance with classical order effect experiments . another interpretation of the conditional probabilities could be that of an answer given some new piece of evidence , but this is not what is considered in this paper . quantum - like models of section [ subsec_ql - models ] can predict an order effect , that is , predict that agents give different answers to the question @xmath4 followed by question @xmath5 , and to the question @xmath5 followed by question @xmath4 ( cf . figure [ fig_order - effect ] ) . this comes from the projection postulate that modifies the state of belief when an answer is given to a question . this order effect property of the quantum - like models is well - known , and it has actually been used to provide a quantum - like account of order effect ( see for example conte et al . 2009 , busemeyer et al . 2009 , busemeyer et al . 2011 , atmanspacher and rmer 2012 , pothos and busemeyer 2013 , wang and busemeyer 2013 and wang et al . 2014 , boyer - kassem et al . 2016 ) thus , the same models are at the basis of the account of order effect and of the conjunction fallacy . ( 80,70)(-30,-10 ) ( 0 , 0)(1,0)50 ( 52 , -2)@xmath9 ( 0 , 0)(0,1)50 ( -5 , 55)@xmath10 ( 0 , 0)(3,1)47.4 ( 50 , 15)@xmath7 ( 0 , 0)(-1,3)15.8 ( -25 , 50)@xmath8 ( 0 , 0)(4,3)40 ( 42 , 33)@xmath17 ( 40,0)(40,15)(40,30 ) ( 40,0)(38,6)(36,12 ) ( 45,15)(42.5,22.5)(40,30 ) ( 45,15)(45,7.5)(45,0 ) more importantly , it can be shown that only models that display an order effect are able to account for the conjunction fallacy ( cf . busemeyer et al . 2011 , busemeyer and bruza 2012 , bruza et al . 2015 p. 388 , busemeyer et al . 2015 ) . in other words , the quantum - like models of section 2 that do not present an order effect can not predict @xmath65 , and thus can not account for the conjunction fallacy . the reason is , in short , the following : questions @xmath5 and @xmath4 are either compatible or incompatible in the standard quantum sense . in the latter case , the hilbert space is ( in the simplest case ) 2d , with basis vectors like on figure [ fig_bases - b - and - f ] , and there is an order effect . in the former case , the hilbert space is ( in the simplest case ) 4d , with basis vectors ( @xmath66 , @xmath67 , @xmath68 , @xmath69 ) , where the vector @xmath70 stands for answer @xmath28 to question @xmath5 and answer @xmath71 to question @xmath4 , in whatever order . and such a model displays no order effect : whatever the order of the questions , the probability of an answer @xmath28 to question @xmath5 and of an answer @xmath71 to question @xmath4 will be @xmath72 , where @xmath73 is the coordinate along the @xmath70 vector ( @xmath74 ) . can such a model predict a conjunction fallacy to occur ? on the one side , consider the evaluation of the conjunction : the agent first considers @xmath4 ; if she answers yes " , the state vector is projected onto the plane @xmath75 . if she now answers yes " to @xmath5 , the resulting vector is projected onto @xmath66 . so , the probability to answer yes " to both questions is given by the square modulus of the @xmath66 component , i.e. @xmath76 . on the other side , consider the evaluation of @xmath1 , for which the agent considers @xmath5 . if she answers yes " , the state vector is projected onto the plane @xmath77 . the probability of such an answer is given by the squared modulus of the length of this projection , namely @xmath78 ( remember that the basis vectors are orthogonal ) . this quantity is at least larger than @xmath76 , so a conjunction fallacy can not occur . to sum up , any quantum - like model of the kind considered in section 2 which claims to account for the conjunction fallacy , be it non - degenerate or degenerate , has to display an order effect on the corresponding questions . this provides our second test ( cf . section [ statanalysis ] for a discussion of the mathematical expression of the test ) . the proponents themselves of the quantum - like account of the conjunction fallacy consider that the use of incompatible concepts ( or questions ) is the key feature of their model . as incompatible questions straightforwardly imply an order effect , our order effect test is actually a direct test of the core feature of the quantum - like account . and @xmath5 are compatible or incompatible . but the easiest way to do so is actually to test the order effect on these two questions . ] as for the gr equations , note that the order effect is here understood as an experimental situation with two successive yes - no questions , posed in one order or in the other after a text has been read , and that no new piece of evidence is provided between the two questions . to sum up , three features are essential for the quantum - like models under study to account for the conjunction fallacy : the born rule ( eq . [ eq_born - rule ] ) , the projection postulate ( eq . [ eq_projection - postulate ] ) , and the presence of incompatible questions entailing order effects . the quantum - like models of section 2 , whether degenerate or not , have recently been shown to entail new testable empirical predictions ( wang and busemeyer 2013 ) : a quantum question " ( qq ) equality . noting @xmath79 the probability of answering first @xmath28 to question @xmath22 _ and then _ @xmath71 to question @xmath80 ( this is a joint probability , not a conditional probability ) , the qq equality reads : @xmath81 this equality is of prime importance . as busemeyer et al . ( 2015 , 241 ) put it , it is an a priori , _ precise _ , _ quantitative _ , and _ parameter - free _ prediction about the pattern of order effects " . it has served as a test of the quantum - like models that claim to account for order effect . it turns out that it has been statistically supported across a wide range of 70 national field experiments ( containing 651 to 3,006 nationally representative participants per field experiment ) that examined question - order effects ( wang et al . , 2014 ) " ( _ ibid . _ ) . similarly , the qq equality can be empirically tested in the case of the quantum - like models that account for the conjunction fallacy , as the models are the same . this constitutes our third test ( further statistical details about the test are given in section 6 ) . the three tests presented in the previous section ( gr equations , order effect , qq equality ) require to carry out an order effect experiment that shows the description of linda and then asks the questions @xmath4 and @xmath5 in both orders , ( @xmath62 ) or @xmath63 . the former order somehow forces the agent to follow the cognitive process supposed by the quantum - like models when evaluating a conjunction . we propose here its first experimental realization , in order to test the quantum - like models of section 2 . the order effect experiment we are considering here is different from the original conjunction fallacy experiment . if we want to claim that it tests anyway the quantum - like account of the conjunction fallacy , do we need to make some extra hypothesis ? for instance , do we need to suppose that the quantum - like model for the conjunction fallacy also applies to another kind of experiment ? or do we need to assume that forcing an agent to explicitly answer the two questions will give the same results as when she answers them for herself ? we need not , because these assumptions are already made in the papers we are considering . first , the simple fact that the quantum - like account of the conjunction fallacy relies on models " that have a general and universal form and not only on _ ad hoc _ rules that apply to a limited number of situations , allows anyone to use these models _ ad libitum _ in any experimental situation that the model may represent . the order effect situation , in which two questions are asked , clearly falls within that range . so , we are allowed to apply ( and thus to test ) the quantum - like models of the conjunction fallacy in an order effect experiment . this amounts to testing experimental predictions of the models that they make because they have a general form . as the proponents of the models write : the basic quantum model underpinning the conjunction fallacy [ ... ] makes new _ a priori _ predictions . foremost among them is the consequence that incompatible judgments and decisions must entail order effects . " ( bruza et al . 2015 , p. 388 ) . ( recall that incompatible judgments are required in the quantum - like model of the conjunction fallacy . ) in other words , the conjunction fallacy model entails order effects , and thus can be tested on them . this is all the more true than the authors actually claim that the quantum - like models used for the conjunction fallacy are the same as those used to explain other fallacies or phenomena , like order effect itself or similarity judgments . all models belong to a family that are often called a theory " of quantum cognition , and they are meant to make predictions on a wide range of phenomena , in diverse experimental situations and the authors rightly claim that this is a strength of their approach . this supports the generality of the quantum - like models used for the conjunction fallacy . thus , it is legitimate to use them in other situations like the order effect one . besides , these models _ have been _ applied to question order effect ( wang and busemeyer 2013 , wang et al . 2014 ) , and it is clear that no extra hypothesis than the ones presented in section 2 is needed for that . in sum , the literature claims that the very same models can be used for the conjunction fallacy and for question order effect , so we are justified in testing them on new order effect cases as linda s . finally , recall that we consider here two successive yes - no questions , asked in both orders . thus , the conditional probabilities are interpreted as probabilities of a second answer given a first answer . this is fully in line with the models of the conjunction fallacy themselves . consider for instance : in this problem there are two questions : the feminism question and the bank teller question . for each question , there are two answers : yes or no . " ( busemeyer and bruza 2012 , p. 15 ) ; we consider two dichotomous questions a and b , as for example a : is linda a feminist ? and b : is linda a bank teller ? " ( franco 2009 p. 416 ) . what we propose here is to explicitly pose these two questions . in order to strengthen our experimental tests , we have considered four scenarios that have been shown in the literature to give rise to conjunction fallacies , from which we have built four experimental tasks a task consists for an agent in reading a text and then sequentially answering two yes - no questions . the first task is drawn from the case of * linda * ( tversky and kahneman 1983 ) : for the french version that was actually used in the experiments . ] * text : linda is 31 years old , single , outspoken , and very bright . she majored in philosophy . as a student , she was deeply concerned with issues of discrimination and social justice , and also participated in anti - nuclear demonstrations . " * @xmath4 : according to you , is linda a feminist ? " * @xmath5 : according to you , is linda a bank teller ? " the second task is drawn from the case of * bill * ( tversky and kahneman 1983 ) : * text : bill is 34 years old . he is intelligent , but unimaginative , compulsive , and generally lifeless . in school , he was strong in mathematics but weak in social studies and humanities . " * @xmath82 : according to you , is bill an accountant ? " * @xmath83 : according to you , does bill play jazz for a hobby ? " the third task is drawn from the case of * mr . f. * ( tversky and kahneman 1983 ) : * text : a health survey was conducted in a representative sample of adult males in france of all ages and occupations . mr . f. was included in the sample . he was selected by chance from the list of participants . " * @xmath84 : according to you , has mr . f. already had one or more heart attacks ? " * @xmath85 : according to you , is mr . f. over 55 years old ? " the fourth task is drawn from the case of * k. * , a russian woman ( tentori et al . 2013 ) : * text : k. is a russian woman " . * @xmath86 : according to you , does k. live in new - york ? " * @xmath87 : according to you , is k. an interpreter ? " so as to increase the robustness of our results , we have chosen these four tasks as they display different kinds of conjunction fallacies , in the sense of tversky and kahneman ( 1983 ) who have distinguished between m a and a b paradigms . in the former , a model m ( the text describing the person ) is positively associated with an event a ( one of the two sentences forming the conjunction ) and negatively with the other event b. this is the case of the linda scenario : the introductory text m is positively associated with the event linda is a feminist " and negatively with the other one linda is a bank teller " . also , bill s scenario is of type m a . differently , in the a b paradigm , a is positively associated with b , but not with the model m. for instance , mr . f. is over 55 years old " is positively associated with mr . f. already had one or more heart attacks " , but not with the text . the scenario of the russian woman seems to correspond to neither paradigm : the positive association occurs between the text m and the conjunction of the two constituents a and b , and not with only one of them , so we might call it m(ab ) the fact that the woman is russian is strongly associated with the fact that she lives in new york and is also an interpreter . conjunction fallacies and quantum - like models have been studied by scholars of various fields , and in particular by psychologists and economists ( cf . section [ sec_intro ] ) . to keep with these two traditions , we have chosen not to limit ourselves to one experimental protocol which also has the advantage of increasing the robustness of the experimental findings . we have varied the administration method , with paper questionnaires like in the psychological tradition and with computer implementations like in the economical tradition , with and without payment . we have carried out three experiments ( cf . table [ tab_listexp ] for a summary ) . in the first experiment , two tasks were successively presented to the subjects : that of mr . f. and that of bill . the experiment was conducted in march and april 2015 at the university of tours and of nice sophia antipolis ( france ) , with a total of 496 students in medicine , economics and management . in the psychological tradition , the tasks were implemented with paper questionnaires , in the lecture hall at the end of classes . because of the improvised recruitment without appointment , and because of the short length of the task , the students were not paid , like in the psychological tradition . these tasks are noted @xmath88 and @xmath89 , with an index @xmath90 " for paper " . the second experiment successively featured the 4 tasks introduced above in the following order : k. the russian woman , mr . f. , bill and linda . the experiment was conducted on april 2015 at the lameta , the experimental economics laboratory of the university of montpellier 1 ( france ) , in 19 sessions , with a total of 302 students possibly from any discipline . in the economics tradition , the tasks were implemented on computers ( created with the z - tree program , fischbacher 2007 ) , and students were recruited online and received a show - up fee ( 5 or 9 euros , according to their campus of origin ) to remunerate their attendance and to reduce the effect of selection bias . these tasks are noted @xmath91 , @xmath92 , @xmath93 and @xmath94 , with an index @xmath95 " for computer " and a euro for the payment . a third experiment involved the task of linda , in a mixed methodology . it was conducted on october 2014 in the leen , the experimental economics laboratory of the university nice sophia antipolis , with a computerized questionnaire . 354 students were recruited on the fly at the end of the classes , and were not paid for the short task . this task is noted @xmath96 , with an index @xmath95 " . .experimental tasks that were carried out , together with their administration methods , the location , and the number of subjects involved . two dashed horizontal lines separate into three groups the seven experimental tasks , corresponding to three distinct experiments.[tab_listexp ] [ cols="<,<,<,<,<,<",options="header " , ] on the basis of the above experimental results , we now would like to test three research hypotheses that have motivated the quantum - like modeling literature on conjunction fallacy , and that correspond to the building blocks of the current models presented in section 2 . this shall provide some interpretation of the bare statistical results obtained in sections [ subsec_test - gr - equations ] to [ subsec_test - qq - equality ] . the first two hypotheses have already been presented in the introduction and concern the validity of quantum - like models , while the third one is larger and goes beyond quantum - like models : * * hyp . # 1 : * non - degenerate quantum - like models ( presented in section 2 ) can account for the conjunction fallacy , * * hyp . # 2 : * non - degenerate or degenerate quantum - like models ( presented in section 2 ) can account for the conjunction fallacy , * * hyp . # 3 : * the conjunction fallacy account can rely on a question order effect account . the first hypothesis is the simplest and less general one . it restricts accounts of the conjunction fallacy to the simplest versions of the quantum - like models , i.e. non - degenerate ones , where answers are represented by 1-d subspaces . this is the hypothesis made in franco ( 2009 ) , who only considers non - degenerate models . this hypothesis implies that the gr equations are empirically verified . as section [ subsec_test - gr - equations ] has shown that the gr equations are never verified in our experiments , we can safely say that the first hypothesis is empirically refuted by our data . in other words , non - degenerate quantum - like models can not account for order effects . this refutes the proposal by franco ( 2009 ) , who has only considered non - degenerate models all other quantum - like models cited in section 2 are not refuted , since they also consider degenerate models . the rejection of the first hypothesis echoes recent debates . the empirical inadequacy of non - degenerate models for the conjunction fallacy has already been discussed , although the question had not been definitely settled ( cf . tentori and crupi 2013 and pothos and busemeyer 2013 , p. 315 - 316 ) . in a similar vein , it has been shown that non - degenerate models for order effect are not empirically adequate ( boyer - kassem et al . overall , our result is in line with previous suggestions that degenerate models should be preferred to non - degenerate models , as the latter should be considered as toy models " only ( e.g. busemeyer and bruza 2012 , busemeyer et al . 2015 ) . the second research hypothesis extends the first one by considering also degenerate models , that is , models in which an answer is represented by a @xmath97-d subspace , e.g a plane . this hypothesis is shared by all papers cited in the beginning of section 2 , except franco ( 2009 ) : the conjunction fallacy can be accounted for by quantum - like models in general , be they non - degenerate or degenerate . as argued in section [ sec_empirical - tests ] , non - degenerate and degenerate models have ( i ) to display an order effect and ( ii ) to respect the qq equality . thus , the second hypothesis is testable by means of the test of the order effect and that of the qq equality . table [ stathyp2final ] summarizes the findings on these matters . both tests results are reported , the satisfaction of the qq equality in the second column and the presence of order effect in the third one . the last column reports the joint outcomes of the two tests , that is , the outcome of the logical operator and " , because either one test or the other one are sufficient to refute the quantum - like models of conjunction fallacy considered in this paper . recall that we have adopted a very conservative approach on the error of type i , so as to be conclusive with a high degree of certainty . so , we can be quite sure that the second research hypothesis is rejected in at least three out of seven tasks . our conclusion here is that the quantum - like models can not account for the general phenomenon of the conjunction fallacy . it is the first time that such a strong result is obtained experimentally . l | c ; 0.5pt/4pt c | c task i d & * qq * & * oe * & * qq and oe * + @xmath88 & - & yes & - + @xmath89 & - & - & - + @xmath91 & - & no & * no * + @xmath98 & - & no & * no * + @xmath93 & - & - & - + @xmath94 & - & - & - + @xmath96 & no & yes & * no * + the third hypothesis is not restricted to quantum - like models , but is concerned with the general idea that the conjunction fallacy is related to a question order effect between suitable questions ( for instance in the linda scenario between the questions @xmath99 and @xmath4 ) . it implies that an order effect must be observed in our experiments , and thus this hypothesis is testable by means of the order effect test . two out of seven tasks exhibit no ( or insignificant ) order effect , as shown in section [ subsec_test - order - effect ] . and yet , the corresponding scenarios ( k. and mr . f. ) do exhibit a conjunction fallacy . these results suggest that the third hypothesis , according to which the conjunction fallacy can be accounted for from an order effect , seems to be experimentally refuted . note that the consequences of the rejection of this hypothesis have an even much broader impact than the ones deriving from the rejections of previous hypotheses : not only are we rejecting the original modeling strategy exploited by the quantum - like literature based on the introduction of an order effect to explain the conjunction fallacy , but we are also preventing its adoption for any other alternative theory ( bayesian , heuristics ... ) . the conjunction fallacy can not be reduced , in terms of mental acts , to the order effect phenomenon . this finding sheds some new light into an important modeling issue . we have considered the quantum - like accounts of the conjunction fallacy that have been proposed or defended by franco ( 2009 ) , busemeyer et al . ( 2011 ) , busemeyer and bruza ( 2012 ) , pothos and busemeyer ( 2013 ) and busemeyer et al . ( 2015 ) which common trait is to represent the belief of the decision - maker with the quantum state . we have tested three empirical predictions of these models : the gr equations ( boyer - kassem et al . 2016 ) , that apply to non - degenerate versions only of the models , the existence of an order effect and the qq equality ( wang and busemeyer 2013 ) , which apply to both non - degenerate and degenerate versions of the models , hence to the most general version of the papers . such tests can not be performed in traditional conjunction fallacy experiments , in which subjects have to rank propositions , but require an order effect experiment , in which two yes - no questions are asked in either order . so , the tests concern empirical predictions that are not the data that the models were supposed to explain in the first place , but are predictions of the models anyway , and are directly related to the core feature of the models , namely the incompatibility between questions . we have performed such order effect experiments , by using a robust protocol that varies the stories ( linda , bill , mr . f. , k. ) , the administration method ( paper questionnaires or computer ) , and a possible payment , with seven tasks in total and several hundreds of subjects . our empirical results clearly reject the hypothesis that non - degenerate models can account for the conjunction fallacy ( which is the hypothesis made in franco 2009 ) . this confirms the recent tendency from the advocates of the quantum - like approach to consider non - degenerate models as toy models only . most importantly , our results also reject the more general hypothesis that non - degenerate or degenerate models can account for the conjunction fallacy , which is the hypothesis made in all other papers . as we have used very conservative statistical tests , we can safely say this general hypothesis is refuted in at least three tasks out of seven . so the present paper provides the first clear experimental rejection of the quantum - like explanation of the conjunction fallacy . now , it may be possible that not _ all _ instances of the conjunction fallacy can be accounted for in a quantum - like fashion , but that _ some _ instances can . for instance , our experimental results have not formally excluded that bill s scenario could be amenable to a quantum - like account . there is room for possible future experimental research here a possible line of division to be investigated could be between ab and ma scenarios of conjunction fallacies . but thus , the quantum - like account would loose its generality , which was its strength . moreover , if quantum - like models were to apply to some cases of conjunction fallacies , it seems very likely that it should be degenerate versions , since non - degenerate one have been strongly ruled out . this comes with possible drawbacks or specific duties , as argued in boyer - kassem et al . in particular , a degenerate model resorts to some extra dimensions in the hilbert space that should receive theoretical and experimental justifications so as not to be just _ ad hoc_. and more general tests on elementary dimensions can also be considered . as our experimental results speak against the quantum - like models of the conjunction fallacy , they can be interpreted as indirect support in favor of alternative accounts of the conjunction fallacy , like bayesian ones ( e.g. tentori et al . 2013 ) , or other kinds of quantum - like models for the conjunction fallacy that have not been tested in this paper , like yukalov and sornette ( 2010 , 2011 ) . however , our results also provide some conclusions well beyond quantum - like modeling : they show that the conjunction fallacy can not be accounted for by any model or mechanism that relies on order effect , or entails an order effect , between the two characteristics at play ( feminist " and bank teller " in linda s case ) . quantum - like models are well - known such examples , but it must be clear that any existing or future alternative explanation that involves a question order effect is ruled out . after the failure of quantum - like models , this places a hard constraint on alternative explanations of the conjunction fallacy . we suggest that future works should try to theoretically inquire whether alternative explanations predict an order effect , and to experimentally test it . even if the quantum - like models studied in this paper are not able to account for our data , a possible research strategy could be not to abandon the quantum - like modeling of the conjunction fallacy altogether , but instead to try to modify and improve it so that it finally agrees with the experimental data . in this spirit , one could investigate whether the use of a more general measurement theory or generalized observables could be adequate . for instance , the use of positive operator valued measures ( povms ) , from quantum physics , has been recently applied to quantum - like models of cognition ( cf . khrennikov and basieva 2014 ) . however , it seems to face some new challenges like response replicability ( cf . khrennikov et al . 2014 , basieva and khrennikov 2015 ) . another quantum - like line of research that does not face this problem considers a modification of the born rule ( aerts and sassoli de bianchi , 2015 ) . as a last remark , our methodology has been here to test quantum - like models of the conjunction fallacy with new experimental predictions . we think this methodology could be fruitfully extended to quantum - like models that address other fallacies , such as the disjunction fallacy or the inverse fallacy . many thanks to corrado lagazio for suggestions on the statistical part . we would like to thank jerome busemeyer , vincenzo crupi , dorian jullien , andrei khrennikov , katya tentori , vyacheslav yukalov and participants at the 2015 conference quantum probability and the mathematical modelling of decision making " ( fields institute , toronto ) , for useful comments and suggestions . experimental economics laboratory at the university of nice ( leen - 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( 1983 ) , extensional versus intuitive reasoning : the conjunction fallacy in probability judgment " , _ psychological review _ 90(4 ) : 293315 . von sydow , momme ( 2011 ) , the bayesian logic of frequency - based conjunction fallacies " , _ journal of mathematical psychology _ 55 : 119139 . walker , esteban and amy s. nowacki ( 2011 ) , understanding equivalence and noninferiority testing " , _ journal of general internal medicine _ 26(2 ) : 192 - 196 . wang , zheng j. and jerome r. busemeyer ( 2013 ) , a quantum question order model supported by empirical tests of an a priori and precise prediction " , _ topics in cognitive science _ 5(4 ) : 689710 . wang , zheng j. , tyler solloway , richard m. shiffrin and jerome r. busemeyer ( 2014 ) , context effects produced by question orders reveal quantum nature of human judgments " , _ proceedings of the national academy of science _ 111(26 ) : 9431 - 9436 . yukalov , vyacheslav i. and didier sornette ( 2010 ) , mathematical structure of quantum decision theory " , _ advances in complex systems _ 13 : 659698 . yukalov , vyacheslav i. and didier sornette ( 2011 ) , decision theory with prospect interference and entanglement " , _ theory and decision _ 70 : 283328 . the statistical test is to compare two conditional relative frequencies @xmath100 and @xmath101 , with the null hypothesis that they are equal . the test is therefore @xmath102 where both @xmath100 and @xmath101 are observed as conditional relative frequencies . + testing equation [ yxtest ] is equivalent to test @xmath103 given that @xmath100 and @xmath101 are not equal to zero . + alternatively , we can formulate the test in terms of the log odds ratio ( or ) @xmath104 consider the first statistical test , @xmath105 we can thus test the following condition : @xmath106 or @xmath107 by expressing the conditional relative frequencies in terms of joint frequencies , that is , @xmath108 with @xmath109 and @xmath110 the @xmath100-components of the marginal frequencies of @xmath111 and @xmath112 , we obtain @xmath113 or simplifying @xmath114 we can thus test indifferently eq . [ condprobtest ] or [ logortest ] . + given condition [ logortest ] , to perform the statistical test we suppose here that @xmath115 where @xmath116 is the standard error of the log odds ratio . it is estimated as the square root of the sum of the inverse of all the joint frequencies that are considered in the estimation of the @xmath117 : + @xmath118 + finally , we also apply the continuity correction to the estimation of or , because the normal approximation to the binomial is used , which is effective in particular for small values of @xmath119 or @xmath120 : @xmath121 + * linda " * : * text : linda a 31 ans , elle est clibataire , franche , et trs brillante . elle est diplme en philosophie . lorsquelle tait tudiante , elle se sentait trs concerne par les questions de discrimination et de justice sociale et avait aussi particip des manifestations anti - nuclaires . " * @xmath4 : selon vous , linda est - elle fministe ? " * @xmath5 : selon vous , linda est - elle employe de banque ? " * text : bill a 34 ans . il est intelligent , mais na pas dimagination , il est compulsif , et gnralement plutt teint . lcole , il tait fort en mathmatiques , mais faible dans les sciences humaines et sociales . " * @xmath82 : selon vous , bill est - il comptable ? " * @xmath83 : selon vous , bill joue - t - il du jazz pour ses loisirs ? " * text : une enqute de sant a t mene en france sur un chantillon reprsentatif dhommes adultes de tous ges et de toutes professions . dans cet chantillon , on a choisi au hasard monsieur f. * @xmath84 : selon vous , monsieur f. a - t - il dj eu une ou plusieurs attaques cardiaques ? " * @xmath85 : selon vous , monsieur f. a - t - il plus de 55 ans ? "

human agents happen to judge that a conjunction of two terms is more probable than one of the terms , in contradiction with the rules of classical probabilities this is the conjunction fallacy . one of the most discussed accounts of this fallacy is currently the quantum - like explanation , which relies on models exploiting the mathematics of quantum mechanics . the aim of this paper is to investigate the empirical adequacy of major quantum - like models which represent beliefs with quantum states . we first argue that they can be tested in three different ways , in a question order effect configuration which is different from the traditional conjunction fallacy experiment . we then carry out our proposed experiment , with varied methodologies from experimental economics . the experimental results we get are at odds with the predictions of the quantum - like models . this strongly suggests that this quantum - like account of the conjunction fallacy fails . future possible research paths are discussed .

You are an expert at summarizing long articles. Proceed to summarize the following text: initially formulated by john wheeler@xcite , the program of deriving quantum formalism from information - theoretic principles has been receiving lately much attention . thus , jozsa@xcite promotes a viewpoint which `` attempts to place a notion of information at a primary fundamental level in the formulation of quantum physics '' . fuchs @xcite presents his program as follows : `` the task is not to make sense of the quantum axioms by heaping more structure , more definitions ... on top of them , but to throw them away wholesale and start afresh . we should be relentless in asking ourselves : from what deep _ physical _ principles might we _ derive _ this exquisite mathematical structure ? .. i myself see no alternative but to contemplate deep and hard the tasks , the techniques , and the implications of quantum information theory . '' in a similar fashion , rovelli@xcite distinguishes a philosophical problem of interpretation from a mathematical problem of derivation of quantum mechanical formalism from the first principles . he writes , `` ... quantum mechanics will cease to look puzzling only when we will be able to _ derive _ the formalism of the theory from a set of simple physical assertions ( postulates , principles ) about the world . therefore , we should not try to append a reasonable interpretation to the quantum mechanics _ formalism _ , but rather to _ derive _ the formalism from a set of experimentally motivated postulates '' . rovelli refers to his own work as a point of view and not as interpretation : `` from the point of view discussed here , bohr s interpretation , consistent histories interpretations , as well as many worlds interpretation , are all correct '' . rovelli s _ point of view _ , i.e. informational treatment of quantum mechanics , thus serves a formal criterion or a filter that permeates certain interpretations and not others . in other words , treatment of quantum mechanics on information - theoretic grounds entails that some interpretations of quantum theory will be with certainty inapplicable but a number of other interpretations will all remain possible . such a result can be naturally expected from any novel formal development of quantum theory that remains in the area of science as opposed to philosophy . any formal derivation of quantum mechanics , in particular those using bayesian methods@xcite and quite promising for someone who believes in information - theoretic foundations of physics , requires a definite conceptual background on which such a derivation will further operate . as it is often the case , to give a rigorous axiomatic system that could provide the necessary background , is a difficult task . below we analyze some three proposed solutions , by rovelli@xcite , by fuchs @xcite and by brukner and zeilinger @xcite . elementary act of measurement is understood by rovelli as yes - no question . brukner and zeilinger use the term `` proposition '' which generalizes the notion of binary question . still , if one looks into where from the term `` proposition '' appears , one finds in @xcite two formulations of zeilinger s fundamental principle for quantum mechanics : fp1 : : an elementary system represents the truth value of one proposition . fp2 : : an elementary system carries one bit of information . it seems that zeilinger s choice of these two principles strongly suggests that the following phrase in bz reflects the view of the authors on the fundamental issue and thus puts them very close to rovelli s position : `` yes - no alternatives are representatives of basic fundamental units of all systems . '' fuchs starts directly with the hilbert space and the full structure of quantum mechanics . he desribes measurements not by projectors but by positive operator - valued measures . this allows one to think that he will not agree with a definition of primitive measurements as consisting of exclusive yes - no alternatives , where the word `` exclusive '' leads to mathematically representing yes - no questions as orthogonal projectors . still , fuchs mentions some of the basic assumptions that he makes in his derivation . rovelli and bz each then pose two axioms . axiom 1 : : : + * rovelli : `` there is a maximum amount of relevant information that can be extracted from a system . '' * fuchs : does nt follow the axiomatic approach ; states that `` there is maximal information about a system . '' * brukner and zeilinger : `` the information content of a quantum system is finite . '' axiom 2 : : : + * rovelli : `` it is always possible to acquire new information about a system . '' * fuchs : does nt follow the axiomatic approach ; states that `` there will always be questions that we can ask of a system for which we can not predict the outcomes . '' * brukner and zeilinger : introduce the notion of total information content of the system ; state that there exist mutually complementary propositions ; state that total information content of the system is invariant under a change of the set of mutually complementary propositions . in spite of a quite striking analogy between the axioms chosen by different authors , as for the following derivation of quantum mechanics , they do not proceed in the same manner . we shall now have a closer look at the axioms and derivation techniques . axiom 1 marks a crucial point of departure from classical physics . newtonian physics employs mathematics of continuum to represent the world and , therefore , any calculation of complete information about , say , a particle position would require an infinitely long computation . this fact has profoundly disturbed many physicists , with most prominently feynman saying , `` it always bothers me that , according to the laws as we understand then today , it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space and no matter how tiny a region of time , ... why should it take an infinite amount of logic to figure out what one tiny piece of space - time is going to do ? '' axiom 1 also goes in line with wheeler s `` no continuum '' principle@xcite . while it seems intuitively plausible to accept axiom 1 , its interpretation is not straightforward . each author imposes his own interpretation by choosing a suitable translation into his language ; for rovelli it means that there exist complete sets of yes - no questions that could provide one , abstractly , with complete information about a system ; for brukner and zeilinger it means that everything that is there to a system is represented by a complete set of mutually complementary propositions . however , it appears that axiom 1 can raise yet a different issue . our intuition is that essential finiteness applies , not to the system to which we address yes - no questions but to the system plus the observer who asks these questions . formal development of this idea will appear in section [ sect3 ] . philosophical argument goes as follows : it is not true , that in order to know a newtonian coordinate we need the knowledge of infinitely many decimal digits . the latter should not make us worry , for we are endowed with an ability to create a special code ( a new concept ) , which will substitute in the thinking the undesired infinity . the same works for computation : feynman s argument from the infinity of logical operations must include the possibility of `` hiding '' newtonian infinities under the `` and so forth '' concept , for which one may specify operational rules . consequently , the requirement of finiteness applies to the observer - observed system . in a similar manner , in bz terms , essential finiteness applies to system plus the one who chooses the propositions to be tested on the system , i.e. the observer . we do not have the intuition that `` one can not know the infinity '' but , rather , that one can not have infinite knowledge . how is this understanding of the finiteness axiom related to the one adopted by rovelli and other authors ? with the assumption of universality of quantum theory , one can deduce the old point of view . indeed , universality allows us to treat the border line between theory and meta - theory in quantum mechanics as flexible @xcite . any given observer ( meta - theoretical entity ) can be included in the theory proper by taking the point of view of an observer external to the one in question . if the amount of information remains finite in spite of the arbitrary choice of the frontier between the system and the observer , we can eliminate from consideration any previously given observer at the price of redefining the question - asking party . imagine , for example , a computer solving in maple software some field theory renormalization problem . renormalization is about removing infinities , so if a computer were let to solve this task without conceptualizing infinities by means of a previously learned renormalization technique , it would have never arrived at any result . abstract amount of information in the system is infinite in this example , but the amount of relevant information is finite . what is relevant , decides the observer who translates relevancy into concepts that he employs for the computation on a system , or equivalently into a specific manner to ask some yes - no questions and not others ( see section [ sect44 ] ) . going to the extreme of definitions , what is information and what is not decides the observer , and it is because of this that the amount of information is finite . had we had the liberty to call information anything we want , there would be no intuitively clear argument showing why this `` anything '' must be finite . finiteness thus has to include the observer , and thanks to the universality of quantum mechanics can be in the limit reduced to finiteness in the sense of rovelli . axiom 2 beautifully corresponds to wheeler s dictum ( adopted after philip anderson ) `` more is different '' . if one wants to get more information , this will be different information ; or one can always get more information , and it will be different information . though in the original `` more is different '' was used in the context of complexity theory , it can as well , as a basic principle , apply to information - based quantum theory . at it will be seen below , for rovelli axiom 2 allows to introduce probabilities and to deduce an analogue of the born rule ; for bz axiom 2 leads to imposing a certain structure on the information space . this axiom is responsible for the departure from classicality , which is not yet fully accommodated by axiom 1 alone . fuchs uses as _ a priori _ given the structure of hilbert space ; his task is to deduce some of the operational structure of quantum mechanics , namely , density matrices . rovelli , on the contrary , is interested in deducing quantum mechanics from the axioms and does not show a way to deduce most of further structure , to start with the superposition principle ( apart from introducing it as an axiom ) . fuchs uses a decision - theoretic ( bayesian ) approach to derive the superposition principle . he refers to rovelli s paper in his own , and one is left free to suggest that many of his axiomatic assumptions , on which he does nt clearly comment , might be similar to rovelli s ones , apart from the key issue of how to define measurement . indeed , fuchs insists on the fundamental character of positive operator - valued measures . this may not seem intuitively evident . but because fuchs leaves the axiomatic foundations of the bayesian approach open , even if we dismiss the necessity to define measurement as povm , there still remains an opportunity to introduce the latter in the theory . povm have a natural description as conventional von neumann measurements on an ancilla system@xcite , and thus to rovelli s axiomatic derivation of the hilbert space structure one may try to add an account of inevitability of ancilla systems and naturally obtain from this the needed povm description . this will be attempted in section [ sect3 ] . brukner and zeilinger proceed differently . if information is primary , they argue , then any formalism should be a formalism dealing with information and not with some other notions . therefore bz construct an _ information space _ where they apply the axioms and use the formalism to deduce testable predictions . bz do not refer to physical space or to hilbert space in their construction . thus they do not have access to algebra allowing a reconstruction of the state space out of the operator hilbert space . therefore , because of this change of scenery , they are bound to postulate more properties of mutually complementary propositions than rovelli or fuchs . namely , they postulate the homogeneity of parameter space . bz s self - imposed terminological limitation to abstract information space does not seem viable for philosophical , i.e. extra - scientific , reasons : it renders the formalism less transparent in use , while introduction of supplementary axioms does not make it conceptually clearer than traditional formalisms . to continue , the question is how to extract a useful approach from the juxtaposition of rovelli s and fuchs s proposals . rovelli , as said before , shows a way to construct the hilbert space structure from two axioms . unlike brukner and zeilinger , we do not call this hilbert space information space but simply physical space , for there is no other space in the whole construction that would be the physical space . bz s information space is what the physical space _ _ is__. next , with hilbert space in hand , we use fuchs s derivation based on gleason s theorem to deduce density matrices and their properties . this requires essentially one more step : we need to introduce povm as measurements , as discussed above . once we ve introduced ancilla systems , we can operationally redefine measurement as described by povm . to move further in combining rovelli s and fuchs s proposals , after the axiomatic stage , we either need a sort of decision - theoretic approach to derive the formal consequences of the necessary intersubjective accord of measurement results ( for fuchs , for example , via a version of the de finetti theorem ) , or we need to use an algebraic approach so that the constructed hilbert space be treated as space of operators corresponding to observables . this latter option will be investigated below . we are guided by the computer metaphor . indeed , the strategic task is to give a reformulation of quantum theory in information - theoretic terms . a theory that operates with the notion of information can be compared to software as opposed to a theory that operates with the notion of energy which can be compared to hardware . ideally one would wish to see all `` hardware '' or energetic language disappear from the formulation of the theory , so that only `` software '' or informational language remain . we are usually interested in information about ( knowledge of ) the chosen system and we disregard particular ways in which we have obtained this information . all that counts is knowledge that can be useful in future or , in other words , relevant knowledge or relevant information . this is why one usually does not pay attention to the very process of interaction between the system being measured and the measuring system , and one treats measuring system as a meta - theoretic , i.e. non - physical , apparatus . to give an example , for some experiment a physicist may need to know the proton mass but he will not at all be interested in how this quantity was measured , unless he is a narrow specialist whose interest is in measuring particle masses . particular ways to gain knowledge are irrelevant , while knowledge itself is highly relevant and useful . some of the experiments where one is interested in the measurement as a physical process are discussed in@xcite . from now on we assume that measurement details are irrelevant , perhaps at the price of redefining what is measurement . in a practical setting , though , information is _ always _ this is to say that there always is some physical support of information , some hardware . the necessity of the physical support requires that we proceed in the following manner : first , treat the measurement interaction as physical ; then , disregard the fact that it was physical and reformulate the theory in terms of measurement results only . to start , make a distinction between two parts of the world : quantum system @xmath0 , which is the system of interest , and the observer . the observer , in the spirit of the software - hardware metaphor , consists of an informational agent ( `` i - observer '' ) and of the physical realization of the observer ( `` p - observer '' ) . there is no i - observer without p - observer . reciprocally , there is no sense in calling p - observer an observer unless there is i - observer ( otherwise p - observer is just a physical system as any ) . hence , the two components of the `` larger observer '' are not in any way separate or orthogonal to each other ; on the contrary , these are merely two viewpoints , and the difference is but descriptive . p - observer interacts with the quantum system and thus provides for the physical basis of measurement . i - observer is only interested in the measurement result , i.e. information per se , and he gets information by reading it from p - observer . the act of reading or getting information is here a common linguistic expression and not a physical process since i - observer and p - observer are not physically distinct . in fact , the concept of `` being physical '' only applies to p - observer , and by definition the physical content of the `` larger observer '' is all contained in p - observer . i - observer as informational agent is meta - theoretic , and hence the fact that its interaction with p - observer , or the act of `` reading information '' , is unphysical . to give a mathematical meaning to this act , we assume that getting information is described as yes - no questions asked by i - observer to p - observer . to follow rovelli s construction@xcite , the set of questions will be denoted @xmath1 . according to axiom 1 , there is a finite number @xmath2 that characterizes p - observer s informational capacity . the number of questions in @xmath3 , though , can be much larger than @xmath2 , as some of these questions are not independent . in particular , they may be related by implication ( @xmath4 ) , union ( @xmath5 ) , and intersection ( @xmath6 ) . one can define an always false ( @xmath7 ) and an always true question ( @xmath8 ) , negation of a question ( @xmath9 ) , and a notion of orthogonality as follows : if @xmath10 , then @xmath11 and @xmath12 are orthogonal ( @xmath13 ) . equipped with these structures , and under the non - trivial assumption that union and intersection are defined for every pair of questions , @xmath14 is an orthomodular lattice . one needs to make a few more steps to obtain the hilbert space structure . as follows from axiom 1 , one can select in @xmath14 a set @xmath15 of @xmath2 questions that are independent from each other . in the general case , there exist many such sets @xmath15 , @xmath16 , etc . if i - observer asks the @xmath2 questions in the family @xmath15 then the obtained answers form a string @xmath17_c.\ ] ] this string represents the `` raw '' information that i - observer got from p - observer as a result of asking the questions in @xmath15 . note that this is not yet information about the quantum system @xmath0 that the i - observer ultimately wants to have , but only a process due to functional separation within the `` larger observer '' . the string @xmath18 can take @xmath19 values and , since these outcomes are by construction mutually exclusive , we can define new questions @xmath20 such that the yes answer to @xmath21 corresponds to the string of answers @xmath22 : @xmath23 to these questions we refer as to `` complete questions '' . by taking all possible unions of sets of complete questions @xmath21 of the same family @xmath15 one constructs a boolean algebra that has @xmath21 as atoms . alternatively , one can consider a different family @xmath16 of n independent yes - no questions and obtain another boolean algebra with different complete questions as atoms . it follows , then , from axiom 1 that the set of questions @xmath14 that can be asked to p - observer is algebraically an orthomodular lattice containing subsets that form boolean algebras . this is precisely the algebraic structure formed by the family of linear subsets of hilbert space . it is interesting to note that in approaches that start with an abstract @xmath24-algebra of operators one needs to use the gelfand - naimark - segal construction to obtain a representation of this algebra as algebra of operators on a hilbert space . in the present approach , information - theoretic axioms are evoked to obtain a similar result , namely , to show that operators form a hilbert space . from the second rovelli s axiom it follows immediately that there are questions such as answers to these questions are not determined by @xmath18 . define , in general , as @xmath25 the probability that a yes answer to @xmath26 will follow from the string @xmath22 . given two complete strings of answers @xmath18 and @xmath27 , we can then consider the probabilities @xmath28 from the way it is defined , the @xmath29 matrix @xmath30 can not be completely arbitrary . first , we must have @xmath31 then , if information @xmath32 is available about the system , one and only one of the outcomes @xmath33 may result . therefore @xmath34 if we assume that @xmath35 then we also get @xmath36 if pursued further in an attempt to deduce probability amplitudes , this derivation , however , encounters some difficulties . to get the result , rovelli postulates explicitly the superposition principle . we , too , introduce a new assumption to obtain more of the structure of quantum theory . namely , we postulate non - contextuality and use gleason s theorem to deduce density matrices . it remains an open question if non - contextuality as an intuitively made assumption is welcome or must be rejected as too strong@xcite . in mathematical terms , it states that probabilities can be defined for a projector independently of the family of projectors of which it is a member , or that in @xmath37 with fixed @xmath38 probability will be the same had the fixed question belonged not to the family @xmath39 but to some other family @xmath16 . one can then prove a theorem due to gleason@xcite : * theorem ( gleason ) * _ let @xmath40 be any function from 1-dimensional projections on a hilbert space of dimension @xmath41 to the unit interval , such that for each resolution of the identity @xmath42 then there exists a unique density matrix @xmath43 such that @xmath44 _ one last step before we move to quantum theory of the system @xmath0 is to obtain unitary dynamics . following rovelli , any question can be labelled by the time variable @xmath45 indicating the time at which it is asked . denote as @xmath46 the one - parameter family of questions defined by the same procedure performed at different times . assume that time evolution is a symmetry in the theory . in the context of our approach the latter word `` theory '' includes theory of p - observer _ and _ of the quantum system @xmath0 . then recall that the set @xmath14 has the structure of a set of linear subspaces in the hilbert space , and the set of all questions at time @xmath47 to the p - observer part of the physically interacting conjunction of two systems , must be isomorphic to the set of all questions at time @xmath48 . therefore , the corresponding family of linear subspaces must have the same structure ; it follows that there must be a unitary transformation @xmath49 such that @xmath50 it is straightforward to see that these unitary matrices form an abelian group and @xmath51}$ ] , where @xmath52 is a self - adjoint operator in the hilbert space , the hamiltonian . in a practical setting , it is from the past or the future of a given experiment , in particular from the intentions of the experimenter , that one can learn which information about the experiment is relevant and which is not . what is relevant can either be encoded in the preparation of the experiment or selected by the experimenter _ a posteriori_. in all cases , the notion of relevance does not enter into the formalism which solely describes the measurement within the context of the experiment . all that is `` allowed to be known '' inside the formal framework is that there ( a ) _ is _ ( b ) _ some _ relevant information . what is relevant is reflected in the choice of questions that are asked by i - observer . interaction between p - observer and the quantum system should be viewed as physical interaction between just any two physical systems . still , because i - observer then reads information from p - observer and because we are nt interested in the posteriority of relations between p - observer and the quantum system , we can treat p - observer as an ancillary system in course of its interaction with @xmath0 . such an ancillary system would have interacted with @xmath0 and then would be subject to a standard measurement described mathematically on its hilbert space via a set of orthogonal yes - no projection operators . so far , for p - observer we have the hilbert space and the standard born rule . the fact that p - observer is treated as ancillary system allows to transfer some of this structure on the quantum system @xmath0 . a new non - trivial assumption has to be made , that the time dynamics that has previously arisen in the context of p - observer alone , also applies to the i - observer and to @xmath0 . in other words , there is only one time in the system . time of i - observer is the one in which one can grasp the meaning of the words `` past '' and `` future '' as used above in relation with the experimental setting and the notion of relevant information : it is in this time that there is a `` before the experiment '' and an `` after the experiment '' . times of physical systems , such as @xmath0 or p - observer , are times in which their dynamics takes place . now , both the physical interaction of p - observer with @xmath0 and the process of asking questions by i - observer to p - observer take place in one and the same time . since ( a ) until i - observer asks the question that he chooses to ask , sets of questions at different times are isomorphic and evolution is unitary , and ( b ) time at which i - observer asks the question only depends on i - observer and considerations of relevance that must not enter into the formalism , then one concludes that the interaction between the quantum system and p - observer must respect the unitary character all until the decoupling of the ancilla . now write , @xmath53 after asking a question corresponding to a projector @xmath54 , probability of the yes answer will be given by @xmath55 because the systems decouple , trace can be decomposed into @xmath56 where all presence of the ancilla is hidden in the operator @xmath57 which acts on the quantum system @xmath0 alone . this operator is positive - semidefinite , and a family of such operators form resolution of identity . they are not , however , mutually orthogonal . such operators form positive operator - valued measures ( povm)@xcite . we have shown how to obtain a description of quantum measurement via povm at the condition of disregarding completely the physical interaction during measurement and the existence of p - observer . if one is only interested in a formal description of how i - observer acquires information about the quantum system @xmath0 , this is done via povm and the born rule following from gleason s theorem . to be mentioned here , gleason s theorem also admits a generalization from von neumann s orthogonal projector measures to povm @xcite . one gets therefore a description of measurement as used in quantum information theory , and one can now continue the development of the theory in the conventional way@xcite . in agreement with the intuition expressed in the key metaphor , all `` hardware '' language is eliminated and the theory can be formulated in the `` software '' language alone . formal deduction of the results concerning the hilbert space , however , was not completely rigorous . rovelli@xcite acknowledges it in his disclaimer , `` i do not claim any mathematical nor philosophical rigor '' . indeed , the fact that yes - no questions form an orthomodular lattice containing subsets that form boolean algebras only commits one to the structure of union of hilbert spaces and not of a single hilbert space . thus , this can happen to be the union of primitive hilbert spaces , which allow for a classical and not a quantum interpretation . generally speaking , the structure will be the one of the hilbert space with superselection rules . one needs then to use axiom 2 to show that the possibility to ask in every situation some new informative question excludes classicality . completion of this program remains an open problem . introduction of space and time `` by hand '' in any algebraic approach to quantum mechanics is certainly quite unsatisfactory . one would wish to see how space and time arise naturally from the formalism this may require a fully rigorous algebraic approach involving von neumann algebras and the gns construction for the hilbert space . however , the author is only aware of one way to introduce time in this framework@xcite . this leaves open the question of link between time and space , and of the possibility to use the two notions together to obtain the evolution equation . the author would like to thank carlo rovelli and christopher fuchs for discussion and suggestions . 10 j.a . wheeler , _ information , physics , quantum : the search for links _ , in anthony j.g . hey , editor , _ feyman and computation : exploring the limits of computers . _ , perseus books , reading , massachusets , 1998 .

information - theoretic derivations of the formalism of quantum theory have recently attracted much attention . we analyze the axioms underlying a few such derivations and propose a conceptual framework in which , by combining several approaches , one can retrieve more of the conventional quantum formalism .

You are an expert at summarizing long articles. Proceed to summarize the following text: intensive multiwavelength monitoring campaigns have shown that variability studies provide an excellent tool to investigate the innermost region of active galactic nuclei ( agns ) . detailed studies of the continuum and emission - line variations have revealed new insights about the size , structure , and dynamics of the broad - line region ( blr ) in these sources ( see peterson 1993 for a review ) . over the last decade , a number of large space - based and ground - based agn monitoring programs have been undertaken . our own group , the international agn watch consortium ( alloin et al . 1994 ) , has undertaken monitoring programs on several seyfert galaxies , including programs on ngc 5548 ( clavel et al . 1991 ; peterson et al . 1991 , 1993 , 1994 ; maoz et al . 1993 ; dietrich et al . 1993 ; korista et al . 1995 ) , ngc 3783 ( reichert et al . 1994 ; stirpe et al . 1994 ; alloin et al . 1995 ) , ngc 4151 ( crenshaw et al . 1996 ; kaspi et al . 1996 ; warwick et al . 1996 ; edelson et al . 1996 ) , fairall 9 ( rodrguez - pascual et al . 1997 ; santos - lle et al . 1997 ) , and ngc 7469 ( wanders et al . 1997 ; collier et al . 1997 ) . in late 1994 , the international agn watch began a multiwavelength monitoring campaign on the broad - line radio galaxy ( blrg ) 3c390.3 , a prominent nearby ( @xmath13 ; osterbrock et al . 1975 ) agn with broad double - peaked emission - line profiles ( sandage 1966 ; lynds 1968 ) . 3c390.3 has a well - known variability history ( e.g. selmes , tritton , & wordsworth 1975 ; barr et al . 1980 ; penston & prez 1984 ; veilleux & zheng 1991 ; zheng 1996 ; wamsteker et al . 1997 ) . zheng ( 1996 ) and wamsteker et al . ( 1997 ) analysed the iue spectra of 3c390.3 which have been taken from 1978 until 1992 . the variable broad ly@xmath5 and iv civ emission are delayed by @xmath1060 days with respect to the uv continuum variations . 3c390.3 is an extended double - lobed fr ii radio source ( leahy & perley 1995 ) with a 60one - sided narrow jet at @xmath14 and it is one of the rare lobe - dominated radio galaxies which show superluminal motions , with @xmath15 ( alef et al . this is the first time that variations in a radio - loud agn have been studied over a broad energy range ( from radio to x - ray energies ) for an extended period ( one year ) the observations obtained for this program cover over eight decades in frequency . the results of the x - ray and ultraviolet monitoring campaigns performed with rosat and the iue satellite will be presented in leighly et al . ( 1997 ) and obrien et al . ( 1998 ) , respectively . in this contribution , we present the optical photometric and spectroscopic observations that were obtained as part of this monitoring program ; observations in other wavelength bands will be presented elsewhere . in 2 , we describe the optical observations and outline intercalibration procedures by which a homogeneous set of the photometric and spectroscopic measurements is achieved . in 3 , we highlight emission - line profile variations based on mean and rms spectra and present the results of some preliminary time - series analysis . we summarize our results in 4 . spectra and broad - band photometric measurements of 3c390.3 were obtained by a large number of observers between 1994 october and 1995 october . table 1 gives a brief overview of the various sources of the data we report here . each group ( column 1 ) was assigned an identification code given in column ( 2 ) which will be used throughout this paper . column ( 3 ) gives the aperture of the telescope used . columns ( 4 ) ( 7 ) list the focal - plane apertures used in various observations ; generally , fixed instrument apertures were used , although two groups ( f , n ) adjusted their aperture to compensate for changes in seeing . in most cases the size of the apertures corresponds to three times the average seeing at the observing site . those large apertures cover the agn as well as the entire galaxy . in column ( 8) , the spectrograph slit widths ( in the dispersion direction ) and extraction widths ( cross - dispersion dimension ) , respectively , of the spectra are listed . complete logs of the photometric and spectroscopic observations ( table 2 and 3 ) are available as electronic files on the www at the url given in the following . the photometric observations were made through different combinations of broad - band filters . generally , the brightness of 3c390.3 was scaled with respect to standard stars in the photometric sequence ( stars a , b , and d ) defined by penston , penston , & sandage ( 1971 ) , plus a hst guide star located at 208 from 3c390.3 at @xmath16 ( hst gs 4591:731 ) . in some cases , star a could not be used because of saturation . the colors of the comparison stars ( b , d , and hst gs 4591:731 ) differ by up to 0.1 mag . while star a is @xmath172.5 mag . brighter than the mean values of these three stars . the mean colors of 3c390.3 are similiar to the mean colors of the comparison stars within 0.2 mag . since the stars and the agn are within the same field of view of the ccd images , effects of different spectral energy distributions and airmasses on the internal calibration can be neglected . in the following , the procedure we use to derive the @xmath2-band magnitudes will be described in some detail . the light curves of the other broad - band measurements were obtained in a similar way . first , a subsample was selected which covered the entire monitoring campaign with a good temporal sampling rate . for the @xmath2-band measurements , the observations recorded with the 0.7-m telescope of the landessternwarte heidelberg ( l ) provide an appropriate data set . the used aperture was a 15 rectangle to ensure that the light loss can be neglected even in the case of bad seeing . the frames were bias and flat - field corrected in the standard way . in the next step , the brightness of 3c390.3 was measured with respect to the comparison stars in the same field , yielding the difference in apparent magnitude between 3c390.3 and the comparison stars . in the @xmath2-band , stars b and d from penston et al . ( 1971 ) and hst gs 4591:731 mentioned above were used for calibration . the @xmath2-band magnitudes that were derived for these stars are presented in table 4 together with the @xmath0 and @xmath1 magnitudes from penston et al.(1971 ) . finally , the @xmath2-band light curves of the other subsamples were shifted to the @xmath2-band light curve derived from the heidelberg sample by applying a constant additive magnitude offset to all of the measurements in a given subset . the additive factor was derived by comparison of @xmath2-band magnitudes which were observed within @xmath183 days . the additive factors for the individual subsamples are presented in table 5 . the additive scaling factors given in table 5 are quite large in some cases . generally , the brightness variations of 3c390.3 were provided in magnitudes using the standard stars in the field for calibration . but a few groups ( m , r , u ) provided the brightness of 3c390.3 as differential magnitudes relative to the calibration stars in the field . the @xmath0-band magnitudes are scaled with respect to the measurements taken at wise observatory ( o ) . the offset for the steward observatory sample ( f ) was derived by comparison of epochs which were simultaneous to within @xmath183 days . this restriction could not be used for the special astrophysical observatory ( j ) ; in this case , the offset is based on epochs separated by no more than 6 days . the resulting additive scaling factors are given in table 5 . the sample recorded at behlen observatory ( q ) was used as the standard in the @xmath1-band because the temporal sampling is good and the measurements were made through a large aperture ( 13 . @xmath197 ) so that uncertainties due to seeing variations are small . the @xmath1-band measurements are made relative to star b , which is close in brightness to the nucleus of 3c 390.3 . in order to scale the @xmath1-magnitudes to a common flux level , we compared measurements at epochs separated by no more than three days . in table 5 , the differential magnitudes are given which were applied to the individual light curves to produce a common light curve . for the calibration of the @xmath3-band magnitudes we used observations taken at calar alto observatory . the @xmath3-band frames were recorded in december 1994 . the calibration was based on the globular cluster ngc2419 using the stars given by christian et al.(1985 ) . the resulting brightnesses of star a , b , d , and hst gs 4591:731 are given in table 4 . the additive factors for the individual subsamples are presented in table 5 . the measured brightnesses of the stars used for calibration are in good agreement with the values provided by penston et al . however , star a deviates from this trend in @xmath2 and @xmath3 in comparison to stars b and d. it might be that this is due to the brightness of star a causing non - linearity effects and an underestimate of the brightness of star a. after combining the measurements of the individual groups to common broad - band light curves , the apparent magnitudes were transformed into flux . the conversion has been performed using the following equations ( allen 1973 ; wamsteker 1981 ) : @xmath20 the resulting time - binned light curves from the @xmath0 , @xmath1 , @xmath2 , and @xmath3 broad - band measurements in units of @xmath21ergs s@xmath12@xmath22@xmath12 are displayed in figure 1 . the flux calibration of agn spectra can be accomplished in several ways . in variability studies , it is common practice to normalize the flux scale to the fluxes of the narrow emission lines , which are assumed to be constant over time scales of at least several decades ( peterson 1993 ) . this assumption is justified by the large spatial extent and low gas density of the narrow - line region ( nlr ) , since light travel - time effects and the long recombination time scale ( @xmath23years for @xmath24@xmath25 ) damp out short time - scale variability . however , 3c390.3 is apparently a special case in this regard , as narrow - line variability has been reported in this object ( clavel & wamsteker 1987 ) . on the basis of spectra obtained between 1974 and 1990 , zheng et al . ( 1995 ) present evidence that the [ oiii]@xmath26 , 5007 fluxes follow the variations in the continuum , although on a longer time scale than the broad emission lines . during the period of decreasing and increasing [ oiii ] flux there might also be periods of several months or one year of nearly constant [ oiii ] flux . however , we need to examine the data closely to test for variability of the [ oiii ] lines before we use them for flux calibration . two data sets , the ohio state sample ( a ) and the lick observatory sample ( b ) , were selected to measure the absolute flux in the [ oiii]@xmath27 line . the spectra were photometrically calibrated by comparison with the broad - band photometric measurements . the spectra were convolved with the spectral response curve of the johnson @xmath1 and @xmath2 filters ( schild 1983 ) , and the flux of the convolved spectra was measured for the wavelength range of the filter curves and compared with photometric data points . photometric and spectroscopic data obtained no more than 3 days apart were used in this comparison . the spectra were then scaled by multiplicative factors to achieve the same total flux ratio as the intercalibrated photometric measurements in @xmath1 and @xmath2 . for the lick data ( b ) , the @xmath2-band measurements were used to scale the entire spectrum , and the [ oiii]@xmath27 flux was then measured . spectra from the ohio state sample ( a ) do not cover the entire wavelength range of the @xmath1-band filter , and therefore the missing contribution to the @xmath1-band measurement was estimated from the lick spectra , which cover the entire wavelength range of the @xmath1 bandpass . the ratio of @xmath1-band flux measured from the ohio state spectra to that measured from the lick spectra was found to be @xmath28 , i.e. , a constant contribution . the fluxes from the ohio state spectra were thus corrected by this constant factor . the [ oiii]@xmath27 flux was then measured from the photometrically scaled ohio state ( a ) and lick observatory ( b ) spectra by integrating the spectrum over the range 52585320 . therefore , a linear pseudo - continuum was fitted beneath the [ oiii]@xmath27emission line . in fig . 2 , we show the [ oiii]@xmath27 flux measured from the ohio state and lick subsets normalized to a mean value of unity and displayed as a function of time . no time - dependent trend is detected , and the rms variation about the mean is @xmath29% . we thus conclude that it is safe to assume that the [ oiii]@xmath27 flux is constant within a few percent over the duration of this monitoring program , and that the [ oiii]@xmath27 flux can be used to calibrate all of the spectra . the [ oiii]@xmath27 flux is taken to be @xmath30\,\lambda5007 ) = 1.44 \times 10^{-13}$]ergs s@xmath12@xmath22 , the mean value of the data points plotted in fig . 2 . it is also important to take aperture effects into account ( peterson & collins 1983 ) . the seeing - dependent uncertainties which are introduced by the aperture geometry can be minimized by using large apertures . it has been shown that apertures of 5 @xmath31 7 . @xmath195 can reduce seeing - dependent photometric errors to no more than a few percent in the case of nearby agns ( peterson et al . 1995 ) . in the case of 3c390.3 , the blr and the nlr can be taken to be point sources ( cf . baum et al . 1988 ) , which means that no aperture correction needs to be made for the agn continuum / narrow - line flux ratios or the broad - line / narrow - line ratios , since seeing - dependent light losses at the slit will be the same for each of these components . however , the amount of host - galaxy starlight that is recorded is still aperture dependent , and systematic corrections need to be employed . in order to estimate how sensitive the measurements are to seeing - dependent light loss from the host galaxy , we carried out simulated aperture photometry on the @xmath2-band frame obtained at calar alto with the 2.2-m telescope under good seeing conditions ( @xmath32 ) . the host galaxy has been separated from the point - like agn component by fitting a de vaucouleurs @xmath33 profile to the observed surface - brightness distribution . this image was convolved with gaussians of various width to simulate various seeing conditions up to 4 . @xmath190 . the flux of the host galaxy and of the point - like agn were measured for a fixed aperture of 10 . @xmath195x10 . the flux ratio of the point - line agn to the host galaxy for the @xmath2-band is @xmath34 . this is similar to the result of smith & heckman ( 1989 ) , who found @xmath35 for @xmath1-band measurements . by using different aperture geometries in these simulations , we find that the ratio of agn light to starlight from the host galaxy changes over the full range of observed seeing values by less than 1% for the larger apertures ( i.e. , slit width greater than 4 ) . for intermediate slit widths ( 2 . @xmath195 3 . @xmath196 ) , seeing variations introduce uncertainties of @xmath36% . for the smallest slit widths ( @xmath37 ) , seeing effects can alter the nucleus to starlight ratio by as much as @xmath38% , with the largest uncertainties occurring for seeing worse than @xmath39 . since the data that constitute the various samples were taken with different instruments in different configurations , the spectra have to be intercalibrated to a common flux level . as we have shown above , the [ oiii]@xmath27 line flux was constant to better than 3% during this campaign , so we can safely use the narrow emission lines as flux standards . in order to avoid any wavelength - dependent calibration errors , each spectrum was scaled in flux locally over a limited wavelength range prior to measurement . the h@xmath6 spectral region was scaled with respect to the [ oiii]@xmath26 , 5007 line fluxes , while the h@xmath5 region was scaled with respect to the fluxes of the [ oi]@xmath40 and [ nii]@xmath41 , 6584 emission lines . the spectra were intercalibrated using the method described by van groningen & wanders ( 1992 ) . this procedure corrects the data for different flux scales , small wavelength shifts , and different spectral resolutions by minimizing the narrow - line residuals in difference spectra formed by subtracting a `` reference spectrum '' from each of the observed spectra . the rescaled spectra are used to derive integrated emission - line fluxes as well as the optical continuum flux . the continuum fluxes are then adjusted for different amounts of host - galaxy contamination ( see peterson et al . 1995 for a detailed discussion ) through the relationship @xmath42}\,\lambda5007 ) } \right]_{\rm obs } - g,\ ] ] where @xmath43 is the adopted absolute [ oiii]@xmath27 flux , the quantity in brackets is the observed continuum to [ oiii]@xmath27 flux ratio measured from the spectrum , and @xmath44 is an aperture - dependent correction for the host - galaxy flux . the ohio state sample ( a ) , which uses a relatively large aperture ( 5@xmath31 7 . @xmath195 ) , was adopted as a standard ( i.e. , @xmath45 by definition ) , and other data sets were merged progressively by comparing measurements based on observations made no more than @xmath183 days apart . this means that any real variability that occurs on time scales this short tends to be somewhat suppressed by the process that allows us to combine the different data sets . the additive scaling factor @xmath44 for the various samples are given in table 6 . the average interval between measurements is about @xmath46days for @xmath47 @xmath48 and @xmath3 while the @xmath0-band variations have been measured with an average sampling interval of @xmath49days ( table 7 ) . the time - binned broad - band continuum light curves are shown in fig . 1 . the variations can be characterized as a nearly monotonic increase of the flux with smaller - scale variations superposed on this general trend . from jd2449760 to jd2449800 , the flux rose in the v , r , i bands faster than during the previous interval . after this period , the flux level stays nearly constant for nearly 3 months and then a second strong increasing episode follows . the @xmath1-band light curve appears to have more complicated structure , and we note in particular apparently rapid variations during the interval jd2449800 to jd2449920 . figure 3 shows two spectra of 3c390.3 that represent the low ( jd2449636 ) and high ( jd2450007 ) flux states observed during this campaign . the most obvious variation is the strong increase of the small blue bump , the very broad feature shortward of 4000 , which is usually ascribed to balmer continuum emission and a blend of several thousand feii emission lines ( wills , netzer , & wills 1985 ) . the broad balmer lines show also evidence for increasing flux . underneath the low - state spectrum in fig . 3 we show the integration ranges for the various features that have been measured in these spectra . the optical emission - line fluxes were integrated over a common range of @xmath50 ^ -1 km s@xmath12 for the strong balmer emission lines ( h@xmath5 , h@xmath6 , and h@xmath7 ) and i5876 hei@xmath8 . the ii4686 heii@xmath9 line flux was integrated over a range corresponding @xmath51 ^ -1 km s@xmath12 to @xmath52 ^ -1 km s@xmath12 to reduce contamination by h@xmath6(cf.table 8) . a local linear continuum fit was interpolated under each emission line . in the case of the h@xmath6 region , the continuum was defined by the flux measured in two narrow ( 10 width ) windows at 4400 and 5475 in the observed frame . although the region of the long wavelength window might be contaminated by several weak emission lines and the mgb absorption feature , this is still the most line - free region near h@xmath6 and is thus an appropriate place to estimate the optical continuum flux . the continuum window at 4400 is displaced from the h@xmath7 line center by @xmath53 ^ -1 km s@xmath12 , and thus possible contamination by broad emission line flux can be safely ignored ( see fig . 3 ) . contamination of h@xmath6 by [ oiii ] emission has been corrected by subtracting the constant [ oiii]@xmath27 flux given earlier plus the [ oiii]@xmath54 flux which we account for by assuming a [ oiii]@xmath27/[oiii]@xmath54 flux ratio of 3 . in the case of the h@xmath5 region , two narrow ( 10 in width ) windows at 5960 and 7495 were used to define the continuum underlying the lines . no attempt has been made to correct any of the measured emission - line fluxes for their respective narrow - line contributions . in each spectrum , the optical continuum flux measured is the average value in the range 54605470 . the extracted range of the emission lines and of the optical continuum is given in table 8 . the time - binned light curves for the strong emission lines and the optical continuum flux @xmath4(5177 ) measured from the calibrated spectra are plotted in fig . the corresponding flux measurements are available on the www at the url given above ( table 9 ) . a final check of the uncertainty estimates was performed by examining the ratios of all pairs of photometric and spectroscopic observations which were separated by 2 days or less . the error estimates were also calculated using maximum intervals of 3 days and 4 days , and the resulting errors are identical to within 3% . generally , there are at least a few hundred independent pairs of measurements within 2 days of one another for the photometric measurements . only the @xmath0-band curve yields fewer independent pairs . for the spectroscopic data the number of independent pairs is of the order of several dozen with the exception of h@xmath7 . the dispersion about the mean ( unity ) , divided by @xmath55 , provides an estimate of the typical uncertainty in a single measurement ( @xmath56 ) . the observational uncertainties ( @xmath57 ) assigned to the spectral flux measurements were estimated from the error spectra which were calculated within the intercalibration routine , as well as from the signal - to - noise ratio within the spectral range near the individual emission lines . for the continuum the mean fractional error ( @xmath57 ) in a given measurement is 0.041 . the average fractional uncertainty from the quoted estimate ( @xmath56 ) for the same measurements ( table 10 ) is 0.040 which implies that the error estimates are probably quite good . generally , the estimated errors ( @xmath58 ) are of the same order as the observational uncertainties ( @xmath59 ) derived directly from the measurements . a comparison of the flux of the broad band measurements and the broad emission lines is given in table 11 . the mean spectroscopic continuum flux is lower than the mean broad band @xmath1- and @xmath2-flux . this can be explained by the fact that the broad band flux measurements contain in addition to the continuum flux emission line contributions . the variability parameter @xmath60 and @xmath61 have been calculated for the broad - band flux variations ( cf . clavel et al . 1991 ; rodrguez - pascual et al . the quantity @xmath61 is simply the ratio of the maximum to the minimum flux . the quantity @xmath60 is an estimation of the fluctuations of the intrinsic variations relative to the mean flux . therefore , the rms of the light curves has been corrected with respect to the uncertainties introduced by the observations . @xmath61 and @xmath60 of the broad - band variations are given in table 11 . we calculated the mean and root - mean - square ( rms ) spectra from the flux - scaled spectra , and the h@xmath5 and h@xmath6 regions are shown in fig . the velocity scale is set by adopting a redshift @xmath62 given by the narrow components of h@xmath5 and h@xmath6 and the [ oiii]@xmath63 emission lines , i.e. the restframe of the nlr . the rms spectrum is useful for isolating the variable parts of the line profile . the full - width at zero intensity ( fwzi ) of the mean h@xmath5 and h@xmath6profiles is ( 25000@xmath182000)^-1 km s@xmath12 , while the fwzi of the rms profiles is only ( 14000@xmath18500)^-1 km s@xmath12 ( cf.fig.5 ) . thus , the variations in the broad emission line profiles are strong at lower radial velocities while the flux originating in high radial - velocity gas varies little , if at all . this behavior is similar to what has been seen in mrk 590 ( ferland , korista , & peterson 1990 ; peterson et al . 1993 ) . the line profiles of h@xmath5 and h@xmath6 show clear asymmetric structure . at least three substructures can be identified in both line profiles . a blue hump in the mean spectra is located near @xmath64 ^ -1 km s@xmath12 with respect to the line peak , and a red hump is seen near @xmath65 ^ -1 km s@xmath12 . the red and blue humps can be seen clearly in the rms spectra , and in addition a broad central component appears . if these three components are modeled as gaussians , the fwhm of the components in the wings is of order @xmath66 ^ -1 km s@xmath12 , while for the central component the fwhm is @xmath67 ^ -1 km s@xmath12 . the components in the profile wings are nearly symmetrically located with respect to the line center . in the rms - spectrum of the h@xmath6 emission line a strong broad feature at @xmath1010000 ^ -1 km s@xmath12 is clearly visible . the fwhm of the structure can be estimated to @xmath103500 ^ -1 km s@xmath12 . the residuals of the narrow [ oiii ] lines are located at @xmath106000 ^ -1 km s@xmath12 and @xmath108900 ^ -1 km s@xmath12 . since this feature is also visible in the rms spectrum of h@xmath5 and of h@xmath7 it can not be caused entirely by inaccurate scaling of the nlr contribution of the narrow [ oiii ] lines . furthermore , in the difference spectrum of the low and high state of 3c390.3 during this monitoring campaign ( cf . fig . 3 ) a strong broad feature is clearly visible at the red side of h@xmath5 ( @xmath107150 ) and a weaker structure at the red side of h@xmath6 ( @xmath10 5300 ) . the analysis of the simultaneous ultraviolet campaign reveals the existence of a feature at @xmath108500 ^ -1 km s@xmath12 in the outer red wing of the iv civ@xmath681548 emission line ( obrien et al.1998 ) . in order to quantify any possible time delay between the various light curves shown in figs . 1 and 4 , we perform a simple cross - correlation analysis . three methods that are commonly used in agn variability studies have been used to compute cross - correlation functions ( ccfs ) the interpolated cross - correlation function ( iccf ) of gaskell & sparke ( 1986 ) and gaskell & peterson ( 1987 ) , the discrete correlation function ( dcf ) of edelson & krolik ( 1988 ) , and the @xmath69-transformed discrete correlation function method ( zdcf ) of alexander ( 1997 ) . the iccf and dcf algorithms and the limitations of the methods have been discussed in detail by robinson & prez ( 1990 ) and by white & peterson ( 1994 ) , and the specific implementation of the iccf and dcf used here are as described by white & peterson ( 1994 ) . the emission lines are expected to change in response to variations in the far - uv continuum , primarily to the unobservable wavelengths just shortward of 912 . we must therefore assume that the observable continuum can approximate the behavior of the ionizing continuum . it is generally assumed in this type of analysis that the shortest observed uv wavelength provides the best observable approximation to the `` driving '' ( ionizing ) continuum . for the first time , however , we have a well - sampled simultaneous soft x - ray light curve ( leighly et al . 1997 ) , so we can compare the x - ray and uv continua ( obrien et al . 1998 ) directly . in this analysis , we used the x - ray light curve as the driving continuum ( fig.6 ) since the observational uncertainties are smaller than for the uv light curve . as will be discussed elsewhere ( obrien et al . 1998 ) , the relative amplitudes of the ultraviolet variations at a wavelength of 1370 are nearly identical to those in the _ rosat _ hri light curve , so the results presented here do not depend critically on our choice of the x - ray light curve in preference to the uv light curve . since the sampling of the @xmath0-band light curve is very poor , we have excluded the @xmath0-band data from the cross - correlation analysis . operationally , the temporal coverage of the optical light curves was restricted to the period covered by the x - ray measurements . all of the light curves were binned into time intervals of 0.5 days to avoid unnecessary structure in the iccf . for each light curve , we also computed the sampling - window autocorrelation function ( acf@xmath70 ) , which is a measure of how much of the width of the acf is introduced by the interpolation process rather than by real correlation of the continuum values at different times . the acf@xmath70 is computed by repeatedly sampling white - noise light curves in exactly the same way as the real observations and then computing the autocorrelation function ( acf ) . the average of many such autocorrelations , the acf@xmath70 , has a peak at zero lag whose width depends on how much interpolation the iccf has to do on short time scales . the width ( fwhm ) of the acf@xmath70 is @xmath71 days for the emission lines and broad - band flux measurements for the x - ray restricted time period . these values are negligible compared to the widths of any of the emission - line acfs or ccfs found here ( see figs . 79 ) , and thus interpolation of the light curves is justified . the general sampling characteristics of each of the restricted and rebinned light curves are given in table 12 . the name of the feature is given in column ( 1 ) , and column ( 2 ) gives the total number of points @xmath72 in the light curve that are used in computing the cross - correlation functions . the width ( fwhm ) of the acf is given in column ( 3 ) , and the width of iccf computed by cross - correlation with the x - ray continuum is given in column ( 4 ) . column ( 5 ) gives the width ( fwhm ) of the corresponding acf@xmath70 . uncertainties in the iccf results for the cross - correlation maxima and centroids , @xmath73 and @xmath74 , respectively , were computed through monte carlo techniques as follows : for both time series , each flux value in the light curves was modified with gaussian deviates based on the quoted uncertainty for that point . each light curve of @xmath72 points was then randomly resampled @xmath72 times in a `` bootstrap '' fashion , specifically allowing points to appear more than one time ( the effect of which is only to remove at random certain points from the light curve ) . the iccf was computed , and the values of @xmath75 and @xmath76 were recorded if they were statistically significant at a confidence level higher than 95% and not clear outliers ( e.g. , lags larger than 100 days ) . by repeating this process 5001000 times , distributions of @xmath75 and @xmath76 were built up . the means of these distributions were always very close to the values of @xmath75 and @xmath76 obtained from the original series , and the standard deviations of these distributions are taken to be the uncertainty associated with a single realization , i.e. , the quoted uncertainties @xmath77 and @xmath74 . the acf , acf@xmath70 , iccf , and zdcf are shown for the broad - band variations in @xmath1 , @xmath2 , and @xmath3 ( fig . 7 ) , the broad balmer emission lines h@xmath5 , h@xmath6 , and h@xmath7 ( fig . 8) , the helium lines i5876 hei@xmath8 and ii4686 heii@xmath9 , and the optical continuum @xmath4(5177 ) ( fig the acfs of the light curves are broad since the shape is dominated by the nearly monotonic increase in the light curves . the fwhms of the acfs of the well - sampled light curves are of the order of 50100days ( table 12 ) . within the uncertainties , the dcfs and zdcfs are identical ; therefore , to avoid confusion only the iccfs and the corresponding zdcfs are displayed in figs . the time delay derived from the centroid of a cross - correlation function provides a more robust estimate of the lag than does the peak , as evidenced by the consistently smaller widths of the monte carlo distributions for the centroid compared to the peak . also , in the case of the emission lines , the centroid is readily identified with a physically meaningful quantity , the luminosity - weighted radius of the line - emitting region ( koratkar & gaskell 1991 ) . the delays expressed by @xmath78 of the three methods used here are nearly identical within the uncertainties , although the dcf method tends to yield smaller delays than the iccf and zdcf method . the results of the cross - correlation analysis are given in table 13 . column ( 1 ) indicates the `` responding '' light curve ( i.e. , the light curve that is assumed to be responding to the driving light curve ) , and column ( 2 ) gives the peak value of the correlation coefficient @xmath79 for the iccf . the position of the peak of the cross - correlation functions @xmath80 was measured by fitting a gaussian curve to the upper 85% of the iccf , zdcf , and dcf ; these values are given in columns ( 3 ) , ( 4 ) , and ( 5 ) , respectively , and column ( 6 ) gives the error estimate for the position of the cross - correlation peak @xmath77 . the centroids @xmath78 of the iccf , zdcf , and dcf , in each case computed using the points in the cross - correlation function with values greater than @xmath81 , are given in columns ( 7 ) , ( 8) , and ( 9 ) , respectively , and the uncertainty in the iccf centroid @xmath82 is given in column ( 10 ) . the broad - band ( @xmath1 , @xmath2 , and @xmath3 ) and the optical continuum flux variations appear to be delayed by a few days , relative to the x - ray or ultraviolet continuum variations however , the measured lags are in no case different from zero at any reasonable level of statistical significance if the monte - carlo based error estimates are reliable . somewhat smaller , but marginally statistically significant , wavelength - dependent continuum lags have been reported in the case of ngc 7469 ( wanders et al . 1997 ; collier et al . 1997 ) , although in the case of ngc 7469 the mean spacing between observations of the driving continuum is much smaller than for the observations reported here . the emission - line time delays are similar for all of the optical emission lines , about 20 days , although in each case the uncertainties are somewhat larger than we have obtained in similar experiments on account of the vagaries of the continuum behavior and sampling . within the uncertainties , the measured time delays for h@xmath5 , h@xmath6 , h@xmath7 , and i5876 hei@xmath8 are indistinguishable . however , the uncertainties are sufficiently large that ionization stratification , as detected in other well - studied sources , also can not be ruled out . only ii4686 heii@xmath9 appears to respond more rapidly than the other lines , but the relatively low value of @xmath79 and relatively narrow width of the cross - correlation function ( about 30 days as compared to @xmath83 days for the other lines ) cast some doubt on the significance of this result . in contrast to the agn studied before , the delay of the ly@xmath5 and iv civ line variations of the current 3c390.3 campaign is significantly larger than the delay of the optical lines . the analysis of the ultraviolet spectra yields a delay of 35 70 days for ly@xmath5 and iv civ ( obrien et al . 1998 ) . in order to study the response of individual parts of the line profile to the x - ray continuum variations and thus search for evidence of an organized radial - velocity field in the blr , we have divided the profiles of the strongest and least contaminated lines , h@xmath5 and h@xmath6 , into three parts : the blue wing ( @xmath51 to @xmath84kms@xmath12 ) , the core ( @xmath85 to @xmath86kms@xmath12 ) , and the red wing ( @xmath87 to @xmath88kms@xmath12 ) . the light curves of the different profile sections ( cf.figs . 10,11 ) were rebinned onto 0.5-day intervals ( available in electronic form on the www at the url given above as table 14 ) and again cross - correlations were performed restricting the data to the x - ray monitoring period . since we are concerned at this point with _ differential _ lags between different sections of the line profiles , we cross - correlate the light curves for the different line sections against each other rather than against the driving continuum ; we arbitrarily chose the line cores as the first ( driving ) series , and computed the lags of the red and blue wings of the lines relative to the core . these results are shown in table 15 and in figs . 12 and 13 . neither h@xmath5 nor h@xmath6 shows any evidence for any differences in the response of the wings relative to the core . in the case of h@xmath5 , the uncertainties are especially large on account of the low amplitude of variability , but the h@xmath6 results seem to exclude the possibility that the blr velocity field is characterized by primarily radial motions , either infall or outflow . the results of a year - long ( 1994 october to 1995 october ) optical monitoring campaign on the blrg 3c390.3 are presented in this paper . the principal findings are as follows : 1 . the broad - band ( @xmath0 , @xmath1 , @xmath2 , and @xmath3 ) fluxes , the optical continuum measured from spectrophotometry @xmath4(5177 ) , and the integrated emission - line fluxes of h@xmath5 , h@xmath6 , h@xmath7 , i5876 hei@xmath8 , and ii4686 heii@xmath9 showed significant variations of order 50% in amplitude . 2 . the parameter @xmath60 , which is essentially the rms variation about the mean , increased with decreasing wavelength for the broad - band measurements as well as for the balmer emission lines . 3 . the variations of the broad - band and emission - line fluxes are delayed with respect to the x - ray variations . cross - correlation functions were calculated with three different methods ( iccf , dcf , and zdcf ) . the time delays of the optical continuum variations expressed by the centroid of the cross - correlation functions are typically about 5 days , but with uncertainties of @xmath105 days . therefore , zero - time delay between the high - energy and low - energy continuum variability can not be ruled out . the delays of the balmer lines h@xmath5 , h@xmath6 , and h@xmath7 and of i5876 hei@xmath8are typically around 20 days @xmath18 8 days . there is some evidence that ii4686 heii@xmath9 responds somewhat more rapidly with a time delay of @xmath1010 days , but again the uncertainties are quite large ( @xmath108 days ) . 4 . the simple cross - correlation analysis of the line core with the line profile wings of the h@xmath5 and h@xmath6 emission might indicate that the wings vary in the same way with respect to the line core . but there might be a weak indication that the variations of the blue wing are delayed by @xmath10 4 days with respect to the red wing . the mean and rms h@xmath5 and h@xmath6 line profiles reveal the existence of at least three substructures a central component , plus strong blue and red components at about @xmath11 ^ -1 km s@xmath12 relative to line center . the rms spectra show that the broad - line variations are much stronger at line center than in the outer wings . alef , w. , wu , s.y . , preuss , e. , kellermann , k.i . , & qui y.h . 1996 , a&a , 308 , 376 alexander , t. 1997 , in astronomical time series , ed d.maoz , a.sternberg , & e.leibowitz ( dordrecht : kluwer ) , in press allen , c.w . 1973 , astrophysical quantities , the athlone press alloin , d. , clavel , j. , peterson , b.m . , reichert , g.a . , & stirpe , g.m . , 1994 , in frontiers of space and ground - based astronomy , ed . w.wamsteker , m.s . longair , & y. kondo ( dordrecht : kluwer ) , p. 423 alloin , d. , et al . 1995 , a&a , 293 , 293 barr , p. , et al . 1980 , mnras , 193 , 549 baum , s.a . , heckman , t. , bridle , a. , van breugel , w. , & miley , g. 1988 , apjs , 68 , 643 christian , c.a . , et al . 1985 , pasp , 97 , 363 clavel , j. , et al . 1991 , apj , 366 , 64 clavel , j. , & wamsteker , w. 1987 , apj , 320 , l9 collier , s. , et al . 1997 in prep . crenshaw , d.m . , et al . 1996 , apj , 470 , 322 dietrich , m. , et al . 1993 , apj , 408 , 416 edelson , r.a . , & krolik , j.h . 1988 , apj , 333 , 646 edelson , r.a . , 1996 , apj , 470 , 364 ferland , g.j , korista , k.t . , & peterson , b.m . 1990 , apj , 363 , l21 gaskell , c.m . , & sparke , l.s . 1986 , apj , 333 , 646 gaskell , c.m . , & peterson , b.m . 1987 , apjs , 65 , 1 kaspi , s. et al . 1996 , apj , 470 , 336 koratkar , a.p . , & gaskell , c.m . 1991 , apjs , 75 , 719 korista , k.t . , et al . 1995 , apjs , 75 , 719 leahy , j.p . , & perley , r.a . 1995 , mnras , 277 , 1097 leighly , k.m . , 1997 , apj , 483 , 767 lynds , c.r . 1968 , aj , 73 , 888 maoz , d. , & netzer , h. 1989 , mnras , 236 , 21 maoz , d. , et al . 1993 , apj , 404 , 576 obrien , p.t . , et al . 1998 , apjs , submitted osterbrock , d.e . , koski , a.t . , & phillips , m.m . 1975 , apj , 197 , l41 penston , m.j . , penston , m.v . , & sandage , a. 1971 , pasp , 83 , 783 penston , m.v . , & prez , e. 1984 , mnras , 211 , 33 peterson , b.m . , & collins , g.w . , ii . 1983 , apj , 270 , 71 peterson , b.m . , et al . 1991 , apj , 368 , 119 peterson , b.m . , et al . 1993 , apj , 402 , 469 peterson , b.m . 1993 , pasp , 105 , 247 peterson , b.m . , et al . 1994 , apj , 425 , 622 peterson , b.m . , pogge , r.w . , wanders , i. , smith , s.m . , & romanishin , w. 1995 , pasp , 107 , 579 reichert , g.a . , et al . 1994 , apj , 425 , 582 robinson , a. , & prez , e. 1990 , mnras , 244 , 138 rodrguez - pascual , p.m. , et al . 1997 , apjs , 110 , 9 sandage , a.r . 1966 , apj , 145 , 1 santos - lle , m. , et al . 1997 , apjs , in press schild , r.e . 1983 , pasp , 95 , 1021 selmes , r.a . , tritton , k.p . , & wordsworth , r.w . 1975 , mnras , 170 , 15 smith , e.p . , & heckman , t.m . 1989 , apj , 341 , 658 stirpe , g.m . , 1994 , apj , 425 , 609 van groningen , e. , & wanders , i. 1992 , pasp , 104 , 700 veilleux , s. , & zheng , w. 1991 , apj , 377 , 89 wanders i. , et al . 1997 , apj , submitted wamsteker , w. 1981 , a&a , 97 , 329 wamsteker , w. , ting - gui , w. , schartel , n. , & vio , r. 1997 , mnras , 288 225 warwick , r. , et al . 1996 , apj , 470 , 349 wills , b.j . , netzer , h. , & wills , d. 1985 , apj , 288 , 94 white , r.j . , & peterson , b.m . 1994 , pasp , 106 , 879 zheng , w. , prez , e. , grandi , s.a . , & penston , m.v . 1995 , aj , 109 , 2355 zheng , w. 1996 , aj , 111 , 1498 lccccccc perkins reflector , lowell obs . & a&1.8&&&&&5x7.5shane reflector , lick obs.&b&3.0&&&&&4x10,2x5.6mt . hopkins observatory & c & 1.6&&&&&3x3,2x3,1x3shajn reflector , crimean obs . & d & 2.6&&&&&3x11beijing observatory & e & 2.2&&&& & 4x7.2 , 4x8.4 , & & & & & & & 4x12 , 4x10.3 & & & & & & & 4x3.6 , 2.5x9.6 & & & & & & & 3.5x9.6steward observatory & f&2.3&2.5 - 3.3 & 2.5 - 3.3 & 2.5 - 3.3&&2.5x12calar alto observatory & g1 & 1.2&&&25.0&25.0&calar alto observatory & g2 & 2.2&14.0&14.0&14.0&14.0&2x14.1,2x10mmt observatory & h & 4.5&&&&&2x10,1x7.2,2x6isaac newton telescope & i & 2.5&&&&&1.62x6.5special astrophysical obs . & j & 1.0&4.0,4.4&5.3,4.8&5.4,8.0&&special astrophysical obs . & j & 6.0&&&&&3x3.6,3.6x3.6mcdonald observatory & k & 2.7&&&&&2x7.2landessternwarte heidelberg&l & 0.7&&&15.0&15.0&james gregory tel . , st . andrews & m & 0.9&&47.0&47.0&47.0&roboscope , indiana univ . & n & 0.4& & var.&&&wise observatory&o & 1.0&10.0&10.0&12.0&12.0&vainu bappu observatory & p1&2.3&&7.85&&&vainu bappu observatory & p2&1.0&&6.06&&&behlen observatory&q & 0.8&&13.7&&&center for basement astrophys . & r & 0.7& & 6.0 & 6.0 & 6.0 & hoher list obs . , bonn & s1 & 0.6&&3.0 & 3.0&&hoher list obs . , bonn & s2 & 0.4&&5.0 & 5.0&&sara telescope , kitt peak & t & 0.9&&&&20.0 & shanghai observatory & u & 1.6&&6 - 7.5&6 - 7.5&6 - 7.5& lcccc n & & @xmath89 & & f & @xmath90&@xmath91 & @xmath92&g1 & & & @xmath93&@xmath94g2 & & & @xmath95&0.305j & @xmath96 & @xmath97&@xmath98 & m & & @xmath99&@xmath100&@xmath101o & & @xmath102&@xmath103 & @xmath104p1& & @xmath105 & & p2& & @xmath106 & & r & & @xmath107&@xmath108&@xmath109s1& & @xmath110&@xmath111&s2& & @xmath112&@xmath113&t & & & & @xmath114u & & @xmath115&@xmath116&@xmath117 lcclcc @xmath0 & 21&15.6@xmath1817.8&h@xmath7 & 55 & 8.2@xmath1814.0 @xmath1 & 244 & 1.4@xmath18 2.5&heii@xmath9&100 & 4.8@xmath18 9.1 @xmath2 & 206 & 1.8@xmath18 2.7&h@xmath6 & 104 & 5.3@xmath1811.2 @xmath3 & 149 & 2.4@xmath18 3.4&hei@xmath8 & 60 & 9.3@xmath1817.0 @xmath4(5177 ) & 97 & 5.7@xmath1812.8&h@xmath5 & 84 & 6.6@xmath1815.5 lcc @xmath0 & 0.036 & 0.027 @xmath1 & 0.048 & 0.043 @xmath2 & 0.020 & 0.015 @xmath3 & 0.019 & 0.018 h@xmath7 & 0.062 & 0.055 heii@xmath9 & 0.145 & 0.137 h@xmath6 & 0.025 & 0.031 @xmath4(5177 ) & 0.040 & 0.041 hei@xmath8 & 0.106 & 0.099 h@xmath5 & 0.028 & 0.029 lcccc @xmath0 & @xmath133 & @xmath134&@xmath135&@xmath136@xmath1 & @xmath137 & @xmath138&@xmath139&@xmath140@xmath2 & @xmath141 & @xmath142&@xmath143&@xmath144@xmath3 & @xmath145 & @xmath146&@xmath147&@xmath148h@xmath7 & @xmath149&@xmath150&@xmath151&@xmath152heii@xmath9 & @xmath153&@xmath154&@xmath155&@xmath156h@xmath6 & @xmath157&@xmath158&@xmath159&@xmath160@xmath4(5177 ) & @xmath161&@xmath162&@xmath163&@xmath164hei@xmath8 & @xmath165&@xmath166&@xmath167&@xmath168h@xmath5 & @xmath169&@xmath170&@xmath171&@xmath172 lcccc @xmath1 & 126 & 89 & 53 & 1.5@xmath2 & 104&121 & 50 & 2.0@xmath3 & 82&132 & 53 & 2.5@xmath4(5177 ) & 55 & 69 & 53 & 1.8h@xmath7 & 42 & 36 & 49 & 1.7 heii@xmath9 & 58 & 8 & 33 & 1.8h@xmath6 & 59 & 64 & 52 & 1.8hei@xmath8 & 40 & 6 & 55 & 1.5h@xmath5 & 54&105 & 56 & 1.7 lccccccccc @xmath1 & 0.62 & 3.5 & 5.3 & 4.9&4.5 & 3.5 & 5.0 & 5.0&3.0 @xmath2 & 0.63 & 3.5 & 4.8 & 3.8&5.6 & 5.5 & 5.8 & 4.0&3.9@xmath3 & 0.61 & 3.5 & 4.5 & 3.7&6.2 & 7.9 & 7.0 & 4.7&4.2 @xmath4(5177 ) & 0.73&@xmath173 & 8.3&11.4&6.2 & 5.3 & 8.5 & 8.5&2.8 h@xmath7 & 0.66 & 24.5 & 23.7&24.1&6.2 & 16.8&13.8&20.8&5.5 heii@xmath9&0.37 & 13.0 & 5.9&10.4&8.3 & 8.6 & 4.6&17.8&7.8 h@xmath6 & 0.64 & 24.5 & 24.4&12.5&5.3 & 22.9&23.2&15.4&3.9 hei@xmath8 & 0.69 & 25.0 & 23.4&11.4&9.5 & 18.6&22.6&17.2&8.5 h@xmath5 & 0.53 & 29.0 & 18.7&11.2&10.3 & 19.3&20.6&15.9&9.5 llccccccccc h@xmath174 & h@xmath175 & 0.89 & 0.0 & 1.0 & -4.0 & 5.7 & 5.6 & 3.1 & 0.0 & 5.6 h@xmath174 & h@xmath176 & 0.84 & 0.5 & -1.2 & 0.0 & 8.6 & 0.3 & 0.4 & 8.8 & 8.7 h@xmath177 & h@xmath178 & 0.86 & 0.0 & 4.4 & -4.0 & 18.4 & 3.9 & 2.1 & -4.0 & 17.8 h@xmath177 & h@xmath179 & 0.87 & 0.0 & 1.2 & 0.0 & 22.6 & @xmath180 & -0.9 & -1.9 & 19.0

results of a ground - based optical monitoring campaign on 3c390.3 in 199495 are presented . the broad - band fluxes ( @xmath0 , @xmath1 , @xmath2 , and @xmath3 ) , the spectrophotometric optical continuum flux @xmath4(5177 ) , and the integrated emission - line fluxes of h@xmath5 , h@xmath6 , h@xmath7 , i5876 hei@xmath8 , and ii4686 heii@xmath9 all show a nearly monotonic increase with episodes of milder short - term variations superposed . the amplitude of the continuum variations increases with decreasing wavelength ( 4400 9000 ) . the optical continuum variations follow the variations in the ultraviolet and x - ray with time delays , measured from the centroids of the cross - correlation functions , typically around 5 days , but with uncertainties also typically around 5 days ; zero time delay between the high - energy and low - energy continuum variations can not be ruled out . the strong optical emission lines h@xmath5 , h@xmath6 , h@xmath7 , and i5876 hei@xmath8respond to the high - energy continuum variations with time delays typically about 20 days , with uncertainties of about 8 days . there is some evidence that ii4686 heii@xmath9 responds somewhat more rapidly , with a time delay of around 10 days , but again , the uncertainties are quite large ( @xmath108 days ) . the mean and rms spectra of the h@xmath5 and h@xmath6 line profiles provide indications for the existence of at least three distinct components located at @xmath11 and 0 ^ -1 km s@xmath12 relative to the line peak . the emission - line profile variations are largest near line center .

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Proceed to summarize the following text: the ideas of zero range potential ( zrp ) approach were recently developed to widen limits of the traditional treatment by demkov and ostrovsky @xmath4 and albeverio et al . @xmath5 . the advantage of the theory is the possibility of obtaining an exact solution of scattering problem . the zrp is conventionally represented as the boundary condition on the matrix wavefunction at some point . alternatively , the zrp can be represented as pseudopotential ( breit @xmath6 ) . on the other hand , darboux transformation ( dt ) allows to construct in natural way exactly solvable potentials . general starting point of the theory goes up to matveev theorem ( see @xcite ) . the transformation can be also defined on the base of covariance property of the schrdinger equation with respect to a transformation of wavefunction and potential ( matveev and salle @xmath7 ) . darboux formulas in multi - dimensional space could be applied in the sense of andrianov , borisov and ioffe ideas @xcite . in the circumstances , dt technique can be used so as to correct zrp model . we attempt to dress the zrp in order to improve the possibilities of the zrp model . we use notations and some results from @xcite . dt modifies the generalized zrp ( gzrp ) boundary condition ( section @xmath8 ) and creates a potential with arbitrarily disposed discrete spectrum levels for any angular momentum @xmath9 . in the section @xmath10 we consider @xmath11-representation for a non - spherical potential so as to dress a multi - centered potential , which includes @xmath0 zrps . as an important example , we consider electron scattering by the @xmath1 and @xmath2 structures within the framework of the zrp model ( section @xmath12 ) . in section @xmath13 we present the our calculations for the electron-@xmath3 scattering and discuss them . let us start from the simplest case of a central field . then angular momentum operator commutates with hamiltonian and therefore wavefunction @xmath14 can be expanded in the spherical waves @xmath15 where @xmath16 , @xmath17 is initial particle direction , @xmath18 are partial waves , and @xmath19 are phase shifts . consider the radial schrdinger equation for partial wave with angular momentum @xmath9 . the atomic units are used throughout the present paper , i.e. @xmath20 and born radius @xmath21 . @xmath22 @xmath23 @xmath24 denotes differential operator , and @xmath25 are hamiltonian operators of the partial waves . this equations describe scattering of a particle with energy @xmath26 . the wavefunctions @xmath18 at infinity have the form @xmath27 let us consider gzrp in coordinate origin . this potential is conventionally represented as boundary condition on the wavefunction ( see @xmath28 ) @xmath29 where @xmath30 are inverse scattering lengths . the potential @xmath31 and therefore wavefunctions @xmath18 can be expressed in terms of the spherical functions @xmath32 where spherical functions @xmath33 are related to usual bessel functions as @xmath34 , @xmath35 . in the vicinity of zero they have the asymptotic behavior @xmath36 , and @xmath37 . to substituting the equation @xmath38 into the boundary condition we obtain the elements of @xmath11-matrix @xmath39 the bound states correspond to the poles of the @xmath11-matrix ( i.e the zeros of the denominator @xmath40 ) , which lie on the imaginary positive semi - axis of the complex @xmath41-plane . it is obvious that bound state , with orbital momentum @xmath9 , exists only if @xmath42 ( elsewise an antibound state exists ) and has the energy @xmath43 . thus , spectral problem for gzrp is solved for any value @xmath41 . on the other hand , the equations ( [ e ] ) are covariant with respect to dt that yields the following transformations of the potentials ( coefficients of the operator @xmath25 ) @xmath44 and the wavefunctions @xmath18 @xmath45 where @xmath46 are some solutions of the equations @xmath47 at @xmath48 , and @xmath49 are real parameters , which can be both positive or negative . the dt @xmath50 combines the solutions @xmath18 and a solution @xmath46 that corresponds to another eigen value @xmath51 . repeating the procedure we obtain a chain of the integrable potentials @xmath52 . in general , dressed potential @xmath53 is real for real function @xmath46 . the next step in the dressing procedure of the zero - range potential ( @xmath31 ) is a definition of the free parameters of the solutions @xmath46 . suppose the prop functions @xmath46 satisfy the boundary conditions @xmath54 with @xmath55 . in the simplest case of @xmath56 we have @xmath57 and @xmath58 the dt @xmath50 gives rise to the following requirement on dressed wavefunction @xmath59 the dressed potential @xmath60 is given by @xmath61 it is regular on semiaxis only if @xmath62 . in the limiting case at @xmath63 we obtain long - range interaction @xmath64 , which can be regular on semiaxis only if @xmath65 . assuming @xmath66 we get @xmath67 ( trivial transformation ) , and boundary condition can be obtained by the substitution : @xmath68 to dress free wave @xmath69 we obtain zrp at the coordinate origin . thus , zrp can be also introduced in terms of dt . to consider transformation with parameter @xmath70 we obtain regular solution @xmath71 and tangent of phase shift is @xmath72 in the other cases asymptotic of the functions @xmath73 at zero is given by @xmath74 it is clear that the each dt introduces short - range core of centrifugal type ( which depends on angular momentum @xmath9 ) in the potential . in this situation the boundary conditions on the dressed wavefunctions @xmath75 $ ] require modification . thus , in the case @xmath76 the boundary conditions become @xmath77 and in the case @xmath78 we obtain @xmath79 in the generalized case , zrp with angular momentum @xmath9 generates also @xmath80 complex poles of the @xmath11-matrix , which correspond the quasi - stationary states ( resonances ) . the dts @xmath50 with the parameters @xmath49 results in the @xmath11-matrix elements for dressed gzrp @xmath81 we can use darboux transformation in order to add ( or remove ) poles of the @xmath11-matrix . the principal observation allows to built a zero - range potential eigen function in the multi - center problem . let us consider @xmath0 zrps at the points @xmath82 and interaction @xmath83 . the wavefunction @xmath14 can be expressed in terms of the ( outgoing - wave ) green function , defined by the equation @xmath84 the second ( ingoing - wave ) green function is defined by @xmath85 . the partial waves @xmath86 , defined by @xmath87 for multi - centered target which includes @xmath0 zrps and interaction @xmath83 can be expressed as a superposition of the green functions @xmath88 in which @xmath89 are phase shifts , @xmath90 denote @xmath11-matrix orthonormal eigenfunctions , @xmath91 are real numbers . they naturally generalize the spherical partial waves @xmath92 for a non - spherical potential @xmath93 . expanding the partial waves @xmath86 at the infinity we obtain the expressions for @xmath90 @xmath94 where @xmath95 is wavefunction for potential @xmath83 , and @xmath96 is scattering amplitude . scattering amplitude for potential @xmath83 and @xmath0 zrp is given by @xmath97 by the imposition of the boundary conditions @xmath54 the calculation of the partial waves is reduced to the solution of the following system @xmath98 in which we use @xmath99 where @xmath100 , @xmath101 . the tangents @xmath102 can be found from compatibility condition of this system . in simplest case @xmath103 @xmath104 the system @xmath105 is reduced to the usual equations of zrp theory @xmath93 . in order to construct functions @xmath106 and @xmath107 for dressed potential , i.e. for @xmath83 , we need to write down green function as single - center expansion over spherical harmonics . in the simplest case , when the prop function is @xmath108 , the green function is given by @xmath109 where @xmath110 the function @xmath111 has the following asymptotic at infinity @xmath112 the integral cross section can be readily derived using the optical theorem : @xmath113 thus , averaged integral cross section is given by @xmath114 for purpose of illustration we consider scattering problem for a dressed multi - center potential . the multi - center scattering within the framework of the zrp model was investigated by demkov and rudakov @xmath93 ( 8 centers , cube ) , drukarev and yurova @xmath116 ( 3 centers in line ) , szmytkowski @xmath117 ( 4 centers , regular tetrahedron ) . let structure @xmath1 contains @xmath0 identical scatterers , which involve only @xmath119 waves . denote position vector of the scatterer @xmath120 by the @xmath121 . suppose , for the sake of simplicity , the distance between any two scatterers @xmath122 and @xmath120 is @xmath123 . there are three such structures in three - dimensional space - dome @xmath124 , regular trihedron @xmath125 , regular tetrahedron @xmath126 . the partial waves @xmath127 and phases @xmath89 can be classified with respect to symmetry group representation , degeneracy being defined by the dimension of the representation @xmath93 . the structures @xmath124 , @xmath125 , @xmath126 belong to the @xmath128 point groups respectively . the equation @xmath105 leads to algebraical problem @xmath129 the phases can be readily found from compatibility condition of this system . to factorize the determinant we derive the expressions for the phases @xmath130 @xmath131 thus , assuming @xmath132 we obtain the phases of a regular tetrahedron @xmath117 . the partial and integral cross sections can be expressed as @xmath133 the structures @xmath2 can be used , for instance , to study a slow electron scattering by the polyatomic molecules like @xmath135 , @xmath136 , @xmath137 , etc . let @xmath138 denote the position vectors of the scatterers @xmath139 and @xmath140 denotes the position vector of the scatterer @xmath141 . the scatterers @xmath139 are situated in vertices of a regular structure @xmath1 , i.e. @xmath123 . suppose , for the sake of simplicity , the distance between the scatterer @xmath141 and any scatterer @xmath139 is @xmath142 . therefore the position of the scatterer @xmath141 perfectly fixed only if @xmath132 ( geometric center of the tetrahedron , @xmath143 ) . the partial waves are given by the equation @xmath127 , where the summation should be performed over the @xmath144 . the constants @xmath91 and phases can be derived analytically . thus , we obtain the phases @xmath145 where @xmath146 is inverse scattering length of the scatterers @xmath139 . the @xmath147 obey the quadratic equation @xmath148 where @xmath149 indicates inverse scattering length of the scatterer @xmath141 . in the limiting case when the distance between @xmath139 and @xmath141 scatterers is very large , i.e. @xmath150 , the expression for @xmath151 becomes @xmath152 and @xmath153 . this situation corresponds to independent scattering on the structure @xmath1 and scatterer @xmath141 . the tangent of @xmath154 also reduces to @xmath155 for structure @xmath1 in the limit @xmath156 . the integral cross sections for @xmath3 ( closed - shell ground - state @xmath157 ) are plotted in fig . @xmath158 for a number of values of @xmath159 , which are regarded as constant in the range of interest . the present calculations were carried out with @xmath160 . our calculations were made within the framework of the zrp model and hence they are not expected to be correct for low impact energies ( i.e for energies @xmath161 ev ) where polarization effects ( @xmath3 spherical polarizability @xmath162 ) are known to be important . because induced polarization potentials are always attractive , including polarization algebraically increases the computed phases ( @xmath163 at @xmath164 ) and therefore decreases the integral cross section . thus , fig . @xmath158 clearly shows that polarization effects reproduce the deep minimum ( ramsauer - townsend minimum near @xmath165 ev ) seen in the experimental data @xmath166 and numerical calculations @xmath167 , which are incomparably smaller in the our calculations . the ics can be corrected by the zrps dressing . fig . @xmath168 show ics for dressed @xmath169 structure . in our calculation , the parameters @xmath170 , @xmath171 were used . in the higher collision energies our cross section ( fig . @xmath158 ) differs from the experimental results in size but coincides in shape . it is known that role of the higher partial waves is important at the higher collision energies . thus , taking into consideration the @xmath172 waves for the scatterers @xmath139 and @xmath141 we can add in partial waves and improve the agreement both in size at the higher energies and in the position of shape resonance . we demonstrate the posibilities of dt in multi - center scattering problem . thus , these transformations allow to correct the zrp model at low energies . in the limiting case @xmath63 dt induces the long - range forces @xmath173 depending on angular momentum and leads to singularity in the cross section at zero energy . albeverio s , gesztesy f , hegh - krohn r and holden h ( 1988 ) solvable models in quantum mechanics , springer - verlag , new york breit g ( 1947 ) phys . 71:215 matveev v b ( 1979 ) lett . 3:213 matveev v b , salle m a ( 1991 ) darboux transformations and solitons , springer , new york andrianov a , borisov n , ioffe m ( 1984 ) phys . a 105:19 - 22 baltenkov a s ( 2000 ) phys . a 286:92 - 99 szmytkowski r , szmytkowski c ( 1999 ) j. math . chem . 26:243 - 254 wan h x , moore j h , tossel j a ( 1989 ) j. chem . 91:1340 zaitsev a , leble s ( 1999 ) preprint 12.01.1999 math - ph/9903005 ; ( 2000 ) romp 46:155

a dressing of a nonspherical potential , which includes @xmath0 zero range potentials , is considered . the dressing technique is used to improve zrp model . concepts of the partial waves and partial phases for non - spherical potential are used in order to perform darboux transformation . the problem of scattering on the regular @xmath1 and @xmath2 structures is studied . the possibilities of dressed zrp are illustrated by model calculation of the low - energy electron - silane ( @xmath3 ) scattering . the results are discussed .

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Proceed to summarize the following text: the properties of a giant resonance in nuclei are commonly determined from the distorted wave born approximation ( dwba ) analysis of its excitation cross - section by inelastic scattering of a certain projectile . the transition potential required in actual implementation of dwba calculation is usually obtained by convoluting the projectile - nucleus interaction with the transition density associated with the giant resonance . the relevant transition density can be obtained from a microscopic theory of the giant resonance , such as the hartree - fock ( hf ) based random phase approximation ( rpa ) . however , the use of a macroscopic transition density @xmath3 greatly simplifies the application of the giant multipole resonance theory to the analysis of the experimental data . the simple form of the transition density @xmath4 obtained by the scaling approximation , is a well - known example of the macroscopic transition density @xmath5 commonly used in the case of the isoscalar giant monopole resonance ( isgmr ) @xcite . the transition density of eq . ( [ tr7 ] ) nicely agrees with the isgmr transition density obtained in microscopic hf - rpa calculations . it has a one - node structure , satisfying the condition of particle number conservation @xmath6 unfortunately the scaling consideration can not be extended to the overtone of the isgmr , where @xmath3 has a two - node structure . to derive the macroscopic transition density @xmath3 in this more general case , one can use the well - known method @xcite of determining @xmath7 from the local sum rule which is exhausted by one collective state with the appropriate choice of the transition operator @xmath8 . however , in the quantum random phase approximation , the highly excited collective modes are strongly fragmented over a wide range of energy and a special averaging procedure must be employed to determine the macroscopic transition density corresponding to an average collective excitation . in this respect , the semiclassical fermi - liquid approach ( fla ) @xcite is more appropriate . both the main isgmr and its overtone are well - defined within the fla as single resonance states . this fact enables us to derive the transition operator @xmath9simply by maximizing the fraction of the energy - weighted sum rule ( fewsr ) exhausted by the single overtone . in the present work we suggest a procedure to derive the macroscopic transition density for the isgmr overtone using both the hf based rpa and the fermi - liquid approaches . we remark that some preliminary results of this investigation were presented in ref . @xcite ( see also ref . * * the transition density @xmath10 for a certain eigenstate @xmath11 of a nucleus with @xmath12 nucleons is given by @xmath13 where @xmath14 is the particle density operator and @xmath15 represents the ground state of the nucleus . the transition density reflects the internal structure of the nucleus and does not depend on the external field . however , a problem arises if one intends to derive the transition density @xmath16 for a group of the thin - structure resonances in the giant multipole resonance ( gmr ) region . an appropriate averaging procedure is necessary in this case and @xmath16 can be evaluated if the nucleus is placed in an external field @xmath17 where the transition operator @xmath18 is so chosen that it provides a preferable excitation of the above mentioned thin - structure resonances . let us introduce the local strength function @xmath19 and the energy smeared local strength function @xmath20 defined near the gmr energy @xmath21 by @xmath22 the corresponding strength functions are given by @xmath23 and @xmath24 let us assume , for the moment , that the operator @xmath25excites only a single state @xmath26 , within the energy interval @xmath27 . the corresponding transition density @xmath28 is then given by the following _ exact _ expression @xmath29 we will extend expression ( [ tr1 ] ) to the case of a group of the thin - structure resonances in the gmr region which are excited by the operator @xmath25 and define the smeared transition density @xmath30 as @xmath31 note that eq . ( [ tr2 ] ) is associated with the strength in the region of @xmath27 and is consistent with the smeared strength function @xmath32 for a single resonance state . that is ( see also eq . ( [ se0 ] ) ) , @xmath33 we also point out that with the lorentz s function @xmath34 the energy smeared @xmath35 is given by @xmath36 and the smeared transition density @xmath30 is obtained from @xmath37 the consistency condition , eq . ( [ se1 ] ) , then reads @xmath38 in the quantum rpa , the local strength function @xmath39 is related to the rpa green s function @xmath40 by @xcite @xmath41 \,d{\bf r}\,^{\prime } \ . \label{se4}\ ] ] for the isoscalar monopole and dipole excitations , the transition operator @xmath42 is taken in the form of @xmath43 and @xmath44 with an appropriate choice of the radial functions @xmath45 and @xmath46 , see below . in the following , the quantum transition density for the main isgmr and its overtone is evaluated using the eq . ( [ tr2 ] ) with @xmath47 taken separately for the isgmr and the overtone regions . let us consider the local energy - weighted sum @xmath48 given by ( see eq . ( [ sre ] ) ) @xmath49 the continuity equation provides the following sum rule @xcite @xmath50 let us assume that only one state @xmath26 exhausts the sum rule eq . ( [ m1r2 ] ) . then for the corresponding ( macroscopic ) transition density , @xmath51 , we have from eqs . ( [ m1r1 ] ) and ( [ m1r2 ] ) the following expression @xmath52 where the normalization coefficient @xmath53 can be found using the energy - weighted sum rule ( ewsr ) @xmath54 taking into account that @xmath55 we obtain @xmath56 thus , the macroscopic transition density @xmath57 of eq . ( [ tr5 ] ) coincides with the quantum transition density for a certain state @xmath26 if the single state @xmath26exhausts all the ewsr , eq . ( [ m1 ] ) , associated with the transition operator @xmath42 . in the case of the transition operator @xmath58 from eqs . ( [ fl0 ] ) and ( [ f1 ] ) , the ewsr is given by @xcite @xmath59 . \label{m1l}\ ] ] assuming that the energy - weighted transition strength ( ewts ) @xmath60 fully exhausts the ewsr associated with @xmath61 , we obtain the expression for the macroscopic transition density for the state at @xmath62 from eq . ( [ tr5 ] ) @xcite @xmath63 \rho _ { { \rm eq}}(r)y_{lm}({\bf \hat{% r } } ) . \label{trl}\ ] ] for @xmath64 we have from eq . ( [ trl ] ) the commonly used isgmr result of eq . ( [ tr7 ] ) . the main ( lowest ) isgdr is the spurious state with the eigenenergy @xmath65 . the next isgdr is the overtone . assuming that the ewts for the @xmath66 overtone equals the ewsr associated with the operator @xmath67 we obtain from eq . ( [ tr5 ] ) the macroscopic transition density as @xmath68 for the isoscalar dipole mode , the translation invariance condition is used for the derivation of @xmath69 . this condition implies that the center of mass of the system can not be affected by internal excitation . we thus have , @xmath70 from eqs . ( [ tr8 ] ) and ( [ cm1 ] ) one obtains , see also refs . @xcite , @xmath71 where @xmath72 is the mean square radius . the free ( mixing ) parameter appearing in the transition operator @xmath73 ( similar to @xmath74 of eq . ( [ f1 ] ) for the @xmath75 case ) can be determined by an appropriate condition leading to a general method for the evaluation of the transition density for the overtone mode . in this work we present this method for the case of the monopole mode , @xmath76 let us introduce the transition operator @xmath9 as @xmath77 the corresponding macroscopic transition density @xmath5 is obtained from eqs . ( [ tr5 ] ) and ( [ f4 ] ) as @xmath78 \rho _ { { \rm eq}}(r)y_{00}(% { \bf \hat{r}})\qquad { \rm for\qquad } l=0\quad ( { \rm overtone } ) . \label{tr9}\ ] ] the determination of the parameter @xmath79 in eq . ( [ tr9 ] ) requires an additional consideration since for the @xmath64 case we have no fundamental condition such as eq . ( [ cm1 ] ) for the @xmath75 mode . we note , however , that if we assume that the isgmr has the transition density of eq . ( [ tr7 ] ) and require that @xmath80 i.e. , the isgmr is not excited by the scattering operator of eq . ( [ f4 ] ) we have @xmath81 similar result is obtained by imposing the condition that the scattering operator @xmath82 does not excite the overtone of the isgmr , assuming the transition density of eq . ( [ tr9 ] ) . following the general requirement for the proper use of eq . ( [ tr5 ] ) in the derivation of the macroscopic transition density @xmath3 , we can determine the parameter @xmath79 from the condition that the transition operator @xmath9 provides for the single overtone the maximum fraction of the energy - weighted sum rule @xmath83 of eq . ( [ m1 ] ) . the transition density @xmath16 and the strength function @xmath84 can also be evaluated within the semiclassical fermi - liquid approach . for a given multipolarity @xmath85 and overtone @xmath86 the fla transition density is given by @xcite @xmath87 @xmath88 \rho _ { 0}y_{l0}({\bf \hat{r } } ) , \label{tr4}\ ] ] where @xmath89 is the bulk density , @xmath90 is the equilibrium nuclear radius and the parameter @xmath91 is determined by the translation invariance condition ( eq . ( [ cm1 ] ) ) in the case of the isoscalar dipole compression mode and is given by @xmath92 the wave numbers @xmath93 are derived from the boundary conditions of the fla model : the normal component of the tensor pressure @xmath94 , created by a sound wave , on the free surface of the nucleus should be equal to zero @xmath95 note that for the case of compression sound modes , the contribution from the surface tension pressure is negligible and it was omitted in eq . ( [ b1 ] ) . the boundary condition ( eq . ( [ b1 ] ) ) leads to the following secular equation ( see ref . @xcite ) @xmath96_{r = r_{0}}=0,\quad \,d_{\mu } = { % \frac{4\,\mu } { m\,\rho _ { 0}c_{0}^{2}},}\qquad \label{sec2}\ ] ] where @xmath97 is the zero sound velocity and the coefficient @xmath98 determines the contribution from the dynamical fermi surface distortion associated with the collective motion in a fermi liquid . in the case of a quadrupole distortion of the fermi surface , one has @xcite @xmath99 here , @xmath100 is the equilibrium pressure of a fermi gas , @xmath101 is the fermi energy , and @xmath102 and@xmath103 are the eigenfrequency and the relaxation time for sound excitations in the fermi liquid , respectively . the relaxation time @xmath104 is assumed to be frequency dependent because of the memory effect in the collision integral @xcite . following refs . @xcite we take @xmath105 where @xmath106 is the constant related to the differential cross section for the scattering of two nucleons in the nuclear interior . in the case of isoscalar sound mode , we will adopt @xmath107 mev @xcite . the eigenfrequency @xmath102 is obtained from the dispersion relation @xmath108 where @xmath109 is given by @xmath110 here @xmath111 is the nuclear incompressibility coefficient and @xmath112 is the friction coefficient @xmath113 the smeared fla strength function @xmath114 can be obtained in a way similar to @xmath115 , obtained within the quantum approach of eq . ( [ se3 ] ) . that is , @xmath116 the smearing function @xmath117 in eq . ( [ sfla ] ) is given by ( @xmath118 ) @xmath119 here , @xmath120 is the damping parameter due to the viscosity of the fermi liquid and @xmath121 , where the eigenfrequency @xmath122 is obtained as a solution to both the dispersion equation ( [ disp ] ) and the secular equation ( [ sec2 ] ) . we point out that the amplitude @xmath123 in eq . ( [ tr4])@xmath124for the fla transition density @xmath125 is derived as the amplitude of the quantum oscillations @xmath126 where @xmath127 is determined by eq . ( [ sec2 ] ) and @xmath128 is the corresponding mass coefficient with respect to the density oscillations . the collective mass coefficient @xmath128 can be found from the collective kinetic energy @xmath129 for the particle density oscillations . the collective kinetic energy is derived as @xmath130 for the compression modes @xmath64 , the mass coefficient @xmath131 is given by @xcite @xmath132 _ { x = x_{0}},\qquad { \rm % for\qquad } l=0 , \label{b10}\ ] ] where @xmath133 . we have carried out calculations for the isgmr in the frameworks of hf based rpa and the semiclassical fermi liquid approach as briefly outlined in the preceding sections . we have evaluated the smeared fla strength function @xmath114 of eq . ( [ sfla ] ) and the hf - rpa smeared strength function @xmath115 of eq . ( [ se3 ] ) for the isgmr in several nuclei , for @xmath134 from eq . ( [ f4 ] ) . in the subsequent discussions , the hf - rpa results presented are obtained using the skyrme force skm@xmath135 @xcite . for the rpa calculations to be highly accurate we discretized the continuum in a large box of size 90 fm and use a smearing parameter @xmath136 mev , in evaluating the rpa green s function ( see eq . ( [ se4 ] ) ) , and allow particle - hole excitations up to 500 mev ( see ref . @xcite for the details ) . in case of the fla calculations we have adopted the values of @xmath137 @xmath138 , @xmath139 and @xmath140 fm . in table [ fla - rpa ] we compare the fla and rpa results for the centroid energies @xmath141 and @xmath142 corresponding to the main and overtone mode of the isgmr , respectively . we see from this table that the fla and rpa results are in qualitative agreement . the small differences ( @xmath143 ) can be understood by the fact that the centroid energy mainly depends on the size of the system . the values of the mean square radii and the higher moments of the ground state density distribution are smaller in the fla than the ones obtained from hf calculations . note also that the ratio @xmath144 in both models considered is greater than two ( @xmath145 ) . we point out that @xmath146 is due to the fermi - surface distortion effect as noted earlier in ref . @xcite . we have performed a comparison of the macroscopic transition density @xmath147 with the ones obtained within the hf - rpa , @xmath148 , and the fla , @xmath149 , approaches for the main resonance @xmath150 and its overtone . the fla transition density is given by eq . ( [ tr4 ] ) with the wave number @xmath151 obtained from the secular equation ( [ sec2 ] ) . the contribution of the isgmr overtone to the ewsr for the case of the transition operator @xmath9 is given by @xmath152 where @xmath153 . the eigenfrequency @xmath154 is obtained from the dispersion equation ( [ disp ] ) and the wave number @xmath155 is the second ( overtone ) solution to the secular equation ( [ sec2 ] ) . the ewsr for the transition operator @xmath9 reads @xmath156 now , we will determine the parameter @xmath79 from the condition that the value of the ratio of the partial sum @xmath157 to the total sum@xmath158 is the maximum . in fig . 1 , we have plotted the ratio @xmath159 as a function of the parameter @xmath79 for the nucleus @xmath160pb obtained in the fla ( dashed line ) and the hf - rpa ( solid line ) models . as seen from fig . 1 , the maximum value of the ratio of @xmath157 to @xmath158 for the fla model is achieved for @xmath161 @xmath162 where the overtone exhausts about @xmath163 of the ewsr . in the case of hf - rpa , the maximum ratio is achieved for @xmath164 @xmath162 where the overtone , in the energy range of 25 - 50 mev , exhausts about @xmath165 of the ewsr . it is interesting to note that if we use the fla and hf ground - state densities to calculate @xmath1 from eq . ( [ xi02 ] ) , we get @xmath166 and 79.0 @xmath167 , respectively . these values are close to the corresponding ones ( 68.3 and 78.6 @xmath167 ) obtained by the condition of maximizing the ratio @xmath168 . this means that the mixing parameter @xmath79 in the transition operator @xmath169 of eq . ( [ f4 ] ) can also be derived from the condition that the main isgmr gives a minimal contribution to the energy - weighted sum rule @xmath158 . the fla as well as hf - rpa calculations show that the difference in the values of @xmath79 obtained in both conditions does not exceed @xmath170 . we have also calculated the dependence of the parameter @xmath79 on the nuclear mass number @xmath12 . following the same procedure we find for @xmath171zr , @xmath172sn and @xmath173sm nuclei the value of mixing parameter @xmath174 38.6 ( 48.5 ) , 45.9 ( 56.7 ) and 53.1 ( 66.1 ) @xmath167 from the fla ( rpa ) calculations , respectively . these values can be well approximated by @xmath175 and 2.36@xmath176 @xmath167 for the fla and rpa approaches , respectively . in fig . 2 , we plot the fla and rpa results for the fraction energy - weighted transition strength as a function of the excitation energy obtained for the transition operator @xmath134 for the @xmath160pb nucleus . we use @xmath177 68.3 and 78.6 @xmath167 in the fla and rpa calculations , respectively . one can clearly see that the rpa calculation yields a wide resonance of width of @xmath178 mev around the excitation energy @xmath179 mev which corresponds to the isgmr overtone . whereas , in the case of fla , the transition operator @xmath180 ( eq . ( [ f4 ] ) ) with an appropriate value of @xmath1 gives rise to a well defined resonance for the overtone mode . we also notice that the rpa results have the reminiscence of the isgmr main tone but it is practically eliminated in the fla calculations . in figs . 3@xmath91 and 3@xmath181 , we compare the radial macroscopic transition density @xmath182 of eq . ( [ tr9 ] ) , obtained using the hf ground - state density , and the corresponding fla and hf - rpa ones for the overtone of the isgmr . the radial transition density @xmath183 for a certain multipolarity @xmath85 is given by @xmath184 the macroscopic transition densities for the overtone in figs . 3a and 3b are not the same . because , for an appropriate comparison , in fig . 3a , we have plotted @xmath185 obtained using @xmath186 and it is normalized to @xmath165 of the ewsr . on the other hand , @xmath185 plotted in fig . 3b corresponds to @xmath187 and is normalized to @xmath163 of the ewsr . further , the rpa transition density is calculated by averaging over the energy range of 25 - 50 mev . notice that the shift of the nodes of @xmath188 to the left with respect to the ones of @xmath189 is caused by the fact that in contrast to eq . ( [ tr9 ] ) used for the macroscopic transition density , in the fla we use sharp nuclear surface , see eq . ( [ tr4 ] ) . we also looked into the energy dependence of the rpa transition density for the operator @xmath73 over the range of energy employed for the averaging . we see that over the entire range considered , the transition density has two nodal structure and the distance between the nodes decreases with the increase in energy . for example , averaging over @xmath190 mev range , we find that at the excitation energies of 30 , 40 and 50 mev , the distances between the two nodes are 3.1 , 2.7 , and 2.4 fm , respectively , which reflects the fact that the microscopic transition density is state dependent . we have used the microscopic transition densities for the operator @xmath73 to evaluate the cross - section for the isgmr overtone mode excited via inelastic scattering of @xmath191-particles with energies 240 and 400 mev . we have used the folding model ( fm)-dwba to calculate the excitation cross - section ( see ref . @xcite for details ) . we find that for the @xmath191-particles with 400 mev energy , the calculated cross - section is about 7 - 10 times higher than the one obtained for @xmath191-particles with 240 mev . note that for a monopole resonance the cross - section is maximum at @xmath192 . the values of the cross - section at @xmath192 for the peak energy of the isgmr overtone are 0.5 and 3.5 mb/(sr mev ) for the case of 240 and 400 mev , respectively . we point out that the maximum cross - section for the case of 240 mev @xmath191-particles is below the current experimental sensitivity of about 2 mb/(sr mev ) @xcite . it may be possible to identify the isgmr overtone mode with 400 mev @xmath191-particles . the transition density @xmath193 for the compression modes is distributed over the nuclear interior and has a node close to the nuclear surface for both the main isgmr and its overtone . the transition density @xmath194 of the overtone has an additional node in the nuclear interior . this feature of @xmath193 can be tested by evaluating the strength distribution @xmath195 of the electromagnetic operator @xmath196 . the strength function @xmath197 for a certain eigenstate @xmath198 is given by @xmath199 where @xmath200 the strength function @xmath201 is related to the excitation function of electron - nucleus scattering in the born approximation . we use eqs . ( [ slk1 ] ) and ( [ slk2 ] ) to calculate the energy - weighted sums @xmath202 and @xmath203 for the main isgmr and its overtone , respectively . in fig . 4 , we display the @xmath204 dependence of the fraction energy - weighted sums @xmath205 and @xmath206 for the @xmath160pb nucleus obtained from the fla ( dashed line ) and hf - rpa ( solid line ) approaches . it can be seen from fig . 4 that @xmath205 and @xmath207 depend strongly on @xmath204 . a shift of the maximum of the ratio @xmath206 for the overtone to the higher value of wave number @xmath204 is due to the more complicated nodal structure of the transition density associated with the overtone as compared with the main resonance . this shift can be exploited to separate the modes in electron - nucleus scattering . in fig . 5 we plot the surface and the volume contributions to the integral in eq . ( [ slk2 ] ) for the transition density associated with overtone mode ( see eq . ( [ tr4 ] ) ) . for smaller @xmath204 , there is a cancellation between the surface and the volume contributions leading to a peak structure for the overtone response as shown in fig . starting from the local strength function @xmath39 and using the smearing procedure , we have extended the quantum expression for the transition density @xmath10 to the case of a group of the thin - structure resonances which are localized in the gmr region and are excited due to the specifically chosen transition operator @xmath208 . our approach was applied to the study of the transition density of the isgmr overtone . in this case , an appropriate form of the transition operator @xmath58 is given by @xmath209 , see eq . ( [ f4 ] ) . the mixing parameter @xmath79 was determined from the condition that the transition operator @xmath210 provides for the single overtone the maximum fraction of the energy - weighted sum rule @xmath83 of eq . ( [ m1 ] ) . the mixing parameter @xmath79 depends on the nuclear mass number @xmath12 . this dependence is well approximated by @xmath211 . we have applied our smearing procedure ( using @xmath212 associated with the maximum fewsr of the overtone ) to the evaluation of the smeared out transition density @xmath30 of eq . ( [ tr2 ] ) within the hf - rpa . we have shown that the smearing procedure for the isgmr overtone region provides a simple two nodal structure of @xmath30 ( see the solid line in fig . 3@xmath91 ) , as expected for the @xmath64 overtone . moreover , the transition density @xmath213 , obtained by the averaging over many quantum states , resembles its macroscopic counterpart . this fact is well illustrated in fig . 3@xmath91 by comparing the quantum smeared transition density @xmath30 with the macroscopic one @xmath51 of eq . ( [ tr9 ] ) . an independent derivation of the smeared out transition density @xmath214 can be also obtained using the semiclassical approaches.in sec . iv , we have applied a simple semiclassical fermi - liquid approximation to the evaluation of the smeared out ( in quantum mechanical sense ) transition density @xmath215 . we have used the same form of the transition operator @xmath9 as in the case of quantum hf - rpa calculation to provide an additional check of the derivation of the mixing parameter @xmath79 from the energy - weighted sum @xmath83 . we found a good agreement between the values of parameter @xmath79 obtained in both the quantum and the semiclassical approaches . it is important to emphasize that equation ( [ xi02 ] ) together with eq . ( [ tr5 ] ) provides a simple expression for the macroscopic transition density that can be employed in the folding model - dwba analysis of excitation cross - section of the isgmr overtone . the nodal structure of the semiclassical transition density @xmath216 is similar to that of both the quantum , @xmath217 , and the macroscopic , @xmath218 , cases , see figs . 3@xmath91 and 3@xmath181 . a discrepancy occurs in the surface region , where the particular behavior of @xmath219 is due to the assumption of the sharp surface of the nucleus in the fla model . this discrepancy is not so significant in the integral quantities like the strength functions . this is illustrated in fig . 4 for the case of the nuclear response to the electromagnetic - like external field @xmath220 . the ratios @xmath221 and @xmath222 for the isgmr and its overtone , respectively , show an distinct feature in the @xmath204-dependence . namely , for a certain value of the wave number @xmath204 , the strength function for the overtone reaches a maximum whereas the contribution of the main resonance to the strength function is strongly suppressed . this fact can be exploited to separate the isgmr and the overtone modes in electron - nucleus scattering by varying the electron s momentum transfer @xmath204 . this work was supported in part by the us department of energy under grant # doe - fg03 - 93er40773 . one of us ( v.m.k . ) thank the cyclotron institute at texas a&m university for the kind hospitality . h. s. khler , nucl . phys . * a378 * , 159 ( 1982 ) . j. bartel , p. quentin , m. brack , c. guet and h. -b . hakansson , nucl . phys . * a386 * , 79 ( 1982 ) . d. h. youngblood , h. l. clark and y. w. lui , phys . . lett . * 82 * , 691 ( 1999 ) . a. kolomietz , o. pochivalov and s. shlomo , phys . c * 61 * , 034312 ( 2000 ) . d. h. youngblood ( private communication ) . 1 . the ratio @xmath223 of the partial contribution of the overtone to the ewsr as a function of the parameter @xmath79 in the transition operator @xmath9 ( eq . ( [ f4 ] ) ) obtained within the fla ( dashed line ) and the hf - rpa ( solid line ) approaches , for the monopole mode @xmath224 in the nucleus @xmath225 fig . 3@xmath91 . the hf - rpa transition density , @xmath227 multiplied by @xmath228 for the overtone of the isgmr in the nucleus @xmath229 ( solid line ) and the corresponding macroscopic transition density @xmath3 taken at @xmath230 @xmath231 ( dotted line ) . 3@xmath181 . the fla transition density , @xmath232 , multiplied by @xmath228 for the overtone of the isgmr in the nucleus @xmath233 ( dashed line ) and the corresponding macroscopic transition density @xmath3 taken at @xmath161 @xmath234 ( dotted line ) . fig . the ratio @xmath235 for the main ( @xmath236 ) isgmr and its overtone ( @xmath237 ) , respectively , as a function of the wave number @xmath238 obtained for electromagnetic operator @xmath239 for the @xmath240pb nucleus . the dashed and the solid lines represent the fla and hf - rpa results , respectively . fig . partial contributions of the volume ( vol ) and surface ( surf ) terms of the fla transition density of the isgmr overtone ( see eq . ( [ tr4 ] ) ) to the integral in eq . ( [ slk2 ] ) ( dashed lines ) . the solid line shows the sum of both the volume and the surface terms . .[fla - rpa ] comparison of the fla and rpa results for the centroid energies ( in mev ) for the main ( @xmath141 ) and the overtone ( @xmath142 ) modes of isgmr . in the case of rpa calculations the values of @xmath141 are obtained by integrating the strength function for the operator @xmath241 over the energy range of 0 - 60 mev and the values of @xmath142 are obtained using the operator @xmath242 ) and the energy ranges of 35 - 60 , 28 - 60 , 27 - 55 and 25 - 50 mev for @xmath243 , @xmath244 , @xmath245 and @xmath246 nuclei , respectively . the experimental data for the main tone is taken from the ref . [ cols="^,^,^,^,^,^,^,^ " , ]

we calculate the transition density for the overtone of the isoscalar giant monopole resonance ( isgmr ) from the response to an appropriate external field @xmath0 obtained using the semiclassical fluid dynamic approximation and the hartree - fock ( hf ) based random phase approximation ( rpa ) . we determine the mixing parameter @xmath1 by maximizing the ratio of the energy - weighted sum for the overtone mode to the total energy - weighted sum rule and derive a simple expression for the macroscopic transition density associated with the overtone mode . this macroscopic transition density agrees well with that obtained from the hf - rpa calculations . we also point out that the isgmr and its overtone can be clearly identified by considering the response to the electromagnetic external field @xmath2 .

You are an expert at summarizing long articles. Proceed to summarize the following text: in the current literature of complex systems , aging is frequently reported as a phenomenon observed in spin glasses and other disordered systems , rather than in biological systems @xcite , though some works have been devoted to biological aging of populations @xcite . relevant contributions have been brought presenting this phenomenon from a general perspective , revealing that temporal autocorrelation functions in the bak - sneppen model display aging behavior similar to glassy systems@xcite . to our knowledge those papers give , up to now , the more relevant contribution to the relation between soc and aging . up to our knowledge , a detailed study of individual aging from the viewpoint of self - organized critical systems has not yet been performed . it is well known that biological aging manifests itself in an individual as the slowing down of many processes as , _ e.g_. , slower growing of tumors and manifestation of senescence in the slowing down of reflex behavior . due to its characteristics , relaxation phenomena in complex systems are good candidates to the study of aging . about 150 years ago , weber and gauss carried out a simple experiment demonstrating that relaxation in complex systems is not exponential . investigating the contraction of a silk thread they found that it does not contract so quickly but it relaxes slowly following a power law @xmath1 . this behavior is not particular of mechanical relaxation , it has been also observed in experiments of magnetic relaxation in spin glasses and high critical temperature superconductors , transient current measurements in amorphous semiconductors , dielectric relaxation , and more @xcite . in relaxation dynamics , aging means that the properties of a system depends on its age @xcite . for instance , consider a glass quenched at time @xmath2 below its glass transition temperature under an external stress . at time @xmath3 the stress is released . if the system were near equilibrium , then its response measured at certain time @xmath4 will be a function of the difference @xmath5 and , therefore , normalized responses taken at different initial times @xmath6 will collapse in a single curve . however , this is not the behavior in glass materials , and , as was pointed above , in live beings . aging is a consequence of the nonequilibrium dynamics and may be considered as a characteristic of the dynamics of complex systems far from equilibrium . in the following sections we demonstrate that these manifestations of individual aging are present in sandpile avalanches modeled with a bethe lattice , particularly , the process of relaxation . contrary to @xcite , we do not consider deterministic toppling ( like that present in the bak - sneppen model ) but a probabilistic one , and avalanches are not regarded as infinite , as is @xcite , but the finite size of the system ( sandpile ) is explicitly included . sandpiles seem to be simplest systems which lead to complex behavior and are a paradigm for the study of all phenomena manifested in complex systems out of equilibrium , like relaxation under external perturbations . they have been taken as the paradigm of self - organization since the introduction of these ideas by bak , tang , and wiesenfeld @xcite . the evolution of a sandpile under an external perturbation has been extensively studied experimentally @xcite , theoretically @xcite , and by computer simulations @xcite . in general , those works analyze the sandpile dynamics close to the critical state , i.e. , close to the critical angle . if a sandpile is subjected to an external perturbation , like , _ e.g. _ , small amplitude vibrations , large amplitude grain motions are rare events but from time to time , a grain may jump from its quasiequilibrium position . if a surface grain jumps , it will fall through the slope of the pile until it collides with another sand grains down the slope . after this collision the initial grain may be trapped with those grains or some of them may fall through the slope . if the last possibility happens then each grain surviving the collision will fall through the slope as the initial grain did . this image of an avalanche as an initial object that consecutively drags another resembles a branching process for which the bethe lattice representation seems to be natural @xcite . the representation of the avalanche dynamics as a bethe lattice is a mean field approach to the problem . it neglects correlations between branches . notwithstanding , if the pile angle is below is critical value the avalanches will be rare events and , in case of occurrence , will be very sparse . thus , such an approximation will be acceptable for analyzing the long time relaxation of the pile angle below its critical value , which is the subject of this paper . away from the critical point the mean field approximation works quite well . let us represent the avalanche as a cascade in the bethe lattice as follows . firstly , we start with a single node , which could represent in this case a grain . in a further step @xmath7 will emerge with probability @xmath8 , depending on the pile angle @xmath9 . this operation of generation of @xmath7 identical particles starting from one is repeated in the next step to each node of the new group , and so on . if the percolation process overcomes a given length ( that of the border of the pile ) those nodes beyond the limit constitute the avalanche . if it does not , there would be no avalanche since the cascade was stopped before reaching the base of the pile ( frustrated avalanche ) . by avalanche size we take the number of nodes , of the corresponding bethe lattice , in the last step . after an avalanche the sandpile _ autoorganizes itself _ with the new number of grains ( i.e. , a new slope is calculated with the remaining grains ) . the occurrence of avalanches will carry as consequence a decrease in the number of grains in the pile and , therefore , a decrease of @xmath10 . each time an avalanche occurs , the occurrence of a new avalanche is less probable . thus , to characterize this feedback mechanism a relation between @xmath11 and the number of grains in the pile @xmath12 is needed . the drag probability @xmath8 is a function of the slope . its value is determined by the competition of two contrary forces : the gravity , which conspires against the stability of the slope , and the friction which favors the slope stability . since the slope forms an angle @xmath9 with the horizontal plane the component of the gravity force in the slope direction will be larger with the increase of this angle , varying from zero to a maximum value when @xmath9 goes from zero to @xmath13 . therefore it is plausible to assume that the contribution of the gravity and , therefore , the tendency of falling down the slope is proportional to @xmath14 . on the contrary , the resistance to this tendency given by the static friction decreases with decreasing the pile angle , varying from a maximum value to zero when @xmath9 goes from zero to @xmath13 . hence , it is also plausible to assume that the resistance to the falling down is proportional to @xmath15 . thus the slope dependence of @xmath16 can be expressed through the ratio of both tendencies @xmath17 . based on this hypothesis we may propose the exponential relation @xmath18 where @xmath19 is a parameter determined by the gravitational field , the friction , and vibration intensity . notice that @xmath20 and @xmath21 . this selection for @xmath8 can not be regarded as a sophisticated trick or something imposed _ ad hoc _ to the model in order to obtain the desired results , since it can be verified that _ any _ function satisfying the very general conditions already stated is adequate to our model . incidentally , it must be said that this reveals the robustness of the model . on the other hand , the number of grains in a pile with slope angle @xmath9 is given by @xmath22 where @xmath23 is a geometrical factor and @xmath24 is the ration between a characteristic size of the pile base @xmath25 and sand grain @xmath26 . combining equations ( [ eq:1 ] ) and ( [ eq:2 ] ) it is obtained @xmath27 where @xmath28 . thus equation [ eq:1 ] relates the dragging probability with the number of grains in the pile . the parameter @xmath29 should not depend on the size of the system . it must be a function of parameters describing the vibrations , amplitude @xmath30 and frequency @xmath31 , and of the gravitational field acceleration @xmath32 . the only nondimensional combination of these magnitudes is given by the ratio between the vibration acceleration @xmath33 and @xmath32 and , therefore , @xmath34 . this conclusion , obtained from dimensional analysis , is corroborated by experiments on sandpiles under vibrations @xcite which show that the ratio @xmath35 is the relevant parameter . for a given value of @xmath29 and @xmath36 , there is a critical number of grains in the pile @xmath37 . this value can be found recalling that in the bethe lattice the critical value of @xmath16 for percolation exists is @xmath38 , resulting @xmath39 in a static sandpile ( i.e. , no vibrations ) this value corresponds with the number of grains in the pile at the angle of repose @xcite . another magnitude of interest is the penetration length , the number of steps @xmath40 in the bethe lattice for an avalanche to take place . this magnitude must be proportional to the length of the pile slope and , therefore , should be given by the expression @xmath41 notice that the possible existence of a geometrical factor in equation [ eq:4a ] may be absorbed in @xmath36 , redefining @xmath23 in equation [ eq:2 ] . moreover , the relation between @xmath42 and @xmath12 may be easily obtained using equation [ eq:2 ] . equations [ eq:3 ] and [ eq:4a ] relate the parameters of the pile with those of its bethe lattice representation . they were obtained here in a different way than that proposed by zaperi _ et al _ @xcite . other dependencies between the dragging probability and the number of grains in the sandpile may be proposed . notwithstanding , as it is discussed below , the precise form of this functional dependence is not relevant . the numerical experiment of relaxation is performed as follows . we start with a certain number of @xmath12 . in a first step , we test if an avalanche takes place using the bethe lattice representation . if it does , then we simulate the process and then recalculate the value of @xmath43 by simply substracting the size of the avalanche to @xmath12 and using equation [ eq:3 ] . the size of the avalanche is the number of nodes generated in the bethe lattice which surpass the size of the pile whose measure is given by ( [ eq:4a ] ) . then this step is repeated again and again , whereas avalanche sizes and times are registered . if avalanches are considered as instantaneous the number of steps is a measure of time . this approximation is valid for low vibration intensities . in this case grain jumps which trigger avalanches are rare events and , therefore , the time between two successive grain jumps will be much larger than the duration of avalanches . to describe the behavior of our system , we define the relaxation function @xmath44 of the sandpile as : @xmath45 where we include the time dependence of the number of grains in the sandpile . @xmath6 is the waiting time , i.e. , the instant at which we begin the counting of the number of grains since the start of the relaxation . ( avalanche ) , as in @xcite . the existence of aging in our model is illustrated in figure 1 . we have plotted the normalized relaxation @xmath46 at different ages ( steps ) of the system ( simulation ) by taking as initial time @xmath47 for @xmath48 and @xmath49 . as it can be seen the relaxation becomes slower with increasing the age of the system , in agreement with experiments in structural and spin glasses @xcite ( and , unfortunately , with human life ) . the system exhibits a delay in the relaxation , after which the relaxation function decreases as a power law ( long time tail ) with an exponent @xmath50 thus , since the angle relaxation becomes slower with the age of the pile then it will never reach an equilibrium angle and , therefore , properties like translational invariance and the fluctuation dissipation theorem do not hold @xcite . associated with these slow relaxation dynamics and aging phenomena we expect to observe a wide distribution of time between avalanches . as time increases the time between two consecutive avalanches @xmath51 becomes larger , because after an avalanche the occurrence of a new avalanche becomes smaller . therefore , it is expected that the mean time between avalanches diverges as @xmath52 . hence , the distribution of time between avalanches @xmath53 should satisfy the asymptotic behavior for large @xmath54 @xmath55 with @xmath56 . this hypothesis is confirmed in our simulations . figure 2 shows the distribution of time between avalanches for @xmath57 and @xmath49 . it is approximately constant for small values and then it decays following a power law in more than two decades . the plateau at small values of @xmath51 is associated to the rapid decay observed at short times while the tail for large @xmath51 should be related to the long time tail . here we observe that @xmath58 thus , there should be some connection between the distribution of time between avalanches and the long time relaxation . finally we want to mention that these simulations were also carried out assuming other functional dependences between the dragging probability and the number of grains in the pile @xmath43 . in all cases the results where qualitatively similar to those presented here using equation [ eq:3 ] , reflecting that the precise dependence is not relevant . a sandpile model for aging was presented revealing that this phenomenon is manifested in relaxation of sandpile avalanches . biological systems and individuals show similar behavior . phenomena related to some kind of relaxation in live beings deserve more quantitative research . the introduction of a bethe lattice representation for the avalanches and a feedback mechanism describes quite well the principal features of the relaxation in sandpiles under low intensity vibrations . the proposed representation leads to long time tails relaxation , aging and lvy ( fractal ) distributions of time constants , which are characteristic properties of the dynamics of complex systems out of equilibrium . this work was partially supported by the _ alma mater _ prize , given by the university of havana , and the direccin general de investigacin cientfica y tcnica dgicyt , spain , ministerio de educacin y cultura .

aging in complex systems is studied @xmath0 the sandpile model . relaxation of avalanches in sandpiles is observed to depend on the time elapsed since the beginning of the relaxation . lvy behavior is observed in the distribution of characteristic times . in this way , aging and self - organized criticality appear to be closely related .

You are an expert at summarizing long articles. Proceed to summarize the following text: graphical models , such as factor graphs and normal factor graphs @xcite @xcite , can provide a concise description of the probabilistic assumption of an inference problem and they are indispensable for analyzing message passing inference algorithms such as bp ( belief propagation ) . for example , the relationship between `` codes and graphs '' is one of key concepts of modern coding theory . al - bashabsheh and mao @xcite recently shed a new light to normal factor graphs . they clearly showed that _ holographic transformations _ to normal factor graphs are versatile tools for deriving non - trivial identities on the partition function of a normal factor graph @xcite . a holographic transformation is a local graphical transformation that preserves the partition function . it should be remarked that the holographic transformation has been used in many research fields . the prominent example is the class of holographic algorithms invented by valiant @xcite . he showed that several combinatorial enumeration problems defined on planer graphs can be transformed into perfect matching problems via appropriate holographic transformations . such a planar perfect matching problem is solvable in polynomial time . another example is duality theorems @xcite @xcite @xcite for codes defined on graphs . the main contribution of this paper is a non - trivial expression , that is called _ dual expression _ , of the posterior values for a non - adaptive group testing problem . the derivation is based on a holographic transformation to the normal factor graph representing a group testing inference problem . the derivation process has similarities to the proof of macwilliams identity @xcite and a bitwise map decoding algorithm by hartmann and rudolph @xcite for binary linear codes . however , in our case , we can not rely on the standard fourier ( _ i.e. , _ hadamard ) transformation because we need to treat or constraints instead of even parity constraints . a local linear transformation matched to or constraints plays a key role for the following discussion . the research of group testing started from the celebrated work by dorfman @xcite and has been extensively studied @xcite . we here suppose the following simplest setting for a non - adaptive group testing . assume that we have @xmath0-_objects _ and some groups of these objects . each object can take value 1 ( positive ) or 0 ( negative ) according to the prior probability for each object . test _ can be applied for each predetermined group . the result of a test is positive if the group contains a positive object ; otherwise the test result becomes negative . our inference problem is to evaluate the posterior probabilities for objects from @xmath1-disjunctive test results and from our knowledge on the prior probabilities . development of fast inference algorithms evaluating posterior probabilities ( or their estimates ) is an active area of research ; for example , see approximate inference algorithms based on bp @xcite @xcite . let @xmath5)$ ] be a binary ( zero or one ) independent random variable representing the state of the @xmath6-th object ( _ i.e. , _ negative or positive ) . the notation @xmath7 $ ] represents consecutive integers from @xmath8 to @xmath9 . the vector of the random variables @xmath10 is thus distributed according to the joint distribution : @xmath11 where @xmath12 . we suppose that an inference algorithm perfectly knows these prior probabilities @xmath13 . assume that an undirected bipartite graph @xmath14 , called a _ pooling graph _ , is given where @xmath15 and @xmath16 are sets of vertices , and @xmath17 is the set of edges connecting a vertex in @xmath18 and a vertex in @xmath19 , namely @xmath20 . the set @xmath21 is defined by @xmath22 \mid ( v^{(2)}_i , v^{(1)}_{j } ) \in e \}\ ] ] for @xmath23 $ ] . the boolean function @xmath24 is just the logical or function with @xmath25-inputs defined as @xmath26 , x_k=1 ] .\ ] ] the indicator function @xmath27 $ ] takes the value one if the condition is true ; otherwise it takes the value zero . a binary random variable @xmath28)$ ] representing a test result is defined by @xmath29 , $ ] where the notation @xmath30 represents a sequence of variables @xmath31 . the vector composed from @xmath28)$ ] is denoted by @xmath32 . it is evident that there is one - to - one correspondence between @xmath33 and @xmath34 and also between @xmath35 and @xmath36 . the index set @xmath21 represents a group corresponding to the @xmath6-th test result . assume that we observed @xmath37 as a realization of @xmath38 . our goal is to evaluate the log posterior probability ratio defined by @xmath39.\ ] ] the probability @xmath40 is the posterior probability on @xmath41-th object . from @xmath42 , we can obtain an estimate vector @xmath43 defined by @xmath44 ( \ell \in [ 1,n ] ) $ ] where this estimation rule can be seen as the bitwise map estimation rule . it may be reasonable to consider the bitwise map estimation for this group testing problem because bitwise map estimation minimizes the bitwise estimation error probability . by using bayes theorem , the posterior probability can be rewritten as @xmath45,\ ] ] where @xmath46 is just a normalization constant and @xmath47 . as a simplified notation , if the domain of the variable is missing in a summation , all the possible values in the domain is taken to evaluate the sum . as in many similar bitwise map estimation problems , naive evaluation according to ( [ exponential ] ) requires exponential time with the number of variables @xmath0 to marginalize all the variables @xmath48 ; namely computation time is @xmath2 if the maximum size of @xmath21 is bounded by a constant . this prohibitive time complexity is the high burden to exploit the bitwise map estimation on this problem . although it is still exponential time , the exponent of computation time can be greatly reduced if we are aware of the following simple fact . if @xmath49 , then we have @xmath50 for any @xmath51 . proof : if @xmath49 , then @xmath52 should be 0 for any @xmath51 because of the relation @xmath53 . this means that @xmath54 is exactly one . in other words , the lemma states that all the objects in the group @xmath21 have the value zero only if @xmath6-th test result @xmath55 is zero . this trivial but useful lemma can significantly reduce the problem size if the number of negative objects are small . therefore , it might be better to redefine the reduced size problem for a given observation vector @xmath56 as follows . let @xmath57 be the induced subgraph of @xmath58 where the vertices of @xmath59 and @xmath60 are given by @xmath61 : t_i = 0}\ { v_k^{(1 ) } \in v_1 \mid k \in \sigma(i ) \ } \right ) , \\ v_2^ * & { \stackrel{\triangle}{= } } & \{v^{(2)}_i \in v_2 \mid t_i = 1 \}.\end{aligned}\ ] ] in other words , we can exclude the groups whose test result is zero in @xmath19 and its incident nodes in @xmath18 for evaluating the posterior probabilities . for the following analysis , it would be convenient to rename the vertices in @xmath59 and @xmath60 as @xmath62 and @xmath63 . the random variable corresponding to @xmath64 and @xmath65 are denoted by @xmath66)$ ] and @xmath67)$ ] , respectively . figure [ ex1 ] illustrates an example of a pair @xmath58 and @xmath68 . the original pooling graph is depicted in fig . [ ex1](a ) . in this case , we have the test result @xmath69 which defines the induced subgraph @xmath68 illustrated in fig . [ ex1](b ) . and @xmath68 ] as in the cases of @xmath71 , we introduce similar notation such as @xmath72 , @xmath73 . in this problem setting , our goal can be recast as the evaluation of the log posterior probability ratio @xmath74 $ ] for given @xmath68 . the symbol @xmath75 represents the vector that all its components are ones . in this subsection , we will rewrite the posterior probabilities in sum - product form which is the foundation of the following discussion . as in the derivation of ( [ exponential ] ) , the posterior probabilities @xmath76 can be expressed as @xmath77 \\ \nonumber & = & \frac{1}{z ' } \sum_{x_1,\ldots , x_n } \left(\prod_{i=1}^m or(x_k|_{k \in \alpha(i ) } ) \right ) \\ & \times & \left(\prod_{j=1}^n p_{x_j}(x_j ) \right ) \bbb i[x_\ell = b ] \end{aligned}\ ] ] for @xmath78 , @xmath79 $ ] and @xmath80 . the symbol @xmath81 represents a normalization constant which is independent of the value of @xmath9 . note that the two sets , @xmath82)$ ] and @xmath83)$ ] , are defined by @xmath84 \mid ( z_i^{(2 ) } , z_j^{(1 ) } ) \in e^ * \ } , \\ \beta(j ) & { \stackrel{\triangle}{= } } & \{i \in [ 1,m ] \mid ( z_i^{(2 ) } , z_j^{(1 ) } ) \in e^ * \ } , \end{aligned}\ ] ] respectively . for the following argument , it is useful to define the quantity @xmath85)$ ] by @xmath86,\end{aligned}\ ] ] that is called a _ posterior value_. by using these posterior values , the log posterior probability ratio @xmath87 can be evaluated by taking the ratio between the posterior values for zero and one : @xmath88 we will further decompose @xmath89 in ( [ posterior ] ) into a finer sum - product form which will be more suitable for a normal factor graph representation to be described in the next section . as building blocks of the finer representation , we here introduce @xmath90 , and @xmath91 functions as follows . the boolean equality function @xmath92 with @xmath25-inputs is defined by @xmath93.\ ] ] the weight function @xmath94 , \ell \in [ 1,n ] , b \in \{0,1\})$ ] is given by @xmath95 , & j = \ell . \end{array } \right.\ ] ] by using these set of functions , @xmath89 can be represented as @xmath96 where @xmath97 is the set of new binary variables defined as @xmath98 , j \in [ 1,n ] , ( z_i^{(2)},z_j^{(1 ) } ) \in e^ * \}\ ] ] and @xmath99 are also binary variables . the main contribution of this paper is the next theorem which gives another expression of the posterior value @xmath89 . it will be the foundation of a novel map algorithm described later . [ dualexpression ] the posterior value @xmath89 can be expressed as @xmath100 where @xmath79 , b \in \{0,1\}$ ] and variables @xmath101 are binary variables . the notation @xmath102 represents the cardinality of @xmath103 . the function @xmath104 , b \in \{0,1\})$ ] is defined by @xmath105 if @xmath106 . if @xmath107 , then @xmath108 is defined as @xmath109,\hfill \mbox{otherwise}. \end{array } \right.\ ] ] in the expression of the posterior value ( [ finerrep ] ) , the indicator variables @xmath99 corresponding to @xmath90 nodes take all the possible binary @xmath110-tuples in the summation . on the other hand , in the expression of posterior values in theorem [ dualexpression ] , the indicator variables @xmath101 appeared in the summation correspond to @xmath111 nodes . we therefore call this expression the dual expression . the proof of this theorem heavily relies on a holographic transformation to the normal factor graph of the posterior value @xmath89 . in the next section , we will discuss an appropriate holographic transformation to derive the dual expression . the normal factor graph ( nfg ) is a graphical representation of a function composed from a product of many functions . the precise definition of the nfg can be found in @xcite @xcite but we here introduce a simplified definition enough for this paper . the nfg of a sum - product form is an undirected graph with vertices corresponding to factor functions and edges corresponding to the variables . the nfg of the posterior value ( [ finerrep ] ) , denoted by @xmath112 , is defined as follows . for each factor of ( [ finerrep ] ) such as @xmath113 a _ factor node _ is associated . in the following , we do not strictly distinguish a factor function from the corresponding factor node if there are no fear of confusion . in a similar way , the variables in ( [ finerrep ] ) such as @xmath114 and @xmath99 are assigned to edges . the rule for the edge connections is simple ; if and only if a variable @xmath115 ( resp . @xmath116 ) is an argument of a factor function @xmath117 , the edge @xmath115 ( resp . @xmath116 ) is connected to the factor node @xmath117 . in other words , if and only if @xmath117 depends on @xmath115 ( resp . @xmath116 ) , the factor node @xmath117 connects to the edge @xmath115 ( resp . @xmath116 ) . according to the semantics for nfgs introduced by al - bashabsheh and mao @xcite , all the edge variables are assumed to be marginalized . figure [ nfg1 ] illustrates the nfg for the posterior value in ( [ finerrep ] ) . we will refer the factor nodes corresponding to the or ( resp . eq ) function as _ or ( resp . eq ) nodes_. in our context , an nfg corresponds to a posterior value in sum - product form . a holographic transformation is a transformation of an nfg that preserves the marginal generating function . in the following discussion , we will insert a pair of _ dual factor nodes _ into each edge connecting an or node and an eq node ( _ i.e. , _ @xmath115 ) . the pair of factor nodes is carefully designed not to change the posterior value . figure [ nfg3 ] is our blueprint that shows how we will proceed in this subsection . in fig.[nfg3 ] ( left ) , for each edge @xmath115 , a pair of dual nodes , @xmath118 and @xmath119 , is inserted . these function nodes @xmath118 and @xmath119 are designed to satisfy the duality condition described later . due to the duality condition on @xmath118 and @xmath119 , the posterior value of this transformed nfg is the same as that of the original nfg . by grouping an eq node and @xmath119 nodes connected to it , a new factor node @xmath120 is created ( fig.[nfg3 ] ( right ) ) . in a similar way , combining an or node and its incident @xmath118-nodes , we obtain a new factor nodes that is called a _ skewed eq ( seq ) function_. it should be emphasized that seq function has almost the same truth table as that of eq function . this fact is important to reduce the computational complexity to evaluate the posterior values . in the following subsections , we will follow this blueprint and present details of dual factor nodes and new factor nodes . these will be the basis of the proof of theorem [ dualexpression ] . let us define @xmath121 by @xmath122 and @xmath123 by @xmath124 it is trivial to check that these two functions @xmath118 and @xmath119 satisfies the duality condition @xmath125 this condition guarantees that the posterior values of the nfg are unchanged if we inserted these function nodes into an edge corresponding to @xmath115 in fig.[nfg1 ] @xcite . this is because a pair of function nodes is equivalent to an eq function which does not affect the consequence of the marginalization . the next lemma tells that a sum - product form of an or function and @xmath118 functions produces an seq function . [ orlemma ] for any @xmath126 , the following equality @xmath127 holds . proof : from the definition of @xmath118 , the right - hand side of ( [ thetaeq ] ) can be evaluated as @xmath128_{y_1,\ldots , y_r } \\ \label{ttt } & = & \left [ ( 1,0)^{\otimes r}- ( 0,-1)^{\otimes r } \right]_{y_1,\ldots , y_r}. \end{aligned}\ ] ] note that @xmath129 represents the tensor power ( kronecker power ) of a matrix @xmath130 . the row vector @xmath131 represents the truth table of @xmath111 function as a row vector . the notation @xmath132_{a_1,\ldots , a_r}$ ] denotes the @xmath9-th component of row ( or column ) vector @xmath133 where @xmath134 . if @xmath25 is odd , then the right - hand side of ( [ ttt ] ) equals @xmath135_{y_1,\ldots , y_r}. $ ] in this case , the claim of the lemma holds because @xmath135_{y_1,\ldots , y_r } $ ] is the truth table of @xmath90 function . if @xmath25 is even , the right - hand side of ( [ ttt ] ) becomes @xmath136_{y_1,\ldots , y_r } , $ ] which is equivalent to the right - hand side of ( [ thetaeq ] ) . it can be seen that only simple tensor calculations are required to show the main claim of this lemma . the right - hand side of ( [ thetaeq ] ) , @xmath137 , is referred to as the skewed eq function that is denoted by @xmath138 . the next lemma plays an crucial role for grouping factor nodes around an eq node . [ eqlemma ] the function @xmath139 can be expressed as @xmath140 for any @xmath141 , \ell \in [ 1,n ] , b \in \{0,1\}$ ] . proof : the truth table of @xmath90 function of @xmath142-inputs is given by the column vector @xmath143 from the definition of the function @xmath119 , we have the following tensor expression : @xmath144_{u , y_1,\ldots , y_r}. \end{aligned}\ ] ] we here define @xmath145 as @xmath146 from the definition of @xmath147 , if @xmath106 , we have @xmath148 otherwise ( _ i.e. , _ @xmath107 ) , the equality @xmath149 , \mbox{otherwise}. \end{array } \right.\ ] ] is obtained . it is clear that @xmath150 holds . we are now ready to prove theorem [ dualexpression ] . as described in subsection [ holographic ] , the nfg illustrated in fig . [ nfg3 ] ( left ) corresponds to the original posterior value due to the duality condition on @xmath118 and @xmath119 . by lemmas [ orlemma ] and [ eqlemma ] , the posterior value @xmath89 can be rewritten as @xmath151 where @xmath152 is a set of variables defined by @xmath153 , j \in [ 1,n ] , ( z_i^{(2)},z_j^{(1 ) } ) \in e^ * \}\ ] ] and @xmath154 note that the expression ( [ otherrep ] ) follows the new factor node grouping described in fig . [ nfg3 ] ( right ) . in the non - vanishing summand of ( [ otherrep ] ) , eq function enforces that @xmath155 takes the same value for any @xmath6 . in other words , all the edges emitted from a skewed eq factor node should take the same value if the product in ( [ otherrep ] ) is not zero . this observation leads to the claim of theorem [ dualexpression ] . the next example presents how this dual expression works . assume that @xmath156 and the pooling graph defined by @xmath157 ( see . [ g1fig ] ) . the prior probabilities of objects are given as @xmath158)$ ] . table [ g1 ] shows the prior probabilities @xmath159 and the indicator values @xmath160 \bbb i[or(x_3,x_3)=1]$ ] . we here focus on the case where @xmath161 . from table [ g1 ] , it is straightforward to evaluate the posterior values as @xmath162 .prior probabilities and indicator value [ cols="^,^,^,^,^",options="header " , ] from engineering point of view , the primal advantage of theorem [ dualexpression ] is that it provides an efficient bitwise map estimation algorithm . if the degrees of a node in a pooling graph is bounded by a constant , exhaustive evaluation of posterior values based on theorem [ dualexpression ] requires @xmath163-time ( a simple implementation trick can reduce this time complexity down to @xmath164 ) . if @xmath165 , this bitwise map algorithm achieves exponential speedup compared with a naive bitwise map algorithm based on ( [ posterior ] ) with time complexity @xmath166 . in a typical use of a non - adaptive group testing , the number of tests is much smaller than the number of objects ; _ i.e. _ , @xmath4 . this implies that a situation satisfying @xmath165 is fairly common . the authors would like to express their sincere appreciation to ryuhei mori for directing our interest towards holographic transformation . this work was supported by jsps grant - in - aid for scientific research ( b ) grant number 25289114 . d. sejdinovic and o. johnson , `` note on noisy group testing : asymptotic bounds and belief propagation reconstruction , '' _ forty - eight annual allerton conference on communication , control , and computing _ , pp . 998 - 1003 , 2010 .

in this paper , an exact bitwise map ( maximum a posteriori ) estimation algorithm for group testing problems is presented . we assume a simplest non - adaptive group testing scenario including @xmath0-objects with binary status and @xmath1-disjunctive tests . if a group contains a positive object , the test result for the group is assumed to be one ; otherwise , the test result becomes zero . our inference problem is to evaluate the posterior probabilities of the objects from the observation of @xmath1-test results and from our knowledge on the prior probabilities for objects . if the size of each group is bounded by a constant , a naive inference algorithm requires @xmath2-time for computing the posterior probabilities for objects . our algorithm runs with @xmath3-time , which is exponentially faster than the naive inference algorithm under a common situation with @xmath4 . the heart of the algorithm is the _ dual expression _ of the posterior values . the derivation of the dual expression can be naturally described based on a holographic transformation to the normal factor graph ( nfg ) representing the inference problem . in order to handle or constraints in the nfg , we introduce a novel holographic transformation that converts an or function to a function similar to an equal function .

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Proceed to summarize the following text: the k1v star hd 189733 hosts the transiting gas giant planet hd 189733b , at an orbital distance of 0.03 au @xcite . in some ways this system is nearly optimally suited for study with transmission spectroscopy . its short planetary orbital period ( 2.2 days ) , its great stellar brightness ( k = 5.5 mag ) , and its large planet - to - star radius ratio ( 0.15 ) combine for convenient observations of high sensitivity and high contrast . on the other hand , a significant negative is its intrinsic photometric variability , @xmath4% peak to valley , due to rotational modulation of star spots @xcite . as such , it is photometrically more variable than @xmath395% of stars in the kepler sample ( @xcite ; @xcite ; cf . also figure 5 of @xcite ) . as this paper will discuss , star spots complicate the interpretation of transmission spectra , including those of hd 189733b . transmission spectra provide a powerful means to detect molecular species in exoplanetary atmospheres due to the long path length of the stellar light through the planetary atmosphere ( e.g. @xcite ; @xcite ; @xcite ) . the differential transit depth of a molecular feature is directly proportional to the scale height of the atmosphere . for hot jupiters , with high temperatures and low mean molecular weight ( i.e. h@xmath5-rich ) , the differential transit depth in specific molecular bands is large enough to be observable with existing instruments . in particular , water vapor has bands with large cross sections observable in the near infrared . upper limits or marginal detections have been reported of water in transmission for exoplanets hd 209458b @xcite , xo-1b ( @xcite ; @xcite ) , and xo-2b @xcite . from _ irac transit photometry at 3.6 @xmath6 m , 5.8 @xmath6 m , and 8 @xmath6 m , @xcite reported evidence of water vapor in the atmosphere of hd 189733b ( but cf . the near - infrared transmission spectrum of hd 189733b in particular has garnered considerable attention , with the same nicmos data being re - analyzed multiple times , with @xcite being the latest . the latter meta - analysis of the transmission spectra of @xcite , @xcite , and @xcite , predicted the amplitude of hd 189733b s 1.4 water vapor feature to be 300 ppm to 400 ppm . other meta - analyses have been performed also ( e.g. @xcite ) . similarly , from ground - based _ irtf _ near - infrared transmission spectra of hd 189733b , @xcite reported water vapor features including ones at 1.15 and 1.4 with amplitudes at least twice those reported here . significant improvement has come with the _ hst _ instrument wfc3 . recently , @xcite reported wfc3 observations of transits of wasp-12b , wasp-17b , and wasp-19b . @xcite reported wfc3 observations of wasp-19b alone . made under _ program 12181 ( p.i . deming ) , those observations preceded the development of spatial scanning and like prior observations , produced marginal detections or upper limits on the 1.4 @xmath6 m water vapor feature . similar wfc3 spectra have been reported for wasp-12b observed in staring mode under _ hst _ programs 12230 ( p.i . swain ; @xcite ) and 12473 ( p.i . sing ; @xcite ) . coupling wfc3 to the new technique of spatially scanning the hubble space telescope ( _ hst _ ) has enabled detections of water vapor in transmission spectra for hd 209458b , xo-1b , and hat - p-1b under _ hst _ program 12181 ( p. i. deming ) and now hd 189733b ( program 12881 ; p. i. mccullough ) . for some planets , clouds and/or haze can greatly affect our view of water vapor even if it exists in their atmospheres . such is the case apparently for the smaller exoplanets gj 1214b @xcite and gj 436b @xcite that both exhibit featureless , flat spectra . planets with no atmospheres , or insufficiently large scale heights of their atmospheres , would also present undetectably weak diagnostic spectral features in their atmospheres @xcite . even if molecular features are observed in a clear atmosphere , a degeneracy exists between the composition and the characteristic temperature of the region probed by the data , since the scale height is proportional to temperature . high resolution ( r@xmath7 ) , high sensitivity data can place joint constraints on the composition and temperature simultaneously ( e.g. @xcite ) . due to their geometry of light rays grazing the terminator region @xcite , transmission spectra are more sensitive to clouds / haze and/or small atmospheric scale heights than day - side emission spectra , which have their own limitations . @xcite attributed to water a 10 absorption feature in the emission spectrum of hd 189733b , and recently , @xcite have done so at 3.2 using phase curves rather than observations at secondary eclipse . such measurements indicate potential for comparison of spectra obtained by the complimentary techniques of transits , eclipses , and phase curves ( cf . @xcite and @xcite ) . the spectral observations reported here were obtained in spatial scanning mode @xcite with the _ hst _ in order to rapidly collect a large number of photons from hd 189733 without it also saturating the detector . in scanning mode , the _ hst _ scans across a small segment of a great circle on the sky while the detector is integrating photo - generated charge from objects as they trail across the detector . since the first such observation of a transiting exoplanet , gj 1214b on april 18 , 2011 ( program 12325 , p.i . mackenty ; @xcite ) , spatial scanning has become the nominal method of spectroscopic observation of bright exoplanet host stars . @xcite scanned one transit each of hd 209458b and xo-1b ; @xcite and @xcite each scanned multiple transits of gj 1214b and gj 436b respectively . @xcite scanned one transit of hat - p-1b . _ hst _ program 12181 ( p.i . deming ) recorded one transit of hd 189733b using spatial scans . however , the earth occulted the telescope s view of the star during the main part of the transit event between contacts 2 and 3 . @xcite observed transits of hd 189733b in the near - infrared with wfc3 , under program 11740 ( p.i . pont ) , although they did so prior to the advent of spatial scanning on _ hst_. in those observations , the very bright star saturated the detector even before each exposure began , because the detector does not have a mechanical shutter . they salvaged light curves from the insensitive blue and red ends of the spectra . based upon comparison of those measurements at @xmath31.1 and @xmath31.6 with measures by @xcite between 0.5 and 1.0 , gibson et al . tentatively suggested that haze dominates the transmission spectrum of hd 189733b both in the visible and the near - infrared . however , they concluded that future observations of the sort reported here would provide the evidence to resolve the issue of whether exoplanetary hazes or clouds obscure spectral features in the near - infrared . this paper is organized as follows . in section [ sec : obs ] we describe the observations , which we analyze in section [ sec : analysis ] . results are in section [ sec : results ] . we separately discuss star spots ( [ sec : spots ] ) and the exoplanet s atmospheric spectrum ( [ sec : planet ] ) , and the combination of the two ( [ sec : combined ] ) . we draw conclusions and summarize in section [ sec : summary ] . _ hst _ program 12881 ( mccullough , p.i . ) was allocated five hst orbits to observe the transit of hd 189733b and five orbits to observe its eclipse . the eclipse observations are reported by @xcite . in both cases , the event ( transit or eclipse ) occurs in the fourth of the five hst orbits . by design and common practice , the data from the first hst orbit is considered unreliable , as the telescope settles to its new thermal environment associated with a new orientation to the sun after slewing to the target , and charge traps in the hgcdte detector equilibrate to the unusual voltage associated with strong illumination . discarding the data from the first orbit would leave two reliable hst orbits pre - transit , one in - transit , and one post - transit . ) , the n. c. one discarded data from the entire first orbit whereas the d. d. one discarded only the first eight scans of the first orbit . ] we observed hd 189733b in transit with hst wfc3 on june 5 , 2013 ( table [ tbl-1 ] ) . we used the g141 grism , to obtain slitless spectroscopy with spectral coverage from 1.1 to 1.7 at a resolution r = @xmath8130 and a dispersion of 4.7 nm pixel@xmath9 @xcite . in each hst orbit of the visit , we obtained a set of thirty - two 5.97-s exposures , each with the rapid sample sequence of seven samples , and each with a 512 pixel by 512 pixel subarray . as noted in the introduction , the spectra were obtained in spatial scanning mode . we interleaved scanning first forward and then reversing direction , obtaining sixteen forward - reverse pairs of exposures in each orbit . we had intended to locate the target on the detector such that its first - order spectrum would be entirely within a region read from a single amplifier ; however we miscalculated and the 1st order spectrum crosses an amplifier boundary ( at physical column 512 ) corresponding to @xmath10 . a similar miscalculation caused the spectrum to scan off the top of the subarray and hence some of the samples were lost . in the forward direction , samples 1 through 5 occur when hd 189733 s spectrum is on the subarray but during samples 6 , 7 , and 8 the spectrum is off the subarray and hence only the first four differences between samples are useful . in the reverse direction , samples 3 through 8 recorded hd 189733 s spectrum , and samples 1 , and 2 were lost ( figure [ fig1 ] ) . the loss of 5 differences between samples of 14 total implies a loss in potential , poisson - limited , signal - to - noise ratio of 20% . we also obtained a direct image as the first exposure of the visit ( figure [ fig1 ] ) ; the purpose of the direct image is to provide a high - resolution image to verify the location of the target with respect to any potentially contaminating stars in the same field of view . hd 189733 is a double star ; the secondary is 4.0 magnitudes fainter in h band and is 11.2 away from the primary at a pa of 246 @xcite . there are a few stars closer to hd 189733a in projection than hd 189733b , but they are much fainter ; in h band each is 1% to 2% of the brightness of hd 189733b , which itself is 2.5% of the brightness of hd 189733a . because the nearby stars are so faint , and because our analysis technique isolates the spectrum of the target star in space and references it in time by dividing the in - transit spectrum by the out - of - transit spectrum , we expect any contributions of nearby stars to the transmission spectrum of the planet hd 189733b to be entirely negligible . we needed to scan at @xmath32s@xmath9 to properly expose the detector and to avoid saturation . also we wanted samples often enough to avoid overlapping the two spectra of the primary star hd 189733a and the secondary star hd 189733b , 11.2 apart . combined , those requirements imply sampling every @xmath11 s , or faster . of multiaccum sequences currently available for wfc3 , only the rapid sequence sufficed . the observational data are interleaved in time ; 7 forward scan differences , each a 0.85-s exposure , followed by a 51 second gap as the scanning telescope reverses and the detector is prepared for the next exposure , followed by the 7 reverse scan - differences each one also 0.85-s long , and then a 51 second gap to return to the beginning of the pattern and to prepare the detector for the next forward - scan exposure . with two 6-s exposures every 114 s , the observational duty cycle would be 10.5% , although because only 9 of the 14 multiaccum differences were useful , the duty cycle was 6.7% . in total , we obtained @xmath12-s of exposure in transit , which is only 3% of the transit s duration . the latter 3% is approximately half of the 6.7% observational duty cycle , because the earth occulted the target during the second half of the transit . rather unsatisfactorily , a 3% duty cycle with the 2.4-m _ hst _ would be equivalent to observing at 100% duty cycle with a 0.4-m diameter telescope of the same end - to - end throughput . further improving wfc3 s duty cycle by any practical operational changes would be beneficial for observations of stars such as hd 189733b . two of us ( n.c . and d.d . ) analyzed the data nearly independently . as planned from the outset to maintain independence , no computer code was shared between the two efforts , but both analysts had read or written the relevant literature , particularly descriptions by @xcite @xcite , @xcite , @xcite , and @xcite . the two resulting exoplanetary transmission spectra were compared and found to be very similar . after discussions each analyst slightly revised his algorithms , which resulted in two final spectra . in this section , we described one analysis , that of n.c . , the other method has been described already in its application to hd 209458 and xo-1 @xcite . table [ tbl - lc ] lists the transit light curve of the n.c . analysis . we describe the differences between the two analyses at the end of this section and list the difference between the two resulting spectra in the last column of table [ tbl - spectra ] . the r.m.s . of the differences , 52 ppm , is 0.75 times the median poisson noise associated with the 4-column binning , indicating that the differences in the resulting binned spectra are statistically less significant than the physical limit imposed by poisson statistics . we begin the analysis with the set of intermediate multiaccum flexible image transport system ( ima.fits ) files , which are produced by the calwfc3 pipeline at stsci and have a number of corrections applied , including dark current subtraction and nonlinearity correction @xcite . as described in section 2 , we discard the data from the visit s first hst orbit . we sort the files into two sets , forward scans ( keyword postarg2 = 54.9 ) and reverse scans ( postarg2 = -2.0 ) , and analyze each separately until combining them near the end of our analysis . the separate analyses of forward- and reverse - scans are very similar , differing only in the parameters defining the useful readouts and the pixels of interest , both of which differ slightly due to the forward and reverse scans not overlapping exactly . from each 3-d multi - extension ima.fits file , we extract the difference images @xmath13 corresponding to the @xmath14th difference of consecutive nondestructive reads of the detector subarray . we divide the 2-d difference images by a 2-d flat field obtained through the f139 m filter . because the in - transit spectrum of hd 189733 will be divided by its out - of - transit spectrum to form the planetary transmission spectrum , i.e. because the flat field is self - calibrated , the choice of particular flat - field is unimportant . flat - fielding makes bad pixels and cosmic ray hits easier to identify . we perform such identification and accordingly correct the images by interpolation as described by @xcite . we define fixed regions on the 2-d images corresponding to the target s signal area and a corresponding sky area . we subtract the sky s value from each image . next we sum each column over the stellar region , yielding a 1-d spectrum of hd 189733 for each difference image . we average those into in - transit spectra ( in , from the fourth _ hst _ orbit ) and out - of - transit spectra ( out , from three other _ hst _ orbits ) . we form each of nine planetary transit spectra , ( out - in)/out , associated with the nine difference images ( four _ forward _ and five _ reverse _ ) , which we then average to form a single planetary spectrum , subtract the wavelength - integrated mean , and bin every four columns to produce the final , differential planetary spectrum ( column 2 of table [ tbl - spectra ] ) . we calibrated the wavelength as a function of detector column by comparing the ( unbinned ) wfc3 spectrum of hd 189733 to the spectrum of a k1 v comparison star @xcite.spex / irtf_spectral_library / data / k1v_hd10476.txt ] . we adjusted the comparison star s spectrum to match hd 189733 s spectrum by multiplying the comparison spectrum by the telescope and instrument throughput @xcite , transforming it from energy units to photo - electron units , smoothing it to 4.7 nm ( 1 column ) resolution , high - pass filtering it and the wfc3 spectrum to emphasize weak spectral features in each , and finally , match the two spectra by linearly transforming the comparison star spectrum with an offset and dispersion that together constitute our wavelength calibration , which we estimate to be accurate to better than 4.7 nm ( 1 column ) . additional details of this procedure are in @xcite . the position of the spectrum shifts slightly throughout each exposure in response to feedback from the fine guidance sensors . variations in the scan rate perpendicular to dispersion introduce apparent photometric variations due to each row having a slightly shorter ( longer ) exposure time because the telescope s scan rate is faster ( slower ) than nominal ( cf . figure 1 of @xcite ) . because such variations are achromatic , they do not affect the differential exoplanetary spectrum . we also examined the shifts in telescope pointing parallel to dispersion by methods similar to those described in @xcite . the peak - to - valley variation of the shifts throughout the visit of program 12881 was typical for _ hst _ @xcite , @xmath30.1 column ( @xmath30.014 ) , similar to that for xo-1 and much less than the @xmath31.0 column shift of hd 209458b , both observed in program 12181 ( cf . figure 4 of @xcite ) . adjusting the spectra of hd 189733 for its small ( @xmath15 column ) shifts made negligible difference in the final exoplanet spectrum compared to not making the adjustments . an important difference between the two analyses is their smoothing . the analysis of d. d. smoothed its spectrum with a gaussian of fhwm=4 columns whereas the other one , of n. c. , used a box - car smoothing of full width 4 columns . purely as a consequence of the choice of smoothing kernels , for an idealized , featureless , flat power spectrum of noise in the unbinned spectrum , the ratio of the noise of the resulting smoothed spectra should be @xmath16 . stated another way , in terms of noise power , the d. d. spectrum has an effective bin width of 6 columns , or 1.5 times the n. c. spectrum s 4-column bin width . the uncertainties in table [ tbl - spectra ] match this expectation . we subdivided the available wfc3 data into nine separate wavelength - integrated `` white light '' curves , corresponding to the nine differences between multiaccum samples in which hd 189733 was on the subarray . figure [ fig2 ] illustrates one example of the nine available . we removed trends in each light curve as follows , for each of the nine multiaccum differences separately . we fit a straight line by least - squares to the first points of the second , third , and fifth _ hst _ orbits , and divided the first point of each _ hst _ orbit ( including the in - transit fourth orbit ) by the fit . we repeated the procedure for the second point , and so on , for all thirty - two scans per hst orbit . table [ tbl - lc ] lists the nine light curves , both before and after de - trending . from the light curve of hd 189733b s transit we obtain the mean level of the transit depth @xmath17 , which is important for comparison with measurements at other wavelengths from other instruments . otherwise , however , we are not as interested in the light curve as the differential transit spectrum described in the next section . we adopted hd 189733b s ephemeris , orbital inclination , and normalized semi - major axis from @xcite . we fit the light curve with the procedure of @xcite , using quadratic limb darkening coefficients @xmath18 and @xmath19 , which we derived by fitting two parameters of a quadratic law to the three - parameter limb - darkening law adopted by @xcite for @xmath20 . the resulting mean transit depth , appropriate for the integrated wfc3 bandpass , is @xmath21 , where the estimated uncertainty ( 100 ppm ) is the standard deviation of the fits to the nine light curves . the observed value of @xmath17 is an upper bound in so far as it neglects unocculted star spots ( cf . [ sec : spots ] ) . the exoplanet transmission spectrum is the transit depth @xmath17 as a function of wavelength . in column 2 of table [ tbl - spectra ] we report the differential transmission spectrum as ( in - out)/out , with its mean value subtracted , where in is the mean in - transit spectrum ( from the fourth _ hst _ orbit ) and out is the mean out - of - transit spectrum ( from the other _ hst _ orbits , neglecting the first one ) . figure [ fig3 ] illustrates the spectrum extending from 1.13 to 1.64 in 28 bins . each bin corresponds to 19 nm or four detector columns . the simple ( in - out)/out estimate neglects limb darkening . our more complicated analysis @xcite involves subtracting a scaled template spectrum from each exposure s 1-d spectrum and then fitting a transit light curve to the residuals to derive the differential transit depth at each wavelength . in the latter method , limb darkening can be included or not , and we experimented with either choice . although the mean `` white light '' transit depth is sensitive to the limb - darkening correction , applying a wavelength - dependent limb - darkening correction , or not , made negligible difference to the differential transit depth . to emphasize the latter point , we computed one differential spectrum with no limb - darkening correction at all ( column 2 of table [ tbl - spectra ] ) , although we did model limb - darkening for the white - light transit depth in the previous section . we estimated the uncertainties in column 3 of table [ tbl - spectra ] as follows . we formed an individual 1-d stellar spectrum from each difference image . we form a template spectrum from data taken out - of - transit . we then scale and subtract the template from each of the individual spectra in order to minimize the r.m.s . of the resulting residuals . for each spectral channel , we group the residuals into two sets , one in - transit and one out - of - transit , divide by the mean flux , and evaluate their respective standard deviations @xmath22 and @xmath23 . we combine the latter in quadrature , accounting for the number of measurements in - transit or out - of - transit respectively , in order to estimate the uncertainties of each spectral channel , @xmath24 . in figure [ fig4 ] we bin the data from column 2 of table [ tbl - spectra ] into seven points , with a channel width of 75 nm ( 16 detector columns ) . with that smoothing , the local maxima of the transit depths in the 1.15 and 1.4 water vapor features occur at the first and fourth data points , with a local minimum , the third point , in between . from those , we estimate the amplitudes of the 1.15 and 1.4 features respectively are @xmath1 ppm and @xmath2 ppm greater than the local minimum at 1.3 . the stated uncertainties are the quadrature sum of the uncertainties of the two points in each case . the 1.4 feature in particular is detected at much greater statistical significance than @xmath25 , i.e. 200/47 , because the feature extends over two 75-nm wide points and the adjacent baseline extends over two or four points instead of just one . by averaging over four points , two inside the 1.4 feature and the two adjacent to it , the @xmath25 result becomes @xmath26 , commensurate with the feature s appearance in figures [ fig4 ] and [ fig5 ] . the peak - to - valley amplitude of the 1.4 feature in figure [ fig3 ] , which has spectral resolution of 30 nm fwhm in the model or 19 nm ( 4 columns ) in the binned data , is @xmath3400 ppm or @xmath3350 ppm , respectively . the peak - to - valley measurement of a noisy spectrum depends on the smoothing that reduces noise and hence reduces the height of the peak and the depth of the valley . the similarity of the model to the data in figure [ fig3 ] illustrates that the data are consistent with a clear - atmosphere of solar - composition , with a volume mixing ratio for water of @xmath27 and a pressure scale height commensurate with a temperature of 700 k. the @xmath27 mixing ratio has been used in prior models as well ( e.g. @xcite , @xcite , @xcite ) and is consistent with the broad range of values presented in figure 3 of @xcite , @xmath28 to @xmath29 . however , inspection of figure [ fig3 ] shows that slightly reducing our model s amplitude would improve its fit to the data . hence , either the volume mixing ratio or the temperature should be smaller than the assumed values . @xcite explore the fitting of these data in greater detail and conclude that the h@xmath5o volume mixing ratio is sub - solar . hd189733 is an active star ( boisse et al . 2009 ) , and transit observations exhibit ubiquitous star spot crossings ( pont et al . 2007 ) . our wfc3 observations have only partial coverage during transit because of occultation by the earth , and no spot crossings are obvious in our data . in any case , because hd 189733 is a spotted star , we must address two effects _ unocculted _ star spots may have on the transmission spectrum . first , unocculted star spots can make the radius of a transiting planet appear larger in the blue than in the red , because unocculted spots make the star darker and redder than it would appear otherwise . if in fact a star is _ darker _ away from the transit chord than a model assumes , that model will overestimate the size of the transiting planet . if in fact the star is also _ redder _ away from the transit chord than the model assumes , then the overestimate of the planet s size will be even greater in the blue than in the red . second , at @xmath30k , hd 189733 s photosphere is too hot for water to exist , but in very cool star spots , water vapor could exist . water vapor absorption in unocculted star spots will make the star spots darker in spectral features of water than in the continuum , and hence make the radius of the transiting planet appear larger in water - vapor features than it would in the continuum @xcite . quantitatively and specifically for hd 189733 , we show in this section that the first effect is likely to be significant but the second effect is not . @xcite analyze star spots on hd 189733 and their potential effects on the transmission spectrum . in their analysis , they show that star spots modulate the stellar flux primarily due to stellar rotation , and also due to the changing set of spots and their characteristics . however , latitudinal _ bands _ of star spots would not modulate the flux much , because as one spot rotates into view , another one rotates out of view . if hd 189733 has a large polar spot , or one or more latitudinal band(s ) of spots beyond the latitudes at which the transit chord passes , then those unocculted spots will contribute to the apparent rise in the `` planetary '' transmission spectrum seen in the visible with acs @xcite and in the ultraviolet with stis @xcite . @xcite were aware of such possibilities but chose to interpret that rise in the spectrum as evidence of rayleigh - scattering dust in the planet s upper atmosphere instead of spots on the star . in this paper we match all of the transit data assembled by @xcite and the wfc3 spectrum derived here with a combination of a clear planetary atmosphere and a spotted stellar photosphere . as in prior work ( e.g. , pont et al . 2008 , 2013 ; sing et al . 2011 ) , we model star spots in aggregate and ignore limb darkening . in the spotted - star model , a fraction @xmath31 of the star s projected surface area is covered with spots , emitting with radiance @xmath32 , less than that of the photosphere , @xmath33 . such a model predicts an apparent transit depth , @xmath34 where the first factor on the right is the transit depth that would be measured in the absence of star spots . @xcite used simulated , rotating spotted stars modeled on hd 189733 to estimate the product of its spots area coverage @xmath31 and ( 1-contrast ) , @xmath35 where @xmath36 is the difference in the stellar flux from maximum to minimum , and @xmath37 is the scatter in the light curve , also due to rotational modulation ( not observational uncertainty ) . from six years of photometry at @xmath38 of hd 189733 illustrated in figure 3 of @xcite , we estimate @xmath39 and hence @xmath40 can be comparable to , or larger than , @xmath41 . because @xmath40 represents the product , spot area times ( 1-contrast ) , and contrast must be less than unity , in general , the fraction @xmath31 of the star s disk covered with spots must be greater than @xmath40 . thus , for hd 189733 , we expect @xmath31 at times is greater than approximately @xmath41 . a caveat worth noting is that the analysis of @xcite is most appropriate for stars for which a single active region dominates the photometric variability . also , as noted by @xcite , geometric foreshortening de - emphasizes the contribution to the rotational modulation of spots near the stellar limb . with those caveats in mind , we admit the possibility that equation [ eq - f ] underestimates @xmath40 and hence @xmath31 could be even larger than 0.04 , if for instance hd 189733 has many active regions spread uniformly in longitude , i.e. bands of star spots . the values of @xcite , which we use as a starting point in our analysis , have been corrected already for unocculted star spots that modulate the light curve , with time - dependent `` ac '' values for @xmath31 corresponding to flux changes in the optical between @xmath42% and @xmath43% ( table 3 of @xcite ) . the most important corrections are to the acs and stis data , because the transmission spectrum at wavelengths longer than @xmath44 are less affected by the unocculted spot correction and/or by the rayleigh - scattering dust model ( figures [ fig4 ] and [ fig5 ] ) . the unocculted spot corrections derived by the procedure of @xcite produced a discontinuity between the stis and acs transmission spectra in the region of overlap at @xmath30.5 . to eliminate the discontinuity , they applied a smaller correction than their method had indicated for unocculted spots at the epoch of the stis observations . while @xcite have corrected for an ac component , here we investigate an underlying `` dc '' component and its potential effects on the transmission spectrum . the dc component is an additional , hypothetical set of unocculted spots in bands or a polar cap . with either such configuration , they would not modulate the light curve or precision radial velocities either . polar spots have been deduced from doppler imaging of rapidly rotating stars , e.g. the k0 dwarf ab dor has a rotation period of 0.5 days and exhibits a prominent , long - lived polar spot @xcite , but we know little about the spots of more slowly - rotating stars such as hd 189733 . before proceeding to the more realistic phoenix atmosphere models , we begin with the approximation of black bodies for the stellar photosphere and the star spots , at temperatures @xmath45 and @xmath46 respectively . in this approximation , equation [ eq - r - tilde ] becomes @xmath47 where @xmath48 is planck s constant , @xmath49 is boltzmann s constant , and @xmath50 is the observing frequency . the spots multiplicative effect on the apparent transit depth as a function of wavelength asymptotically approaches one value in the wien limit ( in the ultraviolet ) and another in the rayleigh - jeans limit ( in the far infrared ) : @xmath51 the difference between the two limits is @xmath52 where the first expression is generic and the second is specific for hd 189733b s nominal transit depth of 2.40% . intuitively , the greater the fractional area of star spots , @xmath31 , the greater the increase in apparent transit depth in the ultraviolet relative to the thermal infrared . counter - intuitively ( perhaps ) , the cooler the spots , the less their effect on the difference in apparent transit depth @xmath53 . this is because in the wien limit , the radiance of cool spots is arbitrarily smaller than that of the photosphere , i.e. the spots are essentially black silhouettes , whereas in the rayleigh - jeans limit , radiance is proportional to temperature , so the _ difference _ in radiances ( rayleigh - jeans minus wien ) is proportional to the ratio of temperatures , and then so is the difference in apparent transit depth . these characteristics of blackbodies inform our expectations for the effect of unocculted spots on the apparent transit depth . pont et al . ( 2008 ) estimate that @xmath544000k , and @xcite find @xmath55 k for hd 189733 . if the typical spot covering fraction @xmath31 is approximately 4% , then equation [ eq - delta ] predicts the difference in apparent transit depth between the ultraviolet and the infrared should be @xmath56 ppm , entirely due to unocculted star spots and not due to any variation in planetary radius with wavelength . from table 6 of @xcite , the observed difference in transit depth between 0.33 and 24 is 1100 ppm . evidently , a majority fraction of the increased apparent transit depth in the ultraviolet compared to the infrared could be due to _ more _ unocculted star spots than that assumed by @xcite . if so , then there would be no need for rayleigh - scattering , non - absorbing dust in the atmosphere of hd 189733b in order to match the slope in the apparent transit depth with wavelength from 0.3 to 1.0 as in the interpretation of @xcite . to achieve greater fidelity than the black body models permit , we have modeled the effect of unocculted star spots using phoenix nextgen model atmospheres ( hauschildt et al . we use a 5000/4.5/0.0 ( teff / log(g)/[m / h ] ) model for the star , and for the spots we use nextgen models whose temperature we vary between 3000 and 5000k , in steps of 50 k via interpolation in the nextgen grid . we constrain the log(g ) and metallicity of the spots to be the same as the stellar model , and we ignore center - to - limb effects on the star spot spectrum . this procedure is consistent with similar previous work ( e.g. , pont et al . 2008 , 2013 ; sing et al . 2011 ) . at each adopted spot temperature , we vary the assumed fractional area covered by the spots , and we calculate the effect on the transit depth as a function of wavelength ( equation [ eq - r - tilde ] ) . we calculate the @xmath57 of the difference between the modeled transit depth as a function of wavelength , using the observations in table 5 of @xcite . to start , we adopt the limiting - case in which all of the observed planetary radius increase in the blue and uv is due to unocculted star spots and the spot - free radius ratio @xmath58 is 0.15459 , from the longest wavelength transit measurement at 24@xmath6 m ( @xcite ; @xcite ) . in that case we find a best - fit spot temperature of 4250k and a best - fit area coverage of 4.3% . this fitting process is well - posed in the sense that the wavelength of the upward bend in the curve is determined by the temperature adopted for the star spots ( the cooler the spots , the longer the wavelength ) , and the difference in apparent transit depth from the rayleigh - jeans end to the wien end of the curves is proportional to the product of the spots temperature and area coverage ( equation [ eq - delta ] ) . our best - fit temperature agrees well with spot temperatures inferred from the amplitudes of spot crossings by this planet ; @xcite find that @xmath59 k , and @xcite find @xmath60k . our best - fit model predicts @xmath61 , similar to the value @xmath62 that we estimated from the spot modulation of hd 189733 s light curve at @xmath63 m . in this case , the contamination of the 1.4 water - vapor feature is negligible , as we now discuss . the model exhibits no spectral feature due to water absorption in the unocculted star spots , since 4250k is too warm for significant water absorption to be present . hence even the worst - case assumption that all of the transit radius measurements are due to star spots does not produce significant contamination of the wfc3 water spectrum . that remains true if a much cooler spot temperature ( 3200k ) is adopted , for example . such a spot is much cooler than the estimates noted in the previous paragraph , and is also cooler than much larger spots seen on other stars ( oneal et al . 1998 ) . moreover , even spots of t = 3200k do not produce sufficient water absorption to contaminate our wfc3 measurement ; the predicted amplitude of false water absorption is @xmath340 ppm - much less than the @xmath3200 ppm feature at 1.4 that we measure . we conclude that our wfc3 measurement of water absorption is due to the planetary atmosphere , not to an effect of unocculted star spots . we constrain the h@xmath5o abundance from our observed transmission spectra of hd 189733b using model atmospheric spectra of its terminator . we use the atmospheric modeling and retrieval technique of madhusudhan & seager ( 2009 ) and madhusudhan ( 2012 ) . the model involves computing line - by - line radiative transfer in a 1-d plane - parallel atmosphere , assuming hydrostatic equilibrium . the temperature profile and chemical composition are free parameters in the model , and , as such , has no constraints of radiative or chemical equilibrium . this modeling approach allows one to compute large ensembles of models ( @xmath64 ) , and explore the parameter space of molecular compositions and temperature structure in search of the best - fitting models . typically , the model includes opacity contributions from the major molecular species expected in hot jupiter atmospheres ( i.e. h@xmath5-h@xmath5 collision - induced absorption and line absorption due to h@xmath5o , co , ch@xmath65 , and co@xmath5 ) , though h@xmath5o is the most dominant molecule spectroscopically in the wfc3 bandpass . the transmission spectrum is less sensitive to the detailed temperature structure , compared to a thermal emission spectrum , but does depend on the characteristic temperature in the upper - atmosphere above the infrared photosphere when viewed through the terminator . in the present work , we assume a characteristic 1-d temperature profile that has been derived from thermal emission measurements in the past ( madhusudhan & seager 2009 ) , and explore the range of chemical compositions that explain the transit data . we find that our wfc3 transmission spectrum of hd 189733b can be explained very well by h@xmath5o absorption at the terminator . the wfc3 spectrum exhibits an absorption feature at 1.4 with an amplitude of @xmath66 ppm that coincides with the expected features of gaseous h@xmath5o in the same band . as shown in fig . [ fig3 ] , the data can be fit well by a solar composition atmosphere in chemical equilibrium . such an atmosphere has a h@xmath5o mixing ratio of @xmath27 , where the mixing ratio is defined as number density relative to h@xmath5 . at the pressures probed by the observations ( @xmath67 bar ) , the temperature profile has typical isothermal temperature of @xmath3700 k , slightly cooler than prior p - t curves used to model nicmos transmission spectra of hd 189733b s terminator and its day - side emission spectrum from _ spitzer _ ( c.f . figures 5 and 8 of @xcite ) . in principle , allowing for higher temperatures ( e.g. 1000 k ) leads to larger scale heights thereby constraining the h@xmath5o abundance to be slightly sub - solar ( @xmath68 , @xcite ) . nevertheless , the evidence of an absorption feature in the h@xmath5o band is unambiguous , despite the degeneracy between the characteristic temperature and the h@xmath5o mixing ratio . a featureless flat spectrum does not fit the data , ruling out the possibility of thick clouds fully obscuring the atmosphere observable along the planet s terminator . on the other hand , a fully - obscuring cloud deck at lower altitudes ( i.e. higher pressures , @xmath69 bar ) in the atmosphere is not ruled out by these wfc3 data . combining our wfc3 spectrum of hd 189733b with previously reported spectra in other bandpasses reveal new constraints on its atmospheric composition . the sum - total of previous data have led to an interpretation of hazes or clouds in the atmosphere of the planet ( pont et al . 2008 ; sing et al . 2011 ; pont et al . in figure [ fig5 ] we present a model that combines the effects a clear planetary atmosphere and unocculted star spots . for the gas giant planet , we model a clear atmosphere of solar composition , a mixing ratio for water of @xmath70 , and zero alkali metal lines ( na and k ) . the rayleigh scattering of h@xmath5 in the planetary atmosphere contributes significantly at wavelengths less than 0.5 microns . incidentally , that scattering may be responsible for the higher geometric albedo for the day side of hd 189733b observed at short wavelengths , @xmath71 , than at longer wavelengths , @xmath72 @xcite . because the stis spectrum ( @xmath73 ) has a rayleigh - scattering slope , the star spot contribution must saturate in order that the model spectrum not rise too quickly into the ultraviolet . that is accomplished with a cool , 3700 k spot temperature . the star s model is a phoenix atmosphere model for the 5000 k stellar photosphere with unocculted spots of temperature @xmath74 k and spot fractional area @xmath75 . although it fits the available data well , the combined model presented here is not unique : it is one example of many possibilities . as another example that could fit the transmission spectrum of hd 189733b is an `` enhanced rayleigh '' model proposed for wasp-12b by @xcite in which the planet s atmosphere has both rayleigh scattering from aerosols and water vapor absorption . a main purpose of this paper is to emphasize that fitting any transmission spectrum with a planetary atmosphere model must also consider the potential contribution and complication of unocculted star spots . in prior work , the case for haze / clouds in the atmosphere of the planet has been based on three key arguments : ( 1 ) the monotonic rise in the spectrum from the near - infrared to the uv consistent with strong rayleigh and mie scattering , ( 2 ) the lack of strong na or k features in the optical spectrum , i.e. presumably masked by a thick haze layer , and ( 3 ) the lack of striking @xmath76o absorption in previous datasets . we are able to explain the monotonic blue - ward rise in the transmission spectrum by considering the effect of star spots on the measured transit depths in the relevant wavelengths ( section [ sec : spots ] ) . secondly , the lack of na and k features in optical transmission spectra could be explained by either a low metallicity in the atmosphere @xcite and/or by condensation of na and k either in the low - temperature upper atmosphere or on the night side . the latter model is a distinct possibility , given that hd 189733b is one of the least irradiated hot jupiters ( @xmath77 k ) . @xcite and @xcite show the condensation curves of various species in pressure - temperature phase space . at @xmath78 bar , condensation temperatures of kcl and na@xmath5s are @xmath56k and @xmath79k , respectively , which can be higher than the temperatures in hd 189733b at the corresponding pressure on the terminator . consequently , na and k could be condensed out of the observable atmosphere by arbitrarily large factors into their corresponding compounds ( e.g. kcl , na@xmath5s ) . finally , the @xmath66 ppm amplitude of the 1.4 feature reported here ( and also in other exoplanets ; see 1 ) explains why it was not as obvious in previous datasets as had been anticipated . we report detection of two water vapor features in the transmission spectrum of hd 189733b , a strong one at 1.4 and a weaker one at 1.15 . their shapes and amplitudes are matched well by a solar - composition planetary atmosphere with a water mixing ratio of @xmath80 . we investigate the possibility that the water vapor could exist in star spots , but even for very cool spots ( @xmath81 k ) , the amplitude of the predicted 1.4 feature is much too small to produce the observed feature . the ensemble , polychromatic transit spectrum of hd 189733b , from the ultraviolet to the thermal infrared , has been interpreted by @xcite , and references therein , as evidence for raleigh scattering in the planetary atmosphere . that interpretation requires specific adjustments for unocculted star spots in order to match the transmission spectrum across the entirety of the ultraviolet and visible . we re - interpret the polychromatic transit spectrum using a clear planetary atmosphere and unocculted star spots . in the model presented here , rayleigh scattering in a clear planetary atmosphere contributes significantly to the slope in the planetary radius as function of wavelength for 0.3@xmath82 . in the range 0.5 @xmath83 , the observed slope is caused by unocculted star spots , not the planetary atmosphere . since rayleigh scattering by transparent dust grains in the planetary atmosphere has been central to prior interpretations of hd 189733b s transmission spectra , our modeling of a clear planetary atmosphere and a spotted stellar atmosphere is a significant revision in the interpretation . given the complexities of stitching together data from different instruments observing transits of a variable star at different epochs , and given the potential of omnipresent star spots to further confuse matters , either the star spot interpretation or the rayleigh scattering interpretation , or some combination of the two , remain viable . prior work has rejected clear planetary atmosphere models based upon the predicted large - and - broad alkali metal lines in transmission spectra , na i ( 0.589 ) and k i ( 0.769 ) @xcite . in our model , the strengths of those features are reduced by arbitrarily large amounts because na and k are expected to precipitate out ( @xcite ; @xcite ) at the low temperatures ( @xmath3700 k ) that we used to model the @xmath3200 ppm amplitude of the water vapor feature at 1.4 . the authors gratefully acknowledge everyone who has contributed to the _ hubble space telescope _ and the wfc3 , and particularly those responsible for implementing the spatial scanning , which was critical to these observations . we thank in particular john mackenty and merle reinhart . we acknowledge conversations with suzanne aigrain , david ciardi , suzanne hawley , leslie hebb , veselin kostov , rachel osten , frederic pont , neill reid , and david sing . this research used nasa s astrophysics data system bibliographic services , and the simbad database , operated at cds , strasbourg , france , and was funded in part by _ hst _ grant go-12881 and origins of solar systems grant nnx10ag30 g . lc hst program ( p.i . ) & 12881 ( mccullough ) + number of hst orbits & 5 + number of scans per orbit & 16 forward & 16 reverse + duration of scan ( s ) & 5.97 + scan rate ( arcsec s@xmath9)[pixels s@xmath9 ] & ( 2.00)[16.5 ] + peak signal on detector ( electrons per pixel ) & @xmath85 + grism ( @xmath86 ) & g141 ( 1.1@xmath6 m to 1.7@xmath6 m ) + detectory subarray size ( pixels ) & 512x512 + sample sequence & rapid + samples per scan & 8 + start of first scan ( hjd ) & 2456448.983024 + corresponding planetary orbital phase & -0.09948 + start of last scan ( hjd ) & 2456449.272364 + corresponding planetary orbital phase & 0.03094 + cccrcccc expstart mjd & expstart hjd & orbit & scan & sample & flag & counts & flux + 2456448.541735 & 2456449.044294 & 2 & 1 & 1 & 0 & 55426193 & 0.99996 + 2456448.541745 & 2456449.044304 & 2 & 1 & 2 & 0 & 55452872 & 1.00014 + 2456448.541755 & 2456449.044314 & 2 & 1 & 3 & 0 & 55434284 & 1.00011 + 2456448.541765 & 2456449.044324 & 2 & 1 & 4 & 0 & 55335761 & 0.99988 + 2456448.542414 & 2456449.044973 & 2 & -1 & 3 & 0 & 61415662 & 0.99990 + 2456448.542424 & 2456449.044983 & 2 & -1 & 4 & 0 & 61552322 & 1.00005 + 2456448.542434 & 2456449.044993 & 2 & -1 & 5 & 0 & 61495951 & 0.99995 + 2456448.542444 & 2456449.045003 & 2 & -1 & 6 & 0 & 61571866 & 1.00017 + 2456448.542454 & 2456449.045013 & 2 & -1 & 7 & 0 & 61610879 & 1.00067 + 2456448.543054 & 2456449.045613 & 2 & 1 & 1 & 0 & 55466554 & 0.99990 + 2456448.543064 & 2456449.045623 & 2 & 1 & 2 & 0 & 55489200 & 0.99993 + notes : the printed table is a truncated version of the electronic table , to illustrate the format . columns , left to right , are modified julian date of the start of the exposure , the associated heliocentric julian date , the hst orbit in the visit , the scan direction ( 1 = forward ; -1 = reverse ) , the multiaccum sample , a data - analysis flag , the total number of photoelectrons from hd 189733 , and the associated normalized flux after detrending . orbit 1 was not detrended and was ignored . rrrrrrr @xmath87 & @xmath88 & @xmath37 & column & @xmath88 & @xmath37 & @xmath89 + ( @xmath6 m ) & ( ppm ) & ( ppm ) & & ( ppm ) & ( ppm ) & ( ppm ) + 1.1279 & -70 & 73 & 171.5 & -58 & 55 & -12 + 1.1467 & 15 & 67 & 175.5 & -25 & 54 & 41 + 1.1655 & 46 & 105 & 179.5 & 36 & 53 & 9 + 1.1843 & 3 & 87 & 183.5 & 13 & 52 & -9 + 1.2031 & -71 & 80 & 187.5 & -36 & 51 & -35 + 1.2218 & -77 & 70 & 191.5 & -63 & 50 & -14 + 1.2406 & -148 & 56 & 195.5 & -136 & 50 & -11 + 1.2594 & -32 & 62 & 199.5 & -91 & 50 & 59 + 1.2782 & -169 & 61 & 203.5 & -108 & 50 & -61 + 1.2970 & -45 & 69 & 207.5 & -21 & 49 & -23 + 1.3157 & -71 & 60 & 211.5 & -86 & 49 & 15 + 1.3345 & -50 & 66 & 215.5 & 20 & 49 & -71 + 1.3533 & 102 & 55 & 219.5 & 106 & 49 & -4 + 1.3721 & 117 & 61 & 223.5 & 75 & 49 & 42 + 1.3909 & 59 & 63 & 227.5 & 53 & 49 & 5 + 1.4096 & 183 & 77 & 231.5 & 195 & 49 & -11 + 1.4284 & 167 & 64 & 235.5 & 165 & 50 & 2 + 1.4472 & 76 & 71 & 239.5 & 73 & 50 & 2 + 1.4660 & -14 & 67 & 243.5 & -41 & 50 & 26 + 1.4847 & 156 & 75 & 247.5 & 85 & 51 & 70 + 1.5035 & -91 & 62 & 251.5 & 32 & 65 & -123 + 1.5223 & 65 & 61 & 255.5 & 60 & 51 & 4 + 1.5411 & -30 & 62 & 259.5 & -53 & 52 & 22 + 1.5599 & -22 & 72 & 263.5 & -27 & 52 & 5 + 1.5786 & 68 & 87 & 267.5 & 61 & 53 & 7 + 1.5974 & -69 & 75 & 271.5 & 12 & 54 & -82 + 1.6162 & -116 & 98 & 275.5 & -111 & 55 & -5 + 1.6350 & 30 & 84 & 279.5 & -131 & 56 & 162 + notes . units are as indicated ; parts per million is abbreviated ppm . the tabulated uncertainties apply to the differential transit depths ; an additional uncertainty applies to the overall depth - see text . the first three columns refer to the analysis of n. c. ; columns 5 and 6 refer to the analysis of d. d. ; the last column contains the difference of the differential spectra , column 2 minus column 5 .

we report near - infrared spectroscopy of the gas giant planet hd 189733b in transit . we used the _ hubble space telescope _ wide field camera 3 ( _ hst _ wfc3 ) with its g141 grism covering 1.1 to 1.7 and spatially scanned the image across the detector at 2@xmath0 . when smoothed to 75 nm bins , the local maxima of the transit depths in the 1.15 and 1.4 water vapor features respectively are @xmath1 ppm and @xmath2 ppm greater than the local minimum at 1.3 . we compare the wfc3 spectrum with the composite transit spectrum of hd 189733b assembled by @xcite , extending from 0.3 to 24 . although the water vapor features in the wfc3 spectrum are compatible with the model of non - absorbing , rayleigh - scattering dust in the planetary atmosphere @xcite , we also re - interpret the available data with a clear planetary atmosphere . in the latter interpretation , the slope of increasing transit depth with shorter wavelengths from the near infrared , through the visible and into the ultraviolet is caused by unocculted star spots , with a smaller contribution of rayleigh scattering by molecular hydrogen in the planet s atmosphere . at relevant pressures along the terminator , our model planetary atmosphere s temperature is @xmath3700 k , which is below the condensation temperatures of sodium- and potassium - bearing molecules , causing the broad wings of the spectral lines of na i and k i at 0.589 and 0.769 to be weak .

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Proceed to summarize the following text: the space density of galaxy clusters as a function of cluster mass is a well - known cosmological probe ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , and ranks among the best observational tools for constraining @xmath8 , the normalization of the matter power spectrum in the low redshift universe ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? is formally defined as the variance of the linear matter density averaged over spheres with radius @xmath9 . ] the basic idea is this : in the high mass limit , the cluster mass function falls off exponentially with mass , with the fall - off depending sensitively on the amplitude of the matter density fluctuations . observing this exponential cutoff can thus place tight constraints on @xmath8 . in practice , however , the same exponential dependence that makes cluster abundances a powerful cosmological probe also renders it susceptible to an important systematic effect , namely uncertainties in the estimated masses of clusters . because mass is not a direct observable , cluster masses must be determined using observable mass tracers such as x - ray emission , sz decrements , weak lensing shear , or cluster richness ( a measure of the galaxy content of the cluster ) . of course , such mass estimators are noisy , meaning there can be significant scatter between the observable mass tracer and cluster mass . since the mass function declines steeply with mass , up - scattering of low mass systems into high mass bins can result in a significant boost to the number of systems with apparently high mass @xcite . if this effect is not properly modeled , the value of @xmath8 derived from such a cluster sample will be overestimated . one approach for dealing with this difficulty is to employ mass tracers that have minimal scatter , thereby reducing the impact of said scatter on the recovered halo mass function . for instance , @xcite introduced a new x - ray mass estimator , @xmath10 , which in their simulations exhibits an intrinsic scatter of only @xmath11 , independent of the dynamical state of the cluster . use of a mass estimator with such low scatter should lead to improved estimates of @xmath8 from x - ray cluster surveys @xcite . such tightly - correlated mass tracers are not always available . in such cases , determination of the scatter in the mass - observable relation is critical to accurately inferring the mass function and thereby determining cosmological parameters . of course , in practice , it is impossible to determine this scatter to arbitrary accuracy , but since the systematic boost to the mass function is proportional to the square of the scatter @xcite ( i.e. the variance ) , even moderate constraints on the scatter can result in tight @xmath8 constraints . in this paper , we use optical and x - ray observations to constrain the scatter in the mass richness relation for the maxbcg cluster catalog presented in @xcite . specifically , we use observational constraints on the mean mass richness relation , and on the mean and scatter of the @xmath12richness relation , to convert independent estimates of the scatter in the @xmath13 relation into estimates of the scatter in the mass richness relation . an interesting byproduct of our analysis is a constraint on the correlation coefficient between mass and x - ray luminosity at fixed richness . to our knowledge , this is the first time that a correlation coefficient involving multiple cluster mass tracers has been empirically determined . the layout of the paper is as follows . in section [ sec : notation ] we lay out the notation and definitions used throughout the paper . section [ sec : data ] presents the data sets used in our analysis . in section [ sec : rough ] we present a pedagogical description of our method for constraining the scatter in the richness - mass relation , while section [ sec : formalism ] formalizes the argument . our results are found in section [ sec : results ] , and we compare them to previous work in section [ sec : other_work ] . in section [ sec : mf ] , we use our result to estimate the halo mass function in the local universe at @xmath7 , the median redshift of the maxbcg cluster sample , and we demonstrate that our recovered mass function is consistent with the latest cosmological constraints from wmap @xcite . a detailed cosmological analysis of our results will be presented in a forthcoming paper ( rozo et al . , in preparation ) . our summary and conclusions are presented in section [ sec : conclusions ] . we summarize here the notation and conventions employed in this work . given any three cluster mass tracers ( possibly including mass itself ) @xmath14 and @xmath15 , we make the standard assumption that the probability distribution @xmath16 is a bivariate lognormal . the parameters @xmath17 , @xmath18 , and @xmath19 are defined such that @xmath20 note the slopes of the mean and logarithmic mean are the same , as appropriate for a log - normal distribution . the scatter in @xmath21 at fixed @xmath15 is denoted @xmath22 , and the correlation coefficient between @xmath21 and @xmath23 at fixed @xmath15 is denoted @xmath24 . _ we emphasize that all quoted scatters are the scatter in the natural logarithm , not in dex . _ note these parameters are simply the elements of the covariance matrix specifying the gaussian distribution @xmath25 . under our lognormal assumption for @xmath16 , the parameters @xmath17 and @xmath18 are related via @xmath26 in this work , the quantities of interest are cluster mass @xmath27 , x - ray luminosity @xmath28 , and cluster richness @xmath29 . unless otherwise specified , cluster mass is defined as @xmath30 , the mass contained within an overdensity of 500 relative to critical . @xmath28 is the total luminosity in the rest - frame @xmath31 band , and @xmath29 is the maxbcg richness measure @xmath3 , the number of red sequence galaxies with luminosity above @xmath32 within an aperture such that the mean density within said radius is , on average , @xmath33 times the mean galaxy density assuming @xmath34 . likewise , unless otherwise stated all parameters governing the relations between @xmath27 , @xmath28 , and @xmath29 assume that @xmath27 is measured in units of @xmath35 , @xmath28 is measured in units of @xmath36 , and @xmath29 is measured in `` units '' of @xmath37 galaxies . for instance , including units explicitly , the mean relation between cluster mass and richness reads @xmath38 a hubble constant parameter @xmath39 is assumed through out . , our weak lensing masses scale as @xmath40 and the x - ray luminosities as @xmath41 . ] in addition , the weak lensing data presented in this analysis assumed a flat @xmath42cdm cosmology with @xmath43 . the recovered mass function has the standard hubble parameter degeneracy . in this work we use the public maxbcg cluster catalog presented in @xcite , which is an optically selected volume limited catalog of close to @xmath44 clusters over the redshift range @xmath45 $ ] . these clusters were found in @xmath46 of imaging data from the sloan digital sky survey ( sdss , * ? ? ? * ) using the maxbcg cluster finding algorithm @xcite . this algorithm identifies clusters as overdensities of red sequence galaxies . all clusters are assigned a redshift based on the sdss photometric data only , and these redshifts are known to be accurate to within a dispersion @xmath47 . every cluster is also assigned a richness measure @xmath3 , which is the number of red sequence galaxies above a luminosity cut of @xmath32 and within a specified scaled aperture , centered on the brightest cluster galaxy ( bcg ) of each cluster . only clusters with @xmath48 are included in the final catalog . interested readers are referred to @xcite and @xcite for further details . in the interest of economy of notation , from now on we denote the maxbcg richness measure simply as @xmath29 . the relationship between cluster richness and various well known mass tracers has been studied in large , homogeneous samples , such as 2mass @xcite and sdss @xcite . of particular interest to us are the weak lensing measurements of the mean mass as a function of richness , and the x - ray measurements of the mean and scatter of the x - ray luminosity as a function of richness . the former analysis has been carried out by @xcite based on the weak lensing data presented in @xcite , and independently by @xcite . in short , @xcite stacked maxbcg clusters within narrow richness bins , and measured the average weak lensing shear profile of the clusters . these shear profiles were turned into surface mass density contrast profiles using the redshift distribution of background sources estimated with the methods of @xcite and the neural net photometric redshift estimators described in @xcite . then , @xcite fit the resulting profiles using a halo model scheme to obtain tight constraints on the mean mass of maxbcg clusters for each of the richness bins under consideration . the @xcite analysis is very similar in spirit to the one described above . the main differences are the way the source redshift distribution is estimated , and the details of the model fitting use to recover the masses . the differences in the results between these two analysis are discussed in appendix [ sec : m - n_priors ] , where we use them to set priors on the mass richness relation . the measurement of the mean x - ray luminosity of maxbcg clusters has been carried out by @xcite following an approach similar to that pioneered in @xcite . the necessary x - ray data is readily available from the rosat all - sky survey ( rass , * ? ? ? in short , @xcite stacked the rass photon maps @xcite centered on maxbcg clusters in narrow richness bins . the background subtracted stacked photon counts within a @xmath49 aperture were used to estimate the mean x - ray luminosity @xmath28 in the @xmath50 rest frame of the clusters . in addition , @xcite measured the scatter in x - ray luminosity at fixed richness by individually measuring @xmath28 for all maxbcg clusters with @xmath51 . it is worth noting that due to the shallowness of rass , many of the maxbcg clusters are not x - ray luminous enough to be detected individually . however , non - detection and upper limits for @xmath28 for individual systems were properly taken into consideration using the bayesian approach detailed in @xcite , and the recovered mean x - ray luminosity from this baysian analysis was fully consistent with the stacked means . in addition to the data sets above , we use the constraints on the @xmath13 relation from @xcite . these constraints are based on the 400d cluster x - ray survey , a flux limited cluster survey based on rosat pointed observations with an effective sky coverage of 397 @xmath52 @xcite . briefly , @xcite measured both the total soft band x - ray luminosity and the cluster mass for each cluster in the sample . x - ray luminosities are estimated from rosat data , and measure the luminosity in the rest - fram @xmath31 band , extrapolated to infinity assuming standard @xmath53 profiles . cluster masses are estimated based on the values of @xmath54 derived from followup chandra observations , though they note that the results they obtain using different mass tracers such as x - ray temperature and total gas mass are very similar . the @xmath55 relation is itself calibrated based on hydro - static mass estimates . importantly , @xcite explicitly correct for the malmquist bias expected for a flux limited cluster sample , so the @xmath13 relation they derive can be interpreted as the relation one would obtain using a mass limited cluster sample . for this work , we have repeated the analysis in @xcite with a slightly different definition for @xmath28 . in particular , we measure the x - ray luminosity in the rest - frame @xmath31 band within a @xmath56 aperture . the change in band is tailored to match the energy band used by @xcite , which we used to place priors on the @xmath13 relation . it is worth noting that @xcite do not use a @xmath57 aperture , as we do . we have , however , carefully calibrated the scaling between our @xmath28 definition and that of @xcite so as to be able to use their results in our analysis.end a detailed description of our measurements can be found in appendix [ app : lx - n_priors ] . the problem we are confronted with is the following : we have four pieces of observational data , namely * the abundance of galaxy clusters as a function of richness . * the mean relation between cluster richness and mass . * the mean and variance of the relation between cluster richness and x - ray luminosity . * the mean and variance of the relation between cluster x - ray luminosity and mass . from this data , we wish to determine the scatter in mass at fixed richness for the cluster sample under consideration . the basic idea behind our analysis is as follows . consider the probability @xmath58 , which we take to be gaussian in @xmath5 and @xmath4 . this probability distribution is completely specified by the mean and variance of both @xmath27 and @xmath28 at fixed richness , and by the correlation coefficient between @xmath27 and @xmath28 . of these , there are only two quantities that are not already observationally constrained : @xmath59 , the scatter in mass at fixed richness , and @xmath60 , the correlation coefficient between mass and @xmath28 at fixed richness . suppose now that we guessed values for these two quantities , so that the probability distribution @xmath61 is fully specified . given the abundance function @xmath62 , we can use @xmath61 to randomly assign a mass and an x - ray luminosity to every cluster in the sample . we can then select a mass limited sub - sample , and measure the corresponding @xmath13 relation , comparing it to the @xmath13 measurement from @xcite . since the @xmath13 relation we predict depends on our assumptions about @xmath61 , there should only be a small region in parameters space where our predictions are consistent with independent observational constraints on the @xmath13 relation . figure [ fig : contours ] illustrates this idea . to create the figure , we have set every observed parameter of the distribution @xmath61 to the central value of the priors described in appendix [ app : priors ] and summarized in table [ tab : priors ] . we then defined a grid in the two dimensional space spanned by @xmath59 and @xmath60 , and carried through the argument described above . the resulting predictions for the amplitude , slope , and scatter of the @xmath13 relation as a function of @xmath59 and @xmath60 are shown in the figure . we plot contours of constant amplitude , slope , and scatter of the @xmath13 relation as solid , dashed , and dotted lines respectively . the thicker curves correspond to the central values of the priors , while thinner curves demark the corresponding @xmath1 confidence limits . as we can see , all three contours intersect in a finite region of parameter space , indicating good agreement between our weak lensing and x - ray data , and the independent determination of the @xmath13 relation . based on figure [ fig : contours ] , we expect a detailed analysis should constrain our parameters to @xmath63 , and @xmath64 . the rest of this paper is simply a way of formalizing the argument described above in order to place errors on both @xmath59 and @xmath60 . we wish to formalize the above argument in order to place quantitative constraints on the scatter in mass at fixed richness . details of how we go about doing so are presented below . readers interested only in our results can move directly to section [ sec : results ] . as we mentioned above , the key point in our analysis is our ability to compute the amplitude and slope of the mean relation @xmath65 , and the scatter about this mean , as a function of our two parameters of interest : the scatter in mass at fixed richness and the correlation coefficient between @xmath27 and @xmath28 at fixed @xmath29 . let us define @xmath66 , and let @xmath67 denote our parameters of interest . our predictions for the @xmath13 relation as a function of our parameters of interest can be summarized simply as @xmath68 . now , adopting a bayesian framework , a set of priors on @xmath69 is simply a probability distribution @xmath70 . since @xmath69 is a function of @xmath71 , the priors immediately define a probability distribution over @xmath71 given by @xmath72 since we know how to compute both @xmath70 and @xmath68 , we can find any confidence regions for our parameters of interest . the problem we are confronted with , however , is slightly more complicated , in that the functions @xmath69 depend not only on @xmath71 , but also on additional nuisance parameters @xmath73 . indeed , our predictions for the observable parameters of the @xmath13 relation depend on both the abundance function of clusters and @xmath61 . the abundance function can be accurately described by a schechter function ( we explicitly checked a schechter function is statistically acceptable ) , @xmath74 given a schechter fit , our prediction for the @xmath13 relation will also depend on the value of the parameters @xmath75 and @xmath76 . likewise , the distribution @xmath61 also depends on the amplitude and slope of the means @xmath77 and @xmath78 , as well as the scatter in @xmath28 at fixed @xmath29 . all in all , we have six additional nuisance parameters @xmath79 . let @xmath80 denote the full set of parameters . the priors from the @xmath13 relation define a probability distribution over @xmath81 given by @xmath82 since we have a total of 8 parameters , and only three observables from the @xmath13 relation , it is obvious that the above likelihood function will result in large degeneracies because the parameters are under - constrained . if one has priors @xmath83 in the nuisance parameters , however , the probability distribution @xmath84 in the parameters of interest is given by @xmath85 this equation allows us to compute @xmath84 , and therefore place constraints on our parameters of interest . in practice , we will ignore the determinant term in the probability distribution defined in equation [ eq : prob ] . this is because the function @xmath86 is estimated using a monte carlo approach , implying that accurate numerical estimates of the jacobian @xmath87 would be too computationally intensive to be performed . fortunately , the determinant typically introduces only slight modulations of the likelihood , so we do not expect our results to be adversely affected by this . we estimate the probability distribution @xmath84 using a monte carlo approach . ignoring an overall normalization constant and setting @xmath88 , we have @xmath89 where @xmath90 for @xmath91 through @xmath92 are random draws of the nuisance parameters @xmath90 , drawn from the prior distribution @xmath93 . we set @xmath94 as our default value ( see below for further discussion ) . the prior distributions for our nuisance parameters are characterized by a statistical and a systematic error . the former is modeled as gaussian and the latter using a top - hat distribution . thus , given a prior of the form @xmath95 a random draw is obtained by setting @xmath96 where @xmath97 is drawn from a gaussian of zero mean with a covariance matrix defined by the statistical errors , and @xmath98 is drawn from a top hat distribution that is non - zero only for @xmath99 . the probability distribution @xmath100 used in equation [ eq : probestimator ] is the product of the likelihoods @xmath101 for each of the @xmath13 parameters @xmath102 . the probability for each @xmath13 parameter is given by the convolution of the top - hat and gaussian distributions defined by the statistical and systematic errors of @xmath103 , so that @xmath104\ ] ] where @xmath105 note that the above equations are appropriate only when the various @xmath13 parameters are uncorrelated , so it is important to place the priors at the pivot point of the @xmath13 relation ( @xmath106 ) . this explains why table [ tab : priors ] quotes a prior on @xmath107 rather than on @xmath108 alone . we also need to specify how the function @xmath109 is evaluated . we do this using a monte carlo approach . given @xmath71 and @xmath73 , we generate @xmath110 mock clusters in the richness range @xmath111 $ ] . we then randomly draw mass and x - ray luminosity values for each of these clusters based on the distribution @xmath61 , and select a mass limited subsample of clusters using a mass cut @xmath112 with @xmath113 ( the reason for this particular value is explained below ) . using a least squares fitting routine , we find the best fit line between @xmath4 and @xmath5 . this defines both @xmath114 and @xmath115 . the scatter @xmath116 is defined as the root mean square fluctuation about the best fit line . using equation [ eq : probestimator ] and the function @xmath109 defined above , we evaluate the probability distribution @xmath84 along a grid of points in @xmath117 $ ] and @xmath118 $ ] with @xmath119 grid points per axis . a full run of our code then requires we perform @xmath120 monte carlo integrals with @xmath94 points in each integration . each draw also requires us to evaluate the function @xmath109 , which in turn requires generating a mock catalog with @xmath110 clusters , so the procedure as a whole is computationally expensive . to increase computational efficiency , for each monte carlo evaluation of @xmath84 we generate a single cluster catalog that is used to estimate the likelihood at every grid point . this correlates the values of @xmath121 along our grid , but does not otherwise adversely affect our results . our monte carlo approach requires that both the number of clusters in the random catalogs @xmath122 and the number of times the likelihood function is evaluated @xmath92 is sufficiently large to achieve convergence . our default values for @xmath122 and @xmath92 were selected to ensure the recovered likelihood is accurate to within a dispersion of @xmath123 inside high likelihood regions . the error in the recovered likelihood increases with decreasing likelihood , but even in the tails of the distributions our estimates are accurate to about @xmath124 . this was explicitly tested by running a coarse grid with our default values for @xmath92 and @xmath122 , and by repeating the analysis with both of these parameters increased by a factor of two . , one needs to generate cluster catalogs with @xmath125 clusters in order for the contours to appear smooth by eye . however , @xmath110 is a sufficient number of clusters for our analysis , since we only require that the noise in the likelihood be much smaller than the width of the priors . since the latter are quite wide , even relatively noisy estimates of the @xmath13 relation are sufficient for constraining the marginalized distribution . ] finally , we emphasize that it is necessary to explicitly check whether our results are sensitive to the @xmath126 cut applied to the maxbcg clusters sample . in particular , when selecting a mass limited subsample of clusters , we need to ensure that the mass limit @xmath127 be sufficiently large that the number of clusters with @xmath128 and @xmath112 is insignificant . we have explicitly checked that for our adopted low mass cut @xmath129 our results are robust to the richness cut @xmath126 by repeating the analysis in a coarse grid using an @xmath130 richness cut instead . we find that the likelihood estimates in both cases are in agreement to within the expected accuracy of our monte carlo approach . the priors used in our analysis are summarized in table [ tab : priors ] . we follow the notation @xmath131 where @xmath132 is the central value , @xmath133 is the 1@xmath134 statistical error on the parameter @xmath135 marginalized over all other parameters , and @xmath136 is the systematic error . in all cases , we model statistical errors as gaussian , and we include known covariances between different parameters . systematic errors are assumed to follow top - hat distributions , and the final prior distribution is given by the convolution of these two functions . @xmath137 & @xmath138 + @xmath75 & @xmath139 + @xmath140 & @xmath141 + @xmath142 & @xmath143 + @xmath144 & @xmath145 + @xmath146 & @xmath147 + @xmath148 & @xmath149 + @xmath150 & @xmath151 + @xmath152 & @xmath153 + @xmath154 & @xmath155 [ tab : priors ] we believe that the priors contained in table [ tab : priors ] are fair , that is , they are neither overly aggressive nor overly conservative . a detailed discussion of our priors can be found in appendix [ app : priors ] . figure [ fig : lkhd ] shows the @xmath156 and @xmath1 probability contours for the parameters @xmath59 and @xmath60 . the likelihood peak occurs at @xmath157 and @xmath158 . the marginalized means are @xmath159 and @xmath160 . we wish to determine whether the breadth of the likelihood region in figure [ fig : lkhd ] is limited by uncertainties in the scaling relations of maxbcg clusters , or by uncertainties in the @xmath13 relation . to do so , we repeat our analysis with two new sets of priors : for the first , we use a tight @xmath161 statistical prior on all nuisance parameters , but let the @xmath13 parameters float . the second set of priors uses a tight @xmath161 prior on each of the @xmath13 parameters , but floats all other nuisance parameters with the original priors . we find that using tight priors on our nuisance parameters has negligible impact on the likelihood regions recovered from our analysis . on the other hand , the confidence regions obtained with the tight @xmath13 priors , shown in figure [ fig : lkhd ] as dashed curves , are tighter than those derived from our original analysis . thus , the dominant source of error in our analysis is the uncertainty in the values of the @xmath13 parameters . this can be easily understood based on figure [ fig : contours ] . we can see from the figure that the uncertainty in @xmath60 is largely due to the prior on the scatter in @xmath28 at fixed @xmath27 , which is already tight and thus does not change between our fiducial prior and our tight priors . on the other hand , we can see that both the amplitude and slope priors cut - off regions with high scatter . tightening these priors excludes a larger section of parameter space , and results in the tighter contours observed in figure [ fig : lkhd ] . figure [ fig : marg ] shows the marginalized probability distributions for @xmath59 and @xmath60 . the solid curves correspond to our original analysis , while the dashed curves illustrate the results one expects assuming our hypothetical tight priors for the @xmath13 relation parameters . we find that the logarithmic scatter in mass at fixed richness and the correlation coefficient between @xmath5 and @xmath4 are @xmath162 assuming our hypothetical tight @xmath13 priors , the constraints become @xmath163 and @xmath6 . we emphasize that these latter constraints are only meant as a guide to the accuracy one could achieve with this method if the @xmath13 relation were known to about @xmath164 accuracy . it is evident from our results that cluster richness is not as effective a mass tracer as x - ray derived masses . indeed , even total ( i.e. not core - core excluded ) x - ray luminosity is a more faithful mass tracer than the adopted richness measure of the maxbcg catalog , as demonstrated both by the smaller scatter and the very large correlation coefficient . note that the latter indicates that , at fixed richness , over - luminous clusters are almost guaranteed to also be more massive than average . this is an important result which forms the basis for a concurrant paper in which we improve our richness estimates by demanding tighter correlations in the @xmath12richness relation @xcite . there are not many previous results against which our measurements of scatter in mass at fixed richness may be compared . one possible reference point is the upper limit based on the error bar in the weak lensing mass estimates of @xcite . more specifically , assuming that the error in @xmath77 is entirely due to the intrinsic scatter in @xmath27 at fixed @xmath29 , it follows that the error in the mass is simply @xmath165 where @xmath166 is the observed error and @xmath62 is the number of clusters with richness @xmath29 . for the richest bin , which provides the tightest constraint , @xcite find @xmath167 . the bin contains @xmath168 clusters , so an upper limit to the scatter in mass at fixed richness is @xmath169 . figure [ fig : marg ] shows that our results easily satisfy this upper limit on the scatter . the only other measurement of the scatter in mass at fixed richness for maxbcg clusters is that found in @xcite . these scatter estimates are obtained as follows : first , @xcite select all maxbcg clusters whose central galaxy has a spectroscopic redshift . they then bin the clusters in richness , and compute the velocity relative to the bcg of every galaxy member with spectroscopic data . the recovered velocity distribution of galaxies is found to be non - gaussian . assuming that the velocity distribution of galaxies of halos of fixed mass is exactly gaussian , and that the observed non - gaussianity is entirely due to mass - mixing within a richness bin , @xcite estimate the scatter in mass at fixed richness based on the observed non - gaussianity of the velocity distribution . an updated version of the results from @xcite can be seen in figure [ fig : compare ] . the only difference between this plot and the corresponding figure in @xcite is that here we have made used of the additional spectroscopic data from the sdss data release 6 @xcite , which results in tighter error bars . also shown in the figure as a horizontal band is the @xmath1 confidence region from our analysis . as we can see , our scatter estimate appears to be systematically lower than that of @xcite , a discrepancy first noted in ( * ? ? ? * more on the relation between our work and theirs below ) . such a bias is not entirely unexpected , as we now know that a significant fraction of cluster have their bcgs miss - identified , a problem that was not yet known and was therefore unaccounted for at the time the @xcite results came out . to get a better understanding of how our results and those of @xcite compare , we can use our results along with the miscentering probability model from @xcite to predict the scatter that @xcite observed given this miscentering systematic . we proceed as follows . first , we use our best fit model for the abundance distribution to generate a mock catalog with @xmath170 clusters with @xmath126 . each of these clusters is assigned a mass by drawing from the @xmath171 distribution defined by the values of @xmath59 corresponding to the two @xmath1 confidence limits on @xmath59 . these assigned masses are then turned into velocity dispersions using the scaling relation from @xcite . at this point , we have a cluster catalog where each cluster has a richness and a velocity dispersion . if a cluster is miscentered , we expect that in most cases the new center will be a cluster galaxy . assuming this is the case , and that bcgs are at rest at the center of a cluster , the velocity dispersion of cluster galaxies relative to random satellites will be a factor of @xmath172 high than relative to the bcg . using the miscentering model described in @xcite for @xmath173 , the probability that a cluster of richness @xmath29 be correctly centered , we randomly label clusters as properly centered or miscentered , and boost their `` observed '' velocity dispersion for those clusters labeled as miscentered by the expected amount . the clusters are assigned a new mass based on their `` observed '' velocity dispersions , and the corresponding scatter in the @xmath174 relation is estimated . we repeat this procedure @xmath175 times in order to compute the mean systematic correction due to miscentering . our predictions for the scatter values observed by @xcite are shown in figure [ fig : compare ] with dashed lines , and correspond to the @xmath1 confidence interval from our analysis . we see that miscentering introduces a richness dependent correction that boosts the scatter in the recovered velocity dispersion and places it in significantly better agreement with the data from @xcite . the agreement with the @xcite data is an interesting result . perhaps the single most difficult systematic effect that had to be addressed in the @xcite analysis is the validity of the assumption that non - gaussianities in the velocity distribution of stacked clusters are entirely due to mass - mixing is a valid . the reasonable agreement between our results and those of @xcite suggests that their assumption is indeed justified , though a robust conclusion will have to wait until a more detailed analysis is performed , especially given the possibility of velocity bias of the galaxy population ( i.e. if satellite galaxies have a velocity dispersion different from that of the dark matter ) . the analysis in this work is also very closely related to that of @xcite . @xcite sought to constrain the @xmath13 relation of clusters by fitting the scaling of @xmath78 with @xmath77 . however , as recognized in @xcite , in order to fully interpret their result in terms of the traditional definition of the @xmath13 relation , i.e. the mean x - ray luminosity at fixed mass , one needs to know both the scatter in mass at fixed richness , and the corresponding correlation coefficient with @xmath28 . given that these two quantities are unknown , but that the @xmath13 relation is already constrained from x - ray surveys , it seems reasonable to suggest that a better use of the lensing and x - ray data of maxbcg clusters is to use our knowledge of the @xmath13 relation to constrain the scatter in mass at fixed richness and the corresponding correlation coefficient , as was done in this work . our work differs from the ideas presented in @xcite in another significant way . while our analysis employs only @xmath176 and @xmath62 , @xcite used the halo mass function @xmath177 and the probability distribution @xmath178 to interpret their measurements . this has the important drawback that in doing so , one needs to assume a cosmological model in order to compute the halo mass function , rendering their interpretation cosmology dependent . by focusing on the quantities that are directly observable , i.e. @xmath62 and @xmath176 , we are able to avoid this difficulty . the price we pay for this is that rather than constraining the scatter in richness at fixed mass , which is the more directly relevant quantity from a cosmological perspective , we constrain instead the scatter in mass at fixed richness . while this makes implementing such a constraint a little more cumbersome in a cosmological analysis , the fact that the constraint itself is cosmology independent is obviously of paramount importance . as mentioned in the introduction , to obtain an unbiased estimate of the halo mass function based on the observed cluster richness function requires that we understand the scatter between cluster richness and halo mass . given our lognormal assumption , and the fact that the mean mass richness relation is already known from weak lensing , our measurement of the scatter in this scaling relation fully determines the probability distribution @xmath171 . thus , we are now in a position to determine the halo mass function of the local universe with the maxbcg cluster catalog . let us define then @xmath179 as the number of halos within a logarithmic mass bin of width @xmath180 centered about @xmath181 , @xmath182 given our cluster catalog and @xmath171 , we construct an estimator @xmath183 for @xmath184 by randomly drawing a mass from @xmath171 for each halo in the cluster catalog , and then counting the number of halos within the logarithmic mass bin centered about @xmath181 . note that since the mass of each cluster is a random variable , our mass function estimator @xmath183 is itself a random variable . the mean and correlation matrix of @xmath183 can easily be obtained by making multiple realizations of @xmath183 , and averaging the resulting mass functions . in practice , we also need to marginalize our results over uncertainties in @xmath171 and over uncertainties in the richness function @xmath62 . to do so , we randomly draw the parameters @xmath185 , and then resample of the cluster richness function to obtain a new estimate of @xmath184 . the whole procedure is iterated @xmath186 times , and the mean and covariance matrix of the number counts in each of our logarithmic mass bins is computed . is large enough for our results to be insensitive to the maxbcg richness cut @xmath126 . ] figure [ fig : mf ] shows the mass function recovered through our analysis . to turn our number counts into a density , we assumed a wmap5 cosmology @xcite , with @xmath43 and @xmath187 , and a photometric redshift error @xmath188 ( used for computing the effective volume of the sample ) . the diamonds correspond to our estimated means , and the error bars are the square root of the diagonal elements of the correlation matrix . we emphasize that the error bars are heavily correlated . the mean and covariance matrix of the recovered halo mass function can be found in appendix [ app : mfdata ] . also shown in figure [ fig : mf ] with dotted lines are the halo mass functions at @xmath7 predicted by wmap5 assuming the @xcite mass function . for both curves , we set all cosmological parameters to the central values reported in @xcite , except for @xmath8 , which is set to @xmath189 for the upper curve and @xmath190 for the lower curve . these two values define the @xmath1 confidence interval for @xmath8 in @xcite . as we can see , the mass function recovered from our analysis is fully consistent with the wmap5 cosmology , though it seems to push for values of @xmath8 on the high end of their allowed region . a detailed cosmological analysis of our data will be presented in a subsequent paper ( rozo et al , in preparation ) . we have shown that by combining the information in the maxbcg richness function , the mean richness - mass relation , the mean and scatter of the @xmath12richness relation , and the mean and scatter of the @xmath13 relation , we can constrain both the scatter in mass at fixed richness for maxbcg clusters , as well as the correlation coefficient between mass and @xmath28 at fixed richness . we find @xmath162 these constraints are dominated by uncertainties in the @xmath13 relation , and can be significantly tightened if our understanding of the @xmath13 relation improves . we also found our results are consistent with those presented in @xcite once miscentering of maxbcg clusters is taken into account . our lower limit on the correlation between @xmath27 and @xmath28 at fixed richness constitutes the first observational constraint on a correlation coefficient involving two different halo mass tracers . note that the large correlation between @xmath28 and @xmath27 implies that @xmath28 - even without core exclusion - is a significantly better mass tracer than the maxbcg richness estimator ( i.e. at fixed richness , over - luminous cluster are nearly always more massive ) . this is an important result , which we use in a concurrent paper to help us define new richness estimators that are better correlated with cluster mass @xcite . using our results , and assuming @xmath43 and @xmath39 , we have estimated the halo mass function at @xmath7 , corresponding to the median redshift of the cluster sample . we find that our recovered mass function is in good agreement with the mass function predicted by @xcite for the wmap5 cosmology @xcite . a detailed cosmological analysis will be presented in a forthcoming paper ( rozo et al , in preparation ) . our work sheds new light on the interrelationship of bulk properties of massive halos . we have used weak lensing , x - ray luminosities , and optical richness estimates to constrain the scatter in the richness - mass relation , which can lead to improved cosmological constraints . in principle , one could also turn this question around , and , assuming cosmology , we could constrain the scatter in the richness - mass relation , which would then allow us to place constraints on the amplitude , slope , and scatter of the @xmath13 relation . such an analysis would be interesting in that , by doing so , one could compare the predicted amplitude of the @xmath13 relation to that derived from hydrostatic mass estimates , thereby directly probing the amount of non - thermal pressure support in galaxy clusters . note that even though this question can also be directly addressed by comparing weak lensing and x - ray mass estimates of individual clusters , the analysis suggested here would benefit from having small uncertainties , whereas projection effects result in rather noisy weak lensing mass estimates for individual systems . the authors would like to thank alexey vikhlinin for providing them with the full covariance of the @xmath13 parameters from their analysis of the 400d cluster sample . er thanks david weinberg and chris kochanek for useful discussions and their careful reading of the manuscript . esr would like to thank the tabasgo foundation . tm and jh gratefully acknowledge support from nsf grant ast 0807304 and doe grant de - 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likelihood function using an amoeba routine . to estimate our errors , we follow a monte carlo approach and resample the observed richness function @xmath198 times . we find that the parameters @xmath76 and @xmath75 are significantly correlated , with the probability distribution being gaussian in @xmath75 and @xmath137 . the best fit parameters are @xmath199 with a correlation coefficient @xmath200 to assess goodness of fit , we generate @xmath198 mock catalogs with as many clusters as the real data from the probability distribution specified by @xmath201 and @xmath202 . we compute the likelihood for each of these mock catalogs , and compare the corresponding likelihood distribution to that observed in the real data . we find that our fit is statistically acceptable . the most significant systematic error affecting our measurements of the shape of the richness function are completeness and purity variations in the cluster catalog . @xcite have shown that the maxbcg catalog is over @xmath203 pure and complete for @xmath126 . here , we take a conservative approach , and consider the change in the best fit parameters assuming the observed counts are rescaled by a completeness / purity correction factor @xmath204 given by @xmath205 this corresponds to a @xmath124 decrease in the observed counts at @xmath206 while holding the counts at @xmath207 constant . upon refitting the data after this correction we find systematic offsets @xmath208 which we adopt as our systematic error . note the systematic offsets are allowed to be both positive and negative , since the correction multiplier @xmath204 above could easily be larger than unity rather than smaller than unity . our priors on the @xmath174 relation are based on the results presented in @xcite , @xcite , and @xcite . to assign our priors , we first compare the results of these two works as a means of assessing systematic uncertainties in the mass parameters . we then focus exclusively on the @xcite results to place our final priors on the @xmath174 relation . the latter choice reflects the fact that @xcite report weak lensing mass estimates for several mass definitions , among them @xmath30 , the relevant quantity in the @xmath13 relation of @xcite . let us then begin by discussing the @xcite results first . while @xcite quote a power - law fit for the mean mass at fixed @xmath77 , this fit is based a non - public version of the maxbcg catalog that extends to a richness of @xmath209 ( the catalog for clusters with @xmath210 is not public ) . since the maxbcg catalog is only known to be highly complete and pure in the range @xmath126 , we have refit the @xcite masses restricting ourselves to the range @xmath211 . this slightly lower cut is necessary due to the richness binning in @xcite . we find that the mass @xmath212 within a 180 overdensity threshold relative to mean matter density is @xmath213 with a correlation coefficient @xmath214 between the amplitude and slope parameters . @xcite preformed a similar but independent weak lensing analysis of the maxbcg clusters , though using @xmath215 as their mass variable . they find @xmath216 to compare against the @xcite values , we use the @xcite mass conversion formulae to find an approximate power law relation between @xmath215 and @xmath212 over the range @xmath217 . we find @xmath218 , which is only a @xmath219 correction . applying this correction , we find that the corresponding @xmath174 parameters from @xcite are @xmath220 we find that the slopes of the @xcite and @xcite results are nearly identical , but that the masses of @xcite are systematically higher by @xmath221 . this difference can be traced back to how the lensing critical surface density for each of the two works is estimated . in general , lensing masses are proportional to the quantity @xmath222 , where @xmath223 is the lensing critical surface density , and the average is to be computed over the source redshift distribution . given multi - band photometric data @xmath224 for each galaxy , one way to compute @xmath225 is to use a photometric redshift estimator @xmath226 , and then assume that the true source redshift distribution is identical to the photometric redshift distribution . @xcite have shown that such a simple approach typically results in biased lensing mass estimates , but they also demonstrate that it is possible to achieve unbiased results using the probability distribution @xmath227 . the weak lensing analysis in @xcite , on which the results from @xcite are based , falls somewhere in between these two approaches . while @xcite does in fact make use of photometric redshifts , they do not simply assume that the source redshift distribution is identical to the photometric redshift distribution . rather , they construct a probability distribution @xmath228 , and use this probability to estimate @xmath225 . as it turns out , evaluating @xmath225 in this way leads to results that are nearly identical to those obtained by simply setting @xmath229 . thus , even though the approach used in @xcite is more sophisticated than the simple case considered in @xcite , we expect the @xcite results to be biased but correctable as prescribed in @xcite . this correction amounts to a boost of the lensing masses by a factor of @xmath230 . the statistical error bar in the correction is added in quadrature to the statistical error bar from our fit , which results in @xmath231\times ( n/20)^{1.18\pm 0.09 } \label{eq : johnston_mass}\ ] ] these new values for the @xcite data are in considerably better agreement with those of @xcite . there remains , however , a systematic @xmath164 difference between the two amplitudes , as well as a small difference @xmath232 between the two slopes . a possible culprit for this systematic @xmath164 offset is the difference in how miscentering is accounted for in the data models . the word miscentering refers to the fact that when finding clusters , one will inevitably find clusters that are improperly centered , either due to failures of the cluster finding algorithm , or simply because there is no obvious center of the cluster based on its optical image . such offsets between the true and assigned centers are problematic because if a cluster is miscentered , the corresponding lensing signal is weakened , resulting in systematically low mass estimates . to determine whether the remaining offset between @xcite and @xcite is consistent with differences in the miscentering model , we refit our data assuming no errors on the miscentering corrections . we find @xmath233 with a correlation coefficient @xmath234 . note that these errors are smaller than the errors quoted before , as they should be , given that this new fit does not marginalize over a wide range of miscentering models . by subtracting the two sets of errors in quadrature , we find that the miscentering priors adopted in @xcite correspond to an error @xmath235 in the amplitude and @xmath161 in the slope . thus , the @xcite mass measurements are well within the centering error included in the analysis of @xcite . nevertheless , it is unclear whether miscentering can in fact account for the difference between the @xcite and @xcite results . more specifically , @xcite also performed their analysis including the @xcite model for miscentering , and find after applying the centering correction their best fit @xmath236 relation becomes @xmath237 comparing this to equation [ eq : johnston_mass ] , we find including a miscentering correction in the @xcite analysis increases the tension between the two results . moreover , it suggests that the difference between the two results is due to some other form of systematic difference between the two analysis pipelines . in light of this , we opt for introducing a systematic correction to the @xcite results of @xmath238 and @xmath239 for the amplitude and slope respectively . we also introduce systematic errors of the same magnitude as this systematic correction , so that our final result is @xmath240(n/20)^{1.13\pm 0.09\ ( stat ) \pm 0.05\ ( sys)}.\ ] ] note the central values of the original @xcite analysis ( corrected for photometric redshift bias ) as well as the @xcite results both with and without miscentering corrections are all encompassed by our systematic error . now , in this work we are interested more in the @xmath241 ( henceforth simply @xmath174 ) relation than in the @xmath242 relation , since it is the former mass which is accessible to x - ray studies . to constrain the @xmath174 relation we use the quoted @xmath30 mass measurements from @xcite , re - scaling their @xmath243 errors to @xmath30 by assuming the relative errors are constant . a fit to the data results in @xmath244 ( n/40)^{1.11 \pm 0.08}\ ] ] with a correlation coefficient @xmath245 . we now boost this expression by factor @xmath246 due to the photometric redshift bias correction , and add the systematic corrections @xmath238 and @xmath239 to the amplitude and slope respectively as per our discussion of the @xmath236 relation . we also include a systematic error on the amplitudes and slopes of this same magnitude . we obtain @xmath247 the final systematics we consider here are the purity and completeness of the sample . now , as long as the completeness is not correlated with mass , completeness should not in any way bias the recovered parameters of the @xmath174 relation , though it obviously affects the error bars due to lower statistics . the same can not be said of purity . if only a fraction @xmath248 of the clusters are actually good matches to real halos in the universe , then a fraction @xmath249 of the clusters will have a lensing signal that is significantly different from the mean signal . as an extreme case , we can consider what happens if a fraction @xmath249 of the clusters had no mass associated with them . in that case , the observed mean mass is simply @xmath250 where @xmath251 is the true mean , so one should boost the observed masses by a factor of @xmath252 to obtain an unbiased estimate . for @xmath253 , this amounts to an increase in @xmath140 of magnitude @xmath254 . now , @xcite showed that the purity of the maxbcg cluster sample is expected to be above @xmath203 over the range or richnesses considered here , and the increase in @xmath140 quoted above is undoubtedly an overestimate of the necessary correction since even false cluster detections will have excess mass associated with them . in light of this , we have adopted a one - sided systematic error bar @xmath255 to take into account the impact of purity in the recovered @xmath174 relation . the error bar is one sided since we expect impurities will tend to decrease the observed mean mass . we can , however , turn this prior into a normal double - sided prior by including a systematic correction @xmath256 to the central value , and setting the systematic error bar to the same magnitude as the central value shift . we can also get a rough estimate for the systematic error on the purity by assuming that the quoted systematic error in the amplitude should be made only in the limit of high or low richness . if that were the case , using the fact the slope is measured over a decade of richness values , the corresponding slope would be @xmath257 which amounts to a systematic offset @xmath258 . these systematic error bars are added linearly to our previous systematic error . our final set of priors for the @xmath174 relation is @xmath259 with a correlation coefficient @xmath245 between the two statistical errors . the priors in the @xmath260 relation come from repeating the analysis described in @xcite , but with @xmath28 defined as the x - ray luminosity in the 0.5 - 2.0 kev band , and corrected for aperture effects . as in @xcite , we restrict this analysis to clusters with @xmath261 . we begin by measuring the stacked mean @xmath260 relation and scatter on a fixed @xmath262 scale @xmath263 where we have measured @xmath28 in units of @xmath264 , with a pivot point of @xmath265 . we emphasize that the scatter determined above is the total scatter in the observed @xmath260 relation that can not be attributed to poisson uncertainties in the rosat photon counts . in particular , the quoted scatter is affected by possible point source contamination , agn activity , cool cores , cluster mergers , etc . there are multiple systematic errors that can affect the derived parameters for the @xmath260 relation . these include photometric redshift errors , evolution of the richness parameter @xmath29 , uncorrelated point sources , cluster mis - centering , and cluster agn and cool cores . in addition , we need to account for the fraction of cluster flux lost due to our finite aperture and the rass psf , in order to compare our results with the luminosity measurements of @xcite . we shall now discuss each of these possible systematic effects . @xcite find that the accuracy of the maxbcg photo - z estimates is high enough such that any biases are insignificant relative to the statistical uncertainty of the parameter determinations , and can thus be safely ignored . however , @xcite did find significant redshift evolution in the @xmath260 relation , well above the expected self - similar evolution . similar redshift evolution is found in @xcite ; the reason for the systematic undercounting of cluster members at high redshift is explained in @xcite . we have estimated the effect of this redshift evolution on our derived scatter parameter via a simple monte carlo , and confirm that although the apparent evolution is strong , it is insignificant relative to the intrinsic scatter . therefore , we may also safely ignore this possible systematic effect . we now take a combined approach to the systematic effects due to cluster mis - centering , a finite aperture , the rass psf and uncorrelated point sources . the first three effects are strongly related , in that they all tend to scatter cluster photons out of our initial fixed @xmath262 aperture , and these may affect the normalization , slope , and scatter in the @xmath260 relation . uncorrelated point sources should not affect the mean relation because the large number of stacked sources smooths out the foreground and background . however , when uncorrelated point sources are aligned with individual clusters they may increase the measured scatter by boosting the apparent @xmath28 . we have estimated the effects of these systematics by running a monte carlo with simulated rass data on top of random backgrounds selected from the area of the rass photon map that overlaps with the maxbcg mask . we first resample the maxbcg richness function 100 times . each cluster is given a redshift drawn from the maxbcg redshift distribution , as well as a random postion on the sky selected from the area of the rass survey that overlaps with the maxbcg mask . after we select the richest 1000 clusters in each realization , each cluster is given a luminosity based on the mean relation from eqn . [ eqn : lxnmean ] and an input intrinsic scatter , @xmath266 . each cluster luminosity is then converted to a number of photon counts according to the rass exposure at the given point , and scattered by poisson uncertainties . then , each cluster is given a position offset according to the maxbcg miscentering distribution described in ( * ? ? ? * see 4.3 ) . the cluster profiles are assumed to follow a @xmath53 model , @xmath267 . to ensure we are on similar footing as @xcite , we randomly assign each cluster @xmath53 model parameters uniformly in the range @xmath268 and @xmath269 . finally , the photons are scattered according to the rass psf , following the method of ( * ? ? ? we then calculate the stacked mean relation and scatter as described in @xcite . figure [ fig : lnsystematics ] summarizes the results from our systematic tests . the x - axis shows the input intrinsic scatter , @xmath270 . the y - axis shows the ratio of the input parameter to output parameter for the normalization @xmath144 ( circles ) , slope @xmath146 ( diamonds ) , and scatter @xmath148 ( squares ) . we note that when @xmath271 then @xmath272 , which can not be displayed on the plot . this is consistent with our expectation that uncorrelated sources may boost the observed scatter , especially with low intrinsic scatter . overall , we find that ( a ) the slope @xmath146 is not significantly biased ; ( b ) at moderate to large scatter ( @xmath273 ) the intrinsic scatter @xmath148 is not significantly biased ; and ( c ) the output normalization @xmath144 must be boosted by a factor of @xmath274 to account for the flux lost to miscentering , the finite aperture , and rass psf effects . our priors become then @xmath275 in addition to these corrections , we also need to take into account systematic uncertainties due to purity and completeness in the sample . just as with the weak lensing mass estimates , completeness should not affect the measured @xmath260 relation , whereas purity will tend to suppress the x - ray luminosity at fixed richness . following the same procedure as in appendix [ sec : m - n_priors ] , we derive systematic errors @xmath276 and @xmath277 , which we add linearly to our previous systematic error estimates . finally , we have repeated our scatter analysis using not just the 1000 richest clusters , but also the 2000 richest clusters , in which case we find @xmath278 . to take into account this variation in our analysis , we also introduce a systematic error @xmath279 . our final set of priors is @xmath280 as discussed in section [ sec : rough ] , our analysis hinges on the fact that we can use prior knowledge about the @xmath13 relation to constrain the @xmath174 relation . here , we use the results of @xcite to put priors on the @xmath13 relation , which may be summarized as , as appropriate for the maxbcg sample . ] @xmath281 we report a prior on @xmath107 because at @xmath282 the @xmath13 parameters derived from the @xcite sample are correlated . to decouple them , one needs to shift to the statistical pivot point @xmath283 and introduce the scatter dependence quoted above ( vikhlinin , private communication ) . these constraints are derived from chandra observations of clusters in the 400d cluster catalog @xcite , which allowed @xcite to measure @xmath54 and thereby infer cluster mass using the @xmath55 relation . this relation was itself calibrated on a cluster subsample for which masses were derived using the standard hydrostatic equilibrium argument . this last point is very important , since simulations suggest that hydrostatic mass estimates of clusters may be biased low by @xmath284 ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? one way to calibrate such uncertainties is to compare weak lensing mass estimates to hydrostatic mass estimates . there are several examples of this type of approach . for instance , @xcite have performed such an analysis using the weak lensing mass estimates of @xcite , and find @xmath285 a similar analysis has been carried out by @xcite , who used the weak lensing mass estimates of @xcite and their own analysis of chandra public data to obtain @xmath286 . finally , using xmm x - ray observations and the weak lensing data of @xcite , @xcite , and @xcite , @xcite find @xmath287 . @xcite also note , however , that a histogram of @xmath288 peaks at a ratio of @xmath289 , and that clusters in the tails of the distribution tend to have tight error bars , possibly biasing the error weighted ratio . in light of this , we have opted for a `` middle of the road '' approach , and introduce a correction factor @xmath290 . our corresponding prior is @xmath291 estimating systematic errors in @xmath152 and @xmath154 is difficult . for instance , comparisons with weak lensing masses are not an effective way of assessing systematics because weak lensing mass estimates are so noisy : trying to fit a power law relation between @xmath292 and @xmath293 results in very large errors for the slope of the relation . one alternative is to consider multiple studies of the @xmath13 relation in order to asses how sensitive the recovered parameters are to the analysis pipeline . unfortunately , such an excercise is far from trivial . one difficulty is the fact that there is very little agreement on the meaning of @xmath28 , with many works focusing on core - excised and/or core - corrected bolometric x - ray lumunisoties ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? even among those works that also explore the @xmath13 relation when @xmath28 is a soft x - ray band luminosity ( e.g. * ? ? ? * ; * ? ? ? * ) , there are still important differences in the aperture used to estimate @xmath28 . in principle , we could attempt to convert between the various definitions of @xmath28 to try to compare the works against each other , but many of these @xmath13 measurements are affected by malmquist bias , making comparisons to the @xcite results difficult . one work that does constrain the the soft x - ray band , non - core excised , malmquist bias corrected @xmath13 relation is @xcite . unfortunately , the energy band they use is slightly different from that of of @xcite , so even here comparison is not trivial . we expect , however , that at least the scatter and slopes of the @xmath13 relation will not be strongly affected by the minor differences between the two @xmath28 definitions . given our purposes , the interesting thing about the @xcite results is that they use a very different methodology for constraining the @xmath13 relation . in particular , they assume knowledge of cosmological parameters , and then use the observed cluster x - ray luminosity function to constrain @xmath294 . assuming their `` compromise cosmology '' , which they argue gives the best results , they find @xmath295 and @xmath296 . these values are in excellent agreement with those of @xcite , and suggest that placing additional systematic errors in the @xmath13 parameters is not really necessary at this point . table [ tab : mfdata ] presents the mean and covariance matrix of the mass function data derived from our analysis . these results represent the state of the art mass function measurements at low redshift from optically derived cluster catalogs . we emphasize we assumed @xmath43 and @xmath39 , so appropriate rescaling must be applied if the results are to be compared against significantly different cosmologies . note that the covariance matrix data in table [ tab : mfdata ] is normalized such that the diagonal entries are the fractional error @xmath297 , while the off diagonal entries are the correlation coefficients @xmath298 . we present the data in this way since it is easier to understand when expressed this way . the actual values for the covariance matrix are easily reconstructed from the data in the table . @xmath299 & 3.22 & 3.70 & 4.26 & 4.91 & 5.65 & 6.50 & 7.49 & 8.62 & 9.92 & 11.42 & 13.14 & 15.12 & 17.41 & 20.04 & 23.07 & 26.55 & 30.56 + 3.22 & 7.90e-7 & 0.22 & 0.82 & 0.77 & 0.72 & 0.66 & 0.61 & 0.55 & 0.50 & 0.44 & 0.39 & 0.35 & 0.30 & 0.25 & 0.21 & 0.18 & 0.15 & 0.12 + 3.70 & 5.61e-7 & 0.82 & 0.24 & 0.79 & 0.76 & 0.71 & 0.67 & 0.62 & 0.57 & 0.52 & 0.46 & 0.41 & 0.36 & 0.31 & 0.27 & 0.23 & 0.19 & 0.16 + 4.26 & 3.92e-7 & 0.77 & 0.79 & 0.27 & 0.77 & 0.74 & 0.70 & 0.66 & 0.61 & 0.57 & 0.52 & 0.46 & 0.41 & 0.36 & 0.31 & 0.26 & 0.22 & 0.18 + 4.91 & 2.70e-7 & 0.72 & 0.76 & 0.77 & 0.30 & 0.75 & 0.72 & 0.68 & 0.64 & 0.59 & 0.55 & 0.50 & 0.44 & 0.38 & 0.34 & 0.29 & 0.24 & 0.20 + 5.65 & 1.82e-7 & 0.66 & 0.71 & 0.74 & 0.75 & 0.35 & 0.72 & 0.69 & 0.65 & 0.61 & 0.57 & 0.52 & 0.46 & 0.41 & 0.36 & 0.30 & 0.26 & 0.22 + 6.50 & 1.21e-7 & 0.61 & 0.67 & 0.70 & 0.72 & 0.72 & 0.41 & 0.68 & 0.65 & 0.62 & 0.57 & 0.52 & 0.47 & 0.42 & 0.37 & 0.32 & 0.27 & 0.22 + 7.49 & 7.93e-8 & 0.55 & 0.62 & 0.66 & 0.68 & 0.69 & 0.68 & 0.47 & 0.64 & 0.60 & 0.57 & 0.52 & 0.47 & 0.42 & 0.37 & 0.32 & 0.27 & 0.23 + 8.62 & 5.11e-8 & 0.50 & 0.57 & 0.61 & 0.64 & 0.65 & 0.65 & 0.64 & 0.55 & 0.59 & 0.55 & 0.51 & 0.47 & 0.41 & 0.37 & 0.32 & 0.27 & 0.23 + 9.92 & 3.24e-8 & 0.44 & 0.52 & 0.57 & 0.59 & 0.61 & 0.62 & 0.60 & 0.59 & 0.65 & 0.53 & 0.49 & 0.45 & 0.40 & 0.36 & 0.31 & 0.26 & 0.22 + 11.42 & 2.03e-8 & 0.39 & 0.46 & 0.52 & 0.55 & 0.57 & 0.57 & 0.57 & 0.55 & 0.53 & 0.76 & 0.47 & 0.43 & 0.38 & 0.34 & 0.30 & 0.25 & 0.21 + 13.14 & 1.25e-8 & 0.35 & 0.41 & 0.46 & 0.50 & 0.52 & 0.52 & 0.52 & 0.51 & 0.49 & 0.47 & 0.92 & 0.40 & 0.36 & 0.32 & 0.28 & 0.24 & 0.20 + 15.12 & 7.61e-9 & 0.30 & 0.36 & 0.41 & 0.44 & 0.46 & 0.47 & 0.47 & 0.47 & 0.45 & 0.43 & 0.40 & 1.11 & 0.33 & 0.30 & 0.26 & 0.22 & 0.19 + 17.41 & 4.53e-9 & 0.25 & 0.31 & 0.36 & 0.38 & 0.41 & 0.42 & 0.42 & 0.41 & 0.40 & 0.38 & 0.36 & 0.33 & 1.36 & 0.26 & 0.24 & 0.20 & 0.17 + 20.04 & 2.63e-9 & 0.21 & 0.27 & 0.31 & 0.34 & 0.36 & 0.37 & 0.37 & 0.37 & 0.36 & 0.34 & 0.32 & 0.30 & 0.26 & 1.74 & 0.21 & 0.19 & 0.16 + 23.07 & 1.48e-9 & 0.18 & 0.23 & 0.26 & 0.29 & 0.30 & 0.32 & 0.32 & 0.32 & 0.31 & 0.30 & 0.28 & 0.26 & 0.24 & 0.21 & 2.22 & 0.17 & 0.14 + 26.55 & 8.29e-10 & 0.15 & 0.19 & 0.22 & 0.24 & 0.26 & 0.27 & 0.27 & 0.27 & 0.26 & 0.25 & 0.24 & 0.22 & 0.20 & 0.19 & 0.17 & 2.88 & 0.12 + 30.56 & 4.41e-10 & 0.12 & 0.16 & 0.18 & 0.20 & 0.22 & 0.22 & 0.23 & 0.23 & 0.22 & 0.21 & 0.20 & 0.19 & 0.17 & 0.16 & 0.14 & 0.12 & 3.88

we measure the logarithmic scatter in mass at fixed richness for clusters in the maxbcg cluster catalog , an optically selected cluster sample drawn from sdss imaging data . our measurement is achieved by demanding consistency between available weak lensing and x - ray measurements of the maxbcg clusters , and the x - ray luminosity mass relation inferred from the 400d x - ray cluster survey , a flux limited x - ray cluster survey . we find @xmath0 ( @xmath1 cl ) at @xmath2 , where @xmath3 is the number of red sequence galaxies in a cluster . as a byproduct of our analysis , we also obtain a constraint on the correlation coefficient between @xmath4 and @xmath5 at fixed richness , which is best expressed as a lower limit , @xmath6 . this is the first observational constraint placed on a correlation coefficient involving two different cluster mass tracers . we use our results to produce a state of the art estimate of the halo mass function at @xmath7 the median redshift of the maxbcg cluster sample and find that it is consistent with the wmap5 cosmology . both the mass function data and its covariance matrix are presented .

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Proceed to summarize the following text: a small fraction of massive stars end their lives with spectacular explosions one or two orders of magnitude more luminous than normal supernovae ( sne ) . after the initial puzzling discoveries of the luminous sne 2005ap @xcite and 2006gy @xcite , modern wide - field surveys over the past decade began to uncover these superluminous sne ( slsne ) in greater numbers . the energy scales involved in these explosions challenge our understanding of conventional sn explosions . normal sne resulting from iron core collapse have characteristic energy scales of @xmath310@xmath15 erg of kinetic energy and @xmath310@xmath16 erg emitted as optical radiation ( @xmath310@xmath17 erg s@xmath6 for @xmath310@xmath18 s ) . the slsne are far off this scale they peak at optical luminosities of up to @xmath34@xmath19 erg s@xmath6 @xcite and emit a total of up to 4@xmath410@xmath15 erg optically @xcite . this large energy scale motivates the question of what physics powers these sne , and how to accommodate these objects within the standard understanding of massive star evolution . theorists have proposed a number of exotic power sources , including the pair instability mechanism ( e.g. , @xcite ) and reprocessed spindown energy released by a newly formed magnetar @xcite . another possibility is interaction with a dense circumstellar medium ( csm ) @xcite , requiring extreme csm masses and densities whose origin remains unexplained ( see @xcite for one possibility ) . all of these models require additional ingredients beyond the normal stellar evolutionary processes . @xcite has attempted to impose order on the menagerie of objects achieving sufficient peak luminosities to be classified as slsne ( @xmath20 mag is a typical requirement ) by sorting them into three categories . all of the hydrogen - rich objects were classified as slsne - ii and all exhibit signs of being powered by dense csm interaction , with the possible exception of sn 2008es @xcite . he split the objects lacking hydrogen into two classes , the rare slsne - r that have slow photometric decline rates consistent with being powered by the radioactive decay of a very large synthesized mass of @xmath21ni , and the relatively homogeneous class of slsne - i , whose power source is still mysterious . a few caveats have been raised . the slsne - r are interpreted to be the results of pair - instability sne . however , existing models for the pair instability process prefer extremely low metallicity , and may be in conflict with the observed spectrum and spectral energy distribution ( sed ) of slsne - r ( e.g. , @xcite ) . also , it is not clear how homogeneous the slsne - i class really is . although the spectra of most appear to be similar to those of sn 2005ap and scp06f6 @xcite , the rise times and peak luminosities of published objects vary by factors of @xmath35 @xcite . all slsne - i to date have had hot spectra and been bright in the rest - frame near - ultraviolet ( nuv ) relative to normal sn seds . in this paper , we present the discovery of ps1 - 10afx , an extreme slsn at redshift @xmath0 that does not fit into this classification scheme and is distinct from all previous slsne . the peak luminosity is comparable to the highest known and the rise time is the fastest measured . the spectra show no evidence for hydrogen and lack any analog in the existing sample of slsne . instead , they most closely resemble those of line - blanketed normal sne ic . in section 2 , we present the suite of optical and near - infrared ( nir ) observations . the host galaxy is described in section 3 . we compare our observations of ps1 - 10afx to known sne in section 4 . in section 5 , we construct the sed and bolometric light curve . we then compare ps1 - 10afx to existing slsn models in section 6 . all calculations in this paper assume a flat @xmath22cdm cosmology with @xmath23=74 km s@xmath6 mpc@xmath6 , @xmath24=0.27 , and @xmath25=0.73 @xcite . [ [ section ] ] the pan - starrs1 ( ps1 ) telescope has a 1.8 m diameter primary mirror that images a field with a diameter of 3.3 @xcite onto a total of sixty @xmath26 pixel detectors , with a pixel scale of 0.258@xcite . a more complete description of the ps1 system , hardware and software , is provided by @xcite . the ps1 observations are obtained through a set of five broadband filters , designated as @xmath27 , @xmath28 , @xmath29 , @xmath30 , and @xmath31 . although the filter system for ps1 has much in common with that used in previous surveys , such as the sloan digital sky survey ( sdss ; @xcite ) , there are differences . most important for this work , the @xmath30 filter is cut off at 9300 , giving it a different response than the detector response defined @xmath32 , and sdss has no corresponding @xmath31 filter . further information on the passband shapes is described by @xcite . photometry is in the `` natural '' ps1 system , @xmath33 , with a single zeropoint adjustment @xmath34 made in each band to conform to the ab magnitude scale @xcite . photometry from all other sources presented in this paper is also on the ab scale . ps1 magnitudes are interpreted as being at the top of the atmosphere , with 1.2 airmasses of atmospheric attenuation being included in the system response function . the ps1 medium deep survey ( mds ) consists of 10 fields across the sky that are observed nightly when in season ( @xmath35 months per year ) with a typical cadence of 3 d between observations in @xmath27@xmath28@xmath29@xmath30 in dark and gray time , while @xmath31 is used near full moon . ps1 data are processed through the image processing pipeline ( ipp ; @xcite ) on a computer cluster at the maui high performance computer center . the pipeline runs the images through a succession of stages , including flat - fielding ( `` de - trending '' ) , a flux - conserving warping to a sky - based image plane , masking and artifact removal , and object detection and photometry . transient detection using ipp photometry is carried out at queen s university belfast . independently , difference images are produced from the stacked nightly mds images by the ` photpipe ` pipeline @xcite running on the odyssey computer cluster at harvard university . the discovery and data presented here are from the ` photpipe ` analysis . ps1 - 10afx was first detected in mds imaging on 2010 august 31.35 ( ut dates are used throughout this paper ) at a position of @xmath35 = 22@xmath3611@xmath3724162 , @xmath38 = @xmath390009@xmath404349 ( j2000 ) , with an uncertainty of 0@xmath411 in each coordinate . the strong detections in @xmath29 and @xmath30 combined with non - detections in @xmath27 and @xmath28 over the next few nights immediately garnered our attention . the unusual colors are evident in the color images of the field presented in figure [ hostfig ] . we constructed deep template stacks from all pre - explosion images and subtracted them from the ps1 observations using ` photpipe ` @xcite . details of the photometry and generation of ps1 sn light curves will be given by rest et al . ( 2013 , in prep . ) and scolnic et al . ( 2013 , in prep . ) . the typical spacing between mds observations in the same filter corresponds to only 1.3 d in the rest frame of ps1 - 10afx , so in some cases we have co - added the photometry from adjacent observations to increase the significance of marginal detections or the depth of the non - detections . the final ps1 - 10afx photometry , after correction for @xmath42 mag of galactic extinction @xcite , is given in table [ phottab ] and shown in figure [ multiplot ] . we fit polynomials to the @xmath30 data points within 15 rest - frame days of peak to determine the time of maximum light , for which we adopt a modified julian date ( mjd ) of 55457.0 ( = 2010 september 18.0 ) . all phases referred to subsequently are in rest - frame days referenced from this date . in addition to the ps1 observations , we obtained two epochs of multicolor photometry using the gemini multi - object spectrographs ( gmos ; @xcite ) on the 8-m gemini - north and south telescopes and one epoch of imaging using the inamori - magellan areal camera and spectrograph ( imacs ; @xcite ) on the 6.5-m magellan baade telescope . the images were processed using standard tasks and then archival fringe frames were subtracted from the gmos images using the ` gemini ` iraf package . we used several sdss stars in the field to calibrate the gemini images , while the imacs zeropoints were checked with observations of standard star fields obtained the same night . we subtracted the deep ps1 templates in the corresponding filter from each image using the isis software package @xcite to correct for host contamination . the final photometry is listed in table [ phottab ] . the @xmath43 and @xmath44 magnitudes agree well with the ps1 observations at similar epochs . however , the @xmath45 observations exhibit an offset from the @xmath30 light curve due to the filter response differences noted above ( gmos and imacs are closer to sdss ) . we therefore add 0.2 mag to the @xmath45 photometry points for consistency with @xmath30 whenever we refer to the combined @xmath46-band light curve . finally , we obtained late - time observations of the host galaxy in @xmath47 and @xmath43 on 2011 october 21.1 ( phase @xmath39167 d ) using the low dispersion survey spectrograph-3 ( ldss3 ) on the 6.5-m magellan clay telescope ( table [ hosttab ] ) . lrcccc 55440.5 & @xmath486.9 & @xmath27 & @xmath4923.50 & & ps1 + 55445.0 & @xmath485.0 & @xmath27 & @xmath4923.91 & & ps1 + 55450.9 & @xmath482.6 & @xmath27 & @xmath4923.77 & & ps1 + 55455.4 & @xmath480.7 & @xmath27 & @xmath4923.22 & & ps1 + 55470.3 & 5.6 & @xmath27 & @xmath4923.68 & & ps1 + 55480.7 & 9.9 & @xmath27 & @xmath4923.85 & & ps1 + 55439.0 & @xmath487.5 & @xmath28 & @xmath4923.13 & & ps1 + 55445.0 & @xmath485.0 & @xmath28 & @xmath4923.47 & & ps1 + 55450.2 & @xmath482.8 & @xmath43 & 24.18 & 0.30 & gmos - n + 55450.9 & @xmath482.5 & @xmath28 & 23.87 & 0.41 & ps1 + 55455.4 & @xmath480.7 & @xmath28 & @xmath4923.42 & & ps1 + 55465.0 & 3.4 & @xmath43 & 23.73 & 0.20 & gmos - s + 55470.3 & 5.6 & @xmath28 & @xmath4923.63 & & ps1 + 55480.8 & 9.9 & @xmath28 & @xmath4923.55 & & ps1 + 55425.0 & @xmath4813.4 & @xmath29 & @xmath4924.21 & & ps1 + 55438.5 & @xmath487.7 & @xmath29 & @xmath4923.84 & & ps1 + 55441.5 & @xmath486.5 & @xmath29 & 23.10 & 0.16 & ps1 + 55444.4 & @xmath485.3 & @xmath29 & 22.57 & 0.12 & ps1 + 55447.4 & @xmath484.0 & @xmath29 & 22.32 & 0.06 & ps1 + 55450.4 & @xmath482.8 & @xmath29 & 22.15 & 0.11 & ps1 + 55450.2 & @xmath482.8 & @xmath44 & 22.18 & 0.11 & gmos - n + 55453.4 & @xmath481.5 & @xmath29 & 22.00 & 0.05 & ps1 + 55456.4 & @xmath480.3 & @xmath29 & 21.99 & 0.05 & ps1 + 55465.0 & 3.4 & @xmath44 & 22.25 & 0.16 & gmos - s + 55465.5 & 3.5 & @xmath29 & 22.28 & 0.20 & ps1 + 55468.3 & 4.7 & @xmath29 & 22.39 & 0.12 & ps1 + 55477.2 & 8.5 & @xmath29 & 23.20 & 0.19 & ps1 + 55480.2 & 9.7 & @xmath29 & 23.28 & 0.42 & ps1 + 55483.2 & 11.0 & @xmath29 & 23.45 & 0.22 & ps1 + 55486.3 & 12.2 & @xmath29 & @xmath4922.90 & & ps1 + 55499.8 & 17.9 & @xmath29 & @xmath4923.87 & & ps1 + 55508.8 & 21.7 & @xmath29 & @xmath4924.08 & & ps1 + 55513.1 & 23.5 & @xmath44 & @xmath4925.01 & & imacs + 55426.0 & @xmath4813.0 & @xmath30 & @xmath4923.47 & & ps1 + 55436.4 & @xmath488.6 & @xmath30 & @xmath4922.79 & & ps1 + 55439.3 & @xmath487.4 & @xmath30 & 23.06 & 0.37 & ps1 + 55442.5 & @xmath486.1 & @xmath30 & 22.83 & 0.37 & ps1 + 55445.4 & @xmath484.8 & @xmath30 & 22.24 & 0.15 & ps1 + 55448.4 & @xmath483.6 & @xmath30 & 21.88 & 0.12 & ps1 + 55450.2 & @xmath482.8 & @xmath45 & 21.59 & 0.13 & gmos - n + 55454.4 & @xmath481.1 & @xmath30 & 21.69 & 0.10 & ps1 + 55457.3 & 0.1 & @xmath30 & 21.76 & 0.11 & ps1 + 55465.0 & 3.4 & @xmath45 & 21.44 & 0.19 & gmos - s + 55466.4 & 3.9 & @xmath30 & 21.82 & 0.10 & ps1 + 55475.2 & 7.6 & @xmath30 & 22.11 & 0.17 & ps1 + 55478.2 & 8.9 & @xmath30 & 22.13 & 0.17 & ps1 + 55481.2 & 10.1 & @xmath30 & 21.98 & 0.19 & ps1 + 55484.2 & 11.4 & @xmath30 & 22.77 & 0.30 & ps1 + 55499.2 & 17.7 & @xmath30 & @xmath4923.04 & & ps1 + 55513.1 & 23.5 & @xmath45 & 23.83 & 0.30 & imacs + 55463.4 & 2.7 & @xmath31 & 21.09 & 0.16 & ps1 + 55492.8 & 15.0 & @xmath31 & @xmath4921.95 & & ps1 + 55523.3 & 27.8 & @xmath31 & @xmath4921.82 & & ps1 + 55463.4 & 2.7 & @xmath50 & 20.96 & 0.10 & niri + 55463.3 & 2.6 & @xmath51 & 20.99 & 0.17 & mmirs + 55463.4 & 2.7 & @xmath51 & 21.19 & 0.10 & niri + 55481.1 & 10.1 & @xmath51 & 22.22 & 0.26 & mmirs + 55485.0 & 11.7 & @xmath51 & 22.28 & 0.30 & mmirs + 55496.0 & 16.3 & @xmath51 & 22.19 & 0.10 & hawk - i + 55515.1 & 24.3 & @xmath51 & 22.84 & 0.14 & hawk - i + 55552.0 & 39.8 & @xmath51 & @xmath4921.90 & & hawk - i + 55561.0 & 43.6 & @xmath51 & @xmath4921.90 & & hawk - i + 55463.4 & 2.7 & @xmath52 & 21.20 & 0.19 & niri + 55485.0 & 11.7 & @xmath52 & 21.05 & 0.44 & mmirs + 55490.1 & 13.9 & @xmath52 & 21.94 & 0.24 & hawk - i + 55515.1 & 24.3 & @xmath52 & 22.28 & 0.39 & hawk - i + 55463.4 & 2.7 & @xmath53 & 21.46 & 0.26 & niri + 55485.0 & 11.7 & @xmath54 & @xmath4921.35 & & mmirs [ phottab ] the high redshift and red optical colors of ps1 - 10afx indicate that nir photometry is required to constrain the sed and bolometric luminosity . we received director s discretionary ( dd ) time at gemini - north to obtain @xmath55 photometry near maximum light using the near infrared imager and spectrometer ( niri ; @xcite ) . the observations were acquired on 2010 september 24.4 , at a phase of @xmath392.7 d. additional observations were obtained over the subsequent month in @xmath56 with the mmt & magellan infrared spectrograph ( mmirs ; @xcite ) on the magellan clay telescope . to follow the light curve in the rest - frame optical to late times , we obtained a series of @xmath57 observations using dd time on the 8-m very large telescope ( yepun ) with the high - acuity wide - field k - band imager ( hawk - i ; @xcite ) . finally , we used the fourstar infrared camera @xcite on the magellan baade telescope to obtain late - time @xmath56 observations of the host galaxy of ps1 - 10afx on 2011 september 18 , 2011 september 21 , and 2011 december 7 ( phases @xmath39153 d , @xmath39154 d , and @xmath39186 d ) , respectively ( table [ hosttab ] ) . the images were flat fielded , sky subtracted , and stacked using standard tasks in iraf , including the ` gemini ` package for the niri data , except for the hawk - i data , which were processed using the instrument pipeline . the images from each instrument were then calibrated using the same 2mass stars in each field , except for the @xmath50 image , which was calibrated using archival niri zeropoints . the @xmath52 and @xmath54 host galaxy fluxes were then subtracted numerically from each datapoint in those filters . for the @xmath50 data point , we interpolated the host galaxy flux between the measured values in @xmath31 and @xmath51 to subtract off the host galaxy contribution ( only a small correction ) . we were able to perform image subtraction on most of the @xmath51 images using the isis software package with the late - time fourstar image as a template , but the others had the host @xmath51 flux subtracted numerically . the final host - corrected photometry points are shown in figure [ nirplot ] and listed in table [ phottab ] , including the calibration uncertainty in the errors . after noticing the unusually red @xmath58 colors of ps1 - 10afx , we immediately triggered optical spectroscopy using gmos - n . a pair of dithered 1800 s observations were taken on 2010 september 11.26 ( phase @xmath59 d ) with the r400 grating and og515 order - blocking filter to cover the wavelength range of 5400@xmath489650 . we used standard tasks in iraf to perform basic two - dimensional image processing and spectral extraction . we used our own idl procedures to apply a flux calibration and correct for telluric absorption based on archival observations of spectrophotometric standard stars . the observations were performed at a mean airmass of 1.5 , but the 1-wide long slit was oriented within 20 of the parallactic angle @xcite , so the overall spectral shape is reliable at these red wavelengths . a second epoch of spectroscopy was obtained on 2010 september 26.04 ( phase @xmath393.4 d ) using gmos - s . the observations and instrumental setup were similar to the first epoch , except that a redder grating tilt was used to cover the range 5880@xmath4810100 . this spectrum has a lower signal - to - noise ratio ( s / n ) than the first one , but the main sn features are still present . we attempted a final epoch of spectroscopy using gmos - s in nod - and - shuffle mode on 2010 november 5 , but ps1 - 10afx had faded significantly , so we only detected a faint continuum with a hint of sn features . we do not consider this spectrum further . we observed ps1 - 10afx with the folded - port infrared echellette ( fire ; @xcite ) spectrograph on the magellan baade telescope on 2010 september 18.17 ( phase @xmath390.1 d ) , using the high throughput , low - resolution prism mode to cover the range of 0.8@xmath482.5 @xmath60 m . the spectral resolution is a strongly decreasing function of wavelength , but is @xmath61 in @xmath51 . six 150 s dithered exposures were obtained for a total of 900 s of on - source time . despite the short exposure time , the continuum of ps1 - 10afx is well detected blueward of @xmath31.6 @xmath60 m . we reduced and combined the spectra using the firehose package , including a correction for telluric absorption obtained from observations of the a0v star hd 208368 using the algorithm of @xcite . we obtained a mmirs spectrum on 2010 september 28 ( phase @xmath394.2 d ) , using the @xmath62 filter and @xmath51 grism to cover the range 1.0@xmath481.34 @xmath60 m . a total of 90 minutes were spent on source in a series of dithered 300 s exposures . we used standard tasks in iraf to combine and extract the spectra . the final optical and nir spectra are shown in figure [ specplot ] . all of our optical spectra exhibit an emission line near 8903 , which we identify as [ ] @xmath633727 emission from the host galaxy , as well as absorption from the @xmath632800 doublet at the same redshift of @xmath0 , which we adopt as the sn redshift . popular models for slsne do not predict detectable radio emission at high redshift , as discussed by @xcite . however , given its unusual properties , we observed ps1 - 10afx on 2010 september 12.2 ( phase @xmath482.4 d ) with the karl g. jansky very large array @xcite as part of our nrao key science project `` exotic explosions , eruptions , and disruptions : a new transient phase - space , '' with 256 mhz of bandwidth centered at 4.96 ghz . the data were reduced with standard tasks in the astronomical image processing system ( aips ; @xcite ) , using j2212 + 0152 as the gain calibrator and 3c48 as the absolute flux density calibrator . these observations yielded a non - detection of 19@xmath6418 @xmath60jy . we measure a host redshift of @xmath65 from fits to the [ ] line in our highest s / n spectrum ( phase @xmath482.8 d ) . a double - gaussian fit to the @xmath632800 doublet absorption in the same spectrum ( inset in figure [ specplot ] ) is blueshifted by 120@xmath6412 km s@xmath6 relative to the [ ] rest frame . such an offset is typical of rest - frame uv selected star - forming galaxies at this redshift @xcite and was also seen in observations of the slsn ps1 - 11bam @xcite . this effect has been interpreted to be caused by absorption occurring in galactic - scale outflows driven by star formation . the rest - frame equivalent widths ( @xmath66 ) of the two lines are @xmath66(@xmath632796)=1.8@xmath640.2 and @xmath66(@xmath632803)=1.6@xmath640.2 . these values are larger than for ps1 - 11bam @xcite , but slightly lower than the median of the intrinsic absorbers from gamma - ray burst ( grb ) host galaxies @xcite . in addition , @xmath632852 absorption is present , but the uv spectral slope of ps1 - 10afx is so red that the s / n rapidly decreases to the blue , making it hard to confirm the presence of other expected strong interstellar absorption lines , such as @xmath632600 . we obtained photometry of the host galaxy of ps1 - 10afx in eight filters from @xmath47 to @xmath54 ( table [ hosttab ] ) to probe the host stellar population from the rest - frame nuv ( @xmath32000 ) to the nir ( @xmath30.9 @xmath60 m ) . the host is well detected in our deep @xmath29@xmath30@xmath31 ps1 template images , which we supplemented with the observations from other facilities described above . host photometry was performed with a consistent aperture of radius 1.7 in all filters . we fit a suite of single stellar population age models from @xcite to the data , assuming a red horizontal branch morphology , a salpeter initial mass function , and a metallicity of @xmath67 . the best - fit model is shown in figure [ hostplot ] and has an age of 10@xmath14 yr and requires a small amount of internal extinction ( a@xmath68 mag ) . the derived stellar mass is @xmath31.8 @xmath4 10@xmath13 m@xmath69 . the @xmath47 magnitude of the host galaxy corresponds to a nuv continuum luminosity of 1.3@xmath410@xmath70 erg s@xmath6 ( after correction for internal extinction ) , which implies a star formation rate ( sfr ) of @xmath318 m@xmath69 yr@xmath6 using the calibration of @xcite . we also estimate a consistent value of @xmath313 m@xmath12 yr@xmath6 from the observed [ ] flux of @xmath7110@xmath72 erg @xmath73 s@xmath6 @xcite . these sfrs would be a factor of @xmath32 lower without any correction for internal extinction . ccccc @xmath47 & 4825/2020 & 24.01 & 0.10 & ldss3 + @xmath43 & 6170/2580 & 23.69 & 0.10 & ldss3 + @xmath29 & 7520/3150 & 23.43 & 0.13 & ps1 + @xmath30 & 8660/3625 & 22.75 & 0.10 & ps1 + @xmath31 & 9620/4030 & 22.29 & 0.28 & ps1 + @xmath51 & 12500/5230 & 22.02 & 0.14 & fourstar + @xmath52 & 16500/6910 & 21.93 & 0.22 & fourstar + @xmath54 & 21490/9000 & 21.53 & 0.25 & fourstar [ hosttab ] this galaxy is significantly more massive than the previous hosts of slsne . @xcite examined the host galaxies of a sample of luminous sne and found them to have generally low stellar masses and high specific sfrs . in particular , all of the slsne - i in their sample had dwarf hosts , with a median stellar mass of @xmath32@xmath410@xmath74 m@xmath12 . by comparison , ps1 - 10afx has a significantly more massive host with a lower specific sfr ( @xmath38@xmath410@xmath75 yr@xmath6 ) than their sample ( with a median of 3@xmath410@xmath76 yr@xmath6 ) . only the luminous host galaxy of the slsn - ii 2006gy exceeded our estimate for the stellar mass of the host of ps1 - 10afx . objects discovered subsequently have continued this trend , having either dwarf host galaxies or no host at all detected to date despite deep observations @xcite . while it has been argued that the association of slsne with dwarf galaxy hosts is driven by their low metallicity ( e.g. , @xcite ) , the massive host of ps1 - 10afx appears to be inconsistent with a low metallicity . for example , the relationships of @xcite imply a super - solar metallicity of @xmath77(o / h ) = 8.85 for our derived stellar mass and sfr , although we caution that we can not measure the metallicity directly with the available data . an additional point of contrast is provided by the sample of long - duration grbs , whose occurrence is also widely believed to be associated with low - metallicity environments ( e.g. , @xcite ) . the host galaxy of ps1 - 10afx is brighter in @xmath54 than any of the grb host galaxies at similar redshifts in the optically - unbiased sample of @xcite , although a few of the `` dark '' grb hosts studied by @xcite are similarly luminous . the absolute magnitudes of ps1 - 10afx s host prior to any internal extinction correction , @xmath78 mag and @xmath79 mag , are @xmath30.4 mag brighter than @xmath80 , the characteristic magnitudes in @xcite function fits to the field galaxy luminosity functions at this redshift @xcite . the @xmath54 magnitude of the host is also near the median of the distribution of field galaxies at this redshift weighted by star formation rate @xcite . combined , these points suggest that the environment of ps1 - 10afx is representative of typical star - forming galaxies at this redshift , while being distinct from the other hydrogen - poor slsne . the host is marginally resolved in our template images , with a full width at half - maximum ( fwhm ) of @xmath31@xmath412 , or 10 kpc at this redshift . astrometric alignment of difference images taken near maximum light with the templates shows that ps1 - 10afx is aligned with the centroid of the host to within 0@xmath411 ( 0.8 kpc ) . the other slsn with a massive host galaxy , sn 2006gy @xcite , is also close to the nucleus , perhaps implying an unusual star formation environment . despite its extraordinary luminosity ( see section 5 ) , ps1 - 10afx has spectra that more closely resemble those of a normal sn ic than any known slsn . we show our highest s / n spectrum ( phase @xmath482.8 d ) in figures [ uvcomp ] and [ uvcomplog ] along with several comparisons drawn from the literature . relatively few uv spectra of core - collapse sne exist , but the available data allow us to sample a wide variety of phenomena . the most prominent feature in the ps1 - 10afx spectrum is a broad p - cygni feature with an absorption minimum near 3730 , which we identify as the typical h&k absorption seen in most types of sne , blueshifted by about @xmath81 km s@xmath6 . there are several other weaker features in the spectrum in the range 3000@xmath483500 . the @xmath393.4 d gmos spectrum is noisier , but similar , with some evidence that the broad absorption minimum decreased in velocity , although the overlapping [ ] emission and strong night sky residuals make this uncertain . these spectra are in strong contrast to the sn 2005ap - like class of slsne @xcite . ps1 - 10ky is a high - redshift example of this class @xcite and it has a much bluer nuv continuum with no feature ( figure [ uvcomp ] ) . a reasonable concern is that ps1 - 10afx only appears to have redder colors due to host - galaxy extinction . however , matching the uv spectral slope of ps1 - 10afx to ps1 - 10ky requires @xmath82 mag of extinction ( gray line in figure [ uvcomp ] ) , assuming a galactic reddening curve @xcite . this is unsatisfactory for two reasons . first , ps1 - 10afx is already more luminous in the nuv than the sn 2005ap - like objects ( see below ) , so @xmath83 mag would imply an upward correction of two orders of magnitude to an already extreme peak luminosity . second , the strongest uv spectral features of the sn 2005ap - like objects are the trio of features blueward of 3000 first seen in scp06f6 @xcite , with the continuum redward of that being largely smooth except for features @xcite . ps1 - 10afx has fundamentally different spectral features with no correspondence in the ps1 - 10ky spectrum , so the simplest explanation is that it has an intrinsically cooler photosphere . the sn iin 1998s is shown in figure [ uvcomp ] as a well - studied example of a sn undergoing strong csm interaction . sn 1998s at this phase showed no broad p - cygni features in the uv , but had a blue continuum with narrow absorption lines from interstellar and circumstellar material @xcite . the slsne 2006gy and 2006tf are higher - luminosity versions of sne dominated by hydrogen - rich csm interaction @xcite . these objects have redder spectra than sn 1998s , but still exhibit balmer lines and deep narrow absorptions that have no analog in the ps1 - 10afx spectrum . the slsn 2008es had almost featureless blue spectra near maximum light @xcite , but the plotted spectrum is from a later epoch when balmer lines were beginning to develop strength in the rest - frame optical spectrum . the nuv spectrum of the broad - lined sn ic 2002ap is included because the engine - driven sne associated with grbs have similar spectra , even if 2002ap itself was not unusually luminous @xcite . sn 2002ap has much broader and more blended features than ps1 - 10afx . thus , none of these objects in figure [ uvcomp ] resembles ps1 - 10afx . the slsn 2007bi has been identified as a potential pair - instability sn @xcite . it is more similar to ps1 - 10afx , with a definite feature , but is still significantly bluer and has several additional spectral features that do not match . instead , the set of spectra from normal sne plotted in figure [ uvcomplog ] provide a much better basis for comparison . the sne ia and sn 1993j ( type iib ) spectra have been blueshifted by the indicated amounts to approximately match the absorption minima to ps1 - 10afx . over the range 2800@xmath484000 , the spectra in figure [ uvcomplog ] are all very similar , with the dominant features being due to in all objects . a notch near 3950 is from @xmath634130 . an emission peak at 2900 also appears to be common to these objects , along with a bluer peak near 3150 . shortward of @xmath32800 , the sne ia diverge from the core - collapse objects . this dropoff in flux , along with a pair of deep absorption features near 2450 and 2600 , is caused by iron - group elements , primarily @xcite . the lack of these features in ps1 - 10afx is evidence that the abundance of newly - synthesized @xmath21ni ( which decays to @xmath21co ) near the photosphere is much lower than in a sn ia . one purpose of these comparisons is to demonstrate the apparent broad similarity of the nuv spectra of ps1 - 10afx to those of sne coming from three very different progenitors : a bare core of a massive star ( sn 1994i : @xcite ) , a partially - stripped massive star in a binary system that retained part of its hydrogen envelope ( sn 1993j : @xcite ) , and the thermonuclear explosions of white dwarfs ( sne 2011fe and 2011iv : @xcite ) . sne 2011fe and 2011iv have some spectral differences from each other near 3000 , exhibiting the potential for spectroscopic variation in objects with similar progenitors . other than the lack of cobalt near the photosphere , the remaining differences between ps1 - 10afx and these other objects are relatively minor . in addition , the overall uv spectral slope of ps1 - 10afx is very similar to those of sne 1993j and 1994i , after correction of both by @xmath84 mag of reddening @xcite . the fact that the only sne in figures [ uvcomp ] and [ uvcomplog ] that have similar spectral features to ps1 - 10afx also have similar sed shapes provides further evidence that the extinction of ps1 - 10afx is not large . if the ps1 - 10afx spectra were corrected for a large amount of reddening , they would be bluer and we would expect different spectral features to be present at the higher implied temperatures . we further explore the spectra using the sn spectrum synthesis code ` syn++ ` with ` synapps ` @xcite to fit the near - maximum - light gmos / fire spectra . these codes are based on the same assumptions as the original ` synow ` ( e.g. , @xcite ) treatment of resonant - scattering lines in the sobolev approximation above a sharp photosphere that emits as a blackbody ( bb ) . the atomic and ionic species are assumed to have levels populated in local thermodynamic equilibrium at some excitation temperature . ` synow ` is a tool designed for line identifications and is not a full self - consistent spectral model . our best ` synow ` model is shown in figure [ synplot ] and contains seven ions . we assumed equal bb continuum and excitation temperatures of @xmath85 k and a photospheric velocity of @xmath8 km s@xmath6 . the deep absorption trough near 3700 is dominated by h&k . and combine to produce several of the wiggles observed in the range @xmath86 . naturally explains the observed absorption near 4250 , while also producing absorption shortward of @xmath32700 . contributes to both the blue wing of the deep feature and the notch near 3950 . the feature near 3950 is present in the overlap between the gmos and fire data and has a consistent strength that is significantly larger than in the synthetic spectrum . with ` synow ` , we have to be careful not to overproduce absorption by the @xmath636355 doublet because no strong feature is seen in the data , although the s / n and the spectral resolution become very poor in the observed - frame @xmath52 . we introduce to help reproduce the wiggle near 5000 , but the identification is not unique and its presence is not required . in addition , we include to help suppress the uv flux . that ion does not contribute sufficiently strongly to any specific feature for us to make a positive identification , but it has been included in previous ` synow ` models of sne ic @xcite because of its expected strength in either helium or carbon / oxygen - rich sn ejecta at these temperatures @xcite . we also consider several other species that were not included in the final fit . as discussed above , produces strong nuv absorptions in sne ia that are not present in ps1 - 10afx . d is commonly seen in all types of sne , but at this redshift it would fall in the strong telluric absorption between @xmath51 and @xmath52 , so we have no constraint on its presence from the observations . we also tested because of its use in prior ` synow ` analyses of sne ib @xcite . scandium results in an additional absorption minimum near @xmath33500 similar to one seen in our data , but does not produce a sufficiently strong improvement in the model to justify its inclusion , particularly given the expectation that it should only become important at lower ejecta temperatures @xcite . there is also an apparent absorption feature near 5300 that is not fit by the model . can produce a feature near that wavelength , but the identification is not certain . lines of hydrogen or helium would be very important if present , but there is no evidence for either , although the low s / n of our nir data precludes strong statements about helium . the sn 2005ap - like slsne have lines of singly - ionized carbon and oxygen in their spectra at early times ( e.g. , @xcite ) , with possible contributions from doubly - ionized cno elements @xcite , but those lines do not appear to be present in our ps1 - 10afx spectra . overall , we can reproduce the shape of the spectrum in the rest - frame optical reasonably well , but the nuv is more problematic . our synthetic spectrum has features at many of the same wavelengths as the real data , but the overall shape in the nuv is less well fit . adding additional ions to the fit might potentially help the uv shape of the synthetic spectrum , but would not be well motivated or add any additional insight . alternatively , the radiative transfer in the uv is known to be complex and we could be encountering the limitations of ` synow ` . the most important point is that the ions we have used are typical for generic fits to sne ia and ic spectra ( e.g. , @xcite ) , but differ from those identified in most slsne - i spectra near maximum light @xcite . existing light curves of slsne are heterogeneous , and the objects have been found at a wide range of redshifts , making direct comparisons difficult . following previous work @xcite , in figure [ uplot ] we compare the rest - frame @xmath87 light curve of ps1 - 10afx ( corresponding to observed - frame @xmath30 ) to several of the most luminous sn 2005ap - like sne in the literature . in addition , we include the hydrogen - rich sn 2008es because its light curve is similar to the others and it did not develop strong balmer lines until well after maximum light @xcite . for comparison , we also plot the @xmath88 light curves of the broad - lined sne ic 1998bw and 2002ap , two objects of normal luminosities @xcite . given the uncertain seds of many of these objects , we do not perform a full @xmath89-correction , but instead correct the observed ab magnitudes in filters whose effective wavelengths are close to @xmath87 in the observed frame by @xmath90d@xmath91(@xmath46)/10 pc ) @xmath48 2.5 @xmath92(1+@xmath46 ) , where d@xmath91 is the luminosity distance . to be conservative , we assume that ps1 - 10afx is unaffected by reddening here and in all following discussion , despite the strong absorption along the line of sight and the inference that the host galaxy sed requires some degree of internal extinction . we presented some evidence above that the reddening to the sn is not large , but it is possible that we have underestimated the peak luminosity of this object . it is apparent from figure [ uplot ] that at peak ps1 - 10afx was slightly more luminous in @xmath87 ( @xmath93 mag ) than any previous slsne . however , by @xmath310 d on either side of maximum light , ps1 - 10afx falls below all the others . this emphasizes the high peak luminosity and fast evolution of the light curve . these objects have rather different seds , so comparisons of the energetics require constructing a bolometric light curve . to proceed further , we first construct seds at three epochs near the times of our nir observations , at phases of @xmath94 , @xmath95 , and @xmath96 d. we fit third - order polynomials to the observed photometry in each @xmath28@xmath29@xmath30 filter around each sed epoch to estimate the flux . we then estimate the error bars on these fitted fluxes by generating monte carlo realizations of the data where we repeatedly randomly adjust the observed fluxes by drawing from a normal distribution having a width of the measured errors , refitting , and determining the variance of the interpolated flux at the desired epoch . we perform a similar procedure whenever we need to interpolate fluxes to a common epoch . for the seds at phases of @xmath94 and @xmath96 d , we use the actual measured @xmath31 and nir fluxes . for the @xmath95 d sed , the observed nir fluxes were quite noisy , so we again interpolate the fluxes in @xmath51 and @xmath52 by fitting second - order polynomials to the observed light curves to generate the points on the sed . the resulting sed evolution is shown in figure [ sedplot ] . we then fit bb spectra to the seds at each epoch . our @xmath94 d sed has the most complete information . the dashed line in figure [ sedplot ] shows the result of a bb fit to the full sed . the derived bb temperature is @xmath97 k , but clearly the curve is a poor fit to the data at rest wavelengths longward of @xmath35000 . if instead we restrict the fit to @xmath31 and the nir bands , the bb temperature is not well constrained ( @xmath98 k ) due to a lack of points blueward of the peak . the formal best fit is plotted as the solid blue line in figure [ sedplot ] and provides a much better fit to the nir data points ( by construction ) , but significantly overestimates the rest - frame uv flux . this is to be expected if line blanketing in the nuv provided by iron - peak elements supresses the uv flux , as is typical for sne ic . for example , the shape of the sed of the normal sn ic 1994i near maximum light ( figure [ sedplot ] ) is similar to that of ps1 - 10afx , but the detailed model fits by @xcite find an underlying bb temperature of 9230 k at the same epoch . the data at phases @xmath95 and @xmath96 d are less complete , so we fit a simple bb to all of the points on the sed and do not correct for line blanketing , which would only be expected to increase as the ejecta cool . the derived bbs show the ejecta clearly cooling from @xmath3@xmath99 k at a phase of @xmath94 d to @xmath3@xmath100 k at @xmath96 d , although @xmath101 should only be regarded as lower limits to the effective temperature ( @xmath102 ) on the later two dates . independent of the details of line blanketing , we can directly see the sed becoming redder , and hence implying cooler temperatures , by looking at the color evolution . in figure [ colorplot ] , we construct @xmath29@xmath48@xmath30 and @xmath30@xmath103 color curves . in each case , we interpolate the light curve in the bluer band to the epochs of observation for the noisier redder band . the @xmath29@xmath48@xmath30 color is consistent with a constant ( 0.30@xmath640.05 mag ) before maximum light , but both colors became redder with time after maximum . we use the bb fits to construct a bolometric light curve , starting with the near - maximum - light sed ( phase @xmath393 d ) shown in figure [ sedplot ] . a trapezoidal integration of the observed fluxes from @xmath28 to @xmath53 gives a minimum bolometric luminosity ( @xmath104 ) near peak of ( 3.6@xmath640.2)@xmath105 erg s@xmath6 . to account for flux emitted redward of our observations , we add the tail of the bb and extend the trapezoidal integration to long wavelengths , which provides only a 14% upward correction to the total flux . we repeat the process for the other two sed epochs . after including the bb tails , the derived bolometric luminosities are ( 4.1@xmath640.2 ) , ( 2.2@xmath640.2 ) , and ( 1.1@xmath640.3)@xmath105 erg s@xmath6 at phases of @xmath393 , @xmath3912 , and @xmath3924 d , respectively . although we lack nir data at other epochs , we can use these fits to estimate the bolometric light curve in combination with the well - sampled ps1 light curves . for each of our three sed epochs , we define a multiplicative bolometric correction factor to obtain @xmath104 from @xmath106@xmath107 in the @xmath30 band . motivated by the lack of strong color evolution before maximum light , we apply the bolometric correction factor from @xmath393 d to all pre - maximum @xmath30 data points . for the post - maximum data , we use a smoothly varying time - dependent correction factor to evolve from the @xmath393 d value to the @xmath3924 d value . we repeat the process for the @xmath29 band , except that we use a smaller ( but constant ) bolometric correction factor before maximum light than the derived @xmath393 d value to better match the @xmath30 results and account for the small amount of @xmath29@xmath48@xmath30 color evolution between @xmath393 d and the pre - maximum data . the combined bolometric light curve is plotted in figure [ boloplot ] . a third - order polynomial fit to the peak of the bolometric luminosity curve gives a maximum value of ( 4.1@xmath640.1)@xmath19 erg s@xmath6 , equivalent to 1.1@xmath410@xmath108 l@xmath12 , not including the uncertainty in the bolometric correction factors ( or potential extinction ) . to check that our derived maximum bolometric luminosity of ps1 - 10afxis realistic and that our procedure for extrapolating the sed to the nir has not introduced any significant error , we can take advantage of the fact that the shapes of the spectra we have of ps1 - 10afx and the derived seds are very similar to those of a sn ia near maximum light ( cf . figure [ uvcomplog ] ) . one of the most distant spectroscopically confirmed sne ia known , hst04sas , is at @xmath109 @xcite , consistent with the redshift of ps1 - 10afx . interpolating the f850lp ( @xmath3@xmath46 band ) light curve of hst04sas to a phase of zero days produces an estimate of the observed magnitude of @xmath110 mag ( ab ) , while fits to ps1 - 10afx show that @xmath30@xmath111 mag . assuming that the similarity of the seds requires no further correction factors and a typical sn ia peak bolometric luminosity of @xmath31.2@xmath410@xmath112 erg s@xmath6 ( e.g. , @xcite ) , the 3.8 mag difference between the objects corresponds to a peak bolometric luminosity for ps1 - 10afx of @xmath11310@xmath5 erg s@xmath6 , in excellent agreement with our value derived above . in fact , the peak of @xmath114 mag is more luminous than any of the hydrogen - poor objects collected by @xcite mag . ] . this statement is somewhat uncertain due to the heterogeneous and fragmentary nature of the available data for seds and bolometric correction factors for the objects in the literature . sne 2008es @xcite , 2005ap @xcite , and scp06f6 @xcite all have similar peak bolometric luminosities to ps1 - 10afx , but the exact comparison depends on the treatment of the bolometric correction for these uv - luminous sources . the only slsn of any type with a clearly higher peak luminosity is the peculiar hydrogen - rich transient css100217 @xcite . however , the sn nature of that object is still questionable due to possible confusion with the active galactic nucleus of its host galaxy . in addition to the high peak luminosity of ps1 - 10afx , another striking fact about the light curve is the rapid time evolution . despite the rolling survey nature of ps1 , our first detection is at a phase of only @xmath115 d , with non - detections prior to that ( figure [ boloplot ] ) . it is traditional in sn studies to find the rise time by fitting a fireball model of the form @xmath116 for @xmath117 at early times , where @xmath118 is the explosion time . for ps1 - 10afx , this gives a @xmath119 d ( figure [ boloplot ] ) . allowing the power - law index to vary gives a best fit with @xmath120 and @xmath121 d. regardless of the exact parameterization , the rise in @xmath30 is a factor of 3.5 in flux in the 7.4 d before peak . this apparent rise time is much faster than the fastest known slsne ( figure [ uplot ] ; @xcite ) . the decay timescale is also unusually fast . the fwhm of the bolometric light curve is only 18 d. the usual parameterization of sn light curve shapes , @xmath122@xmath123 , is the decline in magnitudes in 15 d after maximum light . the bolometric light curve of ps1 - 10afx has @xmath122@xmath123=0.95 mag . by comparison , only sn 1994i has a faster @xmath122@xmath123(@xmath124 ) in the collection of sne ib / c light curves of @xcite , although several additional objects had @xmath122@xmath123(@xmath125 ) that are comparable . figure [ uplot ] demonstrates that in @xmath87 , ps1 - 10afx also has a faster light curve decay than slsne of similar peak luminosity . integrating the observed bolometric light curve from the first detection at a phase of @xmath487.4 d to the last detection at @xmath3924 d provides a lower limit on the emitted energy of @xmath37@xmath410@xmath126 erg ( over only 31 d ) . reasonable extrapolations of the light curve to later times will increase the total to @xmath310@xmath15 erg , comparable to most other hydrogen - deficient slsne , but not notably high ( e.g. , @xcite ) . the fast timescale of ps1 - 10afx compensates for its high peak luminosity . the difficult constraints placed by the rise time on any model can be seen from the inferred emitting radius . assuming @xmath127 k as derived above , the bb radius ( @xmath128 ) of the photosphere at maximum light is 5.5@xmath129 cm . this is larger than for other hydrogen - poor slsne , which typically have inferred radii of @xmath310@xmath10 cm @xcite , as expected from the higher luminosity and lower @xmath101 of ps1 - 10afx . if @xmath102 is actually as low as the color temperature of @xmath3@xmath130 k , then the inferred radius at maximum light increases to 1.6@xmath131 cm . the most conservative set of numbers at maximum light ( @xmath128=5.5@xmath129 cm , @xmath118=@xmath4811.8 d ) requires an expansion velocity of @xmath350,000 km s@xmath6 if @xmath118 is the true explosion time , in obvious conflict with the observed photospheric velocities seen in the spectrum . material moving at the ` synow`-derived velocity of @xmath8 km s@xmath6 will take 58 d to reach that radius , implying an explosion @xmath350 d before our first detection . there is an observed example of such an effect with the slsn 2006 oz . it exhibited a precursor plateau with a luminosity of @xmath32@xmath410@xmath112 erg s@xmath6 prior to rising to the main peak of the light curve , which @xcite interpreted as being due to a recombination wave in an oxygen - rich csm . a weighted average of the ps1 - 10afx non - detections between phases @xmath132 and @xmath133 d produces a mean luminosity of ( 0.2@xmath641.2)@xmath410@xmath112 erg s@xmath6 . at the 2@xmath134 level , this non - detection is consistent with a hypothetical plateau being present at the level seen in sn 2006 oz , but unobserved due to the limiting magnitudes of our survey . at phases @xmath3912 and @xmath3924 d , the @xmath104 quoted above and @xmath101 shown in figure [ sedplot ] combine to produce bb radii of 1.4 and 1.3@xmath131 cm , respectively , although these are likely overestimated due to the underestimate of @xmath102 caused by line blanketing . notably , these values are consistent with the photosphere begining to recede through the ejecta ( in both the physical and comoving senses ) as they become optically thin . by contrast , the bb radii inferred for the interacting slsne 2006tf and 2006gy increased linearly with time at the same rate as the velocities measured from the spectra , consistent with an expanding optically - thick shell @xcite . although the measurement errors make it less certain , this also appears to be true for at least some sn 2005ap - like sne ( e.g. , @xcite ) . the kinetic energy of a powerful sn explosion provides an attractive reservoir of energy to power slsn light curves , but only if that energy can be efficiently converted to radiation . for ps1 - 10afx , this already requires an initial sn energy of at least a @xmath13510@xmath15 erg in order for the object to radiate 10@xmath15 erg and still have ejecta velocities as high as those observed . this generally requires a csm with a mass comparable to that of the ejecta and a radius comparable to the photospheric radius at maximum light in order to maximize efficiency and not lose energy to adiabatic expansion ( e.g. , @xcite ) . another attraction of interaction models is that the sn can explode well before the start of the optical rise of the sn if the csm is sufficiently dense to trap radiation from the shock @xcite . this could greatly alleviate the conflict described above between the rise time , ejecta velocity , and photospheric radius . the required csm densities in these models are much larger than for well - studied examples at lower luminosity . @xcite have identified two regimes of interest when the csm is this dense . if the radiation from the shock can escape while the shock is still interacting with the csm , then the light curve will be broad and exhibit a slow decay after maximum light . their model works well for the light curves of slsne like 2006gy . an additional consequence of this model is that there should be spectral signatures of the ongoing csm interaction , such as emission lines from the unshocked csm ( e.g. , @xcite ) . other luminous sne iin in the literature , such as sne 2008fz @xcite and 2008am @xcite , with their blue continua and strong balmer emission lines , can also be fit by similar models . the much faster light curve of ps1 - 10afx and lack of strong emission lines imply a different regime , one where the outer radius of the csm is comparable to or smaller than the diffusion length of photons from the shock . in this framework , the optical light we detect is then associated with the breakout of the shock from the outer radius of the csm . in the context of a dense wind with a density profile that scales with radius as @xmath136 out to a maximum radius of @xmath137 , where @xmath138 is a density normalization parameter , the total mass of the wind is @xmath139 . in the models of @xcite , @xmath138 can be estimated from the diffusion timescale as @xmath140 ( @xmath141/days ) g cm@xmath6 , if we assume an ionized he - rich composition . we can approximate the diffusion timescale @xmath141 by the rise time and set @xmath137 to the bb radius at maximum light to produce an estimate of the wind mass of @xmath143 m@xmath12 . these values are only approximate , as @xcite have shown that in the regime of interest , the analytic approximations are outside the range of validity and require true hydrodynamic calculations . models with shock breakout occurring in a dense , truncated wind can produce bolometric light curves with the desired peak luminosities and time scales , but it is unclear if this can be done consistently with the other information about the object . @xcite were able to approximately match the light curve shapes of the slsne 2010gx and 2005ap , but the derived @xmath101 were hard to match to the observations . a theoretical caveat is that the observed color temperature could deviate from @xmath102 if the shock is not in equilibrium when it breaks out @xcite . this deviation is caused by the nature of the opacity , and usually implies that the color temperature is hotter than the true @xmath102 ( e.g. , see @xcite for models of 2006gy ) , which would in turn require that the photosphere be at even larger radii than we have found . this effect might then conflict with the fast timescale of the light curve , which otherwise is indicative of a more compact csm @xcite , but detailed modeling with radiative transfer is needed to correctly evaluate these concerns . further , a key component of the rising sn luminosity in the shock breakout models of both @xcite and @xcite is that the temperature rises to a maximum near the peak of the light curve and declines thereafter . although we lack full sed information on the rise of the light curve for ps1 - 10afx , the observed @xmath144 color shows no evolution prior to the peak of the light curve ( figure [ colorplot ] ) . despite the interpretation uncertainties introduced by the possibilities of line blanketing or deviations from equilibrium , the evolution of @xmath101after maximum light results in a noticeable reddening of the @xmath145 color . if @xmath101 were rising by a similar amount while the light curve rose to its peak , we would have been able to detect it . a separate uncertainty is the origin of this csm , which could not be produced by steady stellar winds due to the high required mass - loss rates and the unexplained truncation at @xmath137 . most theoretical work has focused on winds , but the material could also be in a shell @xcite . @xcite invoked a pulsational pair instability to produce shells of matter at the appropriate distances , but these require very massive stars . other possibilities include common envelope ejection @xcite and instabilities in the late stages of stellar evolution @xcite . another challenge for csm interaction being the dominant contributor to the luminosity of ps1 - 10afx is the appearance of the spectra . interaction - powered sne convert the kinetic energy of the explosion into uv / optical continuum radiation , frequently reprocessing some of the continuum into strong emission lines from the dominant constituents of the csm external to the expanding shock . the spectral comparisons in figure [ uvcomp ] show several examples of objects consistent with this picture . however , ps1 - 10afx is even more luminous than those objects and yet shows p - cygni features originating in the sn ejecta , including a deep absorption comparable to that seen in sn 1994i . the lack of strong emission lines from the csm can be explained if the material has a sharp outer radius , but interaction sufficiently luminous to dominate the light curve should dilute and weaken spectral features from the ejecta due to the `` top - lighting '' effect @xcite , if not completely hide emission from the unshocked ejecta . for example , ptf 09uj is a candidate for being a lower - luminosity version of a shock breakout through a dense ( hydrogen - rich ) wind and , as expected , its spectra are very blue and lack strong p - cygni features @xcite . it is not clear how strong csm interaction could dominate the luminosity of ps1 - 10afx without leaving spectral signatures . in summary , existing csm interaction models can match the gross photometric properties of ps1 - 10afx ( peak luminosity , timescales ) , but only if the models have the freedom to assume the necessary csm structure . the origin of such unusual csm is not explained _ a priori _ in these models . in addition , the details of the sed , color evolution , and emitted spectra appear to be in conflict with the observations , although more detailed radiative transfer calculations are necessary to be confident in the model predictions , which are largely based on bb assumptions . normal hydrogen - deficient sne have light curves powered by the diffusion of energy deposited by the radioactive decay of @xmath21ni and @xmath21co . the light curves of most slsne - i appear to fade too rapidly relative to their peak luminosities to have radioactively - powered ejecta @xcite , and ps1 - 10afx is no exception . with a rise time of 12 d , the peak bolometric luminosity requires a nickel mass of @xmath314 m@xmath12 . however , making standard assumptions about the structure of the ejecta @xcite allows us to use the photospheric velocity of 11,000 km s@xmath6 and light curve timescale of 12 d ( assuming that the diffusion timescale is comparable to the rise time ) to estimate the ejecta mass as @xmath32 m@xmath12 , clearly in contradiction of the high nickel mass . another potential power source suggested for slsne is the spindown energy from a newly - born magnetar @xcite . the high luminosity and fast rise of ps1 - 10afx stretch the existing models to the bounds of plausibility . inspection of figures 4 and 5 of @xcite shows that the light curve of ps1 - 10afx requires a magnetar solution to have a very fast initial spin period ( @xmath146 ms , near the breakup speed of a neutron star ) , a low ejecta mass ( @xmath31 m@xmath12 ) , and a high magnetic field ( @xmath14710@xmath148 g ) . this is because the high luminosity requires a high spindown power , while the fast light curve requires both fast spindown times and short diffusion timescales . we examined magnetar - powered light curves using the formalism of @xcite and found parameters that produced light curves that approximately matched the shape of the bolometric light curve of ps1 - 10afx . one such model is plotted in figure [ boloplot ] . however , it is not clear whether this fit should be regarded as more than a numerical curiosity because the physical assumptions underlying the model break down . in the parameter range appropriate for ps1 - 10afx , the magnetar dipole spindown energy ( 5@xmath410@xmath15 ( @xmath149/2 ms)@xmath150 erg ) dominates over the initial sn kinetic energy ( assumed to be 10@xmath15 erg ) . the @xcite models make an ansatz that the entire spindown energy of the magnetar is simply thermalized spherically at the base of the sn ejecta and deposited into the internal energy . because this process accelerates and heats the ejecta , the final kinetic energy , the peak luminosity , and the temperature are coupled . the model plotted in figure [ boloplot ] predicts expansion velocities of @xmath925,000 km s@xmath6 , in contradiction of the observed spectrum . in addition , the model predicts a radius at maximum light ( from the ejecta velocity and rise time ) that when combined with the luminosity implies a bb temperature in excess of @xmath151 k ( with strong evolution at other times ) . this contradicts the observed @xmath101 near maximum light and the constant red @xmath29@xmath48@xmath30 color during the rise of the light curve . we find that increasing the assumed sn kinetic energy relative to the spindown energy does not alleviate these issues because of the difficulty in simultaneously satisfying the constraints from the expansion velocity seen at maximum light ( which implies a normal energy - to - mass ratio , and hence high mass if the initial sn energy is high ) with the fast diffusion timescale ( which requires a low mass ) . the models of @xcite start from massive star progenitors and proceed through the explosions , but they only probe slower rotating , and hence less energetic , magnetars than those required for ps1 - 10afx . @xcite also examined explosion models for type ib / c sne with the addition of a central energy source that mimicked the effect of a magnetar . they followed the effects of a central energy source on the ejecta produced by the explosion models and generated light curves and spectra . if the central energy source deposited @xmath310@xmath15 erg of energy , the resulting sne had high velocities and were blue at maximum light , which appears promising for explaining the sn 2005ap - like class of slsne , but is very different from ps1 - 10afx . as noted above , the light curve of ps1 - 10afx requires even more energy input from the magnetar . therefore , we disfavor magnetar models at this time . @xcite have proposed a scenario that shares some similarities with the magnetar model in that their model has a post - explosion internal energy source , in this case provided by fallback accretion onto a newly formed compact remnant , likely a black hole . their suite of models can only match the fast timescale and high peak luminosity of ps1 - 10afx if the progenitor is a blue supergiant and the initial sn explosion energy is extremely low with an ejecta mass of @xmath31 m@xmath12 . their hydrogen - poor models do not reach the same peak luminosities , which is a problem for ps1 - 10afx . in addition , these models have the same problem as the magnetar ones in simultaneously satisfying the high peak luminosity and relatively cool photospheric temperature . in all of the preceding discussion , we have necessarily assumed spherical symmetry in order to convert our observations into intrinsic quantities . however , core - collapse sne are all known to be aspherical at some level ( e.g. , @xcite ) . the most basic effect of asphericity is to increase the uncertainty in @xmath104 and dilute the connection between observables such as the rise time and physical quantities such as the diffusion timescale . the broad - lined sne ic associated with grbs have been extensively studied in the context of aspherical models , with different studies coming to conflicting conclusions for the same object ( e.g. , sn 1998bw : @xcite ) . the models of @xcite for sn 1998bw predict that lines of sight along the major axis of a bipolar outflow will see light curves with faster rise times and higher luminosities than average , perhaps indicating a general trend with relevance for ps1 - 10afx . however , despite the high explosion energies ( @xmath4910@xmath152 erg ) and fairly high nickel masses ( @xmath30.4 m@xmath12 ) assumed in their models , the maximum peak luminosity was still an order of magnitude lower than for ps1 - 10afx . in addition , the spectral features of ps1 - 10afx are not as blended as those of broad - lined sne ic ( figure [ uvcomp ] ) . we conclude that an aspherical sn 1998bw - like model is unlikely to produce an object that resembles ps1 - 10afx unless some other ingredient is added to the models . another possibility in a csm interaction scenario is that the material with which the sn is interacting is distributed in an aspherical fashion , perhaps allowing a distant observer to see both the luminous continuum emission from the interaction region and the sn ejecta . however , such models still require the combination of the covering fraction of the csm , the radiative efficiency factor , and the sn kinetic energy to be sufficiently high to produce the @xmath310@xmath15 erg of optical emission that we measure . the slsn models of @xcite invoke remnant protostellar disks around massive stars , but do not simultaneously fit the high peak luminosity and the fast timescale of ps1 - 10afx . in addition , such a model has similar difficulties as spherical csm interaction models in explaining the strength of the p - cygni features in the observed spectra if there is continuum emission from an interaction region that dilutes the strength of the spectral features from the ejecta . we now consider whether gravitational lensing could result in an object with the spectrum and light curve shape of a normal sn , but artificially boosted in luminosity . however , the peak luminosity of ps1 - 10afx is a factor of @xmath950 higher than normal sne ic , requiring an extremely high magnification factor despite the lack of an obvious lens . while this work was being refereed , a preprint by @xcite expanded on this hypothesis further and proposed that ps1 - 10afx was instead a normal sn ia and therefore had to be lensed by a magnification factor of @xmath320 . there are several constraints against the lensing hypothesis , starting with the lack of an apparent lens . in standard @xmath22cdm cosmologies , the optical depth to lensing with amplification factors @xmath4910 is very small , 10@xmath153 @xmath48 10@xmath154 for sources at the redshift of ps1 - 10afx ( e.g. , @xcite ) , and the mass distribution of the lenses for those extreme magnifications is peaked at cluster - scale haloes of @xmath310@xmath148 m@xmath12 @xcite . however , there is no foreground galaxy cluster visible in the ps1 images , as @xcite also concluded on the basis of a clustering analysis of independent canada - france hawaii telescope images of the field . the source present along the line of sight to ps1 - 10afx exhibits [ ] emission and the sed ( including a balmer break ) of a galaxy at the same redshift as the sn . the [ ] and rest - frame uv continuum luminosities produce similar estimates of the sfr . these observations are all consistent with the continuum flux from the source along the line of sight being dominated by , if not entirely due to , emission from the host galaxy of ps1 - 10afx , and leave little room for a contribution from a hypothetical lens galaxy at lower redshift . these constraints led @xcite to propose the existence of a dark lens from a previously unknown population , such as a free - floating black hole or a dark matter halo with few baryons . this provocative conclusion is based on the premise that ps1 - 10afx has to be classified as a normal sn ia and therefore the intrinsic luminosity is known . we demonstrated above that the spectra of ps1 - 10afx are similar to those of both sne ia and ic near maximum light ( section 4 ) . we also note that the presence of an absorption feature from @xmath636355 does not uniquely classify an object as a sn ia because that line is also present in some sne ic . as an experiment , we took near - maximum light spectra of an unambiguous nearby sne ic that exhibits features , sn 2004aw @xcite , resampled the data , and added noise to match the s / n of our ps1 - 10afx fire data that show the possible absorption feature near 6100 . these noisy spectra were then run through the supernova identification ( snid ) code of @xcite , using the updated templates of @xcite , with care taken to exclude templates from the same object . we found that typically all of the top 20 best matching templates are sne ia . this is in part due to the biased sample of spectral templates used by sn classification codes , where sne ia are largely overrepresented . @xcite also claim that the light curve of ps1 - 10afx is a good match for sne ia . however , we note that in order to get a good reduced-@xmath155 for their fit , they had to artificially increase the error bars in the nir and exclude the @xmath51-band data . furthermore , all of the ps1 - 10afx observations prior to a phase of @xmath156 d fall below the sn ia prediction , including the @xmath30 non - detection at @xmath115 d that is @xmath30.7 mag below their model . this is a consequence of the fast rise time of ps1 - 10afx , which we measure above to be @xmath157 d if we assume that the rise scales as @xmath116 , and @xmath158 d if we relax that constraint . although spectroscopically normal sne ia show a small dispersion in their rise times , objects with normal light curve shapes have rise times of 16@xmath4818 d in @xmath159 ( e.g. , @xcite ) . @xcite demonstrated that the rise in @xmath88 ( closer in rest wavelength to our @xmath30 observations ) was 2.3 d shorter , but still significantly longer than observed for ps1 - 10afx . at phases of @xmath160 to @xmath161 d , normal sne ia emit fluxes about 20@xmath4840% of the peak @xcite , but that is about 4@xmath134 above our non - detections ( figure [ boloplot ] ) . in summary , although sne ia have reasonably similar spectra and light curve shapes to ps1 - 10afx , the details of the rise time are not a good fit for sne ia and the spectra of the unambiguous sn ic 2004aw are also similar to sne ia when degraded to our low s / n ratio . therefore , an identification of ps1 - 10afx with normal sne ia is not required by the data , and there is no need to invoke a new population of unseen dark gravitational lenses . instead , we accept the lack of an apparent lens at face value and conclude that ps1 - 10afx was actually an unusually luminous hydrogen - deficient sn . we present multiwavelength observations of ps1 - 10afx at redshift @xmath0 , perhaps the most luminous sn yet discovered . the combination of observables presents strict constraints on any theoretical interpretation . these are : * a peak bolometric luminosity of 4.1@xmath410@xmath5 erg s@xmath6 * an observed rise time of @xmath312 d * a fast light curve decay , with @xmath122@xmath123= @xmath162 mag * at least 7@xmath410@xmath126 erg of optical radiation emitted , with a total likely closer to 10@xmath15 erg * red color near maximum light ( @xmath163 k ) , although @xmath102 may be closer to @xmath99 k * constant uv color before maximum light * photospheric velocities near maximum light of @xmath3@xmath8 km s@xmath6 * spectra that most closely resemble normal sne ic , with deep p - cygni absorption * photometric and spectroscopic evidence for some line blanketing in the nuv from iron - peak elements , although not as much as for sne ia * a massive ( @xmath32@xmath410@xmath13 m@xmath12 ) host galaxy that is unlikely to have an extremely low metallicity we surveyed existing models for slsne and found that none were acceptable . in particular , the large inferred @xmath128 near maximum light is very hard to reconcile with the fast observed rise time and the measured photospheric velocities because the sn ejecta need too much time to reach such large radii . the magnetar models that match the peak luminosity and rise time do so by producing a much higher temperature at a smaller radius . these flaws are generic to models powered by internal energy sources , although if the onset of the internal energy source can be sufficiently delayed after the initial explosion , then maybe the conflict between the velocities and radii can be avoided . shock breakout scenarios invoking dense csm provide a promising solution by allowing the sn to explode well before the optical emission is detectable . if the csm is sufficiently dense out to @xmath35@xmath410@xmath10 cm , then the light curve will only rise after the ejecta have had time to reach that radius . however , these models leave the normal sn ic - like spectrum unexplained and are in apparent conflict with the observed color evolution before maximum light . more detailed radiative transfer calculations of the models are necessary to know whether these flaws can be avoided . in addition , it is not clear how to produce the special csm structure that has to be assumed ( a truncated wind or a shell ) . ps1 - 10afx was initially identified as an object of interest due to its unusual colors ( @xmath164 2 mag at peak ) and would not have been found without the ps1 observation strategy , which includes regular @xmath29 and @xmath30 observations as part of the search . at redshifts below @xmath31 , an object like ps1 - 10afx would have a strong detection in @xmath28 and would not stand out without knowing the redshift . unlike many of the sn 2005ap - like sne , the time - dilated light curve of a ps1 - 10afx - like sn at lower redshift would not be unusually long and would not be associated with the attention - grabbing lack of a visible host galaxy . however , ps1 - 10afx was more than a magnitude brighter than its host galaxy , which potentially provides a selection criterion to increase the odds of finding similar objects in the future . in the sample of slsne found by ps1 , sn 2005ap - like objects @xcite outnumber ps1 - 10afx by at least an order of magnitude . our poor understanding of spectroscopic incompleteness and the relevant selection effects precludes a more precise estimate of rates at this time . an additional complication to be considered is the role of metallicity . the massive host of ps1 - 10afx is in contrast with the low - mass @xcite and low - metallicity @xcite hosts of known hydrogen - poor slsne . the natal composition of the progenitor could have an indirect effect on the explosion through stellar evolutionary processes or it could directly affect the appearance through the opacity in the outer ejecta ( could sn 2005ap - like events with higher metal abundances exhibit line - blanketed spectra like ps1 - 10afx ? ) . most of the published slsne that lack hydrogen in their spectra are similar to sn 2005ap and scp06f6 @xcite . sn 2007bi @xcite and a couple of as - yet unpublished objects @xcite were the only known exceptions . they exhibited rather different spectra and had very slow light curve decays , which were interpreted by @xcite to be the result of decay of a large amount of radioactive @xmath21ni produced in a pair - instability explosion . this result has been challenged on theoretical grounds by @xcite , who show that the observations do not match theoretical expectations for pair - instability sne . with the discovery of ps1 - 10afx , we now have further evidence that the most luminous sne are a heterogeneous lot , even when just considering the objects lacking hydrogen . a key question for future investigations is whether these diverse outcomes of stellar evolution can be produced by variations on a single underlying physical model or whether multiple pathways exist to produce sne of these extraordinary luminosities . ps1 - 10afx differs from existing slsne - i in almost every observable except for the peak luminosity . it is hard to understand how all of these differences could be produced with only small modifications to a single model , and may indicate that ps1 - 10afx is the first example of a different channel for producing slsne . this paper includes data gathered with the 6.5-m magellan telescopes located at las campanas observatory , chile . we thank the staffs at ps1 , gemini , magellan , and the vlt for their assistance with performing these observations . we are grateful for the grants of dd time at gemini and the vlt . we acknowledge j. strader for obtaining some fourstar observations , as well as the mmirs observers who helped obtained some of the data here during the fall 2010 queue observations : m. kriek , i. labbe , j. roll , d. sand , and s. wuyts . sjs acknowledges funding from the european research council under the european union s seventh framework programme ( fp7/2007 - 2013)/erc grant agreement n@xmath165 291222 . the pan - starrs1 surveys ( ps1 ) have been made possible through contributions of the institute for astronomy , the university of hawaii , the pan - starrs project office , the max - planck society and its participating institutes , the max planck institute for astronomy , heidelberg and the max planck institute for extraterrestrial physics , garching , the johns hopkins university , durham university , the university of edinburgh , queen s university belfast , the harvard - smithsonian center for astrophysics , the las cumbres observatory global telescope network incorporated , the national central university of taiwan , the space telescope science institute , and the national aeronautics and space administration under grant no . nnx08ar22 g issued through the planetary science division of the nasa science mission directorate . some observations were obtained under program ids gn-2010b - q-5 ( pi : berger ) , gs-2010b - q-4 ( pi : berger ) , and gn-2010b - dd-2 ( pi : chornock ) at the gemini observatory , which is operated by the association of universities for research in astronomy , inc . , under a cooperative agreement with the nsf on behalf of the gemini partnership : the national science foundation ( united states ) , the science and technology facilities council ( united kingdom ) , the national research council ( canada ) , conicyt ( chile ) , the australian research council ( australia ) , ministrio da cincia , tecnologia e inovao ( brazil ) and ministerio de ciencia , tecnologa e innovacin productiva ( argentina ) . some observations were collected at the european organisation for astronomical research in the southern hemisphere , chile under ddt programme 286.d-5005 ( pi : smartt ) . the national radio astronomy observatory is a facility of the national science foundation operated under cooperative agreement by associated universities , inc . some of the archival data presented in this paper were taken under programs go-5623 and go-9114 ( pi : kirshner ) and were obtained from the mikulski archive for space telescopes ( mast ) . stsci is operated by the association of universities for research in astronomy , inc . , under nasa contract nas5 - 26555 . some of the computations in this paper were run on the odyssey cluster supported by the fas science division research computing group at harvard university . tonry , j. , & onaka , p. 2009 , advanced maui optical and space surveillance technologies conference , proceedings of the advanced maui optical and space surveillance technologies conference , ed . : s. ryan , p.e40 .

we present the discovery of ps1 - 10afx , a unique hydrogen - deficient superluminous supernova ( slsn ) at redshift @xmath0 . the light curve peaked at @xmath1 mag , making ps1 - 10afx comparable to the most luminous known sne , with @xmath2 mag . our extensive optical and near - infrared observations indicate that the bolometric light curve of ps1 - 10afx rose on the unusually fast timescale of @xmath312 d to the extraordinary peak luminosity of 4.1@xmath410@xmath5 erg s@xmath6 ( @xmath7 mag ) and subsequently faded rapidly . equally important , the sed is unusually red for a slsn , with a color temperature of @xmath36800 k near maximum light , in contrast to previous hydrogen - poor slsne , which are bright in the ultraviolet ( uv ) . the spectra more closely resemble those of a normal sn ic than any known slsn , with a photospheric velocity of @xmath3@xmath8 km s@xmath6 and evidence for line blanketing in the rest - frame uv . despite the fast rise , these parameters imply a very large emitting radius ( @xmath95@xmath410@xmath10 cm ) . we demonstrate that no existing theoretical model can satisfactorily explain this combination of properties : ( i ) a nickel - powered light curve can not match the combination of high peak luminosity with the fast timescale ; ( ii ) models powered by the spindown energy of a rapidly - rotating magnetar predict significantly hotter and faster ejecta ; and ( iii ) models invoking shock breakout through a dense circumstellar medium can not explain the observed spectra or color evolution . the host galaxy is well detected in pre - explosion imaging with a luminosity near @xmath11 , a star formation rate of @xmath315 m@xmath12yr@xmath6 , and is fairly massive ( @xmath32@xmath410@xmath13 m@xmath12 ) , with a stellar population age of @xmath310@xmath14 yr , also in contrast to the young dwarf hosts of known hydrogen - poor slsne . ps1 - 10afx is distinct from known examples of slsne in its spectra , colors , light - curve shape , and host galaxy properties , suggesting that it resulted from a different channel than other hydrogen - poor slsne .

You are an expert at summarizing long articles. Proceed to summarize the following text: special relativity ( sr ) and general relativity ( gr ) are at the basis of our understanding of space and time and thus are fundamental for the formulation of physical theories . without sr we can not explain the phenomena in high energy physics and in particle astrophysics , without gr there is no understanding of the gravitational phenomena in our solar system , of the dynamics of galaxies and of our universe , and , finally , of the physics of black holes . both theories are linked by the fact that the validity of st is necessary for gr . due to the overall importance of these theories , a persistent effort to improve the experimental tests of their foundations is mandatory . modern tests of sr and gr using ultrastable oscillators have been performed on earth @xcite , and an experiment is planned on the international space station @xcite . furthermore , new results from quantum gravity theories predict small deviations from sr and gr giving additional motivation to improve tests on sr and gr . for example , loop gravity and string theory predict modifications of the maxwell equations @xcite . these modifications lead to an anisotropic speed of light and to a dispersion in vacuum , thus violating the postulates of sr . quantum gravity also predicts , besides a violation of the weak equivalence principle @xcite , a violation of the universality of the gravitational red shift @xcite . although the amount of sr violations predicted by quantum gravity are too small to be in the range of experimental capabilities in the near future , these predictions open up the window for violations for such basic principles . since we do not know the true quantum gravity theory we also do not know the true parameters of the theory and therefore the predicted range of violations of sr and gr is no final prediction , but merely a hint . ( -5,-3)(7,5)(-4.6,-3.5)(7,5 ) ( -3,0)0.47 ( -3,0),title="fig:",width=37 ] ( -0.5,1.5)(0,0)(4.5,1.7 ) ( -3,-1.5)perigee @xmath0 10000 km ( 3.7,4.4)apogee @xmath0 40000 km ( -0.5,3.4)(4.8,2.2 ) ( -0.5,3.3)(1.5,-1.7 ) ( -1.7,3.3)(0,0.3)(0.6,0.2 ) ( -0.3,0.3)(0.3,1 ) ( 0.4,0.42)(0.3,0.45 ) ( -0.5,3.3 ) ( -1.5,0.5)(-0.5,1.5)(-3.5,1.5)(-4.5,0.5)(-1.5,0.5 ) ( 0,0)(1,0.5 ) ( -1,0)(1,2 ) ( -1,0)(-1,2 ) ( 1,0)(1,2 ) ( 0,2)(1,0.5 ) ( 0.5,0.5)(1.5,1.5)(4.5,1.5)(3.5,0.5)(0.5,0.5 ) ( 0,2)(0,9 ) ( -1.4,3.5)@xmath1 ( -2,4.3)(-4,4.3 ) ( -3,4.6)to sun ( 4,-0.45)2.8 ( 4.75,-0.15 ) ( 3.2,4.2)(3.2,-1.2)(1.2,-3.2)(-4.2,-3.2)(-6.2,-1.2)(-6.2,4.2)(-4.2,6.2)(1.2,6.2)(3.2,4.2 ) ( 0,-0.8)(0.8,0 ) ( 1.2,-0.8)(2,0 ) ( 0,-2)(0.8,-1.2 ) ( 1.2,-2)(2,-1.2 ) ( 0,-0.8)(0,0)(0.8,0 ) ( 1.2,0)(2,0)(2,-0.8 ) ( 0,-1.2)(0,-2)(0.8,-2 ) ( 1.2,-2)(2,-2)(2,-1.2 ) ( 2,-1.2)(2,-0.8 ) ( 0.8,-2)(1.2,-2 ) ( 0,-1.2)(0,-0.8 ) ( 0.8,0)(1.2,0 ) ( -5.5,-1.3)(-4,-0.7 ) ( -4,-0.9)(-4,-0.7)(-5.5,-0.7)(-5.5,-1.3)(-4,-1.3)(-4,-1.1 ) ( 0.7,3)(1.3,4.5 ) ( 0.9,3)(0.7,3)(0.7,4.5)(1.3,4.5)(1.3,3)(1.1,3 ) ( -6,0.7)(-4.5,1.3 ) ( -6,2.7)(-4.5,3.3 ) ( -5.25,1.3)(-5.25,2.7 ) ( -1,4.2 ) ( 0,0)0.30180 ( 0,0)0.30180 ( 0,0.3)(0,1.2 ) ( -1.9,1 ) ( 0,0)0.30180 ( 0,0)0.30180 ( 0.5,0.5)(1.5,1.5 ) ( -1.5,-1.5)(-0.5,-0.5 ) ( -0.5,0.5)(-1.5,1.5 ) ( -1.5,2.5)(-0.5,3.5 ) ( -4,-1)(-2.5,-1 ) ( -2.5,-1)(1.6,-1 ) ( 2,-1)(0.4,-1 ) ( -1,-1)(-1,0 ) ( -1,0)(-1,2 ) ( -1,2)(-1,4.2 ) ( -4.5,3)(-2.75,3 ) ( -2.75,3)(-1,3 ) ( 1,3)(1,1.25 ) ( 1,2.5)(1,-1.6 ) ( 1,-2)(1,-0.4 ) ( 1,1)(0,1 ) ( 0,1)(-1,1 ) ( -1,1)(-1.9,1 ) ( -2.2,1)(-3,1 ) ( -4.5,1)(-3.64,1 ) ( -3.34,1.3)(-3.34,2.9 ) ( -3.34,3.1)(-3.34,5.4 ) ( -3.34,1)@xmath2 ( -4.5,-1.6)laser 1 ( 1.6,3.75)laser 2 ( 0,-2.3 ) ( 0,-2.7 ) ( -2.2,5.7)to data recording ( 2.7,1.5)thermal isolation ( -5.1,0.4)atomic ( -5.2,-0.1)clock ( -5,4.1)comb ( -4.9,3.6)generator by means of the proposed mission optis an improvement of three tests of sr and gr by up to three orders of magnitude is projected : llc experiment & present accuracy & projected accuracy + [ 1ex ] ' '' '' michelson morley experiment ( mm ) & @xmath3 @xcite & @xmath4 + kennedy thorndike experiment ( kt ) & @xmath5 @xcite & @xmath6 + universality of gravitat . red shift ( lpi ) & @xmath7 @xcite & @xmath8 + [ 1ex ] the main features of the optis mission can be seen in fig.[optisprinzip ] : a spinning drag free satellite orbits the earth . the satellite payload consists of two lasers , two orthogonal optical cavities , a femtosecond laser comb generator , and an atomic clock . the two cavities are used for the mm experiment which searches for differences in the velocity of light in different directions . the atomic clock represents an independent clock of different physical nature . a comparison between the atomic clock and an optical cavity can be performed by means of the comb generator . a search for a dependence of their frequency ratio with respect to a change of the velocity of the satellite or with respect to a change of the gravitational potential amounts to a kt test and to a test of the universality of the gravitational red shift , respectively . the most precise experiments testing the constancy of the speed of light use cavities . the wave vector magnitude @xmath9 of an electromagnetic wave in a cavity of length @xmath10 is given by @xmath11 and the frequency @xmath12 of an outcoupled wave by @xmath13 . if the velocity of light depends on the orientation of the cavity and on the velocity @xmath14 of the laboratory , @xmath15 , so will the frequency @xmath16 . fig.[fig : schemetest ] shows a schematic setup for a search of an orientation and velocity dependence of the frequency . since sr is based on an orientation and velocity _ in_dependent speed of light , the search for an orientiation and velocity dependence of the frequency amounts to a test of sr . ( 1,-2)(9,7)(5,1.4)(0,0.02)10(0,0)(4,1 ) ( 4.4,1.6)(5.2,1.4)(5.7,2.3)(4.9,2.5)(4.4,1.6 ) ( 5.2,1.4)(5.7,2.3)(5.7,1.7 ) ( 2.7,1.3)(5,2 ) ( 5,2)(5,4 ) ( 5,4)(5,5 ) ( 3.7,1.6)(-0.05,-0.015)20(0,0)(0.25,0.4 ) ( 5,-1)(5,0.36 ) ( 5,5)(0,0)0.30180 ( 0,0)0.30180 ( 0,0.3)(0,1.8)(0.7,-0.2)(0.7,1.3 ) ( 7.3,5.1 ) ( 6.5,1.8)mirror ( 3,2.8)cavity ( -0.3,1.2)turn table ( 5,0.8)(0.5,-0.992157)(1,-0.968246)(1.5,-0.927025)(2,-0.866025)(2.5,-0.780625)(3,-0.661438)(3.5,-0.484123)(3.7,-0.379967)(3.9,-0.222205 ) ( 8.3,-0.2)@xmath17 ( 4,-2)(7,-1 ) ( 7,-1.4)@xmath18 according to common test theories @xcite , the orientation and velocity dependence of the velocity of light is parametrized according to @xmath19 this means that all anomalous terms vanish for vanishing velocity @xmath14 . here @xmath17 is the angle between the velocity with respect to the cosmic preferred frame @xmath18 and the cavity axis . in addition , an expansion with respect to @xmath20 has been used , for simplicity . if special relativity is valid , then @xmath21 . the parameters @xmath22 and @xmath23 in the mansouri sexl and robertson test theories are given by [ cols="<,<,<",options="header " , ] in order to test the isotropy of space , or the isotropy of the velocity of light , one has to mount the cavity on a turn table and look for a variation of the frequency as the turn table is rotated . this setup has been used in experiments since 1955 , see fig.[fig : historymm ] . in terms of the relative variation of the velocity of light , of the robertson parameter , and of the mansouri sexl parameter , the most accurate result is @xcite @xmath24 for more interpretation see @xcite . a hypothetical dependence of the velocity of light on the velocity @xmath14 of the laboratory can be tested by changing the velocity of the cavity and looking for a variation of the frequency . in earth based experiments , the rotation of the earth around its axis ( @xmath25 ) or around the sun ( @xmath26 ) can be used , where @xmath27 is the velocity with respect to the cosmic microwave background , the cosmologically preferred frame . because of technical reasons only the rotation of the earth around its own axis has been used so far . in terms of the same parameters as above , the most accurate result is @xcite @xmath28 in the framework of einstein s gr , the comparison of two identical clocks of frequency @xmath29 located in different gravitational potentials @xmath30 and @xmath31 yields @xmath32 . the gravitational red shift does not depend on the type of clock . this is the universality of the gravitational red shift , an aspect of local position invariance . if einstein s theory is not correct , then the red shift may depend on the clock @xmath33 with @xmath34 . in einstein s gr @xmath35 . if two different clocks are displaced together in a gravitational potential , a relative frequency shift @xmath36 may occur , which is proportional to the difference of the gravitational potential relative to the uinitial position . ( 0,-1)(16,11)(1,-1.5)(16,-1.5 ) ( 1,-1.5)(1,10 ) ( 0.1,10)@xmath37 ( 1,-1.5)(1,0)15(0,0)(0,0.2 ) ( 2,-2)1880 ( 3,-2)1890 ( 4,-2)1900 ( 5,-2)1910 ( 6,-2)1920 ( 7,-2)1930 ( 8,-2)1940 ( 9,-2)1950 ( 10,-2)1960 ( 11,-2)1970 ( 12,-2)1980 ( 13,-2)1990 ( 14,-2)2000 ( 0.3,-1)@xmath38 ( 1,-1)(15,-1 ) ( 0.3,0)@xmath39 ( 1,0)(15,0 ) ( 0.3,1)@xmath40 ( 1,1)(15,1 ) ( 0.3,2)@xmath41 ( 1,2)(15,2 ) ( 0.3,3)@xmath42 ( 1,3)(15,3 ) ( 0.3,4)@xmath43 ( 1,4)(15,4 ) ( 0.3,5)@xmath44 ( 1,5)(15,5 ) ( 0.3,6)@xmath45 ( 1,6)(15,6 ) ( 0.3,7)@xmath46 ( 1,7)(15,7 ) ( 0.3,8)@xmath47 ( 1,8)(15,8 ) ( 0.3,9)@xmath48 ( 1,9)(15,9 ) ( 2,8)(0,0 ) ( 4.4,7)(0,0)@xmath49 morley & miller 1904 ( 6.4,7.2)(0,0)@xmath49 tomaschek 1924 ( 6.6,5.5)(0,0)@xmath49 kennedy 1926 ( 6.7,6.8)(0,0)@xmath49 illingworth 1927 ( 7,6.5)(0,0)@xmath49 joos 1930 ( 9.5,4)(0,0)@xmath49 essen 1955 ( 9.8,3.2)(0,0)@xmath49 cedarholm et al 1958 ( 10.4,3.2)(0,0)@xmath49 jaseja et al 1964 ( 11.9,2.5)(0,0)@xmath49 brillet & hall 1979 ( 14.3,-1)(0,0)@xmath49 optis ( proposed ) for the hydrogen maser @xmath50 ( gp - a experiment , @xcite ) , verifying the gravitational red shift . for a test of the universality of the gravitational red shift by means of a comparison between a microwave cavity and an atomic cesium clock , the best result is @xmath51 @xcite . the advantages of a sattelite mission for doing experiments for testing sr and gr are the following : * the high orbital velocity ( see eq.([genansatz ] ) . for the proposed ( elliptic ) orbit , @xmath14 varies between @xmath52 and @xmath53 over half the orbit period @xmath54 . this value is 20 times larger than the change of the velocity of the earth s surface over a period of 12 h. * the shorter period @xmath55 of the velocity modulation as compared to 24 h on earth . it relaxes the requirements on the optical resonators and is directly relevant for the kt and lpi tests . it is indirectly useful also for the elimination of systematic effects in the mm test . * the difference of the gravitational potential in the highly eccentric orbit is @xmath56 . this is about three orders of magnitude larger than the difference of the potential of the sun which an earth bound observer experiences . this is relevant for the lpi test . * the microgravity environment minimizes distortions of the optical resonators . this is relevant for the mm test . * a variable spin frequency of the satellite permits elimination of sytematic effects . * long integration times ( longer than 6 months ) . in this and the following section we discuss the science payload of the satellite , the relevant performance specifications and the resulting requirements for the orbit and the satellite bus structure . in order to perform the mm test , the satellite has to spin around its axis . a typical rotation period is @xmath57 . for the elimination of systematic errors , the rotation frequency @xmath1 can be varied . for the kt test the timescale is the orbit time @xmath58 . the main subsystems of the experimental payload are optical resonators , ultra stable lasers , an optical frequency comb generator and an ultra stable microwave oscillator . these components are interconnected and supplemented by the locking and stabilization electronics , the optical bench , the drag free control system ( discussed in the next section ) , and an advanced thermal control system . the lisa pre phase a report @xcite showed that a 3stage passive thermal isolation is sufficient to achieve a level of thermal fluctuations below @xmath59 at @xmath60 . this performance ( assuming no shadow phases ) would be sufficient for the optis random noise requirements . to supress the fluctuations correlated with the rotation of the satellite , an improvement of the thermal stability by one order of magnitude would be required . this could be achieved either by an additional stage of passive isolation , or by adding an active temperature stabilization . the optical resonators are the central part of the experimental setup . in the baseline configuration , two crossed standing wave resonators are implemented by optically contacting four highly reflective mirrors to a monolithic spacer block with two orthogonal holes made from a low expansion glass ceramic ( ule , zerodur ) . with a technically feasible finesse of @xmath61 and a length of @xmath62 each , these resonators should exhibit linewidths of @xmath63 . low expansion glass ceramics are designed for a minimal thermal expansion coefficient ( @xmath64 ) at room temperature and can be manufactured in many different shapes and dimensions . resonators made from these materials are well suited for laser stabilization , although aging effects cause a continuous shrinking of the material and thereby frequency drifts of typically @xmath65 . the monolithic construction of the resonator block serves to strongly reduce the effect of shrinking and of temperature fluctuations on the mm experiment : a high degree of common mode rejection ( two orders of magnitude ) in the differential frequency measurement of the lasers locked to the resonators is expected . for the kt and the redshift experiment , on the other hand , the aging related frequency drift is critical . it has to be modelled well enough to keep the unpredictable residual part below @xmath66 ( i.e. @xmath67 ) over the signal half period @xmath68 . this is feasible @xcite . for the projected measurement sensitivity the temperature stability requirements are as follows : a level @xmath69 for random fluctuations over the spin period time scale and a level @xmath70 for a temperature modulation correlated with the rotation of the satellite . for the kt as well as for the redshift experiment the requirements are @xmath71 for random temperature fluctuations on a time scale of the orbit period and @xmath72 for temperature modulation correlated with the orbital motion . the length of the reference resonators is affected by accelerations , with a typical sensitivity of @xmath73 for a resonator length of @xmath74 @xcite . this leads to a requirement of @xmath75 for random residual accelerations and @xmath76 for residual accelerations correlated with the rotation of the satellite . the drag - free control system ( see next section ) is designed to meet these requirements . the lasers used for the optis mission should have high intrinsic frequency stability , narrow linewidth and high intensity stability . these requirements are best fulfilled by diode pumped monolithic nd : yag lasers , which are also used in gravity wave detectors ( geo600 , ligo , virgo , lisa ) and many other high precision experiments @xcite . such lasers are already available in space qualified versions . locking the lasers to the reference resonators using the pound drever hall frequency modulation method requires fast , low noise photodetectors and an optimized electronic control system . the intrinsic noise of the photodetectors has to be sufficiently low to allow shot noise limited detection . the residual amplitude fluctuations of the lasers have to be actively suppressed . to prevent thermally induced length changes of the resonators by absorbed laser radiation , the intensity of the laser beams has to be actively stabilized to a relative level of @xmath77 in the frequency range @xmath78 to @xmath79 . the whole optical setup should be stable and isolated from vibrations to prevent frequency fluctuations caused by vibration induced doppler shifts . even very small displacements ( less then @xmath80 ) of the laser beams relative to the reference resonators are known to cause substantial frequency shifts @xcite . these requirements can be fulfilled by using a well designed monolithic optical bench on which all optical components are stably mounted as close together as possible . the kt and the universality of red shift experiments require an independent frequency reference . this reference should be based on the difference of two energy levels in an atomic or molecular system . atomic clocks based on hyperfine transitions in cesium or rubidium atoms are suited for this task . they are available in space qualified versions with relative instabilities of better than @xmath81 for the relevant time scale @xmath82 of several hours . the atomic clock can also serve as the reference for the microwave synthesizer required to mix down the beat signal between the two stabilized lasers in the mm experiment from typically 1 ghz to a lower frequency for data acquisition and analysis . the required relative instability @xmath83 on the timescale @xmath84 of @xmath85 is thereby satisfied . for the frequency comparison between the atomic clock and the lasers stabilized to the reference resonators it is necessary to multiply the microwave output of the atomic clock ( @xmath86 ) into the optical range ( @xmath87 ) . thanks to recent important progress in the field of frequency metrology , this can now be done reliably amd simply by using femtosecond optical comb generators @xcite . here the repetition rate of a mode locked femtosecond laser of @xmath88 is locked to an atomic clock . its optical spectrum ( a comb of frequencies spaced at exactly the repetition rate ) is broadened to more than one octave by passing the pulses through a special optical fiber . by measuring and stabilizing the beatnote between the high frequency part of the comb and the frequency doubled low frequency part , it is then possible to determine the absolute frequency of each component of the comb relative to the atomic clock . in a final step the frequency of the cavity stabilized lasers is then compared to the closest frequency component of the comb by measuring their beatnote with a fast photodetector . compact diode pumped comb generators with very low power consumption , as required for space applications , are already under development . the experimental requirements define an optimum mission profile . in particular , the requirements of attitude control , maximum residual acceleration , and temperature stability result in the following specifications : * the satellite needs a drag - free attitude and orbit control system for all 6 degrees of freedom . thrust control must be possible down to @xmath89n by means of ion thrusters ( field emission electical propulsion , feep ) . * for drag - free control the satellite needs an appropriate reference sensor . * feeps are not effective for orbit heights less than 1000 km , because of the high gas density in lower orbits . * to avoid charging of the capacitive reference sensor by interactions with high energy protons , highly eccentric orbits , where the satellite passes the van belt , are not appropriate . * the kt experiment requires a low orbit , because the experimental resolution depends directly on the satellite orbit velocity . * the precise attitude control requires a star sensor with a resolution of 10 arcsec . * mechanical components for attitude control ( e.g. fly wheel or mechanical gyros ) can not be used , because of the sensitivity of the experiment to vibrations . * temperature stability requirements during integration times of more than 100 s can only be realized on orbits without or with rare eclipse intervals . * the thermal control of the satellie structure must achieve a stabilioty of @xmath90 . * the mission time is 6 months minimum . optis is designed to be launched on a micro satellite with limited technological performance . considering all experimental requirements , technological feasibility , launch capability , and design philosophy the most feasible solution is to launch the satellite in a high elliptical orbit , attainable via a geo transfer orbit ( gto ) by lifting the perigee . in this scenario , the satellite is first launched as an auxiliary payload ( asap5 ) by an ariane 5 rocket into the gto with an apogee of 35800 km and a perigee of only 280 km . an additional kick motor on the satellite will lift the satellite in its final orbit with a perigee of about 10000 km , corresponding to @xmath91 . a minimum height of 10000 km enables one to use ion thrusters ( feeps ) and avoids flying through the van allen belt . also , the orbit eccentricity of @xmath92 is high enough to attain sufficient velocity differences for the kt experiment . although the relatively high orbit reduces the in orbit velocity of the satellite by a factor of 2.8 compared to a low earth circular orbit , it is still 20 times higher than for an earth based experiment . the satellite has cylindrical shape , with a height and a diameter of about 1.5 m. it spins around its cylindrical axis which is always directed to the sun . the front facing the sun is covered by solar panels . behind the front plate serving as thermal shield a cylindrical box for on board electronics and thruster control is located . the experimental box whose temperature is actively controlled is placed below the electronics . a stringer structure below the experimental box carries the kick motor and the fuel tank . 4 clusters of ion thrusters for fine ( drag free ) attitude control as well as 3 clusters of cold gas thrusters for coarse attitude control and first acquisition operation are mounted circumferentially . the structure elements of the entire bus have to satisfy the extreme requirement for passive thermal control . their thermal expansion coefficient has to be less than @xmath93 . the total satellite mass is about 250 kg including 90 kg of experimental payload . the total power budget is estimated to be less than 250 w. during experimental and safe mode coarse attitude and orbit control are based on a sun sensor ( 1 arcsec resolution ) and a star sensor ( 10 arcsec resolution ) . fiber gyros are used for spin and de spin maneuvers and serve to control the cold gas thrusters . the fine attitude control , also called the drag free control , must be carried out with an accuracy of @xmath94 within the signal bandwidth of @xmath95 to @xmath96 hz , depending on the satellite s spin rate . the general principle of drag free control is to make the satellite s trajectory as close as possible a geodesic . therefore , a capacitive reference sensor @xcite is used . the sensor unit consists of a test mass whose movements are measured capacitively with respect to all 6 degrees of freedom . apart from the sensing electrodes , electrodes for servo control surround the test mass and compensate its movement relative to the satellite structure . the signal is also used to control the satellite s movement via the ion thrusters . thus , the test mass falls quasi freely and is shielded against all disturbances by the satellite , in particular against solar pressure and drag . because the servo - control is influenced by back action effects , the test mass and the servo control form a spring mass system whose spring constant and eigenfrequnecy must be adapted to the signal bandwidth . therefore , electrode surfaces , charging by external sources as well as the precision of the test mass and the electrode alignments influence the measurement of @xmath97 directly and make necessary repeated in orbit calibration maneuvers @xcite . the chosen orbit avoids charging effects as much as possible . feep ion thrusters ( field emission electric propulsion ) must be used to overcome ( 1 ) solar radiation pressure acting on the satellite and disturbing its free fall behaviour , and ( 2 ) to control the residual acceleration down to @xmath98 in the signal bandwidth . the first requirement sets an upper limit for the thrust : linear forces acting on the satellite are less than @xmath99 in all 3 axes and maximum torques are about @xmath100 . the second condition determines the resolution of thrust control which have to be done with an accuracy of about @xmath101 . a feep able to satisfy these requirements is the indium liquid metal ion source ( lmis ) of the austrian research centers seibersdorf ( arcs ) @xcite . thrust is produced by accelerating indium ions in a strong electrical field . a sharpened tungsten needle is mounted in the centre of a cylindrical indium reservoir bonded to a ceramic tube which houses a heater element for melting the indium . ion emission is started by applying a high positive potential between the tip covered with a thin indium film and an accelerator electrode . to avoid charging a neutralizer emitting electones is installed . the feeps have a length of 2 cm , a diameter of 4 mm , and weigh only several grams sufficient for the entire mission . a continuous thrust of @xmath102 per thruster is available . for a satellite diameter of about 1.5 m ( the maximum asap 5 size ) , the solar radiation pressure of ca . @xmath103 and the radiation pressure of the earth albedo of @xmath104 sum up to a total drag of about @xmath105 . considering thruster noise , misalignments and other disturbing effects , a continuous thrust of @xmath106 would be sufficient . to control all 6 degrees of freedom a minimum of 3 clusters of 4 thrusters is necessary . to guarantee a continuous thrust with some redundancy 4 clusters are desirable . the power consumption is less than 3 w on average with peaks up to 25 w. the proposed optis mission is capable to make considerable improvements , up to three orders of magnitude , in the tests of sr and gr . it is designed to be a low cost mission . it is based on using ( i ) recent laboratory developments in optical technology and ( ii ) the advantages of space conditions : quit environment , long integration time , large velocities and large potential differences . the optical technology includes an optical comb generator , stabilized lasers and highly stable cavities . we remark that an alternative optical cavity system consisting of a monolithic silicon block operated at @xmath107 , the temperature where the thermal expansion coefficient vanishes , should be studied @xcite . the advantage of this approach could be a significantly reduced level of creep and therefore a corresponding improvement of the kt test .

a new satellite based test of special and general relativity is proposed . for the michelson morley experiment we expect an improvement of at least three orders of magnitude , and for the kennedy thorndike experiment an improvement of more than one order of magnitude . furthermore , an improvement by two orders of the test of the universality of the gravitational red shift by comparison of an atomic clock with an optical clock is projected . the tests are based on ultrastable optical cavities , an atomic clock and a comb generator .

You are an expert at summarizing long articles. Proceed to summarize the following text: gamma ray bursts ( grbs ) have been deeply studied in the kev mev energy range , but only little information in the gev range has been provided in the past decade by egret measurements . only 3 bursts have been detected at energies @xmath21 gev , with one photon reaching 18 gev @xcite . the study of the high energy emission of grbs could provide extremely useful data able to constrain the emission models and the value of the ambient parameters . at these high energies the detection from space is hampered by the very low fluxes , requiring large collection areas . from ground the last generation cherenkov telescope magic @xcite , designed also to point at the detected grb direction in a very short time , is still making efforts to lower the threshold at energies @xmath3100 gev . its duty cycle and field of view are however very small : a wide field detector , able to cover simultaneously and continuously a significant ( @xmath01 steradian ) fraction of the sky , is thus necessary . with such a detector the milagro collaboration reported evidence for tev emission from grb 970417a with a significance slightly greater than 3@xmath4 @xcite . the study of the high energy spectrum of grbs is perhaps the strongest motivation for an all - sky vhe detector . the study of transient phenomena can be successfully performed at energies down to 1 gev by air showers arrays working in `` single particle mode '' @xcite , i.e. , counting all the particles hitting the individual detectors during fixed time intervals . the observation of an excess in coincidence with a grb detected by satellites would be an unambiguous signature of the nature of the signal . both the sensitivity and the energy threshold improve with larger detection areas and higher observation levels , making air shower detectors at very high altitude the most suitable . the argo - ybj experiment , located at the yangbajing cosmic ray laboratory ( 4300 m a.s.l . ) with a detection area of @xmath0 6700 @xmath5 , is an air shower array exploiting the full coverage approach at very high altitude , with the aim of studying the cosmic radiation with a low energy threshold . in this paper we present results on the search for grbs in coincidence with satellite detections performed in the december 2004 - may 2006 period with the argo - ybj experiment . the argo - ybj detector is constituted by a single layer of resistive plate chambers ( rpcs ) with @xmath093@xmath6 of active area . this carpet has a modular structure , the basic module being a cluster ( 5.7@xmath77.6 m@xmath1 ) , divided into 12 rpcs ( 2.8@xmath71.25 m@xmath1 each ) . each chamber is read by 80 strips of 67.5@xmath7618 mm@xmath1 , logically organized in 10 independent pads of 55.6@xmath761.8 cm@xmath1 which are individually acquired and represent the high granularity pixel of the detector @xcite . the carpet is composed by 154 clusters for a total surface of @xmath06700 m@xmath1 . the detector is connected to two different daq systems , which work independently : in shower mode , for each event which fulfill the trigger conditions the position and time of each detected particle is recorded , allowing the reconstruction of the lateral distribution and of the arrival direction @xcite ; in scaler mode , where there is no trigger , the counting rate of each cluster is measured every 0.5 s , with no measurement of the space distribution and arrival direction of the detected particles . in the scaler mode daq , for each cluster the signal coming from the 120 pads is added up and put in coincidence in a narrow time window ( 150 ns ) , giving the rate of counts @xmath91 , @xmath92 , @xmath93 , @xmath94 , read by four independent scaler channels . the corresponding measured rates are , respectively , @xmath0 40 khz , @xmath0 2 khz , @xmath0 300 hz and @xmath0 120 hz for each cluster . the counting rates for a given multiplicity are then obtained with the relation n@xmath10 = n@xmath11 - n@xmath12 for i = 1 , 2 , 3 . the use of four different scalers may give an indication of the source spectrum in case of signal detection . in order to correctly handle the data , it is very important to evaluate the response to particles hitting the detector . for scaler mode operations , the most important effect is the strip cross - talk , i.e. , the probability of having more than one strip fired by a single particle , giving fake coincident counts . due to the front - end logic , this can happen only for strips belonging to different pads , since the maximum number of counts for each pad is 1 independently of the number of particles hitting simultaneously the pad . an analytical calculation based on the measured `` occupancy '' , i.e. , the mean number of strips fired by 1 particle , has been made and checked experimentally . from the experimental point of view , it is important to take into account the background counting rate variations due to changes in environmental parameters such as the atmospheric temperature and pressure ( which modify the shower development in the atmosphere ) and the detector temperature ( instrumental effect ) . more troublesome are other possible instrumental effects , such as the electronic noise , that could simulate narrow signals in time , producing spurious increases in the background rate . working in single particle mode requires very stable detectors , and a very careful and continuous monitoring of the experimental conditions . by comparing the counting rate of the single detectors and requiring simultaneous and consistent variations in all of them , it is possible to identify and reject most of the fake excesses due to instrumental effects . short variations in the single particle counting rate have been measured in coincidence with strong thunderstorms and have been ascribed to the effects of atmospheric electric fields on the secondary particles flux @xcite . the static electric field is measured on the roof of the argo - ybj building with an efm100 boltek atmospheric field monitor . anyway , we note that the occurrences of these events are very rare and even in this case the observed time scales ( @xmath010 - 15 minutes ) are longer than the typical grb duration . the study of the counting distribution for each cluster is important in order to monitor the stability of the detector and its statistical ( poissonian ) behaviour . in fig . [ fig : csum ] the total counting rate of a typical cluster , added up on the 4 multiplicity channels during a period of 30 minutes , follows a poissonian distribution with a @xmath13 given by : @xmath14 in order to study the detector response to extensive air showers ( eass ) , a detailed mc simulation has been carried out for both protons and photons with fixed energies in the range 1 gev - 1 tev and zenith angles @xmath15 = 0@xmath16 , 10@xmath16 , 20@xmath16 , 30@xmath16 , 40@xmath16 . the corsika / qgsjet code 6.204 @xcite has been used with a full electromagnetic component development down to e@xmath17= 0.05 mev for both electrons and photons and 50 mev for muons and hadrons . a detailed description of the detector has been carried out to correctly simulate the `` cluster size '' , i.e. , the correlation between the number of particles hitting the detector and the number of signals generated in the different multiplicity channels . since the actual efficiency depends essentially on the shower particle lateral distribution , a huge quantity of showers must be simulated over a very large area to completely contain it . to save the computing time the shower sampling can be performed by means of the _ `` reciprocity technique '' _ @xcite . the sampling area a@xmath18 ( @xmath0 5000@xmath75000 m@xmath1 ) is uniformly filled with replicas of the same argo - ybj carpet , one adjacent to the other . following the reciprocity concept , we sample the shower axis only over the area covered by the carpet located at the center of the array , with the prescription of considering the response of all the detector replicas . on an event - by - event basis we calculate the number of clusters which contain more than 1 , 2 , 3 , 4 fired pads , summed on the entire grid . fig.[fig : aeff_20 g ] shows the effective areas for primary photons and protons with zenith angle @xmath19 in the four multiplicity channels for the complete argo - ybj detector constituted by 154 clusters ( @xmath06700 m@xmath1 sensitive area ) . we note that for a multiplicity @xmath20 the detector sensitivity does not depend on its geometrical features , like the area of the single counters or their relative positions , but only on the total sensitive area @xcite . therefore , the effective areas for any carpet dimension can be scaled from the plotted values . the effective areas for primary protons are then convoluted with the following spectrum : @xmath21 with @xmath22 @xcite and taking into account the local geomagnetic cutoff @xcite . the resulting counting rates , considering an opening angle of 60 degrees around the zenith , are the following : 21 khz for n = 1 , 1.7 khz for n = 2 , 180 hz for n = 3 and 80 hz for n @xmath23 4 . comparison with the measured rates , i.e. , 38 khz for n = 1 , 1.7 khz for n = 2 , 180 hz for n = 3 and 120 hz for n @xmath23 4 , shows that the values obtained by our simulations are lower in the multiplicity channels n = 1 and n @xmath23 4 . the discrepancy for n = 1 is expected because of dark counting and natural radioactivity . since from both of them we expect mostly single counts , these effects are expected to influence only the @xmath91 scaler channel . the search for emission from grbs started with the first grb detection by the swift satellite on december 17 , 2004 , when only 16 clusters ( @xmath0693 m@xmath1 of sensitive area ) out of the total 154 were in data taking . up to may 2006 , 28 grbs detected by satellites were within the argo - ybj field of view ( for this search , @xmath24 ) . because of detector installation and debugging operations , the duty cycle of data taking has been reduced and reliable data are available only for 16 of these grbs ( see table [ tab : grb ] ) . for every grb , the number of counts @xmath25 , recorded in each of the four multiplicity channels during the duration time @xmath26 measured by the satellites , is compared with the number @xmath27 expected from the background ( obtained from the average counting rate in @xmath28 around the burst ) . the difference @xmath29 in units of standard deviations , i.e. , @xmath30 , gives the statistical significance @xmath31 of the excess , which we report in column 8 of table [ tab : grb ] for @xmath20 . the data analysis of 3 grbs ( grb051114 , grb060105 and grb060510a ) gives 2.8 , 3.6 and 3.7 as the statistical significance of the signal , respectively . taking into account that we considered a sample of 16 grbs , these values correspond to a post - trial probability p(@xmath32 ) , p(@xmath33 ) and p(@xmath34 ) , respectively , of being a background fluctuation . as a consequence , no convincing excess in the scaler counts was observed in the duration time measured by the satellites . therefore @xmath35 upper limits to the fluence of these events were calculated in the 1 100 gev energy range using the spectral indices determined at lower energies by satellites . our results are reported in the last column of table [ tab : grb ] . for those grbs whose redshift has been also determined , the upper limit was calculated including a model for @xmath36 absorption by the extragalactic background light ( ebl ) @xcite and the corresponding values printed in bold . for the other grbs z = 0 was assumed ( below 300 gev the @xmath36 absorption is almost negligible for z@xmath370.2 ) . llccccccc grb & sat . & t90/dur . & @xmath38 & redshift & spectral & carpet & @xmath39 & ul@xmath40 + & & ( s ) & ( deg ) & & index & area ( m@xmath1 ) & & ( fluence ) + 041228 & swift & 62 & 28.1 & & 1.56 & 693 & -1.3 & 3.3@xmath4110@xmath42 + 050408 & hete & 15 & 20.4 & 1.24 & 1.98 & 1820 & -2.2 & * 9.6@xmath4110@xmath43 * + 050509a & swift & 12 & 34.0 & & 2.1 & 1820 & 0.29 & 1.6@xmath4110@xmath42 + 050528 & swift & 11 & 37.8 & & 2.3 & 1820 & -0.012 & 6.5@xmath4110@xmath42 + 050802 & swift & 20 & 22.5 & 1.71 & 1.55 & 1820 & 0.74 & * 1.0@xmath4110@xmath42 * + 051105a & swift & 0.3 & 28.5 & & 1.33 & 3379 & 0.90 & 1.4@xmath4110@xmath43 + 051114 & swift & 2 & 32.8 & & 1.22 & 3379 & 2.8 & 1.9@xmath4110@xmath43 + 051227 & swift & 8 & 22.8 & & 1.31 & 3379 & 0.93 & 2.5@xmath4110@xmath43 + 060105 & swift & 55 & 16.3 & & 1.11 & 3379 & 3.6 & 5.9@xmath4110@xmath43 + 060111 & swift & 13 & 10.8 & & 1.63 & 3379 & 0.82 & 2.5@xmath4110@xmath43 + 060115 & swift & 142 & 16.6 & 3.53 & 1.76 & 4505 & -2.2 & * 2.3@xmath4110@xmath42 * + 060421 & swift & 11 & 39.3 & & 1.53 & 4505 & -0.46 & 1.6@xmath4110@xmath42 + 060424 & swift & 37 & 6.7 & & 1.72 & 4505 & 1.9 & 4.1@xmath4110@xmath43 + 060427 & swift & 64 & 32.6 & & 1.87 & 4505 & -1.8 & 1.8@xmath4110@xmath42 + 060510a & swift & 21 & 37.4 & & 1.55 & 4505 & 3.7 & 2.3@xmath4110@xmath42 + 060526 & swift & 14 & 31.7 & 3.21 & 1.66 & 4505 & 0.75 & * 1.2@xmath4110@xmath42 * + @xmath44 zenith angle . + @xmath45 significance of the signal for the single event . + @xmath46 upper limit on the fluence ( 1 100 gev ) in erg @xmath47 . the numbers in bold take into account absorption by the ebl . a search for vhe emission from grbs has been performed with an increasing detector area of the argo - ybj experiment . a total of 16 satellite - triggered grbs in the field of view ( @xmath48 ) of argo - ybj in the dec . 2004 - may 2006 period has been analyzed . no significant emission was detected and typical fluence upper limits of @xmath49 erg @xmath47 in the 1 100 gev energy range were obtained using the measured counting rates and grb parameters determined by the satellite observations . we expect to increase the sensitivity by a factor @xmath02 converting the secondary photons with a 0.5 cm thick layer of lead . catelli , j.r . et al . : in : c.a . meegan ( ed . ) , aip conf . proc . 428 , p. 309 ( aip , new york , 1998 ) . albert , j. et al . : apj lett . * 641 * l9 ( 2006 ) . atkins , r. et al . : apj lett . * 553 * l119 ( 2000 ) . vernetto , s. : astrop . * 13 * 75 ( 2000 ) . aielli , g. et al . : nim * a562 * 92 ( 2006 ) . di sciascio , g. et al . : 29th icrc , pune , * 2 * , 33 - 36 ( 2005 ) . aglietta , m. et al . : 26th icrc , salt lake city , * 7 * , 351 - 354 ( 1999 ) . heck , d. et al . : report * fzka 6019 * forschungszentrum karlsruhe , 1998 . battistoni , g. et al . : astrop . phys . * 7 * 101 ( 1997 ) . gaisser , t.k . , honda m. : ann . nucl . part . sci . * 52 * 153 ( 2002 ) . storini m. , smart d.f . , shea m.a . : in : 27th icrc , hamburg , * 10 * 4106 - 4109 ( 2001 ) . kneiske , t.m . a@xmath50a * 413 * 807 ( 2004 ) .

the argo - ybj experiment is almost completely installed at the yangbajing cosmic ray laboratory ( 4300 m a.s.l . , tibet , p.r . china ) . the lower energy limit of the detector ( e @xmath0 1 gev ) is reached with the scaler mode , i.e. , recording the single particle rate at fixed time intervals . in this technique , due to its high altitude location and large area ( @xmath0 6700 m@xmath1 ) , this experiment is the most sensitive among all present and past ground - based detectors . in the energy range under investigation , signals due to local ( e.g. solar gles ) and cosmological ( e.g. grbs ) phenomena are expected as significant enhancements of the counting rate over the background . results on the search for grbs in coincidence with satellite detections are presented . 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: we discuss specific aspects of the core of a vortex line in layered high t@xmath1 superconductors . the physics of these vortices is governed by two distinct length scales , the london penetration depth in the planes , @xmath2 , and the coherence length in the planes , @xmath3 . the penetration depth is the electromagnetic length scale of a vortex . the physics on this length scale is well described by a combination of macroscopic electromagnetism , london s theory for supercurrents along the layers , and interlayer josephson coupling . this description breaks down in the core of the vortex , _ i.e. _ at distances of order @xmath4 from the center of the vortex . thus , physical properties of the core carry information on the microscopic physics of high t@xmath1 superconductivity . the small coherence length of high t@xmath1 superconductors makes the vortex core a good potential sensor for microscopic mechanisms of superconductivity . our discussion of the vortex core in high t@xmath1 superconductors is based on the fermi - liquid model of superconductivity . the physical properties of the vortex core predicted by this model are spectacular , unique , and could serve as fingerprints of the traditional pairing theory of superconductivity . the vortex core of traditional high-@xmath5 superconductors is well described by the bardeen - stephen model@xcite which represents the core by a region of normal electrons . the bardeen - stephen model is justified as long as the mean free path , @xmath6 , is much shorter than the core size , so that the motion of an electron gets randomized before it leaves the core . this condition is not fulfilled in high t@xmath1 superconductors which are generally clean superconductors with @xmath7 . the core of a vortex in a clean superconductor was first studied in the classic papers of caroli , matricon , and de gennes.@xcite these authors calculated the spectrum of quasiparticle states in the core , and showed that electrons and holes form bound states at energies below the bulk energy gap . further early studies of the excitation spectrum in the core can be found in refs . . more recent theoretical work was stimulated by the direct observation of core states in @xmath8 by scanning tunnelling spectroscopy ( sts).@xcite the recent report of sts in ybco @xcite provides new information on the excitation spectrum of vortices in the high @xmath9 cuprates . consequently , theoretical efforts focused on the tunnelling density of states of bound states in isolated vortices and vortex lattices.@xcite these calculations show that the bound states in the core have a different nature compared with the usual quantum mechanical bound states in a potential well . the core states are coherent superpositions of particle states and hole states and are formed by repeated andreev scattering from the pair potential ( order parameter ) in the core . andreev scattering is a process of `` retroreflection '' of excitations : spatial variations of the amplitude or the phase of the order parameter induce branch conversion of electron - like excitations into hole - like excitations , and vice versa . bound states occur at energies at which the phases of multiply reflected electron - like and hole - like states interfere constructively . the charge current carried by an incoming electron and an outgoing andreev reflected hole is identical because the reversal of the velocity in an andreev reflection process is compensated by the reversal of the charge due to electron - hole conversion . consequently , andreev bound states can transport a charge current , unlike bound states in a potential well . charge conservation requires that the current carried by the bound states inside the core is transported outside the core by bulk supercurrents . this leads to an interplay between supercurrents flowing past the core and the bound states in the core . hence , the physics of the ` normal core ' in clean superconductors is basically the physics of the bound states in contact and intimate exchange with the superconducting environment outside the core . consider a stack of `` pancake '' vortices forming an isolated vortex line whose axis is oriented perpendicular to the layers . we investigate the current distribution in the core of a pancake vortex , and show how this distribution changes if the vortex is exposed to a bulk supercurrent , or the circulation is changed from @xmath10 to @xmath11 . we calculate the _ spectral current density _ , which carries the information on the contribution of the states in a given energy interval to the total current density . a supercurrent in homogeneous superconductors is distributed over all continuum states . these states exhibit doppler shifts of their energies , @xmath12 , in the presence of a phase gradient in the order parameter ( or superfluid momentum , @xmath13 ) . the total current is obtained by adding the contributions of states with positive shifts from quasiparticles co - moving with the flow and the contributions with negative shifts from quasiparticles that are counter - moving relative to the flow field . we find that the currents in the core have a very different spectral distribution from bulk supercurrents . the continuum states ( scattering states ) show smeared out doppler shifts , and contribute very little to the total current . the dominant contributions to the circulating currents around the vortex center , as well as the currents through the core , come from andreev bound states . hence , the physics of vortex cores in clean superconductors ( @xmath14 ) is very different from the physics of the vortex core in a dirty superconductor ( @xmath15 ) , which is well described by a continuum of normal electronic states . the calculations presented in this paper concentrate on stationary properties of the vortex core of clean layered superconductors . we expect more spectacular effects in the dynamic properties . the bound states react sensitively to the environment outside of the core . this leads to a coupling of the collective degrees of freedom in the london range of the vortex and the bound states in the core , which will produce a rich spectrum of largely unexplored dynamical phenomena . below we present analytical and numerical calculations for the states in the vortex core . we use two versions of a quasiclassical formulation of the bcs theory of superconductivity : a ) _ andreev s theory_@xcite which represents the quasiclassical limit of bogolyubov s equations,@xcite and b ) the _ quasiclassical theory _ of eilenberger @xcite , larkin , and ovchinnikov@xcite which represents the quasiclassical limit of gorkov s green s function theory . andreev s theory and the quasiclassical theory are essentially equivalent for clean superconductors , and in this limit the choice of approach is largely a matter of taste . however , the quasiclassical theory has a wider range of application . it is the generalization of landau s fermi liquid theory to the superconducting state , and is capable of describing a broader range of superconducting materials and phenomena , such as dirty superconductors or superconductors with short inelastic lifetimes ( strong - coupling superconductors).@xcite section ii contains analytical results for the bound states and the spectral current density for a pancake vortex with a superimposed bulk supercurrent . these results are obtained from andreev s hamiltonian@xcite by the methods described in ref . . in section iii we discuss the numerical results , which are obtained using the quasiclassical theory of fermi - liquid superconductivity . we solve the quasiclassical transport equations to obtain self - consistently the pair amplitude ( order parameter ) for pancake vortices . given the pair amplitude we calculate the excitation spectrum in the core of the vortex , and deduce from it the spectral current density . the numerical calculations are done for layered superconductors with s - wave pairing . the analytical and numerical results confirm and complement each other , and they establish the important role of the bound states for the currents in the core region of a vortex . in this section , we investigate the spectrum of current carrying states of a two - dimensional pancake vortex in equilibrium at temperature @xmath16 , in the presence of an externally imposed supercurrent . we ignore the spin degree of freedom of a quasiparticle excitation;@xcite in this case it is sufficient to work in the two - dimensional space of particle - hole degrees of freedom . operators in this space are @xmath17 matrices , and we use the notation @xmath18 , @xmath19 , @xmath20 , for the three pauli matrices in particle - hole space ( nambu space ) , and @xmath21 for the unit matrix . the hamiltonian for the quasiparticle excitations , @xcite the bogolyubov hamiltonian , then reads : @xmath22 where @xmath23 and @xmath24 are the momentum and position operators appropriate to a particle moving in two dimensions , @xmath25 is the fermi momentum and @xmath26 is the electromagnetic vector potential . the order parameter , @xmath27 is an off - diagonal matrix and is generally represented by a linear combination of @xmath18 and @xmath19 . in the absence of an externally imposed supercurrent , we write the order parameter of the vortex as @xmath28 where @xmath29 is the magnitude of the order - parameter of a bulk superconductor at temperature @xmath16 , @xmath30 is the normalized profile of the vortex , which is a monotonically increasing function of @xmath31 obeying @xmath32 , and @xmath33 is the angular coordinate of @xmath34 with @xmath35 , @xmath36 the main assumption we make in this section is that the order - parameter in the presence of a superflow @xmath37 has the form @xmath38 we assume throughout this section that @xmath39 is small compared to the bulk critical current , @xmath40 . the principal physical quantities with which we shall concern ourselves are the _ spectral current density _ , and the total equilibrium current density which is related to @xmath41 by @xmath42 we shall also make reference to the local density of states , @xmath43 . the quantities @xmath44 and @xmath45 may be expressed in terms of the one - particle greens function , @xmath46 or , equivalently , the `` spectral function '' @xmath47 . using the spectral function , we find that in dirac notation , @xmath48 @xmath49 where the subscript @xmath50 denotes the upper left element of the @xmath17 matrices , thereby selecting out the _ particle sector _ of the spectral function and the factor @xmath51 takes into account both spin projections of the quasiparticles . most calculations of the properties of superconductors with inhomogeneous order parameters are simpler in the quasiclassical limit , where one takes advantage of the separation in the scales of the wavelength of quasiparticles near the fermi energy and the characteristic scale for spatial variations of the pair potential , i.e. @xmath52 . the quasiclassical limit of the bogolyubov hamiltonian ( [ bogoliubov ] ) is the andreev hamiltonian in which the kinetic energy in ( [ bogoliubov ] ) is replaced by an operator that is _ linear _ in the gradient.@xcite let us define the normal - state density of states at the fermi level , @xmath53 , and introduce the directions , @xmath54 and @xmath55 , which are , respectively , parallel and perpendicular to trajectories of a quasiparticle wavepacket in the quasiclassical description , _ i.e. _ @xmath56 . the coordinates along these directions are defined by @xmath57 . in addition we work in the limit @xmath58 , in which case the vector potential is approximately constant in the vicinity of the vortex core and can be neglected . the order parameter in ( [ delta ] ) can be written as @xmath59 .\ ] ] by performing a gauge transformation that removes the factor of @xmath60 in eq . ( [ op ] ) we obtain the spectral current density and the local density of states in terms of the andreev hamiltonian for an isolated vortex , @xmath61 @xmath62)|\zeta \rangle _ { 1,1 } \,,\ ] ] @xmath63)|\zeta \rangle _ { 1,1}\,,\ ] ] where @xmath64 is an eigenvector of the `` one - dimensional '' trajectory coordinate operator , @xmath65 : @xmath66 . the operators @xmath67 and @xmath68 appearing in @xmath69 are canonically conjugate : @xmath70=i\hbar$ ] . the quasiclassical interpretation given to ( [ andreev ] ) is as follows : quantum - mechanical evolution in particle - hole space takes place along classical trajectories parallel to @xmath71 having a fixed value of @xmath72 . thus , @xmath73 is identified as a c - number impact parameter.@xcite let us write the current density at temperature @xmath16 as @xmath74 where @xmath75 is a high energy cutoff that serves to make manipulations of @xmath76 well defined ; large positive energies are automatically cut off by the fermi function , @xmath77 . where no ambiguity arises , we shall take @xmath78 . defining @xmath79 , and using ( [ andreev ] ) we have @xmath80 next , we split up the energy integrals into the following terms @xmath81 @xmath82 @xmath83 \,,\ ] ] @xmath84 the three contributions to the current have different interpretations . 1 . since @xmath85 is large , @xmath86 may be replaced by its high energy , normal - state limit , @xmath87 and @xmath88 this term coincides with the @xmath89 current of a uniform superconductor . the term @xmath90 contributes to `` backflow '' , since it always yields a current with a component in the @xmath91 direction.@xcite the term @xmath90 contains the current carried by the bound states and also a correction to the @xmath89 current due to the thermal breaking of pairs . to appreciate these points we look at this term in two limits , assuming @xmath92 . ( i ) for @xmath89 we have @xmath93 the small size of @xmath94 ensures that the energy integral only selects states in the gap , thus @xmath95 only obtains contributions from the bound states . ( ii ) for @xmath96 and assuming @xmath97 to be that of a uniform system , @xmath98 we write @xmath99@xmath100@xmath101 and obtain @xmath102= -ev_f^2n_f{\bf p}_sy(\beta \delta_{0 } ) \,,\ ] ] where @xmath103 is the yosida function which gives a quantitative measure of the thermal breaking of cooper pairs . the term @xmath104 is independent of @xmath105 and is simply the current density of a vortex in the absence of an externally imposed supercurrent . consider the spectral properties at the center of the vortex . for @xmath106 @xmath107 with @xmath108 accounting for the @xmath109 phase change across the vortex core . this special case of the andreev hamiltonian is identical in form to the continuum hamiltonian used to describe _ trans_-polyacetylene containing a single topological soliton.@xcite it is known that this hamiltonian always has a non - degenerate bound state at zero energy.@xcite whether or not it has other bound states depends on the form of the profile , @xmath110 . for the single quantum vortex and trajectories through the center there are no other bound states . the eigenfunction for the zero - energy bound state , @xmath111 , is found by solving @xmath112 \psi_0(\zeta ) = 0.$ ] the normalized solution is @xmath113 @xmath114 where @xmath115 is a profile dependent quantity with the dimensions of length , @xmath116 . analytical estimates of the bound states at distances far from the vortex are given in appendix [ far ] . for energies @xmath117 only the bound state of @xmath118 will contribute to the spectral current density ( and the local density of states ) , @xmath119_{1,1}\ ] ] @xmath120 \hat{{\bf p}}_s\ , , \quad there is a simple relation between @xmath121 and @xmath122 when @xmath123 in ( [ jo ] ) , the delta function in the integrand of @xmath121 effectively replaces @xmath71 by @xmath124 . taking this factor outside the integral leaves an integral identical to that of the local density of states . consequently , @xmath125 note that the contribution of negative energy ( bound ) states to the total current density lies in the @xmath126 direction , i.e. _ opposite _ to the externally imposed supercurrent . at zero temperature the total current density originates from the bound states having energies @xmath127 , @xmath128 the current density of an isolated vortex with @xmath129 vanishes at the center of the vortex , _ i.e. _ @xmath130 . we can combine the result in ( [ j_bound ] ) with @xmath131 given in ( [ j1 ] ) to obtain the total current density at @xmath89 : @xmath132 thus , for sufficiently small * p*@xmath133 the bound - state contribution dominates ( [ center ] ) and @xmath134 will point in the @xmath135 direction . for equilibrium conditions the divergence of the current density vanishes . from ( [ j1 ] ) , @xmath131 has a vanishing divergence , and for an undisturbed vortex we have @xmath136 . thus , @xmath137 . at @xmath89 , @xmath138 in eq . ( [ andreevspectrum ] ) of appendix [ simplification ] we show that @xmath139 \,,\ ] ] yielding latexmath:[\[\label{sso}s({\bf r})=4\pi ev_f\delta_{0}n_f\int_0^{2\pi } \frac { d\varphi _ k}{2\pi } \int_0^{-{\bf q\cdot}\hat{\bf k}}d\epsilon \text{\/}\frac { f(r)}r\mbox{tr}\left [ ( \zeta \hat\tau_{2}-\eta \hat\tau_{1})\langle \zeta @xmath92 , only the bound state , @xmath141 , contributes to the @xmath142 integral . thus , @xmath143 where @xmath144 is the bound - state energy for an impact parameter @xmath73 . at small distances from the center of the vortex ( @xmath145 ) @xmath146 , with @xmath147 , and the lowest energy bound state is then @xmath148 @xmath149 substituting @xmath150 and @xmath151 into ( [ s ] ) and setting @xmath152 yields @xmath153 which is non - zero , indicating that the ansatz ( [ delta ] ) is not physically correct for any value of @xmath154 . this failure to satisfy the conservation law is due to the lack of self - consistency of the order parameter in eq . ( [ delta ] ) in the presence of the flow field . in the absence of pinning , the vortex will move in response to a flow field , even one of arbitrarily small strength . the results in section ii implicitly assume a _ pinned _ vortex . thus , there will be distortion of the vortex away from its cylindrically symmetric equilibrium form ( [ delta ] ) . in the following section we show that the self - consistently determined vortex order parameter , which includes the deformation by the flow field , restores the conservation law . a versatile and efficient method for calculating local spectral properties of superconductors is the quasiclassical theory of superconductivity.@xcite this theory is well adapted for a numerical approach to microscopic problems in superconductivity , such as the calculation of the structure and the excitation spectrum of vortex cores for superconductors with isotropic , anisotropic or unconventional order parameters . the quasiclassical theory is the only theoretical formulation which can handle equally well clean and dirty superconductors , as well as more complicated geometries than that of an isolated vortex with cylindrical symmetry or a perfect vortex lattice . it can be interpreted as the generalization of landau s theory of normal fermi liquids to the superconducting state . the quasiclassical theory shares with landau s theory the semiclassical description of the orbital degrees of freedom of quasiparticle excitations . on the other hand , the internal degrees of freedom , _ i.e. _ the spin and the particle - hole degrees of freedom , are described by quantum mechanics . quantum mechanical coherence of particle and hole excitations is the basis of all superconducting effects such as persistent supercurrents , flux quantization , josephson effects and andreev reflection . here we use the quasiclassical theory for our investigations of the vortex core . numerical work on vortices in superconductors using the quasiclassical theory started with a series of publications by kramer , pesch , and watts - tobin . @xcite more recent work includes pinning of vortices at small defects,@xcite vortices in superfluid @xmath155he and other systems with unconventional pairing,@xcite the excitation spectrum of bound quasiparticles,@xcite and the spectrum of moving pancake vortices.@xcite we use in this article the notation of refs . . the central objects of the quasiclassical theory of superconductors in equilibrium are the quasiclassical propagators @xmath156 , which are @xmath17 matrices in the particle - hole index , @xmath157 the variables @xmath158 and @xmath159 stand for the energy of an excitation and its momentum ( on the fermi surface ) . the momentum variable reduces to @xmath160 for an isotropic fermi surface in two dimensions ( see section ii ) . general symmetries lead to the following fundamental relation between @xmath161 and @xmath162 , @xmath163 we use , as described in section ii , the notation @xmath18 , @xmath19 , @xmath20 , for the three pauli matrices in particle - hole space , and @xmath21 for the unit matrix . the off - diagonal terms @xmath164 in eq . ( [ matrx ] ) are the pair amplitudes . they vanish in the normal state , and measure the amount of particle - hole mixing in the superconducting state . the diagonal elements of the propagators determine the density of states , @xmath165 and the equilibrium current density . the most detailed information on the current distribution is obtained from the _ spectral current density _ , @xmath166 where @xmath167 is the dimensionless density of states for co - moving ( @xmath168 ) and counter - moving ( @xmath169 ) excitations along the trajectory line defined by @xmath159 , and @xmath170 is the fermi velocity at the point @xmath159 on the fermi surface . this spectral density measures the contributions of quasiparticle states with energy @xmath158 and momentum near the fermi surface point @xmath159 to the current density at position @xmath171 . the full current density is obtained by weighting the spectrally resolved current density by the occupation probability of the quasiparticle states , then integrating over fermi momenta and energies . for equilibrium states , @xmath172 where @xmath173 is the fermi distribution function . the symbol @xmath174 denotes a weighted integral over the fermi surface . the weight at @xmath159 is @xmath175 , and the integral is normalized , @xmath176 . the spectral current density is particularly well suited for our study of the importance of andreev bound states for the current flow in a vortex core . these bound states appear as delta functions in the spectral current density at energies below the bulk energy gap . the spectral weight of the delta function , combined with the occupation of the bound state , determine its contribution to the total current density . we calculate @xmath177 from eilenberger s transport equation@xcite @xmath178 + i\hbar{\bf v}_f\cdot\mbox{\boldmath $ \nabla$}\hat g^r({\bf p}_f,{\bf r};\epsilon)=0\ , , \ ] ] supplemented by the condition of analyticity in the upper half of the complex @xmath158-plane , and the normalization condition @xmath179 for a fixed fermi momentum @xmath159 this is a first order differential equation along a straight - line classical trajectory in the direction of the fermi velocity @xmath170 . the propagator @xmath180 at a chosen point of interest , @xmath181 , is determined by the solution of ( [ eilen ] ) along the trajectory through @xmath171 in the direction @xmath170 . complete information on the local physical properties at point @xmath171 , such as the current density , is obtained by sampling all trajectories through @xmath171 . the propagator @xmath182 is intimately related to the @xmath183 density matrix of the particle - hole degrees of freedom of a quasiparticle moving along the classical trajectory specified by @xmath159 , @xmath171 . thus , @xmath177 describes the state of the internal degrees of freedom of the excitation . the internal state , _ i.e. _ the amount of particle - hole mixing , may change along the trajectory as a consequence of the off - diagonal pair potential , @xmath184 , which acts as a driving term that ` rotates ' the particle - hole pseudo spin . the pair potential couples particle and hole excitations , and is the origin of particle - hole coherence . it depends on the real space position , @xmath171 , and , for anisotropic superconductors , on the fermi surface position , @xmath159 , @xmath185 the pair potential must be calculated self - consistently from the gap equation , @xmath186 where @xmath187 is the pairing interaction , which determines the orbital symmetry of the pair potential , its magnitude and @xmath9 . our procedure for numerical calculation of the currents in the core of 2d pancake vortices is the following . we first solve self - consistently the gap equation and eilenberger s equation at matsubara energies . this allows us to determine the pair potential and the supercurrent density . we then insert the pair potential into eilenberger s differential equation at real energies , and obtain from its solution the excitation spectrum : the density of states and the spectral current density . the differential equations are solved by standard 4@xmath188 order runge - kutta and multiple shooting methods , and self - consistency is achieved iteratively by using alternatively a relaxation method and the mbius - eschrig algorithm.@xcite we consider three examples of pancake vortices : isolated , i ) singly - quantized and ii ) doubly - quantized @xmath189-wave vortices , and iii ) a pinned @xmath189-wave vortex in the presence of a uniform transport supercurrent . we choose a temperature of @xmath190 , unless otherwise noted , and assume @xmath191 . in this limit the vector potential is essentially constant in the core region , and can be neglected . figure 1 shows the amplitude of the order parameter of a singly - quantized @xmath189-wave vortex . the amplitude is isotropic and vanishes linearly in the core . the variation of the amplitude and phase along two trajectories are also shown in fig . 1 . for trajectory @xmath192 passing through the center of the vortex , the phase changes discontinuously and the amplitude vanishes linearly at the vortex center . for trajectory @xmath193 , with impact parameter @xmath194 , there is only a small change in the amplitude of @xmath195 . for singly - quantized vortices the phase of the order parameter is the more important factor determining the spectrum of bound states . figure 2 shows the angle - resolved local density of states for the two trajectories shown in fig . 1 . for trajectory ( a ) through the center of the vortex , the spectrum shows a zero - energy bound state separated from the continuum that begins at the bulk gap . the bound state results from constructive interference of particle- and hole - like quasiparticles that undergo andreev reflections from the vortex order parameter . this bound state corresponds to the zero angular momentum bound state found by caroli , de gennes and matricon.@xcite a zero - energy bound state is always present for trajectories in which the order parameter is real ( up to a constant phase factor ) and has different signs when going to @xmath196 along the trajectory.@xcite bound states with non - zero energies are found for trajectories with a nonzero impact parameter measured from the vortex center . these bound states correspond to the spectrum of bound states with non - zero angular momenta obtained by caroli et al.@xcite figure 2b shows the spectrum for a trajectory with an impact parameter of @xmath197 and @xmath198 measured at the point of closest approach to the vortex center . the bound state is shifted down in energy to @xmath199 , and the continuum states are shifted and inhomogeneously broadened by the doppler energy , @xmath200 . the spectrum near the onset at point @xmath201 in fig . 2b has low weight and corresponds to the continuum edge at @xmath202 far from the impact point , while the peak in the spectrum at point @xmath51 corresponds to the maximum doppler shift , @xmath203 at the impact point @xmath204 . note the development of the bcs coherence peak as the density of states is sampled further from the vortex center . 2 = = the density of states of an @xmath189-wave vortices has been investigated by several authors.@xcite our emphasis is on the importance of the andreev bound states for the current distribution in the vortex core . we show in fig . 2c the spectral current density for the trajectory with @xmath205 and @xmath206 . the net current carried by the states at the point @xmath207 on the fermi surface is obtained by weighting this spectrum by the equilibrium distribution and integrating over all energies . thus , for @xmath208 only the negative energy states contribute . it is clear from fig . 2c that the current in the vicinity of the vortex core is carried almost entirely by the bound states with @xmath209 . the continuum states give almost no net contribution to the current in the core . figure 2d shows the spectral current density of the set of bound states with trajectories @xmath210 as a function of the impact parameter @xmath73 for @xmath211 . the peak at @xmath212 corresponds to the trajectory with impact parameter @xmath213 . the bound state energy decreases with increasing distance from the core . for small @xmath73 we obtain , @xmath214 , in reasonable agreement with the analytic estimate in eq . ( [ eig ] ) . as indicated in fig . 2d the contribution of the bound state to the current density decreases as the impact parameter increases . however , even at a relatively large distance , @xmath215 , the bound state still contributes significantly to the circulating current density of the vortex . the evolution of the bound state energy for small impact parameters can be written in terms of the angular momentum of an excitation about the vortex center , @xmath216 ; _ i.e. _ @xmath217 , where @xmath218 . this spectrum was originally obtained by caroli et al.@xcite by solving the bogolyubov equations . in the bogolyubov or gorkov formulation the spectrum is discrete : @xmath219 with @xmath220 and @xmath221 defining the level spacing of the low - lying bound states in the core . the lowest energy bound state in the core has a zero - point energy of @xmath222 which is outside the resolution of the quasiclassical or the andreev theory . the discrete spectrum of the bogolyubov theory corresponds to the continuous andreev spectrum in the limit where the level spacing is small compared to all other relevant energy scales , _ i.e. _ @xmath223 , etc . this is generally an excellent approximation in conventional type ii superconductors . for the high @xmath9 cuprates the discrete level structure is expected to play a more important role , particularly in the transverse response of vortices in the ultra - clean limit , _ i.e. _ @xmath224 , where @xmath225 is the mean scattering time.@xcite = it is interesting to compare the single - quantum vortex with the axially symmetric , @xmath11 vortex , @xmath226 . the double quantum vortex has higher energy than a pair of isolated single - quantum vortices ; however , once created the double - quantum vortex is metastable against dissociation into singly - quantized vortices . the amplitude of the order parameter for the double - quantum vortex decreases as @xmath227 for @xmath228 as shown in fig . = in contrast to the @xmath10 vortex there is no sign change of the order parameter for trajectories passing through the center of the vortex core . this difference has a profound effect on the spectrum of andreev bound states in the core . 3b shows the excitation spectrum of the doubly - quantized vortex at the center of a trajectory passing through the center of the vortex core . a symmetric spectrum of two bound states at @xmath229 are separated from the continuum . figure 3 also shows the current density of the doubly - quantized vortex . the remarkable feature is the _ reversal _ of the current direction in the core , i.e. for @xmath230 ( see fig . 3c and 3d ) . this current anomaly is associated with the appearance of a counter - moving andreev bound state below the fermi level ( @xmath231 ) . the evolution of the spectral current density is shown in fig . the trajectories are parallel to @xmath232 and the spectral current density is shown as a function of the impact parameter . at distances greater than @xmath233 two bound states lie below zero energy , and both states are co - moving with the circulating phase gradient , @xmath105 . as the vortex core is approached the co - moving bound state nearest the fermi level moves to higher energy , and a counter - moving bound state above the fermi level ( not shown ) moves to lower energy . these two states cross the fermi energy ( @xmath231 ) at approximately @xmath234 , leading to a reversal of the integrated current density inside the core . the cumulative current density for each trajectory is shown as the thick solid line in each panel of fig . 4 . = finally , consider the current and excitation spectrum of a @xmath10 vortex in the presence of a uniform supercurrent @xmath235 . in the absence of pinning the vortex will move in the direction ( @xmath236 ) in order to reduce the kinetic energy . thus , in order to investigate the excitation spectrum in the presence of a transport current we must pin the vortex to the lattice . our model for the pinning center is a normal metal inclusion where the pairing interaction ( or the local @xmath9 ) vanishes . 5a shows the order parameter of a pinned vortex for an @xmath189-wave superconductor and a pinning center with a radius of @xmath237 . in the presence of a transport current the amplitude of the order parameter deforms ; it is suppressed on the high current side of the vortex as shown in fig . 5a . for relatively small transport currents , e.g. @xmath238 , the center for the phase winding lies within the normal inclusion . however , as shown in fig . 5b , the vortex current no longer vanishes at the center of the vortex core ; there is substantial current _ through _ the vortex core region , including the normal inclusion . the current density inside the normal inclusion is carried by the andreev bound states and is a consequence of the proximity effect . the bound state spectrum at the center of the vortex is shown for a trajectory parallel to the transport current and passing through the vortex center . the negative energy bound state carries the transport current inside the normal inclusion . 5d shows the spectral current density measured at the center of the normal inclusion for the trajectories with @xmath239 . note that the bound state dominates the current and that this current is opposite to the applied transport current . this result was also obtained in section [ analyt ] , without taking into account the distortion of the vortex core order parameter . this led to a violation of charge conservation in the core . our numerical calculation shows that the main features of the analytic model for the bound state spectrum and the self - consistent determination of the order parameter for the pinned vortex in the presence of a transport current guarantees that charge is conserved . we have discussed the current carried by the excitations of s - wave vortices in clean layered superconductors . the _ spectral current density _ was introduced in order to identify the excitations that determine the transport and circulating currents of a vortex . the bound states of the vortex carry most of the current in the vicinity of the core , including transport currents that flow through the core of a pinned vortex . far from the vortex core currents are carried primarily by the bound - pair continuum that forms the condensate . for currents flowing through a pinned vortex , current conservation is maintained by `` spectral transfer '' of the current carried by the andreev bound states to the continuum states outside the core . a novel example of the evolution of the spectral current density is provided by the double quantum vortex which shows the connection between the spectrum of bound states and the symmetry or topology of the order parameter . at low temperatures ( @xmath240 ) the double quantum vortex exhibits a ` current reversal ' relative to the asymptotic direction of the circulation . the counter - circulating current in the core is due to a counter - moving bound state that appears below the fermi level and dominates the current for distances of order @xmath241 . at high temperatures , @xmath242 , this counter - moving bound state is thermally depopulated with the result that the current reversal in the core disappears in the ginzburg - landau limit . in summary , we find that the andreev bound states dominate the current of vortices on the scale of a few coherence lengths . the nonequilibrium properties of vortices on this scale are expected to be dominated by the spectral evolution and dynamics of these bound states . this work was initiated when the authors were participants at a workshop at the institute for scientific interchange , villa gualino , torino . the research of dw was supported in part by the engineering and physical sciences research council , while that of dr and jas was supported in part by nsf grant dmr 91 - 20000 through the science and technology center for superconductivity , the max planck gesellschaft and the alexander von humboldt stiftung . we also thank dr . m. j. graf for his comments on the manuscript . in this appendix we derive a form for a matrix element used in section [ div ] . the matrix element in question is @xmath243|\zeta \rangle _ { 1,1}$ ] where we write @xmath244 we have @xmath245 = i\left ( \hat\tau_{3 } ( \hat{h}_a-\hat{\delta } ) \delta ( \epsilon -\hat{h}_a)-\delta ( \epsilon -\hat{h}_a)(\hat{h}_a-\hat{\delta } ) \hat\tau_{3}\right)\ ] ] @xmath246-i\{\hat\tau_{3}% \hat{\delta } , \delta ( \epsilon -\hat{h}_a)\}\,.\ ] ] then @xmath247|\zeta \rangle _ { 1,1 } = i \mbox{tr}\left [ \frac 12(1+\hat\tau_{3})\langle \zeta |([\hat\tau_{3}\epsilon , \delta ( \epsilon -\hat{h}_a)]- \{\hat\tau_{3 } \hat{\delta } , \delta ( \epsilon -\hat{h}_a)\})|\zeta \rangle \right]\ ] ] @xmath248 \,.\ ] ] substituting the explicit form for @xmath249 yields the relation @xmath250|\zeta \rangle _ { 1,1}=\delta _ { 0}\frac{f(r)}r\mbox{tr}\left [ ( \zeta \hat\tau_{2}-\eta \hat\tau_{1})\langle \zeta |\delta ( \epsilon -\hat{h}% _ a)|\zeta \rangle \right ] .\ ] ] the parameter @xmath252 appearing in the above equation has the semiclassical interpretation as a c - number impact parameter . in this appendix we present approximations to the above equation for large values of the impact parameter , @xmath252 . for @xmath253 we are justified to replace @xmath254 by its asymptotic value of unity . furthermore , making the somewhat crude approximation @xmath255 yields for the bound state @xmath256 this indicates that at large impact parameters , the bound states of the andreev equation are found very close to the threshold of the continuum . when required , more refined estimates may be obtained by writing @xmath257\ ] ] and then performing a unitary transformation @xmath258 , @xmath259 , with @xmath260 $ ] to remove the phase from the order parameter . the transformed andreev equation is @xmath261 \widetilde{\psi } ( \zeta ) = e\widetilde{\psi } ( \zeta ) \,,\ ] ] this is a one dimensional dirac equation with a weak scalar potential , which has weakly bound states with energies near @xmath262 . a `` non - relativistic '' treatment is appropriate in this case and we approximate the dirac equation by a schrdinger equation . for example , for @xmath263 we write @xmath264 with @xmath265 scalars . straightforward manipulations indicate that @xmath266 approximately obeys the schrdinger equation @xmath267 \psi _ l(\zeta ) = ( e-\delta _ { 0})\psi _ l(\zeta ) .\ ] ] all the machinery of schrdinger theory may be used on this equation to estimate e.g. the bound states . we can put a lower limit on the bound state energy . this may be obtained from the fact that the eigenvalues of the schrdinger operator are @xmath268 , the minimum of the potential . thus , @xmath269 $ ] , i.e. @xmath270 note that @xmath272 is proportional to an @xmath273 integral of @xmath274 \langle \zeta |\delta ( \epsilon-\hat{h}_a)|\zeta \rangle _ { 1,1}$ ] . since @xmath275 and@xmath276 \leq 0 $ ] it immediately follows that @xmath277 for a recent review see n. kopnin , _ theory of flux flow hall effect in clean type ii superconductors _ , to be published in `` quasiclassical methods in the theory of superconductivity and superfluidity '' , eds . d. rainer and j. a. sauls , springer - verlag , 1996 .

we calculate the spectrum of quasiparticle excitations in the core of isolated pancake vortices in clean layered superconductors . we show that both the circular current around the vortex center as well as any transport current through the vortex core is carried by localized states bound to the core by andreev scattering . hence the physical properties of the core are governed in clean high-@xmath0 superconductors ( e.g. the cuprate superconductors ) by the andreev bound states , and not by normal electrons as it is the case for traditional ( dirty ) high-@xmath0 superconductors .

You are an expert at summarizing long articles. Proceed to summarize the following text: in their papers @xcite , @xcite , @xcite , ozsvth and szab constructed a decorated topological quantum field theory ( tqft ) in @xmath0 dimensions , called heegaard floer theory . ( strictly speaking , the axioms of a tqft need to be altered slightly . ) in its simplest version ( called hat ) , to a closed , connected , oriented three - manifold @xmath1 and a @xmath2 structure @xmath3 on @xmath1 one associates a vector space @xmath4 over the field @xmath5 also , to a connected , oriented four - dimensional cobordism from @xmath6 to @xmath7 decorated with a @xmath2 structure @xmath8 , one associates a map @xmath9 the maps @xmath10 can be used to detect exotic smooth structures on @xmath11-manifolds with boundary . for example , this can be seen by considering the nucleus @xmath12 of the elliptic surface @xmath13 i.e. a regular neighborhood of a cusp fiber and a section , cf . @xcite . let @xmath14 be the result of a log transform with multiplicity @xmath15 ( @xmath16 , odd ) on a regular fiber @xmath17 , cf . * section 3.3 ) . then @xmath18 and @xmath19 are homeomorphic 4-manifolds ( with @xmath20 ) , having as boundary the brieskorn sphere @xmath21 however , they are not diffeomorphic : this can be shown using the donaldson or seiberg - witten invariants ( see @xcite , @xcite , @xcite ) , but also by comparing the hat heegaard floer invariants @xmath22 and @xmath23 , where @xmath24 and @xmath25 are the cobordisms from @xmath26 to @xmath27 obtained by deleting a @xmath11-ball from @xmath18 and @xmath19 , respectively . indeed , the arguments of fintushel - stern @xcite and szab - stipsicz @xcite can be easily adapted to show that @xmath24 and @xmath25 have different hat heegaard floer invariants ; one needs to use the computation of @xmath28 due to ozsvth - szab @xcite , and the rational blow - down formula of roberts @xcite . ( it is worth noting that the maps @xmath29 give no nontrivial information for closed 4-manifolds , cf . @xcite ; exotic structures on those can be detected with the mixed heegaard floer invariants of @xcite . ) the original definitions of the vector spaces @xmath30 and the maps @xmath29 involved counting pseudoholomorphic disks and triangles in symmetric products of riemann surfaces ; the riemann surfaces are related to the three - manifolds and cobordisms involved via heegaard diagrams . in @xcite , sarkar and the third author showed that every three - manifold admits a heegaard diagram that is nice in the following sense : the curves split the diagram into elementary domains , all but one of which are bigons or rectangles . using such a diagram , holomorphic disks in the symmetric product can be counted combinatorially , and the result is a combinatorial description of @xmath31 for any @xmath32 as well as of the hat version of heegaard floer homology of null homologous knots and links in any three - manifold @xmath1 . a similar result was obtained in @xcite for all versions of the heegaard floer homology of knots and links in the three - sphere . the goal of this paper is to give a combinatorial procedure for calculating the ranks of the maps @xmath33 when @xmath24 is a cobordism between @xmath6 and @xmath7 with the property that the induced maps @xmath34 and @xmath35 are surjective . note that this case includes all cobordisms for which @xmath36 is torsion , as well as all those consisting of only 2-handle additions . roughly , the computation of the ranks of @xmath33 goes as follows . the cobordism @xmath24 is decomposed into a sequence of one - handle additions , two - handle additions , and three - handle additions . using the homological hypotheses on the cobordism and the @xmath37-action on the heegaard floer groups we reduce the problem to the case of a cobordism map corresponding to two - handle additions only . then , given a cobordism made of two - handles , we show that it can be represented by a multi - pointed triple heegaard diagram of a special form , in which all elementary domains that do not contain basepoints are bigons , triangles , or rectangles . in such diagrams all holomorphic triangles of maslov index zero can be counted algorithmically , thus giving a combinatorial description of the map on @xmath38 we remark that in order to turn @xmath30 into a fully combinatorial tqft ( at least for cobordisms satisfying our hypothesis ) , one ingredient is missing : naturality . given two different nice diagrams for a three - manifold , the results of @xcite show that the resulting groups @xmath30 are isomorphic . however , there is not yet a combinatorial description of this isomorphism . thus , while the results of this paper give an algorithmic procedure for computing the rank of a map @xmath39 the map itself is determined combinatorially only up to automorphisms of the image and the target . in fact , if one were to establish naturality , then one could automatically remove the assumption on the maps on @xmath40 , and compute @xmath33 for any @xmath24 , simply by composing the maps induced by the two - handle additions ( computed in this paper ) with the ones induced by the one- and three - handle additions , which are combinatorial by definition , cf . @xcite . the paper is organized as follows . in section [ sec : triangles ] , we define a multi - pointed triple heegaard diagram to be nice if all non - punctured elementary domains are bigons , triangles , or rectangles , and show that in a nice diagram holomorphic triangles can be counted combinatorially . below ) in @xcite , using slightly different methods . ] we then turn to the description of the map induced by two - handle additions . for the sake of clarity , in section [ sec : two ] we explain in detail the case of adding a single two - handle : we show that its addition can be represented by a nice triple heegaard diagram with a single basepoint and , therefore , the induced map on @xmath30 admits a combinatorial description . we then explain how to modify the arguments to work in the case of several two - handle additions . this modification uses triple heegaard diagrams with several basepoints . in section [ sec : last ] , we discuss the additions of one- and three - handles , and put the various steps together . finally , in section [ sec : example ] we present the example of @xmath41 surgery on the trefoil . throughout the paper all homology groups are taken with coefficients in @xmath42 unless otherwise noted . we would like to thank peter ozsvth and zoltn szab for helpful conversations and encouragement . in particular , several key ideas in the proof were suggested to us by peter ozsvth . this work was done while the third author was an exchange graduate student at columbia university . he is grateful to the columbia math department for its hospitality . he would also like to thank his advisors , robion kirby and peter ozsvth , for their continuous guidance and support . finally , we would like to thank the referees for many helpful comments , and particularly for finding a critical error in section [ sec : last ] of a previous version of this paper . the goal of this section is to show that under an appropriate condition ( `` niceness '' ) on triple heegaard diagrams , the counts of holomorphic triangles in the symmetric product are combinatorial . we start by reviewing some facts from heegaard floer theory . a triple heegaard diagram @xmath43 consists of a surface @xmath44 of genus @xmath45 together with three @xmath46-tuples of pairwise disjoint embedded curves @xmath47 @xmath48 in @xmath44 such that the span of each @xmath46-tuple of curves in @xmath49 is @xmath45-dimensional . if we forget one set of curves ( for example @xmath50 ) , the result is an ( ordinary ) heegaard diagram @xmath51 by the condition on the spans , @xmath52 , @xmath53 and @xmath54 each has @xmath55 connected components . by a multi - pointed triple heegaard diagram @xmath56 , then , we mean a triple heegaard diagram @xmath57 as above together with a set @xmath58 of @xmath55 points in @xmath44 so that exactly one @xmath59 lies in each connected component of @xmath52 , @xmath53 and @xmath54 . to a heegaard diagram @xmath60 one can associate a three - manifold @xmath61 . to a triple heegaard diagram @xmath62 , in addition to the three - manifolds @xmath61 , @xmath63 and @xmath64 , one can associate a four - manifold @xmath65 such that @xmath66 ; see @xcite . associated to a three - manifold @xmath1 is the heegaard floer homology group @xmath31 . this was defined using a heegaard diagram with a single basepoint in @xcite . in @xcite , ozsvth and szab associated to the data @xmath67 , called a multi - pointed heegaard diagram , a floer homology group @xmath68 by counting holomorphic disks in @xmath69 with boundary on the tori @xmath70 and @xmath71 it is not hard to show that @xmath72 ( here , @xmath73 is the @xmath74-torus , and @xmath75 means ordinary ( singular ) homology . ) the decomposition is not canonical : it depends on a choice of paths in @xmath44 connecting @xmath59 to @xmath76 for @xmath77 . the heegaard floer homology groups decompose as a direct sum over @xmath2-structures on @xmath61 , @xmath78 more generally , there is a decomposition @xmath79 associated to the triple heegaard diagram @xmath80 together with a @xmath2-structure @xmath8 on @xmath65 , is a map @xmath81 the definition involves counting holomorphic triangles in @xmath82 with boundary on @xmath83 , @xmath84 and @xmath85 , cf . @xcite and @xcite . two triple heegaard diagrams are called _ strongly equivalent _ if they differ by a sequence of isotopies and handleslides . it follows from ( * ? ? ? * proposition 8.14 ) , the associativity theorem ( * ? ? ? * theorem 8.16 ) , and the definition of the handleslide isomorphisms that strongly equivalent triple heegaard diagrams induce the same map on homology . call a @xmath86-pointed triple heegaard diagram _ split _ if it is obtained from a singly - pointed heegaard triple diagram @xmath87 by attaching ( by connect sum ) @xmath74 spheres with one basepoint and three isotopic curves ( one alpha , one beta and one gamma ) each , to the component of @xmath88 containing @xmath89 . we call @xmath87 the _ reduction _ of the split diagram . the following lemma is a variant of ( * ? ? ? * proposition 3.3 ) . [ lemma : splitdiagrams ] every triple heegaard diagram @xmath80 is strongly equivalent to a split one . reorder the alpha circles so that @xmath90 are linearly independent . let @xmath91 be the connected component of @xmath52 containing @xmath92 . since @xmath90 are linearly independent , one of the curves @xmath93 , @xmath94 , must appear in the boundary of @xmath91 with multiplicity exactly @xmath95 . by handlesliding this curve over the other boundary components of @xmath91 , we can arrange that the resulting @xmath93 bounds a disk containing only @xmath92 . repeat this process with the other @xmath59 , @xmath96 , being sure to use a different alpha curve in the role of @xmath93 at each step . we reorder the curves @xmath97 so that @xmath93 encircles @xmath98 . now repeat the entire process for the beta and gamma curves . finally , choose a path @xmath99 in @xmath44 from @xmath59 to @xmath76 for each @xmath100 . move the configuration @xmath101 along the path @xmath99 by handlesliding ( around @xmath93 , @xmath102 or @xmath103 ) the other alpha , beta and gamma curves that are encountered along the path @xmath99 . the result is a split triple heegaard diagram . note that its reduction is obtained from the original diagram @xmath80 by simply forgetting some of the curves and basepoints . the maps from are compatible with the isomorphism , in the following sense . given a triple heegaard diagram @xmath80 , let @xmath87 be the reduction of a split diagram strongly equivalent to @xmath80 . then the following diagram commutes : @xmath104^ { } \ar[d]^{\cong } & \hf\left(y_{\alpha,\gamma } , { \mathfrak{t}}|_{y_{\alpha,\gamma}}\right)\otimes h_*\left(t^k\right)\ar[d]^{\cong}\\ \hf(\sigma,\alphas,\betas,{\mathbf{z}},{\mathfrak{t}}|_{y_{\alpha,\beta } } ) \otimes\hf(\sigma,\betas,\gammas,{\mathbf{z } } , { \mathfrak{t}}|_{y_{\beta,\gamma } } ) \ar[r]^(.62){{\hat{f}}_{\sigma,\alphas,\betas,\gammas,{\mathbf{z}},{\mathfrak{t } } } } & \hf(\sigma,\alphas,\gammas,{\mathbf{z}},{\mathfrak{t}}|_{y_{\alpha,\gamma } } ) . } \ ] ] here , the map in the first row is @xmath105 on the @xmath30-factors and the usual intersection product @xmath106 on the @xmath75-factors . the vertical isomorphisms are induced by the strong equivalence . the proof that the diagram commutes follows from the same ideas as in @xcite : each of the @xmath74 spherical pieces in the split diagram contributes a @xmath107 to the @xmath108 factors above ; moreover , a local computation shows that the triangles induce intersection product maps @xmath109 which tensored together give the intersection product on @xmath110 it is tempting to assert that the map @xmath111 induced by a singly - pointed triple heegaard diagram depends only on @xmath65 and @xmath8 . however , this seems not to be known . fix a triple heegaard diagram @xmath112 as above . the complement of the @xmath113 curves in @xmath44 has several connected components , which we denote by @xmath114 and call _ elementary domains_. the _ euler measure _ of an elementary domain @xmath115 is @xmath116 a _ domain _ in @xmath44 is a two - chain @xmath117 with @xmath118 its euler measure is simply @xmath119 as mentioned above , the maps @xmath120 induced by the triple heegaard diagram @xmath121 are defined by counting holomorphic triangles in @xmath122 with respect to a suitable almost complex structure . according to the cylindrical formulation from @xcite , this is equivalent to counting certain holomorphic embeddings @xmath123 where @xmath124 is a riemann surface ( henceforth called the source ) with some marked points on the boundary ( which we call corners ) , and @xmath125 is a fixed disk with three marked points on the boundary . the maps @xmath126 are required to satisfy certain boundary conditions , and to be generically @xmath46-to-@xmath95 when post - composed with the projection @xmath127 more generally , we will consider such holomorphic maps @xmath128 which are generically @xmath129-to-@xmath95 when post - composed with @xmath130 these correspond to holomorphic triangles in @xmath131 where @xmath129 can be any positive integer . we will be interested in the discussion of the index from @xcite . although this discussion was carried - out in the case @xmath132 , @xmath133 , it applies equally well in the case of arbitrary @xmath74 and @xmath129 with only notational changes . in the cylindrical formulation , one works with an almost complex structure on @xmath134 so that the projection @xmath135 is holomorphic , and the fibers of @xmath136 are holomorphic . it follows that for @xmath137 holomorphic , @xmath138 is a holomorphic branched cover . the map @xmath139 need not be holomorphic , but since the fibers are holomorphic , @xmath139 is a branched map . fix a model for @xmath125 in which the three marked points are @xmath140 corners , and a conformal structure on @xmath44 with respect to which the intersections between alpha , beta and gamma curves are all right angles . since @xmath126 is holomorphic , the conformal structure on @xmath124 is induced via @xmath138 from the conformal structure on @xmath125 . it makes sense , therefore , to talk about branch points of @xmath139 on the boundary and at the corners , as well as in the interior . generically , while there may be branch points of @xmath139 on the boundary of @xmath124 , there will not be branch points at the corners . suppose @xmath128 is as above . denote by @xmath141 the projection to @xmath142 there is an associated domain @xmath143 in @xmath144 where the coefficient of @xmath145 in @xmath143 is the local multiplicity of @xmath146 at any point in @xmath147 by @xcite , the _ index _ of the linearized @xmath148 operator at the holomorphic map @xmath126 is given by @xmath149 for simplicity , we call this the index of @xmath150 note that , by the riemann - hurwitz formula : @xmath151 where @xmath152 is the ramification index ( number of branch points counted with multiplicity ) of @xmath153 ( here , branch points along the boundary count as half an interior branch point . ) from and we get an alternate formula for the index : @xmath154 note that it is not obvious how to compute @xmath155 from @xmath143 . a combinatorial formula for the index , purely in terms of @xmath143 , was found by sarkar in @xcite . however , we will not use it here . fix a multi - pointed triple heegaard diagram @xmath156 . recall that a domain is a linear combination of connected components of @xmath157 . the _ support _ of a domain is the union of those components with nonzero coefficients . if the support of a domain @xmath158 contains at least one @xmath59 then @xmath158 is called _ punctured _ ; otherwise it is called _ unpunctured_. [ def : nice ] an elementary domain is called * good * if it is a bigon , a triangle , or a rectangle , and * bad * otherwise . the multi - pointed triple heegaard diagram @xmath56 is called * nice * if every unpunctured elementary domain is good . this is parallel to the definition of nice heegaard diagrams ( with just two sets of curves ) from @xcite . a multi - pointed heegaard diagram @xmath159 is called _ nice _ if , among the connected components of @xmath160 all unpunctured ones are either bigons or squares . note that a bigon , a triangle , and a rectangle have euler measure @xmath161 and @xmath162 respectively . since @xmath163 is additive , every unpunctured positive domain ( not necessarily elementary ) in a nice diagram must have nonnegative euler measure . a quick consequence of this is the following : [ lemma : forget ] if @xmath80 is a nice triple heegaard diagram , then if we forget one set of curves ( for example , @xmath50 ) , the resulting heegaard diagram @xmath159 is also nice . in order to define the triangle maps it is necessary to assume the triple heegaard diagram is weakly admissible in the sense of ( * ? ? ? * definition 8.8 ) . in fact , nice diagrams are automatically weakly admissible , cf . corollary [ lemma : admissible ] below . our goal is to give a combinatorial description of the holomorphic triangle counts for nice triple diagrams . [ easternorthodox ] let @xmath164 be a nice multi - pointed triple heegaard diagram . fix a generic almost complex structure @xmath165 on @xmath134 as in ( * ? ? ? * section 10.2 ) . let @xmath166 be a @xmath165-holomorphic map of the kind occurring in the definition of @xmath167 . in particular , assume @xmath126 is an embedding , of index zero , and such that the image of @xmath146 is an unpunctured domain . then @xmath124 is a disjoint union of @xmath129 triangles , and the restriction of @xmath146 to each component of @xmath124 is an embedding . since the image of @xmath146 is unpunctured and positive , we have @xmath168 by , we get @xmath169 this means that at least one component of @xmath124 is topologically a disk . let @xmath170 be such a component . it is a polygon with @xmath171 vertices . we will show that @xmath172 and that @xmath173 is an embedding . let us first show that @xmath170 is a triangle . the index of the @xmath174 operator at a disconnected curve is the sum of the indices of its restrictions to each connected component . therefore , in order for an index zero holomorphic curve to exist generically , the indices at every connected component , and in particular at @xmath175 must be zero . applying to @xmath176 we get @xmath177 if @xmath178 then by we have @xmath179 hence , the map @xmath146 has no interior branch points . if @xmath180 then @xmath170 is mapped locally diffeomorphically by @xmath139 to @xmath44 . the image must have negative euler measure , which is a contradiction . so , suppose @xmath181 . the preimages of the alpha , beta , and gamma curves cut @xmath170 into several connected components . without loss of generality , assume that the boundary branch point is mapped to an alpha circle . then , along the corresponding edge of @xmath170 there is a valence three vertex @xmath182 , as shown in figure [ figure : hexagon ] . let @xmath183 denote the edge in the interior of @xmath170 meeting @xmath182 . since there is only one boundary branch point , the other intersection point of the edge @xmath183 with @xmath184 is along the preimage of a beta or gamma circle . it follows that one of the connected components @xmath185 of @xmath186 is a hexagon or heptagon . smoothing the vertex @xmath182 of @xmath185 we obtain a pentagon or hexagon which is mapped locally diffeomorphically by @xmath139 to @xmath44 . the image , then , has negative euler measure , again a contradiction . therefore , @xmath172 so @xmath170 is a triangle . furthermore , by , @xmath187 which means that there are no ( interior or boundary ) branch points at all thus , just as in the hypothetical hexagon case above , the preimages of the alpha , beta , and gamma curves must cut @xmath170 into 2-gons , 3-gons , and 4-gons , all of which have nonnegative euler measure . since the euler measure of @xmath170 is @xmath188 there can be no bigons ; in fact , @xmath170 must be cut into several rectangles and exactly one triangle . it is easy to see that the only possible tiling of @xmath170 of this type is as in figure [ fig : tiling ] , with several parallel preimages of segments on the alpha curves , several parallel beta segments , and several parallel gamma segments . we call the type of a segment ( @xmath189 , @xmath190 or @xmath191 ) its color . the tiling consists of one triangle and six different types of rectangles , according to the coloring of their edges in clockwise order ( namely , @xmath192 , @xmath193 , @xmath194 , @xmath195 , @xmath196 , and @xmath197 ) . we claim that the images of the interiors of each of these rectangles by @xmath146 are disjoint . because of the coloring scheme , only rectangles of the same type can have the same image . suppose that two different @xmath192 rectangles from @xmath170 have the same image in @xmath142 ( the cases @xmath193 , @xmath194 are exactly analogous . ) let @xmath198 and @xmath199 be the two rectangles ; suppose that @xmath198 is closer to the central triangle than @xmath200 and @xmath199 is closer to the @xmath189 boundary of @xmath201 because of the way the rectangles are colored , the upper edge of @xmath198 must have the same image as the upper edge of @xmath202 hence the @xmath192 rectangle right above @xmath198 has the same image as the one right above @xmath202 iterating this argument , at some point we get that the central triangle has the same image as some @xmath192 rectangle , which is impossible . now suppose that two different @xmath195 rectangles , @xmath198 and @xmath200 have the same image . ( the cases @xmath196 and @xmath197 are exactly analogous . ) there are two cases , according to whether the upper edge of @xmath198 has the same image as the upper edge of @xmath200 or as the lower edge of @xmath202 suppose first that the upper edge of @xmath198 has the same image as the upper edge of @xmath199 . by the _ @xmath190-height _ of @xmath203 we mean the minimal number of beta arcs that an arc in @xmath204 starting in @xmath203 , going right , and ending at a gamma arc must cross . ( the diagram is positioned in the plane as in figure [ fig : tiling ] . ) since @xmath139 is a local homeomorphism , and @xmath198 and @xmath199 have the same image , it is clear that the @xmath190-height of @xmath198 and the @xmath190-height of @xmath199 are equal . by the _ @xmath189-height _ of @xmath203 we mean the minimal number of alpha arcs that an arc in @xmath205 starting in @xmath206 going up , and ending at a gamma arc must cross . again , it is clear that the @xmath189-heights of @xmath198 and @xmath199 must be equal . but this implies that @xmath198 and @xmath199 are equal . now , suppose that the upper edge of @xmath198 has the same image as the lower edge of @xmath199 . there is a unique rectangle @xmath91 in @xmath124 with boundary contained in @xmath207 , containing @xmath198 and @xmath199 , and with one corner the same as a corner of @xmath198 and the opposite corner the same as a corner of @xmath199 . it is easy to see that @xmath139 maps antipodal points on the boundary of @xmath91 to the same point in @xmath44 . it follows that @xmath208 is a two - fold covering map . but then @xmath139 must have a branch point somewhere inside @xmath91 a contradiction . see figure [ fig : grid ] . finally , suppose some arc @xmath209 on @xmath210 has the same image as some other arc @xmath211 in @xmath124 . if @xmath211 is in the interior of @xmath124 then any rectangle ( or triangle ) adjacent to @xmath209 has the same image as some rectangle ( or triangle ) adjacent to @xmath211 . we have already ruled this out . if @xmath211 is on @xmath210 , then either any rectangle ( or triangle ) adjacent to @xmath209 has the same image as some rectangle ( or triangle ) adjacent to @xmath211 or there is a branch point somewhere on @xmath210 . we have already ruled out both of these cases . we have thus established that @xmath170 is an embedded triangle . by forgetting @xmath170 , we obtain a holomorphic map to @xmath212 still of index zero , but such that its post - composition with @xmath135 is generically @xmath213-to-@xmath95 rather than @xmath129-to-@xmath214 the result then follows by induction on @xmath215 observe that , in proposition [ easternorthodox ] above , even though each of the @xmath129 triangles is embedded , some of their domains may overlap . it turns out that they may do so only in a specific way , however : [ allow ] suppose @xmath209 is an index zero homology class represented by a union of embedded holomorphic triangles , in a nice triple diagram . suppose the union of triangles corresponds to an embedded holomorphic curve in @xmath216 . then any two triangles in @xmath209 are either disjoint in @xmath44 or overlap in @xmath44 `` head to tail '' as shown in figure [ fig : overlap ] . let @xmath217 and @xmath218 be two of the triangles in the domain @xmath209 . for a generic representative of @xmath209 to exist , the pair must also have index zero , and be embedded in @xmath134 . we already know that @xmath217 and @xmath218 are tiled as in figure [ fig : tiling ] . this strongly restricts how @xmath217 and @xmath218 can overlap . one way for @xmath217 and @xmath218 to overlap is for @xmath217 to be entirely contained inside @xmath218 . in this case , it is not hard to see that the two holomorphic triangles in @xmath134 intersect in one interior point . indeed , the intersection number of two holomorphic curves in a @xmath11-manifold is invariant in families . if we deform the heegaard diagram so that the boundary of @xmath217 in @xmath44 is a single point ( i.e. , the alpha , beta and gamma circles involved in @xmath219 intersect in an asterisk , with vertex the `` triangle '' @xmath217 ) then obviously @xmath220 is a single point . it follows that the same is true for the original triangles @xmath217 and @xmath218 . another way that @xmath217 and @xmath218 might overlap is `` head to head '' as shown on the left side of figure [ fig : headhead ] . it is then possible to decompose @xmath221 into a pair of rectangles @xmath91 and @xmath222 , and two new embedded triangles @xmath217 and @xmath218 , as shown in figure [ fig : headhead ] . an immersed rectangle in @xmath44 has index at least @xmath95 , since it admits a generic holomorphic representative . so , each of @xmath223 and @xmath224 has index at least @xmath95 . similarly , the pair of triangles @xmath225 has index at least @xmath226 . so , by additivity of the index , the whole domain has index at least @xmath227 a contradiction . using these two observations , and the rulings of @xmath217 and @xmath218 , it is then elementary to check that the only possible overlap in index zero is `` head to tail '' as in figure [ fig : overlap ] . it follows that , for a nice heegaard diagram , we can combinatorially describe the generic holomorphic curves of index @xmath226 . if @xmath158 is the domain of a generic holomorphic curve of index @xmath226 then @xmath228 has @xmath129 components , each of which bounds an embedded triangle in @xmath44 . each pair of triangles must either be disjoint or overlap as shown in figure [ fig : overlap ] . any such @xmath158 clearly has a unique holomorphic representative with respect to a split complex structure @xmath229 on @xmath134 . further , it is well known that these holomorphic curves are transversally cut out , and so persist if one takes a small perturbation of @xmath229 . in summary , to count index zero holomorphic curves in @xmath134 with respect to a generic perturbation of the split complex structure , it suffices to count domains @xmath158 which are sums of @xmath129 embedded triangles in @xmath44 , overlapping as allowed in the statement of lemma [ allow ] . in ( * ? ? ? * definition 4.2 ) , ozsvth and szab associate to a four - dimensional cobordism @xmath24 consisting of two - handle additions certain kinds of triple heegaard diagrams . the cobordism @xmath24 from @xmath6 to @xmath7 corresponds to surgery on some framed link @xmath230 denote by @xmath231 the number of components of @xmath232 . fix a basepoint in @xmath6 . let @xmath233 be the union of @xmath232 with a path from each component to the basepoint . the boundary of a regular neighborhood of @xmath233 is a genus @xmath231 surface , which has a subset identified with @xmath231 punctured tori @xmath234 , one for each link component . a singly - pointed triple heegaard diagram @xmath235 is called _ subordinate to @xmath233 _ if * @xmath236 describes the complement of @xmath233 , * @xmath237 are small isotopic translates of @xmath238 , * after surgering out the @xmath239 , the induced curves @xmath102 and @xmath103 ( for @xmath240 ) lie in the punctured torus @xmath234 . * for @xmath240 , the curves @xmath102 represent meridians for the link components , disjoint from all @xmath241 for @xmath242 , and meeting @xmath103 in a single transverse intersection point . * for @xmath240 , the homology classes of the @xmath103 correspond to the framings of the link components . a related construction is as follows . given @xmath6 and @xmath232 as above , choose a multi - pointed heegaard diagram @xmath243 for @xmath244 as in @xcite , of some genus @xmath45 ; here , @xmath245 and @xmath246 . precisely , the sets @xmath247 and @xmath248 are collections of distinct points on @xmath44 disjoint from the alpha and the beta curves , with the following two properties : first , each connected component of @xmath249 and @xmath250 contains a single @xmath59 and a corresponding @xmath251 . second , if @xmath252 is an @xmath231-tuple of embedded arcs in @xmath250 connecting @xmath59 to @xmath251 ( @xmath253 ) , and @xmath254 is an @xmath231-tuple of embedded arcs in @xmath249 connecting @xmath59 to @xmath251 ( @xmath253 ) then the link @xmath232 is the union of small push offs of @xmath252 and @xmath254 into the two handlebodies ( induced by the beta and alpha curves , respectively ) . next , we attach handles @xmath255 to @xmath256 connecting @xmath59 to @xmath251 ( for @xmath253 ) , and obtain a new surface @xmath44 . we choose a new @xmath102 ( @xmath257 ) to be the belt circle of the handle @xmath258 , and a new @xmath93 to be the union of the core of @xmath258 with @xmath259 , so @xmath93 intersects @xmath102 in one point . choose @xmath103 ( @xmath260 ) to be a small isotopic translate of @xmath102 , intersecting @xmath102 in two points . let @xmath261 ( @xmath257 ) be the union of @xmath262 with a core of the handle @xmath258 . obtain @xmath103 ( @xmath257 ) by applying dehn twists to @xmath261 around @xmath102 ; the framing of the link is determined by the number of dehn twists . in this fashion we obtain a multi - pointed triple heegaard diagram @xmath263 with @xmath264 curves of each kind . note that @xmath67 is an @xmath231-pointed heegaard diagram for @xmath6 , @xmath265 is a @xmath231-pointed heegaard diagram for @xmath7 , and @xmath266 is a @xmath231-pointed heegaard diagram for @xmath267 . the new circles @xmath93 , @xmath257 , are part of a maximal homologically linearly independent subset of @xmath268 , and similarly for @xmath102 and @xmath103 ( @xmath257 ) . consequently , by the proof of lemma [ lemma : splitdiagrams ] , there is a split diagram strongly equivalent to @xmath269 whose reduction @xmath87 is obtained from @xmath270 by forgetting @xmath271 of the @xmath93 ( respectively @xmath102 , @xmath103 ) , @xmath272 , as well as @xmath273 . it is then not hard to see that @xmath87 is a triple heegaard diagram subordinate to a bouquet for @xmath232 . in this section we will show that for any two - handle addition , one can construct a _ triple heegaard diagram strongly equivalent to a diagram @xmath274 as above . this involves finessing the diagram for the link @xmath232 inside @xmath1 and then , after adding the handles and the new curves , modifying the diagram in several steps to make it nice . for the most part we will focus on the case when we add a single two - handle . in the last subsection we will explain how the arguments generalize to several two - handles . to keep language concise , in this section we will refer to elementary domains as _ let @xmath159 be a multi - pointed heegaard diagram . recall that the diagram is called nice if all unpunctured regions are either bigons or squares . [ lemma : bad_region_adjacent ] on a nice heegaard diagram @xmath159 , for any alpha circle @xmath93 with an arbitrary orientation , there exists a punctured region @xmath158 which contains an edge @xmath163 belonging to @xmath93 , and such that @xmath158 is on the left of @xmath93 . the same conclusion holds for each beta circle . suppose a half - neighborhood on the left of the alpha circle @xmath93 is disjoint from all the punctured regions . then immediately to the left of @xmath93 we only have good regions . there are two possibilities as indicated in figure [ fig : bad_region_adjacent ] . if there is a bigon region on the left of @xmath275 then the other edge is some beta edge @xmath276 . the region on the other side of @xmath276 must be a bigon region or a square since otherwise we would have a punctured region on the left of @xmath93 . if we reach a square , we continue to consider the next region . eventually we will reach a bigon region since the number of regions are finite and we will not reach the same region twice . all regions involved form a disk bounded by @xmath93 ( as in figure [ fig : bad_region_adjacent ] ( a ) ) . in particular , this means @xmath93 is null homologous . this contradicts the fact that the @xmath277 alpha circles represent linearly independent classes in @xmath278 in the second case , there are no bigon regions . then on the left of @xmath93 , we see a chain of squares , as in figure [ fig : bad_region_adjacent ] ( b ) . the opposite edges on these squares give another alpha circle , say @xmath279 . then @xmath93 and @xmath279 are homologous to each other in @xmath278 this contradicts the same fact as in the previous case . recall that in order to define the triangle maps it is necessary for the triple heegaard diagram to be weakly admissible in the sense of ( * ? ? ? * definition 8.8 ) . [ lemma : admissible ] if @xmath80 is a nice multi - pointed triple heegaard diagram then @xmath80 is weakly admissible . by definition , the diagram is weakly admissible if there are no nontrivial domains @xmath158 supported in @xmath280 with nonnegative multiplicity in all regions , and whose boundary is a linear combination of alpha , beta , and gamma curves . suppose such a domain @xmath158 exists , and consider a curve appearing with a nonzero multiplicity in @xmath281 without loss of generality , we can assume this is an alpha curve , and all regions immediately to its left have positive multiplicity in @xmath282 by lemma [ lemma : forget ] , the diagram @xmath67 is nice . lemma [ lemma : bad_region_adjacent ] now gives a contradiction . let @xmath283 be a three - manifold together with a knot @xmath284 . we choose a singly pointed heegaard diagram @xmath285 for @xmath1 together with an additional basepoint @xmath286 such that the two basepoints determine the knot as in @xcite . after applying the algorithm from @xcite to the heegaard diagram , we can assume that the heegaard diagram is nice , with @xmath287 the ( usually bad ) region containing the basepoint @xmath288 furthermore , the algorithm in @xcite also ensures that @xmath287 is a polygon . we denote by @xmath289 the region containing @xmath290 note that either @xmath291 or @xmath289 is good . throughout this section , we will suppose that @xmath289 and @xmath287 are two different regions , and that @xmath289 is a rectangle . the case when @xmath291 corresponds to surgery on the unknot , which is already well understood . the case when @xmath289 is a bigon can be avoided by modifying the original diagram by a finger move . ( alternately , this case can be treated similarly to the case that @xmath289 is a rectangle . ) let @xmath24 be the four manifold with boundary obtained from @xmath292 $ ] by adding a two handle along @xmath293 in @xmath294 , with some framing . @xmath24 gives a cobordism between @xmath1 and @xmath295 , where @xmath295 is obtained from @xmath1 by doing the corresponding surgery along @xmath293 . now we are ready to describe our algorithm to get a nice triple heegaard diagram for the cobordism @xmath296 let @xmath252 be an embedded arc in @xmath44 connecting @xmath297 and @xmath298 in the complement of beta curves , and @xmath254 be an embedded arc connecting @xmath297 and @xmath298 in the complement of alpha curves . the union of @xmath252 and @xmath254 is a projection of the knot @xmath284 to the surface @xmath144 where @xmath44 is viewed as a heegaard surface in @xmath299 for convenience , we will always assume that @xmath252 and @xmath254 do not pass through any bigon regions , and never leave a rectangle by the same edge through which they entered ; this can easily be achieved . in this step , we modify the doubly pointed heegaard diagram @xmath300 to make @xmath301 embedded in @xmath144 while preserving the niceness of the heegaard diagram . typically , @xmath252 and @xmath254 have many intersections . we modify the diagram inductively by stabilization at the first intersection @xmath302 on @xmath254 ( going from @xmath297 to @xmath298 ) to remove that intersection , while making sure that the new diagram is still nice . a neighborhood of @xmath252 and the part on @xmath254 from @xmath297 to @xmath15 are shown in figure [ fig : knot_embedded_before ] . in the same picture , if we continue the chain of rectangles containing @xmath252 , we will end up with a region @xmath303 which is either a bigon or the punctured region @xmath287 . to get rid of the intersection point @xmath15 , we stabilize the diagram as in figure [ fig : knot_embedded_after ] . more precisely , we do a stabilization followed some handleslides of the beta curves and an isotopy of the new beta curve . after these moves , the number of intersection points decreases by one and the diagram is still nice . if we iterate this process , in the end we get a nice heegaard diagram in which @xmath252 and @xmath254 only intersect at their endpoints . furthermore , the bad region @xmath287 is still a polygon . our goal in steps 2 and 3 is to describe a particular triple heegaard diagram for the cobordism @xmath24 . starting with the alpha and the beta curves we already have , for each beta curve @xmath102 we will add a gamma curve @xmath103 ( called its twin ) which is isotopic to @xmath102 and intersects it in exactly two points . ( after this , we will add some more curves in the next step . ) for any beta curve @xmath102 , by lemma [ lemma : bad_region_adjacent ] we can choose a region @xmath145 so that @xmath145 is adjacent to the punctured region @xmath287 with the common edge on @xmath102 . if @xmath304 , then we add @xmath103 close and parallel to @xmath102 as in figure [ fig : adding_gamma_curve_simple ] ( a ) , and make a finger move as in figure [ fig : adding_gamma_curve_simple ] ( b ) . here and after , without further specification , we make the convention that the thick arcs are alpha arcs , the thin ones are beta arcs , and the interrupted ones are gamma arcs . suppose now that @xmath145 is different from @xmath287 . then @xmath145 has to be a good region . if @xmath145 is not a bigon , since the complement of the beta curves in @xmath44 is connected , we can connect @xmath145 with @xmath287 without intersecting beta curves , via an arc traversing a chain of rectangles , as indicated in the figure [ fig : adding_gamma_curve_before ] . then we do a finger move of the curve @xmath102 as indicated in figure [ fig : adding_gamma_curve_after ] . note that the knot remains embedded in @xmath44 . now we have a bigon region . we then add the gamma curve @xmath103 as shown in figure [ fig : adding_gamma_curve_after ] . note that for each pair @xmath102 and @xmath103 , we either have one sub - diagram of the form in figure [ fig : adding_gamma_curve_simple ] ( b ) , or one sub - diagram of the form in figure [ fig : beta_gamma_bigon ] . observe also that during this process , no bad region other than @xmath287 is created . after step 2 , the knot is still embedded in the heegaard diagram . in other words , we can use arcs to connect @xmath297 to @xmath298 by paths in the complement of alpha curves , and in the complement of beta curves so that the two arcs do not intersect except for the end points @xmath298 and @xmath297 , and do not pass through any bigons . we will see two chains of squares , as indicated in figure [ fig : stabilization ] . we do a stabilization of the heegaard diagram by adding a handle with one foot in each of @xmath287 and @xmath289 . we add the additional beta circle @xmath305 to be the meridian of the handle , which we push along @xmath252 until it reaches @xmath306 we also push @xmath305 through the opposite alpha edge of @xmath289 , into the adjacent region . then , we connect the two feet in the complement of alpha curves along @xmath254 and get a new alpha circle @xmath307 . finally , we add the surgery gamma circle @xmath308 as in figure [ fig : stabilization ] . the result is a triple heegaard diagram ( with @xmath309 curves ) which represents surgery along the knot @xmath284 , with a particular framing ; the framing is the sum of the number of twists of @xmath308 around the handle and a constant depending only on the original heegaard diagram . note that , depending on the framing , the local picture around the two feet of the handle may also look like figure [ fig : stabilization_anti ] , in which case instead of the octagon region @xmath310 from figure [ fig : stabilization ] we have two hexagon bad regions @xmath311 and @xmath312 . after the stabilization , @xmath307 and @xmath308 separate @xmath287 into several regions ; among these , @xmath313 and @xmath314 are ( possibly ) bad but all other regions are good . we end up with a diagram with four ( or five ) bad regions : @xmath313 , @xmath314 , @xmath310 ( or @xmath311 and @xmath312 ) , and @xmath303 . ( in some cases , @xmath313 or @xmath314 might be good , or , if there is little winding of @xmath315 some of @xmath310 , @xmath316 and @xmath314 might coincide . the argument in these cases is a simple adaptation of the one we give below . ) we will kill the badness of @xmath317 @xmath318 and @xmath310 ( or @xmath311 and @xmath312 ) , while the region @xmath313 will be the one containing the basepoint @xmath297 for our final triple heegaard diagram . we push the finger in @xmath303 across the opposite alpha edge until we reach a bigon , @xmath319 or a region of type @xmath320 as in figure [ fig : beta_gamma_bigon ] . in this case ( figure [ fig : dprime_bigon ] ( a ) ) , our finger move will kill the badness of @xmath303 , as indicated in figure [ fig : dprime_bigon ] ( b ) , and does not create any new bad regions . this is completely similar to case 1 . the finger move kills the badness of @xmath303 , and does not create any new bad regions . let us suppose the topmost region in figure [ fig : beta_gamma_bigon ] is @xmath313 . the case when the topmost region is @xmath314 is completely similar . the regions involved look like figure [ fig : dprime_ds_bad ] . if on the left @xmath313 is on top of @xmath314 , we isotope the diagram to look as in figure [ fig : dprime_ds_nice ] . the case when on the left of figure [ fig : beta_gamma_bigon ] @xmath314 is on top of @xmath313 is similar , except that we do the double finger move on the other side of @xmath305 . we have now killed the badness of @xmath303 . if there are any bigons between beta and gamma curves adjacent to @xmath314 as in figure [ fig : beta_gamma_bigon ] or figure [ fig : adding_gamma_curve_simple ] ( b ) , also shown in figures [ fig : dw2_special_handleslide ] ( a ) resp . ( c ) , we do a `` handleslide '' ( more precisely , a handleslide followed by an isotopy ) of @xmath308 over each @xmath103 ( @xmath322 ) involved as indicated in figures [ fig : dw2_special_handleslide ] ( b ) resp . the intersection of @xmath308 and @xmath307 has the pattern as figure [ fig : dw2_two_patterns ] ( a ) or ( b ) . in case ( a ) , we do nothing . in case ( b ) , we do the finger move as in figure [ fig : dw2_two_patterns ] ( c ) . now among the possibly bad regions generated from @xmath314 , we have a unique one whose boundary has an intersection of a beta curve with a gamma curve , namely the one near @xmath305 as in figure [ fig : dw2_beta_gamma_crossing ] ( a ) . ( see also figures [ fig : stabilization ] and [ fig : dprime_ds_nice ] . ) we then do a finger move as in figure [ fig : dw2_beta_gamma_crossing ] ( b ) . note that this finger move will not create any badness other than that of @xmath313 . after these special handleslides and finger moves , the region @xmath314 is divided into several possibly bad regions @xmath323 these bad regions are all adjacent to @xmath313 via arcs on @xmath308 and , furthermore , there are no intersection points of beta and gamma curves on their boundaries . we seek to kill the badness of @xmath324 using the algorithm in @xcite . the algorithm there consisted of inductively decreasing a complexity function defined using the unpunctured bad regions . in our situation , we apply a simple modification of the algorithm to the heegaard diagram made of the alpha and the gamma curves ; the modification consists of the fact that we do not deal with the bad region(s ) @xmath310 ( or @xmath311 and @xmath312 ) , but rather only seek to eliminate the badness of @xmath325 thus , in the complexity function we do not include terms that involve the badness and distance of @xmath310 ( or @xmath311 and @xmath312 ) . since all the @xmath326 s are adjacent to the preferred ( punctured ) region @xmath313 via arcs on @xmath315 the algorithm in @xcite prescribes doing finger moves of @xmath308 through alpha curves , and ( possibly ) handleslides of @xmath308 over other gamma curves . we do all these moves in such a way as not to tamper with the arrangements of twin beta - gamma curves , i.e. as not to introduce any new intersection points between @xmath308 and @xmath327 for any @xmath328 . ( in other words , we can think of fattening @xmath329 before applying the algorithm , so that they include their respective twin beta curves . ) in particular , regions of type @xmath320 are treated as bigons . the fact that the algorithm in @xcite can be applied in this fashion is based on the following two observations : * our fingers or handleslides will not pass through the regions adjacent to @xmath305 , except possibly @xmath313 itself . ( this is one benefit of the modification performed in figure [ fig : dw2_beta_gamma_crossing ] . ) * we will not reach any squares between @xmath102 and @xmath330 nor the `` narrow '' squares created in figure [ fig : dw2_special_handleslide ] . in the end , all the badness of @xmath324 is killed . we arrive at a heegaard diagram which might still have some bad regions coming from regions of type @xmath331 as in figure [ fig : dw2_special_bad_region ] ( a ) . we kill these bad regions using the finger moves indicated in figure [ fig : dw2_special_bad_region ] ( b ) . after these moves , the only remaining bad regions are @xmath310 ( or @xmath311 and @xmath312 ) , and the preferred bad region @xmath332 our remaining task is to kill the badness of @xmath310 or @xmath333 . recall that depending on the pattern of the intersection of @xmath307 and @xmath308 ( cf . figure [ fig : dw2_two_patterns ] ) , there are two cases : either we have an octagon bad region @xmath310 , or two hexagon bad regions @xmath311 and @xmath312 . in the first case , one possibility is that a neighborhood of @xmath334 looks as in figure [ fig : dstar_bad ] . we then do the finger moves indicated in figure [ fig : dstar_nice ] . it is routine to check that the new diagram is isotopic to the one in figure [ fig : dstar_bad ] . similarly , in the second case , one possibility is that a neighborhood of @xmath334 looks as in figure [ fig : dstar_bad_case2 ] . in this case , we do the finger moves indicated in figure [ fig : dstar_nice_case2 ] . however , the actual picture on the heegaard diagram may differ from figure [ fig : dstar_bad ] or [ fig : dstar_bad_case2 ] in several ( non - essential ) ways . one possible difference is that at the bottom of the figure [ fig : dstar_bad_case2 ] , the extra gamma curve on top of @xmath313 might be on the right rather than on the left ; however , we can still push the two fingers starting from @xmath313 on each side of @xmath307 . another possible difference is that at the very left of figures [ fig : dstar_bad ] and [ fig : dstar_bad_case2 ] , the curve @xmath308 may have an upward rather than a downward hook , i.e. look as in figure [ fig : dw_1and2_switched](c ) rather than ( a ) . if so , instead of the beta finger from the left in figures [ fig : dstar_bad ] and [ fig : dstar_bad_case2 ] ( cf . also figure [ fig : dw_1and2_switched](b ) ) , we push a beta - gamma finger as in figure [ fig : dw_1and2_switched](d ) . finally , instead of the situations shown in figures [ fig : dstar_bad ] and [ fig : dstar_bad_case2 ] , we might have the same pictures reflected in a horizontal axis . if so , we apply similar finger moves and arrive at the reflections of figures [ fig : dstar_nice ] and [ fig : dstar_nice_case2 ] . in all cases , the finger moves successfully kill the badness of all regions other than @xmath316 in which we keep the basepoint @xmath288 the result is a nice triple heegaard diagram for the cobordism @xmath24 . we now explain how the arguments in this section can be extended to a cobordism @xmath24 which consists of the addition of several two - handles . we view @xmath24 as surgery along a link @xmath335 of @xmath231 components . we start with a multi - pointed heegaard diagram @xmath336 together with another set of basepoints @xmath248 describing the pair @xmath337 as in @xcite . each of the two sets of curves ( @xmath268 and @xmath338 ) has @xmath339 elements . applying the algorithm in @xcite we can make this diagram nice , i.e. such that all regions not containing one of the @xmath297 s are either bigons or rectangles . for @xmath340 we denote by @xmath341 the region containing @xmath342 as in step 1 of section [ sec : boss ] , we inductively remove intersection points between the various components of the projection of @xmath232 to @xmath44 . this projection consists of arcs @xmath262 and @xmath343 with endpoints at @xmath59 and @xmath344 , such that each @xmath262 is disjoint from the beta curves , and each @xmath343 is disjoint from the alpha curves . instead of figure [ fig : knot_embedded_before ] we have the situation in figure [ fig : link_embedded_before ] . again , we stabilize and perform an isotopy to obtain a good diagram with one fewer intersection point , as in figure [ fig : knot_embedded_after ] . iterating this process ( on all link components ) , we can assume that the projection of @xmath232 is embedded in the heegaard surface . we then add twin gamma curves as in step 2 of section [ sec : boss ] . for this we need to do several isotopies of the beta curves as in figure [ fig : adding_gamma_curve_after ] . in that figure , if the region on the top left is @xmath345 the one on the right might be @xmath346 for @xmath347 ; however , the isotopy can be done as before . next , we stabilize the heegaard diagram @xmath231 times ( once for each link component ) to obtain a triple diagram for the cobordism , as in step 3 of section [ sec : boss ] . we then do the analogue of step 4 by pushing @xmath231 fingers to kill the badness of the regions of type @xmath348 . since the fingers only pass through rectangles , they do not intersect each other . the only change is that in figures [ fig : dprime_ds_bad ] and [ fig : dprime_ds_nice ] , the region on the very right may contain a different puncture @xmath59 than the one on the left . by contrast , in case 2 of step 4 , the region on the right in figure [ fig : dprime_bigon ] contains the same puncture as the region on the left , since they lie in the same connected component of the complement of the beta circles . at the end of step 4 , the beta curves split the heegaard surface into @xmath231 connected components @xmath349 . we then do the analogue of step 5 in section [ sec : boss ] . note that this step ( except for the very last bit , figure [ fig : dw2_special_bad_region ] ) only involves moving gamma curves through alpha curves . ( here , we think of the move in figure [ fig : dw2_special_handleslide ] as a single step , rather than as a handleslide followed by an isotopy . ) therefore , we can perform the moves in this step once for each connected component of @xmath232 , independently of each other , because the moves take place in the corresponding component @xmath350 in the situation considered in figure [ fig : dw2_special_bad_region ] , the gamma curves cross a beta curve ; however , the special region @xmath321 is part of a unique @xmath351 , so we can perform the isotopy of the gamma curves as before , without interference from another @xmath352 finally , for step 6 , note that in all the previous steps we have not destroyed the property that the projection of @xmath232 to the heegaard surface is embedded . more precisely , in the part of the stabilized heegaard diagram shown in figure [ fig : stabilization ] , we take a component of the link projection to be a loop starting in @xmath310 , near the upper foot of the handle , going down along @xmath307 until it reaches @xmath313 , then going inside @xmath313 until it reaches the intersection of @xmath308 and @xmath353 and then going along a sub - arc of @xmath308 to its original departure . these paths remain embedded , and disjoint from each other , throughout steps 4 and 5 . ( indeed , neither @xmath305 nor this sub - arc of @xmath308 is moved during these steps . ) it then suffices to note the finger moves in step 6 take place in a neighborhood of the projection of the corresponding link component ( the path considered above ) . therefore , these finger moves can be done without interfering with each other . the result is a nice multi - pointed triple heegaard diagram for @xmath24 . let @xmath24 be a @xmath11-dimensional cobordism from @xmath354 to @xmath355 , and @xmath8 a @xmath2-structure on @xmath24 . choose a self - indexing morse function on @xmath24 . this decomposes @xmath24 as a collection of one - handle additions which together form a cobordism @xmath356 , followed by some two - handle additions forming a cobordism @xmath357 , and three - handle additions forming a cobordism @xmath358 , in this order . let @xmath6 and @xmath7 be the intermediate three - manifolds , so that @xmath359 the map @xmath360 from @xcite is defined as the composition @xmath361 of maps associated to each of the pieces @xmath356 , @xmath357 and @xmath358 . we will review the definitions of these three maps in sections [ section : onethreehandles ] and [ sec : twohandlemaps ] . first we review a few facts about the @xmath362-action on the hat heegaard floer invariants . in ( * ? * section 4.2.5 ) , ozsvth and szab constructed an action of the group @xmath363 on @xmath31 . in @xcite ( see also ( * ? ? ? * section 2 ) ) , they also showed that the cobordism maps @xmath22 extend to maps @xmath364 moreover , if @xmath24 is a cobordism from @xmath6 to @xmath7 ( endowed with a @xmath2 structure @xmath8 ) and we denote by @xmath365 the natural inclusions , then for any @xmath366 of the form @xmath367 one has @xmath368 this equality has the following immediate corollaries : [ l1 ] if @xmath369 then @xmath370 for any @xmath371 [ l2 ] if @xmath372 then @xmath373 for any @xmath374 consider now a 3-manifold of the form @xmath375 , and let @xmath376 denote the torsion @xmath2-structure on @xmath377 . then for any @xmath2-structure @xmath3 on @xmath1 there is an isomorphism @xmath378 as @xmath379$]-modules , cf . * theorem 1.5 ) . here , the action of @xmath380 on @xmath4 is trivial , as is the action of @xmath362 on @xmath381 . further , the action of @xmath382 on @xmath383 is exactly the given by cap product @xmath384 . next , we review the definition of the heegaard floer maps induced by one- and three - handle additions , cf . * section 4.3 ) . suppose that @xmath356 is a cobordism from @xmath354 to @xmath6 built entirely from @xmath95-handles . let @xmath8 be a @xmath2-structure on @xmath356 . the map @xmath385 is constructed as follows . if @xmath386 are the @xmath95-handles in the cobordism , for each @xmath387 pick a path @xmath388 in @xmath354 , joining the two feet of the handle @xmath389 this induces a connected sum decomposition @xmath390 , where the first homology of each @xmath391 factor is generated by the union of @xmath388 with the core of the corresponding handle . further , the restriction of @xmath8 to the @xmath392-summands in @xmath6 is torsion . it follows that @xmath393 . let @xmath394 be the generator of the top - graded part of @xmath395 . then the heegaard floer map induced by @xmath356 is given by @xmath396 it is proved in ( * ? ? ? * lemma 4.13 ) that , up to composition with canonical isomorphisms , @xmath397 does not depend on the choices made in its construction , such as the choice of the paths @xmath398 dually , suppose that @xmath358 is a cobordism from @xmath7 to @xmath355 built entirely from @xmath399-handles . let @xmath8 be a @xmath2-structure on @xmath358 . the map @xmath400 is constructed as follows . one can reverse @xmath358 and view it as attaching @xmath95-handles on @xmath355 to get @xmath401 after choosing paths between the feet of these @xmath95-handles in @xmath402 we obtain a decomposition @xmath403 ( where @xmath129 is the number of @xmath399-handles of @xmath358 ) . further , the restriction of @xmath8 to the @xmath392-summands in @xmath7 is torsion . it follows that @xmath404 . let @xmath405 be the generator of the lowest - graded part of @xmath406 . then the heegaard floer map induced by @xmath358 is given by @xmath407 and @xmath408 for any homogeneous generator @xmath409 of @xmath406 not lying in the minimal degree . again , the map is independent of the choices made in its construction . let @xmath357 be a two - handle cobordism from @xmath6 to @xmath7 , corresponding to surgery on a framed link @xmath232 in @xmath6 , and let @xmath8 be a @xmath2-structure on @xmath357 . let @xmath87 be a triple heegaard diagram subordinate to a bouquet @xmath233 for @xmath232 , as in the beginning of section [ sec : two ] . then , in particular , @xmath410 is a heegaard diagram for @xmath6 , @xmath411 is a heegaard diagram for @xmath7 , and @xmath412 is a heegaard diagram for @xmath413 . ( here , @xmath45 is the genus of @xmath414 and @xmath231 the number of components of @xmath232 . ) the @xmath2-structure @xmath8 induces a @xmath2-structure ( still denoted @xmath8 ) on the four - manifold @xmath415 specified by @xmath416 ( note that @xmath415 can be viewed as a subset of @xmath357 . ) consequently , there is an induced map @xmath417 as discussed in section [ sec : prels ] . the @xmath2-structure @xmath418 is necessarily torsion , so @xmath419 let @xmath394 denote the generator for the top - dimensional part of @xmath420 . then we define ( cf . * section 4.1 ) ) the map @xmath421 by @xmath422 . now , consider instead the nice , @xmath231-pointed triple heegaard diagram @xmath80 constructed in section [ sec : two ] . as discussed in the beginning of section [ sec : two ] , @xmath80 is strongly equivalent to a split triple heegaard diagram whose reduction @xmath87 is subordinate to a bouquet @xmath233 as above . let @xmath423 be the generator for the top - dimensional part of @xmath424 . then , by diagram , with @xmath425 , we have @xmath426 in light of corollary [ lemma : admissible ] and proposition [ easternorthodox ] , the rank of the map @xmath120 can be computed combinatorially . further , since the triple heegaard diagram @xmath80 is nice , so are each of the three ( ordinary ) heegaard diagrams it specifies . consequently , by @xcite , the element @xmath427 can be explicitly identified ( as can a representative for @xmath423 in @xmath428 ) . therefore , the rank of @xmath429 can be computed combinatorially . recall that @xmath24 is a @xmath11-dimensional cobordism from @xmath354 to @xmath355 , @xmath8 a @xmath2-structure on @xmath24 , and that @xmath24 is decomposed as a collection of one - handle additions @xmath356 , followed by some two - handle additions @xmath357 , and three - handle additions @xmath358 , with @xmath6 and @xmath7 the intermediate three - manifolds , so that @xmath359 as in section [ sec : action ] , we consider the maps @xmath430 [ onto1 ] if @xmath431 is surjective , then @xmath432 the cobordism @xmath356 consists of the addition of some @xmath95-handles @xmath433 as in section [ section : onethreehandles ] , we choose paths @xmath388 in @xmath354 joining the two feet of the handle @xmath389 the union of @xmath388 with the core of @xmath255 produces a curve in @xmath356 , which in turn gives an element @xmath434 since @xmath24 is obtained from @xmath435 by adding @xmath399-handles , we have @xmath436 , so the hypothesis implies that the map @xmath437 is surjective . hence there exist disjoint , embedded curves @xmath262 in @xmath354 ( disjoint from all the @xmath438 ) such that @xmath439 ) = -e_i , i=1 , \dots , n.$ ] we can connect sum @xmath388 and @xmath262 to get new paths @xmath440 in @xmath354 between the two feet of @xmath389 using the paths @xmath440 we get a connected sum decomposition @xmath441 as in section [ section : onethreehandles ] , with the property that the inclusion of the summand @xmath442 in @xmath443 is trivial . since we can view @xmath435 as obtained from @xmath357 by adding @xmath399-handles ( which do not affect @xmath444 ) , it follows that the inclusion of @xmath380 in @xmath445 is trivial . lemma [ l1 ] then says that @xmath446 for any @xmath447 thus the kernel of @xmath448 contains all elements of the form @xmath449 , where @xmath450 and @xmath451 is any homogeneous element not lying in the top grading of @xmath452 on the other hand , from section [ section : onethreehandles ] we know that the image of @xmath453 consists exactly of the elements @xmath454 where @xmath394 is the top degree generator of @xmath455 therefore , @xmath456 this gives the desired result . [ onto2 ] if @xmath457 is surjective , then @xmath458 this is similar to the proof of lemma [ onto1 ] . a suitable choice of paths enables us to view @xmath7 as @xmath459 such that the inclusion of the summand @xmath460 in @xmath445 is trivial . lemma [ l2 ] then says that @xmath461 for any @xmath462 and @xmath463 in other words , every element in the image of @xmath448 must be of the form @xmath464 where @xmath465 and @xmath405 is the lowest degree generator of @xmath455 on the other hand , from section [ section : onethreehandles ] we know that the kernel of the map @xmath466 does not contain any nonzero elements of the form @xmath467 let @xmath24 be a cobordism from @xmath354 to @xmath355 , and @xmath8 a @xmath2-structure on @xmath24 . assume that the maps @xmath431 and @xmath457 from formula are surjective . then in each ( relative ) grading @xmath468 the rank of @xmath469 can be computed combinatorially . the map @xmath10 is , by definition , the composition @xmath470 . lemmas [ onto1 ] and [ onto2 ] imply that @xmath471 or , equivalently , @xmath472 using lemma [ onto1 ] again , the expression on the right is the same as the rank of @xmath473 thus , the maps @xmath10 and @xmath448 have the same rank . as explained in section [ sec : twohandlemaps ] , the rank of @xmath448 can be computed combinatorially . note that the relative gradings on the generators of the chain complexes are also combinatorial , using the formula for the maslov index in ( * ? ? ? * corollary 4.3 ) . this completes the proof . in fact , using sarkar s remarkable formula for the maslov index of triangles ( * ? ? ? * theorem 4.1 ) , the absolute gradings on the heegaard floer complexes can be computed combinatorially , and so the rank of @xmath10 in each absolute grading can be computed as well . we give a nice triple heegaard diagram for the cobordism from the three - sphere to the poincar homology sphere , viewed as the @xmath41 surgery on the right - handed trefoil . the right - handed trefoil knot admits the nice heegaard diagram shown in figure [ fig : poincare_trefoil ] , which is isotopic to ( * ? ? ? * figure 14 ) . applying the algorithm described in section [ sec : two ] , we obtain the nice triple heegaard diagram shown in figure [ fig : poincare_nice ] . we leave the actual computation of the cobordism map to the interested reader .

in a previous paper , sarkar and the third author gave a combinatorial description of the hat version of heegaard floer homology for three - manifolds . given a cobordism between two connected three - manifolds , there is an induced map between their heegaard floer homologies . assume that the first homology group of each boundary component surjects onto the first homology group of the cobordism ( modulo torsion ) . under this assumption , we present a procedure for finding the rank of the induced heegaard floer map combinatorially , in the hat version .

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Proceed to summarize the following text: the problem of determining the appearance of a fixed _ pattern _ in long sequences of observation is relevant in many scientific problems . for example , in the area of computer network security , the detection of intrusions , which become increasingly frequent , is very important . intrusion detection is primarily concerned with the detection of illegal activities and acquisitions of privileges that can not be detected by information flow and access control models . there are several approaches to intrusion detection , but recently this subject has been studied in relation to pattern matching ( see @xcite ) . this leads to the study of the construction of particular words avoiding a given pattern in an alphabet @xmath2 . the present paper aims to be a contribution in this direction . let @xmath3 be the set of binary words @xmath4 such that @xmath5 , for any @xmath6 , @xmath7 and @xmath8 corresponding to the number of 0 s and 1 s in the word @xmath4 , respectively . in this paper we study the construction of the subset @xmath9 } \subset f$ ] of binary words excluding a given pattern @xmath10 , that is a word @xmath11}$ ] if and only if it does not contain a sequence of consecutive indices @xmath12 such that @xmath13 . if we consider the set of binary words without any restriction , the defined language is regular and we can refer to using classical results ( see , e.g. , @xcite ) . when the restriction to words with no more 0 s than 1 s is valid , the language @xmath14}$ ] is not a regular one and it becomes more difficult to deal with . for example , in order to generate the language @xmath14}$ ] for each forbidden pattern @xmath15 an `` ad hoc '' grammar should be defined . our aim is to determine a constructive algorithm suggesting a more unified approach which makes it possible to generate all binary words in the class @xmath14}$ ] . in this paper we show how to obtain all binary words belonging to @xmath16 and avoiding the pattern @xmath17 , for any fixed @xmath1 . we @xcite introduced an algorithm for the construction of all binary words in @xmath16 having a fixed number of 1 s and excluding those containing the forbidden pattern @xmath18 , for any fixed @xmath19 . that algorithm generates all the words in @xmath16 then eliminates those containing the forbidden pattern . basically , the construction marks in an appropriate way the forbidden patterns in the words and generates @xmath20 copies of each word having @xmath21 forbidden patterns such that the @xmath22 instances containing an odd number of marked forbidden pattern are annihilated by the other @xmath22 instances containing an even number of marked forbidden patterns . for example , the words @xmath23 and @xmath24 , containing two copies of the forbidden pattern @xmath25 , ( the marked forbidden patterns are over - lined ) are eliminated by the words @xmath26 and @xmath27 , respectively . this is possible since no prefix of @xmath28 is also a suffix of @xmath15 , that is the forbidden patterns do not overlap and so they are univocally identified inside the words . then , the algorithm in @xcite can not be used to generate the words in @xmath14}$ ] when @xmath17 since the forbidden patterns may overlap inside the words . for example , in @xmath29 there are two overlapping copies of the forbidden pattern @xmath30 . so , we propose a new algorithm that generates right the words in @xmath16 avoiding the forbidden pattern @xmath17 , for any fixed @xmath19 . the paper is organized as follows . in section [ sec : basic ] we give some basic definitions and notation . in particular , we recall how every binary word can be represented as a path on the cartesian plane . in section [ sec : algo ] we give a construction , according to the number of 1 s , for the set of binary words excluding the pattern @xmath17 , for any fixed @xmath19 , and such that the number of 0 s in each word is inferior to or the same as the number of 1 s . in section [ sec : proof ] we prove that the construction given in section [ sec : algo ] allows us to obtain an exhaustive and univocal generation of such binary words having @xmath31 1 s . let @xmath3 be the set of binary words @xmath4 such that @xmath5 , for any @xmath6 , @xmath7 and @xmath8 corresponding to the number of 0 s and 1 s in the word @xmath4 , respectively . in this paper we study the construction of the subset @xmath9 } \subset f$ ] of binary words excluding a given pattern @xmath32 , for any fixed @xmath19 . given @xmath33 the _ length _ of @xmath6 , we denote by @xmath34 , ( @xmath35 ) , the word with length @xmath36 obtained by linking @xmath4 to itself @xmath37 times , that is @xmath38 and @xmath39 , @xmath40 being the empty word . each word @xmath6 can be naturally represented as a path on the cartesian plane by associating a _ rise _ ( or _ up _ ) _ step _ , defined by ( 1,1 ) and indicated by @xmath41 , with each bit 1 in @xmath4 and a _ fall _ ( or _ down _ ) _ step _ , defined by ( 1,-1 ) and indicated by @xmath42 , with each bit 0 in @xmath4 . for example , the word @xmath43 is represented by the path @xmath44 ( see figure [ fig : path ] ) . an _ up - down _ step is the sequence @xmath45 . from now on , we refer interchangeably to words or their graphical representation on the cartesian plane , that is paths . so by @xmath9}$ ] we denote both the set of pattern @xmath15 avoiding binary words and the set of corresponding paths . in the rest of this paper , a path is defined as : * _ primitive _ if it begins and ends at ordinate 0 and remains strictly above the @xmath41-axis , * _ positive _ if it begins at ordinate 0 and remains above or on the @xmath41-axis , * _ negative _ if it begins and ends at ordinate 0 and remains below or on the @xmath41-axis ( remark that a negative path in @xmath16 necessarily ends at ordinate 0 ) , * _ strongly negative _ if it begins and ends at ordinate -1 and remains below or on the line @xmath46 , * _ underground _ if it ends with a negative suffix . the _ complement _ of a path @xmath47 is the path @xmath48 obtained from @xmath47 by switching rise and fall steps . in this section we show the constructive algorithm to generate the set @xmath9}$ ] , @xmath49 for any fixed @xmath1 , according to the number of rise steps , or equivalently to the number of 1 s . given a path @xmath11}$ ] with @xmath31 rise steps , we generate a given number of paths in @xmath9}$ ] with @xmath50 rise steps , @xmath51 , by means of constructive rules . the number and the shape of the generated paths depend on the ordinate @xmath52 of the endpoint of @xmath4 and on its suffix . with regard to @xmath52 , we can point out three cases : @xmath53 , @xmath54 and @xmath55 , while as for the suffix we consider whether it is equal to @xmath56 or not . when @xmath53 , we must pay attention also to the case in which @xmath4 is an underground path ending with the pattern @xmath57 . as we will show further on , for each @xmath11}$ ] such that @xmath53 or @xmath55 , the generating algorithm produces two or more positive paths and one underground path with @xmath50 rise steps , @xmath51 , while , when @xmath54 , it produces only one positive path with @xmath50 rise steps . let us denote by @xmath58 a path with endpoint at ordinate @xmath52 . the generating algorithm of the class @xmath9}$ ] with @xmath59 , for any fixed @xmath19 , is described in the following sections . the constructive rules related to the special cases in which the suffix of @xmath4 is @xmath56 or @xmath57 are described in sections [ sec : positivesuffix ] and [ sec : negativesuffix ] , respectively , while in section [ sec : simplecase ] we examine all the other simple cases . the starting point of the algorithm is the empty word @xmath40 . in this section we describe the constructive rules to be applied when the suffix of @xmath4 is neither @xmath56 nor @xmath57 . we point out three cases for the ordinate @xmath52 of the endpoint of @xmath4 : @xmath53 , @xmath54 and @xmath55 . : : a path @xmath11}$ ] , with @xmath31 rise steps and such that its endpoint has ordinate 0 , generates , for any @xmath37 , @xmath51 , three paths with @xmath50 rise steps : a path ending at ordinate 1 by adding to @xmath4 a rise step and a sequence of @xmath60 up - down steps ; a path ending at ordinate 0 by adding to @xmath4 a rise step , a sequence of @xmath60 up - down steps and a fall step , and an underground path obtained by the one generated in the previous step and mirroring on @xmath41-axis its rightmost primitive suffix . + figure [ fig : zero1 ] shows the above described operations ; the number above the right arrow corresponds to the value of @xmath37 . both in this figure and in the following ones we consider @xmath61 , that is @xmath62 . + therefore @xmath63 @xmath54 . : : a path @xmath11}$ ] , with @xmath31 rise steps and such that its endpoint has ordinate 1 , generates , for any @xmath37 , a path with @xmath50 rise steps with endpoint at ordinate 2 obtained by adding to @xmath4 a rise step and a sequence of @xmath60 up - down steps ( see figure [ fig : uno1 ] ) . + therefore @xmath64 @xmath55 . : : a path @xmath11}$ ] , with @xmath31 rise steps and such that its endpoint has ordinate @xmath52 , @xmath55 , generates , for any @xmath37 , @xmath65 paths with @xmath50 rise steps : a path ending at ordinate @xmath66 by adding to @xmath4 a rise step and a sequence of @xmath60 up - down steps ; @xmath67 paths ending at ordinate @xmath68 , respectively , by adding to @xmath4 a rise step , a sequence of @xmath69 , @xmath70 , fall steps and a sequence of @xmath60 up - down steps ; a path ending at ordinate 0 by adding to @xmath4 a rise step , a sequence of @xmath52 fall steps , a sequence of @xmath60 up - down steps and a fall step , and an underground path which will be described in section [ sec : under ] . figure [ fig : due1 ] shows the above described operations . + therefore + @xmath71 at this point it is clear that : 1 . when the path @xmath4 ends with the suffix @xmath56 the paths obtained by means of the constructions ( [ alfa ] ) , ( [ beta ] ) and ( [ gamma ] ) contain the forbidden pattern @xmath72 . so , we will act as described in section [ sec : positivesuffix ] ; 2 . when @xmath4 is an underground path ending with the pattern @xmath57 , some paths generated by means of the above constructions might contain the forbidden pattern @xmath72 . so , we will follow a different procedure described in section [ sec : negativesuffix ] . even when the path @xmath4 ends with the suffix @xmath56 , the number and the shape of the generated paths depend on the ordinate @xmath52 of the endpoint of @xmath4 . let @xmath73 be the suffix of @xmath4 . : : a path @xmath11}$ ] , with @xmath31 rise steps and such that its endpoint has ordinate 0 , generates , for any @xmath37 , @xmath74 , three paths with @xmath50 rise steps ( see figure [ fig : zero2 ] ) : a path ending at ordinate 1 , by inserting a sequence of @xmath60 up - down steps and a rise step on the left of @xmath75 ; a path ending at ordinate 0 , by inserting a sequence of @xmath60 up - down steps and a rise step on the left of @xmath75 and adding a fall step at the end of @xmath4 , and an underground path , obtained by mirroring on @xmath41-axis the rightmost primitive suffix of the path generated at the previous step . therefore + @xmath76 @xmath54 . : : a path @xmath11}$ ] , with @xmath31 rise steps and such that its endpoint has ordinate 1 , generates , for any @xmath37 , a path with @xmath50 rise steps with endpoint at ordinate 2 , obtained by inserting a sequence of @xmath60 up - down steps and a rise step on the left of the suffix @xmath75 ( see figure [ fig : uno2 ] ) . therefore + @xmath77 @xmath55 . : : a path @xmath11}$ ] , with @xmath31 rise steps and such that its endpoint has ordinate @xmath52 , @xmath55 , generates , for any @xmath37 , @xmath65 paths with @xmath50 rise steps ( see figure [ fig : due2 ] ) : a path ending at ordinate @xmath66 , by inserting a sequence of @xmath60 up - down steps and a rise step on the left of the suffix @xmath75 ; @xmath67 paths ending at ordinate @xmath68 , respectively , by inserting a sequence of @xmath60 up - down steps , a rise step and a sequence of @xmath69 , @xmath78 , fall steps on the left of @xmath75 ; a path ending at ordinate 0 , by inserting a sequence of @xmath60 up - down steps , a rise step and a sequence of @xmath52 fall steps on the left of @xmath75 , and then adding a fall step at the end of @xmath4 , and an underground path which will be described in section [ sec : under ] . therefore + @xmath79 the paths @xmath80}$ ] ending on the @xmath41-axis with the sequence @xmath57 have the following shape @xmath81 where @xmath82 is a path ending on the @xmath41-axis and @xmath83 is either the empty path @xmath40 or is a strongly negative path . the constructions applied to paths ending at ordinate 0 described in ( [ alfa ] ) ( see figure [ fig : zero1 ] ) can be used even for the paths ending with the sequence @xmath57 , when @xmath84 , or to generate the paths ending at ordinate 1 or on the @xmath41-axis with a positive suffix , when @xmath85 . nevertheless , when @xmath85 , by applying the construction , we obtain an underground path which contains the forbidden pattern @xmath86 . therefore if the path ends with the sequence @xmath57 and @xmath85 , in order to generate the underground path we proceed as follows . two cases must be taken into consideration . 1 ) @xmath82 does not end with a peak @xmath45 . : : the underground path generated from @xmath87 is obtained by adding the path @xmath88 to @xmath89 , mirroring on @xmath41-axis the rightmost suffix @xmath90 of @xmath91 and shifting the sequence @xmath56 between @xmath82 and the sub - path @xmath92 . + so the path @xmath87 generates the underground path @xmath93 ( see figure [ fig : casoa ] ) . it should be noticed that this construction applies to @xmath4 even if @xmath94 . 2 ) @xmath82 ends with a peak @xmath45 . : : when the path @xmath82 ends with a peak @xmath45 , that is @xmath95 , the insertion of the sequence @xmath56 between @xmath82 and the sequence @xmath96 produces the forbidden pattern @xmath86 . let us consider the following subcases : @xmath97 and @xmath98 . + 2.1 ) @xmath99 . ; ; the underground path is obtained by performing on @xmath100 the following operations : shifting the rightmost peak @xmath45 of @xmath82 to the right of the sub - path @xmath92 , mirroring on @xmath41-axis the sequence @xmath101 and adding to such path the steps @xmath102 . + so , when @xmath85 , the underground path with negative suffix generated by @xmath103 is @xmath104 ( see figure [ fig : casob1 ] ) . 2.2 ) @xmath105 . ; ; in this case , the underground path obtained by means of the construction described in 2.1 ) is @xmath106 and it contains the forbidden pattern @xmath86 if @xmath107 ends with the sequence @xmath90 or with the sequence @xmath108 , where @xmath109 is a not empty strongly negative path . let us take the longest suffix of @xmath110 into account so that @xmath111 , where @xmath112 and @xmath113 is the empty path or is a strongly negative path . every sequence @xmath114 , @xmath115 , will be changed into @xmath116 in the following way : + 2.2.1 ) : : if @xmath47 is a path that does not end with a peak @xmath117 , then @xmath118 and the underground path generated by @xmath89 is @xmath119 ( see figure [ fig : casob2i ] ) ; 2.2.2 ) : : if @xmath47 ends with a peak @xmath45 , that is @xmath120 , then @xmath121 and the underground path generated by @xmath89 is @xmath122 ( see figure [ fig : casob2ii ] ) . now let us describe how to obtain the underground path generated by @xmath123 . for any @xmath37 , @xmath51 , let @xmath124 be the path obtained from @xmath58 and ending on the @xmath41-axis with a positive suffix , @xmath47 is the rightmost suffix in @xmath125 which is primitive . if the path @xmath48 does not contain the forbidden pattern @xmath15 , the underground path generated by @xmath58 is @xmath126 . if the path @xmath48 contains the forbidden pattern @xmath15 , we must apply a _ swap _ operation @xmath127 in order to obtain a path @xmath128 avoiding the forbidden pattern . the underground path generated by @xmath58 is @xmath129 . before describing the @xmath127 operation on @xmath48 , let us consider the following proposition . [ prop1]let @xmath130}$ ] a primitive path ; @xmath131 contains the forbidden pattern @xmath86 if and only if @xmath82 contains the pattern @xmath132 . from proposition [ prop1 ] it follows that , if @xmath48 contains the forbidden pattern @xmath15 , then it is preceded and followed by at least a rise step . operation @xmath127 must generate a path @xmath133 avoiding the forbidden pattern @xmath86 and such that @xmath134}$ ] ; in this way @xmath133 is not the complement of any path in @xmath9}$ ] . the path @xmath135 is obtained in the following way : * consider the straight line @xmath136 from the beginning of the pattern @xmath137 and let @xmath138 be the rightmost point in which @xmath136 intersects @xmath48 on the left of @xmath15 such that @xmath138 is preceded by at least two fall steps ; * let @xmath139 , @xmath140 , the subsequence on the right of @xmath138 , followed by at least a fall step ; * _ swap _ the initial subsequence @xmath141 of @xmath15 and @xmath142 . let us remark that @xmath142 can not be equal to @xmath56 as @xmath47 does not contain the forbidden pattern @xmath72 ( see figure [ fig : swap].a ) ) . when @xmath143 , that is @xmath142 is the empty word , we simply insert @xmath144 into @xmath138 ( see figure [ fig : swap].b ) ) . operation @xmath127 is applied to each forbidden pattern in @xmath48 . [ prop2 ] let @xmath145 , then @xmath146}$ ] . * proof . * the @xmath127 operation transforms the subsequence @xmath147 , ( @xmath148 ) , of @xmath48 into the subsequence @xmath149 of @xmath133 . the complement of @xmath150 is @xmath151 so @xmath152 contains the forbidden pattern @xmath72 . @xmath153 [ prop3 ] let @xmath154}$ ] a primitive path such that @xmath155}$ ] . then there exists a path @xmath156}$ ] such that @xmath157 . * proof . * if @xmath154}$ ] and @xmath155}$ ] then @xmath131 contains the pattern @xmath158 ; we apply to @xmath131 the following operation @xmath159 : * consider the straight line @xmath136 from the end of the pattern @xmath56 and let @xmath160 be the leftmost point where @xmath136 intersects @xmath131 on the right of @xmath56 such that @xmath160 is followed by at least two rise steps ; * let @xmath139 , @xmath140 , the subsequence on the left of @xmath160 , preceded by at least a rise step ; * swap the subsequence @xmath56 and @xmath142 . when @xmath143 , that is @xmath142 is the empty word , we simply insert @xmath56 into @xmath160 . @xmath153 figure [ fig : tree ] shows the initial steps of the generating algorithm of the paths corresponding to words in @xmath9}$ ] , @xmath161 . let us remark that , following the above constructions , given a path @xmath4 , the number of generated paths depends only on the ordinate of endpoint of @xmath4 . so , the complete generating algorithm can be briefly described by the succession rule ( [ one ] ) ( for more details on succession rules see @xcite ) @xmath162 where each number corresponds to the ordinate of the endpoint of a path . the zero in the first line in ( [ one ] ) is associated with the empty path . the second line in ( [ one ] ) is associated with operations @xmath163 and @xmath164 , the third line is associated with operations @xmath165 and @xmath166 , and the last line describes the construction when the endpoint has ordinate @xmath167 , underground path included . in this section we prove that the construction described in section 3 allows to generate the class @xmath9}$ ] exhaustively for any fixed forbidden pattern @xmath17 , @xmath19 , in the sense that all the words in @xmath9}$ ] with @xmath31 1 s , @xmath168 , can be generated . [ teo : uno ] given a fixed forbidden pattern @xmath17 , @xmath169 , the construction described in section [ sec : algo ] generates all the paths with @xmath31 , @xmath168 , rise steps representing the binary words in @xmath9}$ ] with @xmath31 1 s . let @xmath11}$ ] , then @xmath170 that is , @xmath4 is made of @xmath171 sub - paths such that : * @xmath172 is the empty path @xmath40 , * @xmath173 , @xmath174 is a path in @xmath9}$ ] beginning from and ending on the @xmath41-axis , * @xmath175 is a path in @xmath9}$ ] beginning from the @xmath41-axis with endpoint at ordinate @xmath176 . the proof of theorem [ teo : uno ] is obtained by induction on the number of sub - paths . * proof . * the empty path @xmath40 is generated as the starting point of the algorithm . let us assume that all the possible sub - paths @xmath177 of @xmath4 are generated . we prove that the algorithm generates the sub - path @xmath178 for any path @xmath179}$ ] . let @xmath173 be a path that does not end with the pattern @xmath56 . in this case the path @xmath180 may be either positive or negative . if @xmath180 is a positive path we have to prove that all the positive paths are generated and this will be demonstrated in section [ sec : positive ] . in the case of a negative path , denoting @xmath181 the largest value of its absolute ordinate , let us remark that : * the negative paths with @xmath182 are @xmath183 , @xmath184 , and they are generated by iterating the construction ( [ alfa ] ) ; * negative paths with @xmath185 are @xmath186 , @xmath187 , and they are generated by means of ( [ alfa ] ) when @xmath188 , or by means of ( [ alfa1 ] ) when @xmath189 ; * let @xmath190 be a negative path with @xmath191 ; if @xmath192}$ ] then @xmath190 is the underground path generated by a positive path with endpoint at ordinate @xmath55 , ( see section [ sec : under ] ) , otherwise @xmath193 for a positive path @xmath83 in @xmath9}$ ] with endpoint at ordinate @xmath194 ( see proposition [ prop3 ] ) . note that when @xmath195 and @xmath173 ends with the pattern @xmath88 , the only possible negative path @xmath180 with @xmath185 is @xmath196 and it is generated by applying the construction ( [ alfa1 ] ) to the path @xmath197 . when the suffix of @xmath173 is the pattern @xmath56 , @xmath180 must be a negative path and the path @xmath198 is the underground path obtained by means of the construction described in case 1 ) in section [ sec : negativesuffix ] ( see figure [ fig : casoa ] ) . in the same way , when the sub - path @xmath199 is of type @xmath200 or @xmath201 where @xmath113 is the empty path @xmath40 or is a strongly negative path , then it is generated by the constructions described in case 2 ) in section [ sec : negativesuffix ] ( see figures [ fig : casob1 ] , [ fig : casob2i ] and [ fig : casob2ii ] ) . then , if we show that all the possible positive paths are generated , then we can claim that theorem [ teo : uno ] is proved . moreover , we observe that for each path @xmath4 in @xmath9}$ ] with @xmath31 rise steps there exists one and only one path @xmath125 in @xmath9}$ ] with @xmath202 rise steps , @xmath51 , such that @xmath4 is obtained from @xmath125 by means of the construction described in section [ sec : algo ] . this assertion is a direct consequence of the construction , since the actions described are univocally determined . @xmath153 in this section we prove that all the positive paths with @xmath31 rise steps are generated by means of the construction described in section [ sec : algo ] . in the sequel of this section we analyze only _ positive _ paths . the proof is obtained by induction on @xmath31 . there are only two paths with @xmath203 rise step , that is @xmath41 and @xmath45 , and they are generated by means of construction ( [ alfa ] ) applied to the empty path @xmath40 . let us assume that all the paths with @xmath204 rise steps are generated ; we will prove that all the paths with @xmath31 rise steps are generated . note that , following the construction given in @xcite , a path with @xmath31 rise steps can be obtained from a dyck path @xmath4 with @xmath205 rise step by inserting one rise step in each point at ordinate @xmath206 of its last descent followed by @xmath207 fall steps , @xmath208 . we will prove that all the paths obtained so are also generated following the constructions given in the above sections . let us denote by @xmath209 a paths ending with @xmath207 fall steps . let @xmath69 be the number of fall steps in the last descent of @xmath4 . first of all , we note that for any value of @xmath69 the paths obtained by inserting a rise step in the point at ordinate 0 are generated by means of the constructions ( [ alfa ] ) or ( [ alfa1 ] ) applied to the path @xmath4 . here we give the proof for the case with @xmath210 , distinguishing three cases : @xmath211 , @xmath212 and @xmath213 . the analogous and simple cases @xmath214 and @xmath215 are left to the reader . * @xmath216 . let @xmath217 . the insertion of a rise step in the point at ordinate 1 gives three paths : * * @xmath218 , which is generated by means of ( [ beta ] ) applied to the prefix @xmath219 of @xmath4 , * * @xmath220 and @xmath221 , which are the paths with endpoints at ordinate 1 and 0 , respectively . if @xmath222 , then @xmath223 and @xmath224 are generated by the construction ( [ gamma ] ) , where @xmath225 , applied to the path @xmath226 with @xmath227 , otherwise , if @xmath195 , they are generated by means of ( [ gamma1 ] ) with @xmath225 and @xmath85 applied to the path @xmath228 . * the insertion of a rise step in the point at ordinate @xmath206 gives @xmath230 paths @xmath209 , @xmath231 . the paths @xmath209 with @xmath232 are all the positive paths generated by means of ( [ gamma ] ) with @xmath233 and @xmath85 applied to the prefix of @xmath4 of length @xmath234 . the path @xmath235 is the path with endpoint at ordinate @xmath206 . when @xmath236 , @xmath223 is generated by means of construction ( [ gamma ] ) with @xmath225 and @xmath227 applied to the path @xmath226 , while , when @xmath195 , it is generated by means of ( [ gamma1 ] ) , with @xmath225 and @xmath85 , applied to the path @xmath228 . * @xmath213 . the insertion of a rise step in the point at ordinate @xmath237 generates @xmath238 paths @xmath209 , @xmath239 . the paths @xmath209 with @xmath232 are all the positive paths generated by means of ( [ gamma ] ) with @xmath225 and @xmath85 applied to the prefix of @xmath4 of length @xmath240 . as far as the generation of @xmath223 is concerned , we have to distinguish three cases : 1 . if @xmath241 , @xmath242 , then @xmath243 . if @xmath244 , then @xmath223 is the path with endpoint at ordinate @xmath66 generated by means of ( [ gamma ] ) with @xmath245 and @xmath246 , applied to the path @xmath247 . if @xmath248 , then @xmath223 is the path with endpoint at ordinate @xmath66 generated by means of ( [ gamma1 ] ) with @xmath245 and @xmath85 applied to the path @xmath249 . note that , when @xmath250 , the endpoint of the prefix @xmath247 has ordinate 1 , and the path @xmath223 is obtained applying the construction ( [ beta ] ) ( or ( [ beta1 ] ) ) instead of ( [ gamma ] ) ( or ( [ gamma1 ] ) ) . if @xmath251 , @xmath252 and @xmath253 , then @xmath254 . if @xmath244 , then @xmath223 is the path with endpoint at ordinate @xmath255 generated by means of ( [ gamma ] ) with @xmath256 and @xmath257 applied to the path @xmath258 . if @xmath248 , then @xmath223 is generated by means of ( [ gamma1 ] ) with @xmath256 and @xmath85 applied to the path @xmath259 . if @xmath260 , @xmath261 and @xmath252 , then @xmath262 . if @xmath244 , then @xmath223 is the path with endpoint at ordinate @xmath66 generated by means of ( [ gamma ] ) , with @xmath245 and @xmath257 applied to the path @xmath263 . if @xmath248 , then @xmath223 is generated by means of ( [ gamma1 ] ) , with @xmath245 and @xmath264 applied to the path @xmath265 . in this paper we propose an algorithm for the construction of particular binary words , according to the number of 1 s , excluding a fixed pattern @xmath266 , @xmath19 . moreover , it would be interesting to study words avoiding patterns which have a different shape , that is not only patterns consisting of a sequence of rise and fall steps . this could be the first step in the study of a possible universal generating algorithm for pattern avoiding words . another interesting field of study is to determine a sort of invariant class of avoiding patterns that is the paths @xmath267 such that @xmath268}|=|f^{[{\mathfrak{p } } _ 2]}|= \dots = |f^{[{\mathfrak{p } } _ l]}|$ ] with consequent bijective problems . one could also consider a forbidden pattern on an arbitrary alphabet and investigate words avoiding that pattern , or study words avoiding more than one pattern and the related combinatorial objects , considering various parameters . s. bilotta , d. merlini , e. pergola , r. pinzani , binary words avoiding a pattern and marked succession rule , _ lattice path combinatorics and applicatons , siena , july 4 - 7 , 2010 _ ( available on line arxiv:1103.5689 ) , ( 2010 ) .

in this paper we propose an algorithm to generate binary words with no more 0 s than 1 s having a fixed number of 1 s and avoiding the pattern @xmath0 for any fixed @xmath1 . we will prove that this generation is exhaustive , that is , all such binary words are generated . * keywords : * binary words , pattern avoiding , exhaustive generation .

You are an expert at summarizing long articles. Proceed to summarize the following text: the great regularity of the tully - fisher relation @xcite has long been thought to originate from a strong mass - velocity relation and a near constancy of mass - to - light ratio . the latter requires a fair but not unreasonable amount of regularity in stellar populations . put simply , ... when it comes to pepper grinders '' ( van den bosch 2001 , private communication ) . ] @xmath0 there have long been indications ( sancisi 1995 , private communication ) that this simple scaling may fail at low luminosities . this has become more clear as data have improved @xcite,@xcite . this breakdown of the tully - fisher relation might arise because of the chaotic star formation histories of low mass galaxies , or as a result of a breakdown in the underlying mass - velocity relation . another possibility is that optical luminosity ceases to trace mass because stars cease to be the dominant mass component in these disks @xcite . it has now become clear that this last possibility is in fact the case . low mass galaxies are often dominated by gas rather than stars . if instead of luminosity or stellar mass , we plot disk ( star + gas ) mass against the flat rotation velocity , a nice mass - velocity relation is recovered over many orders of magnitude ( fig . 1 ) . this ` baryonic tully - fisher relation ' ( btf ) is @xcite @xmath1 for which the data in fig . 1 give a = 50_^-4 ^ 4 + b = 4.0 0.1 . the normalization of the btf is rather uncertain : formally acceptable values fall in the range @xmath2 . the precise value of the slope has been modestly controversial : @xmath3 was given by @xcite while @xmath4 was found by @xcite . this difference can be traced to different assumptions about the ( rather goofy @xcite ) distance to the uma cluster for which some of the better rotation curve @xcite and photometric data @xcite exist . as the distance increases , the gas mass increases faster than the stellar mass ( as @xmath5 and as @xmath6 , respectively ) . this boosts the total mass of gas dominated galaxies by a larger factor than star dominated galaxies . since these reside at opposite ends of the relation , the slope tips to shallower values with increasing @xmath6 nevertheless , the population models of @xcite are consistent with a slope of @xmath3 ( fig . 2 ) . while the calibration of the btf can always be improved , it already provides an excellent indicator of disk mass . moreover , continuity between gas - rich and star - rich galaxies constrains stellar population mass - to - light ratios . the favored values are reasonable in terms of population synthesis models ( fig . 2 ) , but unpleasantly heavy for cuspy dark matter halos . -band and ( * b * ) the @xmath7-band predicted by a slope 4 btf for the uma galaxies @xcite , @xcite , @xcite . these are plotted as a function of @xmath8 color , together with the bruzual & charlot , salpeter imf model from @xcite ( the first model in their table 4 ) . the population synthesis models are in good agreement with the btf , indicating that we have a good handle on @xmath9 and disk masses . ] rotation curves , by themselves , can only give a lower limit on the total halo mass : that enclosed by the last measured point . however , if the functional form of the halo were known , it might be possible to provide some constraint by fitting the observations to the known form . the nfw halo paradigm @xcite,@xcite which has arisen from cosmological n - body simulations in principle gives a way to do this . unfortunately , if not surprisingly , observed rotation curves never extend far enough to constrain the circular velocity at the virial radius , @xmath10 @xcite . there is a great deal of degeneracy between the concentration @xmath11 and @xmath10 . an example is given in fig . 3 , which shows how difficult it can be to distinguish between fits with nfw halos of rather different parameters . and @xmath12 for @xmath13 . another tolerable fit with @xmath14 and @xmath15 is also shown ( lower dotted curve ) to illustrate the degeneracy between parameters . though many models sort of fit , their concentrations are implausibly low for @xmath16cdm . ] matters are made worse by the general failure of the nfw form to provide a good description of the data . the data just do nt look like nfw halos . statistically , halos with constant density cores are almost always preferred over those with cusps @xcite . this is most clear in the best resolved cases @xcite . the most important systematic concern at this point is not observational . resolution has improved by an order of magnitude @xcite , @xcite , @xcite over the original 21 cm data for lsb galaxies @xcite , @xcite . the nfw shape has not become apparent as the data have improved . instead , the systematics pointed out by @xcite,@xcite as problematic for cdm ( independent of the cusp issue ) have only become more clear . concern over slit mispositioning @xcite are misplaced : independent observers reproduce one anothers results @xcite,@xcite . while there are certainly cases in which the error bars are large enough to allow an nfw fit , isothermal fits are inevitably better . simply changing the size of the error bars wo nt change this : a systematic change in the shapes of @xmath17 high resolution rotation curves is required . one can certainly imagine ways in which this might happen @xcite , but it is extremely unlikely that any of these ideas apply to real data , let alone to * all * of the data from various independent sources . the most serious issue is in the mass models : stars have mass . even in the limit of zero stellar mass , which is the most favorable to the nfw case , isothermal halos are statistically preferred @xcite,@xcite . the situation only becomes more grim if stars are allowed to have mass . though lsb galaxies are dark matter dominated down to small radii , plausible @xmath9 models do require that _ some _ of the velocity be attributed to luminous mass . this pulls the inferred dark matter distribution further away from the expected cusp slope . we are hardly unique in reaching these conclusions , which are shared by * all * published analyses of high resolution long slit h@xmath18 data @xcite,@xcite,@xcite,@xcite,@xcite,@xcite,@xcite . high resolution fabry - perot @xcite,@xcite,@xcite and co @xcite data are also inconsistent with cuspy halos , as are a variety of data for the milky way itself @xcite . the only analyses which are favorable to nfw are those of low resolution data with large error bars @xcite,@xcite . when the error bars are large , any model can be driven through them . though it has not been emphasized , constant density cores provide as good or better fits even in these cases . the isothermal halo form , while effective , is an extremely flexible fitting function which lacks the motivation of the nfw halo form . so one might persists that the nfw fits are still more appropriate in that they can be related to cosmology . standard @xmath16cdm makes a clear prediction @xcite for what the concentrations of dark matter halos should be : @xmath19 for @xmath20 . scatter about this value should be modest the largest estimate @xcite finds a lognormal distribution with @xmath21 . the median observed concentration is @xmath22 @xcite which is different from the standard @xmath16cdm prediction by many @xmath23 . the problem with nfw halos is not just a matter of getting fits to individual galaxies , but also of understanding how the observed concentrations can be so low . these low concentrations would be tolerable in a very low density universe with @xmath24 @xcite . even then there exists a significant tail of very low concentration ( @xmath25 ) galaxies which simply should not exist for any plausible cosmology . the debate over halo profiles , while contentious , misses the real point . many halo profiles are nominally viable because they have lots of degenerate free parameters . mass modeling is a bit like fitting a high order polynomial to a few data points : the line goes through the data , but means nothing . one would prefer to have a minimal parameter description of the data . such a prescription exists @xcite . it has long been noted that there is a strong coupling between mass and light . oddly , this coupling persists for dark matter dominated lsb galaxies . one needs only a single parameter per galaxy , the stellar mass - to - light ratio , in order to fit the rotation curve in comparable or greater detail than can be matched by many - parameter halo models . the mass - to - light ratio in the @xmath26-band is close enough to constant that one can make a good zero parameter prediction with such data @xcite . until we come to terms with this observed phenomenology , debating the cusp slope of dark matter halos is rather akin to debating the number of angels that can dance on the head of a pin . + * acknowledgements : * i am most grateful to vera rubin , erwin de blok , and albert bosma for their work on the issues discussed here . i would also like to thank renzo sancisi , marc verheijen , rob swaters , and frank van den bosch for many lively and stimulating conversations . no doubt , i have yet to hear the end of it ! the work of ssm is supported in part by nsf grant ast9901663 . freeman : ` historical introduction ' . in : _ the low surface brightness universe , iau colloquium 171 _ , ed . davies , c. impey , s. phillipps ( astronomical society of the pacific , san francisco 1999 ) pp . 3 - 8 mcgaugh : ` dynamical constraints on disk galaxy formation ' . in : _ galaxy dynamics : from the early universe to the present _ , ed . f. combes , g.a . mamon , v. charmandaris ( astronomical society of the pacific , san francisco 2000 ) pp .

i review what we currently do and do not know about the masses of disk galaxies and their dark matter halos . the prognosis for disks is good : the asymptotic rotation velocity provides a good indicator of total disk mass . the prognosis for halos is bad : cuspy halos provide a poor description of the data , and the total mass of individual dark matter halos remains ill - constrained .

You are an expert at summarizing long articles. Proceed to summarize the following text: since their introduction @xcite , topological field theories have been responsible for many applications @xcite and are object of continuous investigations . nowadays they represent an important chapter of quantum field theory . the original motivation was related to the possibility of describing topological invariants by means of standard field - theory techniques @xcite . in order to give an idea of this framework , let us briefly present here the field - theory characterization of one of the most simple and familiar topological invariants , namely , the linking number @xmath3 of two nonintersecting smooth closed oriented curves in @xmath4@xcite : as is well known , the linking number @xmath5 is an integer which counts the number of times that one curve winds around the other . it is independent from the shape of the curves and can be represented by the gauss integral @xmath6 expression ( [ g - int ] ) is in fact easily seen to be an integer by use of the stokes theorem @xcite . taking a field theory point of view , the linking number @xmath5 may be obtained by introducing the topological abelian chern simons action @xcite @xmath7 and by evaluating the correlation function of two loop variables @xmath8 , _ i .. e . , _ @xmath9 that expression ( [ l - cs ] ) reproduces the linking number follows from the observation that the propagator of the gauge field @xmath10 obtained from the chern simons action ( [ c - s ] ) upon quantization in the landau gauge is precisely the kernel of the gauss integral ( [ g - int ] ) , _ i.e. , _ @xmath11 the correlator ( [ l - cs ] ) may thus be regarded as a field - theory description of the linking @xmath12 the action ( [ c - s ] ) can be suitably extended to higher dimensions , providing a field - theory characterization of the generalizations of the linking number @xcite . moreover , the nonabelian version of the three - dimensional chern simons action ( [ c - s ] ) has been proven to play a very relevant role in knot theory @xcite . although the topological field theories possess their own interests and applications , it is worth underlining here that topological terms appear frequently as parts of more general effective actions useful for the theoretical description of a large number of phenomena in different space - time dimensions . for instance , the effective action corresponding to the bosonization @xcite of relativistic three - dimensional massive fermionic systems at @xmath13 can be written as the sum of the chern - simons term ( [ c - s ] ) and of an infinite series of higher - order terms in the curvature @xmath14 and its derivatives , _ @xmath15 with @xmath16 being a combination of terms of the type @xmath17 this kind of action turns out to be useful in order to study several three - dimensional phenomena such as the fermi bose transmutation @xcite and the quantum hall effect @xcite . a second interesting example is provided by the five - dimensional generalization of ( [ eff - cs ] ) , obtained from the ads / cft _ _ _ _ correspondence @xcite , which relates the conformal @xmath18 super - yang - mills theory to type - iib superstring on ads@xmath19 . in fact , in the conformal case , the dual supergravity on @xmath20 possesses a chern simons term obtained from a @xmath21 doublet of two - forms @xmath22 @xmath23 . in this case , the relevant effective action for @xmath22 @xmath23 looks like @xcite @xmath24 where the term @xmath25 collects all the higher - order terms in the curvatures @xmath26 . the correlation function ( [ l - cs ] ) generalizes now to @xmath27 where @xmath28 @xmath29 are appropriate two - surfaces . in view of these applications , it seems natural to ask ourselves what is the response of a correlator of the type ( [ l - cs ] ) when the corresponding topological field theory is perturbed by the introduction of a nontopological interaction term depending on the curvature . this is the aim of the present paper . more precisely , we shall report on the four - loop computation of the correlator @xmath30 when the three - dimensional chern simons action ( [ c - s ] ) is perturbed by a nontopological interaction term of the kind @xmath31 namely _ _ , _ _ expression ( [ p - c ] ) will be evaluated with an effective action @xmath32 given by @xmath33 with @xmath34 and @xmath35 being an arbitrary parameter with negative mass dimension , reflecting the power - counting nonrenormalizability of the perturbation . in particular , we shall be able to prove that the correlation function ( [ p - c ] ) turns out to be independent from @xmath35 , yielding the linking number @xmath5 of the two curves @xmath1 , @xmath36 although the loop analysis will be worked out only up to the fourth order , this conclusion holds to all orders of perturbation theory and may be easily generalized to any local nontopological interaction term containing arbitrary powers of the curvature @xmath14 as well as to the higher - dimensional cases @xcite as , for instance , the effective action of eq.([ads ] ) . this result means that the loop correlator ( [ p - c ] ) is stable with respect to the perturbations which can be added to the starting topological action . in other words , the expression ( [ p - c ] ) will give the linking number of the two curves @xmath1 , @xmath2 , regardless of any @xmath37-dependent perturbation term that can be introduced and of their power - counting nonrenormalizability character . two remarks are now in order . first , we will limit here ourselves only to effective actions which are abelian . second , we shall consider only @xmath37-dependent terms which can be treated as true perturbations . therefore , we shall avoid in the effective action ( [ seff ] ) the inclusion of a term of the maxwell type @xmath38 where @xmath39 is a mass parameter . the presence of this term would completely modify the original properties of the model . in fact , being expression ( [ max ] ) quadratic in the gauge fields , it can not be considered as a perturbation term , as it will be responsible for the presence of massive excitations in the spectrum of the theory @xcite . rather , the presence of the maxwell term in the effective action ( [ seff ] ) will give rise to the existence of two distinct regimes corresponding to the long and short distance behaviours , respectively . for distances larger than the inverse of the mass parameter @xmath39 ( _ i.e. _ , the low - energy regime ) , the topological term will prevail , _ _ _ _ while the maxwell term will become the relevant one at short distances ( _ i.e. , _ the high - energy regime ) . it is worth mentioning here that these two regimes can be accessed in a very simple way by means of suitable gauge - invariant field redefinitions of the gauge connection @xmath10 @xcite . however , their full understanding is a difficult and delicate task , which is beyond the aim of the present paper , being under investigation . we should also underline here that , in the abelian case , the loop variable @xmath8 is gauge invariant for closed curves , and so there is no need to take into account its exponentiation @xmath40 , as it would be required in the nonabelian case . this feature has a useful consequence . it allows indeed to avoid the case in which the double - line integral ( [ p - c ] ) has to be taken along the same curve . this case , usually referred to as the self - linking , would be automatically generated by the perturbative taylor expansion of the exponential @xmath41 . in other words , as far as the abelian case is concerned , the loop variables in eq.([p - c ] ) do not need to be exponentiated . therefore , the two curves @xmath1 and @xmath2 will always refer to two distinct curves which do not intersect each other . as we shall see in the following , this point will be relevant in order to establish the independence from the parameter @xmath35 of the expression ( [ p - c ] ) . in order to discuss the perturbative expansion of the loop correlator ( [ p - c ] ) , let us first define the gauge - fixed version of the effective action which shall be used throughout the present article , namely , @xmath42 where the lagrange multiplier @xmath43 has been introduced in order to implement the landau gauge . notice that we have wick - ordered the quartic interaction term , which will allow to rule out tadpole diagrams . and @xmath44 of the field strength @xmath45 are each gauge invariant . ] as usual in this kind of problem , we shall make use of the configuration space rather than the momentum space . let us now give the elementary wick contractions which shall be needed for the evaluation of the feynman diagrams . recalling that @xmath46 from eq.([prop ] ) one obtains @xmath47 and @xmath48 concerning now the perturbative loop expansion , it is easily checked that the first feynman diagram which contributes to the correlation function ( [ p - c ] ) is of two - loop order and can be drawn as follows : in the above figure , the wavy and dashed lines refer respectively to the wick contractions @xmath49 and @xmath50 . the feynman integrals corresponding to the diagram of fig.2 are easily written down by means of eqs.([wc1 ] ) , ( [ wc2 ] ) . however , before computing them , let us spend a few words on the mechanism which is responsible for the independence on the parameter @xmath35 of expression ( [ p - c ] ) . from the structure of the diagram of fig.2 , we observe that the gauge fields @xmath51 and @xmath52 lying on the two curves @xmath53and @xmath54will be always contracted with the @xmath55 s present in the interaction term of the expression ( [ effective ] ) . therefore , besides contractions of the type @xmath56 , the corresponding feynman integrals will always contain two contractions of the kind @xmath57 . however , one should remark in the second term of eq.([wc1 ] ) that one of the lorentz indices of the two space - time derivatives corresponds to the vector index of a gauge field lying on either @xmath58or @xmath36 it thus refers to a total derivative with respect to the variable running along one of the closed loops , implying a vanishing contribution . in other words , the second term of eq.([wc1 ] ) may be neglected . as a consequence , all the wick contractions entering the feynman integrals will basically lead to a product of delta functions . after the introduction of a suitable regularization , the latter can be integrated out , finally resulting in a @xmath59 , where , we remind , @xmath60 and @xmath61 run along each of the two curves , respectively . as these variables never coincide , the whole expression vanishes identically , ensuring the independence from the parameter @xmath35 of the correlator ( [ p - c ] ) . the same mechanism can be seen to occur at higher loop orders , as it will be explicitly shown later on . apart from an irrelevant global symmetry coefficient , the diagram of fig.2 therefore corresponds to the following integral : @xmath62 . \nonumber \\ & & \,\ , \label{int2}\end{aligned}\ ] ] let us analyse the first term of the above expression . making use of the propagators ( [ wc1 ] ) and ( [ wc2 ] ) , we obtain @xmath63 \left [ g_{\nu \gamma } \delta ^3(y - z_2)+\partial _ \nu \partial _ \gamma \frac 1{4\pi \left| y - z_2\right| } \right ] \nonumber \\ & & \quad \times \left [ \varepsilon _ { \beta \rho \lambda } \partial ^\lambda \delta ^3(z_1-z_2)\right ] \left [ \varepsilon ^{\alpha \rho \tau } \partial _ \tau \delta ^3(z_1-z_2)\right ] \left [ \varepsilon ^{\beta \gamma \sigma } \partial _ \sigma \delta ^3(z_1-z_2)\right ] \ , . \label{second}\end{aligned}\ ] ] as previously mentioned , the terms containing the derivatives @xmath64 and @xmath65 do not contribute , as they correspond to total derivatives on closed curves . expression ( [ second ] ) then becomes @xmath66 \left [ \varepsilon _ \mu { } ^{\rho \tau } \partial _ \tau \delta ^3(z_1-z_2)\right ] \left [ \varepsilon ^{\beta \sigma } { } _ \nu \partial _ \sigma \delta ^3(z_1-z_2)\right ] \ , . \label{second2}\end{aligned}\ ] ] in spite of the presence of products of delta functions with the same arguments , the above expression is easily seen to vanish . let us show this claim in two ways . first , we observe that there is always a possible order of taking the integrations over the delta functions such that we end up with products of @xmath59 and not of @xmath67 . in the present case , this would amount to integrate out first the two delta functions with arguments @xmath68 and @xmath69 , which would lead to @xmath70 \left [ \varepsilon _ \mu { } ^{\rho \tau } \partial _ \tau \delta ^3(x - y)\right ] \left [ \varepsilon ^{\beta \sigma } { } _ \nu \partial _ \sigma \delta ^3(x - y)\right ] = 0\ , , \label{second3}\ ] ] since @xmath71 never vanishes . it is worth remarking that this possibility exists , in fact , for the higher - order diagrams , as will be shown below . second , we can adopt a more rigorous treatment by regularizing the delta functions with coinciding arguments through the point - splitting procedure already used by polyakov @xcite : @xmath72 more precisely , whenever a product of @xmath73 delta functions with coinciding arguments occurs , it will be understood as @xmath74 ^n=\left [ \delta _ \varepsilon ( z_1-z_2)\right ] ^{n-1}\delta ^3(z_1-z_2)\,,\ ] ] where the limit @xmath75 is meant to be taken at the end of all calculations . accordingly , expression ( [ second2 ] ) will be replaced by its regularized version , @xmath76 \left [ \varepsilon _ \mu { } ^{\rho \tau } \partial _ \tau \delta _ \varepsilon ( z_1-z_2)\right ] \left [ \varepsilon ^{\beta \sigma } { } _ \nu \partial _ \sigma \delta ^3(z_1-z_2)\right ] \ , . \label{second4}\end{aligned}\ ] ] whatever the order of integration , we get , before taking the limit , an expression containing @xmath59 , which leads to a null result . the second term of ( [ int2 ] ) follows analogously , so that the two - loop diagram of fig.2 does not contribute to the correlator ( [ p - c ] ) . concerning the higher - order contributions in the perturbation theory , the results are of a similar nature . the topologically distinct diagrams contributing to the 3- and 4-loop are given in fig.3 and in figs.4 and 5 . it is sufficient to present here just one typical term of each order . a notational simplification is convenient . we define the transverse derivative operator @xmath77 for instance , a typical contraction from fig.3 is proportional to @xmath78 \left [ \tilde{\partial}_{\beta \delta } \delta ^3(z_1-z_3)\right ] \left [ \tilde{\partial}^{\beta \lambda } \delta ^3(z_1-z_3)\right ] \nonumber \\ & & \times \left [ \tilde{\partial}_{\rho \lambda } \delta ^3(z_2-z_3)\right ] \left [ \tilde{\partial}^{\rho \delta } \delta ^3(z_2-z_3)\right ] \nonumber \\ & = & \oint_{\gamma _ 1}dx^\mu \oint_{\gamma _ 2}dy^\nu \int d^3z_3\left [ \tilde{\partial}_{\mu \nu } \delta ^3(x - y)\right ] \nonumber \\ & & \times \left [ \tilde{\partial}_{\beta \delta } \delta _ \varepsilon ( x - z_3)\right ] \left [ \tilde{\partial}^{\beta \lambda } \delta ^3(x - z_3)\right ] \left [ \tilde{\partial}_{\rho \lambda } \delta _ \varepsilon ( y - z_3)\right ] \left [ \tilde{\partial}^{\rho \delta } \delta ^3(y - z_3)\right ] , \nonumber \\ & & \label{third}\end{aligned}\ ] ] while the diagram of fig.4 gives @xmath79 \left [ \tilde{\partial}^{\beta \lambda } \delta ^3(z_1-z_3)\right ] \left [ \tilde{\partial}_{\beta \sigma } \delta ^3(z_1-z_4)\right ] \nonumber \\ & & \times \left [ \tilde{\partial}^{\rho \sigma } \delta ^3(z_2-z_4)\right ] \left [ \tilde{\partial}_{\rho \lambda } \delta ^3(z_2-z_3)\right ] \left [ \tilde{\partial}^{\varphi \omega } \delta ^3(z_3-z_4)\right ] \nonumber \\ & & \times \left [ \tilde{\partial}_{\varphi \omega } \delta ^3(z_3-z_4)\right ] \nonumber \\ & = & \oint_{\gamma _ 1}dx^\mu \oint_{\gamma _ 2}dy^\nu \int d^3z_3\,d^3z_4\,\left [ \tilde{\partial}_{\mu \nu } \delta ^3(x - y)\right ] \nonumber \\ & & \times \left [ \tilde{\partial}^{\beta \lambda } \delta ^3(x - z_3)\right ] \left [ \tilde{\partial}_{\beta \sigma } \delta ^3(x - z_4)\right ] \left [ \tilde{\partial}^{\rho \sigma } \delta ^3(y - z_4)\right ] \nonumber \\ & & \times \left [ \tilde{\partial}_{\rho \lambda } \delta ^3(y - z_3)\right ] \left [ \tilde{\partial}^{\varphi \omega } \delta _ \varepsilon ( z_3-z_4)\right ] \left [ \tilde{\partial}_{\varphi \omega } \delta^3(z_3-z_4)\right ] . \label{fourth}\end{aligned}\ ] ] in order to obtain the above expressions we have followed the same prescription established before for regularizing the delta functions with identical arguments . notice also that we have integrated out first the two @xmath80-functions whose arguments depend on the points @xmath60 and @xmath61 of the two curves . all terms in all possible diagrams may then be seen to be proportional to @xmath59 ( or its derivatives ) . one may easily convince oneself that this mechanism also applies to any order in perturbation theory . as we always have @xmath81 , these diagrams all amount to a null correction to the basic diagram , so that the correlation function ( [ p - c ] ) for two closed smooth nonintersecting curves @xmath82 gives their linking number to all orders : @xmath83 we have been able to show , in the present article , that the correlation function ( [ p - c ] ) is unaffected by the radiative corrections , provided @xmath82 are two nonintersecting closed curves . although we have given explicit expressions for the @xmath84 perturbation , the same result may be achieved for any local interaction term of the type @xmath85 . we may interpret this result as a kind of nonrenormalization property of the linking number , reflecting its stability with respect to any local gauge invariant perturbation of the starting chern - simons action . the conselho nacional de pesquisa e desenvolvimento ( cnpq / brazil ) , the fundao de amparo pesquisa do estado do rio de janeiro ( faperj ) and the sr2-uerj are gratefully acknowledged for financial support . e. c. marino , phys . b * * 263 * * ( 1991 ) 63 * * ; * * + e. fradkin and f.a . schaposnik , phys . b * * 338 * * ( 1994 ) 253 * * ; * * + f. a. schaposnik , phys . lett . b * * 356 * * ( 1995 ) 39 * * ; * * + c. p. burgess , c. a. ltken and f. quevedo , phys . lett . b * * 336 * * ( 1994 ) 18 * * ; * * + d. g. barci , c.d . fosco and l. e. oxman , phys . b * * 375 * * ( 1996 ) 267 * * ; * * + r. banerjee and e.c . marino , nucl . b * * 507 * * ( 1997 ) 501**. * * t.h . hansson , a. karlhede and m. rocek , phys . lett . b * 225 * ( 1989 ) 92**. * * + n. brali and l. vergara , _ fuzzy statistics in covariant quantum field theory , _ int . europhys . conf . on high energy physics , marseille , france , 1993

the abelian chern simons theory is perturbed by introducing local gauge - invariant interaction terms depending on the curvature . the computation of the correlation function @xmath0 for two smooth closed nonintersecting curves @xmath1 , @xmath2 is reported up to four loops and is shown to be unaffected by radiative corrections . this result ensures the stability of the linking number of @xmath1 and @xmath2 with respect to the local perturbations which may be added to the chern simons action .

You are an expert at summarizing long articles. Proceed to summarize the following text: the long - time dynamics of biological evolution have recently attracted considerable interest among statistical physicists @xcite , who find in this field new and challenging interacting nonequilibrium systems . an example is the bak - sneppen model @xcite , in which interacting species are the basic units , and less fit " species change by mutations " that trigger avalanches that may lead to a self - organized critical state . however , in reality both mutations and natural selection act on _ individual organisms _ , and it is desirable to develop and study models in which this is the case . one such model was recently introduced by hall , christensen , and coworkers @xcite . to enable very long monte carlo ( mc ) simulations of the evolutionary behavior , we have developed a simplified version of this model , for which we here present preliminary results . the model consists of a population of individuals with a haploid genome of @xmath1 binary genes @xcite , so that the total number of potential genomes is @xmath2 . the short genomes we have been able to study numerically ( here , @xmath3 ) should be seen as coarse - grained representations of the full genome . we thus consider each different bit string as a separate species " in the rather loose sense that this term is used about haploid organisms . in our simplified model the population evolves asexually in discrete , nonoverlapping generations , and the population of species @xmath4 in generation @xmath5 is @xmath6 . the total population is @xmath7 . in each generation , the probability that an individual of species @xmath4 has @xmath8 offspring before it dies is @xmath9 , while it dies without offspring with probability @xmath10 . the reproduction probability @xmath11 is given by @xmath12 } \;. \label{eq : p}\ ] ] the verhulst factor @xmath13 @xcite , which prevents @xmath14 from diverging , represents an environmental `` carrying capacity '' due to limited shared resources . the time - independent interaction matrix @xmath15 expresses pair interactions between different species such that the element @xmath16 gives the effect of the population density of species @xmath17 on species @xmath4 . elements @xmath16 and @xmath18 both positive represent symbiosis or mutualism , @xmath16 and @xmath18 both negative represent competition , while @xmath16 and @xmath18 of opposite signs represent predator - prey relationships . to concentrate on the effects of interspecies interactions , we follow @xcite in taking @xmath19 . as in @xcite , the offdiagonal elements of @xmath16 are randomly and uniformly distributed on @xmath20 $ ] . in each generation , the genomes of the individual offspring organisms undergo mutation with probability @xmath21 per gene and individual . mc simulations were performed with the following parameters : mutation rate @xmath22 per individual , carrying capacity @xmath23 , fecundity @xmath24 , and genome length @xmath3 . for a system with @xmath25 or only a single species and @xmath26 , the steady - state total population is found by linear stability analysis @xcite to be @xmath27 . in this regime both the number of populated species and the total population @xmath28 are smaller than the number of possible species , @xmath29 . this appears biologically reasonable in view of the enormous number of different possible genomes in nature . an important quantity is the diversity of the population , which is defined as the number of species with significant populations . operationally we define it as @xmath30 $ ] , where @xmath31 is the information - theoretical entropy ( known in ecology as the shannon - weaver index @xcite ) , @xmath32 \ln \left [ { n_i(t)}/{n_{\rm tot}(t ) } \right ] $ ] . results for a run of @xmath33 generations are shown in fig . [ fig : fig1 ] . in fig . [ fig : fig1](*a * ) are shown time series of @xmath34 and @xmath28 . we see relatively quiet periods ( quasi - steady states , qss ) punctuated by periods of high activity . during the active periods the diversity fluctuates wildly , while the total population falls below its typical qss value . a corresponding picture of the species index ( the decimal representation of the binary genome ) is shown in fig . [ fig : fig1](*b * ) , with grayscale indicating @xmath6 . comparison of the two parts of fig . [ fig : fig1 ] show that the qss correspond to periods during which the population is dominated by a relatively small number of species , while the active periods correspond to transitions during which the system is searching for " a new qss . closer inspection of fig . [ fig : fig1 ] suggests that there are shorter qss within some of the periods of high activity . this led us to consider the power - spectral densities ( psd ) of the diversity and total population , measured in very long simulations of @xmath35 generations . the psd of the diversity is shown in fig . [ fig : fig2 ] and indicates that the model exhibits flicker noise with a spectrum near @xmath0 @xcite over at least four to five decades in frequency . it has been much discussed in evolutionary biology whether species evolve gradually or in a succession of qss , punctuated by periods of rapid change . the latter mode has been termed punctuated equilibria " by gould and eldredge @xcite . there is also some indication that flicker noise is found in the fossil record of extinctions , but due to the sparseness of the fossil evidence this is a contested issue @xcite . the model discussed here can at best be applied to the evolution of asexual , haploid organisms such as bacteria , and one should also note that no specific , biologically relevant information has been included in the interaction matrix . nevertheless , we find it encouraging that such a simple model of macroevolution with individual - based births , deaths , and mutations can produce punctuated equilibria and flicker noise reminiscent of current theories of biological macroevolution . we thank b. schmittmann and u. tuber for useful discussions , and p.a.r . thanks the department of physics , virginia polytechnic institute and state university , for its hospitality . this research was supported by u.s . national science foundation grant nos . dmr-9981815 , dmr-0088451 , dmr-0120310 , and dmr-0240078 , and by florida state university through the school of computational science and information technology and the center for materials research and technology . generations with the parameters given in the text . ( * a * ) time series showing the diversity , @xmath34 ( _ black _ ) , and the normalized total population , @xmath36 $ ] ( _ red _ ) . ( * b * ) species index @xmath4 vs time . the symbols indicate @xmath37 ( _ black _ ) , @xmath38 $ ] ( _ blue _ ) , @xmath39 $ ] ( _ red _ ) , @xmath40 $ ] ( _ green _ ) , and @xmath41 ( _ yellow _ ) . , title="fig : " ] generations with the parameters given in the text . ( * a * ) time series showing the diversity , @xmath34 ( _ black _ ) , and the normalized total population , @xmath36 $ ] ( _ red _ ) . ( * b * ) species index @xmath4 vs time . the symbols indicate @xmath37 ( _ black _ ) , @xmath38 $ ] ( _ blue _ ) , @xmath39 $ ] ( _ red _ ) , @xmath40 $ ] ( _ green _ ) , and @xmath41 ( _ yellow _ ) . , title="fig : " ] generations each . the model parameters are those given in the text and used in fig . [ fig : fig1 ] . the @xmath0 like spectrum is indicative of very long - time correlations and a wide distribution of qss lifetimes . ]

we present long monte carlo simulations of a simple model of biological macroevolution in which births , deaths , and mutational changes in the genome take place at the level of individual organisms . the model displays punctuated equilibria and flicker noise with a @xmath0-like power spectrum , consistent with some current theories of evolutionary dynamics .

You are an expert at summarizing long articles. Proceed to summarize the following text: accretion and clustering are the most fundamental aspects of ob star formation ( see garay & lizano 1999 , mckee & ostriker 2007 and zinnecker & yorke 2007 for reviews ) . ob stars are observed to form in very special regions of molecular clouds massive molecular clumps , typically of parsec scale in size . large mass and high density are needed for the accretion flow to feed the forming massive stars as well as the associated stellar cluster at a high enough rate . however , how the massive clumps form , whether the internal conditions of the massive clumps facilitate the accretion process , and how the massive clumps fragment into clusters , are still open questions . in the present work , we improve our understanding of these issues observationally , using the submillimeter array ( sma ; ho , moran , & lo 2004 ) and the iram 30 m telescope . we focus on the well studied uc hii region g10.6 - 0.4 , for which a wealth of complementary data are available to help address these questions . g10.6 - 0.4 is a well studied ob cluster forming region at a 6 kpc distance ( caswell et al . 1975 ; downes et al . 1980 ) . in the central @xmath21 pc region , high bolometric luminosity ( 9.2@xmath310@xmath4 l@xmath1 ) and bright free - free continuum emission ( @xmath52.6 jy within a 0.05 pc radius , at 1.3 cm band ) were detected ( ho & haschick 1981 ; sollins et al . 2005 ; sollins & ho 2005 ) , suggesting that a cluster of o - type stars has formed ( o6.5b0 ; ho and haschick 1981 ) . observations of the 20 cm continuum emission using the nrao very large array ( vla ) unveil a @xmath65 pc scale ionized bubble to the north of the brightest ob cluster , which suggests that the evolution of the cloud can be affected by the impact of the hii region projected close to g10.60.4 ( ho , klein & haschick 1986 ) . interferometric observations with high resolutions detected groups of water and oh masers ( genzel & downes 1977 ; ho & haschick 1981 ; ho et al . 1983 ; hofner & churchwell 1996 ; fish et al . 2005 ) , and multiple high velocity @xmath7co outflows ( liu , ho & zhang 2010 ) , which indicates on going and in situ star formation . observations of various molecular transitions ( ho & haschick 1986 ; keto , ho , & haschick 1987 ; keto , ho , & haschick 1988 ; guilloteau et al . 1988 ; omodaka et al . 1992 ; ho , terebey , & turner 1994 ; klaassen et al . ( 2009 ) ; liu et al . 2010 ; liu , zhang & ho 2011 ; beltrn et al . 2011 ) suggest that the general motion of the molecular gas at the 1 pc scale is dominated by gravity and shows rotation along a flattened geometry . the overall geometry and the excitations of the molecular gas in the central @xmath61 pc region resemble a scaled up low mass star forming core ( liu , zhang , & ho 2011 ) , implying that the global contraction is efficient . a similar geometry in the central 1 pc scale is also resolved in the luminous ( l@xmath66.6@xmath810@xmath4 l@xmath1 ) massive cluster forming region g20.08 - 0.14 n ( galvn - madrid et al . 2009 ) . the high resolution spectral line observations further show the clumpiness and the spatial asymmetry of the mass and the velocity field at a scale of 0.5 pc ( liu et al . 2010 ; liu , zhang & ho 2011 ) . this asymmetric flow appears to have a spin up rotational motion , and might continue to the highly clumpy 0.1 pc scale , flattened , hot toroid , which was first resolved by the nh@xmath9 optical depth studies . the hot toroid has a high temperature and should immediately surround the central 200 m@xmath1 ob cluster ( sollins & ho 2005 ; liu et al . 2010 ; beltrn et al . 2011 ) , suggesting that massive stars may accrete from the very clumpy and geometrically thick molecular materials . observations of the radio recombination lines suggest that part of the material that is ionized by the embedded ob stars can continue across the hii boundary , to feed the embedded ob stars ( keto 2002 ; keto & wood 2006 ) . despite the rich literatures in the studies of the kinematics in the inner 1 pc region , how the parsec scale massive molecular clump connects to the more extended structure , how the massive clumps form , and how the detailed morphology and dynamics lead to the fragmentation and the formation of a cluster , remain unanswered questions . to improve the understanding in these aspects , we observed the 1.2 mm continuum emission in a 10 pc scale area using the iram 30 m telescope . at such a large scale , the 1.2 mm continuum emission is dominated by the thermal dust emission . the large dish of the iram 30 m telescope allows us to resolve the dust emission , and simultaneously provides the short spacing information , to complement the sma data . this yields high resolution and high dynamic range images for our morphological studies . in addition , we observed the dust polarization properties at the 0.87 mm wavelength using the sma , to gauge the role of the magnetic field in the formation of the massive molecular clump . for a few 1.2 mm clumps resolved in the iram 30 m observations , we followed up with the sma at 0.87 mm , in order to resolve their internal structures and subsequent fragmentation . we also compare the large scale 1.2 mm continuum emission with the spitzer mipsgal 24 @xmath0 m map and the glimpse 8 @xmath0 m map . the diffuse 24 @xmath0 m and 8 @xmath0 m emissions mainly trace the ionized gas and the photon dominant regions ( pdr ) at the boundary of the neutral material . these comparisons help to define the locations and the effects of both the embedded and the external ionizing sources . we retrieved the archived vla 20 cm continuum data and the 3.6 cm and 1.3 cm continuum data to trace the diffuse and the confined ionized gas in the observed area . the observations and data reductions are introduced in section [ chap_obs ] . the observational results are presented in section [ chap_result ] . the physical implications of the results are discussed in section [ chap_discussion ] . a brief summary and our plan of the follow up researches are given in section [ chap_summary ] and section [ chap_future ] . we will follow up the studies of the kinematics in a future publication . we use the coordinate system of epoch j2000 throughout the present paper . [ [ iram-30m - telescope ] ] iram 30 m telescope + + + + + + + + + + + + + + + + + + we perform a 7@xmath10@xmath37@xmath10 otf scanning observation using the iram 30 m telescope mambo-2 array receiver , on 2011 january 20 . the observation is carried out at a frequency of 250 ghz ( 1.2 mm ) , with a primary beam size of 10.5@xmath11 . we made three maps ( map1 , 2 , 3 hereafter ) , with one hour on source time for each , centered at r.a.=18@xmath1210@xmath1330@xmath14.14 , decl.=-19@xmath1555@xmath1029@xmath11.70 , using the wobbler switching mode . the wobbler throw is 120@xmath11 ; the scanning directions of these three maps have different position angles relative to the r.a . axis , due to the rotation of the sky . basic data reduction was carried out using the ` mopsic ` software . the map1 is observed in a low sky noise condition , while map2 and map3 showed higher noise and stripes in the image before the sky noise subtraction was performed . without performing the sky noise subtractions50 % . we use the sky noise subtracted maps as a reference for the distribution and the geometry of the faint structures . however , we do not present the sky noise subtracted map to avoid the confusions in the brightness distributions . ] , we consistently obtain the peak flux of 9.5 jybeam@xmath16 for all three maps . this value is comparable with the previous sma measurement of @xmath610 jy at 1.3 mm , in the central 10@xmath11 region ( liu et al . 2010 ) . a few percents of errors in estimating the flux distribution in the brightest central region will couple to the uncertainties in the modeling of the local zero flux level . this can propagate significant ( @xmath6100 mjybeam@xmath16 ) repetitive artifacts in the scanning direction.we formed the final image based on map1 . we used the measurements in map2 and map3 to amend the pixels in map1 which are corrupted by the repetitive artifacts . however , because of the sky subtraction effects noted above , the replaced pixels may still bias the local brightness distribution in certain areas . these effects , however , are limited to a level that does not affect the discussions in the present paper . the rms noise level of map1 ( without sky noise subtraction ) is about 7 mjybeam@xmath16 . [ [ sma ] ] sma + + + we performed the 1.3 mm band observations using the 6 meter dishes of the sma in the subcompact configuration , the compact configuration and the very extended configuration on 2009 february 09 , 2009 june 10 and 2009 july 12 , respectively . in the compact configuration and the very extended configuration observations , the observing frequencies were centered on 231 ghz ( 1.30 mm ) in the lower sideband , and centered on 241 ghz ( 1.24 mm ) in the upper sideband , respectively ; in the subcompact configuration observation , the observing frequencies of the two sidebands were centered on 221 ghz ( 1.36 mm ) and 231 ghz ( 1.30 mm ) . these observations were carried out with 8 antennas . the pointing center of these observations is r.a.=18@xmath1210@xmath1328@xmath14.683 , decl.=-19@xmath1555@xmath1049@xmath11.07 . the primary beam size of these observations is 55@xmath11 . the basic calibrations were carried out using the ` mir idl ` and the ` miriad ` software package ; the self - calibrations of these data were carried out using the ` aips ` package . we constructed the continuum band visibility data at 1.3 mm from the line free channels . the combined sma data sets cover a uv sampling range of 5410 k@xmath17 . we detect @xmath614 jy of stokes i emission in the field of view of the sma observations . the continuum image of this combined sma data set ( @xmath18@xmath3@xmath19 = 0@xmath11.79@xmath30@xmath1158 ) has been published in liu , ho & zhang ( 2010 ) . we performed the follow up 0.87 mm band observations using the sma on 2011 march 15 in the subcompact array configuration with 7 antennas , and on 2011 march 25 in the compact array configuration with 8 antennas . the observing frequencies of the two sidebands were centered on 336 ghz ( 0.89 mm ) and 348 ghz ( 0.86 mm ) . the pointing center of these observations are r.a.=18@xmath1210@xmath1341@xmath14.10 , decl.=-19@xmath1557@xmath1041@xmath11.30 ( p1 region hereafter ) , and r.a.=18@xmath1210@xmath1336@xmath14.80 , decl.=-19@xmath1557@xmath1003@xmath11.20 ( p2 region hereafter ) . the observations were carried out in the snapshot mode , with 20 minutes on source integration per pointing in the subcompact configuration , and 16 minutes on source integration per pointing in the compact configuration ( after flagging the bad data ) . the theoretical rms noise level after combining the subcompact array and the compact array data is about 3.4 mjybeam@xmath16 . we provide 2 versions of the continuum map . in one version , we limit the uv sampling range to 845 k@xmath17 , to optimize the sensitivity and the shape of the synthesized beam ( 4@xmath11.8@xmath33@xmath11.7 ) . without limiting the uv sampling range ( 8 - 80 k@xmath17 ) , we obtain a higher resolution 0.87 mm continuum image with 2@xmath11.9@xmath31@xmath11.9 synthesized beam , to resolve the detailed structures in the brighter p1 region . [ [ iram-30m - telescope - sma ] ] iram 30 m telescope + sma + + + + + + + + + + + + + + + + + + + + + + + + we approximate the sma observations at 1.2 mm using the sma data taken at 1.24 mm and at 1.30 mm . the iram 30 m observations detect @xmath630 jy of stokes i emission within the sma primary beam . we convert the iram 30 m image into a uv visibility data set by using the ` miriad ` tasks ` demos ` , ` uvrandom ` and ` uvmodel ` . we limit the uv sampling range of the iram 30 m visibility to be 04 k@xmath17 , to fill in the missing short spacing information in the sma data . we note that in such a small uv sampling range , the visibility model of the iram data is minimally affected by the attenuation of the single dish primary beam . we have inspected the data and confirm the consistent absolute flux levels of the sma data and the visibility model of the iram 30 m data . we limit the uv sampling range of the sma data to be within 0220 k@xmath17 to optimize the sensitivity to the extended structures and the angular resolution . by applying ` invert ` and ( non box ) ` clean ` to the sma and the iram 30 m visibility data sets with 5000 iterations in ` miriad ` , we obtain a synthesized beam of 3@xmath11.4@xmath33@xmath11.1 ( p.a.=15@xmath15 ) , and an rms noise level of 5 mjybeam@xmath16 . the recovered flux by ` clean ` is 29.93 jy . the final combined 1.2 mm image is formed by the ` miriad ` task ` restor ` . we performed observations in the 0.87 mm band in polarization mode using the sma , in the compact north array on 2010 july 31 , and in the extended array on 2010 september 12 . the observing frequencies of the two sidebands were centered on 336 ghz ( 0.89 mm ) and 348 ghz ( 0.86 mm ) . the pointing center of both observations is r.a.=18@xmath1210@xmath1328@xmath14.683 , decl.=-19@xmath1555@xmath1049@xmath11.07 . the primary beam size of these observations is 40@xmath11 . these observations cover the uv sampling ranges of 10135 k@xmath17 and 20220 k@xmath17 , respectively . the recovered stokes i fluxes are 15 jy and 10 jy , respectively . calibrations and imaging were carried out using the ` miriad ` software package . we averaged the line free channels in the individual observations to generate the continuum channels , and jointly imaged the upper 4 ghz and the lower 4 ghz sideband . for the compact north array observations , we obtain an rms noise level of 2.7 mjybeam@xmath16 , for the stokes q and the stokes u images . for the extended array observations , we obtain an rms noise level of 3.0 mjybeam@xmath16 , for the stokes q and the stokes u images . jointly imaging the data from these observations yield a synthesized beam of 1@xmath11.2@xmath31@xmath11.0 , and the rms noise level of 2.3 mjybeam@xmath16 for the stokes q and the stokes u images . we retrieved the archived 3.6 cm continuum emission toward g10.6 - 0.4 in the vla a configuration including the vlba pie town antenna on 2005 january 2 ; and we observed the 3.6 cm continuum emission in the nrao ( expanded ) very large array ( vla / evla ) c configuration on 2009 july 27 . the pointing center of these observations is r.a.=18@xmath1210@xmath1328@xmath14.683 , decl.=-19@xmath1555@xmath1049@xmath11.07 . the primary beam size of these observations is 330@xmath11 . the basic calibrations , self - calibration , and imaging of these data were carried out using the ` aips ` package . we combined the a array+pie town visibility data with the c - array visibility data , and applied a gaussian taper in the uv domain with a fwhm of 170 k@xmath17 , yielding a 2@xmath11.3@xmath31@xmath11.5 synthesized beam with a position angle of -10.9@xmath15 . the observed rms noise of the 3.6 cm continuum image is about 1 mjybeam@xmath16 ( @xmath65 k in terms of brightness temperature ) . we note that the 3.6 cm continuum emission traces free free emission based on spectral index measurements . we retrieved the archived vla 20 cm continuum data , taken on 1984 june 13 . the primary beam size of this observation is 1833@xmath11 ( @xmath630.5@xmath10 ) . the basic calibrations , self - calibration , and imaging of these data were carried out using the ` aips ` package , yielding an rms noise level of 1 mjybeam@xmath16 , with a 21@xmath11@xmath314@xmath11 synthesized beam ( see also ho , klein & haschick 1986 ) . we retrieved the archived vla 1.3 cm continuum data , taken on 2002 february 1 . the primary beam size of this observation is 2@xmath10 . the basic calibrations , self - calibration , and imaging of these data were carried out using the ` aips ` package , yielding an rms noise level of 0.13 mjybeam@xmath16 , with a 0@xmath11.11@xmath30@xmath11.07 synthesized beam ( see also sollins et al . 2005 ; sollins & ho 2005 ) . we observed the cs ( 1 - 0 ) transition using the nrao ( expanded ) very large array ( vla / evla ) in the dnc configuration , on 2009 september 27 . these observations had a continuous lst duration of 9 hours , with 20 available antennas after flagging . the pointing center is r.a.=18@xmath1210@xmath1328@xmath14.683 , decl.=-19@xmath1555@xmath1049@xmath11.07 . the primary beam size of these observations is 55@xmath11 . the observations cover the uv sampling range of 4245 k@xmath17 . continuum emission are averaged from the line - free channels and then subtracted from the line data . calibrations , self calibrations , and imaging were carried out using the ` aips ` package , yielding a synthesized beam of 1@xmath11.5@xmath31@xmath11.1 , and an rms noise level of 22 mjybeam@xmath16 . this cs dataset was previously published in liu , zhang & ho ( 2011 ) . the observations discussed in this paper cover a tremendous amount of molecular structures , which have a broad range of linear size scales , and are embedded in different ambient environments ( e.g. different molecular volume density , geometry , morphology , gravitational potential , etc ) . to clarify the terminology ( e.g. filaments , clumps , cores , protrusive features / structures , envelope , etc ) , we present a schematic model in figure 1 , which also forecasts parts of the observational results . this model presents a filamentary 10 pc scale , filamentary molecular cloud , which can be embedded in a giant molecular cloud . the local concentrations of mass in the filaments fragment into smaller and denser substructures on a shorter dynamical timescale than the timescale of the global contraction . depending on the physical environments in the parent structures , the processes of fragmentation may also be hierarchical . we note that the scale bars in the schematic model reflect the physical size scales of the structures that we will be referring to . part of the schematic model , the massive molecular envelope region , has been introduced in a previous publication ( liu , zhang & ho 2011 ) . [ chap_result ] figure [ fig_rgb ] shows the 1.2 mm continuum map from the iram 30 m observations . in this figure , we label the significant 24 @xmath0 m emission regions ( a o ) as the suspected ob star forming regions . most of these ob star forming regions already show 20 cm or 6 cm free free continuum emissions ( ho , klein & haschick 1986 ) . the resolved uc hii regions in our 3.6 cm observations are also labeled ( a e ; see also figure [ fig_2panel ] ) . the uc hii region a located at the geometric center , which contains the ob cluster with 200 m@xmath1 , is resolved in the highest resolution vla observations at 1.3 cm ( sollins et al . 2005 ; sollins & ho 2005 ; see also section [ chap_context ] ) . from figure [ fig_rgb ] we see at least 5 extended ( @xmath5 3 pc in the projected scale ) massive filaments in the 1.2 mm continuum map . some of the extended structures may still be blended , and remain to be resolved with higher angular resolution . the most significant feature in figure [ fig_rgb ] is the clear separation between the hii region in the northeast , and the neutral material in the southwest . a @xmath2010 pc scale ionized arc traced by the diffuse 24 @xmath0 m emission is immediately followed by a slightly more extended 8 @xmath0 m bright pdr shell southwest of it . most of the dense neutral material are distributed further southwest of the pdr shell , and is traced by the 1.2 mm continuum emission . the neutral material is filamentary , and is extremely clumpy , with massive stars forming in localized dense structures . multiple 24 @xmath0 m bright hii regions with significant pdr shells are seen over the @xmath2010 pc scale region ( e.g. massive cluster a , c , d , e , f , j , m , n ) . uv photons from the massive cluster o appear to photoionize a filament in the south , and generate a bow shock shaped hii region ( see also section [ chap_bow ] ) . the extended filament in the northwest seems to continue to the massive clusters e , f , and g. how this filament is dynamically associated with the densest structure in the center of this map is uncertain . the rest of the extended filaments appear to connect to the geometrical center , an extremely bright 1.2 mm source with a projected scale of @xmath623 pc . a high concentration of 8 @xmath0 m point sources is resolved in this central region , indicating the active fragmentation and the formation of a stellar cluster . this region has high extinction , and there might be more embedded massive stars which are not visible in the optical and near infrared observations . clusters of water masers ( hofner & churchwell 1996 ) and the high velocity @xmath7co ( 21 ) outflows ( liu , ho & zhang 2010 ) were also reported in this region . we observe the 1.3 mm continuum emission using the sma , and observe the 3.6 cm free free continuum emission and the cs ( 10 ) transitions using the vla / evla in the center of this dense structure ( section [ chap_context ] ) . figure [ fig_contour ] shows the iram 30 m 1.2 mm continuum map with much more detailed contours , overlaid with the combined iram 30m+sma 1.2 mm continuum map in the central region . assuming @xmath21=2 , the gas to dust ratio of 100 ( lis et al . 1998 ) , an average temperature of 3050k , and subtracting the free free continuum flux of 4 jy ( liu et al . 2010 ) , the detected 1.2 mm flux within the central 55@xmath11 sma primary beam corresponds to 1.84@xmath810@xmath22 m@xmath1 of molecular mass . we note that from the cs observations , omodaka et al . ( 1992 ) estimated the mass to be 4@xmath810@xmath22 m@xmath1 in the central 30@xmath11 region . we show the combined iram 30m+sma 1.2 mm continuum map , and the velocity integrated cs ( 10 ) maps in figure [ fig_2panel ] , to demonstrate the structures in the inner region with higher angular resolutions . while figure [ fig_contour ] shows multiple filaments on the scale of a few parsecs , in figure [ fig_2panel ] , we see multiple protrusions in the inner parsec scale region . these protrusions seem to have no preferred alignment , exhibiting instead a radial configuration connecting to a common center . we visually identify the protrusions in the combined 1.2 mm map ( figure [ fig_2panel ] ) . the identified structures are labeled by fl ne , e , se , s , sw2 , sw1 , nw2 , nw1 , respectively . the elongated structure fl nw1 , fl e and fl sw1 are consistent with the more extended molecular filaments , which were previously detected in the nh@xmath9 ( ho & haschick 1986 ) and the cs ( 10 ) observations ( omodaka et al . we note that the velocity gradient of fl e was marginally resolved by ho & haschick ( 1986 ) , and was interpreted as the envelope rotational motion . the fl s might have been detected in the previous c@xmath23o observation ( ho , terebey , & turner 1994 ) . however , we emphasize that the 1.2 mm continuum emission presented in this paper traces the thermal dust emission , without being biased by the excitation conditions and the molecular abundances . some of the protrusions continue to a large scale , which can only be followed in the iram 30 m 1.2 mm continuum map , for example , fl ne , fl . there might be some foreshortened structures blended around the middle of the map ( figure [ fig_contour ] ) , which are only marginally resolved in the iram 30 m continuum map . there is also an apparent filament or protrusion , which continues from the central parsec region to a large scale , and is associated with the massive cluster c. we suspect another filamentary structure that continues to the large scale , which is related to the formation of the massive cluster m and n. these suggestions can be tested with interferometric mosaic observations . comparing the combined 1.2 mm continuum map with the velocity integrated cs ( 10 ) map ( figure [ fig_2panel ] ) suggests that some protrusions might continue directly into the central 0.05 pc , where the fast spinning hot toroid ( sollins & ho 2005 ; liu et al . 2010 ; beltrn 2011 ) is immediately around the most massive ( @xmath6200 m@xmath1 ) ob cluster . we note the filamentary and clumpy nature of the massive molecular envelope traced by the cs ( 10 ) map has been highlighted in liu , zhang & ho ( 2011 ) . as an example of the continuation of the filaments , we zoom in on the fl se region and show the combined 1.2 mm continuum map and cs ( 10 ) map in figure [ fig_filament ] . in the figure , we see that both the 1.2 mm emission and cs ( 10 ) emission show a number of elongated structures . the most prominent one follow the suggested plane of rotation ( keto , ho & haschick 1988 ; sollins & ho 2005 ; liu et al . figure [ fig_freefree ] demonstrates the relation between these elongated cs emission and the 1.3 cm free free continuum emission from the centrally embedded uc hii region . we note that the arc shaped features in the east of the 1.3 cm continuum map are explained as signatures of the externally ionized molecular clumps , which are approaching the embedded ob cluster ( sollins & ho 2005 ) . this is the first time we resolve the geometry of the massive accretion flows in the distant ( few kpc ) ob cluster forming regions , on such a large scale and with such a high spatial dynamic range . however , the resolved structures mentioned here are all projected onto the two dimensional images . high resolution spectral line observations ( liu et al . 2010 ; liu , zhang & ho 2011 ) can be used to study the three dimensional geometrical context and the overall dynamics . this will be the subject in our next paper . the parsec scale filaments have complicated internal structures . as an example , figure [ fig_flrg ] shows the iram 30 m 1.2 mm continuum map in fl e , overlaid with the saturated 24 @xmath0 m and 8 @xmath0 m ` spitzer ` images . from this figure , we see that both the 1.2 mm continuum emission and the near infrared opacity suggest a structure continuing from the southwest , which may still contain substructures . more detailed structures are suggested by the near infrared opacity , although it is hard to distinguish if those structures are physically associated or a mere projection along the line of sight . in such an evolved region , the distribution of the foreground and background extended infrared emissions severely confuse the structure identification . the local structures can best be recognized from the millimeter and submillimeter thermal dust emission . however , the iram 1.2 mm continuum map has much poorer angular resolution as compared to the 24 @xmath0 m and 8 @xmath0 m ` spitzer ` images . here , we define the millimeter clumps as compact structures identified in the iram 1.2 mm map , which have sizescales of @xmath60.5 pc . we define the submillimeter cores as compact structures identified in the sma 0.87 mm stokes i image , which have sizescales @xmath240.2 pc . we identify the millimeter clumps as well as the submillimeter cores in regions where the fluxes are predominantly contributed by fl e and fl s , since structures in other regions , particularly in the east appear to be blended due to the still insufficient angular resolution . in addition , the millimeter and the submillimeter continuum emissions in fl e and fl s are minimally contaminated by the free free continuum emission , and are therefore good tracers of the molecular mass ( figure [ fig_contour ] , [ fig_ionized ] ; see also ho , klein & haschick 1986 for the free free continuum emission ) . note that these observationally defined features are still limited by the insufficient angular resolution and sensitivity of our observations , with likely unresolved substructures . we perform 2dimensional gaussian fit using the ` aips ` task ` imfit ` to obtain the size . we note that the size from the fitting is limited by the ( synthesized ) beam of the observations . the gaussian fits of the fainter cores are biased by the confusion with the emission from the filaments . the profiles of the brightness distribution of the identified structures may be non gaussian due to the internal structures . hence , to obtain the total flux , we simply sum the flux within the ellipses as defined by the gaussian fits . we provide the preliminary mass estimates based on the optically thin dust emission formulation in lis et al . ( 1998 ) , assuming an average temperature of 25k , the gas to dust ratio of 100 , and @xmath21=2 . we approximate the volume of each of the clumps and cores by assuming a spherical geometry , with the radius being the average of their projected major and minor axes . the estimated flux and mass of the clumps and cores can be biased to be factors of 23 higher , due to those uncertainties in the gaussian fits . however , while estimating the density , these biases will be compensated by the similar bias in the volume . the coordinates , size , millimeter or submillimeter flux , mass , and the average density are summarized in tables [ table_mm ] and [ table_submm ] . with the similar assumptions , the overall molecular mass in fl e and fl s , without including the structures in the central 3 pc dense region , are 6@xmath810@xmath25 m@xmath1 and 4.5 @xmath810@xmath25 m@xmath1 , respectively . the identified structures are also labeled ( mm 19 , smm 16 , and smm 15 a , b , c ) in figures [ fig_contour ] , [ fig_ionized ] , and [ fig_condensations ] . the millimeter clumps in fl e have an average projected separation of 0.82 pc ; and the millimeter clumps in fl s have an average projected separation of 0.54 pc . our follow up submillimeter stokes i observations focus on the millimeter clumps mm1 , mm2 and mm4 . we resolved 6 submillimeter substructures in the lower resolution versions of the 0.87 mm stokes i image ( smm 16 , figure [ fig_ionized ] ) . smm 1 and smm 2 have a projected separation of 0.27 pc ; and smm 3 , 4 , 5 have an average projected separation of 0.15 pc . we note that the jeans length is 0.14 pc , assuming an average temperature of 25 k , and an average density of n@xmath26 = 10@xmath4 @xmath27 . without limiting the uv sampling ( section [ chap_0p87 ] ) , we made a higher angular resolution ( 2@xmath11.9@xmath31@xmath11.9 ) 0.87 mm stokes we find that the core smm1 is resolved into three independent submillimeter cores , while smm 2 , 3 , and 5 are also each resolved into two further submillimeter cores ( figure [ fig_condensations ] ) . this implies smaller actual sizes and average separations of submillimeter cores in mm1 and mm2 . we label the resolved substructures by smm 15 with suffix a , b , c. the submillimeter cores smm 1a , 1b , 1c , 2a and 2b seem to follow one elongated distribution ; the submillimeter cores smm 3a , 3b , 4 , 5a , 5b are aligned in another elongated distribution . with the still insufficient angular resolution of the sma 0.87 mm stokes i image ( section [ chap_0p87 ] ) , we may underestimate the volume density ( table [ table_submm ] ) of the submillimeter cores . the submillimeter core smm 6 has an elongated shape ( figure [ fig_ionized ] ) , which may indicate further internal structures , but is too faint to be resolved . we do not detect the stokes q and stokes u emissions at the 3@xmath28 level , in any combination of the compact north array and the extended array data . we note that our polarized intensity maps have achieved comparable resolutions and lower rms noise levels as compared with the previous sma studies of distant ob star formation regions ( g31.41 + 0.31 : girart 2009 ; g5.89 - 0.39 : tang et al . 2009a ; w51 e2/e8 cores : tang et al . our non detection suggests that g10.60.4 is less than 2.8% polarized . possible explanations for the lower level of the polarized emission and the physical implications are discussed in section [ chap_mag ] . the early vla 20 cm observations show an ionized bubble to the north of the central ob cluster . this ionized bubble can be seen in the 24 @xmath0 m emission ( figure [ fig_rgb ] ) , and also with the 8 @xmath0 m bright pdr shell external to it . the 24 @xmath0 m emission additionally shows some patchy structures on the eastern edge of the ionized bubble . those patchy structures have high opacities , and are dark in the 24 @xmath0 m and 8 @xmath0 m maps . the observed area has strong diffuse infrared emission . the structures in the background infrared brightness distributions and the foreground infrared emission can confuse the identifications of structures . we report the patchy dust emission from the same area in the 1.2 mm band . a blow up view of the iram 30 m 1.2 mm continuum map overlaid with the vla 20 cm continuum map is provided in figure [ fig_bubble ] . the distribution of the 20 cm continuum emission suggests that the 1.2 mm emission from the patchy molecular gas are not contaminated by the free free emission . this patchy molecular gas shows rich subparsec scale structures . with our current data , how these patchy structures form , and if they are interacting with the ionized bubble or the hii region , are unclear . we suspect that they are naked dense cores which lost their ambient gas through some interactions ; or they can be some foreground , isolated , subparsec scale molecular cloudlets . these have to be distinguished in future observations . from figure [ fig_rgb ] , we see the formation of the ob stars over the entire @xmath510 pc projected area . the formation of these ob stars seems to be synchronized on a time interval , which is short in the sense that it takes @xmath2910@xmath30 years to propagate the sound wave in this @xmath610 pc cloud . this propagation timescale is much longer than the collapsing timescale plus the life time of the massive stars . the alfvn waves can propagate supersonically . however , they will be restricted by the detailed magnetic field line configurations and the magnetic field strength . are there large scale physical phenomena , which are globally regulating or triggering the local structure formations ( e.g. the formation of the massive core , or the collapsing of the massive cores , etc ) ? the impulse from the hii region in the northeast might be a candidate of this global mechanism , which was suggested by ho , klein & haschick ( 1986 ) with their vla 20 cm continuum observations . this scenario has to be tested by the numerical hydrodynamical simulations . with the 1.2 mm dust continuum maps , one may argue that the filamentary geometry of the neutral material could be a projection effect . in this section , we present the case of the ionizing pattern around the parsec scale filament in the south , and the massive cluster o ( figure [ fig_rgb ] ) , to argue in favor of the elongated geometry of the parsec scale structures . while fragmentation in filamentary infrared dark clouds are ubiquitously seen , we argue that our result provides hints for their evolutionary context . we can not rule out the possibility that there are edge on sheets which in projection appear like filaments . however , it is questionable whether a @xmath205 pc scale sheet is likely at such a late evolutionary stage where clusters of smaller structures and massive stars are already present . the idea is that if there is an ob cluster located close to the parsec scale massive filaments ( with a gradient of volume density along the radius ) , its ionizing photons and stellar wind will be blocked in the direction of the dense neutral material . the uv photons emanating from the ob stars will only efficiently ionize the outer edges of the filament . if the filament presents a small solid angle to the ob cluster , most of the uv photons , the stellar wind and the ionized gas will flow around the filament , and form a * u * shaped free free continuum emission region . such a * u * shape illumination pattern is seen around the massive filament in the south ( figure [ fig_ionized ] ) . if that filament were just filamentary in projection , but is an extended smoothed sheet in reality , we do not expect to see the ionized gas penetrating to the northeast of it ; instead , the free free emission should be limited to the southwest . alternatively , the * u * shape illumination pattern can also be explained by the addition of two photoionized molecular filaments . this scenario can be examined by deeper observations of thermal dust emission with higher angular resolution . we note that the cloud with a filamentary morphology may be self shielded from an ob cluster deeply embedded in the geometrical center of the cloud . while the parsec scale filament in the south is illuminated by the massive cluster o and shows significant 8 @xmath0 m emission ( figure [ fig_rgb ] ) , the filament in the east remains dark at 8 @xmath0 m . if there are smaller and denser filaments continuing to the centrally embedded ob cluster , the small cross sections of the filamentary accretion flow is minimally affected by the radiation from the luminous ob cluster , and the pressure force of the ionized gas . therefore , it is conducive for the accretion of the massive stars . previous observations have suggested that the major , massive filaments can be formed from the merging of the smaller ones ( jimnez - serra et al . the filaments may also collide with each other , which is conducive for the formation of cores and the massive stars ( galvn - madrid et al . 2010 ) . at 10 pc scale , the filamentary morphology has a low volume filling factor , or a high porosity even after being projected onto a 2dimensional observing area . the intensity ratios of molecular gas tracers ( e.g. hcn / co ) can then be easily converted to volume filling factors ( e.g. dense gas volume filling factor ) as in extragalactic studies . from figures [ fig_contour ] , [ fig_ionized ] , and figure [ fig_condensations ] , we see concentration of dense gas in the form of massive molecular clumps and molecular cores . the volume densities of these clumps and cores ( tables [ table_mm ] , [ table_submm ] ) are high enough to excite the dense gas tracers cs , hco@xmath31 , hcn , and h@xmath32co . the parsec scale filaments , especially the two in the south and in the east , show rich marginally resolved 1.2 mm substructures , the millimeter clumps ( figure [ fig_contour ] ) . the clumps in the eastern filament show the generally stronger 1.2 mm emission than those in the southern filament , which can be due to their difference in mass , temperature , or the dust properties . this can be further examined by observing the nh@xmath9 emission with multiple transitions , and by observing the dust continuum emission in multiple frequency bands . at least three massive clumps in the eastern filament may be forming massive stars , and show bright 8 @xmath0 m point sources around the region we label cluster l ( figure [ fig_rgb ] ) . clumps in the southern and the eastern filaments seem to be regularly separated by 0.51 pc , implying a scale length for the local contractions which may be related to the length and width of the filaments . in the clumps mm1 and mm2 , our follow up sma observations have resolved much more submillimeter cores , which have an averaged projected separation comparable to the thermal jeans length ( section [ chap_mm ] , figure [ fig_condensations ] ) . compared with other high resolution interferometric observations of filamentary infrared dark clouds , it is likely that hierarchical fragmentation is common . undoubtedly , there will be more blended substructures in our iram 30 m map due to the still insufficient angular resolution . for example , in the case study of the infrared dark cloud g28.34 + 0.06 ( @xmath334.8 kpc ; wang et al . 2008 ; zhang et al . 2009 ; wang et al . 2011 ) , the 1.3 mm continuum observation of sma resolved five cores within a single local maximum in the iram 30 m 1.2 mm continuum map . we note that the submillimeter cores in the molecular clumps mm 1 and mm 2 ( figure [ fig_condensations ] ) are not randomly distributed , but are aligned linearly , forming the main axis of the filament . that the successive fragmentation at even smaller scales , continue to be aligned along the filaments , suggests a common process . the process of fragmentation into clumps can be regulated by the density , mass concentration , the geometry or the morphology of the clump , the local magnetic field structure , the local feedback and the ambient pressure , the velocity field , interactions , and so on . successive fragmentation takes place as the local conditions change . furthermore , collision or merging of filaments may also be an important process . we postpone the detailed statistical studies of these cores until we obtain follow up observations that achieve the physical resolution of @xmath340.1 pc , and achieve a much better sensitivity to recover the solar mass or smaller substructures . some of the parsec scale filaments appear to converge into the geometric center of a 23 pc scale densest structure . the abundance of protrusions in both the iram 30 m 1.2 mm continuum map , and the combined iram 30 m + sma 1.2 mm continuum map ( figure [ fig_contour ] ) support such a scenario . we note however that the massive stars are distributed over the entire 10 pc area . hence , it appears that the star formation process may be driven by local processes , and may not be related to inflow from large scales via the filaments . we provide simple arguments based on the global and the local free fall timescale , to justify that the filamentary accretion flows with a radial alignment will allow efficient fragmentation during the quick global contraction . theories on how contractions depend on morphology , geometry , and initial conditions , are described by ledoux ( 1951 ) , ostriker ( 1964 ) , larson ( 1985 ) , vishnaic ( 1994 ) , whitworth et al . ( 1994 ) , curry ( 2000 ) , myers ( 2009 ) , and myers ( 2011 ) . our estimates specifically focus on the materials in the central 2 pc region ( i.e. inside the sma primary beam shown in figure [ fig_2panel ] ) , in which we see a concentration of bright infrared point sources ( figure [ fig_rgb ] ) . we do not consider the magnetic field in our estimates , which is consistent with our observations that strong and organized magnetic fields may not be present at the sampled sizescales ( see also section [ chap_mag ] ) . with some rescaling , the same calculations can be applied to the more extended area while a lack of understanding of the large scale initial turbulence brings large uncertainties . the free fall timescale is inversely proportional to the square root of the averaged density @xmath35 . the value of @xmath35 can be estimated by @xmath36 while @xmath37 is the radius around a local center of mass , and @xmath38 is the total mass enclosed in the radius @xmath37 . for the case of g10.60.4 , in the central 2 pc region , we have resolved @xmath68 protrusions ( figure [ fig_2panel ] ) . assuming a filamentary geometry for those protrusions , of which the width is a fraction of a parsec , say 0.10.3 pc , the volume occupied by each filament can be approximately estimated as the volume of a 2 pc rod , which is @xmath39@xmath40@xmath41@xmath42 @xmath43 0.0160.14 pc@xmath25 . the volume filling factor in the inner 2 pc region can be approximately estimated by @xmath44 if we project all structures along the line of sight , we can also estimate the surface filling factor , which is @xmath550% . we note that the smaller beam filling factor of 30% reported in the previous low resolution nh@xmath9 observations ( keto , ho , & haschick ) , suggests that most of the mass are concentrated to filaments or in the central massive core . the estimated small volume filling factor implies potentially a much higher local volume density than the averaged volume density in that region . within the region enclosed by the 1 pc radius , the free fall contraction timescale of the local perturbations can be a factor of 1.95.8 times shorter than the global free fall contraction timescale . depending on the support of the rotational velocities , the timescale of the large scale contraction could be a few times longer than the global free fall contraction timescale ( ho & haschick 1986 ; keto , ho & haschick 1987 , 1988 ; keto 1990 ; liu et al . 2010 ; liu , zhang & ho 2011 ) . thus we suggest that although the system may have a quick global contraction ( see also the _ clump fed _ scenario in wang et al . 2010 ) , the existence of small scale structures implies that efficient fragmentation and subsequent star formation may have already occurred . the accretion flows are clearly highly clumpy , which have been resolved in the high resolution nh@xmath9 absorption experiment ( sollins & ho 2005 ) , and the pv diagram of @xmath45cs ( 54 ) ( liu et al . the fragmentation of the accretion flows is consistent with the observed clusters of uc hii regions , multiple water masers and high velocity @xmath7co outflow sources ( figure [ fig_2panel ] ) , many dusty cores with strong 1.3 mm emission , all external to the centrally located , compact , fast spinning , hot toroid ( liu , ho & zhang 2010 ; liu , zhang & ho 2011 ) . the earliest epoch of star formation will generate turbulence , which will help regulate the subsequent star formation efficiency and the accretion of the centrally embedded massive star ( g10.60.4 : liu , ho & zhang 2010 ; simulations : li & nakamura 2006 ; nakamura & li 2007 ; carroll et al . 2009 ; wang et al . 2010 ) . from the three color image ( figure [ fig_rgb ] ) , we see a higher concentration of the 8 @xmath0 m sources in the central parsec region than in the extended region , which may be explained by the faster fragmentation and contraction of the filamentary structures in the higher density environment . we note that the interferometric observations of the nearby massive cluster forming region , the orion kl central core , have also demonstrated the highly filamentary nature of the molecular gas , as well as the fragmentation in those filaments , in the inner @xmath61 pc region ( wiseman & ho 1996 ; wiseman & ho 1998 ) . note that on such a small scale , the filaments do not need to be primordial nor a continuation of the outer structures . the interactions between the ambient gas with the ( proto)stellar feedback can also lead to the formation of filaments in the later evolutionary stages . of course , the filaments can also be destroyed by the same feedback mechanisms ( wang et al . the process of infall and fragmentation is highly dynamical and chaotic . the _ red excess _ of the velocity field in g10.6 - 0.4 was previously reported in the nh@xmath9 absorption line experiments , which is consistent with a high velocity infall ( ho & haschick 1986 ; keto , ho & haschick 1987 , 1988 ; keto 1990 ; sollins & ho 2005 ) . from 1@xmath11 resolution emission line observations , liu et al . ( 2010 ) confirms the red excess in the central @xmath60.06 pc ( 2@xmath11 ) region , and further sees the high spatial asymmetry and the clumpiness in the mass distribution , from the position velocity ( pv ) diagram . motivated by the resolved protrusions in the present work ( figure [ fig_2panel ] ) , we propose that the spin up rotational motions of the accretion flow with a filamentary morphology may coherently explain the red excess , the spatial asymmetry and the clumpiness in those pv diagrams . the detailed modeling will be provided in our follow up kinematics studies . sma observations of luminous massive star forming regions ( g31.41 + 0.31 : girart 2009 ; g5.89 - 0.39 : tang et al . 2009a ; w51 e2/e8 cores : tang et al . 2009b ) have shown that the polarization percentage of the 0.87 mm thermal dust emission , range from a few percent ( e.g. 4% in g31.41 + 0.31 ) , to up to @xmath68% in the w51 e2/e8 regions , and up to @xmath622% in the g5.89 - 0.39 region . the polarization observations of g10.6 - 0.4 achieved an improved sensitivity over the previous sma studies of similar targets . while those previous cases all show significant detections of polarized flux , in the present work , we do not detect the polarized emission above the 3@xmath28 rms noise level . in g10.6 - 0.4 , the polarization percentage of the dust emission at 0.87 mm is constrained to be less than 2.8% . such a polarization percentage is lower than the reported polarization percentages in those previous cases . if g10.6 - 0.4 is a strongly magnetized case , the low polarized flux can be interpreted by some differences in the dust properties from the other targets , which produce less polarized flux , the relatively inefficient dust alignment in g10.60.4 , and/or the canceling of the polarized emission from structures overlapped along the line of sight . the feedback of the high velocity molecular outflows and the expansions of the hii regions may , but not necessarily disturb the magnetic field and lead to the non detection of the polarized flux . in the case of g5.89 - 0.39 , the embedded uc hii region has already undergone an expansion phase , and multiple molecular outflows have been detected , and the magnetic field structure is strongly affected ( tang et al . . the magnetic field might also be more important on a much smaller scale , which is unresolved in our sma observations . the radial morphology of the filamentary structures in the central core , suggests contraction by gravity to be dominant over all possible sources of support . this needs to be confirmed via modeling of the kinematics . the absence of detectable magnetic field structures is consistent with this scenario . comparing our high resolution 1.2 mm continuum maps and the cs ( 10 ) observations , suggests that the massive molecular clump containing a luminous ( @xmath2910@xmath4 l@xmath1 ) ob cluster is part of a @xmath510 pc scale organized structure . the extended molecular gas has a filamentary geometry , starting from a @xmath65 pc extent in radius , to a geometrical center of a 23 pc scale dense structure . the parsec scale filaments show regularly spaced molecular clumps , each containing massive molecular cores that are associated with the local massive star formations . part of these parsec scale filaments might directly continue into the central 0.05 pc radius hot toroid , which encircles the highest mass stellar cluster . in this central region , the radial filamentary configuration suggests accretion flows which would allow simultaneously a quick global contraction and the efficient local fragmentation . such a model would be consistent with the previously seen centrally located 0.1 pc scale fast spinning hot toroid , and the rich cluster of water masers , millimeter cores , and high velocity @xmath7co ( 21 ) outflows . we propose that the spin up rotational motions of a filamentary accretion flow can explain the red / blue excess of the velocity field , and the spatial asymmetry of the mass distributions , although the interpretation is not unique . in this scenario , the magnetic field may not , and does not necessarily play an important role on the scale that we are observing ( 0.055 pc radius ) . the large numerical hydrodynamics simulations to compare the strongly and the weakly magnetized cases will help to improve the physical insights , and also improve the interpretation of the data . the proposed scenario in this paper might not be the unique mode of ob cluster formation . survey observations of a larger sample will be useful to identify or to classify other modes of ob cluster formation . we emphasize that the subject under the consideration in the present paper is specifically the luminous ( @xmath2910@xmath4 l@xmath1 ) ob cluster . in the present work , we suggest the convergence of the massive filamentary structures is one very important aspect in ob cluster formation . however , how such filaments of a few parsec scale form , is still uncertain , and should be a very fundamental aspect in the study of the interstellar medium . the very sensitive high resolution and high dynamic range spectral line and dust polarization observations might help to address this question . another aspect which can not be addressed by our present research is how the 0.1 pc scale hot toroid converge to form the embedded ob clusters . high resolution molecular and recombination line observations with alma will certainly address this issue . we thank the sma staff for making these observations possible . we also thank the iram staff for help with the observations and for developing the data reduction routines . liu thanks asiaa for supporting this research . this work is based in part on observations made with the _ spitzer space telescope _ , which is operated by the jet propulsion laboratory , california institute of technology . we thank the glimpse team ( pi : e. churchwell ) and mipsgal team ( pi : s. carey ) for making the irac and mips images available to the community . k. w. acknowledges the support from the sma predoctoral fellowship and the china scholarship council . and r@xmath46 . the purple arrows indicate the parsec scale protrusions ( fl ne , e , se , s , sw1 , sw2 , nw1 , nw2 ) . the blue contours are 16 mjybeam@xmath16 @xmath3 [ -2 , -1 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 12 , 14 , 16 , 20 , 60 ] ( 1@xmath28= 5 mjybeam@xmath16 ) ; the orange contours are 3 mjybeam@xmath16 @xmath3 [ 1 , 2 , 4 , 8 , 16 , 32 , 64 ] ( 1@xmath28= 0.3 mjybeam@xmath16 ) ; the magenta contours are 40 mjybeam@xmath16 @xmath3 [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 ] ( 1@xmath28= 7 mjybeam@xmath16 ) . the negative contours are shown in dotted lines . the cs distributed in projection against the bright uc hii regions show negative flux due to strong absorption . we avoid integrating the negative flux while generating the cs map to emphasize the structures in emission . ] @xmath3 [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 12 , 14 , 16 , 20 , 60 ] ( 1@xmath28= 5 mjybeam@xmath16 ) . the green dashed line show the pv cut as well as a proposed plane of rotation in liu et al . 2010 , liu , zhang & ho ( 2011 ) . the dashed line is 25@xmath11 ( 0.75 pc ) in length , and the top right end of this line corresponds to the zero angular offset . we note a continuation of an elongated cs structure to the 1 mm protrusion along the dashed line . we also note two more elongated cs structures in the north of the previously mentioned one . ]

we observe the 1.2 mm continuum emission around the ob cluster forming region g10.6 - 0.4 , using the iram 30 m telescope mambo-2 bolometer array and the submillimeter array . comparison of the spitzer 24 @xmath0 m and 8 @xmath0 m images with our 1.2 mm continuum maps reveals the ionization front of an hii region , the photon dominated layer , and several 5 pc scale filaments following the outer edge of the photon dominated layer . the filaments , which are resolved in the mambo-2 observations , show regularly spaced parsec scale molecular clumps , embedded with a cluster of submillimeter molecular cores as shown in the sma 0.87 mm observations . toward the center of the g10.6 - 0.4 region , the combined sma+iram 30 m continuum image reveals several , parsec scale protrusions . they may continue down to within 0.1 pc of the geometric center of a dense 3 pc size structure , where a 200 m@xmath1 ob cluster resides . the observed filaments may facilitate mass accretion onto the central cluster forming region in the presence of strong radiative and mechanical stellar feedbacks . their filamentary geometry may also facilitate fragmentation . we did not detect any significant polarized emission at 0.87 mm in the inner 1 pc region with the sma .

You are an expert at summarizing long articles. Proceed to summarize the following text: the search for the higgs boson and , hence , for the origin of electroweak symmetry breaking and fermion mass generation , remains one of the premier tasks of present and future high energy physics experiments . fits to precision electroweak ( ew ) data have for some time suggested a relatively small higgs boson mass , of order 100 gev @xcite . this is one of the reasons why the search for an intermediate mass higgs boson is particularly important @xcite . for the intermediate mass range , most of the literature has focussed on higgs boson production via gluon fusion @xcite and @xmath7 @xcite or @xmath8 @xcite associated production . cross sections for standard model ( sm ) higgs boson production at the lhc are well - known @xcite , and while production via gluon fusion has the largest cross section by almost one order of magnitude , there are substantial qcd backgrounds . a search for the very clean four - lepton signature from @xmath9 decay can find a higgs boson in the mass region @xmath10 gev , but due to the small branching fraction of this mode very large integrated luminosities , up to 100 fb @xmath5 or more , are required . one can search for @xmath11 via @xmath12 decays with much lower luminosity @xcite , but with lower signal - to - background ratios . the second largest production cross section is predicted for weak - boson fusion ( wbf ) , @xmath13 . these events contain additional information in their observable quark jets . techniques like forward jet tagging @xcite can then be exploited to significantly reduce the backgrounds . wbf and gluon fusion nicely complement each other : together they allow for a measurement of the @xmath14 coupling ratio . another feature of the wbf signal is the lack of color exchange between the initial - state quarks . color coherence between initial- and final - state gluon bremsstrahlung leads to suppressed hadron production in the central region , between the two tagging - jet candidates of the signal @xcite . this is in contrast to most background processes , which typically involve color exchange in the @xmath15-channel and thus lead to enhanced hadronic activity between the tagging jets . we exploit these features , via a veto of soft jet activity in the central region @xcite . while some attention has been given to intermediate - mass @xmath16 searches at the lhc in the framework of gluon fusion @xcite , production via weak boson fusion for the same decay mode has not yet been discussed in the literature . thus , we provide a first analysis of intermediate - mass @xmath17 at the lhc ( and of the main physics and reducible backgrounds ) which demonstrates the feasibility of higgs boson detection in this channel , with very low luminosity . @xmath16 event characteristics are analyzed for dual leptonic decays to @xmath18 only , to avoid backgrounds from @xmath19 . our analysis is a parton - level monte carlo study , using full tree - level matrix elements for the wbf higgs signal and the various backgrounds . in section [ sec : calc ] we describe our calculational tools , the methods employed in the simulation of the various processes , and important parameters . extra minijet activity is simulated by adding the emission of one extra parton to the basic signal and background processes . generically we call the basic signal process ( with its two forward tagging jets ) and the corresponding background calculations `` 2-jet '' processes , and refer to the simulations with one extra parton as `` 3-jet '' processes . in section [ sec : analysis ] , using the 2-jet programs for the backgrounds , we demonstrate forward jet tagging , a @xmath20 veto and other important cuts which combine to yield an @xmath212/1 to 1/2 signal - to - background ( s / b ) ratio , depending on the higgs mass . in section [ sec : minijet ] we analyze the different minijet patterns in signal and background , using both the truncated shower approximation ( tsa ) @xcite to regulate the cross sections , and the gluon exponentiation model to estimate the minijet multiplicity @xcite . by exploiting the two most important characteristics of the extra radiation , its angular distribution and its hardness , the qcd backgrounds can be suppressed substantially by a veto on extra central jet emission . within the tsa and exponentiation models , probabilities are estimated for vetoing signal and background events , and are combined with the production cross sections of the previous section to predict signal and background rates in table [ summary ] . these rates demonstrate the feasibility of extracting a very low background @xmath16 signal at the lhc . our signal selection is not necessarily optimized yet . the variables we identify for cuts are the most distinctive , but deserve a multivariate analysis with detector simulation . we do construct an additional variable in section [ sec : disc ] which is not used for cuts , but rather can be used to extract the higgs boson mass from the final event sample . we simulate @xmath22 collisions at the cern lhc , @xmath23 tev . all signal and background cross sections are determined in terms of full tree level matrix elements for the contributing subprocesses and are discussed in more detail below . for all our numerical results we have chosen @xmath24 , @xmath25 gev , and @xmath26 , which translates into @xmath27 gev and @xmath28 when using the tree - level relations between these input parameters . this value of @xmath29 is somewhat lower than the current world average of @xmath30 gev . however , this difference has negligible effects on all cross sections , e.g. the @xmath31 signal cross section varies by about @xmath32 between these two @xmath1 mass values . the tree level relations between the input parameters are kept in order to guarantee electroweak gauge invariance of all amplitudes . for all qcd effects , the running of the strong coupling constant is evaluated at one - loop order , with @xmath33 . we employ cteq4l parton distribution functions @xcite throughout . unless otherwise noted the factorization scale is chosen as @xmath34 min(@xmath35 ) of the defined jets . the signal can be described , at lowest order , by two single - feynman - diagram processes , @xmath37 , _ i.e. _ @xmath38 and @xmath39 fusion where the weak bosons are emitted from the incoming quarks @xcite . because of the small higgs boson width in the mass range of interest , these events can reliably be simulated in the narrow width approximation . from previous studies of @xmath40 @xcite and @xmath41 @xcite decays in weak boson fusion we know several features of the signal , which can be exploited here also : the centrally produced higgs boson tends to yield central decay products ( in this case @xmath42 ) , and the two quarks enter the detector at large rapidity compared to the @xmath1 s and with transverse momenta in the 20 to 100 gev range , thus leading to two observable forward tagging jets . for the study of a central jet veto , we utilize the results of previous studies where we simulated the emission of at least one extra parton @xcite . this was achieved by calculating the cross sections for the process @xmath43 , _ i.e. _ weak boson fusion with radiation of an additional gluon , and all crossing related processes . an important additional tool for distinguishing the @xmath44 signal from various backgrounds is the anti - correlation of the @xmath1 spins , as pointed out in ref . this is due to the preservation of angular momentum in the decay of the spin-0 higgs boson . of course , we can observe only the angular distributions of the charged decay leptons , but this is sufficient . the decay rate is proportional to @xmath45 . in the rest frame of the higgs boson , in which the @xmath46 or @xmath47 pairs are emitted back - to - back for @xmath42 production at threshold , this product is a maximum for the charged leptons being emitted parallel . this characteristic is preserved and even enhanced when boosted to the lab frame , as the higgs boson in weak boson fusion is typically emitted with @xmath48 gev . given the h decay signature , the main physics background to our @xmath50 signal arises from @xmath51 production , due to the large top production cross section at the lhc and because the branching ratio @xmath52 is essentially @xmath53 . the basic process we consider is @xmath54 , which can be either @xmath55- or @xmath56-initiated , with the former strongly dominating at the lhc . qcd corrections to this lead to additional real parton emission , _ i.e. _ to @xmath57 events . relevant subprocesses are [ qcd_tt ] g q t |t q , g |q t |t |q , q @xmath58 events can be obtained similarly . for the case of no additional partons , the @xmath20 s from the decaying top quarks may be identified as the tagging jets . in this case , calculating the cross section for @xmath57 where the @xmath20 s are explicitly identified as the tagging jets serves to estimate the effect of additional soft parton emission , _ i.e. _ minijet activity in the central detector ; this is described in detail in sec . [ sec : minijet ] . at the same time , we can identify a distinctly different , perturbative region of phase space , where the final - state light quark or gluon gives rise to one tagging jet , and one of the two decay @xmath20 s is identified as the other tagging jet . in this case , @xmath58 may be used to estimate minijet activity for the hard process @xmath59 . finally , there is a third distinct region of phase space , for the perturbative hard process @xmath60 , where the final state light quarks or gluons are the two tagging jets . thus , the `` @xmath61 '' and `` @xmath62 '' calculations serve a dual purpose : to obtain the cross sections for the contribution of the perturbative processes where light quark or gluon jets lie in the region of phase space where they are experimentally identified as far - forward / backward tagging jets ; and to estimate the additional qcd radiation patterns for the next - lower - order perturbative @xmath51 process . the @xmath63 and @xmath64 matrix elements were constructed using madgraph @xcite , while the @xmath62 matrix elements are from ref . @xcite . decays of the top quarks and @xmath1 s are included in the matrix elements ; however , while the @xmath1 s are allowed to be off - shell , the top quarks are required to be on - shell . energy loss from @xmath65 is included to generate more accurate @xmath66 distributions . in all cases , the factorization scale is chosen as @xmath34 min(@xmath67 ) of the massless partons / top quarks . the overall strong coupling constant factors are taken as @xmath68 , where the product runs over all light quarks , gluons and top quarks ; _ i.e. _ the transverse momentum of each additional parton is taken as the relevant scale for its production , irrespective of the hardness of the underlying scattering event . this procedure guarantees that the same @xmath69 factors are used for the hard part of a @xmath51 event , independent of the number of additional minijets , and at the same time the small scales relevant for soft - parton emission are implemented . the next obvious background arises from real - emission qcd corrections to @xmath42 production . for @xmath71 events these background processes include @xcite [ qcd_ww ] q g q g w^+ w^- , q q q q w^+ w^- , which are dominated by @xmath15-channel gluon exchange , and all crossing related processes , such as q |q g g w^+ w^- , g g q |q w^+ w^- . we call these processes collectively the `` qcd @xmath72 '' background . we do not calculate cross sections for the corresponding @xmath73-jet processes , but instead follow the results of our analysis of the radiation patterns of qcd @xmath74 processes , detailed in sec . [ sec : minijet ] , and apply those results here to estimate minijet veto probabilities . the factorization scale is chosen as for the higgs boson signal . the strong coupling constant factor is taken as @xmath75 , _ i.e. _ , the transverse momentum of each additional parton is taken as the relevant scale for its production . variation of the scales by a factor 2 or @xmath76 reveals scale uncertainties of @xmath77 , however , which emphasizes the need for experimental input or nlo calculations . the @xmath38 background lacks the marked anti - correlation of @xmath1 spins seen in the signal . as a result the momenta of the charged decay leptons will be more widely separated than in @xmath16 events . these backgrounds arise from @xmath42 bremsstrahlung in quark(anti)quark scattering via @xmath15-channel electroweak boson exchange , with subsequent decay @xmath78 : qq qq w^+w^- [ eq : ew_ww ] navely , this ew background may be thought of as suppressed compared to the analogous qcd process in eq . ( [ qcd_ww ] ) . however , it includes electroweak boson fusion , @xmath79 via @xmath80- or @xmath15-channel @xmath81-exchange or via @xmath82 4-point vertices , which has a momentum and color structure identical to the signal . thus , it can not easily be suppressed via cuts . the matrix elements for these processes were constructed using madgraph @xcite . we include charged - current ( cc ) and neutral - current ( nc ) processes , but discard s - channel ew boson and t - channel quark exchange processes as their contribution was found to be @xmath83 only , while adding significantly to the cpu time needed for the calculation . in general , for the regions of phase space containing far - forward and -backward tagging jets , s - channel processes are severely suppressed . we refer collectively to these processes as the `` ew @xmath72 '' background . both @xmath1 s are allowed to be off - shell , and all off - resonance graphs are included . in addition , the higgs boson graphs must be included to make the calculation well - behaved at large @xmath1-pair invariant masses . however , these graphs include our signal processes and might lead to double counting . thus , we set @xmath84 to 60 gev in the ew @xmath72 background to remove their contribution . a clean separation of the higgs boson signal and the ew @xmath72 background is possible because interference effects between the two are negligible for the higgs boson mass range of interest . again we will need an estimate of additional gluon radiation patterns . this was first done for ew processes in ref . @xcite , but for different cuts on the hard process , and again for ew @xmath85 processes in ref . we reanalyze the ew @xmath85 case in sec . [ sec : minijet ] and directly apply the resulting minijet emission probabilities here . the ew @xmath85 and ew @xmath72 backgrounds are quite similar kinematically , which justifies the use of the same veto probabilities for central jets . the leptonic decay of @xmath87 s provides a source of electrons , muons and neutrinos which can be misidentified as @xmath1 decays . thus , we need to study real - emission qcd corrections to the drell - yan process @xmath88 . for @xmath89 events these background processes include @xcite [ qcd_tau ] q g q g ^+ ^- , q q q q ^+ ^- , which are dominated by @xmath15-channel gluon exchange , and all crossing - related processes , such as q |q g g ^+ ^- , g g q |q ^+ ^- . all interference effects between virtual photon and @xmath90-exchange are included . we call these processes collectively the `` qcd @xmath85 '' background . the cross sections for the corresponding @xmath91-jet processes , which we need for our modeling of minijet activity in the qcd @xmath85 background , have been calculated in refs . similar to the treatment of the signal processes , we use a parton - level monte - carlo program based on the work of ref . @xcite to model the qcd @xmath85 and @xmath92 backgrounds . from our study of @xmath41 in weak boson fusion @xcite , we know that the ew ( t - channel weak boson exchange ) cross section will be comparable to the qcd cross section in the phase space region of interest . thus , we consider those processes separately , in a similar manner as for the ew @xmath72 contribution . we use the results of ref . @xcite for modeling the ew @xmath85 background . the dual leptonic decays of the @xmath87 s are simulated by multiplying the @xmath93 cross section by a branching ratio factor of @xmath94 and by implementing collinear tau decays with helicity correlations included as in our previous analysis of @xmath41 @xcite . the qcd processes discussed above lead to steeply falling jet transverse momentum distributions . as a result , finite detector resolution can have a sizable effect on cross sections . these resolution effects are taken into account via gaussian smearing of the energies of jets/@xmath20 s and charged leptons . we use = 5.2 e .009 , for jets ( with individual terms added in quadrature ) , based on atlas expectations @xcite . for charged leptons we use = 2% . in addition , finite detector resolution leads to fake missing - transverse - momentum in events with hard jets . an atlas analysis @xcite showed that these effects are well parameterized by a gaussian distribution of the components of the fake missing transverse momentum vector , @xmath95 , with resolution ( p_x , p_y ) = 0.46 , for each component . in our calculations , these fake missing transverse momentum vectors are added linearly to the neutrino momenta . the @xmath96 dual leptonic decay signal is characterized by two forward jets and the @xmath1 decay leptons ( @xmath97 ) . before discussing background levels and further details like minijet radiation patterns , we need to identify the search region for these hard @xmath98 events . the task is identical to the higgs searches in @xmath99 which were considered previously @xcite . we can thus adopt the strategy of these earlier analyses and start out by discussing a basic level of cuts on the @xmath100 signal throughout this section we assume a higgs mass of @xmath101 gev , but we do not optimize cuts for this mass . the minimum acceptance requirements ensure that the two jets and two charged leptons are observed inside the detector ( within the hadronic and electromagnetic calorimeters , respectively ) , and are well - separated from each other : [ eq : basic ] & p_t_j 20 gev , |_j| 5.0 , r_jj 0.7 , + & p_t _ 20 gev , |_| 2.5 , r_j 0.7 . a feature of the qcd @xmath72 background is the generally higher rapidity of the @xmath1 s as compared to the higgs signal : weak boson bremsstrahlung occurs at small angles with respect to the parent quarks , producing @xmath1 s forward of the jets . thus , we also require both @xmath102 s to lie between the jets with a separation in pseudorapidity @xmath103 , and the jets to occupy opposite hemispheres : [ eq : lepcen ] _ j , min + 0.7 < _ _ 1,2 < _ j , max - 0.7 , _ j_1 _ j_2 < 0 finally , to reach the starting point for our consideration of the signal and various backgrounds , a wide separation in pseudorapidity is required between the two forward tagging jets , [ eq : gap ] _ tags = |_j_1-_j_2| 4.4 , leaving a gap of at least 3 units of pseudorapidity in which the @xmath102 s can be observed . this technique to separate weak boson scattering from various backgrounds is well - established @xcite , in particular for heavy higgs boson searches . line 1 of table [ ww_data ] shows the effect of these cuts on the signal and backgrounds for a sm higgs boson of mass @xmath104 gev . overall , about @xmath105 of all @xmath106 events generated in weak boson fusion are accepted by the cuts of eqs . ( [ eq : basic]-[eq : gap ] ) ( for @xmath104 gev ) . somewhat surprisingly , the ew @xmath72 background rate reaches 2/3 of the qcd @xmath72 background rate already at this level . this can be explained by the contribution from @xmath107 exchange and fusion processes which can produce central @xmath1 pairs and are therefore kinematically similar to the signal . this signal - like component remains after the forward jet tagging cuts . as is readily seen from the first line of table [ ww_data ] , the most worrisome background is @xmath1 pairs from @xmath51 production . of the 1080 fb at the basic cuts level , 12 fb are from @xmath2 , 310 fb are from @xmath61 , and the remaining 760 fb arise from @xmath62 production . the additional jets ( corresponding to massless partons ) are required to be identified as far forward tagging jets . the @xmath62 cross section is largest because the @xmath2 pair is not required to have as large an invariant mass as in the first two cases , where one or both @xmath20 s from the decay of the top quarks are required to be the tagging jets . for the events where one or both of the @xmath20 s are not identified as the tagging jets , they will most frequently lie between the two tagging jets , in the region where we search for the @xmath1 decay leptons . vetoing events with these additional @xmath20 jets provides a powerful suppression tool to control the top background . note that this does _ not _ require a @xmath20-tag , merely rejection of any events that have an additional jet , which in this case would be from a hadronically decaying @xmath20 . we discard all events where a @xmath20 or @xmath108 jet with @xmath109 gev is observed in the gap region between the tagging jets , [ eq : bveto ] p_t_b > 20 gev , _ j , min < _ b < _ j , max . this leads to a reduction of @xmath61 events by a factor 7 while @xmath62 events are suppressed by a factor 100 . this results in cross sections of 43 and 7.6 fb , respectively , at the level of the forward tagging cuts of eqs . ( [ eq : basic]-[eq : gap ] ) , which are now comparable to the other individual backgrounds . this is shown in the second line of table [ ww_data ] . note that the much higher @xmath20 veto probability for @xmath62 events results in a lower cross section than that for @xmath61 events , an ordering which will remain even after final cuts have been imposed ( see below ) . qcd processes at hadron colliders typically occur at smaller invariant masses than ew processes , due to the dominance of gluons at small feynman @xmath110 in the incoming protons . we observe this behavior here , as shown in fig . [ fig : mjj ] . the three @xmath51 backgrounds have been combined for clarity , even though their individual distributions are slightly different . we can thus significantly reduce much of the qcd background by imposing a lower bound on the invariant mass of the tagging jets : [ eq : mjj ] m_jj > 650 gev . another significant difference is the angular distribution of the charged decay leptons , @xmath111 and @xmath112 , relative to each other . in the case of the higgs signal , the @xmath1 spins are anti - correlated , so the leptons are preferentially emitted in the same direction , close to each other . a significant fraction of the various backgrounds does not have anti - correlated @xmath1 spins . these differences are demonstrated in fig . [ fig : angdist ] , which shows the azimuthal ( transverse plane ) opening angle , polar ( lab ) opening angle , and separation in the lego plot . we exploit these features by establishing the following lepton - pair angular cuts : [ eq : ang ] _ e < 105^ , cos _ e > 0.2 , r_e < 2.2 . it should be noted that while these cuts appear to be very conservative , for higher higgs boson masses the @xmath113 and @xmath114 distribution broadens out to higher values , overlapping the backgrounds more . for @xmath115 gev these cuts are roughly optimized and further tightening would require greater integrated luminosity for discovery at this upper end of the mass range . because of the excellent signal - to - background ratio achieved below , we prefer to work with uniform acceptance cuts , instead of optimizing the cuts for specific higgs boson mass regions . we also examine the distributions for lepton - pair invariant mass , @xmath116 , and maximum lepton @xmath35 , as shown in fig . [ fig : mllptl ] for the case @xmath117 gev . as is readily seen , the qcd backgrounds and ew @xmath72 background prefer significantly higher values for both observables . thus , in addition to the angular variables , we find it useful to restrict the individual @xmath35 of the leptons , as well as the invariant mass of the pair : [ eq : adv ] m_e < 110 , p_t_e , < 120 . these are particularly effective against the top backgrounds , where the large top mass allows for very high-@xmath35 leptons far from the tagging jets , and against the ew @xmath72 background , where the leptons tend to be well - separated in the lego plot . again , the cuts are set quite conservatively so as not to bias a lower higgs boson mass . results after cuts ( [ eq : mjj]-[eq : adv ] ) are shown on the third line of table [ ww_data ] , for the case of a 160 gev higgs boson . .signal rates @xmath118 for @xmath104 gev and corresponding background cross sections , in @xmath22 collisions at @xmath119 tev . results are given for various levels of cuts and are labeled by equation numbers discussed in the text . on lines six the minijet veto is included . line five gives the survival probabilities for each process , with @xmath120 gev . the expected tagging jet identification efficiency is shown on the last line . all rates are given in fb . [ cols="<,^,^,^,^,^,^,^",options="header " , ] the high purity of the signal is made possible because the weak boson fusion process , together with the @xmath121 decay , provides a complex signal with a multitude of characteristics which distinguish it from the various backgrounds . the basic feature of the @xmath31 signal is the presence of two forward tagging jets inside the acceptance region of the lhc detectors , of sizable @xmath35 , and of dijet invariant mass in the tev range . typical qcd backgrounds , with isolated charged leptons and two hard jets , are much softer . in addition , the qcd backgrounds are dominated by @xmath1 bremsstrahlung off forward scattered quarks , which give typically higher - rapidity charged leptons . in contrast , the ew processes give rise to quite central leptons , and this includes not only the higgs signal but also ew @xmath72 and @xmath85 production , which also proceed via weak boson fusion . it is this similarity that prevents one from ignoring ew analogs to background qcd processes , which a priori are smaller by two orders of magnitude in total cross section , but after basic cuts remain the same size as their qcd counterparts . for @xmath122 decays , lepton angular distributions are extremely useful for reducing the backgrounds even further . the anti - correlation of @xmath1 spins in @xmath123 decay forces the charged leptons to be preferentially emitted in the same direction , close together in the lego plot . this happens for a small fraction of the background only . we have identified the most important distributions for enhancing the signal relative to the background , and set the various cuts conservatively to avoid bias for a certain higgs boson mass range . there is ample room for improvement of our results via a multivariate analysis of a complete set of signal and background distributions , which we encourage the lhc collaborations to pursue . additional suppression of the @xmath51 background may be possible with @xmath20 identification and veto in the @xmath124 gev region . in addition to various invariant mass and angular cuts , we can differentiate between the @xmath1 s of the signal and @xmath125 backgrounds and the real @xmath87 s in the qcd and ew @xmath85 backgrounds . this is possible because the high energy of the produced @xmath87 s makes their decay products almost collinear . combined with the substantial @xmath35 of the @xmath86 system this allows for @xmath87-pair mass reconstruction . the @xmath1 decays do not exhibit this collinearity due to their large mass , thus the angular correlation between the @xmath126 vector and the charged lepton momenta is markedly different . our real-@xmath87 rejection makes use of these differences and promises to virtually eliminate the @xmath85 backgrounds . we advocate taking advantage of an additional fundamental characteristic of qcd and ew processes . color - singlet exchange in the @xmath15-channel , as encountered in higgs boson production by weak boson fusion ( and in the ew @xmath127 background ) , leads to additional soft jet activity which differs strikingly from that expected for the qcd backgrounds in both geometry and hardness : gluon radiation in qcd processes is typically both more central and harder than in wbf processes . we exploit this radiation , via a veto on events with central minijets of @xmath109 gev , and expect a typical @xmath128 reduction in qcd backgrounds and about a @xmath129 suppression of ew backgrounds , but only about a @xmath130 loss of the signal . beyond the possibility of discovering the higgs boson in the @xmath122 mode , or confirmation of its existence , measuring the cross sections in both weak boson and gluon fusion will be important both as a test of the standard model and as a search for new physics . for such a measurement , via the analysis outlined in this paper , minijet veto probabilities must be precisely known . for calibration purposes , one can analyze @xmath127 events at the lhc . the production rates of the qcd and ew @xmath127 events can be reliably predicted and , thus , the observation of the @xmath131 peak allows for a direct experimental assessment of the minijet veto efficiencies , in a kinematic configuration very similar to the higgs signal . observation of sm @xmath44 at the lhc is possible for very low integrated luminosities , if the higgs boson lies in the mass range between about 130 and 200 gev . weak boson fusion at the lhc will be an exciting process to study , for a weakly coupled higgs sector just as much as for strong interactions in the symmetry breaking sector of electroweak interactions . this research was supported in part by the university of wisconsin research committee with funds granted by the wisconsin alumni research foundation and in part by the u. s. department of energy under contract no . de - 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weak boson fusion promises to be a copious source of intermediate mass standard model higgs bosons at the lhc . the additional very energetic forward jets in these events provide for powerful background suppression tools . we analyze the @xmath0 decay mode for a higgs boson mass in the 130 - 200 gev range . a parton level analysis of the dominant backgrounds ( production of @xmath1 pairs , @xmath2 and @xmath3 in association with jets ) demonstrates that this channel allows the observation of @xmath4 in a virtually background - free environment , yielding a significant higgs boson signal with an integrated luminosity of 5 fb@xmath5 or less . weak boson fusion achieves a much better signal to background ratio than inclusive @xmath6 and is therefore the most promising search channel in the 130 - 200 gev mass range .

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Proceed to summarize the following text: ngc7538irs1 was first detected in the radio at 5 ghz by @xcite , who found three compact radio sources at the se edge of the large ( @xmath3 4 ) region ngc7538 , which is at a distance 2.65 kpc @xcite . the brightest of the three , source b , has later become known as irs1 @xcite . the far - infrared luminosity of the three sources is @xmath31.9 10@xmath7 , completely dominated by irs1@xcite . irs1 was first resolved with the vla at 14.9 ghz by @xcite , who showed that it has a compact ( @xmath302 ) bipolar n - s core with faint extended fan - shaped lobes , suggesting an ionized outflow . this is an extremely well - studied source with numerous masers , a prominent molecular outflow and extremely broad hydrogen recombination lines indicating substantial mass motion of the ionized gas @xcite . although it has been modeled as an ionized jet @xcite or a photo - ionized accretion disk @xcite , the source is usually referred to as an ultra compact or hyper compact region with a turnover at @xmath3 100 ghz . since we have carried extensive observations of irs1 with bima and carma , the desire to determine the contribution from dust emission at mm - wavelengths led us to investigate where the free - free emission from irs1 becomes optically thin . such a determination is not possible from published data , because irs1 has been observed with a variety of vla configurations , some of which resolve out the extended emission and others may only quote total flux . this prevented us from separately estimating the flux for the compact bipolar core and the extended lobes . furthermore the flux density varies with time @xcite , which makes it difficult to derive an accurate spectral index . we have retrieved and re - analyzed a few key observations of ngc7538irs1 from the vla data archive . these are all long integration observations with very high angular resolution , good uv - coverage , and high image fidelity . table [ tbl-1 ] gives the observing dates , synthesized beam width , sensitivity , the integrated flux of the bipolar core , and the observed total flux . we included a bna array observation obtained by us at x - band ( 8.5 ghz)@xcite , because it is more contemporary with the high frequency data sets and provides an additional data point at a frequency where there are no other high angular resolution vla data available . fig [ fig - irs1 - 3 ] shows the irs1 - 3 field at 4.8 ghz , while fig [ fig - vla ] shows the morphology of the compact bipolar core as a function of frequency . we determined flux densities of the compact bipolar core with an accuracy of a few percent . the results are given in table [ tbl-1 ] . table [ tbl-1 ] also gives total flux densities , obtained by integrating over the whole area where we detect emission from irs1 . at 4.9 ghz the total linear extent of irs1 is @xmath3 7 , while it is @xmath3 14 at 43.4 ghz . even though we can reliably determine the flux density of the bipolar core at 43.4 ghz , some of the faint extended emission is filtered out ; therefore the total flux is underestimated . at 4.9 ghz the bipolar core has a total length of @xmath3 11 fwhm ( full width half maximum ) , with a separation between the two peaks of 068 , at 14.9 ghz the size is 044 with a peak separation of 024 ( table [ tbl-1 ] ) . at 43.4 ghz we can still fit the core with a double gaussian with a peak separation of @xmath3 012 , while the linear size ( length ) has a fwhm of 020 . since one of our data sets ( x - band , 8.4 ghz ) has insufficient resolution to resolve the two bipolar cores , we plot the linear size ( length ) as a function of wavelength in fig . [ fig - jet ] , although fitting to the lobe separation gives virtually identical results , but with a larger uncertainty . we find the size to vary with frequency as @xmath8 , while the flux density increases with frequency as @xmath9 . the frequency dependence of the total flux is more difficult to estimate , since some vla configurations are not very sensitive to faint extended emission and will therefore underestimate the total flux . if we chose data sets which give a reliable flux for irs2 , which is a spherical region with a size of @xmath3 8 , we find that the total flux from irs1 has a slightly steeper spectral index , 0.8 @xmath10 0.03 determined from data sets from 4.8 ghz to 49 ghz . the fit to the total flux densities for irs1 is shown as a dotted line in fig . [ fig - jet ] . such a shallow frequency dependence would require a very steep density gradient in the ionized gas , if the emission originates from an region ionized by a central o - star , although it is not impossible , see e.g. @xcite . a high - density region with a steep density gradient should be spherical , not bipolar with a dark central lane as we observe in irs1 . neither can a steep density gradient model explain the extremely broad recombination lines observed in irs1 @xcite . all these characteristics can be explained , if the emission originates in an ionized stellar wind or jet @xcite . the inner part of the jet is optically thick and further out , where the jet expands , the emission becomes optically thin . at higher frequencies the outer part of the jet becomes optically thin and therefore appears shorter and more collimated , exactly what we see in irs1 , see fig [ fig - vla ] . for a uniformly expanding spherical wind , the size of the source varies as a function of frequency as @xmath11 , while the flux density density goes as @xmath12 @xcite . since we resolve the emission from irs1 , we know that it is not spherical , but instead it appears to originate in an initially collimated bipolar jet ( opening angle @xmath13 30 ) approximately aligned with the molecular outflow from irs1 , see section [ accretion ] . for a collimated ionized jet , @xcite showed that it can have a spectral index anywhere between 2 and @xmath140.1 , depending on gradients in jet width , velocity , degree of ionization , and temperature . lccllll 4.86 & 0.43 @xmath15 0.37 @xmath165.4 & 0.30 & 1.06 @xmath15 0.44 @xmath149.6 & 94.3 @xmath10 4.8 & 122.5 & 1984:1123 + 8.46 & 1.21 @xmath15 0.47 @xmath140.5 & 0.05 & 0.76 @xmath15 0.30 @xmath1410.9 & 135.7 @xmath10 13.6 & 148.1 & 2003:1014 + 14.94 & 0.14 @xmath15 0.12 @xmath1647.6 & 0.017 & 0.44 @xmath15 0.20 @xmath140.3 & 185.9 @xmath10 5.6 & 248.7 & 2006:0226,0512 + 14.91 & 0.14 @xmath15 0.10 @xmath1613.7 & 0.081 & 0.50 @xmath15 0.20 @xmath141.2 & 165.2 @xmath10 13.3 & 291.0 & 1994:1123 + 22.37 & 0.08 @xmath15 0.08 @xmath1443.0 & 0.017 & 0.34 @xmath15 0.17 @xmath163.1 & 286.5 @xmath10 11.5 & 370.0 & 1994:1123 + 43.37 & 0.14 @xmath15 0.12 @xmath1614.5 & 0.17 & 0.20 @xmath15 0.14 @xmath141.0 & 429.7 @xmath10 12.9 & 473.7 & 2006:0706,0914 + although the molecular outflow from irs1 in early studies was found to be quite compact and going from se to nw @xcite , we find that the molecular outflow is very extended ( @xmath17 4 ) and approximately aligned with the free - free jet from analyzing large mosaics obtained with carma in @xmath18co and @xmath19co j = @xmath20 , and hco@xmath4 j = @xmath20 . for hco@xmath4 we filled in missing zero spacing with fully sampled single dish maps obtained with fcrao . a thorough discussion of the molecular outflow is beyond the scope of this paper , and will be discussed in a forthcoming paper ( corder et al . 2009 , in prep ) . here we therefore summarize some of the main results . the carma mosaics show that the outflow is very large and has a position angle ( pa ) of @xmath3 @xmath1420 , which is similar to the orientation of the collimated inner part of the ionized jet driven by irs1 on smaller angular scales . the outflow extends several arcminutes to the north and starts as a wide angle limb brightened flow . to the south the outflow is difficult to trace , because of several other outflows in the giant molecular core in which irs1 is embedded , but appears more collimated with a linear extent of @xmath3 2 . analysis of spitzer irac archive data show that the outflow may be even larger to the south . the 5.8 and 8.0 @xmath21 m irac images show a jet - like feature projecting back towards irs1 , extending to @xmath3 3.5 ( 2.6 pc ) from irs1 , which is much further out than what was covered by the carma mosaics @xcite . however , the carma observations also confirm that irs1 is still heavily accreting . in hco@xmath4 j = @xmath20 we observe a clear inverse p cygni profile towards the strong continuum emission from irs1 ( fig [ fig - ipc ] ) . the absorption is all red - shifted and has two velocity components , which could mean that the accretion activity is varying with time ( episodic ) . from these data @xcite estimates @xmath3 6.8 of infalling gas . with the assumption that the accretion time is similar to the free - fall time , @xmath3 30,000 yr , @xcite finds an accretion rate @xmath3 2 10@xmath6 m@xmath22/yr , which will block most of the uv photons from the central o - star , allowing them to escape only at the polar regions . in fig . [ fig - jet ] we also plotted flux densities at 3 and 1 mm from the literature @xcite , supplemented with our own results at 3 mm from bima ( wright et al . 2009 , in prep ) and carma @xcite . all the observed flux densities at 3 mm or even at 1.3 mm can be explained by free - free emission , with at most a marginal excess from dust emission . high angular resolution carma continuum observations at 91.4 and 108.1 ghz confirm what is predicted from the fit to total flux densities in fig [ fig - jet ] , i.e. that the free - free emission dominates at 3 mm . these carma observations resolve irs1 with a size of 05 @xmath2303 pa @xmath141 and 04 @xmath2301 pa 20 at 94 and 109 ghz , respectively , i.e. the 3 mm emission is aligned with the free - free emission , and not with an accretion disk , which is expected to be perpendicular to the jet . since we detect a collimated free - free jet and since there is a strong accretion flow towards irs1 ( section [ accretion ] ) , irs1 must be surrounded by an accretion disk . the morphology of the free - free emission suggests that the disk should be almost edge - on , and centered halfway between the northern and the southern peak of the optically thick inner part of the free - free jet . however , if such an edge - on accretion disk is thin , i.e. the height of the disk is small relative to its diameter , it may not provide much surface area , and is therefore difficult to detect at 3 mm . at 3 mm irs1 and irs2 are very strong in the free - free , and even the free - free emission from irs3 can not be ignored , which makes it very hard to detect dust emission from the accretion disk . since the spectral index for dust is @xmath3 3 - 3.5 , the dust emission may start to dominate at frequencies above 300 ghz , observable only by sma , but even at 1.3 mm we may have a much better chance to detect the accretion disk . there is some evidence for such a disk in 450 and 350 @xmath21 m continuum images obtained at jcmt @xcite . until higher spatial resolution interferometer images are available , it is not clear whether this emission originates in a disk , from the surrounding in - falling envelope , or from a superposition of several nearby sources . free - free emission from ionized jets is very common in low mass protostars ( class 0 and class i sources ) , which all appear to drive molecular outflows . this situation is especially true for young stars , which often have rather well collimated outflows @xcite . jets may also be common in young early b - type high mass stars @xcite , although they have not been studied as well as low - mass protostars , because they are short lived , more distant and harder to identify . although there have not been any detections of jets in young o stars , i.e. stars with a luminosity of @xmath17 10@xmath7 , such objects are likely to exist . broad radio recombination line ( rrl ) objects , many of which are classified as ultra compact or hyper compact regions @xcite offer a logical starting point , because they show evidence for substantial mass motions , which would be readily explained if the recombination line emission originates in a jet . several of them also show evidence for accretion . in the sample analyzed by @xcite , which partly overlaps with the sources discussed by @xcite , they find four sources ( including irs1 ) with bipolar morphology . all appear to be dominated by wind ionized emission , although not necessarily jet driven winds . k3 - 50a , however , at a distance of 8.7 kpc @xcite , has a luminosity of a mid - o star , drives an ionized bipolar outflow @xcite , and has a spectral index of 0.5 in the the frequency range 5 - 15 ghz @xcite . k3 - 50a is therefore another example of an o - star , where the free - free emission appears to be coming from an ionized jet . detailed studies of broad rrl objects will undoubtedly discover more examples of jet - ionized regions . such objects are likely to drive molecular outflows , excite masers and show evidence for strong accretion . to summarize : we have shown that the emission from irs1 is completely dominated by a collimated ionized jet . the jet scenario also readily explains why the free - free emission is variable , because the accretion rate from the surrounding clumpy molecular cloud will vary with time . it also explains why irs1 is an extreme broad rrl object . since most hyper compact regions are broad rrl objects , and show rising free - free emission with similar spectral index to irs1 , other sources now classified as hyper compact regions , may also be similar to irs1 . the national radio astronomy observatory ( nrao ) is a facility of the national science foundation operated under cooperative agreement by associated universities , inc . the bima array was operated by the universities of california ( berkeley ) , illinois , and maryland with support from the national science foundation . support for carma construction was derived from the states of california , illinois , and maryland , the gordon and betty moore foundation , the kenneth t. and eileen l. norris foundation , the associates of the california institute of technology , and the national science foundation . ongoing carma development and operations are supported by the national science foundation under a cooperative agreement , and by the carma partner universities .

analysis of high spatial resolution vla images shows that the free - free emission from ngc7538irs1 is dominated by a collimated ionized wind . we have re - analyzed high angular resolution vla archive data from 6 cm to 7 mm , and measured separately the flux density from the compact bipolar core and the extended ( 15 - 3 ) lobes . we find that the flux density of the core is @xmath0 , where @xmath1 is the frequency and @xmath2 is @xmath3 0.7 . the frequency dependence of the total flux density is slightly steeper with @xmath2 = 0.8 . a massive optically thick hypercompact core with a steep density gradient can explain this frequency dependence , but it can not explain the extremely broad recombination line velocities observed in this source . neither can it explain why the core is bipolar rather than spherical , nor the observed decrease of 4% in the flux density in less than 10 years . an ionized wind modulated by accretion is expected to vary , because the accretion flow from the surrounding cloud will vary over time . bima and carma continuum observations at 3 mm show that the free - free emission still dominates at 3 mm . hco@xmath4 j = @xmath5 observations combined with fcrao single dish data show a clear inverse p cygni profile towards irs1 . these observations confirm that irs1 is heavily accreting with an accretion rate @xmath32 10@xmath6 /yr .

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Proceed to summarize the following text: strong gravitational lensing systems provide valuable tools for studying the properties and evolution of galaxies and quasars , for measuring the distribution of dark matter , and for constraining cosmological parameters . in particular , lensed quasar systems can provide information on , for example , intervening absorption line systems ( e.g. , * ? ? ? * ) , properties of quasar host galaxies ( e.g. , * ? ? ? * ) , dark matter subtructure in the lens ( e.g. , * ? ? ? * ) , and the stellar content of the lensing galaxy ( e.g. , * ? ? ? moreover , when combined with time delay measurements and careful lens modeling , lensed quasar systems can provide powerful cosmological constraints that are complementary to those of other techniques ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? previously , large samples of lensed quasars have been found by surveys such as the cosmic lens all - sky survey ( class ; * ? ? ? * ; * ? ? ? * ) in the radio and the sloan digital sky survey quasar lens search ( sqls ; * ? ? ? * ; * ? ? ? * ) and the sdss - iii boss quasar lens survey @xcite in the optical . the dark energy survey ( des ; * ? ? ? * ; * ? ? ? * ) is an ongoing imaging survey covering 5000 deg@xmath3 of the southern galactic cap in the @xmath4 filters using the dark energy camera @xcite , and it holds the promise of significantly increasing the numbers of lensed quasars . in particular , based on the forecasts of @xcite , we expect to find in des about 120 lensed quasar systems brighter than @xmath5 ( magnitude limit applies to the fainter image for pairs and third brightest image for quadruple systems , or quads ; see figure 1 of * ? ? ? des , specifically with the external collaboration project strides ( pi t. treu ) , aims to use the resulting large lensed quasar sample for the primary science goal of improving constraints on cosmological parameters . this lensed quasar sample is predicted to include about 20 of the very rare quad systems which will provide additional valuable information compared to pair systems for constraining lens models ( specifically extra constraints from 2 more positions and 2 more time delays ) , in particular for cosmology purposes ( e.g. , * ? ? ? * ) and substructure studies ( e.g. , * ? ? ? * ) . to date , we have discovered and spectroscopically confirmed three lensed quasar pair systems @xcite in des . here in this letter we report the discovery and confirmation of the first lensed quasar quad ( or quad - like ) system , , in the dark energy survey . we first describe our lensed quasar search procedure , system discovery , and photometric data in [ sec : search ] . we then describe our spectroscopic observations and present our data in [ sec : spectra ] . we summarize and conclude in [ sec : conclusions ] . detailed photometry analysis and lens modeling for this system are presented in a companion paper , @xcite , to which we will refer the reader where relevant below . the system was discovered during a systematic visual search for candidate strong lensing systems in the dark energy survey year 1 ( y1 ) data @xcite . a number of different search methods have been applied to the des y1 data , but the specific method involved in this case was a `` blue - near - red '' technique , where we first automatically identified candidate systems of red galaxies with multiple neighboring blue objects within some radius , and then used visual inspections of these systems to rate them and to select the best candidates for subsequent spectroscopic follow - up . this method has been used previously in the sloan digital sky survey ( sdss ) data to search for bright strong lensing systems and then to confirm them spectroscopically with good success ( e.g. , * ? ? ? * ; * ? ? ? here we started with a des y1 red galaxy sample selected using the redmagic technique @xcite . to minimize stellar contamination of this sample , we rejected objects with sextractor @xcite spread_model@xcite values @xmath6 . we then identified as our initial set of candidates those redmagic galaxies with three or more blue objects within a radius @xmath7 , where we defined a blue object as one with colors @xmath8 and @xmath9 . we did not apply any star / galaxy separation cut to the blue objects , but did reject objects that are saturated in any of the @xmath10 filters using a sextractor flags @xmath11 cut . we also applied a magnitude cut @xmath12 on the blue objects to keep the number of candidates manageable for the visual inspection step and to have relatively brighter candidates to ease follow - up spectroscopic redshift measurements . applying the above criteria resulted in a list of 6526 systems that one of us ( hl ) then inspected visually by examining their des @xmath13 color composite images . the discovery image of is shown in figure [ fig : images ] , where we see a central red galaxy ( g1 ) surrounded by three blue objects ( a , b , and d ) and a fourth red object ( g2 ) ( objects labeled as in * ? ? ? this system stood out as a potential quadruply lensed quasar system , except that the fourth putative lensed quasar image has a conspicuously redder color than those of the other three images , suggesting that the fourth image may instead be that of a foregound red galaxy , which is possibly also associated with the central red lensing galaxy . here the system would be a triple , where two of the quad s images are blended into one due to a fold configuration . another possibility is that the fourth image is a blend of a foreground red galaxy with the fourth blue lensed quasar image , as the fourth image in figure [ fig : images ] appears not as red as the central galaxy . regardless , the quad - like configuration of made it a very good candidate for subsequent spectroscopy , described in the next section . we present a detailed lensing model for this system in the companion paper @xcite . in the des y1 catalog , is composed of three objects due to the fact that components d and g1 are blended and that b and g2 are also blended . as such , the catalog contains entries from a , d+g1 , and b+g2 . the sextractor segmentation map for the system is shown in figure [ fig : images ] , overplotted on the des @xmath14-band image of the system . we need to point out here that all three catalog objects actually meet our blue object color criteria given before ( see figure [ fig : quadcolor ] , leftmost panel ) , so that the redmagic galaxy for this system is not g1 but rather the red galaxy g0 marked in figure [ fig : images ] . g0 is about @xmath15 away from g1 and has a redmagic photometric redshift of @xmath16 , close to g1 s spectroscopic redshift of 0.597 ( see [ sec : spectra ] ) . therefore in this case our blue - near - red method found the system via another red galaxy likely associated with the lensing galaxy , rather than directly via the red lensing galaxy itself . the discovery was thus likely not coincidental , but was also not by the direct route as intended by the method . in similar des y1 blue - near - red and related searches , we also see other cases in galaxy group and cluster environments where the lensing system is being indirectly found this way ( * ? ? ? * in preparation ) . the coordinates and photometry of the three des catalog objects are given in table [ tab : phot ] , while we present in @xcite a more detailed analysis that provides the deblended positions and magnitudes of all five components in this system . the des sextractor auto magnitudes of the three catalog objects are listed in table [ tab : phot ] , as are the near - infrared kron magnitudes @xcite from the vista hemisphere survey ( vhs ; * ? ? ? * ) and the mid - infrared magnitudes from the wide - field infrared survey explorer ( wise ; * ? ? ? * ) . for the near - infrared catalog , magnitudes were obtained by cross - matching the des and vhs catalogs using a @xmath17 search radius . component d+g1 does not have a @xmath18-band magnitude value due to blending with b+g2 in this band . for the mid - infrared magnitudes , the des and wise catalogs were cross - matched using a @xmath19 search radius . all components of the system are blended into a single wise source , which is why the same values of the @xmath20 and @xmath21 magnitudes are displayed in the table . cccc ra & 62.091333 & 62.090469 & 62.089596 + dec & -53.900266 & -53.899641 & -53.899776 + @xmath22 & 19.99 @xmath23 0.01 & 20.45 @xmath23 0.01 & 19.74 @xmath23 0.01 + @xmath24 & 19.94 @xmath23 0.01 & 20.08 @xmath23 0.01 & 19.51 @xmath23 0.01 + @xmath14 & 19.75 @xmath23 0.01 & 19.57 @xmath23 0.01 & 19.18 @xmath23 0.01 + @xmath25 & 19.51 @xmath23 0.02 & 19.13 @xmath23 0.02 & 18.82 @xmath23 0.01 + @xmath26 & 19.48 @xmath23 0.07 & 19.09 @xmath23 0.05 & 18.69 @xmath23 0.03 + @xmath18 & 19.58 @xmath23 0.06 & & 18.77 @xmath23 0.03 + @xmath27 & 19.62 @xmath23 0.08 & 18.90 @xmath23 0.04 & 18.65 @xmath23 0.03 + @xmath28 & 19.06 @xmath23 0.07 & 18.51 @xmath23 0.04 & 18.15 @xmath23 0.03 + @xmath20 & & 16.78 @xmath23 0.02 & + @xmath21 & & 16.51 @xmath23 0.02 & + after was first announced to the strides collaboration as a lensed quasar candidate from the blue - near - red search , a number of other search methods within strides were examined and also seen to identify the system as a candidate . these other search methods are described in more detail in @xcite . we provide here a brief summary : ( 1 ) the gaussian mixture model ( gmm ) method of @xcite , which uses supervised machine learning in a five - dimensional optical plus infrared color space and identified as a candidate pair ( d+g1 not found separately in @xmath18 band due to blending noted above ) ; ( 2 ) chitah @xcite , which uses pixel - based automatic recognition on @xmath4 cutout images and identified the system as a candidate pair ( not flagged as quad because fourth image g2 is too red ) ; and ( 3 ) the artificial neural network ( ann ) method of @xcite , which uses @xmath29 and @xmath30 magnitudes and identified as a candidate extended quasar . figure [ fig : quadcolor ] shows the positions of des catalog objects a , b+g2 , and d+g1 in the color space used by the gmm method , illustrating how the system was flagged as a candidate due to the quasar - like colors of its components . spectroscopic observations of were carried out using the gemini multi - object spectrograph ( gmos - s ) on the gemini south telescope , as part of a larger gemini large and long program ( llp ) ( pi e. buckley - geer ; program ids gs-2015b - lp-5 , gs-2016a - lp-5 ) of spectroscopic follow - up for des strong lensing systems and for des photometric redshift ( photo - z ) calibrations . gmos - s was used in multi - object spectroscopy ( mos ) mode on 9 dec 2015 ut to take spectra of the three blue objects a , b , and d. the @xmath14-band gmos - s acquisition image is shown in figure [ fig : images ] ( top right ) . a single slit mask was used , with one slit for a and a second slit for b and d together ; see blue slits in figure [ fig : images ] ( bottom right ) . ( another 38 slits were assigned to unrelated des photo - z calibration targets . ) two sets of spectra were taken : blue data using the b600 grating ( dispersion @xmath31 pixel@xmath32 ) and red data using the r400 grating ( dispersion @xmath33 pixel@xmath32 ) . we used 4@xmath34900 second exposures for cosmic ray rejection and processed the data using the iraf gemini reduction package . the seeing was @xmath35 , measured from spectra of mask - alignment stars in the science data . from the extracted ( but not flux calibrated ) 1d spectra shown in figure [ fig : spectra ] ( top panels ) , we clearly see that all three blue images a , b , and d show strong quasar emission lines at the same redshift , specifically ly@xmath36 1216 and civ 1549 in the blue spectra , and ciii ] 1909 and mgii 2800 in the red spectra . from the ciii ] line , which shows the cleanest , most symmetric line profile among these four broad emission features , we use the emsao task in the iraf external package rvsao @xcite to report a redshift @xmath1 for the quasar . to compare the spectra of the three blue images in more detail , we also show in figure [ fig : spectra ] ( bottom panels ) the fractional differences between the blue and red spectra of images a and d relative to those of image b , which has the spectra with the most counts . we first rescale the spectra of a and d so that they each have the same median counts as those of b over the respective blue and red wavelength ranges plotted . the results in figure [ fig : spectra ] show generally good agreement among the spectra , especially for the red spectra . for the blue spectra , we do see an overall slope for the spectrum of a relative to that of b ; we attribute this slope to extinction differences between the lines of sight to these two images , and/or to differences in atmospheric differential refraction between the two differently oriented slits ( figure [ fig : images ] ) used to observe images a and b. in addition , we also see conspicuous differences between the spectra of d and b at the locations of the strong ly@xmath36 and civ emission lines , and to a lesser extent at the ciii ] line . we attribute these differences to the effects of microlensing by stars in the lensing galaxy , manifested as flux ratio differences , in pairs of lensed quasar images , for the continuum vs. the emission line regions in the spectra ( e.g. , * ? ? ? subsequent to these observations that confirmed the three blue images as having the same quasar redshift , we designed a second mos slit mask targeting the central lensing galaxy g1 and the fourth image g2 , each allocated to a single slit . the slits for g1 and g2 are shown in red in figure [ fig : images ] ( bottom right ) . red r400 observations were obtained in the gemini south semester 2016a observing queue , on 9 apr 2016 , under @xmath37 seeing conditions and using the same observing setup as above . blue b600 observations were also put into the queue , but no data were obtained before the system set in 2016a . we show the r400 spectra in figure [ fig : more_spectra ] , where we see that g2 shows broad ciii ] and mgii emission at the same redshift as found for the three blue images , while g1 shows ca h+k absorption lines , indicating that g1 is an early - type lensing galaxy with redshift @xmath2 . however , we also see that the g1 spectrum unexpectedly shows the same ciii ] emission line as in the lensed quasar spectra , leading us to suspect there is contaminating light from the neighboring lensed quasar image d , which is only about @xmath38 away ( cf . the @xmath37 seeing ) . likewise , object b , the brightest quasar image , is only about @xmath37 away from the substantially fainter object g2 , suggesting that the ciii ] line seen in the g2 spectrum is similarly contamination from object b. using the image modeling code galfit @xcite , we simulate image b as a moffat profile with a fwhm of @xmath37 and find that about 12% of its light falls within the 1-wide slit and 2-long spectral extraction aperture used for g2 . though seemingly small , this 12% fraction actually amounts to about 80% of the @xmath24-band light from g2 itself , given that @xmath39 and @xmath40 from table 1 of @xcite . ( we use @xmath24 band as the ciii ] line falls within that filter . ) this estimated contamination is thus substantial and unfortunately we are unable to correct for it as image b lies off the slit for g2 ( figure [ fig : images ] , bottom right ) and we do not have a concurrent spectrum of b. we thus can not rule out contamination from b as the source of the ciii ] line seen in the g2 spectrum ( likewise for d and g1 ) and will need future observations to verify whether g2 indeed shows quasar emission features or not . was identified as a candidate lensed quasar quad system from the des first - year data . subsequent follow - up spectroscopy confirmed the three bright blue objects in this system to be the images of a quasar at redshift @xmath1 , lensed by an early - type red galaxy with redshift @xmath2 . another reddened object in the system , possibly a blend of a perturbing galaxy and the fourth lensed quasar image , was also observed spectroscopically , though without conclusive results . our companion paper @xcite presents a detailed model of the system , from which we expect the longest time delay in the system to be about 80 days , making well suited @xcite for time delay measurements , which we are undertaking via a monitoring campaign within the strides collaboration . the lensing model also constrains the mass of the possible perturbing galaxy and thus provides information about substructure in the lensing mass distribution . further spectroscopic follow - up and high resolution imaging data should provide more needed details about the main lensing galaxy and its environment . we thus expect this quad system to be particularly useful for the application of time delay cosmography ( e.g. * ? ? ? * ) and substructure studies ( e.g. , * ? ? ? heralds a much larger sample of some 20 of these very rare and valuable quad lensed quasar systems anticipated to be discovered by the dark energy survey . funding for the des projects has been provided by the u.s . department of energy , the u.s . national science foundation , the ministry of science and education of spain , the science and technology facilities council of the united kingdom , the higher education funding council for england , the national center for supercomputing applications at the university of illinois at urbana - champaign , the kavli institute of cosmological physics at the university of chicago , the center for cosmology and astro - particle physics at the ohio state university , the mitchell institute for fundamental physics and astronomy at texas a&m university , financiadora de estudos e projetos , fundao carlos chagas filho de amparo pesquisa do estado do rio de janeiro , conselho nacional de desenvolvimento cientfico e tecnolgico and the ministrio da cincia , tecnologia e inovao , the deutsche forschungsgemeinschaft and the collaborating institutions in the dark energy survey . the collaborating institutions are argonne national laboratory , the university of california at santa cruz , the university of cambridge , centro de investigaciones energticas , medioambientales y tecnolgicas - madrid , the university of chicago , university college london , the des - brazil consortium , the university of edinburgh , the eidgenssische technische hochschule ( eth ) zrich , fermi national accelerator laboratory , the university of illinois at urbana - champaign , the institut de cincies de lespai ( ieec / csic ) , the institut de fsica daltes energies , lawrence berkeley national laboratory , the ludwig - maximilians universitt mnchen and the associated excellence cluster universe , the university of michigan , the national optical astronomy observatory , the university of nottingham , the ohio state university , the university of pennsylvania , the university of portsmouth , slac national accelerator laboratory , stanford university , the university of sussex , texas a&m university , and the ozdes membership consortium . the des data management system is supported by the national science foundation under grant number ast-1138766 . the des participants from spanish institutions are partially supported by mineco under grants aya2012 - 39559 , esp2013 - 48274 , fpa2013 - 47986 , and centro de excelencia severo ochoa sev-2012 - 0234 . research leading to these results has received funding from the european research council under the european union s seventh framework programme fp7/2007 - 2013 ) including erc grant agreements 240672 , 291329 , and 306478 . based on observations obtained at the gemini observatory ( acquired through the gemini observatory archive and processed using the gemini iraf package ) , which is operated by the association of universities for research in astronomy , inc . , under a cooperative agreement with the nsf on behalf of the gemini partnership : the national science foundation ( united states ) , the national research council ( canada ) , conicyt ( chile ) , ministerio de ciencia , tecnologa e innovacin productiva ( argentina ) , and ministrio da cincia , tecnologia e inovao ( brazil ) abazajian , k. n. , et al . 2009 , , 182 , 543 agnello , a. , et al . 2015 , , 448 , 1446 agnello , a. , et al . 2015 , , 454 , 1260 agnello , a. , et al . 2017 , , submitted belokurov , v. , et al . 2009 , , 392 , 104 bertin , e. , & arnouts , s. 1996 , , 117 , 393 bonvin , v. , et al . 2017 , , 465 , 4914 bouy , h. , et al . 2013 , , 554 , 101 browne , i. w. a. , et al . 2003 , , 341 , 13 chan , j. h. h. , et al . 2015 , , 807 , 138 dalal , n. , & kochanek , c. s. 2002 , , 572 , 25 dark energy survey collaboration , arxiv : astro - 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we report the discovery and spectroscopic confirmation of the quad - like lensed quasar system found in the dark energy survey ( des ) year 1 ( y1 ) data . this system was discovered during a search for des y1 strong lensing systems using a method that identified candidates as red galaxies with multiple blue neighbors . consists of a central red galaxy surrounded by three bright ( @xmath0 ) blue objects and a fourth red object . subsequent spectroscopic observations using the gemini south telescope confirmed that the three blue objects are indeed the lensed images of a quasar with redshift @xmath1 , and that the central red object is an early - type lensing galaxy with redshift @xmath2 . is the first quad lensed quasar system to be found in des and begins to demonstrate the potential of des to discover and dramatically increase the sample size of these very rare objects .

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Proceed to summarize the following text: -input - multiple - output ( mimo ) communications @xcite have continued to be one of the key technologies of the next generation wireless systems due to their potential of providing higher data rate and better reliability compared to the conventional single - input - single - output ( siso ) systems . when the channel state information ( csi ) is available at both the transmitter and receiver , it is well known that the closed - loop gain can be further acquired by jointly designing the precoder and the equalizer . among these closed - loop transceiver design schemes , singular - value - decomposition ( svd)-based linear transceiver decomposes the mimo channel into multiple parallel subchannels and is known to achieve the channel capacity if proper power allocation @xcite is applied . however , due to the variation of the signal - to - noise - ratio ( snr ) in each subchannel , the bit error rate ( ber ) performance is dominated by the subchannel with the worst snr . consequently , without sophisticated bit - allocation schemes , fundamental trade - off between the ber and capacity can not be avoided in this type of design @xcite . in addition to the svd - based linear design , geometric - mean - decomposition ( gmd)-based nonlinear transceiver design has also been proposed @xcite . with the help of gmd @xcite , the mimo channel is decomposed into multiple subchannels with identical snr , and hence the simple identical bit allocation can be used for all subchannels . it has also been shown that the gmd - based transceiver under the zero - forcing ( zf ) constraint asymptotically achieves both the optimal ber and capacity simultaneously at sufficiently high snr . due to these nice properties , various extensions and generalizations of gmd - based transceivers have been reported in the literature @xcite . as the gmd is the core of many advanced mimo transceiver designs , the associated implementation issues started to draw researchers attention @xcite . in @xcite , a scaled geometric mean decomposition algorithm was proposed in order to simplify the detection logic . in @xcite , the authors presented a constant throughput of gmd implementation which also supports hardware sharing between precoding and signal detection modules . in this correspondence , a new implementation issue of the gmd algorithm is addressed . we noticed that all the existing gmd algorithm requires the computation of the geometric mean @xmath0 of all the positive singular values of the matrix to be decomposed in the initialization step . this requires the capability of computing the @xmath1th root @xmath2{a}$ ] of some positive real number @xmath3 , in which @xmath4 for all @xmath5 . for special cases where @xmath6 with @xmath7 being some positive integer , it is possible to decompose the computation of @xmath0 into successive geometric mean computations of two numbers , @xmath8 where the square root operation can be carried out efficiently using cordic - based computing @xcite . for @xmath9 , while finding @xmath2{a}$ ] can generally be achieved by using newton s type of @xmath1th - root algorithm @xcite , the difficulty lies in the fact that a good initial guess is often required for the algorithm to converge . another possible way of computing @xmath2{a}$ ] is to first transform @xmath10 into logarithmic domain and then convert it back after divided by @xmath1 . cordic - based algorithms can be used to implement algorithmic and exponential functions , but the inherent bounded input range often limits their applications unless extra pre- and post - processing steps are applied . an alternative method is to use lookup tables and/or a piecewise polynomial ( including linear ) algorithms to realize both the logarithmic and exponential functions . a mass of memory and extra computations are therefore required to ensure the accuracy for such high dynamic - range computations . in this correspondence , we propose an iterative gmd ( igmd ) algorithm based on the technique of successive approximation . the proposed algorithm has a regular structure and is applicable to matrices of any dimension . it does not require explicit computation of the @xmath1th root , and hence eliminates the hardware difficulties mentioned above . the convergence proof of proposed iterative gmd algorithm is provided , and numerical results show the performance of the proposed algorithm converges to that of the gmd very quickly within a few iterations . _ notations : _ throughout this paper , matrices and vectors are set in boldface , with uppercase letters for matrices and lower case letters for vectors . the superscripts @xmath11 , @xmath12 denote the transpose and conjugate transpose of a matrix , respectively . @xmath13 denotes the diagonal matrix with diagonal elements @xmath14 . @xmath15_{p , q}$ ] and @xmath15_{m : n , p : q}$ ] denote the @xmath16th component and the submatrix formed by the consecutive @xmath17th to @xmath18th rows and @xmath19th to @xmath20th columns of @xmath21 , respectively . the proposed iterative gmd algorithm can be described as follows . * initialization : * given the matrix @xmath22 of rank @xmath23 , the algorithm starts with some general orthogonal decomposition of @xmath24 @xmath25 where @xmath26 and @xmath27 are both semi - unitary , and @xmath28 is upper - triangular . for general @xmath29 , @xmath30 , and @xmath1 , one can always choose the singular value decomposition for initialization . for special cases where @xmath24 is full column rank with @xmath31 , other orthogonal decompositions such as the qr decomposition @xcite can also be used . the algorithm then initializes by setting @xmath32 , @xmath33 , @xmath34 , and starts with iteration index @xmath35 . * iteration : * in each iteration , the algorithm performs @xmath36 stages of operations as the stage index @xmath37 ranges from @xmath38 to @xmath36 . at stage @xmath37 , the algorithm first computes the singular value decomposition for the @xmath39 submatrix of @xmath40 @xmath41=\mathbf{u}^{(k)}_\gamma \boldsymbol{\sigma}^{(k)}_\gamma \mathbf{v}_\gamma^{(k)\mathrm{h } } , \label{r_tilde_22}\end{aligned}\ ] ] where the singular matrices @xmath42 and @xmath43 are both unitary , and @xmath44 is a diagonal matrix with singular values @xmath45 and @xmath46 . without loss of generality , we assume @xmath47 . after the singular values are obtained , carefully designed planar rotations are then applied to obtain an upper triangular matrix with positive diagonal elements @xmath48 and @xmath49 , where @xmath50 is a continuous mapping from @xmath51 to @xmath52 with some desired property to be discussed in details shortly . in matrix notations , we then have @xmath53 . \label{eq : planar_rotation_v1}\end{aligned}\ ] ] note that the planar rotations @xmath54 and @xmath55 applied in ( [ eq : planar_rotation_v1 ] ) always exist as long as @xmath56^\mathrm{t}$ ] multiplicatively majorizes @xmath57^\mathrm{t}$ ] , or equivalently when @xmath58 holds @xcite . it is easy to verify that the matrices @xmath54 and @xmath55 can be constructed as @xmath59,\label{eq : phi_l}\\ \boldsymbol{\phi}^{(k)}_{\mathrm{r}}&=\left[\begin{array}{cc}c & -s \\ s & c\end{array}\right],\end{aligned}\ ] ] where @xmath60 combining the relations in ( [ r_tilde_22 ] ) and ( [ eq : planar_rotation_v1 ] ) , we then obtain @xmath61 , \label{eq : transformation}\end{aligned}\ ] ] where @xmath62 , and @xmath63 . since @xmath64 and @xmath65 are both products of unitary matrices , they are unitary matrices as well . after @xmath66 and @xmath67 are obtained , @xmath68 , and @xmath69 are then constructed from the identity matrix @xmath70 with the submatrix @xmath71_{k : k+1,k : k+1}$ ] and @xmath72_{k : k+1,k : k+1}$ ] replaced by @xmath73 and @xmath74 , respectively . the matrices @xmath40 , @xmath75 , and @xmath76 are then updated as @xmath77 it is clear that @xmath40 remains upper - triangular , and @xmath75 and @xmath76 both remain unitary after ( [ eq : update_r])-([eq : update_s ] ) are performed at the end of each stage . if the stage index @xmath37 is smaller than @xmath36 , the algorithm set @xmath78 and performs the procedure ( [ r_tilde_22])-([eq : update_s ] ) . otherwise , the algorithm set the iteration index @xmath79 and start a new iteration unless the prescribed number of iterations is attained . for the convenience of the subsequent discussion , we denote @xmath80}$ ] , @xmath81}$ ] , @xmath82}$ ] as the updated @xmath75 , @xmath40 , @xmath76 respectively at the end of @xmath83th stage in the @xmath84th iteration . then the following relations hold for the proposed iterative geometric mean decomposition algorithm : @xmath85}=&\mathbf{q}^{[\ell]}\mathbf{g}^{(1)\mathrm{t}}_\mathrm{l}\ldots\mathbf{g}^{(k-1)\mathrm{t}}_\mathrm{l},\label{eq : q_recursive}\\ \mathbf{s}^{[\ell+1]}=&\mathbf{s}^{[\ell]}\mathbf{g}^{(1)}_\mathrm{r}\ldots\mathbf{g}^{(k-1)}_\mathrm{r},\\ \mathbf{r}^{[\ell+1]}=&\mathbf{g}^{(k-1)}_\mathrm{l}\cdots\mathbf{g}^{(1)}_\mathrm{l}\mathbf{r}^{[\ell]}\mathbf{g}^{(1)}_\mathrm{r}\ldots\mathbf{g}^{(k-1)}_\mathrm{r}. \label{eq : r_recursive}\end{aligned}\ ] ] for all @xmath86 . here @xmath87}$ ] , @xmath88}$ ] , and @xmath89}$ ] are defined as @xmath90 , @xmath91 , and @xmath92 , respectively . the planary rotation matrices @xmath93 and @xmath94 clearly also depend on the iteration index @xmath84 , but the dependency is not denoted explicitly in ( [ eq : q_recursive])-([eq : r_recursive ] ) for simplicity as long as no confusion results . in the following section , we show that one can design the mapping @xmath50 such that @xmath95}\right]_{k , k}=\bar{\sigma}=\left(\prod_{k=1}^k \left[\mathbf{\tilde{r}}\right]_{k , k}\right)^{1/k } , \end{aligned}\ ] ] for all @xmath96 . the geometric mean decomposition is therefore obtained when the algorithm converges . for the ease of following discussions , we introduce several new notations . for given @xmath97 , we define a subset @xmath98 : @xmath99 where @xmath100^\mathrm{t}$ ] . we also define continuous mappings @xmath101 , given by @xmath102 \right)&= \left[\begin{array}{c } \mathbf{x}_{1:j-1 } \\ \omega\left(x_j , x_{j+1}\right)\\ \frac{x_j x_{j+1}}{\omega\left(x_j , x_{j+1}\right ) } \\ \mathbf{x}_{j+2:k } \end{array}\right ] . \label{eq : tj}\end{aligned}\ ] ] if we denote the vector on the main diagonal of @xmath81}$ ] as @xmath103}$ ] , then the diagonal vectors of @xmath104}$ ] and @xmath81}$ ] can be related from ( [ eq : update_r ] ) and ( [ eq : r_recursive ] ) using the new notations @xmath105 } & = t^{(k-1)}\left ( \ldots\left(t^{(2)}\left(t^{(1)}\left(\mathbf{r}^{[\ell]}\right)\right)\right)\ldots\right)\label{eq : definition_t}\\ & = t\left(\mathbf{r}^{[\ell]}\right)=t^{\ell+1}\left(\mathbf{r}^{[0]}\right),\end{aligned}\ ] ] where @xmath106 , @xmath107 is the @xmath108-fold repeated composition of the mapping @xmath109 , and @xmath110}=\mathrm{diag}\left\{\mathbf{\tilde{r}}\right\}$ ] . with the aforementioned notations , we now present the main results for the @xmath50 design . @xmath111}=\bar{\sigma}\mathbf{1}$ ] if the mapping @xmath112 satisfies the following property @xmath113 for all @xmath114 , and the equality holds if and only if @xmath115 . given @xmath110}=\left[r^{[0]}_1,\ldots , r^{[0]}_k\right]=\mathrm{diag}\{\mathbf{\tilde{r}}\}$ ] , we let @xmath116}_k$ ] , and consider the function @xmath117 , @xmath118 . from the arithmetic - geometric inequality , @xmath119 it is clear that @xmath120 attains its absolute minimum in @xmath121 at @xmath122 . from the definition of @xmath109 and ( [ eq : condition_omega ] ) , it is clear that @xmath123 for all @xmath124 , and the equality holds if and only if @xmath125 . it follows readily that @xmath109 , which is a composite mapping of @xmath126 , satisfies @xmath127 with the equality holds if and only if @xmath128 . consequently , @xmath129}=f(t(\mathbf{r}^{[\ell-1]}))$ ] is a monotonically decreasing sequence in @xmath52 , and hence is guaranteed to converge to the greatest lower bound @xmath130 @xcite . furthermore , since @xmath131 is continuous , we have @xmath132}\right)\right)=f\left(\lim_{\ell\rightarrow \infty}t\left(\mathbf{r}^{[\ell-1]}\right)\right)=k\bar{\sigma},\end{aligned}\ ] ] which is attained when @xmath133}\right)=\bar{\sigma}\mathbf{1}$ ] . as a result , we have @xmath134}=\bar{\sigma}\mathbf{1}$ ] , which completes the proof . there exists potentially many functions that satisfy condition ( [ eq : condition_omega ] ) . the geometric mean @xmath135 clearly satisfies ( [ eq : condition_omega ] ) as @xmath136 always holds due to the am - gm inequality , and equality holds if and only if @xmath115 . in addition to @xmath137 , the arithmetic mean @xmath138 is another choice that also satisfies ( [ eq : condition_omega ] ) . this can be observed by squaring both sides of the am - gm inequality @xmath139 as a result , @xmath140 , and the equality holds if and only if @xmath115 . note that @xmath141 is simply the harmonic mean function @xmath142 while @xmath143 , it is clear that @xmath144 also satisfies ( [ eq : condition_omega ] ) from the same relation we obtained in ( [ eq : am_gm_square ] ) . based on @xmath145 , @xmath146 , and @xmath147 , we can then construct our iterative gmd algorithms : igmd - am , igmd - gm , and igmd - hm , respectively . since these mappings are highly nonlinear , it is very challenging to compare the convergence speed of the proposed algorithm in these three constructions . in fact , the convergence behaviour not only depends on the topological property of the mapping but also depends on how the algorithms are initialized . from the implementation point of view , igmd - am and igmd - hm may have some advantages over the igmd - gm as only basic arithmetic operations are required in computing @xmath148 and @xmath144 rather than the square root operations required in computing @xmath137 . in this section , we present some simulation results of the proposed igmd algorithm under three different constructions : igmd - am , igmd - gm , and igmd - hm . throughout the simulation , we assume standard @xmath149 i.i.d . rayleigh fading channel in which every element in the channel matrix @xmath24 is modelled as zero - mean circularly symmetric complex gaussian random variable with unit variance . to highlight the applicability of the proposed algorithm in the challenging large @xmath9 case , we choose @xmath150 in the simulation . each simulation point in the figure is averaged over @xmath151 channel realizations . [ fig : convergence_mse_svd ] and fig . [ fig : convergence_mse_qr ] show the mean - square - error ( mse ) convergence behaviour of the diagonal elements of @xmath40 using svd and qr as initializations , respectively . for svd initialization , we propose an alternative interleaved - svd ( intrlv - svd ) , defined as @xmath152 where @xmath153 \right\}$ ] and @xmath154 to enable more efficient averaging in each stage . for qr initialization , the qr factorization with vblast ordering ( vbqr ) @xcite is also proposed to speed up convergence . as the diagonal elements of @xmath92 in vbqr generally has less spread than those in qr , vbqr initialization provides a smaller mse when used in the initialization as shown in fig . [ fig : convergence_mse_qr ] . due to the highly nonlinear nature of the corresponding mappings @xmath155 , @xmath156 , and @xmath157 , it is very difficult to analytically compare the convergence behaviour of the three igmd constructions . hence we resort to numerical simulations and leave the more challenging theoretical analysis to our future work . the simulation results in fig . [ fig : convergence_mse_svd ] and fig . [ fig : convergence_mse_qr ] show that the igmd - hm has the fastest convergence rate , followed by the igmd - gm , and the igmd - am when qr , vbqr , and svd are used as initialization . on the other hand , when intrlv - svd are used , the igmd - am has the fastest convergence rate , followed by the igmd - gm , and finally the igmd - hm . in the second simulation setting , we investigate the convergence behaviour of the proposed igmd on the error rate performance of a gmd - based mimo system . a @xmath158 gmd - based zero - forcing tomlinson - harashima precoded ( zfthp ) mimo system @xcite using @xmath159-quadrature amplitude modulation is simulated . [ fig : convergence_ber_qr ] and fig . [ fig : convergence_ber_vbqr ] show the error rate of the proposed igmd using qr and vbqr , respectively . by comparing the ber of zfthp - qr in fig . [ fig : convergence_ber_qr ] and zfthp - vbqr in fig . [ fig : convergence_ber_vbqr ] , it is clear that the vbqr provides a better initialization for the proposed igmd , and results in faster convergence . at the @xmath38st iteration , the error rate performance of igmd - am and igmd - hm appears to be similar . however , for iteration number greater than @xmath38 , the igmd - gm and igmd - hm both outperform the igmd - am and perform very close to the optimal gmd after @xmath160 iterations . [ fig : convergence_ber_svd ] and fig . [ fig : convergence_ber_intrlvsvd ] show the error rate of the proposed igmd using regular svd and interleaved svd , respectively . in medium to low snr region , the igmd - hm shows comparable or even better performance than the igmd - gm , while for sufficiently high snr , the igmd - gm performs the best , followed by the igmd - hm , and the igmd - am . on the contrary , the proposed igmd using interleaved svd as initialization shows very different convergence behaviour . the igmd - hm with interleaved svd performs much worse compared to the igmd - am and igmd - gm . for most snr region of practical interests in this setting , the igmd - am is comparable to the igmd - gm for iteration number greater than @xmath38 . from fig . [ fig : convergence_ber_vbqr ] and fig [ fig : convergence_ber_intrlvsvd ] , it is observed that the proposed igmd - intrlv - svd - gm and igmd - intrlv - svd - am achieve even better error rate than the igmd - vbqr - gm and igmd - vbqr - hm after @xmath160 iterations . an iterative geometric mean decomposition algorithm for mimo communications is proposed . the proposed algorithm has a regular structure and can be easily adapted to accommodate problems of different dimensions . through iteratively updating the constituents matrices , the algorithm is able to converge to the true gmd without performing the @xmath1th root computation . the convergence of the algorithm under certain sufficient condition is proved analytically and verified through computer simulations . the authors would like to thank prof . chiu - chu melissa liu for the helpful discussion . zhang , a. kavi , and k. m. wong , `` equal - diagonal qr decomposition and its application to precoder design for successive - cancellation detection , '' _ ieee trans . inf . theory _ 51 , no . 1 , 154172 , jan . 2005 . f. liu , l. jiang , and c. he , `` advanced joint transceiver design for block diagonal geometric mean decomposition based multiuser mimo system , '' _ ieee transactions vehicular technology _ , vol . 692703 , feb . 2010 . liu and p. p. vaidyanathan , `` generalized geometric mean decomposition and dfe transceiver design part i : design and complexity , '' _ ieee trans . signal process . _ , vol . 60 , no . 6 , pp . 31123123 , jun . 2012 . k. meher , j. vallas , t .- b . juang , k. sridharan , and k. maharatna , `` @xmath161 years of cordic : algorithms , architectures , and applications , '' _ ieee trans . circuits syst . i , reg . papers _ , vol . 56 , pp . 18931907 , sep . 2009 . w. wolniansky , 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 , '' in _ proc . ursi international symposium on signals systems and electronics _ , 1998 , pp . 295300 . g. d. golden , g. j. foschini , r. a. valenzuela , and p. w. wolniansky , `` detection algorithm and initial laboratory results using v - blast space time communications architecture , '' _ electronic letters _ , vol . 1416 , jan .

this correspondence presents an iterative geometric mean decomposition ( igmd ) algorithm for multiple - input - multiple - output ( mimo ) wireless communications . in contrast to the existing gmd algorithms , the proposed igmd does not require the explicit computation of the geometric mean of positive singular values of the channel matrix , and hence substantially reduces the required hardware complexity . the proposed igmd has a regular structure and can be easily adapted to solve problems with different dimensions . we show that the proposed igmd is guaranteed to converge to the perfect gmd under certain sufficient condition . three different constructions of the proposed algorithm are proposed and compared through computer simulations . numerical results show that the proposed algorithm quickly attains comparable performance to that of the true gmd within only a few iterations . geometric mean decomposition ( gmd ) , mimo , qr , vblast , tomlinson - harashima precoding ( thp )

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Proceed to summarize the following text: an extreme geomagnetic storm on 20 november 2003 was caused by the interaction of the earth s magnetosphere with an interplanetary magnetic cloud ( mc ) , whose magnetic helicity , @xmath0 , was positive . the very strong magnetic field in the mc of up to @xmath1 nt had a long - lasting negative @xmath2 component ( up to @xmath3 nt ) . these circumstances were crucial in identifying the solar source for the mc . and definitely associated the mc with the filament eruption in the active region ( ar ) 10501 ( we will use henceforth the last three digits for brevity ) on 18 november . these authors considered the direction of the axial magnetic field in the pre - eruptive filament to correspond to the expected projection @xmath4 . with such a direction of the axial field , the current helicity of the filament , @xmath5 , is positive . from the condition @xmath6 , which is valid for a linear force - free magnetic field , it follows that the magnetic helicity is also positive . later found a correspondence between the flare reconnected magnetic flux , measured as the flare ribbon flux , and the poloidal magnetic flux of the mc under the assumption that the mc was a part of a magnetic flux rope with a length of 0.52 au . the studies by , , and gave the impression of an acceptable correspondence between the conditions of the eruption in ar 501 and the parameters of the mc observed in the earth orbit : @xmath7 . the study of changed this situation . from the observed morphological features they found that the large - scale magnetic field in ar 501 had a negative helicity sign . this finding seemingly contradicted what was expected from the magnetic helicity conservation requiring the same sign of the magnetic helicity in the ar and mc . this circumstance has raised a question , why the ar , which had a global negative magnetic helicity , could expel a positive - helicity mc to the interplanetary medium . one possible answer was proposed by , who found a localized positive helicity injection in the southern part of ar 501 and concluded that the right handedness of the observed mc was due to the ejection from this portion of the ar . on the other hand , in their study of 12 interplanetary mcs and related solar active regions have found : _ i _ ) a significant difference between the total twist of the magnetic field inside active regions , @xmath8 , and that in the mc , @xmath9 ; _ ii _ ) the absence of any significant sign relationship between them . the authors used the linear force - free approximation , @xmath10 is a constant . the dipole scale , @xmath11 , was measured as the distance between the centroids of the positive and negative fluxes in the magnetogram of an ar . the magnetic field in an mc was fit with the lundquist magnetic model with @xmath12 au length . findings ( _ i _ ) and ( _ ii _ ) have compelled to conclude that `` _ _ magnetic clouds associated with active region eruptions are formed by magnetic reconnection between these regions and their larger - scale surroundings , rather than simple eruption of preexisting structures in the corona or chromosphere _ _ '' . described a similar phenomenon analytically by using the idealized example of the axially - symmetric reconfiguration of two twisted magnetic fluxes from their unconnected initial state to the connected relaxed state . they have shown that magnetic reconnection can reverse the twist direction of a flux rope emerging into preexisting fields under the conservation of the total relative magnetic helicity . the complex reconnection of a flux rope with the adjacent field in complex magnetic topology has been also described by , _ e.g. _ , , , and . @xcite ( @xcite ) have found observational evidence of magnetic reconnection between the internal field belonging to the eruptive filament and the preexisting coronal field . this is a phenomenon of plasma dispersal from an eruptive filament over the solar surface that is visible as the disintegration of the filament . the whole mass of an eruptive filament or a considerable fraction of its mass does not leave the sun as a part of a cme . the motion of the cool plasma of the eruptive filament continues along new magnetic field lines passing inside the eruptive filament and ending far on the solar surface . clouds of such plasma can screen the emission of compact sources in active regions as well as the emission from quiet solar areas . absorption phenomena can be observed in microwaves and also in the he ii 304 line . events of such a kind are associated with active region eruptions . they have been rarely detected for observational reasons . the analysis of the solar geoeffective event of 18 november 2003 by ( hereafter paper i ) and ( hereafter paper ii ) has revealed that the major eruption in this event , _ i.e. _ , the eruption of the u - shaped filament , which we call f1 , from ar 501 , was also not a simple one . the eruptive filament bifurcated and transformed into a large y - shaped cloud , which moved from the region of bifurcation ( rb ) to the south west across the solar disk toward the limb . figure [ f - shape_transform ] illustrates what has happened to the eruptive filament , as shown by the h@xmath10 images produced by the kanzelhhe solar observatory ( kso ) , the _ extreme - ultraviolet imaging telescope _ ( eit ; ) , on board the _ solar and heliospheric observatory _ ( soho ) , and the _ spectroheliographic x - ray imaging telescope _ ( spirit ; and ) , on board the _ complex orbital near - earth observations of activity of the sun _ ( coronas - f ) satellite . line - center image ( kso ) . the frames denote the fields of view of the four images shown in the lower row ( b e ) . the axes are in arc seconds from the solar disk center . the turquoise oval in all the images denotes the region rb where the filament bifurcated . the yellow curves roughly outline the frontal edge of the filament before and during the eruption . ( c ) soho / eit image in the 195 channel . ( d ) kso h@xmath10 image observed in the far blue wing . ( e ) y - like cloud in the coronas - f / spirit 304 image . ] the masses of the y - like cloud and the pre - eruption filament were similar ( paper i ) ; on the other hand , remnants of the filament were not evident in the southwestern cme observed by the _ large angle and spectrometric coronagraph _ ( lasco ) that was previously regarded as the source of the 20 november geomagnetic storm . the observations analyzed in paper i and paper ii suggest a possible additional eruption in the interval from 08:07 to 08:14 ut above the bifurcation region close to the solar disk center that could be the source of the interplanetary mc on 20 november . these facts disfavor the simple scenario , in which the 20 november mc is considered as a flux rope formed directly from a structure initially associated with the pre - eruptive filament f1 in region 501 . as mentioned , the right - handed mc produced in this event and responsible for the superstorm had a very strong magnetic field near earth of up to @xmath1 nt and @xmath13 nt . according to , its inclination to the ecliptic plane was @xmath14 , and the magnetic flux in this plane was @xmath15 mx ; however , its significant part could be lost by reconnection in the interplanetary space . in ( hereafter paper iv ) , we additionally find that the mc was compact , with a size of about 0.2 au , and had some atypical properties , such as a wide range of proton temperatures , from @xmath16 k to @xmath17 k ; its magnetic structure was closed , disconnected from the sun , and probably had a spheromak configuration . the present paper ( hereafter paper iii ) endeavors to understand how the catastrophe of the eruptive filament could occur and create the right - handed spheromak - like mc . section [ s - outline ] outlines the eruptive event and results of its analysis . section [ s - helicity ] analyzes the helicity in ar 501 . in section [ s - large_scale_config ] we address the causes of the bifurcation of the eruptive filament . in section [ s - mc_formation ] we try to understand how the mc could be formed . section [ s - summary ] briefly summarizes the outcome of this study . paper i and paper ii have revealed a few eruptive episodes in the complex event of 18 november 2003 . here we outline and illustrate them using the images from eit in figures [ f - history]a[f - history]f and the time profiles in figure [ f - history]g produced in hard x - rays ( hxr ) by the _ reuven ramaty high - energy solar spectroscopic imager _ ( rhessi : ) and , for the rhessi nighttime , in microwaves by the us air force _ radio solar telescope network _ radiometers in san vito . the pre - eruptive u - shaped filament f1 rooted in ar 501 was pointing in the plane of the sky toward a small ring - like structure rb ( figure [ f - history]a ) . active region 503 was located north of rb . episode e1 presumably gave rise to a first southeastern cme ( cme1 ) responsible for am elongated dimming d1 , which started to develop during the interval shown in figure [ f - history]b . a jet - like ejection , ej , in figure [ f - history]c ( e2 ) triggered the motion of filament f1 , but did not produce any cme directly . the filament accelerated in a weak episode e3 ( figure [ f - history]g ) . the eruptive filament , whose trajectory crossed a topological discontinuity located at a height of about 100 mm , collided with a coronal structure above rb ( e4a ) and bifurcated ( e4b ) , apparently rolling around it . in response to the interaction , brightenings overlapping rb appeared in figure [ f - history]d . they rotated clockwise and , after e4b , vanished in figure [ f - history]e . episode e4c was presumably related to the onset of the second southwestern cme ( cme2 ) . in figure [ f - history]e , dimmings developed at the previous position of rb and west of it ( d2 ) ; a central brightening ( not mentioned in papers i and ii ) appeared in ar 503 ( e4d ) , indicating its involvement in an eruption . after the chain of eruptions e1e4 , regions d1 , d2 , d3 , and the former rb region dimmed ( figure [ f - history]f ) ; the bifurcated filament f1 apparently transformed into a y - like ` cloud ' moving across the solar disk ( figures [ f - shape_transform]c[f - shape_transform]e ) . one more ejection probably occurred near the solar disk center during the e4 burst ( figures [ f - history]d , [ f - history]e ) . the ejection was detected in paper ii as a faintly visible round feature , which expanded approximately from the position of the bifurcation region in the images produced by the _ solar x - ray imager _ ( sxi ) on board the _ geostationary operational environmental satellite _ ( goes-12 ) . the observed radial expansion was coordinated with the trajectory of a drifting type iv radio burst in the dynamic radio spectrum using the acceleration profile in figure [ f - history]h . the final speed of this radial expansion was @xmath18 km s@xmath19 . the mass of this ejection should be @xmath20 g ( see paper i ) . the properties of the presumable ejection are very different from those of cme2 ejected at the same time , but appear to match the expectations for a source of the 20 november superstorm . this low - mass , weakly expanding ejection presumably moved along the sun earth line , and therefore its meager thomson - scattered light was insufficient to be detected by the lasco coronagraphs . on the other hand , neither cme1 nor cme2 seem to be a promising candidate to be the source of the superstorm , being able to produce , at most , a glancing blow on the earth s magnetosphere ( paper ii ) . according to three reconstructions of the mc responsible for the geomagnetic superstorm @xcite , its dimensions in the ecliptic plane were @xmath21 au . this size corresponds to an expansion angle of @xmath22 , which is similar to that of the presumable cme ( paper ii ) . to meet the earth , the mc expanding in such a narrow angle should be ejected close to the solar disk center . such a weak expansion is favored , if the mc is disconnected from the sun . these speculations will be supported in paper iv . the observational suggestions and the listed conjectures indicate that the ejection responsible for the superstorm probably originated during the e4 burst . the formation of the ejection started at e4a at a height of about 100 mm ( @xmath23 ) and was completed by the end of e4d . the latter is supported by the appearance of the central brightening in region ar 503 ( figure [ f - history]e ) . a possible height , at which the formation was completed , is of the order of @xmath24 , with @xmath25 being the duration of burst e4 , and @xmath26 km s@xmath19 being an average sun earth transit speed of the icme ( estimated from the decrease of the dst index on 20 november ) . this height can be overestimated by a factor of 23 due to the uncertainties in the velocity as the cme can accelerate as it leaves the sun and decelerate during its transit away from the sun . a probable height , at which the formation was completed , is therefore between @xmath27 and @xmath28 . it is difficult to expect to have direct observations of the processes , which occurred at the heights previously estimated close to the solar disk center . therefore , we are forced to involve indirect observational indications and calculations . we use the observational indication provided by the evolution of the dimming regions d1d3 in figure [ f - dimming ] . their configuration is visible in the eit 195 difference image shown in figure [ f - dimming]a . the time profiles of the average brightness in selected regions having the lowest intensity and longest lifetime are presented in figure [ f - dimming]b . unlike , who defined a dimmed region as the brightness decrease below the quiet - sun brightness level ( the horizontal line in figure [ f - dimming]b ) , we select the dimmings simply by a decrease of the brightness with respect to the pre - event level . our different selection criterion indicates that we are considering a different kind of phenomenon . and @xmath29 . the horizontal line presents the average quiet - sun intensity level in the eit 195 images , two of which were used for the difference in panel ( a ) . ] most likely , the major cause of a dimming ( in our definition ) is the density decrease in the coronal structures due to their expansion . the brightness in the extreme - ultraviolet ( euv ) and soft x - ray images is proportional to the column emission measure . the brightness , @xmath30 , of an expanding closed coronal structure of a linear size , @xmath31 , and area , @xmath32 , filled with a fixed total number of emitting particles , @xmath33 , should be @xmath34 . thus , the expansion alone should result in a considerable brightness decrease and a strong pressure gradient , which causes a secondary subsonic plasma outflow in the footprint regions of a cme @xcite . on the one hand , the outflow is responsible for the redistribution of the coronal plasma from footprints into the expanding volume . on the other hand , due to the subsequent plasma supply from the chromosphere - to - corona transition region , the outflow probably becomes the major factor to recover the brightness in the dimmed regions . this simple consideration also explains why the development of dimming is often observed to start before the eruption , when coronal structures gradually expand during the initiation phase . the time profiles in figure [ f - dimming]b show that different parts of the star - like dimming d3 recovered considerably faster than the long - lived dimmings d1 and d2 . the analysis of the magnetic connectivity shows that the plage region , where dimming d2 occurred , was connected to ar 503 associated with dimming d3 . in addition , a fan of long diverging field lines rooted in the plage region connected it to remote regions far to the south west . possibly , d2 shared a footprint of cme2 and had therefore the time profile typical of long - lived core dimmings . the pronouncedly shorter lifetimes of regions @xmath35 , @xmath29 , and 4 of the atypical dimming d3 hints at its involvement into an eruption , in which a structure disconnected from the sun developed . the disconnection produced a stretched magnetic loop . its subsequent evolution is determined by the relation between the magnetic tension , which tends to shrink the loop , and the opposite influence of the plasma pressure and the plasma outflow responsible for the dimming in its bases . the shrinking duration for the loop with a length of @xmath36 mm is presumably comparable to the time required to stop the plasma outflow , @xmath37 s ( @xmath38 km s@xmath19 is a mean sound speed of the plasma outflow originating from the low corona up to the chromosphere - to - corona transition region ) . this estimate is of the order of the observed formation time of dimming d3 . figure [ f - suggestion ] summarizes the listed observational indications of a possible interaction between the eruptive filament ( red ) and a large static coronal structure ( blue ) . one of its ends is rooted in ar 503 , and the opposite - polarity basis corresponds to a plage region and region of bifurcation , rb . the onset of the interaction and bifurcation of the filament in shown in figure [ f - suggestion]a . in a subsequent expansion of the eruptive filament in figure [ f - suggestion]b , its top brakes at the blue structure , while its lateral portions stretch out in a y - like form . in response to the interaction , dimmings develop in the plage region ( d2 ) and rb as well as dimming d3 at the opposite end of the blue structure rooted in ar 503 . most likely , the eruptive filament was far from ar 503 ( _ cf . _ figures [ f - history ] and [ f - dimming ] ) , which nevertheless appears to be implicated . we shall use this presumable scheme as a hint in our considerations . a major step in our study is the analysis of the coronal magnetic field _ via _ the extrapolation of photospheric magnetograms . for ar 501 we have a vector magnetogram that makes possible a non - linear force - free ( nlff ) field extrapolation within its field of view , @xmath39 . however , the region of interest , including rb , d2 , and d3 in figure [ f - dimming]a , is far west from the vector magnetogram . the only possible way to analyze a larger region is a potential extrapolation of the full - disk soho / mdi magnetograms @xcite . the potential approximation is usually considered to be insufficiently accurate to describe realistic configurations under the presence of significant electric currents . nevertheless , in paper i we have used it successfully to visualize the filament , its height , and the topological discontinuity , whose presence accounted for the apparent disintegration of the eruptive filament . let us try to check , how realistic the results of the potential extrapolation are by comparing them with real coronal loops in euv images . figure [ f - f_lines_center ] compares the field lines computed in the potential approximation , using the method and package of and for the full - disk soho / mdi magnetogram observed on 18 november at 09:35 ut . the spatial resolution is determined by 90 spherical harmonics . the starting points for all of these field lines were chosen manually , and their density does not correspond to the real magnetic field strength distribution . the correspondence between the computed field lines and the visible coronal loops is what is relevant to us . figure [ f - f_lines_center]a presents the eit 195 image observed before the event in gray scale overlaid with a few sets of computed magnetic field lines . the same sets of the field lines are overlaid on the post - event eit 284 image in figure [ f - f_lines_center]b . the red lines embrace the pre - eruptive filament f1 . the light - blue lines correspond to open magnetic field lines rooted in the coronal hole east of ar 501 . the green lines cover a southern filament channel ; the corresponding coronal loops appeared after the event in figure [ f - f_lines_center]b . all other structures are traced by the blue lines . although a one - to - one correspondence is not observed , as expected , the figure shows an acceptable similarity of the computed blue field lines to the structures in the euv images . the pre - event eit 195 image in figure [ f - f_lines_center]a presents rather short loops reaching small heights . the computed field lines correspond to the loops diverging from ar 503 and from the plage region slightly south west of rb . the overall correspondence in shapes and directions between the computed and real structures is observed to the south east and north east of ar 501 . the red field lines above the filament correspond to magnetic structures , which are expected to be seen but are not visible . these lines fairly correspond to the post - eruption arcade to the south of ar 501 in figure [ f - f_lines_center]b . the light - blue open field lines are not expected to be visible . the post - event eit 284 image in figure [ f - f_lines_center]b shows somewhat higher loops . for example , connections between ar 501 and ar 503 become detectable here . in addition to the mentioned features , note that region rb is not seen here . dimming d2 resembling a transient coronal hole has decreased in size with respect to its appearance in figure [ f - dimming ] . we can not observe in these images the structures , which reach still larger heights , because of their lower brightness ( due to a lower plasma density ) against the brighter lower loops . they can only be detected against the sky on the limb . however , the on - limb magnetogram is very uncertain . we therefore have computed the field lines extrapolated from the magnetogram observed on 18 november at 09:35 ut and rotated them to 13 november . this day was characterized by ongoing activity on the east limb . we have built a composed off - limb euv image from the ratios of several eit 195 images observed from 00:00 ut to 05:00 ut , bypassing the intervals of activity . the on - disk part is an average of six eit 195 images observed from 00:00 ut to 01:13 ut , in which the solar rotation was compensated . the result is presented in figure [ f - f_lines_limb ] . the light - blue lines correspond to the open field . in spite of the five - day difference and active conditions , there is an overall correspondence between the calculated field lines and the off - limb structures . in summary , figures [ f - f_lines_center ] and [ f - f_lines_limb ] demonstrate that the extrapolation in the potential approximation presents a more or less realistic picture . furthermore , we observe that despite the ongoing activity , the distribution of coronal magnetic field remained relatively stable . computed the maps of the magnetic helicity flux density injection and , in spite of the mixed helicity signs , showed the existence of a localized positive helicity injection in the southern part of ar 501 . they also concluded that the positive helicity was ejected from this portion of the ar leading to the observed positive - helicity mc . the accumulated positive helicity , as considered , was concentrated in one of the portions of the pre - eruptive filaments , while the helicity of other portions was negative and corresponded to the sign of the global helicity of ar 501 . however , the idea of to account for the positive helicity of the near - earth mc does not seem to be convincing . firstly , the helicity injection rate measured for some part of an active region during a rather short time interval does not provide information on the total helicity accumulated during the preceding evolution . secondly , the fresh idea of the authors about the presence of the opposite - helicity portions in the body of the pre - eruptive filament , in our opinion , does not have a convincing visual support . finally , pointed out that the injection rate of the positive helicity flux was not high , so that such an injection was able to accumulate the helicity appropriate for the mc in , at least , six days . used the orientation of the filament barbs as a morphological indication of a section with a positive twist in the pre - eruptive filament ( figures [ f - shape_transform]a and [ f - shape_transform]b ) . these authors found a filament segment consisting of two sections , which had the same direction of the central axis , but opposite twists . the authors indicated , however , that this determination was relatively ambiguous . in our opinion , the identification of the filament barbs and their usage as a morphological indication is questionable for such a broad filament . according to paper i , the filament was tilted by about @xmath40 to the solar surface , and features as the filament barbs could correspond to different parts located at different heights ( see figures 14 and 17 from paper i ) . to confirm the suggestions inferred from the apparent filament barbs , invoked different methods , _ i.e. _ , magnetic field extrapolation based on the linear ( constant @xmath10 ) force - free field approximation , and the computation of the magnetic helicity flux density injection . using a magnetogram before the eruptions ( at 06:23 ut ) , they found a dominant negative @xmath10 by matching the shapes of the computed field lines to that of the observed coronal loops . this finding confirms their conclusion that the large - scale magnetic field of ar 501 had a negative helicity sign . the authors also compared the computed field lines with the post - flare arcade loops in the trace 171 full - resolution image @xcite at 09:43 ut and found that the loops could be modeled by using any small @xmath10 value positive , or negative , or null . they also found that , despite the predominance in the ar 501 of negative @xmath10 values , corresponding to the negative global magnetic helicity , a small @xmath10 tended to be positive in the southeastern part of the ar . the authors also computed an ongoing injection of the positive - helicity flux in this part of ar 501 and conjectured that this localized injection could be sufficient to make the magnetic helicity positive in this area , but its total value remained uncertain . modeled the post - flare loops visible in a half - resolution 195 image obtained with eit at 09:36 ut and found that the best value of @xmath10 was slightly positive along the total length of the visible arcade . this led them to a conclusion that the ar had a global positive magnetic helicity . the authors used the images obtained at the late post - flare stage , when the non - potentiality of the post - flare loops was least pronounced being , therefore , difficult to observe . on the other hand , at the earlier stages of a flare , more favorable in this respect , the loops are usually unresolved because of their small size . we have succeeded in solving this problem . the loops of a compact arcade visible in the trace images in figures [ f - helicity]a[f - helicity]c ( not analyzed by the aforementioned authors ) are skewed counter - clockwise in sunspot n3 and the region around it , indicating negative helicity . the skewness of the loops in region s4 considered by seems to be the same , as expected for both ends of the same loops , connecting these regions in figures [ f - helicity]a[f - helicity]c . the whole flare arcade has an inverse s - shape , which also corresponds to negative helicity . g ) and positive ( continuous , @xmath41 g ) @xmath2 polarities in the bbso magnetogram used in the computations . the top left gray box presents the ratio of the total negative to positive current helicity at the photospheric layer ( surf ) and in the coronal volume ( vol ) for the whole region . the bottom right gray box is related to the framed region . the axes are in arc seconds from the solar disk center . ] to verify this suggestion , we have computed the current helicity , @xmath42 , from a vector magnetogram of ar 501 observed at _ big bear solar observatory _ ( bbso ) on 18 november at 20:26 ut ( courtesy v. yurchyshyn ) . the field of view of the magnetogram , of @xmath39 , was centered near the n1 sunspot ( about @xmath43 $ ] at 08:00 ut ) and covered the active region and its vicinity , but did not reach the region of bifurcation . the line - of - sight component in the bbso magnetogram was considerably saturated in sunspots s1 , n3 , and s3 . our computations were carried out in a spherical box with a photospheric base of @xmath44 and a height of 135 mm from both the raw magnetogram and a saturation - corrected one by using a temporally close soho / mdi magnetogram . we used the method of the nlff extrapolation @xcite to compute two kinds of the current helicity maps . the first kind of map , the photospheric @xmath45 density map , presents the distribution of @xmath45 at the photospheric layer . the second kind of map is the column density distribution of @xmath45 . each pixel of such a map is the photospheric base of a vertical column , whose height is 135 mm , and presents the total @xmath45 over all cells constituting this column . comparison of the maps of the two kinds show that they are not very different , because the major contribution to the total current helicity is due to the lowest layers closest to the photosphere . each attempt has resulted in a significant excess of the negative helicity in both the whole active region and in the framed area considered by , but with somewhat different quantitative results . the saturation - corrected current helicity map is shown in figure [ f - helicity]d . the ratios of the total negative to positive helicity for both the photospheric layer and for the total volume are specified in the gray boxes . the top left box presents the ratios for the whole region , the bottom right box is related to the framed region . indeed , the excess of the total negative helicity is less in the framed region , in agreement with the ongoing injection of the positive helicity found by , but still insufficient to make the total helicity in this region positive . the absence of positive magnetic helicity in the localized area of ar 501 also indicates the absence of segments with the corresponding handedness in the body of the main filament f1 . note also that the central part of the inverse s - shaped structure in figures [ f - helicity]a and [ f - helicity]b passes between sunspots n3 and s3 , where the negative current helicity is the largest in the whole ar 501 . to make the situation clearer , we have additionally calculated magnetic field lines from the same nlff extrapolation , which was used in the computation of the current helicity . it is difficult to reach a perfect correspondence between the field lines and the observed loops , because the rapid evolution of ar 501 during the 12.5 hours separating the euv image and the vector magnetogram could imply a change in the small - scale features , while the strong saturation of the vector magnetogram ( and , to a lesser degree , its limited field of view ) could considerably affect long loops . for example , we have not succeeded to reproduce the long loops in the north east part of the arcade ; however , its negative helicity is undoubted . nevertheless , three sets of field lines with @xmath46 overlaid on the enlarged image of the arcade in figure [ f - helicity]b more or less correspond to the actually observed loops , in particular , the green lines overlapping the region of the questionable handedness . all of these circumstances show that the attempts to reconcile the handedness of the magnetic cloud and active region 501 are not promising . the question how the right - handed mc could be formed in the eruption of the left - handed filament remains unanswered . the scenario proposed by corresponds to the concept of a simple eruption , when the internal magnetic helicity of a pre - eruptive structure , _ i.e. _ , its self - helicity ( or twist helicity ) , @xmath47 , determines the helicity sign of the interplanetary mc . probably , this situation is typical of cmes associated with eruptions of quiescent filaments outside active regions . however , if a filament ( flux rope ) erupts from an active region , then , under certain circumstances , the helicity of the mc can have a different origin . this scenario implies the interaction and reconnection between magnetic fields of the eruptive structure with coronal magnetic fields surrounding the parent ar . in such a case , a new magnetic structure is formed , which is the progenitor of the future interplanetary mc . its helicity is determined by the sum of @xmath47 and the mutual ( or linkage ) helicity , @xmath48 , of the interacting magnetic fluxes . _ via _ magnetic reconnection , the mutual helicity is transformed into the self - helicity of the mc . depending on the sign of @xmath48 and the value of the @xmath49 ratio , the mc helicity can be different from the pre - eruptive structure not only in the value , but also in the sign . such eruptions with mismatching helicity were suggested in the study by ; this scenario is also supported by the observations of the apparent disintegration of eruptive filaments mentioned in section [ s - introduction ] . to understand what could happen on the sun on 18 november , we first analyze the configuration of the magnetic field on spatial scales considerably larger than the size of the vector magnetogram in figure [ f - helicity ] . this is possible by using full - disk magnetograms , of which line - of sight measurements ( _ e.g. _ , by soho / mdi ) are only available for 2003 . thus , we had to use for this purpose mdi magnetograms and the potential field extrapolation . the coronal field in a magnetic complex consisting of regions 501 , 503 , and their environment was computed in the potential approximation using the mentioned package @xcite for the full - disk soho / mdi magnetograms of 18 november observed at 06:23 , 07:59 , and 09:35 ut . the computations used 90 spherical harmonics . the field line distribution computed from each of the magnetograms was similar ; we mainly use here the magnetogram at 09:35 ut . figures [ f - field_lines][f - fil_motion ] present the coronal magnetic configuration . the s n s n quadrupole in figures [ f - field_lines ] , [ f - top_view ] , and [ f - fil_motion ] is its basis . figure [ f - top_view ] shows that the major eastern s - polarity sunspot has a large excess of negative magnetic flux which is unbalanced in this quadrupole . a considerable portion of this flux is connected to remote sites on the solar surface ( figure [ f - fil_motion ] ) separated from ar 501 by the global polarity inversion line . g , continuous @xmath50 g. the axes are in solar radii from the solar disk center . ] a topological particularity of the coronal configuration is a magnetic null point located at a height of about 100 mm above the photosphere . this is the only null point associated to large - scale magnetic fields on the visible side of the sun . figure [ f - field_lines ] shows a side view of the complex of regions 501 , 503 , and their surroundings . this complex was located at the solar disk center on 18 november . figure [ f - field_lines]a shows the location of the null point ( slanted cross ) inside this complex , the pre - eruptive u - shaped filament f1 ( red semicircular arrow ) , and region of bifurcation rb , which had well - pronounced counterparts on the solar disk visible in the h@xmath10 line and in soft x - rays ( see paper i ) . figure [ f - field_lines]b shows the magnetic field lines , which pass close to the null point . according to the classification of , here we are dealing with a negative improper three - dimensional null point with the fan field lines rooted in the n polarities , and the field lines around the spine rooted in the s polarities . the fan is perpendicular to the spine . the region of bifurcation rb is rather close ( but not exactly co - spatial ) to the site , where the spine field line , which leaves the null point , enters the photosphere . the lack of coincidence in our extrapolation , besides their possible actual difference , can be due to : a ) the usage of the potential approximation , and b ) the insufficient spatial resolution because of the limited number of the spherical harmonics . there are indirect indications of a connection between rb and the null point : 1 ) rb firstly appeared as an isolated bright point in the h@xmath10 and sxi images one minute later than the u - shaped filament started to move at 07:41 ut . the rb region and the filament were not connected by field lines . their connection is possible _ via _ reconnection at the magnetic null point located between them . invoking magnetic reconnection with a possible transformation of the null point into a current sheet is not the only possibility here . the mhd disturbances generated by the initial displacement of the eruptive filament can cause local plasma heating in the vicinity of the null point by the accumulation and dissipation of the energy of fast - mode mhd waves or alfvn waves along the spine ( see , _ e.g. _ , the review by ; ) . such heating augments the flux of heat and , possibly , that of non - thermal particles responsible for the increase of the emission intensity from the chromospheric and coronal plasmas above the photospheric base of the spine field line . 2 ) the rb region has appeared as a small ring with a brightening running clockwise ( paper i ) at the same time as the eruptive filament f1 would pass the magnetic null point ( this phenomenon started at 08:08 ut ) . a similar situation with a considerably larger ring was observed in an eruptive event of 1 june 2002 @xcite . in that event , the magnetic configuration was different , it had a funnel - like shape , and the ring was situated above the photospheric footprint of a hemispheric separatrix surface with a magnetic null at its top . the ` magnetic funnel ' confined all the ejections , as suggested by the movies of the event . similar to our case , a brightening running along the ring was observed in the 1 june 2002 event , when the eruptive filament , being transformed into a rotating ejection , passed through the magnetic null . the motion of this ejection through the null must be accompanied by reconnection of the magnetic field lines associated to the ejection with those on the separatrix surface or nearby . a response to this process was a concurrent euv disturbance , which propagated along the ring - like footprint of the separatrix surface . in the 18 november 2003 event , the ring structure of region rb could be a manifestation of the photospheric base of a bundle of magnetic field lines twisting around , or close to the spine field line . the twist indicates the presence of the longitudinal electric current along the spine and the inadequacy of the potential field approximation , which we use . the twist is necessary for the brightening running along the ring that accompanies the concurrent magnetic reconnection of this bundle with an arbitrary magnetic rope or eruptive filament , for example , with a cylindrical shape . a scheme in figure [ f - rotation ] illustrates the interaction . as this figure shows , the bundle should be similar to a non - uniformly twisted cylindrical flux tube to produce a running brightening . hoyle uniformly twisted tube with a non - constant @xmath10 does not satisfy this condition , while the lundquist magnetic cylinder with a constant @xmath10 satisfies it . since rb is located within the zone of the negative magnetic polarity , the clockwise - running brightening corresponds to @xmath51 . in turn , this indicates that the self - helicity of the magnetic flux around the spine is most likely negative . , @xmath52 , and @xmath53 are denoted with the colors presented in the top rectangle . magnetic reconnection leads to a concurrent brightening running clockwise in the chromosphere and low corona above the rb region ( the brown round arrow at the bottom ) . the axial magnetic fields , @xmath54 , of the eruptive flux rope and the bundle correspond to the positions of the thick blue and red loops at @xmath55 in figure [ f - scenario]a , respectively.,scaledwidth=60.0% ] 3 ) in the 1 june 2002 event @xcite , the interaction of the ejection with the magnetic ` funnel ' and its passage through the null appeared to have produced a wave disturbance in the corona observed as an ` eit wave ' rapidly expanding above the limb . in the 18 november 2003 event , the bifurcation of the eruptive filament was also accompanied by the appearance of a coronal wave , whose kinematical center was located high in the corona , above rb ( paper ii ) . this was the second wave , not expected in the event . the kinematical center of the first wave was within ar 501 and corresponded to the impulsive acceleration stage that is typical of impulsive eruptions @xcite . until now , the contribution to the mc magnetic helicity from the relative position of the pre - eruptive filament and the structures outside of ar 501 have not been taken into account . these structures got in contact at the magnetic null point , toward which the top of the expanding filament f1 moved . figure [ f - field_lines ] illustrates the positions of the structures which could be involved in the eruptive process . the values of the fluxes belonging to the west , east , north , and south magnetic dipoles are not known _ a priori_. note that each dipole represents only a fraction of the magnetic flux of a corresponding magnetic domain within the quadrupole configuration in figure [ f - field_lines ] . it is also not known if the interaction between any of these dipoles and the eruptive filament was significant . let us estimate by linking with which of these dipoles the pre - eruptive filament had the largest positive mutual helicity . the presence of the positive mutual helicity is suggested by figure [ f - field_lines]b . the curved blue arrow in the figure indicates the direction of the magnetic flux in the west dipole . the red arrow indicates the direction of the axial magnetic field in the pre - eruptive filament . let us invoke the right - handed screw rule and direct the screw axis along the blue arrow at its descending spine portion ; then the right - handed direction corresponds to the direction of the red axial field in the top of the filament . if we put the screw axis along the top of the filament , then the result is the same ; the descending part of the blue arrow rotates clockwise . therefore , one might expect that the west dipole and the pre - eruptive filament constitute a right - handed system . to estimate the mutual magnetic helicity , @xmath48 , we use the technique of interior angles @xcite . for this purpose we need a top view of the magnetic configuration presented in figure [ f - field_lines]b . such a view is shown in figure [ f - top_view ] , which represents the complex of regions 501 , 503 , and their environment located almost at the solar disk center when viewed from earth . the dotted lines @xmath56 , @xmath57 , @xmath58 , and the blue vector @xmath59 connect the positive and negative footpoints of the magnetic fluxes in the north , south , east , and west magnetic dipoles . the red vector * f1 * connects the western ( negative ) and eastern ( positive ) ends of the erupted portion of the filament . the sign of @xmath48 depends on the relation between the height @xmath60 of the pre - eruptive filament and those of the magnetic loops , @xmath61 , @xmath62 , @xmath63 , and @xmath64 , which constitute each of the dipoles . the estimation shown below corresponds to the real case in which the height of the pre - eruptive filament is less than any of the other magnetic loops . for example , the mutual magnetic helicity of filament f1 and the west dipole @xmath59 is : @xmath65 where the angles @xmath66 are positive counter - clockwise , so that @xmath67 and @xmath68 . the flux of the axial magnetic field in the pre - eruptive filament , @xmath69 , is unknown . the magnetic flux of the west dipole , @xmath70 , will be estimated in section [ s - double_dimming ] . . the red vector * f1 * connects the photospheric bases of the erupted portion of the filament in ar 501 . the yellow dashed lines @xmath56 , @xmath57 , @xmath58 , and the blue vector @xmath71 connect the photospheric bases of the north , south , east , and west magnetic dipoles . @xmath72 and @xmath73 are the angular spans of vector @xmath59 measured from the ends of vector * f1*. the axes are in solar radii from the solar disk center . the contour levels are symmetric relative to zero in steps of 35 g ; dotted @xmath74 g , continuous @xmath75 g. ] the mutual helicities @xmath76 can be estimated in a similar way . in particular , @xmath77 . next , the angles between the lines @xmath78 , @xmath79 , and @xmath80 in figure [ f - top_view ] are small ; therefore , @xmath81 and @xmath82 . small changes of their mutual orientations can change the signs of the corresponding mutual helicities . thus , @xmath83 appears to be the only solar source for a significant positive magnetic helicity of the interplanetary magnetic cloud . if so , then the magnetic flux of the west dipole should be involved in the eruption of the u - shaped filament f1 . this conjecture is confirmed observationally by the development of the double dimmings in the conjugate footpoints of the loops anchored in the west dipole ( see section [ s - double_dimming ] ) . figure [ f - dimming_mag ] presents the development of dimming around 08:14 ut in association with the main eruption on 18 november . the gray scale background in the panels shows three fixed - base ratios of eit 195 images , @xmath84 . the contours outline the positive ( blue ) and negative ( red ) modeled flux concentrations of the coronal @xmath85 magnetogram . here @xmath85 is the radial component of the coronal magnetic field computed at the spherical surface with a radius @xmath86 ( _ i.e. _ , at a height of @xmath87 mm above the photosphere ) from the soho / mdi magnetogram observed at 07:59 ut . magnetogram extrapolated from an mdi magnetogram at a height of 28 mm ( blue n - polarity , red s - polarity ) . the contour levels are @xmath88 g in panels ( a ) and ( b ) and @xmath89 g in panel ( c ) . the yellow contour in panel ( a ) outlines the leading edge of the eruptive filament pouncing on a coronal obstacle above the bifurcation region rb denoted by the small oval . the green contours in panel ( c ) outline the darkest dimming areas , which overlay a dipole outlined with a large oval . the arrow shows the direction of the resulting magnetic field in the dipole . ] figure [ f - dimming_mag]a presents the collision of the eruptive filament f1 ( yellow ) with a high coronal structure . their interaction is indicated by the brightenings turning around the bifurcation region ( rb ) underneath ( see paper i ) . a large dimming south of ar 501 resulted from a preceding eruption and is beyond our scope . figure [ f - dimming_mag]b ( 12 minutes later ) presents the outcome of the collision . both the brightenings and region rb disappeared . two dimming areas developed instead . a new large dimming appeared at the previous position of the bifurcation region , and a weaker star - like dimming developed in ar 503 . figure [ f - dimming_mag]c shows the situation observed still 12 minutes later . the green contours outline the darkest portions of the two major west dimming regions . the star - like dimming in ar 503 and the dimming encompassing the former bifurcation region overlaid a dipole pointing nearly south . the magnetic flux in this dipole probably corresponds to the magnetic flux in a presumable eruptive structure . the value of the magnetic flux computed within the dimming regions depends on the height , @xmath90 , to which the magnetogram is related . this can be a photospheric magnetogram actually observed with mdi or a coronal @xmath85 magnetogram computed from the potential field extrapolation . to estimate the magnetic flux in the eruption , we assume that the dipolar double dimming developed mainly due to the stretching of a closed magnetic flux tube , connecting these areas . one might expect that the positive and negative magnetic fluxes within the conjugate dimming areas are equal in absolute value . the equality should be reached at a proper height above the photosphere . at the photospheric level , @xmath91 , the computation gives a negative imbalance of the magnetic fluxes , @xmath92 mx in the star - like dimming and @xmath93 mx in the large south dimming . while the height increases , the imbalance of the fluxes decreases . it becomes zero at @xmath87 mm , where @xmath94 mx , and then becomes increasingly positive . the height at which the flux is balanced is consistent with a typical height range of the coronal emission in the 195 line . considering each pole of the dipole as the centroid of the magnetic field distribution within each dimming region , we compute an orientation of the dipole of @xmath95 with respect to the north or about @xmath96 with respect to the ecliptic ( in the gse coordinate system ) . the gray arrow in figure [ f - dimming_mag]c connects the centroids . the direction of the arrow and its ends acceptably match the blue arrow of the west dipole in figure [ f - top_view ] . our previous discussion provides a confirmation of the participation of the west dipole in the formation of the earth - directed mc with a positive helicity . the estimated magnetic flux , @xmath97 mx , is adequate to that in the magnetic cloud near earth , @xmath98 mx , estimated by . the estimated magnetic flux in the west dipole is sufficient for the mc , while the two - fold excess could be lost by reconnection in the interplanetary space as proposed by and consistent with the fact that the mc crossed the sector boundary of the interplanetary magnetic field . the inclination of the mc to the ecliptic plane estimated by different authors within a range of @xmath14 ( see ) corresponds to the orientation of the west dipole @xmath99 in figures [ f - top_view ] and [ f - dimming_mag]c . according to our analysis in papers i , ii , and iv , the mc hitting the earth on 20 november had the following characteristics : i ) it was formed close to the solar disk center in the bifurcation of the eruptive filament , ii ) it was compact and disconnected from the sun , and iii ) it had an atypical spheromak - like configuration . the outcome of section [ s - large_scale_config ] suggests that the mc could be formed by the interaction between , at least , two magnetic structures , whose mutual helicity was positive before the eruption . these structures were the west dipole and the pre - eruptive filament . the position of the west dipole on the solar disk meets requirement i ) . an additional requirement follows from the total magnetic helicity conservation @xmath100 , where @xmath101 and @xmath102 are the self - helicities of the pre - eruptive filament and the west dipole , and @xmath103 is their mutual helicity . the final configuration of the eruption depends on the sign of the total helicity . however , it is not possible to determine the sign of @xmath104 from the analysis of and our considerations in section [ s - helicity_absence ] . the preceding estimates of the helicity sign were related to ar 501 only and did not contain information about the mutual helicity between the filament in the active region and magnetic structures outside of ar 501 . the results of sections [ s - mutual_helicity ] and [ s - double_dimming ] show that if the magnetic fluxes of the west dipole , @xmath70 , and the filament , @xmath69 , are related as @xmath105 , then @xmath106 under typical assumptions . thus , if the total helicity of two interacting fluxes is transformed through reconnection into the self - helicity of a single eruptive structure , then a right - handed mc can not be formed . the situation is different , if the interaction and a chain of magnetic reconnections are followed by the formation of two eruptive structures rather than a single one . in this way , redistribution of the magnetic helicity is possible with almost the full transformation of the mutual helicity of two reconnected fluxes into the self - helicity of one of the resulting structures . the second structure can carry away practically the whole negative helicity . the mutual helicity of the two new eruptive structures approaches zero as they separate from each other , and therefore can be neglected . as a next step we try to understand : i ) the possible geometry of the interaction between the eruptive filament f1 and the magnetic loops in the vicinity of the null point ( we associate its projection on the solar disk with the bifurcation region rb ) at their first contact and just after the passage of the null point , and ii ) why this interaction was followed by the visible dispersal over the solar surface of the cool plasma , which initially belonged to the eruptive filament f1 . figure [ f - fil_motion ] shows a side view of the magnetic configuration , in which the main eruption occurred . for further considerations it is convenient to replace the erupting filament by a magnetic flux rope . the rope has a toroidal ( axial ) and poloidal ( azimuthal ) components of the magnetic field . the direction of motion of the middle part of the rope ( the solid red arrow marked 0 ) crosses the magnetic null point ( the green slanted cross ) . the red dashed arrows marked 1 and 2 limit the cross section of the expanding rope . a selected magnetic field line ( thick blue ) denotes a magnetic loop of the west dipole , which was the major partner of the eruptive rope in the creation of the spheromak . the thin contours on the solar surface correspond to the magnetic field values in steps of 34 g ; they are @xmath107 g ( dotted ) and @xmath108 g ( continuous ) away from the stronger - field regions near the limb . the thick white circle denotes the solar limb . the axes are in solar radii from the solar disk center . ] the interaction between the eruptive magnetic flux rope and surrounding magnetic fields can result in two effects . one is the redistribution of masses due to magnetic reconnection between the flux rope and the outer magnetic domain . this domain contains field lines , which reach the solar surface far from the quadrupole . in the projection presented in figure [ f - fil_motion ] , the red solid line 0 is tangent to such field lines . this effect is represented in figure [ f - reconnect_dispersal ] as implied in two - dimensional reconnection models . another effect is a kinematic linkage between the interacting structures . this essentially three - dimensional effect is used in a scheme presented in figure [ f - scenario ] . . right : after magnetic reconnection . the center of the rope displaces across the outer field @xmath109 , and cool plasma continues moving along @xmath109.,scaledwidth=80.0% ] figure [ f - reconnect_dispersal ] shows a cross section of the top of the magnetic flux rope . this section moves along the external magnetic field @xmath110 with a velocity @xmath111 . in figure [ f - fil_motion ] this situation corresponds to the position of the flux rope s center just after passing the magnetic null . the two circles to the left of the figure are poloidal field lines of the flux rope . cool plasma enclosed between them is denoted by the gray shading . the situation after magnetic reconnection is shown to the right of the figure . the center of the flux rope has shifted across @xmath110 , while plasma ( gray ) has departed from the flux rope and moves along the outer field @xmath110 . the scheme in figure [ f - reconnect_dispersal ] is basically similar to figure 6 in . the apparent difference between the schemes is due to different directions of motions of the eruptive filaments through the magnetic null point in the quadrupolar magnetic configuration . the 2d scheme in figure [ f - reconnect_dispersal ] does not change if we introduce an additional homogeneous magnetic field , @xmath112 , perpendicular to the plane of the figure without fixed ends and consider a non - compressive plasma . during magnetic reconnection , the frozen - in @xmath112 components of the total magnetic field are mixed like pens in a box without any effect on the process presented . however , this is not the case . the magnetic field perpendicular to the plane of the figure is strongly inhomogeneous being concentrated in the curved toroidal flux rope , which continues its motion governed almost entirely by the toroidal propelling force . reconnection creates new three - dimensional field lines between the footpoints of the eruptive flux rope and those of the outer field lines involved in the reconnection process . the outcome is as follows : i ) the eruptive filament loses mass , ii ) the propelling toroidal force decreases faster than without reconnection , iii ) the disintegrating eruptive flux rope separates from cool plasma spreading out behind it . we have found that the compact magnetic cloud in the 1820 november 2003 event could be the result of the redistribution of the magnetic helicity after the interaction between the left - handed eruptive filament and the west dipole @xmath113 . the conclusion about the positive mutual helicity between the pre - eruptive filament f1 and the west dipole drawn in section [ s - large_scale_config ] implies the inequality @xmath114 . this means that the height above the photosphere , @xmath60 , of the red loop , which represents filament f1 in figure [ f - field_lines ] , was less than that of the blue loop , which represents the loops belonging to dipole @xmath113 . this was really the case before the eruption . if the inequality would have been the reverse ; then , the sign of the mutual helicity would have changed . in an intermediate case , @xmath115 , the mutual helicity is zero . the conservation of the total magnetic helicity restricts the choice of possible options for the subsequent dynamic reconfiguration of eruptive magnetic structures . in the absence of a clear idea about the formation mechanism of a compact right - handed spheromak - like configuration ( called henceforth the spheromak for brevity ) , we use the following heuristic consideration . let us consider a straight trajectory along which the centroid of the eruptive filament moves . arrow 0 in figure [ f - fil_motion ] corresponds to this trajectory across the magnetic null point . the passage of the centroid through the null point ( whose position is assumed to be fixed ) corresponds to the equality @xmath116 , where @xmath117 is the height of the centroid . this equality means that the mutual helicity between the moving filament and the loops anchored to the west dipole , @xmath118 , becomes zero . to keep @xmath119 until the onset of the interaction between these structures , the trajectory of the eruptive filament should be below the null point . this could correspond to , _ e.g. _ , arrow 2 in figure [ f - fil_motion ] , if the condition @xmath120 , which was valid before the eruption , is also valid in the dynamic regime . in this case , the clockwise angle between arrows 0 and 2 corresponds to a positive mutual helicity . similarly , arrow 1 in figure [ f - fil_motion ] corresponds to a trajectory above the null point . some consequences of this geometry including magnetic reconnection and mass depletion are shown in the two - dimensional figure [ f - reconnect_dispersal ] . the counter - clockwise angle between arrows 0 and 1 corresponds to a negative mutual helicity between the eruptive filament and the loops of the west dipole . figure [ f - scenario ] presents a hypothetical 3d scheme for the formation of a right - handed spheromak , if the trajectory is below the null point . the relative position of the red and blue thick solid loops in figure [ f - scenario]a is the same as in figure [ f - field_lines]b , which shows the situation before the eruption ( @xmath121 ) . the active red loop represents the left - handed eruptive filament during subsequent times @xmath122 . the passive blue loop is anchored to the west dipole involved in the eruption . the red and blue dotted lines represent the modified shapes of the red and blue loops just after the onset of the interaction . the expanding red loop embraces the leg of the blue loop rooted in rb and takes a y - like shape ( _ cf . _ figure [ f - shape_transform ] ) . the stretching of the red and blue magnetic structures in figure [ f - scenario]b leads to the formation of secondary loops which are linked . magnetic reconnection results in the detachment in figure [ f - scenario]c of the secondary magnetic loops from their parent loops . a closed system of the orthogonal red and blue magnetic rings is formed . the blue ring developed from the magnetic flux of the west dipole . we neglect its self - helicity and represent it as a system of non - twisted thin blue rings instead of a single thick blue ring . the magnetic pressure separates these rings ( not interconnected ) from each other , and they get distributed along the red ring . remnants of its negative helicity annihilate in this way ( figure [ f - scenario]d ) . a right - handed spheromak is formed . the direction of its motion does not necessarily coincide with the direction , in which the eruptive filament , which takes back its shape , moves ( the thick red arrow ) . b and [ f - top_view ] for the case when the trajectory ( black dashed line ) of the eruptive filament ( active red loop ) passes below the magnetic null point ( arrow 2 in figure [ f - fil_motion ] ) . the blue loop is anchored to the west dipole involved in the eruption . rb is the bifurcation region shown in figures [ f - shape_transform ] , [ f - field_lines ] , and [ f - dimming_mag ] . ( a ) @xmath121 : denotes the time before the eruption ; the interaction of the loops starts at @xmath55 . the dotted lines at @xmath52 show the loops just after the onset of their interaction . the expanding red loop embraces the leg of the blue loop rooted in rb and takes the y - like shape ( _ cf . _ figure [ f - shape_transform ] ) . ( b ) the stretching of the red and blue magnetic structures results in the formation of secondary loops which are linked . ( c ) magnetic reconnection detaches the secondary loops from the parent ones . a closed system of red and blue magnetic rings is formed . ( d ) the blue rings separate from each other and distribute along the red ring thus destroying remnants of its left helicity . a right - handed spheromak develops . the apparent intersections of the blue and red loops in panels ( a)(d ) are due to the projection effect and do not imply the possibility of reconnection . figure [ f - suggestion ] roughly outlines the dynamic reconfiguration presented in panels ( a ) and ( b ) of this figure , but viewed from a different angle . ] the outcome of the process presented in figure [ f - scenario ] is the formation of two different eruptive structures sharing their positive and negative magnetic helicity . the first structure is a modification of the parent left - handed eruptive filament f1 , which recovers after the bifurcation and moves in the southwestern sector of the solar disk ( figure [ f - shape_transform ] ) . the second structure is the right - handed spheromak , which was created due to the catastrophe of the filament observed as its bifurcation . formally assuming that the effects presented in figures [ f - reconnect_dispersal ] and [ f - scenario ] coexisted in the real eruptive process , we come to the following conclusion . the bifurcation of the eruptive filament f1 was accompanied by the decrease of its poloidal flux and mass depletion . the loss of mass accounts for the absence of a conspicuous core in cme2 , which appeared in lasco images in the same southwestern sector , where the eruptive filament f1 moved . the y - like trace of the mass , which has not left the sun , carries information about the deformation of the magnetic flux rope in the vicinity of the magnetic null point . if the magnetic cloud , which reached earth , would have been associated with cme2 , then its magnetic helicity should be negative rather than positive . this association seemed to be plausible in the first studies of the 1820 november 2003 event , but was ruled out in our paper ii . in our considerations , we called the ` spheromak ' a spheromak - like force - free magnetic configuration . the development of a such a configuration should be accompanied by a transformation of the magnetic energy excess of the interacting pre - spheromak structures into the kinetic energy of chaotic or directed plasma motions . it is possible that some part of this kinetic energy had gone into the formation of a wave , possibly a shock , propagating away from the formation site of the spheromak . an indication of such a wave is presented in paper ii ( see also section [ s - null_and_bifurcation_region ] , last paragraph ) . while constructing the scheme in figure [ f - scenario ] , we pursued to have as a result the compact size and positive helicity of the developing magnetic structure to match the mc , which actually hit earth . the spheromak meets these requirements . however , our hypothetical scheme does not guarantee an earthward direction of the spheromak . there are additional indications supporting that the spheromak has actually reached earth . the formation of the spheromak in figure [ f - scenario ] is due to the interaction of different - temperature plasma structures , _ i.e. _ , the cool eruptive filament and the magnetic flux of the west dipole frozen into the coronal plasma . probably , this circumstance determined the atypically high inhomogeneity of the temperature distribution in the 20 november mc mentioned in section [ s - introduction ] . the axial magnetic field of the spheromak in figure [ f - scenario ] is formed from the magnetic flux of the west dipole . therefore , the orientation of the axial field of the spheromak should be close to the direction of the vector @xmath59 in figure [ f - top_view ] . in turn , the inclination of @xmath59 to the ecliptic plane is reasonably close to the inclination of the mc ( section [ s - double_dimming ] , last paragraph ) . to conclude this section , we note that the scheme of the interaction between the red and blue loops in the case of an upper trajectory 1 in figure [ f - fil_motion ] also provides the possibility to form a spheromak , but with a negative helicity . most likely , this option did not occur in this event , as supported by the downward displacement of the central part of the eruptive flux rope in figure [ f - reconnect_dispersal ] . in the magnetic configuration presented in figure [ f - fil_motion ] this effect works in the same direction both before the passage of the null point by arrow 0 and after it . thus , the lower trajectory appears to be preferential and the only possibly one in this particular event . the scenario of the 18 november 2003 event does not correspond to the concept of a simple eruption directly from ar 501 , in which the twist helicity of an eruptive structure or active region determines the handedness of the interplanetary magnetic cloud . the nlff extrapolation of ar 501 shows an excess of negative twist , which is opposite to the positive sign of twist in the mc . to solve this contradiction , we have used the positive mutual helicity between the pre - eruptive filament and the flux tubes of a magnetic domain of the large - scale quadrupole configuration . the interaction of these magnetic fluxes presumably occurred as the eruptive filament passed in the neighborhood of the coronal magnetic null point . the positive mutual helicity of these two fluxes changed through magnetic reconnections into the positive self - helicity of a spheromak - like structure , whose geometry and parameters correspond to the magnetic cloud , which reached earth . in paper iv , we analyze the interplanetary disturbance responsible for the 20 november superstorm and outline the overall scenario of the whole event . we thank v. yurchyshyn , who kindly supplied the vector magnetogram of ar 501 observed at bbso , and m. temmer for the h@xmath10 data . we thank the co - authors of our papers i , ii , and iv , who are not involved in this study . we appreciate an anonymous reviewer for valuable remarks and comments . we are grateful to the instrumental teams of the kanzelhhe solar observatory ; trace and coronas - f missions ; mdi and eit on soho ( esa and nasa ) for the data used here . this study was supported by the russian foundation of basic research under grants 11 - 02 - 00757 , 11 - 02 - 01079 , 12 - 02 - 00008 , 12 - 02 - 92692 , and 12 - 02 - 00037 , and the ministry of education and science of russian federation , projects 8407 and 14.518.11.7047 . grechnev , v.v . , uralov , a.m. , chertok , i.m . , slemzin , v.a . , filippov , b.p . , egorov , ya.i . , fainshtein , v.g . , afanasyev , a.n . , prestage , n. , temmer , m. : 2014b , _ solar phys . _ _ 289 _ , 1279 . grechnev , v.v . , uralov , a.m. , slemzin , v.a . , chertok , i.m . , filippov , b.p . , rudenko , g.v . , temmer , m. : 2014c , _ solar phys.___289 _ _ , 289 . grechnev , v.v . , uralov , a.m. , slemzin , v.a . , chertok , i.m . , kuzmenko , i.v . , shibasaki , k. : 2008 , _ solar phys . _ * 253 * , 263 . slemzin , v.a . , kuzin , s.v . , zhitnik , i.a . , delaboudiniere , j .- p . , auchere , f. , zhukov , a.n . , van der linden , r. , bugaenko , o.i . , ignatev , a.p . , mitrofanov , a.v . , pertsov , a.a . , oparin , s.n . , stepanov , a.i . , afanasev , a.n . : 2005 , _ solar sys . res . _ * 39 * , 489 . srivastava , n. , mathew , s.k . , louis , r.e . , wiegelmann , t. : 2009 , _ j. geophys . _ * 114 * , a03107 . yurchyshyn , v. , hu , q. , abramenko , v. : 2005 , _ space weather _ , * 3 * , s08c02 . zhang , m. , b. c. low : 2003 , _ astrophys . j. _ * 584 * , 479 . zhitnik , i.a . , bougaenko , o.i . , delaboudinire , j .- , ignatiev , a.p . , korneev , v.v . , krutov , v.v . , kuzin , s.v . , lisin , d.v . , _ et al . _ : 2002 , _ proc . 10th european solar physics meeting , prague ( esa sp-506 ) _ , 915 .

our analysis in papers i and ii ( grechnev _ et al . _ , 2014 , solar phys . 289 , 289 and 1279 ) of the 18 november 2003 solar event responsible for the 20 november geomagnetic superstorm has revealed a complex chain of eruptions . in particular , the eruptive filament encountered a topological discontinuity located near the solar disk center at a height of about 100 mm , bifurcated , and transformed into a large cloud , which did not leave the sun . concurrently , an additional cme presumably erupted close to the bifurcation region . the conjectures about the responsibility of this compact cme for the superstorm and its disconnection from the sun are confirmed in paper iv ( grechnev _ et al . _ , solar phys . , submitted ) , which concludes about its probable spheromak - like structure . the present paper confirms the presence of a magnetic null point near the bifurcation region and addresses the origin of the magnetic helicity of the interplanetary magnetic clouds and their connection to the sun . we find that the orientation of a magnetic dipole constituted by dimmed regions with the opposite magnetic polarities away from the parent active region corresponded to the direction of the axial field in the magnetic cloud , while the pre - eruptive filament mismatched it . to combine all of the listed findings , we come to an intrinsically three - dimensional scheme , in which a spheromak - like eruption originates _ via _ the interaction of the initially unconnected magnetic fluxes of the eruptive filament and pre - existing ones in the corona . through a chain of magnetic reconnections their positive mutual helicity was transformed into the self - helicity of the spheromak - like magnetic cloud .

You are an expert at summarizing long articles. Proceed to summarize the following text: in the last decade we entered the data - intensive era of astrophysics , where the size of data has rapidly increased , reaching in many cases dimensions overcoming the human possibility to handle them in an efficient and comprehensible way . in a very close future petabytes of data will be the standard and , to deal with such amount of information , also the data analysis techniques and facilities must quickly evolve . for example the current exploration of petabyte - scale , multi - disciplinary astronomy and earth observation synergy , by taking the advantage from their similarities in data analytics , has issued the urgency to find and develop common strategies able to achieve solutions in the data mining algorithms , computer technologies , large scale distributed database management systems as well as parallel processing frameworks @xcite . astrophysics is one of the most involved research fields facing with this data explosion , where the data volumes from the ongoing and next generation multi - band and multi - epoch surveys are expected to be so huge that the ability of the astronomers to analyze , cross - correlate and extract knowledge from such data will represent a challenge for scientists and computer engineers . to quote just a few , the esa euclid space mission will acquire and process about 100 gbday@xmath1 over at least 6 years , collecting a minimum amount of about @xmath2 tb of data @xcite ; pan - starrs @xcite is expected to produce more than @xmath3 tb of data ; the gaia space mission will build a @xmath4 map of the milky way galaxy , by collecting about one petabyte of data in five years @xcite ; the large synoptic survey telescope ( @xcite ) will provide about @xmath5tb / night of imaging data for ten years and petabytes / year of radio data products . many other planned instruments and already operative surveys will reach a huge scale during their operational lifetime , such as kids ( kilo - degree survey ; @xcite ) , des ( dark energy survey , @xcite ) , herschel - atlas @xcite , hi - gal @xcite , ska @xcite and e - elt @xcite . the growth and heterogeneity of data availability induce challenges on cross - correlation algorithms and methods . most of the interesting research fields are in fact based on the capability and efficiency to cross - correlate information among different surveys . this poses the consequent problem of transferring large volumes of data from / to data centers , _ de facto _ making almost inoperable any cross - reference analysis , unless to change the perspective , by moving software to the data @xcite . furthermore , observed data coming from different surveys , even if referred to a same sky region , are often archived and reduced by different systems and technologies . this implies that the resulting catalogs , containing billions of sources , may have very different formats , naming schemas , data structures and resolution , making the data analysis to be a not trivial challenge . some past attempts have been explored to propose standard solutions to introduce the uniformity of astronomical data quantities description , such as in the case of the uniform content descriptors of the virtual observatory @xcite . one of the most common techniques used in astrophysics and fundamental prerequisite for combining multi - band data , particularly sensible to the growing of the data sets dimensions , is the cross - match among heterogeneous catalogs , which consists in identifying and comparing sources belonging to different observations , performed at different wavelengths or under different conditions . this makes cross - matching one of the core steps of any standard modern pipeline of data reduction / analysis and one of the central components of the virtual observatory @xcite . the massive multi - band and multi - epoch information , foreseen to be available from the on - going and future surveys , will require efficient techniques and software solutions to be directly integrated into the reduction pipelines , making possible to cross - correlate in real time a large variety of parameters for billions of sky objects . important astrophysical questions , such as the evolution of star forming regions , the galaxy formation , the distribution of dark matter and the nature of dark energy , could be addressed by monitoring and correlating fluxes at different wavelengths , morphological and structural parameters at different epochs , as well as by opportunely determining their cosmological distances and by identifying and classifying peculiar objects . in such context , an efficient , reliable and flexible cross - matching mechanism plays a crucial role . in this work we present @xmath0 ( _ command - line catalog cross - match tool and the user guide are available at the page http://dame.dsf.unina.it/c3.html.]_ , @xcite ) , a tool to perform efficient catalog cross - matching , based on the multi - thread paradigm , which can be easily integrated into an automatic data analysis pipeline and scientifically validated on some real case examples taken from public astronomical data archives . furthermore , one of major features of this tool is the possibility to choose shape , orientation and size of the cross - matching area , respectively , between elliptical and rectangular , clockwise and counterclockwise , fixed and parametric . this makes the @xmath0 tool easily tailored on the specific user needs . the paper is structured as follows : after a preliminary introduction , in sec . [ sec : techniques ] we perform a summary of main available techniques ; in sec . [ sect : c3design ] , the design and architecture of the @xmath0 tool is described ; in sections [ sect : config ] and [ sect : optimization ] , the procedure to correctly use @xmath0 is illustrated with particular reference to the optimization of its parameters ; some tests performed in order to evaluate @xmath0 performance are shown in sec . [ sect : performances ] ; finally , conclusions and future improvements are drawn in sec . [ sect : conclusion ] . cross - match can be used to find detections surrounding a given source or to perform one - to - one matches in order to combine physical properties or to study the temporal evolution of a set of sources . the primary criterion for cross - matching is the approximate coincidence of celestial coordinates ( positional cross - match ) . there are also other kinds of approach , which make use of the positional mechanism supplemented by statistical analysis used to select best candidates , like the bayesian statistics @xcite . in the positional cross - match , the only attributes under consideration are the spatial information . this kind of match is of fundamental importance in astronomy , due to the fact that the same object may have different coordinates in various catalogs , for several reasons : measurement errors , instrument sensitivities , calibration , physical constraints , etc . in principle , at the base of any kind of catalog cross - match , each source of a first catalog should be compared with all counterparts contained in a second catalog . this procedure , if performed in the naive way , is extremely time consuming , due to the huge amount of sources . therefore different solutions to this problem have been proposed , taking advantage of the progress in computer science in the field of multi - processing and high performing techniques of sky partitioning . two different strategies to implement cross - matching tools basically exist : web and stand - alone applications . web applications , like openskyquery @xcite , or cds - xmatch @xcite , offer a portal to the astronomers , allowing to cross - match large astronomical data sets , either mirrored from worldwide distributed data centers or directly uploadable from the user local machine , through an intuitive user interface . the end - user has not the need to know how the data are treated , delegating all the computational choices to the backend software , in particular for what is concerning the data handling for the concurrent parallelization mechanism . other web applications , like arches @xcite , provide dedicated script languages which , on one hand , allow to perform complex cross - correlations while controlling the full process but , on the other hand , make experiment settings quite hard for an astronomer . basically , main limitation of a web - based approach is the impossibility to directly use the cross - matching tool in an automatic pipeline of data reduction / analysis . in other words , with such a tool the user can not design and implement a complete automatic procedure to deal with data . moreover , the management of concurrent jobs and the number of simultaneous users can limit the scalability of the tool . for example , a registered user of cds - xmatch has only @xmath6 mb disk space available to store his own data ( reduced to @xmath3 mb for unregistered users ) and all jobs are aborted if the computation time exceeds 100 minutes @xcite . finally , the choice of parameters and/or functional cases is often limited in order to guarantee a basic use by the end - users through short web forms ( for instance , in cds - xmatch only equatorial coordinate system is allowed ) . stand - alone applications are generally command - line tools that can be run on the end - user machine as well as on a distributed computing environment . a stand - alone application generally makes use of apis ( application programming interfaces ) , a set of routines , protocols and tools integrated in the code . there are several examples of available apis , implementing astronomical facilities , such as stil @xcite , and astroml @xcite , that can be integrated by an astronomer within its own source code . however , this requires the astronomer to be aware of strong programming skills . moreover , when the tools are executed on any local machine , it is evident that such applications may be not able to exploit the power of distributed computing , limiting the performance and requiring the storage of the catalogs on the hosting machine , besides the problem of platform dependency . on the contrary , a ready - to - use stand - alone tool , already conceived and implemented to embed the use of apis in the best way , will result an off - the - shelf product that the end - user has only to run . a local command - line tool can be put in a pipeline through easy system calls , thus giving the possibility to the end - user to create a custom data analysis / reduction procedure without writing or modifying any source code . moreover , being an all - in - one package , i.e including all the required libraries and routines , a stand - alone application can be easily used in a distributed computing environment , by simply uploading the code and the data on the working nodes of the available computing infrastructure . one of the most used stand - alone tools is stilts ( stil tool set , @xcite ) . it is not only a cross - matching software , but also a set of command - line tools based on the stil libraries , to process tabular data . it is written in pure java ( almost platform independent ) and contains a large number of facilities for table analysis , so being a very powerful instrument for the astronomers . on one hand , the general - purpose nature of stilts has the drawback to make hard the syntax for the composition of the command line ; on the other hand , it does not support the full range of cross - matching options provided by @xmath0 . in order to provide a more user - friendly tool to the astronomers , it is also available its graphical counterpart , tool for operations on catalogs and tables ( topcat , @xcite ) , an interactive graphical viewer and editor for tabular data , based on stil apis and implementing the stilts functionalities , but with all the intrinsic limitations of the graphical tools , very similar to the web applications in terms of use . regardless the approach to cross - match the astronomical sources , the main problem is to minimize the computational time exploding with the increasing of the matching catalog size . in principle , the code can be designed according to multi - process and/or multi - thread paradigm , so exploiting the hosting machine features . for instance , @xcite evaluated to use a multi - gpu environment , designing and developing their own xmatch tool , @xcite . other studies are focused to efficiently cross - match large astronomical catalogs on clusters consisting of heterogeneous processors including both multi - core cpus and gpus , ( @xcite , @xcite ) . furthermore , it is possible to reduce the number of sources to be compared among catalogs , by opportunely partitioning the sky through indexing functions and determining only a specific area to be analyzed for each source . cds - xmatch and the tool described in @xcite use hierarchical equal area isolatitude pixelisation ( healpix , @xcite ) , to create such sky partition . @xcite , instead , proposed a combined method to speed up the cross - match by using htm ( hierarchical triangle mesh , @xcite ) , in combination with healpix and by submitting the analysis to a pool of threads . healpix is a genuinely curvilinear partition of the sphere into exactly equal area quadrilaterals of varying shape ( see fig . 3 in @xcite ) the base - resolution comprises twelve pixels in three rings around the poles and equator . each pixel is partitioned into four smaller quadrilaterals in the next level . the strategy of htm is the same of healpix . the difference between the two spatial - indexing functions is that htm partitioning is based on triangles , starting with eight triangles , @xmath7 on the northern and @xmath7 on the southern hemisphere , each one partitioned into four smaller triangles at the next level ( see also fig . 2 in @xcite ) . by using one or both functions combined together , it is possible to reduce the number of comparisons among objects to ones lying in adjacent areas . finally openskyquery uses the _ zones _ indexing algorithm to efficiently support spatial queries on the sphere , @xcite . the basic idea behind the _ zones _ method is to map the sphere into stripes of a certain height @xmath8 , called zones . each object with coordinates ( @xmath9 , @xmath10 ) is assigned to a zone by using the formula : @xmath11 a traditional b - tree index is then used to store objects within a zone , ordered by _ zoneid _ and right ascension . in this way , the spatial cross - matching can be performed by using bounding boxes ( b - tree ranges ) dynamically computed , thus reducing the number of comparisons ( fig . 1 in @xcite ) . finally , an additional and expensive test allows to discard false positives . all the cross - matching algorithms based on a sky partitioning have to deal with the so - called block - edge problem , illustrated in fig . [ fig : block - edge ] : the objects @xmath12 and @xmath13 in different catalogs correspond to the same object but , falling in different pieces of the sky partition , the cross - matching algorithm is not able to identify the match . to solve this issue , it is necessary to add further steps to the pipeline , inevitably increasing the computational time . for example , the zhao s tool , @xcite , expands a healpix block with an opportunely dimensioned border ; instead , the algorithm described by @xcite , combining healpix and htm virtual indexing function shapes , is able to reduce the block - edge problem , because the lost objects in a partition may be different from one to another . and @xmath13 in two catalogs . even if corresponding to the same source , they can be discarded by the algorithm , since they belong to two different blocks of the sky partition.,title="fig:",width=226 ] + @xmath0 is a command - line open - source python script , designed and developed to perform a wide range of cross - matching types among astrophysical catalogs . the tool is able to be easily executed as a stand - alone process or integrated within any generic data reduction / analysis pipeline . based on a specialized sky partitioning function , its high - performance capability is ensured by making use of the multi - core parallel processing paradigm . it is designed to deal with massive catalogs in different formats , with the maximum flexibility given to the end - user , in terms of catalog parameters , file formats , coordinates and cross - matching functions . in @xmath0 different functional cases and matching criteria have been implemented , as well as the most used join function types . it also works with the most common catalog formats , with or without header : flexible image transport system ( fits , version tabular ) , american standard code for information interchange ( ascii , ordinary text , i.e. space separated values ) , comma separated values ( csv ) , virtual observatory table ( votable , xml based ) and with two kinds of coordinate system , equatorial and galactic , by using stilts in combination with some standard python libraries , namely _ _ numpy _ _ @xcite , and _ _ pyfits _ _ ] . + despite the general purpose of the tool , reflected in a variety of possible functional cases , @xmath0 is easy to use and to configure through few lines in a single configuration file . main features of @xmath0 are the following : 1 . _ command line _ : @xmath0 is a command - line tool . it can be used as stand - alone process or integrated within more complex pipelines ; 2 . _ python compatibility _ : compatible with python 2.7.x and 3.4.x ( up to the latest version currently available , @xmath15 ) ; 3 . _ multi - platform _ : @xmath0 has been tested on ubuntu linux @xmath16 , windows @xmath17 and @xmath18 , mac os and fedora ; 4 . _ multi - process _ : the cross - matching process has been developed to run by using a multi - core parallel processing paradigm ; 5 . _ user - friendliness _ : the tool is very simple to configure and to use ; it requires only a configuration file , described in sec . [ sect : config ] . the internal cross - matching mechanism is based on the sky partitioning into cells , whose dimensions are determined by the parameters used to match the catalogs . the sky partitioning procedure is described in [ sect : preproc ] . the fig . [ fig : flowchart ] shows the most relevant features of the @xmath0 processing flow and the user parameters available at each stage . as mentioned before , the user can run @xmath0 to match two input catalogs by choosing among three different functional cases : 1 . _ sky _ : the cross - match is done within sky areas ( elliptical or rectangular ) defined by the celestial coordinates taken from catalog parameters ; 2 . _ exact value _ : two objects are matched if they have the same value for a pair of columns ( one for each catalog ) defined by the user ; 3 . _ row - by - row _ : match done on a same row - id of the two catalogs . the only requirement here is that the input catalogs must have the same number of records . the positional cross - match strategy of the @xmath0 method is based on the same concept of the q - fulltree approach , an our tool introduced in @xcite and @xcite : for each object of the first input catalog , it is possible to define an elliptical , circular or rectangular region centered on its coordinates , whose dimensions are limited by a fixed value or defined by specific catalog parameters . for instance , the two full width at half maximum ( fwhm ) values in the catalog can define the two semi - axes of an ellipse or the couple width and height of a rectangular region . it is also possible to have a circular region , by defining an elliptical area having equal dimensions . once defined the region of interest , the next step is to search for sources of the second catalog within such region , by comparing their distance from the central object and the limits of the area ( for instance , in the elliptical cross - match the limits are defined by the analytical equation of the ellipse ) . + in the _ sky _ functional case , the user can set additional parameters in order to characterize the matching region and the properties of the input catalogs . in particular , the user may define : 1 . the shape ( elliptical or rectangular ) of the matching area , i.e. the region , centered on one of the matching sources , in which to search the objects of the second catalog ; 2 . the dimensions of the searching area . they can be defined by fixed values ( in arcseconds ) or by parametric values coming from the catalog . moreover , the region can be rotated by a position angle ( defined as fixed value or by a specific column present in the catalog ) ; 3 . the coordinate system for each catalog ( galactic , icrs , fk4 , fk5 ) and its units ( degrees , radians , sexagesimal ) , as well as the columns containing information about position and designation of the sources . an example of graphical representation of an elliptical cross - match is shown in fig . [ fig : crossmatch ] . in the _ exact value _ case , the user has to define only which columns ( one for each input catalog ) have to be matched , while in the most simple _ row - by - row _ case no particular configuration is needed . @xmath0 produces a file containing the results of the cross - match , consisting into a series of rows , corresponding to the matching objects . in the case of _ exact value _ and _ sky _ options , the user can define the conditions to be satisfied by the matched rows to be stored in the output . first , it is possible to retrieve , for each source , all the matches or only the best pairs ( in the sense of closest objects , according to the match selection criterion ) ; then , the user can choose different join possibilities ( in fig . [ fig : joins ] the graphical representation of available joins is shown ) : @xmath19 and @xmath20 : : only rows having an entry in both input catalogs , ( fig . [ fig : joins]a ) ; @xmath19 or @xmath20 : : all rows , matched and unmatched , from both input catalogs , ( fig . [ fig : joins]b ) ; all from @xmath19 ( all from @xmath20 ) : : all matched rows from catalog @xmath19 ( or @xmath20 ) , together with the unmatched rows from catalog @xmath19 ( or @xmath20 ) , ( fig . [ fig : joins]c - d ) ; @xmath19 not @xmath20 ( @xmath20 not @xmath19 ) : : all the rows of catalog @xmath19 ( or @xmath20 ) without matches in the catalog @xmath20 ( or @xmath19 ) , ( fig . [ fig : joins]e - f ) ; @xmath19 xor @xmath20 : : the `` exclusive or '' of the match - i.e. only rows from the catalog @xmath19 not having matches in the catalog @xmath20 and viceversa , ( fig . [ fig : joins]g ) . + + any experiment with the @xmath0 tool is based on two main phases ( see fig . [ fig : flowchart ] ) : 1 . _ pre - matching : _ this is the first task performed by @xmath0 during execution . the tool manipulates input catalogs to extract the required information and prepare them to the further analysis ; 2 . _ matching : _ after data preparation , @xmath0 performs the matching according to the criteria defined in the configuration file . finally , the results are stored in a file , according to the match criterion described in sec . [ sect : join ] , and all the temporary data are automatically deleted . this is the preliminary task performed by @xmath0 execution . during the pre - matching phase , @xmath0 performs a series of preparatory manipulations on input data . first of all , a validity check of the configuration parameters and input files . then it is necessary to split the data sets in order to parallelize the matching phase and improve the performance . in the _ exact value _ functional case only the first input catalog will be split , while in the _ sky _ case both data sets will be partitioned in subsets . in the latter case , @xmath0 makes always use of galactic coordinates expressed in degrees , thus converting them accordingly if expressed in different format . when required , the two catalogs are split in the following way : in the first catalog all the entries are divided in groups , whose number depends on the multi - processing settings ( see sec . [ sect : config ] ) , since each process is assigned to one group ; in the second catalog the sky region defined by the data set is divided into square cells , by assigning a cell to each entry , according to its coordinates ( fig . [ fig : partitioning ] ) . we used the python multiprocess module to overcome the gil problem , by devoting particular care to the granularity of data to be handled in parallel . this implies that the concurrent processes do not need to share resources , since each process receives different files in input ( group of object of the 1st catalog and cells ) and produces its own output . finally the results are merged to produce the final output . the partitioning procedure on the second catalog is based on the dimensions of the matching areas : the size of the unit cell is defined by the maximum dimension that the elliptical matching regions can assume . if the `` size type '' is `` parametric '' , then the maximum value of the columns indicated in the configuration is used as cell size ; in the case of `` fixed '' values , the size of the cell will be the maximum of the two values defined in the configuration ( fig . [ fig : partitioning]a ) . in order to optimize the performance , the size of the unit cell can not be less than a threshold value , namely the _ minimum partition cell size _ , which the user has to set through the configuration file . the threshold on the cell size is required in order to avoid the risk to divide the sky in too many small areas ( each one corresponding to a file stored on the disk ) , which could slow down the cross - matching phase performance . in sec . [ sect : optimization ] we illustrated a method to optimize such parameter as well as the number of processes to use , according to the hosting machine properties . once the partitioning is defined , each object of the second catalog is assigned to one cell , according to its coordinates . having defined the cells , the boundaries of an elliptical region associated to an object can fall at maximum in the eight cells surrounding the one including the object , as shown in fig . [ fig : partitioning]b . this prevents the block - edge problem previously introduced . once the data have been properly re - arranged , the cross - match analysis can start . in the _ row - by - row _ case , each row of the first catalog is simply merged with the corresponding row of the second data set through a serial procedure . in the other functional cases , the cross - matching procedure has been designed and implemented to run by using parallel processing , i.e. by assigning to each parallel process one group generated in the previous phase . in the _ exact value _ case , each object of the group is compared with all the records of the second catalog and matched according to the conditions defined in the configuration file . in the _ sky _ functional case , the matching procedure is slightly more complex . as described in sec . [ sect : usecases ] , the cross - match at the basis of the @xmath0 method is based on the relative position of two objects : for each object of the first input catalog , @xmath0 defines the elliptical / rectangular region centered on its coordinates and dimensions . therefore a source of the second catalog is matched if it falls within such region . in practice , as explained in the pre - matching phase , having identified a specific cell for each object of a group , this information is used to define the minimum region around the object used for the matching analysis . the described choice to set the dimensions of the cells ensures that , if a source matches with the object , it must lie in the nine cells surrounding the object ( also known as moore s neighborhood , @xcite , see also fig . [ fig : partitioning]b ) . therefore it is sufficient to cross - match an object of a group only with the sources falling in nine cells . in the _ sky _ functional case , @xmath0 performs a cross - matching of objects lying within an elliptical , circular or rectangular area , centered on the sources of the first input catalog . the matching area is characterized by @xmath21 configuration parameters defining its shape , dimensions and orientation . in fig . [ fig : pa ] is depicted a graphical representation of two matching areas ( elliptical and rectangular ) with the indication of its parameters . in particular , to define the orientation of the matching area , @xmath0 requires two further parameters besides the offset and the value of the position angle , representing its orientation . the position angle , indeed , is referred , by default , to the greatest axis of the matching area with a clockwise orientation . the two additional parameters give the possibility to indicate , respectively , the correct orientation ( clockwise / counterclockwise ) and a shift angle ( in degrees ) . finally , the results of the cross - matching are stored in a file , containing the concatenation of all the columns of the input catalogs referred to the matched rows . in the _ sky _ functional case the column reporting the separation distance between the two matching objects is also included . the tool @xmath0 is interfaced with the user through a single configuration file , to be properly edited just before the execution of any experiment . if the catalogs do not contain the source s designation / id information , @xmath0 will automatically assign an incremental row - id to each entry as object designation . for the _ sky _ functional case , assuming that both input catalogs contain the columns reporting the object coordinates , @xmath0 is able to work with galactic and equatorial ( icrs , fk4 , fk5 ) coordinate systems , expressed in the following units : degrees , radians or sexagesimal . if the user wants to use catalog information to define the matching region ( for instance , the fwhms or a radius defined by the instrumental resolution ) , obviously the first input catalog must contain such data . the position angle value / column is , on the contrary , an optional information ( default is 0@xmath22 , clockwise ) . @xmath0 is conceived for a community as wide as possible , hence it has been designed in order to satisfy the requirement of user - friendliness . therefore , the configuration phase is limited to the editing of a setup file , can also automatically generate a dummy configuration file that could be used as template . ] containing all the information required to run @xmath0 . this file is structured in sections , identified by square brackets : the first two are required , while the others depend on the particular use case . in particular , the user has to provide the following information : 1 . the input files and their format ( fits , ascii , csv or votable ) ; 2 . the name and paths of the temporary , log and output files ; 3 . the match criterion , corresponding to one of the functional cases ( _ sky , exact value , row - by - row _ ) . @xmath0 gives also the possibility to set the number of processes running in parallel , through an optional parameter which has as default the number of cores of the working machine ( minus one left available for system auxiliary tasks ) . the configuration for the _ sky _ functional case foresees the setup of specific parameters of the configuration file : those required to define the shape and dimensions of the matching area , the properties of the input catalogs already mentioned in sec . [ sect : usecases ] , coordinate system , units as well as the column indexes for source coordinates and designation . in addition , a parameter characterizing the sky partitioning has to be set ( see sec . [ sect : preproc ] for further information ) . the parameters useful to characterize the matching area are the following : area shape : : it can be elliptical or rectangular ( circular is a special elliptical case ) ; size type : : the valid entries are _ fixed _ or _ parametric_. in the first case , a fixed value will be used to determine the matching area ; in the second , the dimensions and inclination of the matching area will be calculated by using catalog parameters ; first and second dimensions of matching area : : the axes of the ellipse or width and height of the rectangular area . in case of fixed `` size type '' , they are decimal values ( in arcsec ) , otherwise , they represent the index ( integer ) or name ( string ) of the columns containing the information to be used ; parametric factor : : it is required and used only in the case of parametric `` size type '' . it is a decimal number factor to be multiplied by the values used as dimensions , in order to increase or decrease the matching region , as well as useful to convert their format ; pa column / value : : it is the position angle value ( in the `` fixed '' case , expressed in degrees ) or the name / id of the column containing the position angle information ( in the `` parametric '' case ) ; pa settings : : the position angle , which in @xmath0 is referred , by default , to the main axis of the matching area ( greatest ) with a clockwise orientation . the two parameters defined here give the possibility to indicate the correct orientation ( clockwise / counterclockwise ) and a shift angle ( in degrees ) . the user has also to specify which rows must be included in the output file , by setting the two parameters indicating the match selection and the join type , as described in sec . [ sect : join ] . for the _ exact value _ functional case it is required to set the name or i d of the columns used for the match for both input files . the user has also to specify which rows must be included in the output file , by setting the two parameters indicating the match selection and the join type , as described in sec . [ sect : join ] . for the _ row - by - row _ functional case , no other settings are required . the only constrain is that both catalogs must have the same number of entries . as reflected from the description of @xmath0 , the choice of the best values for its internal parameters ( in particular the number of parallel processes and the minimum cell size , introduced in sec . [ sect : preproc ] ) , is crucial to obtain the best computational efficiency . this section is dedicated to show the importance of this choice , directly depending on the features of the hosting machine . in the following tests we used a computer equipped with an intel(r ) core(tm ) @xmath23 , with one @xmath24 , @xmath25 cpu , @xmath26 gb of ram and hosting ubuntu linux @xmath16 as operative system ( os ) on a standard hard disk drive . we proceeded by performing two different kinds of tests : 1 . a series of tests with a fixed value for the minimum cell size ( @xmath27 ) and different values of the number of parallel processes ; 2 . a second series by using the best value of number of parallel processes found at previous step and different values for the minimum cell size . the configuration parameters used in this set of tests are reported in table [ test1:settings ] . the input data sets are two identical catalogs ( csv format ) consisting of @xmath28 objects extracted from the ukidss gps public data @xcite , in the range of galactic coordinates @xmath29 $ ] , @xmath30 $ ] . each record is composed by @xmath31 columns . the choice to cross - match a catalog with itself represent the worst case in terms of cross - matching computational time , since each object matches at least with itself . by setting `` match selection '' as best and `` join type '' as 1 and 2 ( see table [ test1:settings ] ) , we obtained an output of @xmath28 objects matched with themselves as expected . we also performed all the tests by using a random shuffled version of the same input catalog , obtaining the same results . this demonstrates that the @xmath0 output is not affected by the particular order of data in the catalogs . .@xmath0 settings in the first set of tests performed to evaluate the impact of the number of parallel processes and the minimum cell size configuration parameters on the execution time . the choice of same dimensions for the ellipse axes was due to perform a fair comparison with stilts and cds - xmatch , which allow only circular cross - matching . [ cols="^,^",options="header " , ] the first input catalog has been extracted by the ukidss gps data in the range of galactic coordinates @xmath32 $ ] , @xmath30 $ ] , while the second input catalog has been extracted by the glimpse _ data , ( @xcite and @xcite ) , in the same range of coordinates . from each catalog , different subsets with variable number of objects have been extracted . in particular , data sets with , respectively , @xmath33 , @xmath34 , @xmath28 , @xmath35 and @xmath36 objects have been created from the first catalog , while , from second catalog , data sets with @xmath33 , @xmath34 , @xmath28 and @xmath35 rows have been extracted . then , each subset of first catalog has been cross - matched with all the subsets of the second catalog . for uniformity of comparison , due to the limitations imposed by cds - xmatch in terms of available disk space , it has been necessary to limit to only @xmath37 the number of columns for all the subsets involved in the tests performed to compare c@xmath38 and cds - xmatch ( for instance , i d and galactic coordinates ) . for the same reason , the data set with @xmath39 rows has not been used in the comparison between c@xmath38 and cds - xmatch . the common internal configuration used in these tests is shown in table [ test1:settings ] , except for the match selection there was , in fact , the necessity to set it to _ all _ for uniformity of comparison with the cds - xmatch tool ( which makes available only this option ) . then the _ best _ type has been used to compare @xmath0 with stilts and topcat . furthermore , in all the tests , the number of parallel processes was set to @xmath40 and the minimum cell size to @xmath27 , corresponding to the best conditions found in the optimization process of @xmath0 ( see sec . [ sect : optimization ] ) . finally , we chose same dimensions of the ellipse axes in order to be aligned with other tools , which allow only circular cross - matching areas . concerning the comparison among @xmath0 and the three mentioned tools , in the cases of both _ all _ and _ best _ types of matching selection , all tools provided exactly the same number of matches in the whole set of tests , thus confirming the reliability of @xmath0 with respect to other tools ( table [ tab : matchres ] ) . rows has not been used . ] in terms of computational efficiency , @xmath0 has been evaluated by comparing the computational time of its cross - matching phase with the other tools . the pre - matching and output creation steps have been excluded from the comparison , because strongly dependent on the host computing infrastructure . the other configuration parameters have been left unchanged ( table [ test1:settings ] ) . the complete setup for the described experiments is reported in the appendix . in fig . [ fig : c3vsstrows ] we show the computational time of the cross - matching phase for @xmath0 and stilts , as function of the incremental number of rows ( objects ) in the first catalog , and by varying the size of the second catalog in four cases , spanning from @xmath33 to @xmath35 rows . in all diagrams , it appears evident the difference between the two tools , becoming particularly relevant with increasing amounts of data . in the second set of tests performed on the @xmath0 cross - matching phase and stilts , the computational time has been evaluated as function of the incremental number of columns of the first catalog ( from the minimum required @xmath37 up to @xmath31 , the maximum number of columns of catalog 1 ) , and by fixing the number of columns of the second catalog in five cases , respectively , @xmath37 , @xmath5 , @xmath41 , @xmath42 and @xmath43 , which is the maximum number of columns for catalog 2 . in terms of number of rows , in all cases both catalogs were fixed to @xmath35 of entries . in fig . [ fig : c3vsstcols ] the results only for @xmath37 and @xmath43 columns of catalog 2 are reported , showing that @xmath0 is almost invariant to the increasing of columns , becoming indeed faster than stilts from a certain amount of columns . such trend is confirmed in all the other tests with different number of columns of the second catalog . this behavior appears particularly suitable in the case of massive catalogs . finally , in the case of two fits input files instead of csv files , stilts computational time as function of the number of columns is constant and slightly faster than @xmath14 . in the last series of tests , we compared the computational efficiency between the @xmath0 cross - matching phase and cds - xmatch . in this case , due to the limitation of the catalog size imposed by cds - xmatch , the tests have been performed by varying only the number of rows from @xmath33 to @xmath35 as in the analogous tests with stilts ( except the test with @xmath36 rows ) , fixing the number of columns to @xmath37 . moreover , in this case , the cross - matching phase of @xmath0 has been compared with the duration of the phase _ execution _ of the cds - xmatch experiment , thus ignoring latency time due to the job submission , strongly depending on the network status and the state of the job queue , but taking into account the whole job execution . the results , reported in fig . [ fig : c3vsxmrows ] , show a better performance of @xmath0 , although less evident when both catalogs are highly increasing their dimensions , where the differences due to the different hardware features become more relevant . at the end of the test campaign , two other kinds of tests have been performed : ( i ) the verification of the portability of @xmath0 on different oss and ( ii ) an analysis of the impact of different disk technology on the computing time efficiency of the tool . in the first case , we noted , as expected , a decreasing of @xmath0 overall time performance on the windows versions ( @xmath17 and @xmath18 ) , with respect to same tests executed on linux versions ( ubuntu and fedora ) and mac os . on average @xmath0 execution was @xmath44 times more efficient on linux and mac os than windows . this is most probably due to the different strategy of disk handling among various oss , particularly critical for applications , like cross - matching tools , which make an intensive use of disk accesses . this analysis induced us to compare two disk technologies : hdd ( hard disk drive ) vs ssd ( solid state disk ) . both kinds of disks have been used on a sample of the tests previously described , revealing on average a not negligible increasing of computing time performance in the ssd case of @xmath45 times with respect to hdd . for clarity , all test results presented in the previous sections have been performed on the same hdd . in this paper we have introduced @xmath0 , a new scalable tool to cross - match astronomical data sets . it is a multi - platform command - line python script , designed to provide the maximum flexibility to the end users in terms of choice about catalog properties ( i / o formats and coordinates systems ) , shape and size of matching area and cross - matching type . nevertheless , it is easy to configure , by compiling a single configuration file , and to execute as a stand - alone process or integrated within any generic data reduction / analysis pipeline . in order to ensure the high - performance capability , the tool design has been based on the multi - core parallel processing paradigm and on a basic sky partitioning function to reduce the number of matches to check , thus decreasing the global computational time . moreover , in order to reach the best performance , the user can tune on the specific needs the shape and orientation of the matching region , as well as tailor the tool configuration to the features of the hosting machine , by properly setting the number of concurrent processes and the resolution of sky partitioning . although elliptical cross - match and the parametric handling of angular orientation and offset are known concepts in the astrophysical context , their availability in the presented command - line tool makes @xmath0 competitive in the context of public astronomical tools . a test campaign , done on real public data , has been performed to scientifically validate the @xmath0 tool , showing a perfect agreement with other publicly available tools . the computing time efficiency has been also measured by comparing our tool with other applications , representative of different paradigms , from stand - alone command - line ( stilts ) and graphical user interface ( topcat ) to web applications ( cds - xmatch ) . such tests revealed the full comparable performance , in particular when input catalogs increase their size and dimensions . for the next release of the tool , the work will be mainly focused on the optimization of the pre - matching and output creation phases , by applying the parallel processing paradigm in a more intensive way . moreover , we are evaluating the possibility to improve the sky partitioning efficiency by optimizing the calculation of the minimum cell size , suitable also to avoid the block - edge problem . the @xmath0 tool , @xcite , and the user guide are available at the page http://dame.dsf.unina.it/c3.html . the authors would like to thank the anonymous referee for extremely valuable comments and suggestions . mb and sc acknowledge financial contribution from the agreement asi / inaf i/023/12/1 . mb , am and gr acknowledge financial contribution from the 7th european framework programme for research grant fp7-space-2013 - 1 , _ vialactea - the milky way as a star formation engine_. mb and am acknowledge the prin - inaf 2014 _ glittering kaleidoscopes in the sky : the multifaceted nature and role of galaxy clusters_. 99 agrafioti , i. 2012 , from the geosphere to the cosmos , synergies with astroparticle physics , astroparticle physics for europe ( aspera ) , contributed volume , http://www.aspera-eu.org annis , j. t. , 2013 , in american astronomical society meeting abstracts # 221 , des survey strategy and expectations for early science , 221 , id.335.05 becciani , u. , bandieramonte , m. , brescia , m. , et al . 2015 , in proc . adass xxv conf . , advanced environment for knowledge discovery in the vialactea project , in press . 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[ sect : comparison ] . the text preceded by the semicolon is a comment . .... \textcolor{red}{[i / o files ] } input catalog 1 : \textcolor{olive}{./input / ukidss.csv } format catalog 1 : \textcolor{olive}{csv } \textcolor{blue}{;csv , fits , votable or ascii } input catalog 2 : \textcolor{olive}{./input / glimpse.csv } format catalog 2 : \textcolor{olive}{csv } \textcolor{blue}{;csv , fits , votable or ascii } output : \textcolor{olive}{./output / out.csv } output format : \textcolor{olive}{csv } \textcolor{blue}{;csv , fits , votable or ascii } log file : \textcolor{olive}{./output / out.log } stilts directory : \textcolor{olive}{./libs } working directory : \textcolor{olive}{./tmp } \textcolor{blue}{;temporary directory , removed when completed } \textcolor{red}{[sky parameters ] } area shape : \textcolor{olive}{ellipse } \textcolor{blue}{;ellipse or rectangle } size type : \textcolor{olive}{fixed } \textcolor{blue}{;parametric or fixed } matching area first dimension : \textcolor{olive}{5 } \textcolor{blue}{;arcsec for fixed type - column name / number for parametric type } matching area second dimension : \textcolor{olive}{5 } \textcolor{blue}{;arcsec for fixed type - column name / number for parametric type } parametric factor : \textcolor{olive}{1 } \textcolor{blue}{;multiplicative factor for dimension columns - required for parametric type } pa column / value : \textcolor{olive}{0 } \textcolor{blue}{;degrees for fixed type - column name / number for parametric type } pa settings : \textcolor{olive}{clock , 0 } \textcolor{blue}{;orientation ( clock , counter ) , shift ( degrees ) -empty or default = clock,0 } catalog 2 minimum partition cell size : \textcolor{olive}{100 } \textcolor{blue}{;arcsec } \textcolor{red}{[catalog 1 properties ] } coordinate system : \textcolor{olive}{galactic } \textcolor{blue}{;galactic , icrs , fk4 , fk5 } coordinate units : \textcolor{olive}{deg } \textcolor{blue}{;degrees ( or deg ) , radians ( or rad ) , sexagesimal ( or sex ) } glon / ra column : \textcolor{olive}{l } \textcolor{blue}{;column number or name - required for sky algorithm } glat / dec column : \textcolor{olive}{b } \textcolor{blue}{;column number or name - required for sky algorithm } designation column : \textcolor{olive}{sourceid } \textcolor{blue}{;column number or name - -1 for none } \textcolor{red}{[catalog 2 properties ] } coordinate system : \textcolor{olive}{galactic } \textcolor{blue}{;galactic , icrs , fk4 , fk5 } coordinate units : \textcolor{olive}{deg } \textcolor{blue}{;degrees ( or deg ) , radians ( or rad ) , sexagesimal ( or sex ) } glon / ra column : \textcolor{olive}{l } \textcolor{blue}{;column number or name - required for sky algorithm } glat / dec column : \textcolor{olive}{b } \textcolor{blue}{;column number or name - required for sky algorithm } designation column : \textcolor{olive}{designation } \textcolor{blue}{;column number or name , -1 for none } \textcolor{red}{[output rows ] } match selection : \textcolor{olive}{all } \textcolor{blue}{;all or best } join type : \textcolor{olive}{1 and 2 } \textcolor{blue}{;1 and 2 , 1 or 2 , all from 1 , all from 2 , 1 not 2 , 2 not 1 , 1 xor 2 } ....

modern astrophysics is based on multi - wavelength data organized into large and heterogeneous catalogs . hence , the need for efficient , reliable and scalable catalog cross - matching methods plays a crucial role in the era of the petabyte scale . furthermore , multi - band data have often very different angular resolution , requiring the highest generality of cross - matching features , mainly in terms of region shape and resolution . in this work we present @xmath0 ( command - line catalog cross - match ) , a multi - platform application designed to efficiently cross - match massive catalogs . it is based on a multi - core parallel processing paradigm and conceived to be executed as a stand - alone command - line process or integrated within any generic data reduction / analysis pipeline , providing the maximum flexibility to the end - user , in terms of portability , parameter configuration , catalog formats , angular resolution , region shapes , coordinate units and cross - matching types . using real data , extracted from public surveys , we discuss the cross - matching capabilities and computing time efficiency also through a direct comparison with some publicly available tools , chosen among the most used within the community , and representative of different interface paradigms . we verified that the @xmath0 tool has excellent capabilities to perform an efficient and reliable cross - matching between large data sets . although the elliptical cross - match and the parametric handling of angular orientation and offset are known concepts in the astrophysical context , their availability in the presented command - line tool makes @xmath0 competitive in the context of public astronomical tools .

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Proceed to summarize the following text: since the discovery of the giant magnetoresistance effect,@xcite research in spintronics has been developing at a fast pace . an important requirement for practical applications of this novel technology is the generation , control and manipulation of spin - polarized currents preferably using electric fields only.@xcite spin - orbit interactions in semiconductor materials are promising tools to achieve that goal . in particular , the rashba interaction,@xcite a type of spin - orbit coupling that originates from a lack of inversion symmetry in semiconductor heterostructures ( such as inas or gaas ) , has been experimentally shown to possess a high degree of tunability using gate contacts.@xcite since the spin - orbit interaction couples the electron momentum and its spin , the rashba field behaves as an effective magnetic field that is responsible for spin coherent oscillations , which can be exploited in spintronics . based on this property , datta and das suggested a spin field - effect transistor.@xcite it consists of a one - dimensional ballistic channel sandwiched by two ferromagnetic contacts . their proposal relies on the control of the current along the channel using the rashba interaction via a third terminal ( the gate ) and the relative orientation of the leads magnetizations . the length of the channel and the intensity of the rashba strength determine the flow of the current . realization of the spin transistor was hindered by some limitations , such as the mismatch problem ( which results in poor injection of spin - polarized current between a ferromagnet and a semiconductor)@xcite and the idealization of ballistic transport.@xcite however , recent experiments on quasi - two dimensional structures@xcite , already discussed in refs . , have overcome these obstacles and have obtained a behavior which looks similar to the spin transistor effect . in reality , strictly one - dimensional channels are hard to fabricate and one must deal mostly with _ quasi_-one dimensional systems containing many propagating channels . confinement in the transversal direction is accomplished with potentials leading to subband spacings often smaller than a few mev , the order of magnitude of the fermi energy in low - dimensional systems . as a consequence , multiple subbands are populated and channel mixing effects become relevant in many situations . in fact , the rashba interaction itself includes an intersubband mixing term which couples adjacent subbands with opposite spins . this coupling has been recently demonstrated to give rise to strongly modulated conductance curves,@xcite especially close to the onset of higher - energy plateaus , due to fano interference@xcite between propagating waves and rashba induced localized levels.@xcite in the presence of in - plane magnetic fields , rashba coupling induced intersubband mixing effects are shown@xcite to reduce the visibility of anomalous conductance steps,@xcite and to produce transmission asymmetric lineshapes even in purely one - dimensional systems.@xcite in this paper , we analyze the role of intersubband coupling effects in multichannel quantum wires . our model consists of a quantum wire with a localized rashba spin - orbit interaction coupled to ferromagnetic leads with magnetization perpendicular to the direction of the rashba field . we find that the rashba intersubband coupling term modifies the spin precession effect in a dramatic way . typically , one finds a few oscillation cycles in the conductance curves before arriving at a strongly irregular domain at high values of the rashba parameter in which case the intersubband coupling produces an effective randomization of the injected spins independent of the relative orientation of the leads magnetization . therefore , our results point out a serious limitation of the spin transistor performance , even in the ideal cases of perfect spin injection and fully ballistic propagation . on the other hand , rashba interaction has lately deserved much attention as a generation procedure of spin - polarized currents . several methods have been proposed in different setups ( see refs . , although the list is by no means exhaustive ) . we here consider a simple system : a rashba quantum wire attached to two nonmagnetic leads . we find that the rashba interaction can produce a highly polarized electric current and that the effect is purely due to interchannel coupling . for quantum waveguides supporting a single propagating mode , the polarization effect vanishes.@xcite since the rashba interaction is localized , we calculate the generated polarization as a function of the interface smoothness and show that the highest values of the polarization are obtained when the transition between the regions with and without spin - orbit interaction is abrupt . in sec . ii we discuss the physical system and establish the theoretical model to calculate the linear conductance . section iii is devoted to the numerical results when the contacts are ferromagnetic . the spin polarization effect in the case of normal contacts is analyzed in sec . finally , sec . v contains our conclusions . we consider a quasi - one dimensional system ( a quantum wire ) with a localized rashba interaction ( the rashba dot ) coupled to semi - infinite leads . figure [ fig1 ] shows a sketch of the physical system . transport occurs along the @xmath0 direction . we characterize the rashba dot as a small region of length @xmath1 with strong spin - orbit coupling with strength @xmath2 . the spin polarization in the leads is described using the stoner model for itinerant ferromagnets . due to exchange interaction among the electrons , the electronic bands in the asymptotic regions become spin split with a splitting phenomenologically given by an effective field @xmath3 , which we take as a parameter . this approximation is good at low temperatures ( lower than the curie temperature ) and for electron densities large enough so that strong correlations can be safely neglected.@xcite denoting the stoner field in left and right regions by @xmath4 and @xmath5 , respectively , the parallel configuration is described by @xmath6 while the antiparallel corresponds to @xmath7 . in addition , we assume that a local gate potential @xmath8 is aligning the potential bottom of the successive regions . this way we remove unwanted conductance modifications due to the potential mismatches,@xcite thus focussing on the properties induced purely by the spin - orbit coupling . the system hamiltonian reads @xmath9 the confinement along the direction @xmath10 , perpendicular to the current , is taken as parabolic with oscillator frequency @xmath11 , which defines the length @xmath12 . the inhomogeneous rashba coupling @xmath13 is given by @xmath14 where , as usual , spin is represented by the vector of pauli matrices @xmath15 while @xmath16 and @xmath17 are the cartesian components of the electron s linear momentum . the rashba intensity @xmath18 varies smoothly taking a constant value @xmath2 inside the rashba dot and vanishing elsewhere . the term proportional to @xmath16 is responsible for spin precession of an injected electron.@xcite the intersubband coupling term proportional to @xmath17 couples adjacent subbands with opposite spins . finally , the term with the derivative @xmath19 is added in eq . ( [ eqr ] ) to ensure the hermitian character of the hamiltonian . as mentioned above , the stoner field @xmath20 is constant in the left and right asymptotic regions ( @xmath21 ) and it smoothly vanishes at distances @xmath22 towards the left and right of the rashba dot . these are assumed large enough such that all evanescent states at the interface vanish before reaching the leads . the gate potential aligning the band bottom of the different regions is taken as @xmath23 . an equivalent choice but localized to the rashba dot would be @xmath24 . all spatial transitions in @xmath18 and @xmath20 are described using fermi - like type functions characterized by a small diffusivity @xmath25.@xcite in general , @xmath25 is assumed to be small enough , although we shall also discuss below the dependence with this parameter in some cases . for a given energy @xmath26 the electron wave function fulfills schrdinger s equation @xmath27 with the appropriate boundary conditions . our method of solution combines discretization of the longitudinal variable @xmath0 in a uniform grid with a basis expansion in transverse eigenfunctions @xmath28 and in eigenspinors @xmath29 along a direction given by a unitary vector @xmath30 , @xmath31 where @xmath32 is the spin quantum number while @xmath33 denotes the twofold spin discrete variable . in terms of the polar and azimuthal angles @xmath34 corresponding to the spin quantization axis @xmath30 we can write @xmath35 the transverse eigenfunctions are the solutions of the harmonic 1d oscillator @xmath36 with @xmath37 projecting eq . ( [ eqs ] ) onto the basis we obtain the equations for the unknown _ channel amplitudes _ @xmath38 @xmath39 notice that the rashba interaction is the only source of interchannel coupling since , in general , the matrix element @xmath40 will be non diagonal . using the separation in two spin - orbit contributions introduced in eq . ( [ eqr ] ) we can write @xmath41 equations ( [ eq9 ] ) and ( [ eq10 ] ) clearly show that , in general , both @xmath42 and @xmath43 couple channels with opposite spins through the matrix elements @xmath44 and @xmath45 . of course , if the spin quantization axis @xmath30 is chosen along the @xmath0 or @xmath10 axis then either @xmath44 or @xmath45 become diagonal . regarding the coupling between transverse modes , we notice that @xmath43 is always diagonal ( @xmath46 ) while @xmath42 is connecting modes differing in one subband index ( @xmath47 ) through the oscillator matrix element @xmath48 . if we neglect @xmath42 as in strict one - dimensional systems , eq . ( [ ccm ] ) involves a single mode @xmath49 . if , in addition , the spin axis is chosen along @xmath10 then the two spin modes uncouple and no spin oscillation is allowed ; in other directions ( @xmath0 or @xmath50 ) a rigid spin precession should be expected if all the contribution between parenthesis in eq . ( [ eq10 ] ) is assumed constant . this precession is the underlying working mechanism of the datta - das spin transistor.@xcite below we investigate the solution of eq . ( [ ccm ] ) in the general case in order to analyze the robustness of the _ spin precession _ scenario when @xmath42 is included and when space inhomogeneity in @xmath18 is also taken into account . the appendix contains the details of the employed numerical method to compute the transmission @xmath51 , i.e. , the probability amplitude from a given left incident mode @xmath52 to the right mode @xmath53 . then , using the scattering approach the linear - response conductance is given by , @xmath54 where @xmath55 is the conductance quantum . for later discussion on the polarization of the transmitted current we also define the _ polarized _ conductance @xmath56 , @xmath57 and the relative polarization @xmath58 @xmath59 , @xmath60 we shall pay special attention to the multichannel case considering energies @xmath26 in eq . ( [ ccm ] ) such that up to 10 propagating modes are active in the leads . figure [ fig2 ] shows the results for polarized leads oriented along @xmath0 . when @xmath42 is neglected the conductance for 5 and 10 propagating modes displays an almost sinusoidal behavior with only minor distortions . these deviations , which are enhanced in the single mode case , can be attributed to the quantum interference with the rashba dot.@xcite the present results confirm , therefore , the precession scenario mentioned above but only when the number of modes is large enough and interband coupling is neglected . quite remarkably , however , this scenario is not robust with the inclusion of @xmath42 . when the full rashba interaction is considered only for small values of @xmath2 the conductance behaves in a regular way . very rapidly as @xmath2 increases @xmath61 fluctuates in a staggered way that resembles the conductance fluctuations of disordered systems . the mean value , in units of @xmath62 , is @xmath63 , with @xmath64 the number of active channels , while the amplitude of the fluctuation decreases when @xmath64 increases . the existence of the first conductance minimum has been clearly seen in the experiments of ref . . our results are in agreement with this experiment , but they also predict that successive maxima and minima are heavily distorted or even fully washed out . it is also worth noticing that the first conductance minimum for the black dots occurs at a slightly lower value of @xmath2 than that of the grey ( red color ) data , indicating that the minima @xmath65 are somewhat contracted with respect to the simple prediction from the rashba dot length : @xmath66 , with @xmath67 ( red symbols ) . figure [ fig3 ] contains the results for polarized leads along @xmath0 but in antiparallel directions . in this case , when @xmath68 the conductance vanishes due to the spin valve effect . as @xmath2 increases , however , the conductance rises and the spin valve effect is effectively destroyed by the presence of the rashba dot . for big enough values the system behaves similarly to the case of parallel polarized leads ( fig . [ fig2 ] ) , displaying irregular oscillations around a mean value @xmath69 . for strong spin - orbit couplings and high number of modes no clear distinction between parallel and antiparallel orientations is then to be expected . this is a consequence of the strong subband mixing . in fact , if @xmath42 is neglected ( red symbols ) there is a full correspondence between the conductance nodes of the parallel geometry with the maxima of the antiparallel one ; as could expected from the simplified rigid precession scenario . the above results are not modified if other values of @xmath21 are used , provided they are large enough to ensure full polarization of the leads . the same is true for distances @xmath22 . they should be large enough to allow the decay of evanescent states at the interfaces with the rashba dot and at the points where stoner fields are switched on . we consider next polarized leads along @xmath10 and @xmath50 ; that is , in directions that are perpendicular to the quantum wire . for @xmath50 polarizations the results are very similar to the @xmath0 ones already discussed and thus will not be shown . figures [ fig4 ] and [ fig5 ] contain the results for @xmath10-polarized parallel and antiparallel leads . a first conspicuous difference with the results of figs.[fig2 ] and [ fig3 ] is that the grey symbols ( red color ) do not display wide sinusoidal oscillations . the conductance when @xmath42 is neglected is actually maximal for the parallel case and stays rather constant with some small oscillations at large @xmath70 s that disappear when the number of channels increases . on the other hand , @xmath61 vanishes for the antiparallel orientation . we understand this spin - valve behavior as a complete absence of spin precession , resulting from the fact that @xmath13 is spin diagonal in this approximation [ cf . ( [ eq10 ] ) ] . including @xmath42 in the @xmath10-polarized geometry again yields qualitative modifications of the linear conductance ( black symbols in figs.[fig4 ] and [ fig5 ] ) . except for the antiparallel one - channel case , @xmath61 shows staggering behavior at large @xmath2 s , quite similarly to the @xmath0-polarized results . on average , the conductance is somewhat reduced from the maximal value in the parallel case ( fig . [ fig4 ] ) and , remarkably , takes a finite value in the antiparallel distribution ( fig . [ fig5 ] ) . for @xmath71 the antiparallel conductance has already reached a value close to @xmath72 and to the eventual saturation value . the rashba coupling is thus quite effective in allowing transmission by flipping spins of the polarized incoming electrons towards the opposite spin orientation of the outgoing ones . the single channel limit ( upper panel of fig . [ fig5 ] ) is obviously an exception since even the black symbols vanish in this case . this is easily understood noticing that the incident @xmath73 mode couples in the rashba dot with modes @xmath74 , but not with @xmath75 , which is the only propagating mode in the right lead . therefore , no conduction is possible under this conditions . experimentally , the absence of conductance oscillation in the parallel @xmath10-oriented configuration has been confirmed.@xcite our results reproduce that behavior ( fig . [ fig4 ] ) and they also suggest the antiparallel @xmath10 orientation ( fig . [ fig5 ] ) as an interesting configuration for a spin - orbit - controlled device . indeed , the initial rise of conductance in the multichannel case , interpreted above as a rashba - induced destruction of the spin valve , could be used as the conducting ( on ) state of the device . one should check , however , that the evolution of @xmath76 from zero to the higher values remains smooth for increasing numbers of propagating channels . the present results do not elucidate this point but they seem to indicate that for @xmath77 propagating modes the initial rise of @xmath76 occurs more rapidly than for @xmath78 . in a future work we shall treat the continuum case , having an infinite number of transverse states , using a different approach from the present one . the results shown above are not much modified if the interfaces with the stoner fields at distaces @xmath79 and @xmath80 to the left and right of the rashba dot , respectively ( see fig . [ fig1 ] ) , are smoothed by increasing the corresponding fermi - function parameter.@xcite this confirms that the conductance modifications are an effect of the rashba dot , and not of the stoner field interfaces . indeed , the more diffuse the interface , the more reflectionless and thus more ideal is the description of the contact . in the next section we shall discuss the case of nonpolarized leads ( @xmath81 ) , but we have also calculated some cases of partial polarization by decreasing @xmath3 when both @xmath82 and @xmath83 transverse states are active , although their number is not perfectly balanced . we have found that the conductance is qualitatively similar to the fully polarized case , with irregular behaviour at large values of @xmath2 . it has been recently pointed out@xcite that a rashba dot can act as a current polarizer in such a way that when a non polarized current enters the dot from the left , the transmitted current to the right may attain an important degree of spin polarization in @xmath10 direction . for this to occur , it has been shown that at least two propagating modes of opposite spin must interfere.@xcite in wires with parabolic transverse confinement this means that the energy should at least exceed @xmath84 such that the four modes @xmath85 are active and the interference occurs in subsets @xmath86 and @xmath87 . the resulting spin polarization is very sensitive to the energy ( see fig . 3 of ref . ) and a large enhancement of the polarization @xmath58 , eq . ( [ eqp ] ) is obtained when the energy is such that a fano - type resonance with a quasibound state from a higher evanescent band is formed . this type of resonances which lead to the fano - rashba effect was investigated in ref . . the polarization of the transmitted current is zero if , instead of @xmath10 , other direction for the quantization axis are chosen . the preference for the transverse @xmath10 direction in polarization is an example of _ chirality _ induced by the rashba interaction . this is possible even with a time - reversal invariant hamiltonian like eq . ( [ eqr ] ) because our boundary condition ( left incidence ) is not time reversal invariant . indeed , if we consider the time reversed boundary condition , i.e. , incidence from the right , the current transmitted to the left is polarized in the opposite direction . the superposition of both solutions completely restores the symmetry without any preferred spin direction . the reversal of the polarization for the right - to - left transmission can be seen as a peculiar behavior of rashba polarizers that makes them _ fragile _ in the presence of magnetic barriers like those of sec . indeed , one could naively think that when the rashba dot acts as a current polarizer the left - to - right transmission with @xmath10-magnetized leads should be very high in parallel configuration and very low in antiparallel configuration . this is not the case , however , because of multiple backwards and forwards reflections with their associated inversions of @xmath58 ( see lower panels of figs . [ fig4 ] and [ fig5 ] ) . in this section we assume nonmagnetic leads by taking @xmath88 , i.e. , vanishing stoner fields in fig . [ fig1 ] , and analyze the evolution of the polarization and the conductance when the number of active channels increases . as shown in fig . [ fig6 ] upper panel , high polarizations @xmath58 are obtained for the minimal number of channels @xmath89 and strong spin - orbit intensities @xmath2 . the clear correlation between @xmath61 and @xmath58 , conductance minima correspond to maxima in polarization , indicate that this is an effect connected with the formation of quasibound states that tend to block the current for a given spin direction . when the number of channels is increased ( lower panels of fig . [ fig6 ] ) both @xmath61 and @xmath58 show reduced staggering oscillations with increasing @xmath70 , as in figs . [ fig2]-[fig5 ] . there is also an overall tendency to smoothly reduce @xmath61 and increase @xmath58 in a linear way with @xmath70 . with increasing number of channels the slopes of these straight lines are reduced and for @xmath90 the polarization reaches the values @xmath91 and @xmath92 for 10 and 20 propagating channels , respectively . in almost all cases the polarization is positive , indicating that the transmitted current is preferentially polarized along @xmath93 . in this subsection we discuss how the results are affected by the way in which the rashba field is switched on spatially . for this , we vary the parameter @xmath25 in the fermi functions describing the transitions shown in fig . [ fig1].@xcite for large values of @xmath25 the edges are quite smooth and correspond to an _ adiabatic _ turn - on or turn - of in space . on the contrary , abrupt changes are given by the limit @xmath94 . our method is based on a grid discretization of the variable @xmath0 and its only requirement is that the grid should be fine enough to describe the spatial variations . the results discussed above have been obtained using @xmath95 , a rather small value describing abrupt transitions in space . we have checked that either using a smaller value @xmath96 or a larger value @xmath97 the behaviors of the conductance in the presence of polarized leads discussed in sec . ii , namely the staggering for high values of @xmath2 and the modification due to intersubband coupling , are not qualitatively changed . of course , it should be fulfilled that the rashba dot length @xmath1 is much greater than @xmath25 in order to still allow the transition to reach to the saturation value @xmath2 . more delicate is the polarization @xmath58 discussed in the preceding subsection and fig [ fig6 ] . in fig [ fig7 ] we show the evolution with @xmath25 of @xmath61 and @xmath58 when @xmath78 channels are propagating in the wire . the polarization vanishes when @xmath25 increases , indicating that smooth edges do not favor the appearance of polarized currents . in this diffuse - edge limit the conductance takes the maximal value @xmath98 as in a purely ballistic wire without any rashba dot . the evolution for @xmath99 ( upper panel ) is quite smooth but for @xmath100 ( lower panel ) superimposed to the overall behavior we find irregular maxima and minima as in previous results . recent experiments have proved the feasibility of the spin transistor proposed by datta and das some years ago.@xcite this device , usually presented as a paradigm of spintronics , is expected to open new ways to overcome present limitations of electronics . in this paper we have discussed some specific aspects related to the rashba interaction , including the so - called intersubband coupling , relevant for a better understanding of the physical mechanisms behind the spin transistors and spin polarizers . taking the wire containing the rashba dot oriented along @xmath0 we have analyzed the transmission in the presence of polarized leads along @xmath0 , @xmath10 or @xmath50 , and with increasing number of propagating channels . the cases of parallel and antiparallel polarized leads along @xmath0 and @xmath10 have been explicitly shown . the evolution with rashba intensity shows dramatic modifications when the rashba intersubband coupling is included . these modifications are specially relevant at strong values of @xmath2 , where staggering oscillations of @xmath61 have been found . in general , only a first smooth oscillation of @xmath76 remains when the full rashba interaction is considered , while successive ones are heavily distorted or even fully washed out . the spin - valve behavior is effectively destroyed by the rashba dot and the conductance for both parallel and antiparallel leads is relatively high . the role of rashba dots as spin polarizers has been discussed and explicitly calculated assuming the leads to be nonpolarized . a smooth linear increase in @xmath58 with rashba intensity has been observed in the multichannel case . in the limit of adiabatic transitions the polarization vanishes . these overall smooth behaviors are superimposed by irregular changes for high values of @xmath2 . useful discussions with m .- s . choi are gratefully acknowledged . this work was supported by the micinn ( spain ) grant fis2008 - 00781 . this appendix gives some details of the practical method to solve eq . ( [ ccm ] ) and the corresponding boundary conditions . we use a method based on the quantum transmitting boundary algorithm.@xcite a fictitious partitioning of the system in central and asymptotic regions ( contacts ) is introduced . the boundaries for the left and right contacts are at @xmath101 and @xmath102 , respectively . in the contacts the band amplitudes take the form @xmath103 where @xmath104 is a label referring to left @xmath105 and right @xmath106 contacts , respectively , and we defined @xmath107 and @xmath108 . the incident and reflected amplitudes for a given mode @xmath52 and contact @xmath109 are given by @xmath110 and @xmath111 , respectively . this expression is for a propagating channel in contact @xmath109 , for which @xmath112 and its corresponding wavenumber @xmath113 is a real number . equation ( [ eccm1 ] ) also applies to evanescent modes , @xmath114 , if we assume in this case @xmath115 and a purely imaginary wavenumber @xmath116 notice that the output amplitudes can be obtained from the wave function right at the interface , @xmath117 substituting eq . ( [ eqbcn ] ) in eq . ( [ eccm1 ] ) we obtain @xmath118 that is the quantum - transmitting - boundary equation for the contacts . equations ( [ ccm ] ) and ( [ eccm2 ] ) , for the central and contact regions , respectively , form a closed set that does not invoke the wave function at any external point . of course , this is not true for any of these two subsets separately , since central and contact regions are connected through the derivative in eq . ( [ ccm ] ) and of @xmath119 in eq . ( [ eccm2 ] ) . in practice , we use a uniform grid in @xmath0 with @xmath49-point formulae for the derivatives ( @xmath120 ) and truncate the expansion in transverse bands , eq . 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we investigate intersubband mixing effects in multichannel quantum wires in the presence of rashba spin - orbit coupling and attached to two terminals . when the contacts are ferromagnetic and their magnetization direction is perpendicular to the rashba field , the spin - transistor current is expected to depend in a oscillatory way on the rashba coupling strength due to spin coherent oscillations of the travelling electrons . nevertheless , we find that the presence of many propagating modes strongly influences the spin precession effect , leading to ( i ) a quenching of the oscillations and ( ii ) strongly irregular curves for high values of the rashba coupling . we also observe that in the case of leads magnetization parallel to the rashba field , the conductance departs from a uniform value as the rashba strength increases . we also discuss the rashba interaction induced current polarization effects when the contacts are not magnetic and investigate how this mechanism is affected by the presence of several propagating channels .

You are an expert at summarizing long articles. Proceed to summarize the following text: in context of ads / cft correspondence , it was proposed that variation of cosmological constant corresponds to varying the number of colors in the boundary field theory of yang - mills with chemical potential interpretation @xcite . on the other hand , according to teitelboim and henneaux s mechanism , coupling four - dimensional gravity with antisymmetric gauge field without cosmological constant results into appearance of the cosmological constant as a constant of motion teit , and therefore , the cosmological constant will be a variable . recently , the cosmological constant was considered as a state - dependant parameter in two dimensional dilaton gravity @xcite . it was shown that treating the cosmological constant as a @xmath4 charge with non - minimal coupling leads to confining electrostatic potential . larranaga showed that considering the cosmological constant as thermodynamical variable can be extended to use the smarr formula for inner and outer horizons of btz black hole @xcite . recently , there has been an increasing interest in thermodynamical behavior of black holes in asymptotically ads spacetime . this growing interest comes from the fact that using ads / cft correspondence , one can find answers regarding a conformal field theory of @xmath5-dimensions by solving problems that a gravitational field presents in ( @xmath6)-dimensional anti de sitter spacetime @xcite . hawking and page , in their pioneering work , showed that the similarity of phase transition between the stable large black hole and thermal gas in ads space can be interpreted as the confinement / deconfinement phase transition in the dual strongly coupled gauge theory @xcite . later , in an interesting article , witten showed that using ads / cft correspondence , one can study the thermal phase transition and the interpretation of confinement in gauge theories @xcite . on the other hand , one may consider the cosmological constant as a thermodynamical pressure ( in order to investigate the phase transition of black holes ) to extend the phase space ( see @xcite for more details ) and modify the first law of black holes thermodynamics @xcite . another contribution of this consideration is a renewed interpretation for the mass of black holes which from internal energy becomes enthalpy @xcite . this interpretation indicates that the mass of black holes plays more important role in thermodynamical structure of black holes and contains more information regarding phase structure of black holes @xcite . for a canonical ensemble with a fixed charge , it was found that there exists a phase transition between small and large black holes . this phase transition behaves very like the gas / liquid phase transition in a van der waals system @xcite . on the other hand , phase transition of small / large black holes in ads / cft correspondence may be interpreted as conductor / superconductor regions of the condescend matter systems @xcite . considering the fact that maxwell theory contains some fundamental problems and nonlinear electromagnetic fields solve some of these shortcomings @xcite , one is motivated to study different models of nonlinear electrodynamics ( ned ) . one interesting class of these models is born - infeld ( bi ) type which is acquired in the low energy limit of heterotic string theory @xcite . therefore , one is motivated to study these theories ( which in this paper we have considered logarithmic @xcite and exponential forms @xcite ) and the nonlinearity effects of electromagnetic field on critical values representing phase transition of black holes . on the other hand , einstein gravity is not flawless and has some fundamental problems @xcite . generalization of the einstein gravity to higher orders of lovelock gravity is one way to solve some of these problems lovelock . besides , the lagrangian of lovelock gravity is obtainable through the use of the low energy effective action of string theory bi - st . one can take this fact into account that modification of einstein gravity may change the conserved quantities of black holes and therefore it is inevitable to see that critical values and phase transition may depend on the choice of gravity model . in literature , there have been various studies regarding higher orders of lovelock gravity in presence of different ned @xcite . also , the phase transition and stability conditions of black holes in various gravity models have been studied intensively @xcite . these investigations lead to interesting consequences and phenomenologies @xcite . in this paper , we consider higher orders of lovelock gravity in presence of two classes of ned and study their phase structure . we investigate the effects of both nonlinearity of the electrodynamic models and lovelock parameters on the phase diagrams and the critical values . considering the fact that we are treating black holes as thermodynamical system and interpret the first law of black holes mechanics as the first law of thermodynamics , we expect to see the similar thermodynamical behavior for the black holes and usual thermodynamical systems . therefore , it is crucial to investigate phase transition and critical values of black holes . moreover , lovelock gravity is a generalization of the einstein gravity and it is a theory that solves some of the einstein gravity problems . hence , this generalization also gives a correction to calculated critical values of the einstein gravity . the thermodynamical behavior of this modification should be reasonable and consistent with thermodynamical concepts . furthermore , as it was mentioned before , nonlinear electromagnetic fields are introduced to solve shortcomings of maxwell theory . so it is reasonable to investigate nonlinear effects of ned on the phase transition of black holes . also , modifications of electrodynamics like gravitational parts must have consistent thermodynamical behavior . this paper is constructed in the following form . next section is devoted to introduction to higher orders of lovelock gravity and conserved quantities . then , we extend the phase space by considering cosmological constant as thermodynamical pressure and study the smarr formula of these black holes . after that , we calculate critical values and plot related diagrams for different cases . we give a detailed discussion regarding diagrams , their physical interpretations , and the effects of both ned and gravitational parameters . we finish our paper with closing remarks . here , we restrict ourselves to bi types ned models that were introduced by soleng @xcite and hendi @xcite with the following lagrangians @xmath7where enef and lnef denote _ exponential form of nonlinear electromagnetic field _ and _ logarithmic form of nonlinear electromagnetic field _ , respectively , and @xmath8 is nonlinearity parameter . considering the fact that we are interested in studying the spherically symmetric spacetime , we employ the following metric @xmath9 where @xmath10 denotes the standard metric of @xmath5-dimensional sphere , @xmath11 , with the volume @xmath12 . it is known that the field equations of einstein gravity in the presence of ned are in the following forms @xcite @xmath13 where @xmath14 and @xmath15 are einstein tensor and cosmological constant , respectively . einstein gravity in the presence of mentioned ned has been studied in ref . regardless of gravitational sector , one may consider eqs . ( [ lned ] ) and ( [ maxwelleq ] ) with the metric ( [ metric ] ) to obtain the nonzero components of electromagnetic fields with the following explicit forms @xcite @xmath16 where @xmath17 is an integration constant which is proportional to total electric charge of the black hole solutions ( with regarding @xmath18 as proportionality constant ) extended phase space thermodynamics and phase diagrams of the einstein gravity in the presence of ned were investigated in ref . so , we consider gauss - bonnet ( gb ) and third order of lovelock ( tol ) gravities and investigate extended phase space thermodynamics and critical behavior of these gravities . at first , we take into account the gb gravity . in order to obtain the field equation of gb gravity , one should add the @xmath19 tensor to the left hand side of eq . ( [ einsteineq ] ) , in which @xmath20 is @xmath21 , \label{gbterm}\]]where @xmath22 and @xmath23 is gb parameter . considering eq . ( [ einsteineq ] ) with the extra term ( [ gbterm ] ) , one can obtain the following solutions @xcite @xmath24where @xmath25 and@xmath26 @xmath27 , & \;\;\text{enef } \\ \frac{8(d-2)}{\left ( d-1\right ) } \left [ \frac{\left ( 2d-3\right ) ( \gamma -1)% } { d-2}-\frac{\left ( d-1\right ) \ln \left ( \frac{1+\gamma } { 2}\right ) } { d-2}+% \frac{\left ( d-2\right ) \left ( 1-\gamma ^{2}\right ) \mathcal{h}}{d-3}\right ] , & \;\;\text{lnef } \end{array}% \right.\]]in which @xmath28 and @xmath29 is an integration constant which is related to the total mass @xcite @xmath30 in addition , @xmath31 and @xmath32 are in the following forms @xmath33 , % \left [ \frac{3d-7}{2d-4}\right ] , 1-\gamma ^{2 } \right ) , \nonumber \\ \gamma & = & \sqrt{1+\frac{q^{2}}{\beta ^{2}r^{2d-4}}}. \nonumber\end{aligned}\ ] ] calculating the kretschmann scalar , one finds that it diverges at @xmath34 , so the metric function ( [ ggb ] ) has an essential singularity at @xmath34 @xcite . we should note that these solutions may be interpreted as asymptotically ads black holes as those of einstein case @xcite . now , we take into account the surface gravity interpretation to obtain the hawking temperature of the mentioned black hole solutions , yielding @xcite @xmath35where@xmath36 e^{\frac{-2e^{2}}{\beta ^{2}}}-1\right ) , \;\;\text{enef } \\ \;8r^{3}\beta ^{2}\ln \left [ 1-\left ( \frac{e}{2\beta } \right ) ^{2}\right ] + % \frac{4r^{3}e^{2}}{1-\left ( \frac{e}{2\beta } \right ) ^{2}},\;\;\text{lnef}% \end{array}% \right . , \]]and @xmath37 . since obtained solutions are asymptotically ads , one may obtain the entropy of the black hole solutions by the use of the gibbs - duhem relation . after some calculations , one can obtain @xcite @xmath38which confirms that the area law is recovered for @xmath39 . now , we insert the following tol term , @xmath40 , to the field equation of gb gravity to obtain the solutions of tol gravity . the tensor @xmath41 may be written as @xmath42 , \label{tolterm}\end{aligned}\]]where @xmath43 and @xmath44 are , respectively , the coefficient and lagrangian of tol gravity @xmath45 hereafter , we consider the special case @xmath46 to simplify the calculations . it has been shown that the metric function of tol gravity in the presence of ned can be written as hendidehghani@xmath47 @xmath48 the geometric and thermodynamic properties of the asymptotically ads black holes have been studied before @xcite . the finite mass of these solutions is the same as that of einstein gravity , where @xmath29 can be obtained as a function of @xmath49 from the metric function of tol gravity . the hawking temperature and the entropy of the tol solutions can be calculated as @xcite @xmath50and @xmath51 as we mentioned in introduction , there are some motivations to view the cosmological constant as a variable . in addition , there are various theories in which some physical constants ( such as gauge coupling constants , newton constant , lovelock coefficients and bi parameter ) are not fixed but dynamical ones . in that case , it is logical to consider the variation of these parameters into the first law of black hole thermodynamics ctevariable . in order to investigate the phase structure of these classes of gravities , we employ the approach in which the cosmological constant is a thermodynamical variable ( pressure ) with the following relation @xmath52 this consideration could be justified due to the fact that in quantum context , fundamental fixed parameters could vary . as one can see the conjugating thermodynamical variable to this assumption ( cosmological constant as thermodynamical pressure ) will be volume where in literature the derived volume for different types of black holes are in agreement with the topology of the spacetime @xcite . in order to calculate the volume of these thermodynamical systems we use the following relation @xmath53 also , we should consider the effects of cosmological constant in the first law of thermodynamics and extend our phase space . with doing so the total finite mass of the black hole will play the role of enthalpy and hence the corresponding gibbs free energy will be in form of @xmath54 using the mentioned comments , one can obtain the volume with the following form @xmath55which is consistent with topological structure of spherical symmetric spacetime . equation ( [ g ] ) was obtained in einstein gravity @xcite which indicates that although considering lovelock gravity modifies the metric function and some conserved quantities of the black hole , it does not change the volume of the black hole . in addition , it was shown that the smarr formula may be extended to lovelock gravity as well as nonlinear theories of electrodynamics @xcite . geometrical techniques ( scaling argument ) were used to derive an extension of the first law and its related modified smarr formula . the result includes variations in the cosmological constant , lovelock coefficients and also nonlinearity parameter . in our case , lovelock gravity in the presence of ned , @xmath56 should be the function of entropy , pressure , charge , lovelock parameters and bi coupling coefficient @xcite . regarding the previous section , we find that those thermodynamic quantities satisfy the following differential form @xmath57 where we have achieved @xmath58 , and one can obtain @xmath59 using the redefinition of @xmath60 and @xmath61 with respect to the single parameter , @xmath62 , we can rewrite @xmath63 as a single differential form @xmath64 moreover , by scaling argument , we can obtain the generalized smarr relation for our asymptotically ads solutions in the extended phase space @xmath65where @xmath66 , & \text{ened } \\ -\frac{{2{\beta ^{2}}r_{+}^{d-1}}}{{(d-1)q}}\left ( { { \eta _ { + } } -1}\right ) , & \text{lned}% \end{array}% \right . , \end{aligned}\ ] ] @xmath67 , \end{aligned}\]]in which @xmath68 , \,\left [ { \frac{{d-3}}{{2d-4}}}\right ] , \,1-}\gamma _ { + } ^{2}\right ) , \\ \mathcal{d}_{+}&=&_{1}{f}_{1 } \left ( [ 1],\left[\frac{5d-11}{2d-4}\right],\frac{% l_{w+}}{2d-4}\right ) , \\ \mathcal{h}_{+}&=&_{2}{f}_{1}\left ( \left [ \frac{1}{2},\frac{d-3}{2d-4}\right ] , \left [ \frac{3d-7}{2d-4}\right ] , 1-\gamma _ { + } ^{2 } \right).\end{aligned}\ ] ] next step will be calculating critical values . due to relation between volume and radius of the black hole , we use horizon radius ( specific volume ) in order to investigate the critical behavior of these systems phase - higher . in order to do so , we use the method in which critical values are obtained through the use of @xmath69 diagrams . at first , we use the following relations to obtain the proper equations for critical radius @xmath70 for the economical reasons we will not bring obtained relations for calculating critical horizon radius . we employ numerical method for calculating critical values which result into following diagrams for different classes of lovelock gravity . we present various tables in order to plot @xmath69 , @xmath71 and @xmath2 and study the effects of gravitational parameter which is presented by @xmath72 and ned parameter which is presented by @xmath73 . in this paper , we have considered two classes of ned . studying different phase diagrams for these ned shows that they have similar behavior . therefore , for economical reason , we will regard only lnef branch and calculate related critical values of this ned model . it will be constructive to give a short description regarding to different phase diagrams and the information they contain before presenting tables and phase diagrams . @xmath2 diagrams are representing energy level of different states that phase transition takes place between them and shows the changes in energy level of before and after phase transition states . the characteristic swallowtail that is seen in these diagrams shows the process that we know as phase transition . it also gives interesting information regarding temperature of critical points . for @xmath71 plot , it contains information regarding critical temperature and horizon radius in which phase transition takes place . also , it gives some insight about single state regions which in our case is small / large black holes . finally , studying @xmath74 plot gives us information regarding the behavior of pressure as function of horizon radius , critical pressure and critical horizon radius ( volume ) of phase transition . one of the reasons for studying these diagrams is the similarity between phase structure of black holes and the van der waals thermodynamical systems . in what follows , we present various tables to investigate the effects of electrodynamics and gravity models on the critical values of phase transition . we also plot @xmath69 , @xmath71 and @xmath2 diagrams for gb and tol gravities and interpret them . it is notable to mention that considering the metric function of gb gravity , one finds that there is an upper limit for gb parameter to have a real solution . [ cols="^,^,^,^,^",options="header " , ] + table ( @xmath75 ) : tol gravity with @xmath76 , @xmath77 and @xmath78 . @xmath79 @xmath80 @xmath81 @xmath82 @xmath83 @xmath84 as one can confirm , higher orders of lovelock gravity modify the phase diagrams and critical values of volume , pressure and temperature ( see tables @xmath85 and related figs . of [ fig1 ] - [ fig7 ] ) . it is clear that considering higher orders of lovelock gravity leads to different kinds of thermodynamical systems which was also evident through calculated conserved quantities . in this paper , we consider thermodynamical effects of lovelock gravity up to gb and tol , separately . it is evident that critical temperature and pressure are decreasing functions of @xmath72 and also orders of lovelock gravity whereas the critical volume is an increasing function of these two factors ( see figs . [ fig8 ] , [ fig9 ] , [ fig11 ] and [ fig12 ] ) . on the other hand , the energy gap between two phases increases drastically by transforming from lower order of lovelock gravity to higher one ( see figs . [ fig8 ] , [ fig9 ] , [ fig11 ] and [ fig12 ] right panels ) or by increasing @xmath72 for each order ( see figs . [ fig8 ] and [ fig9 ] right panels ) . in addition , the length of subcritical isobars increases which means that the single phase region of small / large black holes decreases ( see figs . fig9 and [ fig12 ] middle panels ) . also , phase transitions region is an increasing function of @xmath72 and order of lovelock gravity ( see figs . [ fig9 ] and [ fig12 ] left panels ) . to conclude , plotted figures show that the highest critical temperature and pressure , and the lowest critical volume and energy gap belong to einstein gravity . on the other hand , system needs higher temperature to have phase transition in einstein gravity . whereas the critical temperature decreases by considering higher orders of lovelock gravity or increasing lovelock coefficients . considering the fact that increasing lovelock parameters and/or adding higher orders of lovelock gravity increase the power of gravity , one may say that in stronger gravitational regimes , phase transitions take place in lower temperature . as for the effects of ned , we find the following results . it is evident that the critical temperature in which swallow tail is formed ( see figs . [ fig5 ] , [ fig11 ] and [ fig12 ] right panels ) and critical pressure ( see figs . [ fig5 ] , [ fig11 ] and [ fig12 ] left panels ) are decreasing functions of nonlinearity parameter whereas the critical volume is an increasing function . in addition , the length of subcritical isobars is an increasing function of nonlinearity parameter . this means that single phase region of small / large black holes is a decreasing function of @xmath73 ( see figs . [ fig5 ] , [ fig11 ] and [ fig12 ] middle panels ) . therefore , for higher values of nonlinearity parameter ( weak nonlinearity strength ) , black holes need to absorb less mass in order to have phase transition . another issue that must be taken into account is the fact that as @xmath73 increases , the gap between isobars decreases whereas for small @xmath73 this gap is greater . due to the fact that we take into account bi type models , for large values of nonlinearity parameter , they will lead to maxwell theory . obtained results show that the lowest critical temperature and pressure and highest critical volume belong to maxwell theory . on the other hand , one can conclude that the power of the nonlinearity causes the system to need higher critical temperature to have a phase transition . this effect is opposite of what was observed for gravity . in addition , the energy gap between two phases , critical temperature and gibbs free energy are increasing functions of dimensions ( fig . [ fig10 ] ) . in other words , for higher dimensions , system needs to have more energy for having phase transition . it is worthwhile to mention that subcritical isobars ( also critical region ) are increasing functions of dimensions ( see figs . [ fig10 ] middle panels ) . whereas critical volume is a decreasing function of dimensions . finally , the ratio @xmath3 is a decreasing ( an increasing ) function of @xmath72 ( @xmath73 ) . @xmath86 @xmath87 @xmath88 @xmath89 @xmath90 @xmath91 in this paper , we have considered both gb and tol gravities in presence of two classes of bi type ned and studied their phase diagrams . we have considered cosmological constant as pressure and its conjugating quantity as thermodynamical volume . the obtained volume for these cases was consistent with topological structure of black holes and what was obtained previously @xcite . it was shown that although both higher orders of lovelock gravity and ned modify thermodynamical quantities but the volume of the black holes in these cases is independent of these two modifications and only depends on the topology of the solutions . by employing numerical method , we have calculated critical thermodynamical values for different cases and studied the effects of gravitational and nonlinear electromagnetic field parameters on these critical values . it was shown that black holes under consideration have similar behavior as van der waals liquid - gas system . we found that critical temperature and pressure were decreasing functions of orders and/or coefficients of lovelock gravity and critical volume and energy gap were increasing functions of them . in other words , black holes with higher orders of lovelock gravity go under phase transition and acquire stable state with lower temperature comparing to einstein case . on the other hand , the length of the subcritical isobars and region of the phase transition were increasing functions of the orders and/or coefficients of lovelock gravity . orders of the lovelock gravity are denoting different powers of the curvature scalar . from what we have obtained , one can argue that the critical temperature , pressure and the region of the small / large black holes are decreasing functions of the power of the curvature scalar . while , the critical volume , subcritical isobars and region of the phase transition are increasing functions of it . therefore , the power of the curvature scalar indeed has a crucial role in variation of the critical values . it is also regarded that considering the higher orders of lovelock theory , increases the power of gravity . it was shown that critical temperature and pressure were decreasing functions of @xmath73 whereas the critical volume is an increasing function of it . in comparison between bi type and maxwell electrodynamics , it was found that the lowest critical pressure and temperature and the largest critical volume belong to linear ( maxwell ) theory . the gravitational and electromagnetic fields have opposite effects on critical pressure and also phase transition . according to these results , one can say that phase transition is fundamentally related to both gravity and electrodynamics and their powers . as the power of gravitational field ( electromagnetic field ) increases ( decreases ) the critical temperature decreases ( increases ) . interaction between the gravitational and electrodynamic sectors of charged black hole may be found through the metric function . in addition , investigations of the phase diagrams confirm that they are weakening each others effects . as for dimensionality , we found that as it increases , the energy gap , critical temperature and critical pressure increase . the universal ratio of @xmath3 was a decreasing function of @xmath73 and increasing function of dimensions and lovelock parameter . for higher orders of lovelock gravity we have higher value of entropy ( @xmath92 ) which indicates that the thermodynamical system that the gravity describe contains higher value of disorder . in addition , considering higher orders of lovelock gravity will cause the black holes to have higher degree of complexity in their geometrical structure . if one considers the complexity of the geometrical structure as a disorder measurement of the system , it is logical to expect to see higher value of entropy for higher order of lovelock gravity . on the other hand , higher value of entropy means that our systems will have phase transition in lower critical temperature . that is the result that we have derived through our numerical calculations . it was shown that @xcite gb gravity in the presence of maxwell field has no phase transition for arbitrary electric charge in higher than @xmath93-dimensions . while we found that adjusting the nonlinearity parameter , @xmath73 , there is phase transition for various values of @xmath17 and @xmath94 . in other words , the nonlinearity parameter of electrodynamics has modified both electrodynamics and thermodynamical behavior of a black hole system . due to the opposite effects of gravitational power in lovelock gravity and the power of nonlinearity in electrodynamics , it is constructive to find the dominant effect for various domains of thermodynamical systems . also , one can study whether these two different fields for certain values of @xmath72 and @xmath73 , cancel each other effects or not . considering the effects of hawking radiation , we expect to see different behavior for higher orders of lovelock gravity . this indicates that in order to investigate black holes evaporation and phase transition of black holes , one could take both effects simultaneously . another interesting issue is studying the connection between complexity of the spacetime ( topological structure of black hole ) and entropy of the system and the interpretation of entropy as geometrical property . we leave these problems for the future work .

in this paper , we consider lovelock gravity in presence of two born - infeld types of nonlinear electrodynamics and study their thermodynamical behavior . we extend the phase space by considering cosmological constant as a thermodynamical pressure . we obtain critical values of pressure , volume and temperature and investigate the effects of both the lovelock gravity and the nonlinear electrodynamics on these values . we plot @xmath0 , @xmath1 and @xmath2 diagrams to study the phase transition of these thermodynamical systems . we show that power of the nonlinearity and gravity have opposite effects . we also show how considering cosmological constant , nonlinearity and lovelock parameters as thermodynamical variables will modify smarr formula and first law of thermodynamics . in addition , we study the behavior of universal ratio of @xmath3 for different values of nonlinearity power of electrodynamics as well as the lovelock coefficients .

You are an expert at summarizing long articles. Proceed to summarize the following text: valuable information about globular clusters and galaxy formation can be obtained by investigating the extent to which the properties of globular clusters belonging to different galaxies are similar . for example , differences in initial mass function during cluster formation and/or in susbsequent cluster dynamical evolution , which may both depend on galactic environment , would translate into differences in present - day stellar content . the globular cluster stellar content can be characterized by the mass - to - light ( @xmath2 ) ratio , which can now be determined for relatively remote globular clusters . collecting all the available data from the literature , pryor & meylan ( 1993 ) derive @xmath1 ratios for 56 galactic globular clusters using king - michie dynamical models . they obtain _ global _ @xmath1 ratios ranging from about 1 to 5 with a mean of 2.3 ( in solar units ) , and found no significant correlations ( apart from a possible weak one between @xmath1 and cluster mass ) between the _ global _ @xmath1 ratios and other parameters such as metallicity , concentration , half - mass relaxation time , or distance from either the galactic center or the galactic plane . @xmath1 ratios similar to those obtained for galactic clusters have been obtained in studies of globular clusters belonging to the magellanic clouds ( dubath et al . 1996b ) and to the fornax dwarf spheroidal galaxy ( dubath et al . 1992 ) . another way of investigating globular cluster similarities , in terms of structure and @xmath1 ratio , is to look at the correlations between velocity dispersion , luminosity and a physical size scale . these correlations , which are analogous to the fundamental plane correlations for elliptical galaxies , have already been discussed for galactic clusters by several authors ( e.g. , meylan & mayor 1986 , paturel & garnier 1992 , djorgovski & meylan 1994 , djorgovski 1995 ) . the tight correlation between the velocity dispersion , the core radius and the central surface brightness obtained for galactic clusters ( djorgovski 1995 ) is consistent with expectations from the virial theorem assuming that galactic globular cluster cores have a universal and constant @xmath1 ratio to within the measurement errors . because of its relative proximity and large size , the m31 globular cluster system is an obvious target for the study of extragalactic clusters . previous studies of various aspects of the m31 globular cluster system have been reviewed by fusi pecci et al . ( 1993 ) , huchra ( 1993 ) , tripicco ( 1993 ) , and cohen ( 1993 ) . the only velocity - dispersion determinations of m31 clusters published so far are by peterson ( 1988 ) . corresponding @xmath1 ratio estimates are given for two clusters and are found to be similar to those typically obtained for galactic clusters . a limitation in this work , however , arises from the difficulty of measuring m31 cluster structural parameters from the ground . in m31 the angular sizes of core and half - light radii are typically @xmath002 and @xmath01 , respectively . in this work , we present new velocity dispersion and @xmath1 ratio determinations for a sample of m31 globular clusters , for which structural parameters derived from hst observations are available in the literature . the @xmath1 ratio estimates are based on simple relations derived from the virial theorem and from king models . our velocity dispersion estimates are also key observational constraints for more detailed dynamical analyses , e.g. , based on fokker - plank or king - michie multi - mass models , which are beyond the scope of this paper . the spectroscopic observations and the data reduction are presented in sect . 2 . numerical simulations used for deriving velocity dispersions from the integrated - light spectra and the corresponding results are described in sect . section 4 discusses the structural parameters , and @xmath1 estimates are given in sect . the relations between velocity dispersion , luminosity and different physical scales are discussed in sect . 6 for our m31 cluster sample together with samples of clusters belonging to the galaxy , the magellanic clouds , the fornax dwarf spheroidal galaxy , and centarus a. we summarize our findings in sect . we obtained 12 high - resolution integrated - light spectra of 10 globular clusters belonging to m31 . the observations were made september 9 - 11 , 1994 ( see table [ tabsig ] ) , with the hamilton echelle spectrograph ( vogt 1987 ) operated at the coud focus of the 3-m shane telescope of the lick observatory . the detector was a thinned , backside - illuminated ti ccd , with 800 800 pixels of 15 15 @xmath3 m each , and with a readout noise of about 6 electrons . during the first night , a relatively wide entrance slit of 20 50 was used to match poor seeing conditions . a slit of 15 50 was used during the second and the third nights . as a consequence , the instrumental resolution , as estimated from the fwhm of the emission lines of thorium calibration spectra , was slightly lower during the first night ( 31000 or 9.7 ) than during the second and the third ones ( 35000 or 8.6 ) . both slits used have widths larger than the apparent half - light radii ( see table [ tabmol ] ) of all our clusters . table [ tabsig ] gives the date of the observations , the exposure time ( texp ) and the signal - to - noise ( s / n ) ratio of the cluster spectra . during each observing night , many radial velocity standard stars were also measured . spectra of a thorium - argon lamp were taken 6 - 8 times through each of the three nights . the spectra were reduced with inter - tacos , a new software package developed in geneva by d. queloz and l. weber . the first - night spectra contain 50 useful orders ranging from 4800 to 8300 , and those obtain the second and third night contain 49 orders ranging from 4600 to 7500 . the different orders do not overlap , i.e. , the echelle spectra exhibit holes in the wavelength coverage . the orders are never rebinned , nor merged together . the shifts in velocity between the different thorium - argon lamp spectra taken during one particular night are always smaller than 1 . however , a velocity shift as large as 5 is observed between the second and the third nights . as a first step , for each night , one wavelength solution is computed from a thorium - argon spectrum , and applied to all the spectra taken during that particular night . the cluster spectra also contain numerous strong night - sky oh and o2 emission lines , because of the long exposure time on relatively faint objects . the residual velocity zero - point shift of each globular cluster spectrum is then derived by measuring the mean shift ( in velocity ) of the emission lines compared to their accurate rest wavelengths ( taken from osterbrock et al . 1996 ) . as a second step of the wavelength calibration , all the _ globular cluster _ spectra are corrected according to these velocity zero - point shifts . the reduced spectra are cross - correlated with an optimized numerical mask , used as a template ( see dubath et al . 1990 for the details of the cross - correlation technique ) . in short , our template contains lower and upper wavelength limits for each line from a selected sample of narrow spectral lines . for a particular radial velocity , the value of the cross - correlation function ( ccf ) is given by the sum , over all spectral lines , of the integral of the considered spectrum within the lower and the upper line limits shifted according to the given velocity . the ccf is thus built step by step over the velocity range of interest . the ccf is not affected by the fact that the spectra are composed of a succession of independent orders , with holes in the wavelength coverage . the ccf is a kind of mean spectral line over the approximately 1400 useful lines spread over the spectral ranges of the different orders . the lower and upper limits of the lines are computed from a mix of observed and theoretical high - resolution spectra of k2 giants , in such a way as to optimize the ccf ( see dubath et al.1996a ) . when the number of considered lines is large enough ( @xmath4 several hundreds ) , this cross - correlation technique produces a ccf which is nearly a perfect gaussian . a gaussian function is fitted to each derived ccf in order to determine three physical quantities : ( 1 ) the location of the minimum equal to the radial velocity , ( 2 ) its depth @xmath5 , and ( 3 ) its standard deviation , related to line broadening mechanisms . figure [ ccf ] displays the ccfs of the integrated - light spectra obtained for the m31 globular clusters in our sample . we have a total of 32 measurements of 11 standard stars , mostly giant stars of spectral type g8 to k4 , collected during the same observing run . the comparison of the standard - star radial velocities with reference values provided by coravel measurements ( mayor , private communication ) shows that the instrumental radial velocity accuracy is of order 0.5 and that the zero - point shift between the two datasets is not significant . in the case of the m31 clusters , the accuracy of the radial velocity zero - point is very probably even better owing to our use of the night - sky emission lines to calibrate the zero point of each cluster spectrum . the mean value of the widths ( ) of the ccfs obtained for our sample of standard stars is 5.80.3 for the first night , and 5.50.2 for the two others . this difference is the direct consequence of the different slit widths used . we see no dependence of the ccf width on spectral type over the range considered here ( g8 to k4 ) . we also know that the ccf width does not depend on the star metallicity from previous measurements of a large number of various types of metal - deficient stars ( see fig . 6 of dubath et al . the _ projected _ velocity dispersions can be derived from the broadening of the cluster ccfs . this broadening results from the doppler line broadening present in the integrated - light spectra because of the random spatial motions of the stars . since the spectrograph slit used is large compared to the the m31 globular clusters apparent size , a very large number of stars contributes significantly to the integrated light . quantitatively , recent numerical simulations ( dubath et al . 1994 , 1996a ) show that statistical errors , which can be very important for integrated - light measurements of some galactic globular clusters because of the dominance of a few bright stars , are negligible in the present case . an integrated light spectrum of an m31 globular cluster is well approximated by the convolution of the spectrum of a typical globular cluster star with the projected velocity distribution . since the ccf is a kind of mean spectral line , the above property also hold for ccfs . the ccf of an integrated - light spectrum of a globular cluster is the convolution of the ccf of the spectrum of a typical globular cluster star by the projected velocity distribution . since the ccfs have a gaussian shape , an estimate of the projected velocity dispersion in the integration area of a globular cluster is given by the quadratic difference , @xmath6 where is the width of the gaussian fitted to the cluster ccf and is the average width of the obtained for a sample of standard stars , as representative as possible of the cluster stars which contribute most to the integrated light . we do not use this formula directly in the present study . in order to derive the cluster projected velocity dispersion , we carry out a large number of numerical simulations . the results of these simulations confirm , however , the validity of eq . [ sigest ] ( see sect . [ gene ] ) . in order to simulate the integrated - light spectrum and the ccf obtained for globular cluster , we proceed in several steps . in the first row of fig . [ simuill ] , the integrated - light spectrum ( left ) and the ccf ( right ) obtained for the cluster bo 218 are displayed . corresponding simulation results are displayed in the last row , while the intermediate steps of the simulation are illustrated in the middle row of this figure . the simulation input parameter is the cluster velocity dispersion ( ) . the simulation starts with a high signal - to - noise spectrum of a standard star of appropriate spectral type , e.g. , a k2 giant . to simulate the doppler line broadening , this spectrum is convolved with a gaussian function of standard deviation equal to the input velocity dispersion . a portion of the standard - star spectrum , before and after convolution , is shown in the panel ( c ) of fig . [ simuill ] . \2 . in general , the spectral lines of the convolved standard - star spectrum do not have the same depths as those of a given cluster spectrum , mainly because of possible metallicity difference . the next step is thus to adjust the depth of the standard - star spectral lines . this is done by scaling the convolved standard - star spectrum so that its ccf is of the same depth as the cluster ccf . ( with our cross - correlation technique , there is a clear relationship between the average depth of the spectral lines of a spectrum and the depth of the spectrum ccf ) . in fig . [ simuill ] for example , the spectra and the ccfs displayed in the middle row are linearly scaled so that the convolved spectrum ccf ( the broad ccf in panel d ) be of the same depth as the cluster ccf ( in panel b ) . a noiseless template of a cluster spectrum for one particular velocity dispersion results from the second step . random noise is added to this template spectrum to simulate the photon counting and ccd readout noises of observed cluster spectra . a portion of the template spectrum for bo 218 ( c ) and of one example of simulated noisy spectrum are displayed in panel ( e ) of fig . [ simuill ] . we then cross - correlate the simulated noisy spectrum and derive the radial velocity ( ) and the sigma ( ) of the resulting ccf . this third step is repeated a large number of times ( 100 to 150 times ) to observe the influence of the noise on and . panel ( f ) of fig . [ simuill ] shows three examples of simulated ccfs taken at random . the comparison of the upper and the lower panels of this figure shows how well our simulations reproduce the integrated - light spectrum and the ccf obtained for bo 218 . in order to derive the best estimates of the projected velocity dispersion ( ) in the clusters , we proceed as follows . for each cluster , a set of different input velocity dispersions ( ) , varying by step of 0.5 or 1 around the expected cluster `` true '' velocity dispersion ( first evaluated with eq . [ sigest ] ) are considered . for each , 100 or 150 simulations are carried out and the relative number of times that the width of the simulated ccfs is consistent with the width of the observed ccf is reported . consistent means here that the sigma of the simulated ccf must be within 0.2 - 0.5 ( depending on the different cases ) of the sigma of the observed ccf . figure [ simudis ] shows , in the case of bo 218 , the distribution of these numbers as a function of the input velocity dispersions ( ) of the simulations . this distribution is a kind of probability distribution ; the best estimate of the `` true '' cluster velocity dispersion is given by the which leads most often , through the simulations , to a ccf consistent with the observed one . for example , among 100 simulations carried out with = 16 , 47 have a width consistent with the width of the observed ccf obtained for bo 218 , while none of the 100 simulations with = 13 is successful in reproducing the observed ccf . a similar distribution is derived for each cluster , and a gaussian is fitted to each of them . the resulting means and sigmas , which provide the most probable cluster velocity dispersions ( ) and their uncertainties , are given in column ( 11 ) of table [ tabsig ] . this section presents a generalization of the simulation results ( which can be skipped by less concerned readers ) . for each of the 9 clusters , simulations are carried out for a set of input velocity dispersions ( ) . in the simulations , a particular cluster is characterized by the spectrum s / n ratio and by the depth ( @xmath5 ) of the cluster ccf . therefore in general , the input parameters are the spectrum s / n , the ccf depth @xmath5 , and the input velocity dispersion . for each set of input parameters , 100 to 150 simulated ccfs are computed , and distributions of the resulting radial velocities and ccf sigmas are thus produced . the standard deviations of these distributions @xmath7 ( ) for the radial velocities and @xmath7 ( ) for the sigmas provide estimates of the uncertainties due to the spectrum noise . the following formula , @xmath8 where @xmath9 and @xmath10 are two constants , is fitted to the results of the simulations . with @xmath11 and @xmath12 , this equation gives in , ( 1 ) @xmath7 ( ) with an accuracy of order of 10% , and ( 2 ) @xmath7 ( ) ) with an accuracy of about 20% . it can then be used to estimated the uncertainties due to the spectrum noise in a more general way , for any set of parameters s / n , @xmath5 , and . equation [ epsi ] is a generalization of eq . ( 3 ) of dubath et al . ( 1990 ) to broad ccfs . the constant @xmath9 is lower in the present paper than in this previous study because the larger wavelength range , and consequently larger number of spectral lines , taken into account in the cross - correlation process ( @xmath9 scales roughly with the square root of the number of lines ) . figure [ simusig ] displays , for each cluster , the sigma ( ) of the ccfs of the observed spectrum as a function of the corresponding estimates of the velocity dispersion ( ) resulting from the simulations . ( [ sigest ] ) is illustrated in this figure by the continuous line , and the good agreement with the simulation results indicates that this equation is a valid model . rrcccrrrcrr & & [ fe / h ] & date & & s / n & & @xmath5 & & + & & & 94 sept . & & & & ( % ) & & + ( 1)&(2 ) & ( 3 ) & ( 4 ) & ( 5 ) & & ( 7 ) & & ( 9 ) & & + 6 & 58 & 15.8 & @xmath130.57 & 9 & 100 & 13 & @xmath13236.90.2 & 4.9 & 12.10.3 & 10.60.4 + 20 & 73 & 14.6 & @xmath131.07 & 11 & 100 & 18 & @xmath13350.80.3 & 3.2 & 16.20.4 & 15.30.5 + 45 & 108 & 15.8 & @xmath130.94 & 9 & 85 & 11 & @xmath13425.10.3 & 3.9 & 10.40.4 & 8.70.5 + 147 & 199 & 15.5 & @xmath130.24 & 11 & 80 & 26 & @xmath1350.70.1 & 7.3 & 6.00.3 & star ? + 158 & 213 & 14.5 & @xmath131.08 & 10 & 50 & 10 & @xmath13186.51.0 & 2.1 & 22.21.1 & 21.91.3 + 218 & 272 & 14.7 & @xmath131.18 & 11 & 50 & 11 & @xmath13220.20.6 & 2.7 & 17.20.7 & 16.30.8 + 225 & 280 & 14.3 & @xmath130.70 & 9 & 120 & 31 & @xmath13164.80.3 & 2.8 & 26.50.4 & 26.90.5 + 343 & 105 & 16.3 & @xmath131.49 & 11 & 100 & 8 & @xmath13359.61.7 & 1.1 & 11.31.7 & 10.21.7 + " & " & " & " & 10 & 63 & 4 & @xmath13357.54.0 & 0.7 & 7.74.1 & ... + 358 & 219 & 15.1 & @xmath131.83 & 10 & 90 & 12 & @xmath13315.11.8 & 0.6 & 8.81.8 & 7.11.8 + " & " & " & " & 9 & 50 & 6 & @xmath13317.53.1 & 0.8 & 12.03.1 & ... + 384 & 319 & 15.7 & @xmath130.66 & 11 & 100 & 10 & @xmath13363.80.3 & 4.4 & 10.50.4 & 9.10.5 + [ tabsig ] for each observation , table [ tabsig ] lists the cluster identification from battistini et al . ( 1980 ) in column ( 1 ) , and from sargent et al . ( 1977 ) in column ( 2 ) , the cluster apparent v magnitude in column ( 3 ) , the cluster [ fe / h ] in column ( 4 ) , the date of the observation in column ( 5 ) , the exposure time in column ( 6 ) , the signal - to - noise ratio of the integrated - light spectrum in column ( 7 ) , the heliocentric radial velocity in column ( 8) , the depth of the ccf in column ( 9 ) , the sigma of the ccf in column ( 10 ) , and the projected velocity dispersion in column ( 11 ) . the cluster apparent v magnitudes are taken from battistini et al . ( 1987 ) , and [ fe / h ] are taken from huchra et al . ( 1991 ) . the radial velocity errors given in column ( 8) of table [ tabsig ] are computed using eq . ( [ epsi ] ) . this equation provides an estimate of the error due to the spectrum noise ( photon counting and ccd readout noises ) but does not take into account the uncertainty of the zero - point corrections computed from measurements of the night - sky emission lines . consequently , the radial velocity errors smaller than @xmath00.3 are probably underestimated . the errors given in column ( 8) of table [ tabsig ] are the square root of the quadratic sum of the errors due to the noise ( estimated using eq . ( [ epsi ] ) ) , and of an instrumental error of 0.25 , derived from standard - star measurements . two cluster spectra are too noisy to provide useful velocity dispersion estimate . the ccf obtained for bo 147 is as narrow as the ccfs obtained for individual standard stars , and much narrower that the value expected from its absolute magnitude . consequently , bo 147 is almost certainly a foreground star . rrccccccccc & @xmath14 & @xmath15 & @xmath16 & m@xmath17 & @xmath18 & @xmath19 & e(b - v ) & @xmath1 & @xmath20 + & pc & pc & & & m@xmath21 & m@xmath21 & & m@xmath22l@xmath23 & m@xmath22l@xmath23 + ( 1)&(2 ) & ( 3 ) & ( 4 ) & ( 5 ) & ( 6 ) & ( 7 ) & ( 8) & ( 9 ) & ( 10 ) & ( 11 ) + 6 & 58 & 0.62 & 2.8 & 1.76 & -9.0 & @xmath24 & @xmath25 & 0.11 & 1.4 & 2.2 + 20 & 73 & 1.18 & 4.7 & 1.71 & -10.2 & @xmath26 & @xmath27 & 0.11 & 1.7 & 2.5 + 45 & 108 & 0.92 & 3.6 & 1.70 & -9.0 & @xmath28 & @xmath29 & 0.12 & 1.2 & 1.8 + 158 & 213 & 1.54 & 5.0 & 1.59 & -10.2 & @xmath30 & @xmath31 & 0.08 & 3.7 & 5.5 + 218 & 272 & 0.47 & 2.0 & 1.74 & -10.0 & @xmath32 & @xmath33 & 0.09 & 0.9 & 1.4 + 225 & 280 & 1.42 & 4.9 & 1.63 & -10.4 & @xmath34 & @xmath35 & 0.09 & 4.4 & 6.5 + 343 & 105 & 0.34 & 2.9 & 2.02 & -8.3 & @xmath36 & @xmath37 & 0.06 & 2.1 & 4.0 + 358 & 219 & 1.82 & 4.7 & 1.46 & -9.5 & @xmath38 & @xmath39 & 0.06 & 0.7 & 1.0 + 384 & 319 & 0.57 & 2.2 & 1.69 & -8.9 & @xmath40 & @xmath41 & 0.06 & 0.9 & 1.4 + [ tabmol ] m31 is at a distance where the core radii of typical globular clusters are considerably smaller in angular size than the typical , ground - based seeing disk . this makes measurements of @xmath14 difficult , as attested to by the large range of values obtained by different authors for the same clusters . fortunately , six of the clusters in our sample have recently been observed with the hubble space telescope ( hst ) , and accurate values of @xmath14 have been published ( fusi - pecci 1994 [ bo218 , 384 ] ; grillmair 1996 [ bo6 , 45 , 343 , 358 ] ) . the values obtained span a large range but are typical for the brighter globulars in our own galaxy . for the remaining 3 clusters in table 1 , we use the ground - based measurements of battistini ( 1982 ) and crampton ( 1985 ) . for two clusters ( bo6 , 45 ) common to all three samples , the agreement between the ground - based values and those measured by grillmair is typically quite good , with the worst cases differing by no more than 30% . the surface density profiles of these clusters have also been difficult to characterize from the ground owing the very low surface brightness relative to the m31 background and the inability to resolve individual stars . however with the advent of hst we now know that globular clusters in m31 are structurally similar to those found in our own galaxy . these clusters can generally be characterized by king models , though departures from king models in the form of collapsed cores ( bo343 , bendinelli 1993 ; grillmair 1996 ) and tidal tails ( bo6 , 343 , 358 , grillmair 1996 ) have also been found . in all respects , it seems that globular clusters in m31 as a group are of the same breed as the clusters belonging to our own galaxy . for the clusters observed by grillmair ( 1996 ) , we have determined the half - light radii @xmath15 ( the radii within which half the total light of the clusters is contained in projection ) by integrating over king models with appropriate core radii and concentration parameters ( @xmath42 ) . for the 2 clusters observed by fusi pecci ( 1994 ) we have used their core and half - light radii and integrated over a grid of king models to infer the corresponding concentration parameter . for the three clusters for which we have only ground - based measurements of @xmath14 , we follow battistini ( 1982 ) in assuming a uniform value of @xmath43pc . our adopted values of @xmath14 and @xmath15 are tabulated in table 2 . in all cases , the published values of @xmath14 have been scaled in accordance with our adopted distance to m31 of 770kpc ( ajhar 1996 ) . if we assume that the clusters are reasonably well represented by king models then , following queloz ( 1995 ) , we can compute the total mass using @xmath44 values of @xmath45 , and @xmath46 have been tabulated for a range of concentration parameters by king ( 1966 ) and peterson & king ( 1975 ) . the @xmath47 in eq . [ mass ] refers to the _ central _ velocity dispersion of the cluster . given the small angular size of the clusters and our use of 1.5 and 2.0 arcsecond - wide slits , it is clear that our dispersion measurements will have been influenced by light from moderately large radii . we tested the effect of our slit - widths on the measured velocity dispersions by integrating luminosity - weighted , king model velocity dispersion profiles over the area subtended by our slit . we found that , over a large range in @xmath16 , and for both slit - widths used , the measured velocity dispersion would be lower than the central velocity dispersion by about 5% . hence , for the purposes of computing masses , we increased the measured dispersions in table [ tabsig ] accordingly . the resulting cluster masses we obtain using eq . [ mass ] are listed in column 7 of table [ tabmol ] . an alternative to the somewhat model - specific method used above is a straightforward application of the virial theorem : @xmath48 where we have assumed an isotropic velocity distribution and @xmath49 ( spitzer 1987 ) , where @xmath50 is the half - mass radius . the masses computed using this equation are given in column 8 of table [ tabmol ] . owing to m31 s low galactic latitude , obscuration of the globular clusters by foreground galactic dust varies significantly from one side of m31 to the other . using the extinction maps of burstein & heiles ( 1982 ) , e(b - v ) was estimated for each cluster and is given in column 9 of table [ tabmol ] . using the @xmath51-magnitudes given by battistini ( 1987 ) , we adopt @xmath52 = 3.2 e(b - v ) ( dacosta & armandroff 1990 ) , and @xmath53 to compute total cluster @xmath51-band luminosities . the corresponding values for @xmath1 are given in column 10 and 11 of table [ tabmol ] . the @xmath1 ratios given in table [ tabmol ] are remarkably similar to those typically found in galactic globulars ( pryor & meylan 1993 ) . bo158 and 225 might seem a trifle high , but we note that both these clusters have only ground - based measurements of @xmath14 . it is entirely possible that these estimates of @xmath14 suffer from incomplete removal of the effects of seeing and are consequently too high . the largest source of uncertainty in @xmath1 is generally in the estimation of @xmath14 , being of the order of 15% even for the hst - imaged clusters . uncertainties in the magnitude estimates of battistini , in our estimates of the local extinction , and in the velocity dispersion measurements contribute @xmath54 each to the final uncertainty . bo343 and 358 are exceptions to this general rule , having reasonably well - measured core radii , but rather less well - determined velocity dispersions . the formally estimated uncertainties in @xmath1 are @xmath55 for those clusters observed with hst , and probably closer to 50% for those clusters imaged only from the ground . table [ tabmol ] shows that the @xmath1 ratios derived with eq . [ virial ] ( column 11 ) are systematically @xmath050% larger than those obtained with eq . [ mass ] ( column 10 ) . this gives a rough idea on how model dependent are our estimates . another way to investigate similarities among different globular cluster populations is to look at the relation between velocity dispersion , luminosity , and a physical size scale . in such a parameter space , systematic differences between two systems of globular clusters , in terms of structure or @xmath2 ratio , would result in either different mean relations or in different degrees of scatter about a given relation . we do not attempt here to derive mean relations through bivariate fits ( e.g. djorgovski 1995 ) as they turn out to be rather unstable because of both the small number of data points as well as the weak correlation between the physical size parameter and the luminosity and velocity dispersion . instead , we fit to the data the relations expected from the virial theorem , or from the king models . assuming a constant @xmath1 ratio , the virial theorem predicts the relation @xmath56 between the _ global _ velocity dispersion @xmath57 , the half - light radius @xmath15 , and the absolute visual magnitude . similarly , king models predict @xmath58 where @xmath59 is the _ central _ velocity dispersion , @xmath3 a dimensionless parameter which varies with cluster concentration @xmath16 , @xmath14 the core radius , and the absolute visual magnitude . figure [ fp ] shows the relations between @xmath60 vs. @xmath61 ( uppermost row of panels ) , @xmath62 vs. @xmath61 ( second row ) , and @xmath63 vs. @xmath61 ( third row ) , for different data sets . for the galactic clusters we use velocity dispersion measurements based on radial velocities of individual stars , taken from the compilation of pryor and meylan ( 1993 ) . for the other clusters , we use velocity dispersions derived from integrated - light observations , taken from dubath et al . ( 1996a ) for 8 old magellanic clusters , from dubath et al . ( 1992 ) for 3 clusters belonging to the fornax dwarf spheroidal galaxy , from the present study for 9 m31 clusters , and from dubath ( 1994 ) for 10 centaurus a clusters . the continuous lines represent the relation @xmath64 ( uppermost row ) , and the relations ( [ virial_2 ] ) and ( [ king_2 ] ) , in the second and third rows respectively , with constants derived by fitting the _ galactic _ cluster data . the dashed lines show the relations obtained when central velocity dispersions ( extrapolated for these clusters by pryor and meylan [ 1993 ] using king models ) are considered instead of the _ global _ velocity dispersion . it is worth mentioning that correlations resulting from bivariate fits do not differ much from the relations expected from the virial theorem ( djorgovski 1995 ) , and that fig . [ fp ] would look very similar if slightly different projections were used . the scatter of the data points probably results to a large extent from measurement errors , and it is quite remarkable that this scatter is of the same order in all panels of fig . the similarity of the different panels of fig . [ fp ] indicates both that there is no large systematic differences in globular cluster @xmath1 ratios between one galaxy to another , and that measurement errors are comparable , and even smaller , for extragalactic clusters than for galactic clusters . in the first row of fig . [ fp ] , we expect a higher degree of scatter since the physical scale of each cluster ( which acts as a second parameter ) is not taken into account . notably , a few large - size / low - concentration galactic clusters lie well below the other clusters in the upper left panel . the fact that similar large clusters are not present in our m31 and cen a samples is due to a selection bias which favors brighter and generally more compact clusters . in any case , the relatively small spread of the data points around the straight lines in the uppermost panels points to an additional similarity in terms of physical size range between the clusters of these different galaxies , with the possible exception of the old magellanic clusters . this is particularly interesting in the case of the centaurus a clusters since these clusters are brighter than the galactic ones , but their physical scales have not yet been measured . for the old magellanic clusters , the products @xmath3 @xmath14 are systematically _ smaller _ than the galactic average . this can perfectly explain why the magellanic clusters appear above the galactic relation in the upper panel in fig . [ fp ] , while the magellanic and galactic middle panels are similar . the lower panel is , however , rather puzzling . unexpectedly , the data points all lie above the galactic relation , as if the half - light radii used here and taken from van den bergh ( 1994 ) were about 40% too large . this point is further discussed in another paper ( dubath et al . 1996b ) . the two m31 clusters ( bo158 & 225 ) with large @xmath1 ratio estimates from last section stand out above the galactic relation in fig . [ fp ] . as already pointed out , these clusters only have ground - based measurements of @xmath14 which may be overestimates . the scatter of the other clusters is remarkably small compared to the scatter of galactic clusters . as illustrated in fig . [ fp ] , the absolute magnitude of an individual globular cluster can be derived from its velocity dispersion and physical scale with an accuracy of @xmath00.5 magnitude . for a given parent galaxy , providing there is no systematic differences in @xmath1 , an accurate distance modulus can in principle be computed using the mean distance modulus of a moderate number ( 10 20 ) of its associated globular clusters . in other word , plotting apparent instead of absolute magnitude in fig . [ fp ] , the difference between the distance modulii of two parent galaxies is given by the horizontal shift required to bring the two ( dereddened ) data sets in agreement . combining the capabilities of the hst and of 10-m class , ground - based telescopes , this method could be applied for galaxies out as far as the virgo cluster . in this paper , we present projected velocity dispersion measurements from integrated - light spectra for a sample of 9 m31 globular clusters . by means of comprehensive numerical simulations , we show that the typical relative uncertainty of our measurements is @xmath05% . because of the relatively large distance of m31 and the aperture angular size used our integrated - light observations sample a large fraction of these very bright clusters . consequently , statistical uncertainties due to small samples of bright dominant stars which can affect integrated - light measurements of nearby galactic clusters are completely negligible . this is confirmed by numerical simulations presented in dubath et al . ( 1996a ) . paradoxically , it is in some respects easier to measure the _ global _ velocity dispersion of a m31 cluster than that of a galactic clusters . for the brightest m31 clusters , reliable velocity dispersions can be measured with a 3-m - class telescope using single exposures of order one hour long . previous velocity dispersion measurements are available from peterson ( 1988 ) for 3 of our clusters . the agreement is good for the cluster bo20 , while we obtain significantly lower and more accurate values for bo158 and 225 . combining the new velocity dispersions with structural parameters , we compute the cluster @xmath1 ratios using relations derived from king models and the virial theorem . these @xmath1 ratios appear remarkably similar to those found for galactic clusters ( see e.g. , pryor & meylan 1993 ) . two clusters ( bo158 & 225 ) having only ground - based measurements of @xmath14 have @xmath1 ratio estimates somewhat above the range of galactic values . it is entirely possible that these estimates of @xmath14 suffer from incomplete removal of the effects of seeing and are consequently too high . another way to investigate similarities of globular clusters located in different parent galaxies is to look at the relation between velocity dispersion , luminosity and a physical scale ( fig [ fp ] ) . using additional data from previous papers , we find remarkable similarities , in terms of @xmath1 ratios and structures , between the globular clusters located in our galaxy , the magellanic clouds , the fornax dwarf spheroidal , m31 , and centaurus a. it is worth emphasizing that our samples include some of the brightest m31 and centaurus a clusters , which are both brighter and more massive than any galactic cluster , as expected because of the larger number of clusters in both these galaxies . our m31 cluster sample is not large enough to investigate possible correlations between the @xmath1 ratio estimates and other cluster parameters . only a 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1988 , in dynamique of dense stellar system , ed . d. merritt(cambridge : cambridge university press ) , 161 pryor , c. , & melan , g. 1993 , in asp conf . 50 , structure and dynamics of globular clusters , s. djorgovski & g. melan(san francisco : asp ) , 357 queloz , d. , dubath , p. , & pasquini , l. 1995 , a&a , 300 , 310 sargent , w.l . , kowal , c.t . , hartwick , f.d.a . , & van den bergh , s. 1977 , aj , 82 , 947 spitzer , l. 1987 , _ dynamical evolution of globular clusters _ , princeton university press , princeton . tripicco , m. 1993 , in asp conf . 48 , the globular cluster galaxy connection , g. smith & j. brodie(san francisco : asp ) , 432 van den bergh , s. 1994 , aj , 108 , 2145 vogt , s. 1987 , pasp , 99 , 1214 _ note added in proof. _ as this paper was beeing accepted for publication , we became aware of a similar work by djorgovski et al . ( apjl , in press ) . they derive velocity dispersions from keck observations for a sample of 21 m31 globular clusters , which includes our 9 clusters . their results are in good agreement with ours , e.g. , the difference in velocity dispersion values is always @xmath65 1.4 for all clusters . they point out an interesting trend between the ratio mass - to - luminosity in the k band and the cluster metallicity , which has not been observed before .

we present internal velocity dispersion determinations from high - resolution spectroscopic observations of a sample of nine globular clusters in m31 . comprehensive numerical simulations are used to show that the typical uncertainty of our velocity dispersion measurements is @xmath05% . using these new velocity dispersions together with structural parameters derived from hst observations , we estimate the @xmath1 ratios of these clusters and find that they are typical of those measured for galactic clusters . we show relations between velocity dispersion , luminosity and physical scales for globular clusters belonging to the galaxy , the magellanic clouds , fornax , m31 , and centaurus a. the mean relations and the degree of scatter are similar in all galaxies . this reveals remarkable similarities , in term of structure and @xmath1 ratio , between the globular clusters belonging to these different galaxies . we briefly discuss the possible use of individual globular clusters as extragalactic distance indicators . # 1#20=1=0=10by-1.2pt1.300 - 0.701 # 1#20=1=0=10by-1.2pt1.200 - 00.601 # 1#2 # 1 # 1#2=#1 # 1#2=#1 -1.5 cm

You are an expert at summarizing long articles. Proceed to summarize the following text: the information contained by an individual finite object ( like a finite binary string ) is objectively measured by its kolmogorov complexity the length of the shortest binary program that computes the object . such a shortest program contains no redundancy : every bit is information ; but is it meaningful information ? if we flip a fair coin to obtain a finite binary string , then with overwhelming probability that string constitutes its own shortest program . however , also with overwhelming probability all the bits in the string are meaningless information , random noise . on the other hand , let an object @xmath0 be a sequence of observations of heavenly bodies . then @xmath0 can be described by the binary string @xmath1 , where @xmath2 is the description of the laws of gravity , and the observational parameter setting , while @xmath3 is the data - to - model code accounting for the ( presumably gaussian ) measurement error in the data . this way we can divide the information in @xmath0 into meaningful information @xmath2 and data - to - model information @xmath3 . the main task for statistical inference and learning theory is to distil the meaningful information present in the data . the question arises whether it is possible to separate meaningful information from accidental information , and if so , how . in statistical theory , every function of the data is called a `` statistic '' of the data . the central notion in probabilistic statistics is that of a `` sufficient '' statistic , introduced by the father of statistics r.a . fisher @xcite : `` the statistic chosen should summarise the whole of the relevant information supplied by the sample . this may be called the criterion of sufficiency @xmath4 in the case of the normal curve of distribution it is evident that the second moment is a sufficient statistic for estimating the standard deviation . '' for traditional problems , dealing with frequencies over small sample spaces , this approach is appropriate . but for current novel applications , average relations are often irrelevant , since the part of the support of the probability density function that will ever be observed has about zero measure . this is the case in , for example , complex video and sound analysis . there arises the problem that for individual cases the selection performance may be bad although the performance is good on average . there is also the problem of what probability means , whether it is subjective , objective , or exists at all . to simplify matters , and because all discrete data can be binary coded , we consider only data samples that are finite binary strings . the basic idea is to found statistical theory on finite combinatorial principles independent of probabilistic assumptions , as the relation between the individual data and its explanation ( model ) . we study extraction of meaningful information in an initially limited setting where this information be represented by a finite set ( a model ) of which the object ( the data sample ) is a typical member . using the theory of kolmogorov complexity , we can rigorously express and quantify typicality of individual objects . but typicality in itself is not necessarily a significant property : every object is typical in the singleton set containing only that object . more important is the following kolmogorov complexity analog of probabilistic minimal sufficient statistic which implies typicality : the two - part description of the smallest finite set , together with the index of the object in that set , is as concise as the shortest one - part description of the object . the finite set models the regularity present in the object ( since it is a typical element of the set ) . this approach has been generalized to computable probability mass functions . the combined theory has been developed in detail in @xcite and called `` algorithmic statistics . '' here we study the most general form of algorithmic statistic : recursive function models . in this setting the issue of meaningful information versus accidental information is put in its starkest form ; and in fact , has been around for a long time in various imprecise forms unconnected with the sufficient statistic approach : the issue has sparked the imagination and entered scientific popularization in @xcite as `` effective complexity '' ( here `` effective '' is apparently used in the sense of `` producing an effect '' rather than `` constructive '' as is customary in the theory of computation ) . it is time that it receives formal treatment . formally , we study the minimal length of a total recursive function that leads to an optimal length two - part code of the object being described . ( `` total '' means the function value is defined for all arguments in the domain , and `` partial '' means that the function is possibly not total . ) this minimal length has been called the `` sophistication '' of the object in @xcite in a different , but related , setting of compression and prediction properties of infinite sequences . that treatment is technically sufficiently vague so as to have no issue for the present work . we develop the notion based on prefix turing machines , rather than on a variety of monotonic turing machines as in the cited papers . below we describe related work in detail and summarize our results . subsequently , we formulate our problem in the formal setting of computable two - part codes . kolmogorov in 1974 @xcite proposed an approach to a non - probabilistic statistics based on kolmogorov complexity . an essential feature of this approach is to separate the data into meaningful information ( a model ) and meaningless information ( noise ) . cover @xcite attached the name `` sufficient statistic '' to a model of which the data is a `` typical '' member . in kolmogorov s initial setting the models are finite sets . as kolmogorov himself pointed out , this is no real restriction : the finite sets model class is equivalent , up to a logarithmic additive term , to the model class of computable probability density functions , as studied in @xcite . related aspects of `` randomness deficiency '' were formulated in @xcite and studied in @xcite . despite its evident epistemological prominence in the theory of hypothesis selection and prediction , only selected aspects of the theory were studied in these references . recent work @xcite can be considered as a comprehensive investigation into the sufficient statistic for finite set models and computable probability density function models . here we extend the approach to the most general form : the model class of total recursive functions . this idea was pioneered by @xcite who , unaware of a statistic connection , coined the cute word `` sophistication . '' the algorithmic ( minimal ) sufficient statistic was related to an applied form in @xcite : the well - known `` minimum description length '' principle @xcite in statistics and inductive reasoning . in another paper @xcite ( chronologically following the present paper ) we comprehensively treated all stochastic properties of the data in terms of kolmogorov s so - called structure functions . the sufficient statistic aspect , studied here , covers only part of these properties . the results on the structure functions , including ( non)computability properties , are valid , up to logarithmic additive terms , also for the model class of total recursive functions , as studied here . it will be helpful for the reader to be familiar with initial parts of @xcite . in @xcite , kolmogorov observed that randomness of an object in the sense of having high kolmogorov complexity is being random in just a `` negative '' sense . that being said , we define the notion of sophistication ( minimal sufficient statistic in the total recursive function model class ) . it is demonstrated to be meaningful ( existence and nontriviality ) . we then establish lower and upper bounds on the sophistication , and we show that there are objects the sophistication achieves the upper bound . in fact , these are objects in which all information is meaningful and there is ( almost ) no accidental information . that is , the simplest explanation of such an object is the object itself . in the simpler setting of finite set statistic the analogous objects were called `` absolutely non - stochastic '' by kolmogorov . if such objects have high kolmogorov complexity , then they can only be a random outcome of a `` complex '' random process , and kolmogorov questioned whether such random objects , being random in just this `` negative '' sense , can occur in nature . but there are also objects that are random in the sense of having high kolmogorov complexity , but simultaneously are are typical outcomes of `` simple '' random processes . these were therefore said to be random in a `` positive '' sense @xcite . an example are the strings of maximal kolmogorov complexity ; those are very unsophisticated ( with sophistication about 0 ) , and are typical outcomes of tosses with a fair coin a very simple random process . we subsequently establish the equivalence between sophistication and the algorithmic minimal sufficient statistics of the finite set class and the probability mass function class . finally , we investigate the algorithmic properties of sophistication : nonrecursiveness , upper semicomputability , and intercomputability relations of kolmogorov complexity , sophistication , halting sequence . a _ string _ is a finite binary sequence , an element of @xmath5 . if @xmath0 is a string then the _ length _ @xmath6 denotes the number of bits in @xmath0 . we identify @xmath7 , the natural numbers , and @xmath5 according to the correspondence @xmath8 here @xmath9 denotes the _ empty word_. thus , @xmath10 . the emphasis is on binary sequences only for convenience ; observations in any alphabet can be so encoded in a way that is ` theory neutral ' . below we will use the natural numbers and the strings interchangeably . a string @xmath11 is a _ proper prefix _ of a string @xmath0 if we can write @xmath12 for @xmath13 . a set @xmath14 is _ prefix - free _ if for any pair of distinct elements in the set neither is a proper prefix of the other . a prefix - free set is also called a _ prefix code _ and its elements are called _ code words_. an example of a prefix code , that is useful later , encodes the source word @xmath15 by the code word @xmath16 this prefix - free code is called _ self - delimiting _ , because there is fixed computer program associated with this code that can determine where the code word @xmath17 ends by reading it from left to right without backing up . this way a composite code message can be parsed in its constituent code words in one pass , by the computer program . ( this desirable property holds for every prefix - free encoding of a finite set of source words , but not for every prefix - free encoding of an infinite set of source words . for a single finite computer program to be able to parse a code message the encoding needs to have a certain uniformity property like the @xmath18 code . ) since we use the natural numbers and the strings interchangeably , @xmath19 where @xmath0 is ostensibly an integer , means the length in bits of the self - delimiting code of the string with index @xmath0 . on the other hand , @xmath20 where @xmath0 is ostensibly a string , means the self - delimiting code of the string with index the length @xmath6 of @xmath0 . using this code we define the standard self - delimiting code for @xmath0 to be @xmath21 . it is easy to check that @xmath22 and @xmath23 . let @xmath24 denote a standard invertible effective one - one encoding from @xmath25 to a subset of @xmath7 . for example , we can set @xmath26 or @xmath27 . we can iterate this process to define @xmath28 , and so on . for definitions , notation , and an introduction to kolmogorov complexity , see @xcite . informally , the kolmogorov complexity , or algorithmic entropy , @xmath29 of a string @xmath0 is the length ( number of bits ) of a shortest binary program ( string ) to compute @xmath0 on a fixed reference universal computer ( such as a particular universal turing machine ) . intuitively , @xmath29 represents the minimal amount of information required to generate @xmath0 by any effective process . the conditional kolmogorov complexity @xmath30 of @xmath0 relative to @xmath11 is defined similarly as the length of a shortest program to compute @xmath0 , if @xmath11 is furnished as an auxiliary input to the computation . for technical reasons we use a variant of complexity , so - called prefix complexity , which is associated with turing machines for which the set of programs resulting in a halting computation is prefix free . we realize prefix complexity by considering a special type of turing machine with a one - way input tape , a separate work tape , and a one - way output tape . such turing machines are called _ prefix _ turing machines . if a machine @xmath31 halts with output @xmath0 after having scanned all of @xmath2 on the input tape , but not further , then @xmath32 and we call @xmath2 a _ program _ for @xmath31 . it is easy to see that @xmath33 is a _ prefix code_. a function @xmath34 from the natural numbers to the natural numbers is _ partial recursive _ , or _ , if there is a turing machine @xmath31 that computes it : @xmath35 for all @xmath0 for which either @xmath34 or @xmath31 ( and hence both ) are defined . this definition can be extended to ( multi - tuples of ) rational arguments and values . let @xmath36 be a standard enumeration of all prefix turing machines with a binary input tape , for example the lexicographical length - increasing ordered syntactic prefix turing machine descriptions , @xcite , and let @xmath37 be the enumeration of corresponding functions that are computed by the respective turing machines ( @xmath38 computes @xmath39 ) . these functions are the partial recursive functions of effectively prefix - free encoded arguments . the kolmogorov complexity of @xmath0 is the length of the shortest binary program from which @xmath0 is computed by such a function . the _ prefix kolmogorov complexity _ of @xmath0 is @xmath40 where the minimum is taken over @xmath41 and @xmath42 . for the development of the theory we actually require the turing machines to use _ auxiliary _ ( also called _ conditional _ ) information , by equipping the machine with a special read - only auxiliary tape containing this information at the outset . then , the _ conditional version _ @xmath43 of the prefix kolmogorov complexity of @xmath0 given @xmath11 ( as auxiliary information ) is is defined similarly as before , and the unconditional version is set to @xmath44 . from now on , we will denote by @xmath45 an inequality to within an additive constant , and by @xmath46 the situation when both @xmath45 and @xmath47 hold . let @xmath48 be the standard enumeration of turing machines , and let @xmath49 be a standard universal turing machine satisfying @xmath50 for all indices @xmath51 and programs @xmath2 . we fix @xmath49 once and for all and call it the _ reference universal prefix turing machine_. the shortest program to compute @xmath0 by @xmath49 is denoted as @xmath52 ( if there is more than one of them , then @xmath52 is the first one in standard enumeration ) . it is a deep and useful fact that the shortest effective description of an object @xmath0 can be expressed in terms of a _ two - part code _ : the first part describing an appropriate turing machine and the second part describing the program that interpreted by the turing machine reconstructs @xmath0 . the essence of the theory is the invariance theorem , that can be informally stated as follows : for convenience , in the sequel we simplify notation and write @xmath53 for @xmath54 . rewrite @xmath55 here the minima are taken over @xmath56 and @xmath57 . the last equalities are obtained by using the universality of @xmath58 with @xmath59 . as consequence , @xmath60 thus , @xmath29 and @xmath61 differ by at most an additive constant depending on the choice of @xmath49 . it is standard to use @xmath62 instead of as the definition of _ prefix kolmogorov complexity _ , @xcite . however , we highlighted definition to bring out the two - part code nature . by universal logical principles , the resulting theory is recursively invariant under adopting either definition or definition , as long as we stick to one choice . if @xmath31 stands for a literal description of the prefix turing machine @xmath31 in standard format , for example the index @xmath63 when @xmath64 , then we can write @xmath65 . the string @xmath66 is a shortest self - delimiting program of @xmath67 bits from which @xmath49 can compute @xmath63 , and subsequent execution of the next self - delimiting fixed program @xmath68 will compute @xmath69 from @xmath63 . altogether , this has the effect that @xmath70 . if @xmath71 minimizes the expression above , then @xmath72 , and hence @xmath73 , and @xmath74 . it is straightforward that @xmath75 , and therefore we have @xmath76 . altogether , @xmath77 . replacing the minimizing @xmath78 by the minimizing @xmath79 and @xmath80 by @xmath81 , we can rewrite the last displayed equation as @xmath82 expression emphasizes the two - part code nature of kolmogorov complexity : using the regular aspects of @xmath0 to maximally compress . suppose we consider an ongoing time - series @xmath83 and we randomly stop gathering data after having obtained the initial segment @xmath84 we can encode this @xmath0 by a small turing machine representing `` the repeating pattern is 01 , '' and which computes @xmath0 , for example , from the program `` 13 . '' intuitively , the turing machine part of the code squeezes out the _ regularities _ in @xmath0 . what is left are irregularities , or _ random aspects _ of @xmath0 relative to that turing machine . the minimal - length two - part code squeezes out regularity only insofar as the reduction in the length of the description of random aspects is greater than the increase in the regularity description . in this setup the number of repetitions of the significant pattern is viewed as the random part of the data . this interpretation of @xmath29 as the shortest length of a two - part code for @xmath0 , one part describing a turing machine , or _ model _ , for the _ regular _ aspects of @xmath0 and the second part describing the _ irregular _ aspects of @xmath0 in the form of a program to be interpreted by @xmath31 , has profound applications . the `` right model '' is a turing machine @xmath31 among the ones that halt for all inputs , a restriction that is justified later , and reach the minimum description length in ( [ eq.kcmdl ] ) . this @xmath31 embodies the amount of useful information contained in @xmath0 . it remains to decide which such @xmath31 to select among the ones that satisfy the requirement . following occam s razor we opt here for the shortest one a formal justification for this choice is given in @xcite . the main problem with our approach is how to properly define a shortest program @xmath52 for @xmath0 that divides into parts @xmath85 such that @xmath2 represents an appropriate @xmath31 . the following central notions are used in this paper . the _ information in @xmath0 about @xmath11 _ is @xmath86 . by the symmetry of information , a deep result of @xcite , @xmath87 rewriting according to symmetry of information we see that @xmath88 and therefore we call the quantity @xmath89 the _ mutual information _ between @xmath0 and @xmath11 . instead of the model class of finite sets , or computable probability density functions , as in @xcite , in this work we focus on the most general form of algorithmic model class : total recursive functions . we define the different model classes and summarize the central notions of `` randomness deficiency '' and `` typicality '' for the canonical finite set models to obtain points of reference for the related notions in the more general model classes . the model class of _ finite sets _ consists of the set of finite subsets @xmath90 . the _ complexity of the finite set _ @xmath91 is @xmath92the length ( number of bits ) of the shortest binary program @xmath2 from which the reference universal prefix machine @xmath49 computes a listing of the elements of @xmath91 and then halts . that is , if @xmath93 , then @xmath94 . the _ conditional complexity _ @xmath95 of @xmath0 given @xmath91 , is the length ( number of bits ) in the shortest binary program @xmath2 from which the reference universal prefix machine @xmath49 , given @xmath91 literally as auxiliary information , computes @xmath0 . for every finite set @xmath96 containing @xmath0 we have @xmath97 indeed , consider the selfdelimiting code of @xmath0 consisting of its @xmath98 bit long index of @xmath0 in the lexicographical ordering of @xmath91 this code is called _ data - to - model code_. its length quantifies the maximal `` typicality , '' or `` randomness , '' data ( possibly different from @xmath0 ) can have with respect to this model . the lack of typicality of @xmath0 with respect to @xmath91 is measured by the amount by which @xmath95 falls short of the length of the data - to - model code , the _ randomness deficiency _ of @xmath0 in @xmath91 , defined by @xmath99 for @xmath100 , and @xmath101 otherwise . data @xmath0 is _ typical with respect to a finite set _ @xmath91 , if the randomness deficiency is small . if the randomness deficiency is close to 0 , then there are no simple special properties that single it out from the majority of elements in @xmath91 . this is not just terminology . let @xmath102 . according to common viewpoints in probability theory , each property represented by @xmath91 defines a large subset of @xmath91 consisting of elements having that property , and , conversely , each large subset of @xmath91 represents a property . for probabilistic ensembles we take high probability subsets as properties ; the present case is uniform probability with finite support . for some appropriate fixed constant @xmath103 , let us identify a property represented by @xmath91 with a subset @xmath104 of @xmath91 of cardinality @xmath105 . if @xmath106 is close to 0 , then @xmath0 satisfies ( that is , is an element of ) _ all _ properties ( that is , sets ) @xmath107 of low kolmogorov complexity @xmath108 . the precise statements and quantifications are given in @xcite , and we do not repeat them here . the model class of _ computable probability density functions _ consists of the set of functions @xmath109 $ ] with @xmath110 . `` computable '' means here that there is a turing machine @xmath111 that , given @xmath0 and a positive rational @xmath112 , computes @xmath113 with precision @xmath112 . the ( prefix- ) complexity @xmath114 of a computable ( possibly partial ) function @xmath115 is defined by @xmath116 the model class of _ total recursive functions _ consists of the set of functions @xmath117 such that there is a turing machine @xmath31 such that @xmath118 and @xmath119 , for every @xmath120 . the ( prefix- ) complexity @xmath121 of a total recursive function @xmath34 is defined by @xmath122 if @xmath123 is a shortest program for computing the function @xmath34 ( if there is more than one of them then @xmath123 is the first one in enumeration order ) , then @xmath124 . in the definitions of @xmath114 and @xmath121 , the objects being described are functions rather than finite binary strings . to unify the approaches , we can consider a finite binary string @xmath0 as corresponding to a function having value @xmath0 for argument 0 . note that we can upper semi - compute @xmath52 given @xmath0 , but we can not upper semi - compute @xmath125 given @xmath115 ( as an oracle ) , or @xmath123 given @xmath34 ( again given as an oracle ) , since we should be able to verify agreement of a program for a function and an oracle for the target function , on all infinitely many arguments . to explain typicality for general model classes , it is convenient to use the distortion - rate @xcite approach for individual data recently introduced in @xcite . modeling the data can be viewed as encoding the data by a model : the data are source words to be coded , and models are code words for the data . as before , the set of possible data is @xmath126 . let @xmath127 denote the set of non - negative real numbers . for every model class @xmath128 ( particular set of code words ) we choose an appropriate recursive function @xmath129 defining the _ distortion _ @xmath130 between data @xmath0 and model @xmath131 . the choice of distortion function is a selection of which aspects of the data are relevant , or meaningful , and which aspects are irrelevant ( noise ) . we can think of the distortion as measuring how far the model falls short in representing the data . distortion - rate theory underpins the practice of lossy compression . for example , lossy compression of a sound file gives as `` model '' the compressed file where , among others , the very high and very low inaudible frequencies have been suppressed . thus , the distortion function will penalize the deletion of the inaudible frequencies but lightly because they are not relevant for the auditory experience . [ ex.11 ] let us look at various model classes and distortion measures : \(i ) the set of models are the finite sets of finite binary strings . let @xmath90 and @xmath132 . we define @xmath133 if @xmath100 , and @xmath101 otherwise . \(ii ) the set of models are the computable probability density functions @xmath115 mapping @xmath5 to @xmath134 $ ] . we define @xmath135 if @xmath136 , and @xmath101 otherwise . \(iii ) the set of models are the total recursive functions @xmath34 mapping @xmath5 to @xmath7 . we define @xmath137 , and @xmath101 if no such @xmath3 exists . if @xmath128 is a model class , then we consider _ distortion balls _ of given radius @xmath138 centered on @xmath139 : @xmath140 this way , every model class and distortion measure can be treated similarly to the canonical finite set case , which , however is especially simple in that the radius not variable . that is , there is only one distortion ball centered on a given finite set , namely the one with radius equal to the log - cardinality of that finite set . in fact , that distortion ball equals the finite set on which it is centered . let @xmath128 be a model class and @xmath3 a distortion measure . since in our definition the distortion is recursive , given a model @xmath139 and diameter @xmath138 , the elements in the distortion ball of diameter @xmath138 can be recursively enumerated from the distortion function . giving the index of any element @xmath0 in that enumeration we can find the element . hence , @xmath141 . on the other hand , the vast majority of elements @xmath11 in the distortion ball have complexity @xmath142 since , for every constant @xmath103 , there are only @xmath143 binary programs of length @xmath144 available , and there are @xmath145 elements to be described . we can now reason as in the similar case of finite set models . with data @xmath0 and @xmath146 , if @xmath147 , then @xmath0 belongs to every large majority of elements ( has the property represented by that majority ) of the distortion ball @xmath148 , provided that property is simple in the sense of having a description of low kolmogorov complexity . the _ randomness deficiency _ of @xmath0 with respect to model @xmath131 under distortion @xmath3 is defined as @xmath149 data @xmath0 is _ typical _ for model @xmath139 ( and that model `` typical '' or `` best fitting '' for @xmath0 ) if @xmath150 if @xmath0 is typical for a model @xmath131 , then the shortest way to effectively describe @xmath0 , given @xmath131 , takes about as many bits as the descriptions of the great majority of elements in a recursive enumeration of the distortion ball . so there are no special simple properties that distinguish @xmath0 from the great majority of elements in the distortion ball : they are all typical or random elements in the distortion ball ( that is , with respect to the contemplated model ) . continuing example [ ex.11 ] by applying to different model classes : \(i ) _ finite sets : _ for finite set models @xmath91 , clearly @xmath151 . together with we have that @xmath0 is typical for @xmath91 , and @xmath91 best fits @xmath0 , if the randomness deficiency according to satisfies @xmath152 . \(ii ) _ computable probability density functions : _ instead of the data - to - model code length @xmath153 for finite set models , we consider the data - to - model code length @xmath154 ( the shannon - fano code ) . the value @xmath154 measures how likely @xmath0 is under the hypothesis @xmath115 . for probability models @xmath115 , define the conditional complexity @xmath155 as follows . say that a function @xmath156 approximates @xmath115 if @xmath157 for every @xmath11 and every positive rational @xmath112 . then @xmath158 is defined as the minimum length of a program that , given @xmath159 and any function @xmath156 approximating @xmath115 as an oracle , prints @xmath0 . clearly @xmath160 . together with , we have that @xmath0 is typical for @xmath115 , and @xmath115 best fits @xmath0 , if @xmath161 . the right - hand side set condition is the same as @xmath162 , and there can be only @xmath163 such @xmath11 , since otherwise the total probability exceeds 1 . therefore , the requirement , and hence typicality , is implied by @xmath164 . define the randomness deficiency by @xmath165 altogether , a string @xmath0 is _ typical for a distribution _ @xmath115 , or @xmath115 is the _ best fitting model _ for @xmath0 , if @xmath166 . \(iii ) _ total recursive functions : _ in place of @xmath153 for finite set models we consider the data - to - model code length ( actually , the distortion @xmath167 above ) @xmath168 define the conditional complexity @xmath169 as the minimum length of a program that , given @xmath170 and an oracle for @xmath34 , prints @xmath0 . clearly , @xmath171 . together with , we have that @xmath0 is typical for @xmath34 , and @xmath34 best fits @xmath0 , if @xmath172 . there are at most @xmath173- many @xmath11 satisfying the set condition since @xmath174 . therefore , the requirement , and hence typicality , is implied by @xmath175 . define the randomness deficiency by @xmath176 altogether , a string @xmath0 is _ typical for a total recursive function _ @xmath34 , and @xmath34 is the _ best fitting recursive function model _ for @xmath0 if @xmath177 , or written differently , @xmath178 note that since @xmath170 is given as conditional information , with @xmath179 and @xmath180 , the quantity @xmath181 represents the number of bits in a shortest _ self - delimiting _ description of @xmath3 . we required @xmath170 in the conditional in . this is the information about the radius of the distortion ball centered on the model concerned . note that in the canonical finite set model case , as treated in @xcite , every model has a fixed radius which is explicitly provided by the model itself . but in the more general model classes of computable probability density functions , or total recursive functions , models can have a variable radius . there are subclasses of the more general models that have fixed radiuses ( like the finite set models ) . \(i ) in the computable probability density functions one can think of the probabilities with a finite support , for example @xmath182 for @xmath183 , and @xmath184 otherwise . \(ii ) in the total recursive function case one can similarly think of functions with finite support , for example @xmath185 for @xmath186 , and @xmath187 for @xmath188 . the incorporation of te radius in the model will increase the complexity of the model , and hence of the minimal sufficient statistic below . a _ statistic _ is a function mapping the data to an element ( model ) in the contemplated model class . with some sloppiness of terminology we often call the function value ( the model ) also a statistic of the data . the most important concept in this paper is the sufficient statistic . for an extensive discussion of this notion for specific model classes see @xcite . a statistic is called sufficient if the two - part description of the data by way of the model and the data - to - model code is as concise as the shortest one - part description of @xmath0 . consider a model class @xmath128 . a model @xmath139 is a _ sufficient statistic _ for @xmath0 if @xmath189 [ lem.v2 ] if @xmath131 is a sufficient statistic for @xmath0 , then @xmath190 , that is , @xmath0 is typical for @xmath131 . we can rewrite @xmath191 . the first three inequalities are straightforward and the last equality is by the assumption of sufficiency . altogether , the first sum equals the second sum , which implies the lemma . thus , if @xmath131 is a sufficient statistic for @xmath0 , then @xmath0 is a typical element for @xmath131 , and @xmath131 is the best fitting model for @xmath0 . note that the converse implication , `` typicality '' implies `` sufficiency , '' is not valid . sufficiency is a special type of typicality , where the model does not add significant information to the data , since the preceding proof shows @xmath192 . using the symmetry of information this shows that @xmath193 this means that : \(i ) a sufficient statistic @xmath131 is determined by the data in the sense that we need only an @xmath194-bit program , possibly depending on the data itself , to compute the model from the data . \(ii ) for each model class and distortion there is a universal constant @xmath103 such that for every data item @xmath0 there are at most @xmath103 sufficient statistics . _ finite sets : _ for the model class of finite sets , a set @xmath91 is a sufficient statistic for data @xmath0 if @xmath195 _ computable probability density functions : _ for the model class of computable probability density functions , a function @xmath115 is a sufficient statistic for data @xmath0 if @xmath196 for the model class of _ total recursive functions _ , a function @xmath34 is a _ sufficient statistic _ for data @xmath0 if @xmath197 following the above discussion , the meaningful information in @xmath0 is represented by @xmath34 ( the model ) in @xmath121 bits , and the meaningless information in @xmath0 is represented by @xmath3 ( the noise in the data ) with @xmath180 in @xmath198 bits . note that @xmath199 , since the two - part code @xmath200 for @xmath0 can not be shorter than the shortest one - part code of @xmath29 bits , and therefore the @xmath3-part must already be maximally compressed . by lemma [ lem.v2 ] , @xmath201 , @xmath0 is typical for @xmath34 , and hence @xmath202 . consider the model class of total recursive functions . a _ minimal sufficient statistic _ for data @xmath0 is a sufficient statistic for @xmath0 of minimal prefix complexity . its length is known as the _ sophistication _ of @xmath0 , and is defined by @xmath203 . recall that the _ reference _ universal prefix turing machine @xmath49 was chosen such that @xmath204 for all @xmath31 and @xmath3 . looking at it slightly more from a programming point of view , we can define a pair @xmath205 to be a _ description _ of a finite string @xmath0 , if @xmath206 prints @xmath0 and @xmath31 is a turing machine computing a function @xmath34 so that @xmath180 . for the notion of minimal sufficient statistic to be nontrivial , it should be impossible to always shift , if @xmath180 and @xmath207 with @xmath208 , always information information from @xmath34 to @xmath3 and write , for example , @xmath209 with @xmath210 with @xmath211 . if the model class contains a fixed universal model that can mimic all other models , then we can always shift all model information to the data - to-(universal ) model code . note that this problem does nt arise in common statistical model classes : these do not contain universal models in the algorithmic sense . first we show that the partial recursive recursive function model class , because it contains a universal element , does not allow a straightforward nontrivial division into meaningful and meaningless information . assume for the moment that we allow all partial recursive programs as statistic . then , the sophistication of all data @xmath0 is @xmath212 . let the index of @xmath49 ( the reference universal prefix turing machine ) in the standard enumeration @xmath48 of prefix turing machines be @xmath213 . let @xmath214 be a turing machine computing @xmath34 . suppose that @xmath215 . then , also @xmath216 . this shows that unrestricted partial recursive statistics are uninteresting . naively , this could leave the impression that the separation of the regular and the random part of the data is not as objective as the whole approach lets us hope for . if we consider complexities of the minimal sufficient statistics in model classes of increasing power : finite sets , computable probability distributions , total recursive functions , partial recursive functions , then the complexities appear to become smaller all the time eventually reaching zero . it would seem that the universality of kolmogorov complexity , based on the notion of partial recursive functions , would suggest a similar universal notion of sufficient statistic based on partial recursive functions . but in this case the very universality trivializes the resulting definition : because partial recursive functions contain a particular universal element that can simulate all the others , this implies that the universal partial recursive function is a universal model for all data , and the data - to - model code incorporates all information in the data . thus , if a model class contains a universal model that can simulate all other models , then this model class is not suitable for defining two - part codes consisting of meaningful information and accidental information . it turns out that the key to nontrivial separation is the requirement that the program witnessing the sophistication be _ total_. that the resulting separation is non - trivial is evidenced by the fact , shown below , that the amount of meaningful information in the data does not change by more than a logarithmic additive term under change of model classes among finite set models , computable probability models , and total recursive function models . that is , very different model classes all result in the same amount of meaningful information in the data , up to negligible differences . so if deterioration occurs in widening model classes it occurs all at once by having a universal element in the model class . apart from triviality , a class of statistics can also possibly be vacuous by having the length of the minimal sufficient statistic exceed @xmath29 . our first task is to determine whether the definition is non - vacuous . we will distinguish sophistication in different description modes : [ lem.exists ] for every finite binary string @xmath0 , the sophistication satisfies @xmath217 . by definition of the prefix complexity there is a program @xmath52 of length @xmath218 such that @xmath219 . this program @xmath52 can be partial . but we can define another program @xmath220 where @xmath221 is a program of a constant number of bits that tells the following program to ignore its actual input and compute as if its input were @xmath9 . clearly , @xmath222 is total and is a sufficient statistic of the total recursive function type , that is , @xmath223 . the previous lemma gives an upper bound on the sophistication . this still leaves the possibility that the sophistication is always @xmath212 , for example in the most liberal case of unrestricted totality . but this turns out to be impossible . [ h - sophi ] ( i ) for every @xmath0 , if a sufficient statistic @xmath34 satisfies @xmath224 , then @xmath225 and @xmath226 . \(ii ) for @xmath0 as a variable running through a sequence of finite binary strings of increasing length , we have @xmath227 \(iii ) for every @xmath228 , there exists an @xmath0 of length @xmath228 , such that every sufficient statistic @xmath34 for @xmath0 that satisfies @xmath224 has @xmath229 . \(iv ) for every @xmath228 there exists an @xmath0 of length @xmath228 such that @xmath230 . \(i ) if @xmath34 is a sufficient statistic for @xmath0 , then @xmath231 since @xmath224 , given an @xmath194 bit program @xmath232 we can retrieve both @xmath170 and and also @xmath233 from @xmath123 . therefore , we can retrieve @xmath234 from @xmath235 . that shows that @xmath236 . this proves both the first statement , and the second statement follows by ( [ eq.pd ] ) . \(ii ) an example of very unsophisticated strings are the individually random strings with high complexity : @xmath0 of length @xmath237 with complexity @xmath238 . then , the _ identity _ program @xmath239 with @xmath240 for all @xmath3 is total , has complexity @xmath241 , and satisfies @xmath242 . hence , @xmath239 witnesses that @xmath243 . this shows ( [ eq.liminf ] ) . \(iii ) consider the set @xmath244 . by @xcite we have @xmath245 . let @xmath246 . since there are @xmath247 strings of length @xmath228 , there are strings of length @xmath228 not in @xmath248 . let @xmath0 be any such string , and denote @xmath249 . then , by construction @xmath250 and by definition @xmath251 . let @xmath34 be a sufficient statistic for @xmath0 . then , @xmath252 . by assumption , there is an @xmath194-bit program @xmath232 such that @xmath253 . let @xmath3 witness @xmath170 by @xmath180 with @xmath198 . define the set @xmath254 . clearly , @xmath255 . since @xmath0 can be retrieved from @xmath34 and the lexicographical index of @xmath3 in @xmath256 , and @xmath257 , we have @xmath258 . since we can obtain @xmath256 from @xmath235 we have @xmath259 . on the other hand , since we can retrieve @xmath0 from @xmath256 and the index of @xmath3 in @xmath256 , we must have @xmath260 , which implies @xmath261 . altogether , therefore , @xmath262 . we now show that we can choose @xmath0 so that @xmath263 , and therefore @xmath229 . for every length @xmath228 , there exists a @xmath264 of complexity @xmath265 such that a minimal sufficient finite set statistic @xmath91 for @xmath264 has complexity at least @xmath266 , by theorem iv.2 of @xcite . since @xmath267 is trivially a sufficient statistic for @xmath264 , it follows @xmath268 . this implies @xmath269 . therefore , we can choose @xmath270 for a large enough constant @xmath271 so as to ensure that @xmath272 . consequently , we can choose @xmath0 above as such a @xmath264 . since every finite set sufficient statistic for @xmath0 has complexity at least that of an finite set minimal sufficient statistic for @xmath0 , it follows that @xmath263 . therefore , @xmath229 , which was what we had to prove . \(iv ) in the proof of ( i ) we used @xmath273 . without using this assumption , the corresponding argument yields @xmath274 . we also have @xmath275 and @xmath276 . since we can retrieve @xmath0 from @xmath256 and its index in @xmath256 , the same argument as above shows @xmath277 , and still following the argument above , @xmath278 . since @xmath279 we have @xmath280 . this proves the statement . the useful ( [ eq.pcondx ] ) states that there is a constant , such that for every @xmath0 there are at most that constant many sufficient statistics for @xmath0 , and there is a constant length program ( possibly depending on @xmath0 ) , that generates all of them from @xmath52 . in fact , there is a slightly stronger statement from which this follows : there is a universal constant @xmath103 , such that for every @xmath0 , the number of @xmath281 such that @xmath282 and @xmath283 , is bounded above by @xmath103 . let the prefix turing machine @xmath214 compute @xmath34 . since @xmath215 and @xmath284 , the combination @xmath285 ( with self - delimiting @xmath123 ) is a shortest prefix program for @xmath0 . from @xcite , exercise 3.3.7 item ( b ) on p. 205 , it follows that the number of shortest prefix programs is upper bounded by a universal constant . previous work studied sufficiency for finite set models , and computable probability mass functions models , @xcite . the most general models that are still meaningful are total recursive functions as studied here . we show that there are corresponding , almost equivalent , sufficient statistics in all model classes . [ lem.explimpl ] \(i ) if @xmath91 is a sufficient statistic of @xmath0 ( finite set type ) , then there is a corresponding sufficient statistic @xmath115 of @xmath0 ( probability mass function type ) such that @xmath286 , @xmath287 , and @xmath288 . \(ii ) if @xmath115 is a sufficient statistic of @xmath0 of the computable total probability density function type , then there is a corresponding sufficient statistic @xmath34 of @xmath0 of the total recursive function type such that @xmath289 , @xmath290 , and @xmath291 . \(i ) by assumption , @xmath91 is a finite set such that @xmath100 and @xmath292 . define the probability distribution @xmath293 for @xmath294 and @xmath295 otherwise . since @xmath91 is finite , @xmath115 is computable . since @xmath296 , and @xmath297 , we have @xmath298 . since @xmath115 is a computable probability mass function we have @xmath299 , by the standard shannon - fano code construction @xcite that assigns a code word of length @xmath300 to @xmath0 . since by ( [ eq.soi ] ) we have @xmath301 it follows that @xmath302 . hence , @xmath303 . therefore , by ( [ eq.soi ] ) , @xmath304 and , by rewriting @xmath305 in the other way according to ( [ eq.soi ] ) , @xmath288 . \(ii ) by assumption , @xmath115 is a computable probability density function with @xmath136 and @xmath306 . the witness of this equality is a shortest program @xmath125 for @xmath115 and a code word @xmath307 for @xmath0 according to the standard shannon - fano code , @xcite , with @xmath308 . given @xmath115 , we can reconstruct @xmath0 from @xmath307 by a fixed standard algorithm . define the recursive function @xmath34 from @xmath115 such that @xmath309 . in fact , from @xmath125 this only requires a constant length program @xmath232 , so that @xmath310 is a program that computes @xmath34 in the sense that @xmath311 for all @xmath3 . similarly , @xmath115 can be retrieved from @xmath34 . hence , @xmath289 and @xmath312 . that is , @xmath34 is a sufficient statistic for @xmath0 . also , @xmath34 is a total recursive function . since @xmath313 we have @xmath314 , and @xmath314 . this shows that @xmath315 , and since @xmath0 can by definition be reconstructed from @xmath123 and a program of length @xmath316 , it follows that equality must hold . consequently , @xmath317 , and hence , by ( [ eq.soi ] ) , @xmath318 and @xmath291 . we have now shown that a sufficient statistic in a less general model class corresponds directly to a sufficient statistic in the next more general model class . we now show that , with a negligible error term , a sufficient statistic in the most general model class of total recursive functions has a directly corresponding sufficient statistic in the least general finite set model class . that is , up to negligible error terms , a sufficient statistic in any of the model classes has a direct representative in any of the other model classes . let @xmath0 be a string of length @xmath228 , and @xmath34 be a total recursive function sufficient statistic for @xmath0 . then , there is a finite set @xmath319 such that @xmath320 . by assumption there is an @xmath194-bit program @xmath232 such that @xmath253 . for each @xmath321 , let @xmath322 . define @xmath323 . we can compute @xmath91 by computation of @xmath324 , on all arguments @xmath51 of at most @xmath325 bits , since by assumption @xmath34 is total . this shows @xmath326 . since @xmath327 , we have @xmath328 . moreover , @xmath329 . since @xmath100 , @xmath330 , where we use the sufficiency of @xmath34 to obtain the last inequality we investigate the recursion properties of the sophistication function . in @xcite , gcs gave an important and deep result ( [ eq.gacs ] ) below , that quantifies the uncomputability of @xmath29 ( the bare uncomputability can be established in a much simpler fashion ) . for every length @xmath228 there is an @xmath0 of length @xmath228 such that : @xmath331 note that the right - hand side holds for every @xmath0 by the simple argument that @xmath332 and hence @xmath333 . but there are @xmath0 s such that the length of the shortest program to compute @xmath29 almost reaches this upper bound , even if the full information about @xmath0 is provided . it is natural to suppose that the sophistication function is not recursive either . the following lemma s suggest that the complexity function is more uncomputable than the sophistication . the function @xmath334 is not recursive . given @xmath228 , let @xmath335 be the least @xmath0 such that @xmath336 . by theorem [ h - sophi ] we know that there exist @xmath0 such that @xmath337 for @xmath338 , hence @xmath335 exists . assume by way of contradiction that the sophistication function is computable . then , we can find @xmath335 , given @xmath228 , by simply computing the successive values of the function . but then @xmath339 , while by lemma [ lem.exists ] @xmath340 and by assumption @xmath341 , which is impossible . the _ halting sequence _ @xmath342 is the infinite binary characteristic sequence of the halting problem , defined by @xmath343 if the reference universal prefix turing machine @xmath49 halts on the @xmath51th input : @xmath344 , and 0 otherwise . [ lem.compks ] let @xmath123 be a total recursive function sufficient statistic of @xmath0 . \(i ) we can compute @xmath29 from @xmath123 and @xmath0 , up to fixed constant precision , which implies that @xmath345 . \(ii ) if also @xmath346 , then we can compute @xmath29 from @xmath123 , up to fixed constant precision , which implies that @xmath347 . \(i ) since @xmath34 is total , we can run @xmath348 on all strings @xmath349 in lexicographical length - increasing order . since @xmath34 is total we will find a shortest string @xmath350 such that @xmath351 . set @xmath352 . since @xmath353 , and by assumption , @xmath354 , we now can compute @xmath355 . \(ii ) follows from item ( i ) . given an oracle that on query @xmath0 answers with a sufficient statistic @xmath123 of @xmath0 and a @xmath356 as required below . then , we can compute the kolmogorov complexity function @xmath357 and the halting sequence @xmath358 . by lemma [ lem.compks ] we can compute the function @xmath29 , up to fixed constant precision , given the oracle ( without the value @xmath359 ) in the statement of the theorem . let @xmath359 in the statement of the theorem be the difference between the computed value and the actual value of @xmath29 . in @xcite , exercise 2.2.7 on p. 175 , it is shown that if we can solve the halting problem for plain turing machines , then we can compute the ( plain ) kolmogorov complexity , and _ vice versa_. the same holds for the halting problem for prefix turing machines and the prefix turing complexity . this proves the theorem . [ lem.chisoph ] there is a constant @xmath103 , such that for every @xmath0 there is a program ( possibly depending on @xmath0 ) of at most @xmath103 bits that computes @xmath360 and the witness program @xmath34 from @xmath361 . that is , @xmath362 . with some abuse of notation we can express this as @xmath363 . by definition of sufficient statistic @xmath123 , we have @xmath364 . by ( [ eq.pcondx ] ) the number of sufficient statistics for @xmath0 is bounded by an independent constant , and we can generate all of them from @xmath0 by a @xmath212 length program ( possibly depending on @xmath0 ) . then , we can simply determine the least length of a sufficient statistic , which is @xmath360 . there is a subtlety here : lemma [ lem.chisoph ] is nonuniform . while for every @xmath0 we only require a fixed number of bits to compute the sophistication from @xmath361 , the result is nonuniform in the sense that these bits may depend on @xmath0 . given a program , how do we verify if it is the correct one ? trying all programs of length up to a known upper bound , we do nt know if they halt or if they halt they halt with the correct answer . the question arising is if there is a single program that computes the sopistication and its witness program for all @xmath0 . in @xcite this much more difficult question is answered in a strong negative sense : there is no algorithm that for every @xmath0 , given @xmath361 , approximates the sophistication of @xmath0 to within precision @xmath365 . for every @xmath0 of length @xmath228 , and @xmath123 the program that witnesses the sophistication of @xmath0 , we have @xmath366 . for every length @xmath228 , there are strings @xmath0 of length @xmath228 , such that @xmath367 . let @xmath123 witness the @xmath360 : that is , @xmath364 , and @xmath368 . using the conditional version of ( [ eq.soi ] ) , see @xcite , we find that @xmath369 @xmath370 in lemma [ lem.compks ] , item ( i ) , we show @xmath371 , hence also @xmath372 . by lemma [ lem.chisoph ] , @xmath373 , hence also @xmath374 . substitution of the constant terms in the displayed equation shows @xmath375 this shows that the shortest program to retrieve @xmath123 from @xmath0 is essentially the same program as to retrieve @xmath52 from @xmath0 or @xmath29 from @xmath0 . using , this shows that @xmath376 since @xmath123 is the witness program for @xmath368 , we have @xmath377 . a function @xmath34 from the rational numbers to the real numbers is _ upper semicomputable _ if there is a recursive function @xmath378 such that @xmath379 and @xmath380 . here we interprete the total recursive function @xmath381 as a function from pairs of natural numbers to the rationals : @xmath382 . if @xmath34 is upper semicomputable , then @xmath383 is _ lower semicomputable_. if @xmath34 is both upper - a and lower semicomputable , then it is _ computable_. recursive functions are computable functions over the natural numbers . since @xmath384 is upper semicomputable , @xcite , and from @xmath384 we can compute @xmath360 , we have the following : \(i ) the function @xmath360 is not computable to any significant precision . \(ii ) given an initial segment of length @xmath385 of the halting sequence @xmath342 , we can compute @xmath360 from @xmath0 . that is , @xmath386 . \(i ) the fact that @xmath360 is not computable to any significant precision is shown in @xcite . \(ii ) we can run @xmath387 for all ( program , argument ) pairs such that @xmath388 . ( not @xmath6 since we are dealing with self - delimiting programs . ) if we know the initial segment of @xmath358 , as in the statement of the theorem , then we know which ( program , argument ) pairs halt , and we can simply compute the minimal value of @xmath389 for these pairs . `` sophistication '' is the algorithmic version of `` minimal sufficient statistic '' for data @xmath0 in the model class of total recursive functions . however , the full stochastic properties of the data can only be understood by considering the kolmogorov structure function @xmath390 ( mentioned earlier ) that gives the length of the shortest two - part code of @xmath0 as a function of the maximal complexity @xmath68 of the total function supplying the model part of the code . this function has value about @xmath6 for @xmath68 close to 0 , is nonincreasing , and drops to the line @xmath29 at complexity @xmath391 , after which it remains constant , @xmath392 for @xmath393 , everything up to a logarithmic addive term . a comprehensive analysis , including many more algorithmic properties than are analyzed here , has been given in @xcite for the model class of finite sets containing @xmath0 , but it is shown there that all results extend to the model class of computable probability distributions and the model class of total recursive functions , up to an additive logarithmic term . the author thanks luis antunes , lance fortnow , kolya vereshchagin , and the referees for their comments . 99 a.r . barron , j. rissanen , and b. yu , the minimum description length principle in coding and modeling , _ ieee trans . inform . theory _ , it-44:6(1998 ) , 27432760 . cover , kolmogorov complexity , data compression , and inference , pp . 2333 in : _ the impact of processing techniques on communications _ , j.k . skwirzynski , ed . , martinus nijhoff publishers , 1985 . t.m . cover and j.a . thomas , _ elements of information theory _ , wiley , new york , 1991 . r. a. fisher , on the mathematical foundations of theoretical statistics , _ philosophical transactions of the royal society of london , ser . a _ , 222(1922 ) , 309368 . p. gcs , on the symmetry of algorithmic information , _ soviet math . _ , 15 ( 1974 ) 14771480 . correction : ibid . , 15 ( 1974 ) 1480 . p. gcs , j. tromp , and p. vitnyi , algorithmic statistics , _ ieee trans . inform . theory _ , 47:6(2001 ) , 24432463 . q. gao , m. li and p.m.b . vitnyi , applying mdl to learn best model granularity , _ artificial intelligence _ , 121(2000 ) , 129 . m. gell - mann , _ the quark and the jaguar _ , w. h. freeman and company , new york , 1994 . grnwald and p.m.b . vitnyi , shannon information and kolmogorov complexity , manuscript , cwi , december 2003 . kolmogorov , three approaches to the quantitative definition of information , _ problems inform . transmission _ 1:1 ( 1965 ) complexity of algorithms and objective definition of randomness . a talk at moscow math . meeting 4/16/1974 . abstract in _ uspekhi mat . nauk _ 29:4(1974),155 ( russian ) ; english translation in @xcite . kolmogorov , on logical foundations of probability theory , pp . 15 in : _ probability theory and mathematical statistics _ , notes math . 1021 , k. it and yu.v . prokhorov , eds . , springer - verlag , heidelberg , 1983 . kolmogorov and v.a . uspensky , algorithms and randomness , _ siam theory probab . appl . _ , 32:3(1988 ) , 389412 . m. koppel , complexity , depth , and sophistication , _ complex systems _ , 1(1987 ) , 10871091 m. koppel , structure , _ the universal turing machine : a half - century survey _ , r. herken ( ed . ) , oxford univ . press , 1988 , pp . 435452 . m. li and p. vitanyi , _ an introduction to kolmogorov complexity and its applications _ , springer - verlag , new york , 1997 ( 2nd edition ) . the mathematical theory of communication . , 27:379423 , 623656 , 1948 . coding theorems for a discrete source with a fidelity criterion . in _ ire national convention record , part 4 _ , pages 142163 , 1959 . shen , the concept of @xmath394-stochasticity in the kolmogorov sense , and its properties , _ soviet math . _ , 28:1(1983 ) , 295299 . shen , discussion on kolmogorov complexity and statistical analysis , _ the computer journal _ , 42:4(1999 ) , 340342 . vereshchagin and p.m.b . vitnyi , kolmogorov s structure functions and model selection , _ ieee trans . _ , to appear . vereshchagin and p.m.b . vitnyi , rate distortion theory for individual data , draft , cwi , 2004 . vitnyi and m. li , minimum description length induction , bayesianism , and kolmogorov complexity , _ ieee trans . inform . theory _ , it-46:2(2000 ) , 446464 . v.v . vyugin , on the defect of randomness of a finite object with respect to measures with given complexity bounds , _ siam theory probab . appl . _ , 32:3(1987 ) , 508512 . v.v . vyugin , algorithmic complexity and stochastic properties of finite binary sequences , _ the computer journal _ , 42:4(1999 ) , 294317 . paul m.b . vitnyi is a fellow of the center for mathematics and computer science ( cwi ) in amsterdam and is professor of computer science at the university of amsterdam . he serves on the editorial boards of distributed computing ( until 2003 ) , information processing letters , theory of computing systems , parallel processing letters , international journal of foundations of computer science , journal of computer and systems sciences ( guest editor ) , and elsewhere . he has worked on cellular automata , computational complexity , distributed and parallel computing , machine learning and prediction , physics of computation , kolmogorov complexity , quantum computing . together with ming li they pioneered applications of kolmogorov complexity and co - authored `` an introduction to kolmogorov complexity and its applications , '' springer - verlag , new york , 1993 ( 2nd edition 1997 ) , parts of which have been translated into chinese , russian and japanese .

the information in an individual finite object ( like a binary string ) is commonly measured by its kolmogorov complexity . one can divide that information into two parts : the information accounting for the useful regularity present in the object and the information accounting for the remaining accidental information . there can be several ways ( model classes ) in which the regularity is expressed . kolmogorov has proposed the model class of finite sets , generalized later to computable probability mass functions . the resulting theory , known as algorithmic statistics , analyzes the algorithmic sufficient statistic when the statistic is restricted to the given model class . however , the most general way to proceed is perhaps to express the useful information as a recursive function . the resulting measure has been called the `` sophistication '' of the object . we develop the theory of recursive functions statistic , the maximum and minimum value , the existence of absolutely nonstochastic objects ( that have maximal sophistication all the information in them is meaningful and there is no residual randomness ) , determine its relation with the more restricted model classes of finite sets , and computable probability distributions , in particular with respect to the algorithmic ( kolmogorov ) minimal sufficient statistic , the relation to the halting problem and further algorithmic properties . _ index terms_ constrained best - fit model selection , computability , lossy compression , minimal sufficient statistic , non - probabilistic statistics , kolmogorov complexity , kolmogorov structure function , sufficient statistic , sophistication

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Proceed to summarize the following text: charmonium spectroscopy has become a challenging topic in hadron physics and qcd , because of the recent findings of possible new charmonium states ( for recent experimental and theoretical reviews and discussions , see e.g. @xcite and references therein ) . among others , for the puzzling state x(3872 ) , possible assignments of e.g. the @xmath8 and @xmath9 charmonium states and the charm - molecule have been suggested ( see , e.g.@xcite for a comprehensive review ) , and it will be helpful to search for those states in other experiments and to clarify these assignments ; the measured mass splitting between @xmath10 and @xmath11 is about 50 mev , which is smaller than some theoretical predictions , and it is certainly useful to search for the @xmath12 to see what will be the mass splitting between the @xmath13 , which could be the observed @xmath14 , and the @xmath12 . this may be particularly interesting since according to some potential model calculations the @xmath12 could lie above 4040 mev ( see , e.g. in @xcite the mass of @xmath12 is predicted to be 4060 mev ) . and the @xmath13 mass could actually be lowered by coupling to the nearby @xmath15 decay channels ( note that the energy level spacing between @xmath16 and @xmath17 is smaller than that between @xmath18 and @xmath16 , which is in contradiction with potential model calculations unless the coupled channel effects are considered or the assignments for @xmath16 and @xmath18 are incorrect . the mass spectrum of excited charmonium states will certainly provide important information on interquark forces and color confinement . in addition , studies of the decay and production of these states will also be very important in understanding the underlying theory of strong interaction perturbative and nonperturbative qcd in view of many outstanding puzzles in charmonium physics . @xmath2 meson decays have proven to be very useful processes to find new charmonium states . aside from the @xmath2 meson decay , @xmath0 annihilation at @xmath1 gev could also be a very useful process in finding the excited charmonium states , since the recent belle experiments @xcite have found unusually strong signals for the double charmonuim production from the @xmath0 continuum , e.g. , @xmath19 , @xmath20 and @xmath21 , @xmath22 . theoretically , the calculated cross sections for these processes based on the leading order non - relativistic qcd(nrqcd ) ( or more generally perturbative qcd ( pqcd ) ) are about an order of magnitude smaller than the experiments @xcite . this is a big issue in charmonium physics and nrqcd , and it still remains to be further clarified though many considerations are suggested to understand the large production rates in both exclusive and inclusive charmonium production via double charm pairs in @xmath0 annihilation @xcite ( the theoretical predictions for the inclusive @xmath23 production cross section with the color - singlet @xcite as well as color - octet@xcite contributions are also much smaller than the belle data ) . despite of these disagreements , however , we find that the calculated relative rates of the double charmonium production processes are roughly compatible with the belle data ( e.g. the production cross sections of @xmath24 , @xmath11 , and @xmath25 associated with @xmath26 and @xmath10 are much larger than that of @xmath27 and @xmath28 ) . so , we may use the same method as in our previous work to calculate the production rates for the excited charmonium states in @xmath0 annihilation into double charmonia , but mainly pay attention to the relative rates for these production processes . we hope the calculation will make sense in predicting the relative production rates for those excited charmonium states , and can be tested by experiments . this will be useful not only in the search for those excited charmonium states , but also in understanding the production mechanism itself . if the predicted relative production rates turn out to be consistent with experiments , it is likely that the nrqcd factorization treatment for these processes probably still makes sense and only an overall enhancement factor is needed and should be clarified in further theoretical considerations ( including qcd radiative corrections , relativistic corrections , and other nonperturbative qcd effects ) . in the last section we will have a discussion on recent developments in this regard . in the following , we will calculate the leading order production cross sections for various excited charmonium states in @xmath0 annihilation at @xmath1 gev in the same way as in @xcite . following the nrqcd factorization formalism@xcite , the scattering amplitude of double charmonia production can be described as @xmath29 where @xmath30 , @xmath31 , @xmath32 , @xmath33 , @xmath34 and @xmath35 are respectively the color - su(3 ) , spin - su(2 ) , and angular momentum clebsch - gordon coefficients for @xmath36 pairs projecting out appropriate bound states . @xmath37 is the scattering amplitude for double @xmath36 production and @xmath38 is the derivative of the amplitude with respect to the relative momentum between the quark and anti - quark in the bound state . the coefficients @xmath39 and @xmath40 can be related to the radial wave function of the bound states or its derivative with respect to the relative spacing as @xmath41 we introduce the spin projection operators @xmath42 as@xcite @xmath43 expanding @xmath44 in terms of the relative momentum @xmath45 , we get the projection operators and their derivatives , which will be used in our calculation , as follows @xmath46 @xmath47 @xmath48 @xmath49.\ ] ] we then get the following expressions and numerical results for various processes of double charmonium production in @xmath0 annihilation at @xmath1 gev . in the calculation of the short distance coefficients , the quark and anti - quark are all on mass shell , and the meson masses are taken to be @xmath50 . the input parameters are @xmath51 ( corresponding to @xmath52 and @xmath53=338 mev ) , and the wave functions at the origin are taken from a potential model calculation ( see the qcd ( bt(buchmller - tye ) ) model in ref.@xcite ) : @xmath54 , @xmath55 , @xmath56 , @xmath57 , @xmath58 , and @xmath59 . in the two s - wave ( ns and ms ) case , the cross section for @xmath61 is given by @xmath62 where @xmath63 is the scattering angle between @xmath64 and @xmath65 , @xmath66 is as follows @xmath67 here the mandelstam variables are defined as @xmath68 @xmath69 @xmath70 the cross sections for the double s - wave charmonium production are listed as follows @xmath71)=5.5[3.6 , 3.1]~\rm{fb},\ ] ] @xmath72)=3.6[2.3 , 2.0]~\rm{fb},\ ] ] where the values in the brackets are the cross sections respectively for @xmath11 and @xmath12 with recoiling @xmath73 mesons . in the one spin - triplet s - wave ( ns ) and one spin - triplet p - wave ( mp ) case , the cross section for @xmath75 process reads @xmath76 here for the production of spin - triplet states @xmath77 , @xmath66 is given in eq . ( [ kc0 ] ) for @xmath25 , in eq . ( [ kc1])for @xmath27 and eq . ( [ kc2 ] ) for @xmath28 , @xmath78 @xmath79 @xmath80 for the known spin - triplet p - wave states @xmath81 we find @xmath82)=6.7[1.1 , 1.6]~\rm{fb},\ ] ] @xmath83)=4.4[0.74 , 1.1]~\rm{fb},\ ] ] for the exited spin - triplet 2p states @xmath84 , which are to be searched for , we find @xmath85)=9.1[1.6 , 2.2]~\rm{fb},\ ] ] @xmath86)=5.9[1.0 , 1.4]~\rm{fb}.\ ] ] in eqs . ( 1922 ) , we see that the cross sections for the excited @xmath87 states @xmath88 are somewhat larger than that for the corresponding @xmath89 states @xmath90 in the nonrelativistic limit . this numerical result is due to the fact that we have chosen the first derivative of the wave function at the origin for the @xmath87 states to be larger than that for the @xmath89 states , and actually the former could be slightly smaller than the latter , depending on the potentials that are used ( see the values in the qcd ( bt ) model and other models in ref.@xcite ) . furthermore , another important effect comes from the relativistic corrections , which may lower the cross sections for the @xmath88 states . e.g. , if we take the charm quark mass to be @xmath91 for the 2p states , then the cross sections will be substantially lower . despite of these uncertainties , we expect that the cross sections for the @xmath87 states should be comparable to that for the @xmath89 states . to search for the spin singlet p - wave charmonium @xmath7 is certainly interesting . recently , cleo has found evidence for @xmath7 in the @xmath93 decay followed by @xmath94 with a mass of @xmath95 and hyperfine splitting of about 1.0 mev measured in both the @xmath24 exclusive and inclusive analysis@xcite . it will also be interesting to search for the @xmath7 in @xmath0 annihilation at @xmath1 gev in the recoil spectra of charge parity c=+1 states such as @xmath96 . for the production of one spin - singlet s - wave state @xmath97 and one spin - singlet p - wave state @xmath98 , differing from @xcite , we find @xmath66 to be not vanishing but given by eq . ( [ echc ] ) : @xmath99 for the spin - singlet p - wave states @xmath100 and @xmath101 we find @xmath102)=0.73[0.99]~\rm{fb},\ ] ] @xmath103)=0.48[0.65]~\rm{fb},\ ] ] @xmath104)=0.41[0.56]~\rm{fb}.\ ] ] in the two p - wave ( np and mp ) case , the cross section for @xmath106 is @xmath107 for the @xmath108 production @xmath66 reads @xmath109 @xmath110 @xmath111 and the corresponding cross sections are @xmath112)=0.22[0.31]~\rm{fb},\ ] ] @xmath113)=1.0[1.4]~\rm{fb},\ ] ] @xmath114)=0.063[0.085]~\rm{fb}.\ ] ] the cross section for @xmath116 process is formulated as @xmath117 @xmath118 and the numerical result is @xmath119 + ^1d_2)=0.19[0.12]~\rm{fb}.\ ] ] the cross section for @xmath121 process is formulated as @xmath122 @xmath123 @xmath124 @xmath125 @xmath126 @xmath127 @xmath128 the numerical results are listed as follows , where @xmath129 , @xmath130 + \delta_1)=0.080[0.041]~\rm{fb},\ ] ] @xmath130 + \delta_2)=0.099[0.084]~\rm{fb},\ ] ] @xmath130 + \delta_3)=0.041[0.0099]~\rm{fb},\ ] ] as a summary of the above results , we show the differential cross sections ( the angular distribution functions ) for different double charmonium production processes at leading order in nrqcd ( qed contributions are not included ) in table i , and the corresponding graphs in fig . 1 - 7 . since the @xmath132 can be produced via a single photon , the qed contribution may be significant or even comparable to the qcd contribution in some exclusive processes involving one @xmath133 charmonium state . these qed effects are considered in @xcite . there are six feynman diagrams for the qed process at @xmath134 order , and only two represent @xmath135 , which is dominant and has been calculated in @xcite . in this paper we include all the eight diagrams to get the full result at order @xmath134 , though the contributions of other six diagrams are numerically small . using the notation in section ii , we re - express below the analytical formulas of the exclusive processes including both qcd and qed contributions . the cross section for @xmath61 is now changed to @xmath136 where @xmath137 and @xmath63 is the scattering angle between @xmath138 and @xmath139 . and @xmath140is @xmath141 the numerical results become @xmath71)=6.6[4.3 , 3.7]~\rm{fb},\ ] ] @xmath72)=4.3[2.8 , 2.4]~\rm{fb},\ ] ] where the values in the brackets are the cross sections respectively for @xmath11 and @xmath12 with recoiled @xmath73 mesons . the cross section for @xmath61 is changed to @xmath143 and now @xmath66 for @xmath144 production , are given in eq . ( [ kc0e ] ) , eq . ( [ kc1e ] ) , eq . ( [ kc2e ] ) respectively . @xmath146 @xmath147 the numerical results of spin - triplet p - wave states @xmath81 are @xmath148)=6.9[1.0 , 1.8]~\rm{fb},\ ] ] @xmath149)=4.5[0.7 , 1.1]~\rm{fb},\ ] ] and for the exited spin - triplet 2p states @xmath84 , the results turn to be @xmath150)=9.4[1.4 , 2.4]~\rm{fb},\ ] ] @xmath86)=6.2[0.9 , 1.6]~\rm{fb}.\ ] ] the cross section for @xmath152 process is @xmath153 and @xmath154 and the numerical result becomes @xmath119 + ^1d_2)=0.21[0.13]~\rm{fb}.\ ] ] the cross section for @xmath152 process is formulated as @xmath156 and @xmath157 and the numerical result becomes @xmath158 in this section we have considered the qed contribution to the exclusive processes involving one @xmath159 charmonium state ( e.g. , @xmath160 and @xmath161 ) . we find that in general by adding qed contribution the cross section can be changed by the order of ten percent . we list the results below with @xmath1gev , @xmath162 . the cross sections of @xmath163 are increased by 20 percent , that of @xmath164 are increased by 4 , -5 , 8 percent for @xmath165 respectively , that of @xmath166 is increased by @xmath167 percent , and that of @xmath168 is increased by @xmath169 percent . in @xcite the authors also considered the qed process but only the two dominant diagrams were included . our results are in agreement with their results in most processes when we choose the same parameters as theirs , except for the process of @xmath170 production , for which they obtained 41% enhancement with qed effects whereas we get 19% with the same parameters . here our analytical expression also differs from theirs . we show the differential cross sections ( the angular distribution functions ) for different double charmonium production processes including both qcd and qed contributions in table ii , and the corresponding graphs in fig . 8 . ratios of production cross sections of various double charmonia to that of @xmath171 in @xmath172 annihilation at @xmath1 gev are listed in table iii . in this paper , we make predictions for various double charmonia production processes in @xmath0 annihilation at @xmath1 gev with @xmath2 factories , based on a complete leading order calculation including both qcd and qed contributions . in particular , we aim at searching for excited charmonium states in these processes . from the obtained results we make the following observations : the calculated relative production rates for @xmath173 , @xmath174 are roughly compatible with the new belle measurements @xcite ( see also @xcite ) , assuming the decay branching ratios into charged tracks are comparable for @xmath175 and @xmath176 . the calculated relative production rates for @xmath177 and @xmath178 are large , and these two states may be observable in the mass range @xmath179 gev and @xmath180 gev respectively ( see , e.g. @xcite ) . both of them are above the ozi ( okubo - zweig - iizuka ) allowed thresholds , but unlike the @xmath181 , the @xmath12 can not decay to @xmath182 pair , which may distinguish between these two states experimentally . the calculated relative production rate for @xmath183 is not zero . this differs from the result given in @xcite , but agrees with @xcite . hence the @xmath7 might be observable via this channel with high statistics in the future . aside from this process , @xmath184 is also very hopeful in finding the @xmath7 meson , since the @xmath27 has a large branching ratio ( larger than 30% ) decaying into @xmath185 , and the @xmath26 can be easily detected by the @xmath186 signal . moreover , the calculated cross section for @xmath187 is about 1.0 fb , not very small and much larger than that for @xmath188 and @xmath189 production rates . we show the differential cross sections ( the angular distribution functions ) for different double charmonium production processes in table i ( not including qed contribution ) and table ii ( including both qcd and qed contributions ) , and the corresponding graphs in fig . 1 - 7 , and fig . 8 . as a whole , our results agree with those in @xcite and @xcite ( also agree with our previous result for @xmath190@xcite ) , if using the same parameters . for the qed part , in @xcite only two dominant diagrams are taken into account , while in this paper all diagrams are considered , but the numerical contributions of the remaining four diagrams are small . however , there still exists a difference in the result for the @xmath191 production . as for numerical results , since we use a larger value of the strong coupling constant @xmath192 ( corresponding to @xmath193 and @xmath194=338 mev ) than that in @xcite , our predicted cross sections are in general larger than that in @xcite . moreover , we use the charmonium wave functions and their derivatives at the origin ( including the ground state and excited states ) from the bt potential model calculation@xcite but not from the experimental values of leptonic decay widths , etc . , as in @xcite , and this may further enlarge our predicted cross sections . these parameters may not be the best choice , but , at present , since we do not have enough available data for higher excited charmonium states , using the potential model calculation may still be a reasonable and tentative choice . we view these as theoretical uncertainties in our approach for the leading order calculations . as emphasized above , this paper aims at searching for excited charmonium states in @xmath0 annihilation at @xmath1 gev with @xmath2 factories . after the main part of the results were presented in @xcite ( with a more complete leading order calculation including both qcd and qed contributions for various double charmonium production processes being added in its present form ) , a number of new experimental and theoretical results have appeared recently . the double charmonium production in @xmath0 annihilation at b factories has been confirmed by the babar collaboration ( see ref.@xcite ) with comparable cross sections to that observed by belle . theoretically , the large gap between experiment and theory could be largely narrowed by the next to leading order qcd radiative corrections @xcite and relativistic corrections ( see , refs.@xcite and references therein ) in the framework of nonrelativistic qcd , and other possible approaches ( see , e.g.@xcite ) . belle has observed the x(3940 ) , a new charmonium state or charmonium - like state , in @xmath195 with @xmath196@xcite . while its main decay mode is @xmath197 , no signal is found for @xmath198 . this rules out the possibility of x(3940 ) being the @xmath181 state . furthermore , the x(3940 ) is unlikely to be the @xmath199 state , since no signal for the @xmath200 is found . this is in line with our calculation , which shows that the cross section for @xmath201 is much smaller than @xmath202 . finally , the x(3940 ) could be the @xmath12 state . according to our calculation ( see eq.(13 ) ) , the production cross sections for @xmath203 are respectively 5.5 , 3.6 , 3.1 fb , and the relative rates are roughly consistent with the signal yields @xmath204 events for @xmath205@xcite . this might be viewed as a support to interpreting the x(3940 ) as the @xmath12 . however , the remaining problem is how to understand its low mass of x(3940 ) if it is the @xmath12 , which is lower than potential model predictions by 50 - 120 mev . but this could be explained by the coupled channel effects that the coupling of @xmath12 to the @xmath206 and @xmath207 charmed meson pair ( in s - wave ) will lower the mass of @xmath12@xcite . belle has not observed the x(3872 ) in @xmath208 . this implies that the x(3872 ) is unlikely to be a conventional @xmath209 or @xmath210 charmonium , since in our calculation they may have relatively large rates to be observed in double charmonium production . on the other hand , a @xmath211 or @xmath212 charmonium could be possible for x(3872 ) , since they have relatively small production rates . of course , the nature of x(3872 ) needs clarifying by other more relevant experiments ( see , e.g. @xcite ) , aside from the @xmath0 annihilation processes . belle has very recently observed a new state , the x(4160 ) , in the process of double charm production @xmath213 followed by @xmath214@xcite . possible interpretations for the x(4160 ) are discussed in@xcite with emphasized possible assignments of @xmath215 charmonium states and related problems . in conclusion , we find that the double charmonium production processes in @xmath0 annihilation at @xmath2 factories are very useful tools in searching for excited charmonium states or charmonium - like states . the leading order nrqcd calculation might hopefully provide a useful guide for the relative production rates , but not the absolute rates themselves . a systematical study for the qcd radiative corrections and relativistic corrections for different processes are apparently needed . on the other hand , studies of charmonium spectroscopy including charmonium masses , decays , and coupled channel effects , and the new type charmonium - like states are also very desirable . we thank p. pakhlov for discussions on the belle data and possible implications , and e. braaten and j. lee for communications on some calculations . this work was supported in part by the national natural science foundation of china ( no . 10421503 , no . 10675003 ) , the key grant project of chinese ministry of education ( no . 305001 ) , and the research found for doctorial program of higher education of china . belle collaboration , k. abe _ et al . _ , phys . lett . * 89 * , 142001 ( 2002 ) . belle collaboration , k. abe _ et al . _ , hep - ex/0407009 , phys . d70 , 071102 ( 2004 ) . e. braaten and j. lee , phys . rev . * d67 * , 054007 ( 2003 ) . b. aubert _ et al _ , the babar collaboration , phys.rev . * d*72 , 031101(2005 ) . y.j . zhang , y.j . gao and k.t . chao , phys . 96 , 092001(2006 ) ; see also y.j . zhang and k.t . chao , phys . lett.98 , 092003 ( 2007 ) . j.p . ma and z.g . si , arxiv : hep - ph/0608221 ; v.v . braguta , a.k . likhoded , and a.v . luchinsky , phys . d72 , 074019 ( 2005 ) ; arxiv : hep - ph/0602047 ; arxiv : hep - ph/0611021 ; a.e . bondar and v.l . chernyak , phys . b612 , 215 ( 2005 ) ; d. ebert and a.p . martynenko , phys . d74 , 054008 ( 2006 ) ; h .- choi and c .- r . ji , arxiv : 0707.1173 [ hep - ph ] .

we suggest searching for excited charmonium states in @xmath0 annihilation via double charmonium production at @xmath1 gev with @xmath2 factories , based on a more complete leading order calculation including both qcd and qed contributions for various processes . in particular , for the c=+ states , the @xmath3 ( n=2,3 ) and @xmath4 ( m=3,4 ) may have appreciable potentials to be observed ; while for the c=- states , the @xmath5 production and especially the @xmath6 production might provide opportunities for observing the @xmath7 with higher statistics in the future . a brief discussion for the x(3940 ) observed in the double charmonium production is included .

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Proceed to summarize the following text: the first neutron star was discovered by bell and hewish in 1967 . since then , the neutron star has spawned the topics of research in numerous studies in astronomy and nuclear physics . a neutron star can be treated as a huge nucleus composed of neutrons , protons , and leptons @xcite . the properties of neutron stars , like mass and radii , can be precisely measured with the gradual development of astronomical observation technology . this development has promoted theoretical studies on the nuclear many - body theory @xcite , especially on the properties of nuclear matter at high densities , where high density refers to densities above 3 times the nuclear saturation density in the core of a neutron star . in the tolman - oppenheimer - volkoff ( tov ) equation @xcite , which is derived by solving the einstein equations for a spherically symmetric and time invariant system , the properties of a neutron star are fixed by the equation of state ( eos ) of neutron star matter , which is a charge neutral system in @xmath1 equilibrium . the eoss of neutron star matter have been widely studied using various nuclear many - body theories , by the application of density functional theory ( dft ) and _ ab initio _ methods in nuclear physics @xcite . the effective nucleon - nucleon ( @xmath0 ) interactions are adopted in dft , which are determined by fitting the empirical saturation properties of nuclear matter and the ground states of finite nuclei . there are usually two schemes used in dft , i.e. , relativistic and non - relativistic approximation . the most popular nuclear many - body theory in non - relativistic dft is the skyrme - hartree - fock theory @xcite , which was constructed based on the point - coupling @xmath0 interaction . in the relativistic version , covariant dft was proposed based on the one - boson - exchange picture @xcite . the eoss of dft around the saturation density region can be constrained very well and are kept consistent within different effective @xmath0 interactions . however , their behaviors in the high density region , especially in the cores of neutron stars , show variability , which generate widely different predictions of the properties of neutron stars . with the discoveries of massive neutron stars , whose masses are around @xmath4 , many dft interactions are eliminated @xcite . on the other hand , in the state - of - the - art nuclear many - body methods , via _ ab initio _ calculations , a realistic @xmath0 interaction is used that is obtained by reproducing the @xmath0 scattering data . the most popular realistic @xmath0 potentials are constructed by the meson - exchange picture @xcite and chiral effective field theory @xcite . the @xmath0 potentials , based on chiral low - momentum expansions , are soft cores at short distances due to pion exchange ; whereas , in the meson - exchange potential , such as the bonn potential , the @xmath5 and @xmath6 meson exchanges are essential in the core region of the @xmath0 interaction , which generate very strong repulsive core . there are also other high - momentum effects at short distance , such as the strong tensor force , besides strong repulsion @xcite which is related with the nucleon inner structure @xcite in the realistic @xmath0 interaction . since 1950 , many microscopic many - body methods have been proposed to take such high momentum contributions into account . _ used the hyper - netted chain - summation technique in the non - relativistic variational method @xcite and the av18 potential @xcite to calculate the eos of neutron rich matter @xcite with a three - body force . krastev and sammarruca adopted the relativistic brueckner - hartree - fock ( rbhf ) @xcite theory to study the properties of neutron stars with bonn potentials @xcite . et al . _ also compared their results with the calculations made by akmal using the auxiliary field diffusion monte carlo method @xcite . discussed the properties of neutron star with @xmath0 interactions from chiral effective filed theory and constrained the neutron star mass - radius relation in the framework of renormalization group theory @xcite . all of these values of eoss obtained from _ ab initio _ calculations agree well with each other in the high density region and generate similar properties of massive neutron stars . for neutron rich matter , the tensor effect in realistic @xmath0 interaction is suppressed largely in the @xmath7 isospin channel @xcite . therefore , it is enough to explicitly consider only the short - range correlation in neutron rich matter . based on this motivation , a relativistic central variational ( rcv ) method was proposed by including a central correlation jastrow function @xcite , which was inspired by the series works by panda _ et al . _ @xcite . in the rcv method , the solutions of the relativistic hartree - fock ( rhf ) method @xcite were chosen as the trial wave function of the total system . the saturation properties obtained by the rcv method are @xmath8 different from the empirical data , likely because of the lack of tensor correlation in symmetric nuclear matter . however , the eos of pure neutron matter within the bonn potentials can reproduce the results from rbhf theory very well . the solutions of the rhf method in nuclear matter are constructed by spinors with plane wave functions , the summation of spin and isospin for which can be easily and systematically evaluated via the feynman trace technology . therefore , the calculation in the rcv method for neutron - rich matter is essential and economical , when compared with other time - consuming _ ab initio _ methods such as non - relativistic variational methods and the rbhf theory in asymmetric nuclear matter . furthermore , the calculation procedures of the _ ab initio _ aforementioned methods , when they were applied to neutron star matter in @xmath1 equilibrium , are complicated , therefore , the equations of @xmath1 equilibrium are usually solved with various approximations instead of self - consistence treatments @xcite . hence , the intention of this work is to apply the rvc method to calculate the eoss of neutron star matter in @xmath1 equilibrium self - consistently and use these eoss to study the properties of neutron stars with realistic @xmath0 interaction , bonn potentials . in addition , we compare our results with the calculations of other _ ab initio _ methods and discuss the role of central correlation in neutron stars . in sect . 2 , we give a basic theoretical formulation of the rcv method . in sect . 3 , numerical results are presented on the properties of neutron stars and are compared with other calculations . in sect . 4 , we give a summary of the present work . we firstly show the hamiltonian of the bonn potential @xcite , which is constructed based on the one - boson - exchange potential . it is defined as a sum of one - particle amplitudes of six bosons , consisting of @xmath9 and @xmath10 pseudovectors , @xmath11 and @xmath12 scalars , and @xmath6 and @xmath5 vector mesons , h&=&^a_it_i+^a_i , jv_i , j&=&d^3(x)(-i+m_n)(x)&&+_i=,,,,,d^3d^3(x)(x)(x)(x ) , where @xmath13 corresponds the nucleon field . @xmath14 is the transfer momentum between two nucleons and @xmath15 are the masses of nucleons and mesons , respectively . the @xmath16 matrices represent the vertex between nucleons and mesons , which are depicted below : & & _ ( 1,2)=-g^2_,&&_(1,2)=-g^2__1_2,&&_(1,2)=-()^2(/q_5)_1(/q_5)_2,&&_(1,2)=-()^2(/q_5)_1(/q_5)_2_1_2,&&_(1,2)=g^2__(1)^(2),&&^v_(1,2)=g^2__(1)^(2)_1_2,&&^t_(1,2)=()^2q_^(1)q^_(2)_1_2,&&^vt_(1,2)=i()_(2)^q_(1)_1_2 , where @xmath17 $ ] is an antisymmetric tensor gamma matrix , and the tensor coupling part between @xmath5 meson and nucleon has been neglected because the value of @xmath18 is negligible . meanwhile , a monopole form factor should be considered , f_i(q^2)= , for each meson - nucleon vertex denoted by @xmath19 . all coupling constants and cut - off momenta @xmath20 were determined by fitting the @xmath0 scattering data . in the rcv method , one introduces a central correlation function of the wave function of the rhf theory as the trial wave function of the nuclear matter system @xcite , to treat the strong short - range repulsion , |=f| , where @xmath21 is the rhf wave function , and the correlation factor @xmath22 is chosen to be a product of two - body correlation functions @xmath23 , f=^a_i < jf(r_ij ) . here , @xmath23 is jastrow correlation function @xcite . the total energy density with the correlation function is obtained as , _ c&=&=||&=&t+t_c+v . where the explicit form of the correlated hamiltonian @xmath24 appears as & = & ^a_it_i+^a_i , j_ij&=&^a_it_i+^a_i , j\{f^(r_ij)[t_i+t_j+v_ij]f(r_ij)-(t_i+t_j)}. furthermore , the kinetic energy density part is not only related with the original one - body kinetic operator @xmath25 , t=^k_f_0p^2dp[pp+m_nm ] , but also the two - body operator with the jastrow correlation function , @xmath26 , t_c&=&c_b^k_f_0p^2dp[pp+m_nm]&-&^k_f_0p^2dpp^2dp\{[pp(p)+2m_n m(p)]i(p , p)+pp(p)j(p,p ) } , where the first and second terms correspond to the direct ( hartree ) and exchange ( fock ) contributions of @xmath26 , respectively and @xmath27 $ ] . the isospin degeneracy is @xmath28 for symmetric nuclear matter and @xmath29 for pure neutron matter . @xmath30 and @xmath31 are defined as p(p)= , m(p)=. the potential energy density @xmath32 will be split into the direct term @xmath33 and exchange term @xmath34 . the pseudo - vector mesons do not provide their contributions in the hartree approximation . we have the result of @xmath33 as , v_d&=&-(^p_s+^p_s)^2-(^p_s-^n_s)^2&&+(^p_b+^n_b)^2+(^p_b-^n_b)^2 , where @xmath35 and @xmath36 are the scalar and nucleon vector densities , respectively , given as , ^p(n)_s&=&^k^p(n)_f_0p^2dpm(p),^p(n)_b&=&^k^p(n)_f_0p^2dp . the proton and neutron cases are distinguished by the superscript @xmath37 . the exchange contribution of @xmath11 meson @xmath38 as an example is given by , v^_e=^k_f_0pdppdp . here , the moment dependent functions @xmath39 and @xmath40 are various angular integrals related with the meson - exchange potentials , which are listed in the appendix of ref . the contributions of other mesons can be expressed in similar forms . finally , the total energy density is written as , _ c = t+t_c+v_d+_iv^i_e . usually , several free parameters , @xmath41 , will appear in the jastrow function @xmath42 . in the present work , the correlation function is chosen as , f(r)=1-(c_0+c_1r+c_2r^2+c_3r^3)e^-c_4r . where the exponential term makes @xmath42 unity at large distance . a natural choice from the unitary property of the correlation function is a normalization constraint on @xmath23 , [ nc ] d^3 r_ij[f^2(r_ij)-1]=0 , and should go to zero for small @xmath43 because of the repulsive core of @xmath0 interaction , which lead to @xmath44 . furthermore , we should also ensure it is a monotonously increasing property at short distance , [ gcon ] f(0)0 . we can then calculate the binding energy of nuclear matter after determining the remaining parameters with the variational principle . the minimal value of the total energy should appear at @xmath45 and @xmath46 with the constraint ( [ gcon ] ) to make the jastrow correlation function increase at short distance . therefore , we can obtain the relations between @xmath47 , and @xmath48 , c_1=c_4 , c_2=. now , there is only one parameter , @xmath48 , in the actual calculation , because @xmath49 is fixed by the normalization condition of the jastrow correlation function , eq . ( [ nc ] ) . we would like to determine @xmath48 by the variational principle with the energy density , = 0 . more detailed formulas of the rcv method can be found in ref . @xcite . for neutron star matter , there are not only the nucleons , but also the leptons , like electrons and muons . all of them exist in the neutron star with the equilibrium conditions of the chemical potentials for the @xmath1 decay , [ bteq ] _ n&=&_p+_e,_&=&_e , where the chemical potentials @xmath50 , and @xmath51 are determined by the relativistic energy - momentum relation at the fermi momentum @xmath52 , _ i&=&^i_0(k_f)+e^_i(k_f),_&= & , where , @xmath53 and @xmath54 . @xmath55 is the zero component of the self - energy of proton or neutron . furthermore , the nucleon density conservation and charge neutrality are imposed in neutron star matter as , & & = _ n+_p,&&_e+_=_p . the pressure of the neutron star system can be obtained with thermodynamics relation , as p()=^2=_i = n , p , e,_i_i- . the stable configurations of a neutron star then can be obtained from the well known hydrostatic equilibrium equations , by tolman , oppenheimer and volkoff @xcite for the pressure @xmath56 and the enclosed mass @xmath57 , [ tov ] & = & -,&=&4r^2(r ) , where , @xmath58 is the pressure of neutron star at radius , @xmath59 , and @xmath60 is the total star mass inside a sphere of radius @xmath59 . once the eos @xmath61 is specified , @xmath62 being the total energy density ( @xmath63 is the gravitational constant ) , for a chosen central value of the energy density , the numerical integration of eq.([tov ] ) provides the mass - radius relation of neutron star . after solving the equilibrium conditions of chemical potentials for various particles in the neutron star matter , eq . ( [ bteq ] ) , the binding energies per nucleon as functions of density are shown in the left panel of fig . [ re ] with three bonn potentials , bonn a , bonn b , and bonn c , which were fitted by the scattering data of @xmath0 system . the binding energy of the bonn a potential is the smallest of the three bonn potentials , which is in accordance with the conclusion of rbhf theory for symmetric nuclear matter @xcite . the tensor component of bonn a among these three potentials is the weakest , whose @xmath64 state probability , @xmath65 , is the smallest in deuteron . furthermore , the difference of binding energies of these three potentials in neutron star matter is obviously less than that in the symmetric nuclear matter . the effect of the tensor force becomes weaker with increasing neutron fraction and does not play any role in pure neutron matter . similarly , the pressure as a function of energy density for neutron star matter is required as input data , when we calculate the properties of neutron stars in the tov equation . the pressure - energy relations with bonn potentials are given in the right panel of fig . their behaviors are very similar to those of the binding energy and are almost identical with each other . it demonstrates that these eoss will generate similar properties of the neutron stars . the central correlation on kinetic energy per nucleon is presented for the neutron star matter in fig . the variational method can be achieved based on the competition between the correlation on kinetic energy and potential . the central correlation on kinetic energy provides a repulsive effect to prevent two - nucleon approach at high densities , whereas the one on potential gives an attractive contribution to remove the repulsion of realistic @xmath0 interactions in short - range region . finally , the correlations on kinetic and potential energies determine the minimum total energy and explicitly confirm the variational parameters in the jastrow function . it can be found that the correlation on kinetic energy contributes almost half of the binding energy per nucleon for neutron star matter and plays a very essential role in the rcv method . in the rcv method there is only one independent variational parameter , @xmath48 @xcite , which is shown in fig . [ c4 ] as a function of density with bonn a , bonn b , and bonn c potentials . from this figure , we can see that the central correlation strength increases slowly at low density , reaches a maximum value around the normal nuclear saturation density , and starts to decrease with increasing density thereafter . it demonstrates that the central correlation will have to consider more variables to generate the saturation density and that it becomes weaker at high density , since the distance between two nucleons is already sufficiently compressed . as a function of density for neutron star matter with bonn a , bonn b , and bonn c potentials.,width=340 ] with the eoss from the rcv method , we obtain the properties of neutron stars by solving the tov equation with bonn potentials . the mass - radius and mass - density relations of neutron stars within our present framework are illustrated in fig . [ rm ] . in the left panel , we plot the mass - radius relations of neutron stars with bonn a , bonn b , and bonn c potentials . the maximum masses and corresponding radii predicted are almost the same , around @xmath2 and @xmath3 km , respectively . these results are in good agreement with the previous _ ab initio _ calculations . the maximum masses of neutron stars are @xmath66 in non - relativistic variational method by akmal _ _ @xcite . similarly , the maximum masses and corresponding radii of neutron stars were given as @xmath67 and @xmath68 km in the rbhf theory with bonn potentials @xcite . it indicates that our rcv method can economically describe the neutron star matter as well as both the non - relativistic full variational method and the rbhf theory . the maximum masses and the mass - radius relations of neutron stars with three bonn potentials have obvious distinctions , since the neutron star matter includes not only neutrons but also protons . the tensor force should contribute to some effects at the @xmath69 channel here , especially at low density , which is shown clearly in the mass - radius curves at large @xmath70 . in the right panel of fig . [ rm ] , we plot the mass - density relations of neutron stars with bonn potentials . the central densities at maximum neutron star mass are @xmath71 @xmath72 for all of three bonn potentials , which are similar to the calculations of akmal _ et al . _ @xcite , @xmath73 @xmath72 and rbhf theory , @xmath74 @xmath72 @xcite . in the low density region , the masses of neutron stars display distinguishable behaviors . they are almost identical with increasing densities . in table [ pro ] , the properties of neutron stars within the present framework are listed for bonn a , bonn b , and bonn c potentials . they are compared with the ones taken from the rbhf theory @xcite . no matter the maximum mass or the corresponding radii and central density , our results are only @xmath75 different from the ones calculated via the rbhf theory . furthermore , the maximum masses of neutron stars in our calculation satisfy the requirements of recent observation on massive neutron stars @xcite , around @xmath4 . the corresponding radius is also located within the constraint region worked out by hebeler _ et al . _ @xcite . .the properties of neutron stars ( maximum mass , corresponding radii and central density ) within present framework and compared with the ones calculated by rbhf theory for bonn a , bonn b , and bonn c potentials . [ cols="^,^,^,^,^,^,^ " , ] the fractions of various particles appearing in neutron stars , which are neutrons , protons , electrons , and muons are plotted in fig . [ ypabc ] with the three bonn potentials . at the beginning , the muon is absent for the @xmath1 equilibrium conditions . when the electron chemical potential is larger than the muon mass , the muon will appear in the neutron star matter at densities less than @xmath76 @xmath72 , which is smaller than the density of muons in the rbhf theory . the earliest appearance of the muon in the bonn a potential around the normal saturation density . at high density , the fraction of muons will approach that of electrons . the proton fraction in bonn a has the largest magnitude compared to the other two bonn potentials , which should be due to its smaller tensor component . we give the proton fractions with bonn potentials alone in fig . [ urca ] . from this figure , the difference of proton fractions obtained from the three bonn potentials is clearly revealed , which becomes very obviously in the intermediate density region and decreases in the high density region . the proton fraction in our calculation continues to increase with density and is in contrast to the case in rbhf theory , where it decreases at high density and the largest fraction is about @xmath77 @xcite . in rbhf theory , to simplify the calculation , the leptons were treated with the non - relativistic approximation . at high densities , the relativistic effect becomes more important , where the fermi momentum is very high and can be comparable with the light speed . therefore , the direct urca processes related with the cooling mechanism of neutron star could not occur easily with bonn b and bonn c potentials in rbhf theory , which should be satisfied by a proton fraction larger than approximately @xmath78 . however , in our work , the direct urca processes can be produced in all of the bonn potentials and the densities appearing in direct urca processes are located between @xmath79 @xmath72 . these densities will lead the neutron star to cool very rapidly . we also give the dirac effective nucleon masses in neutron star matter in fig . [ emn ] . for proton or neutron dirac effective masses reflecting the nucleon media effect , their magnitudes with different potential are almost equal , which is very similar situation to the energy and pressure cases . if we compare the proton effective mass with the neutron effective mass , it can be found that the proton effective mass is larger than the neutron one at low density . when the density is @xmath80 @xmath72 , the splitting of proton and neutron effective masses is reversed . this behavior is quite different from the effective mass in rhf theory , where the proton effective mass should be larger than the neutron one in all density regions @xcite . for rbhf theory , the splitting of proton and neutron effective masses for asymmetric nuclear matter is strongly dependent on the treatment of the @xmath81matrix . brockmann and machleidt used the single - particle potential to extract the effective nucleon masses , where the neutron effective mass is larger than the proton one in neutron - rich matter @xcite ; whereas dalen _ et al . _ adopted the projection method to distinguish the spin components in the @xmath81matrix and calculated the effective nucleon mass with these interaction components in the rhf model @xcite . in this scheme , the proton effective mass is larger than the neutron one . to discuss the splitting of proton - neutron effective masses in our framework , we show the kinetic and potential contributions of the nucleon effective mass in a neutron star with bonn a potential in fig . [ emc ] . without the central correlation function , the kinetic energy is a one - body operator , which does not provide any contribution to the nucleon effective mass . once the central correlation function is included , the central correlation on kinetic energy becomes a two - body operator and contributes to the nucleon effective mass . from fig . [ emc ] , we can find that the potential generates the negative contribution to the effective nucleon mass . in neutron - rich matter , the neutron effective mass will obtain more negative components compared to the proton one . therefore , in the rhf model , the proton effective mass is larger than the neutron one in neutron - rich matter . meanwhile , the central correlation on kinetic energy has the positive contributions to effective nucleon mass . furthermore , its effect on neutrons is much higher than on protons at high densities . with the competition between the potential and kinetic energy , the proton effective mass is larger than the neutron one at low density and with increasing density , the neutron effective mass is larger than the proton one . therefore , the central correlation on kinetic energy plays a very important role in the splitting of proton - neutron effective masses . for dalen _ et al . _ , they obtained the effective nucleon mass from the potential part ; whereas brockmann and machleidt considered the splitting of effective masses from the single particle potential , which is related to kinetic energy . this may be the reason why there are opposite conclusions between the two groups , in rbhf theory on effective nucleon mass . the rcv method based on the framework of rhf theory was applied to study the properties of neutron stars with one - boson - exchange potentials , i.e. , bonn potentials , which were determined by fitting the nucleon - nucleon ( @xmath0 ) scattering data . in neutron - rich matter , the tensor force has a very small effect of its isospin feature . therefore , it is essential to take the central correlation on the strong repulsion of @xmath0 interaction at short distances for the description of neutron - rich matter . the equation of state ( eos ) of pure neutron matter obtained by a novel _ ab initio _ calculation of rbhf theory was completely reproduced by the present rcv method . the eoss of neutron star matter in @xmath1 equilibrium with nucleons and leptons , were self - consistently solved in the rcv method with bonn a , bonn b , and bonn c potentials . their behaviors were almost identical for the weak effect of tensor force in neutron - rich matter , since the difference among three bonn potentials only appeared in their tensor components . the relativistic central correlation on kinetic energy played a very important role in the process of minimizing the total binding energy , with the variational principle , and gave half of the contribution to total binding energy . furthermore , we found that the strength of the central correlation function was the strongest at the saturation density through the variational parameters , which correspond to the saturation mechanism of symmetric nuclear matter . the properties of neutron stars were studied with the eoss of neutron star matter by solving the tov equation . the maximum neutron star masses and corresponding radii were around @xmath2 and @xmath3 km , respectively , using the rcv method with bonn potentials . the central densities of neutron stars were about @xmath71 @xmath72 . these results are in good agreement with the calculations from rbhf theory and the non - relativistic variational method . it demonstrated that the rcv method can describe neutron - rich matter reasonably and economically , compared with the conventional _ ab initio _ calculation . the proton fractions in neutron star matter with the three bonn potentials showed some differences . the proton fraction in bonn a potential was the largest and bonn c the smallest . the direct urca processes would be generated in all of these potentials above the densities , @xmath79 @xmath72 with a proton fraction larger than approximately @xmath78 . furthermore , the splitting of proton - neutron effective masses was reversed with increasing density of neutron star matter . in the low density region , the proton effective mass was larger than the neutron one ; 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the properties of neutron stars are investigated within the relativistic central variational method by using a realistic nucleon - nucleon ( @xmath0 ) interaction . the strong repulsion of realistic @xmath0 interactions at short distances is treated by a jastrow central correlation function , whose form is completely determined through minimization of the total energy of the nuclear many - body system . the relativistic hartree - fock wave functions are chosen as the trial wave function . in this framework , the equation of state of the neutron star matter in @xmath1 equilibrium is obtained self - consistently . we further determine the properties of neutron stars via the tolman - oppenheimer - volkoff equation using bonn a , b , and c potentials . the maximum masses of neutron stars with these realistic potentials are around @xmath2 and their corresponding radii are around @xmath3 km . these results are in accordance with the calculations of the relativistic brueckner - hartree - fock theory with the same potentials . furthermore , we also find that the splitting of proton - neutron effective masses will be reversed at high density in the neutron star matter , which are caused by the contribution of short - range correlation on kinetic energy .

You are an expert at summarizing long articles. Proceed to summarize the following text: the @xmath0-invariance of the maxwell equation was discovered by cunningham and bateman a century ago . however in order to quantize the maxwell field and due to gauge freedom , a gauge fixing condition is necessary . the lorenz gauge is usually used , which breaks the @xmath0 invariance . nonetheless since such a symmetry mights apear to lack physical meaning , its breaking does not bother many people @xcite . the purpose of the present paper is to demonstrate the benefits of keeping this fundamental symmetry when quantizing the maxwell field in conformally flat spaces ( cfs ) . the starting point is the following . a classical @xmath0-invariant field can not , at least locally , distinguish between two cfss @xcite . so why not maintain the @xmath0-invariance during the quantization process in a cfs ? doing so , a free field living in a cfs might behave like in a flat space and the corresponding wightman two - point functions can be related to their minkowskian counterparts . the work @xcite confirms this assertion in the special case of maxwell field in de sitter space . indeed , a new and simple two - point wightman function @xmath1 was found and which has the same physical ( gauge independent ) content as the two - point function of allen and jacobson @xcite . this is because the faraday propagator @xmath2 is the same . the present work extends to general cfss and clarify the quantum structure of the formalism developed in @xcite . we use dirac s six - cone formalism and realize all cfss as intersections of the null cone and a given surface in a six - dimensional lorentzian space . the introduction of auxiliary fields and the use of the gubta - bleuler quantization scheme are necessary to deal with gauge freedom of the maxwell field . another important ingredient is the use of a well - suited coordinate system . this allows to @xmath0-invariant cfs formulas to get a minkowskian form . the main result is a set of wightman two - point functions for maxwell and auxiliary fields . this paper is organized as follows . [ geom ] sets the coordinates systems and the geometrical construction of cfss . [ fields ] defines the fields and gives their dynamical equations . in sec . [ quantum - field ] , the dynamical system is solved , the quantum field is explicitly constructed and the two - point functions are written down . some technical details are given in appdx . [ details ] . the infinitesimal @xmath0 action on the fields @xmath3 is expanded in appdx . [ action ] and their @xmath0-invariant scalar product is given in appdx . the six - dimensional lorentzian space @xmath4 is provided with the natural orthogonal coordinates @xmath5 and equipped with the metric @xmath6 . quantities related to @xmath4 and its null cone @xmath7 are labeled with a tilde . we define a second coordinate system @xmath8 , @xmath9 where the four components @xmath10 is the so - called polyspherical coordinate system @xcite and @xmath11 . a straightforward calculation yields @xmath12 which means that the component @xmath13 carries alone the homogeneity of the @xmath14 s . using the system @xmath15 , the null cone reads @xmath16 a five - dimensional surface in @xmath4 is defined through @xmath17 where the real and smooth function @xmath18 depends only on @xmath10 and @xmath19 and is then homogeneous of degree @xmath20 . the intersection of @xmath21 and @xmath22 is a four - dimensional space @xmath23 where the index @xmath24 in @xmath25 refers to @xmath26 ) . regarding to its metric inherited from @xmath27 , precisely @xmath28 @xmath25 turns out to be a cfs . a smooth move of the surface @xmath29 , which corresponds to changing the function @xmath30 , amounts to perform a weyl rescaling . this locally relates all cfss and permits to go from one to another . note that for @xmath31 , @xmath25 reduces to minkowski space @xmath32 and accordingly the four components system @xmath10 yields the usual cartesian system . the gradients @xmath33 are extensivelly used in this article . the function @xmath30 does not depend on @xmath13 and thus @xmath34 . the choice of the function @xmath30 , including its @xmath19 dependence has to be done in such a way to ensure the invariance ( in @xmath4 ) of the surface @xmath22 under the action of the isometry group associated to the desired @xmath25 four - dimensional space . since the @xmath4 null - cone is @xmath0-invariant , the resulting @xmath25 will be invariant under its isometry group . let us consider an example : @xmath35^{-1}$ ] , where @xmath36 is a constant . the associated surface @xmath22 and thus the corresponding @xmath25 are left invariant under the action of de sitter group @xcite . also , @xmath25 is a de sitter space . in this section , we explain how to obtain the @xmath0-invariant maxwell field in @xmath25 from a six - dimensional one - form . following dirac @xcite , we consider a one - form @xmath37 defined in @xmath4 homogeneous of degree @xmath38 and which decomposes on the @xmath39 basis as @xmath40 the components @xmath41 are homogeneous of degree @xmath42 and obey to the equation @xmath43 this equation is naturally invariant under the @xmath0 action since this group has a linear action when acting in @xmath4 . we then decompose the one - form @xmath37 on the basis @xmath44 corresponding to the system @xmath45 ( [ coord+muc ] ) , with a slight but capital modification on the @xmath46 component . there are two ways , the first decomposition reads @xmath47 the second is given by @xmath48 now , identifying ( [ eq-1 ] ) with ( [ eq-2 ] ) , one obtains the relation between the fields @xmath49 and @xmath50 through @xmath51 all the fields @xmath52 and @xmath53 are by construction homogeneous of degree @xmath54 . as a consequence , @xmath55 and @xmath56 this amounts to project the fields @xmath52 on @xmath57 and @xmath53 on @xmath58 . then projecting the fields on the null cone @xmath21 yields @xmath59 thus @xmath60 and @xmath61 are respectively @xmath25 and minkowski fields . though in a slightly different maner , this relation was obtained in @xcite in the particular case of de sitter space and was called the `` extended weyl transformation '' . the fields @xmath62 and @xmath63 are auxiliary fields and the field @xmath64 is , up to the condition @xmath65 the maxwell field . this will become clear here after . let us now turn to the dynamical equations . our strategy is to transport minkowskian @xmath0-invariant equations to get @xmath0-invariant equations in the @xmath25 space . the first step is thus to write down the minkowskian equations which are obtained using the equation ( [ equation - a ] ) and the relation ( [ a(a)-m ] ) . this system reads @xmath66 the corresponding system in @xmath25 is obtained using ( [ extendedweyl - bis ] ) , @xmath67 where all contractions are performed using @xmath68 even though we are in the curved space @xmath25 . the field @xmath69 obeying to the system above is not yet the maxwell one . nevertheless , the constraint @xmath70 simplifies the system ( [ syst1m - h ] ) and leads to @xmath71 despite their minkowskian form , these equations are the maxwell equation and a conformal gauge condition on any conformally flat space . this is due to the use of the polyspherical coordinate system ( [ coord+muc ] ) , which makes apparent the flatness feature of the @xmath25 spaces . the constraint @xmath72 reduces the extended weyl transformation ( [ extendedweyl - bis ] ) into the identity @xmath73 recovering the ordinary vanishing conformal weight of the maxwell field @xmath69 . after some algebra , the covariant form of ( [ maxwellh1 ] ) takes the form @xmath74 where @xmath75 . the first line ( resp . the second one ) is the covariant maxwell ( resp . the eastwood - singer gauge @xcite ) equation in an arbitrary @xmath25 space . this conformal gauge was first derived by bayen and flato in minkowski space @xcite . its extension to curved spaces ( even cfss ) is not trivial and can be performed using adapted tools like the weyl - gauging technique @xcite or the weyl - to - riemann method @xcite . note that the system ( [ system - covariant ] ) is valid only if @xmath76 ( an @xmath0-invariant constraint ) . but the latter has to be fixed at the end of the quantization process , not at the begining . indeed , the auxiliary field @xmath77 acts as a faddeev popov ghost field and its retention during the quantization process is necessary . the constraint @xmath76 will be applied on the quantum space to select an invariant subspace of physical states and the wightman functions thus include the whole big space . this is related to the undecomposable group representation ( see appendix [ action ] ) . we now apply the gupta - bleuler quantization scheme @xcite . this can be summarized as follows . we have seen that @xmath78 is interpreted as the maxwell field in the eastwood - singer gauge ( [ system - covariant ] ) on the space @xmath79 when the constraint @xmath80 is applied . the problem is that pure gauge solutions ( @xmath81 , with @xmath82 and @xmath80 ) are orthogonal to all the solutions including themselves . as a consequence , the space of solutions is degenerate and no wightman functions can be constructed . to fix this problem , we consider the system ( [ syst1m - h ] ) , instead of ( [ system - covariant ] ) , for which @xmath83 and thus a causal reproducing kernel can be found . this means that for quantum fields @xmath84 acting on some hilbert ( or krein ) space @xmath85 , we can not impose the operator equation @xmath86 . instead , we define the subspace of physical states @xmath87 which cancels the action of @xmath88 . then the maxwell equation and the eastwood - singer gauge hold in the mean on the space @xmath89 . the task seems complicated at first sight , but thanks to the correspondence ( [ extendedweyl - bis ] ) we only need to solve the minkowskian system ( [ syst1 m ] ) , which is already done in @xcite . indeed , using the weyl equivalence between cfss , the whole structure of an @xmath90-covariant free field theory can be transported from minkowski to another cfs . in the following , we solve the dynamical equations , obtain the modes , determine the quantum field , the subspace of physical states and finally compute the two - point functions . the solutions of the minkowskian system ( [ syst1 m ] ) can be obtained from @xcite and read @xmath91 where @xmath92 are polarization vectors whose components are given by @xmath93 and verifying @xmath94 with respect to the scalar product ( [ scalarps - a ] ) . the matrix @xmath95 relates the fields @xmath96 and @xmath97 ( [ matrix - s ] ) . the scalar modes @xmath98 are solutions of the minkowskian @xmath0-invariant ( or massless ) sclalar field equation @xmath99 , @xmath100 where @xmath101 denotes the usual hyperspherical harmonics . the normalization constant @xmath102 is chosen in order to get @xmath103 with respect to the klein - gordon scalar product . as a consequence , the solutions ( [ modes - a ] ) are normalized with respect to ( [ scalarps - a ] ) , @xmath104 thus the general solution of the system ( [ syst1 m ] ) reads @xmath105 where @xmath106 are real constants . let us now turn to the modes of the system ( [ syst1m - h ] ) . they are obtained thanks to the extended weyl transformation ( [ extendedweyl - bis ] ) applied on the minkowskian modes ( [ modes - a ] ) @xmath107 these modes are normalized like ( [ norm ] ) but according to the scalar product ( [ scalarps - a - k ] ) . the general solution on @xmath79 reads @xmath108 where the @xmath109 are some real constants . note that when @xmath110 the solutions ( [ generalsolution - am ] ) and ( [ generalsolution - ak ] ) solve the maxwell equation in the eastwood - singer gauge . we can now define the quantum fields and construct the fock spaces as usual . the quantum fields corresponding to ( [ generalsolution - am ] ) and ( [ generalsolution - ak ] ) are respectively defined through @xmath111 @xmath112 where the operators @xmath113 and @xmath114 are respectively the annihilators and creators of the modes ( [ modes - a ] ) in @xmath32 and the modes ( [ modes - a - k ] ) in @xmath25 . the use of the same annihilators and creators for all cfss is highly important for our purpose . indeed , this allows to define the the same vaccuum state @xmath115 through @xmath116 for any annihilator . the one - particle states are built by applying the creators on the vacuum state @xmath117 and the multiple particle states of the fock spaces are constructed as usual . moreover , the annihilation and creation operators obey to the following algebra @xmath118 = [ \hat a_{{\scriptscriptstyle l}m ( \alpha)}^{\dag } , \hat a_{{\scriptscriptstyle l}'m ' ( \beta)}^{\dag } ] = 0 \\ & [ \hat a_{{\scriptscriptstyle l}m ( \alpha ) } , \hat a_{{\scriptscriptstyle l}'m ' ( \beta)}^{\dag } ] = -\tilde \eta_{{\scriptscriptstyle \alpha}\beta } \delta_{{\scriptscriptstyle l}l ' } \delta_{{\scriptscriptstyle m}m'}. \end{split}\ ] ] the subset of physical states in both spaces is determined thanks to the classical physical solutions ( [ generalsolution - am ] ) and ( [ generalsolution - ak ] ) verifying ( [ a+=0 ] ) . in quantum language , @xmath119 is a physical state iff @xmath120 where @xmath121 is the annihilator part of @xmath122 . this implies the equality @xmath123 for any physical states @xmath119 and @xmath124 . also , the subspace of physical states is the same in all cfss , which allows to transport physical quantities from minkowski space into the @xmath25 space . as a consequence , one obtains @xmath125 in @xmath25 and the corresponding minkowskian system in @xmath32 . the quantum fields fulfill the maxwell equation together with the eastwood - singer gauge in the mean on the physical states . we show in this part how to get the wightman two - point functions on @xmath25 from their minkowskian counterparts . the wightman functions related to the minkowskian fields @xmath126 are defined through @xmath127 their expressions are given in @xcite and read @xmath128 where @xmath129 with @xmath130 stands for the wightman two - point function related to the minkowskian massless scalar field . the wightman two - point functions related to the field @xmath131 are given by @xmath132 now , using ( [ modes - a - k ] ) , ( [ maxwelds2ptdef ] ) and ( [ wightman - k ] ) , allows to write the following capital formula @xmath133 where the @xmath134 terms read @xmath135 the wightman two - point functions ( [ dk - dm ] ) read @xmath136 where @xmath137 , @xmath138 , @xmath139 and @xmath140 . + to end this paper , let us consider an important particular case , that corresponding to de sitter space . this case is obtained by specifying @xmath141 where @xmath142 is related to the de sitterian ricci scalar through @xmath143 . the gradients ( [ upsilon ] ) read @xmath144 in this case we obtain simple expressions for the two - point functions related to the fields @xmath145 on de sitter space . the three more relevant yield @xmath146 where we have used ( de sitter is a maximally symmetric space ) the standard unit tangent vectors @xmath147 and @xmath148 , the parallel propagator along the geodesic @xmath149 and the usual function @xmath150 of the geodesic distance @xmath151 relating @xmath152 and @xmath153 , @xmath154 . see @xcite for a more precise statement . note that the two - point function @xmath155 has the same physical content with the allen and jacobson two - point function @xcite . an @xmath0-covariant quantization of the maxwell field in an arbitrary conformally flat space was presented . following dirac s six - cone formalism , all conformally flat spaces @xmath25 are realized as intersections of the null cone and a given surface @xmath29 . the quantum field was explicitly constructed using the gupta - bleuler canonical quantization scheme and the wightman two - point functions were given . the price to pay for this simplicity and the maintaining of the @xmath0 invariance during the whole quantization process was the introduction of two auxiliary fields @xmath156 and @xmath77 . as a consequence , the maxwell field @xmath69 does not propagate `` alone '' but together with its two auxiliary fields . the propagation must use all the wightman functions ( [ d - x - k - relevent ] ) and not only the `` purely '' maxwell one @xmath157 . nonetheless , in a recent work @xcite , we have used the functions ( [ d - ds ] ) to propagate the maxwell field generated by two charges of opposite sign placed at the two poles of a de sitter space . the calculations showed that only @xmath157 is involved , which trivialize the problem . one can consider to use the two - point functions ( [ d - x - k - relevent ] ) to propagate the electromagnetic field for some charge distribution given in other cfss , like flrw spaces for instance . one concludes that is much worth to maintain the @xmath0 symmetry during the whole quantization process when dealing with maxwell field in a conformally flat space . the problem then goes back to minkowski and the calculations become much easier . in fact , the classical and quantum structures of the free maxwell field are locally the same in all conformally flat spaces . the remained question is to know if this is true for other free fields and how to deal with @xmath0-invariant interactions ? i would like to thank m. novello , j. renaud and e. huguet for illuminating discussions and the cnpq for financial support . considering ( [ field - a ] ) and ( [ eq-1 ] ) , expressing the basis @xmath159 in terms of @xmath44 and then identifying both sides , one obtains the expression of @xmath160 in terms of @xmath161 . we find , after using the homogeneity properties , @xmath162 which reads @xmath163 this system can be inverted in @xmath164 following the same steps as above , one obtain the matrix linking the @xmath165 to the @xmath166 @xmath167 + a^{{\scriptscriptstyle k}}_4 [ \upsilon_c(1+x^2)-1 ] \\ & & \qquad \qquad \qquad + \upsilon_c a^{{\scriptscriptstyle k}}.x \biggr\ } \\ { \displaystyle a_\mu^{{\scriptscriptstyle k } } } & = & k \biggl\ { a^{{\scriptscriptstyle k}}_5 \left ( ( 1 -x^2 ) \upsilon_{\mu } - \frac{1}{2}x_{\mu } \right ) \\ & & + a^{{\scriptscriptstyle k}}_4 \left ( ( 1 + x^2 ) \upsilon_{\mu } + \frac{1}{2 } x_{\mu } \right ) + a^{{\scriptscriptstyle k}}_\nu \left ( \upsilon_{\mu } x^\nu + \delta_{\mu}^{\nu } \right ) \biggr\ } \\ { \displaystyle a_+^{{\scriptscriptstyle k } } } & = & { \displaystyle k \biggl\{a^{{\scriptscriptstyle k}}_5 ( 1- x^2 ) } + a^{{\scriptscriptstyle k}}_4 ( 1 + x^2 ) + a^{{\scriptscriptstyle k}}.x \biggr\}. \end{array } \right .\ ] ] this system can be obtained using the minkowskian system ( [ a(a)-m ] ) , the relation @xmath168 ( which comes out from the homogeneity properties of the fields ) and the extended weyl transformations ( [ extendedweyl - bis ] ) . this is inverted in @xmath169 the @xmath0 infinitesimal action on the field @xmath3 is given by commutators of the group generators and the field . first , we write down the infinitesimal transformations of the minkowskian fields @xmath96 which can be found in @xcite then we transport the resulting representation into @xmath25 . for any element @xmath170 , the related generator is denoted by @xmath171 and whose action on the field @xmath96 reads @xmath172 \\ & = x_{e } \ a_{{\scriptscriptstyle i}}^{{\scriptscriptstyle m } } + \left ( \sigma_{e } \right)_{{\scriptscriptstyle i}}^{{\scriptscriptstyle j } } \ a_{{\scriptscriptstyle j}}^{{\scriptscriptstyle m } } \end{split}\ ] ] where the first part represents the scalar action and the second the spinorial action . setting @xmath173 the minkowskian infinitessimal action reads @xmath174 } \ a^{{\scriptscriptstyle m}}_\tau \\ & \left ( x_{\mu\nu}^{{\scriptscriptstyle m } } \ a^{{\scriptscriptstyle m}}\right)_+ = x_{\mu\nu } a^{{\scriptscriptstyle m}}_+ , \end{aligned } \right.\ ] ] for the rotations , @xmath175 for the translations ; @xmath176}^\lambda + x^\lambda \eta_{\mu\nu } ) a^{{\scriptscriptstyle m}}_\lambda - 2\eta_{\mu\nu } a^{{\scriptscriptstyle m}}_+ \\ & \left(k_\mu^{{\scriptscriptstyle m } } \ a^{{\scriptscriptstyle m}}\right)_+ = k_\mu a^{{\scriptscriptstyle m}}_+ , \end{aligned } \right.\ ] ] for the special conformal transformations ( sct ) . finally , we have @xmath177 for the dilations . the undecomposable structure of the fields @xmath96 is made clear . under the @xmath0 action , the component @xmath178 overlaps @xmath179 which in turn overlaps @xmath180 . so we have the scheme @xmath181 the second step is to trasport the group action from minkowski to the @xmath79 space using the extended weyl transformation ( [ extendedweyl - bis ] ) @xmath182 a_{+}^{{\scriptscriptstyle k } } \end{split}\ ] ] where we have used @xmath183 for all @xmath184 . also only the second part of the last line has to be computed . note that the constraint @xmath189 ( [ a+=0 ] ) is @xmath0-invariant . this is important since this constraint defines the subset of physical states . the @xmath0-invariant scalar product for the minkowskian field @xmath96 reads @xmath190 where @xmath191 is some cauchy surface in @xmath32 and @xmath192 is a surface element . an important point is that this cauchy surface is common to all the spaces @xmath25 since they are all conformally equivalent @xcite . using ( [ lien - weyl ] ) and ( [ extendedweyl - bis ] ) , the scalar product for the field @xmath166 is obtained from ( [ scalarps - a ] ) and reads @xmath193 where the @xmath25 surface element is related to its minkoskian counterpart by @xmath194 .

we present an @xmath0-covariant quantization of the free electromagnetic field in conformally flat spaces ( cfs ) . a cfs is realized in a six - dimensional space as an intersection of the null cone with a given surface . the smooth move of the latter is equivalent to perform a weyl rescaling . this allows to transport the @xmath0-invariant quantum structure of the maxwell field from minkowski space to any cfs . calculations are simplified and the cfs wightman two - point functions are given in terms of their minkowskian counterparts . the difficulty due to gauge freedom is surpassed by introducing two auxiliary fields and using the gupta - bleuler quantization scheme . the quantum structure is given by a vacuum state and creators / annihilators acting on some hilbert space . in practice , only the hilbert space changes under weyl rescalings . also the quantum @xmath0-invariant free maxwell field does not distinguish between two cfss .

You are an expert at summarizing long articles. Proceed to summarize the following text: chiral symmetry together with its spontaneous breaking is one of the key ingredients in the low - energy hadron or nuclear physics . due to its spontaneous breaking , up and down quarks , whose current masses are of the order of a few mev , acquire the large constituent masses of a few hundreds mev , and are consequently responsible for about 99% of mass of the nucleon and hence that of our world . one would thus consider that chiral condensate @xmath7 , the order parameter of the chiral phase transition , plays an essential role in the hadron - mass genesis in the light quark sector . on the other hand , chiral symmetry gets restored in systems where hard external energy scales such as high - momentum transfer , temperature(@xmath8 ) , baryon density and so on exist , owing to the asymptotic freedom of qcd . then , several questions may arise : are all hadronic modes massless in such systems ? can hadrons be massive even without non - vanishing chiral condensate ? an interesting possibility was suggested some years ago by detar and kunihiro @xcite , who showed that nucleons can be _ massive even without the help of chiral condensate _ due to the possible _ chirally invariant mass terms _ , which give _ degenerated _ finite masses to the members in the chiral multiplet ( a nucleon and its parity partner ) even when chiral condensate is set to zero : in order to demonstrate this possibility for a finite-@xmath8 case , they introduced a linear sigma model which has a nontrivial chiral structure in the baryon sector and a mass - generation mechanism essentially different from that by the spontaneous chiral symmetry breaking . interestingly enough , their chiral doublet model has recently become a source of debate as a possible scenario of _ observed parity doubling in excited baryons _ @xcite , although their original work @xcite was intended for an application to finite-@xmath8 systems . it is thus an intriguing problem to clarify the chiral structure of excited baryons in the light quark sector beyond model considerations . one of the key observables which are sensitive to the chiral structure of the baryon sector is axial charges @xcite . the axial charge of a nucleon @xmath9 is encoded in the three - point function @xmath10 u.\ ] ] here , @xmath11 is the isovector axial current . the axial charge @xmath12 is defined by @xmath13 with the vanishing transferred momentum @xmath14 . it is a well - known fact that the axial charge @xmath15 of @xmath16 is 1.26 . though the axial charges in the chiral broken phase can be freely adjusted with higher - dimensional possible terms and can not be the crucial clues for the chiral structure @xcite , they would reflect the internal structure of baryons and would play an important role in the clarification of the low - energy hadron dynamics . in this paper , we show the first unquenched lattice qcd study @xcite of the axial charges @xmath0 of @xmath1 and @xmath2 , two lowest nucleon resonances in the negative parity channel . we employ @xmath17 lattice with two flavors of dynamical quarks , generated by cp - pacs collaboration @xcite with the renormalization - group improved gauge action and the mean - field improved clover quark action . we choose the gauge configurations at @xmath18 with the clover coefficient @xmath19 , whose lattice spacing @xmath20 is determined as 0.1555(17 ) fm . we perform measurements with 590 , 680 , and 680 gauge configurations with three different hopping parameters for sea and valence quarks , @xmath21 and @xmath22 , which correspond to quark masses of @xmath23 150 , 100 , 65 mev and the related @xmath24-@xmath25 mass ratios are @xmath26 , @xmath27 and @xmath28 , respectively . statistical errors are estimated by the jackknife method with the bin size of 10 configurations . our main concern is the axial charges of the negative - parity nucleon resonances @xmath1 and @xmath2 in @xmath29channel . we then have to construct an optimal operator which dominantly couples to @xmath1 or @xmath2 . we employ the following two independent nucleon fields , @xmath30 and @xmath31 in order to construct @xmath32 correlation matrices and to separate signals of @xmath1 and @xmath2 . here , @xmath33 and @xmath34 are dirac spinors for u- and d- quark , respectively , and @xmath35 denote the color indices . even after the successful signal separations , there still remain several signal contaminations : signal contaminations by _ scattering states _ and _ wraparound effects_. due to the unquenched gauge configurations , the negative parity nucleon states could decay to @xmath24 and n , and their scattering states could inevitably get into the spectrum . the sum of the pion mass @xmath36 and the nucleon mass @xmath37 is however in our setups heavier than the masses of the lowest two states ( would - be @xmath1 and @xmath2 ) in the negative parity channel . we then do not suffer from any scattering - state signals . the other possible contamination is wraparound effects @xcite . since we perform unquenched calculations , the excited nucleon @xmath38 can decay into @xmath9 and @xmath24 , and even when we have no scattering state @xmath39 , we could have another type of `` scattering states '' . the baryonic correlator @xmath40 can still accommodate , for example , the following term . @xmath41 here , @xmath42 denotes the temporal extent of a lattice . such a term is quite problematic and mimic a fake plateau at @xmath43 in the effective mass plot because it behaves as @xmath44 . in order to eliminate such contributions , we impose the dirichlet condition on the temporal boundary for valence quarks , which prevents valence quarks from going over the boundary . ( wraparound effects can be found even in quenched calculations @xcite . ) we employ zero - momentum - projected point - type operators for the sinks , and employ wall - type operators in the coulomb gauge for the sources . after we diagonalize the @xmath32 correlation matrices , we can construct optimized operators for n(1535 ) and n(1650 ) states . once we construct optimized operators , we can easily compute the ( non - renormalized ) vector and axial charges @xmath45}}$ ] for the positive- and negative - parity nucleons via three - point functions with the so - called sequential - source method @xcite . in practice , we evaluate @xmath46}}(t)$ ] defined as @xmath47}}(t ) = \frac { { \rm tr}\ \gamma_{a } \langle b(t_{\rm snk } ) j_\mu^{a}(t ) \overline b(t_{\rm src } ) \rangle } { { \rm tr}\ \gamma_{a } \langle b(t_{\rm snk } ) \overline b(t_{\rm src } ) \rangle } , \ ] ] and extract @xmath46}}$ ] by the fit @xmath46}}=g_{a}^{\pm{\rm [ lat]}}(t)$ ] in the plateau region . @xmath48 denotes the ( optimized ) interpolating field for nucleons . @xmath49 is @xmath50 , and @xmath51 is an axial vector current inserted at @xmath52 . we show in fig . [ 3pointfunc ] the non - renormalized axial charge @xmath53}}(t)$ ] for @xmath1 as a function of the current insertion time @xmath52 . they are rather stable around @xmath54 . the non - renormalized axial charge of @xmath1 , @xmath53}}(t)$ ] , as a function of the current insertion time @xmath52 . ] we finally reach the renormalized charges @xmath55}}_{a}$ ] with the prefactors @xmath56 , which are estimated with the values listed in ref . @xcite . the renormalized axial charges of the positive- and the negative - parity nucleons are plotted as a function of the squared pion mass @xmath57 . the solid line is drawn at @xmath58 and the dashed line is drawn at @xmath59 . ] we first take a stock of the axial charge @xmath60 of the ground - state positive - parity nucleon , which is well known and can be the references . the axial charge @xmath60 of the positive parity nucleon is shown in fig . [ axialvectorc ] as open squares . the axial charge of the positive parity nucleon shows little quark - mass dependence , and they lie around the experimental value 1.26 . we finally show the axial charges of the negative - parity nucleon resonances in fig . [ axialvectorc ] . one finds at a glance that the axial charge @xmath61 of @xmath1 takes quite small value , as @xmath62 and that even the sign is quark - mass dependent . while the wavy behavior might come from the sensitiveness of @xmath61 to quark masses , this behavior may indicate that @xmath61 is rather consistent with zero . these small values are not the consequence of the cancellation between u- and d - quark contributions . the u- and d - quark contributions to @xmath61 are in fact individually small @xcite . on the other hand , the axial charge @xmath63 of @xmath2 is found to be about 0.55 , which has almost no quark - mass dependence . the striking feature is that these axial charges , @xmath64 and @xmath65 , are consistent with naive nonrelativistic quark model calculations @xcite , @xmath66 and @xmath67 . such values are obtained if we assume that the wave functions of @xmath1 and @xmath2 are @xmath68 and @xmath69 neglecting the possible state mixing . ( here , @xmath70 denotes the orbital angular momentum and @xmath71 the total spin . ) in the chiral doublet model @xcite , the small @xmath0 is realized when the system is decoupled from the chiral condensate @xmath7 . the small @xmath61 of @xmath1 then does not contradict with the possible and attempting scenario , the _ chiral restoration scenario in excited hadrons _ @xcite . if this scenario is the case , the origin of mass of @xmath1 ( or excited nucleons ) is essentially different from that of the positive - parity ground - state nucleon @xmath16 , which mainly arises from the spontaneous chiral symmetry breaking . however , the non - vanishing axial charge of @xmath2 unfortunately gives rise to doubts about the scenario . we have performed the first lattice qcd study of the axial charges @xmath0 of @xmath1 and @xmath2 , with two flavors of dynamical quarks employing the renormalization - group improved gauge action at @xmath3=1.95 and the mean - field improved clover quark action with the hopping parameters , @xmath4=0.1375 , 0.1390 and 0.1400 . we have found the small axial charge @xmath61 of @xmath1 , whose absolute value seems less than 0.2 and which is almost independent of quark mass , whereas the axial charge @xmath63 of @xmath2 is found to be about 0.55 . these values are consistent with naive nonrelativistic quark model predictions , and could not be the favorable evidences for the chiral restoration scenario in ( low - lying ) excited hadrons . further investigations on the axial charges of @xmath1 or other excited baryons will cast light on the chiral structure of the low - energy hadron dynamics and on where hadronic masses come from . all the numerical calculations were performed on nec sx-8r at rcnp and cmc , osaka university , on sx-8 at yitp , kyoto university , and on bluegene at kek . the unquenched gauge configurations employed in our analysis were all generated by cp - pacs collaboration @xcite . we thank l. glozman , d. jido , s. sasaki , and h. suganuma for useful comments and discussions . this work was supported by a grant - in - aid for scientific research by monbu - kagakusho ( no . 20028006 and 20540265 ) , the 21st century coe `` center for diversity and university in physics '' , kyoto university and yukawa international program for quark - hadron sciences ( yipqs ) . c. detar and t. kunihiro , phys . d * 39 * , 2805 ( 1989 ) . l. y. glozman , phys . lett . * 99 * , 191602 ( 2007 ) [ arxiv:0706.3288 [ hep - ph ] ] . r. l. jaffe , d. pirjol and a. scardicchio , phys . lett . * 96 * , 121601 ( 2006 ) [ arxiv : hep - ph/0511081 ] . r. l. jaffe , d. pirjol and a. scardicchio , phys . rept . * 435 * , 157 ( 2006 ) d. jido , t. hatsuda and t. kunihiro , phys . lett . * 84 * , 3252 ( 2000 ) d. jido , m. oka and a. hosaka , prog . phys . * 106 * , 873 ( 2001 ) [ arxiv : hep - ph/0110005 ] . b. w. lee , chiral dynamics , gordon and breach , new york , 1972 a. ali khan _ et al . _ [ cp - pacs collaboration ] , phys . rev . d * 65 * , 054505 ( 2002 ) [ erratum - ibid . d * 67 * , 059901 ( 2003 ) ] t. t. takahashi , t. umeda , t. onogi and t. kunihiro , phys . d * 71 * , 114509 ( 2005 ) [ arxiv : hep - lat/0503019 ] . s. sasaki , k. orginos , s. ohta and t. blum [ the riken - bnl - columbia - kek collaboration ] , phys . d * 68 * , 054509 ( 2003 ) [ arxiv : hep - lat/0306007 ] . j. c. nacher , a. parreno , e. oset , a. ramos , a. hosaka and m. oka , nucl . a * 678 * , 187 ( 2000 ) [ arxiv : nucl - th/9906018 ] . l. y. glozman and a. v. nefediev , arxiv:0801.4343 [ hep - ph ] .

we show the first lattice qcd results on the axial charges @xmath0 of @xmath1 and @xmath2 . the measurements are performed with two flavors of dynamical quarks employing the renormalization - group improved gauge action at @xmath3=1.95 and the mean - field improved clover quark action with the hopping parameters , @xmath4=0.1375 , 0.1390 and 0.1400 . we construct 2@xmath52 correlation matrices and diagonalize them so that the signals of @xmath1 and @xmath2 are properly separated . wraparound contributions in the correlator , which can be another source of signal contaminations , are eliminated by imposing the dirichlet boundary condition in the temporal direction . the axial charge of @xmath1 is found to take small values as @xmath6 , whereas that of @xmath2 is approximately 0.5 , which is almost independent of quark masses and consistent with the predictions by the naive nonrelativistic quark model .

You are an expert at summarizing long articles. Proceed to summarize the following text: the cosmic star formation history ( sfh ) is an important test for galaxy formation models . experimentally our knowledge of the sfh comes from @xcite up to @xmath5 and from observations of color - selected lyman break galaxies @xcite , ly@xmath6 emitters @xcite , uv+ir measurements @xcite , and grb observations @xcite at higher @xmath4 ( hereafter , these will be refereed to as h2006 , m2007 , b2008 , o2008 , r2008 , c2007 , y2008 and w2009 , respectively ) . however , direct high-@xmath4 measurements constitute an extreme challenge even for the most powerful telescopes and remain sparse . due to their high luminosity grbs can be used as sfh probes into the very distant universe @xcite , potentially to higher redshifts than allowed by galaxies alone . for example , grb 090429b at @xmath7 @xcite is the current record - holder object , followed by a @xmath8 galaxy @xcite and grb 090423 , at @xmath9 @xcite . in principle , the redshift distribution of grbs can give us important clues on the early stages of cosmic history . in practice , the connection between grbs and the underlying host galaxy star formation mode is far from trivial ( see e.g. @xcite ) , thus making the probe value subject to uncertainties . the purpose of this work is to show how _ principal component analysis _ ( pca ) can be used to map the grb redshift distribution onto the cosmic sfh in a model - independent way . pca has already been applied to other astrophysical and cosmological contexts @xcite , and recognized as a useful tool to reconstruct parameters without the introduction of _ ad hoc _ parameterizationscdm model with wmap7 best fit parameters @xcite , @xmath10 , and @xmath11 km s@xmath2 mpc @xmath2 . ] . we assume that the formation rate of long grbs ( duration longer than 2 sec ) follows closely the sfh ( e.g. , @xcite ) . hence , the comoving rate of observable grbs , @xmath12 is @xmath13 where @xmath14 is the field of view of the experiment , @xmath15 is the grb formation efficiency , @xmath16 is the beaming factor of the burst , @xmath17 is the fraction of grbs with measured redshift and @xmath18 is the luminosity threshold of a given experiment . in the following we discuss in detail each of the terms in eq . ( [ psigrb ] ) . we define telescope - related quantities in eq . ( [ psigrb ] ) after _ swift _ specs , i.e. @xmath19 @xcite and @xmath20 .. ] the overall grb rate depends on @xmath21 , where @xmath22 is the opening angle of the jet . according to @xcite the average value of @xmath23 . using a radio transients survey , @xcite placed the upper limit @xmath24 . we set @xmath25 as a fiducial value . the stellar initial mass function ( imf ) , @xmath26 , determines the fraction of stars massive enough to leave a black hole remnant . current theories indicate that the threshold mass to trigger a grb is @xmath27@xcite . however , only a fraction @xmath28 @xcite of black holes resulting from supernova explosion actually gives rise to a grb ; to be conservative , we considered @xmath29 . hence , the grb formation efficiency factor per stellar mass is @xmath30 for simplicity , we assume a standard " salpeter imf , @xmath31 , with @xmath32 @xcite . the number of detectable grbs depends on the instrument sensitivity and intrinsic isotropic grb luminosity function . for the latter , we adopt a power - law distribution function of @xcite @xmath33 with @xmath34 erg s@xmath2 , @xmath35 and @xmath36 . the luminosity threshold is then @xmath37 , where @xmath38 is the luminosity distance and @xmath39 is the bolometric energy flux limit of the instrument . in what follows we set @xmath40 @xcite . from eq . ( [ psigrb ] ) we can determine the number of observed grbs with redshift in ( @xmath4 , @xmath41 ) over a time interval , @xmath42 , in the observer rest frame : @xmath43 where @xmath44 is the comoving volume element per unit of redshift . our observable , i.e. the cumulative number @xmath45 of grbs up to redshift @xmath4 , is given by @xmath46 so far we have discussed the physical meaning of all terms in eq . ( [ psigrb ] ) except the sfh . since our aim is to build a model which is independent from the specific form of @xmath47 , we avoid making hypothesis about this quantity . instead , our intention is to derive @xmath47 by using pca and a ( mock or real ) data set whose data points are @xmath48 pairs . the main goal of pca is the dimensionality reduction of an initial parameter space through the analysis of its internal correlations . suppose we have a model composed by @xmath49 parameters . if two of them are highly correlated , they are actually providing the same information . this means that it is possible to rewrite the data in a new parameter space consisting of @xmath50 terms , with minimum loss of information ( for a complete review see @xcite ) . this new set of parameters are recognized as the principal components ( pcs ) , or the eigenvectors of the fisher information matrix , . we postulate that the data set is composed by @xmath51 independent observations , each one characterized by a gaussian probability density function , @xmath52 $ ] . in our notation , @xmath53 is a measurement of an independent variable ; @xmath54 represent the measurements of a quantity @xmath55 depending on @xmath53 ; @xmath56 is the uncertainty associated with the measures , and @xmath57 is the parameter vector of the theoretical model . in other words , we investigate a specific quantity , @xmath58 , which can be written as a function of the parameters @xmath59 . in this context , the likelihood function is given by @xmath60 and the fisher matrix is defined as @xmath61 brackets in eq . ( [ eq : fmdefinicao ] ) represent the expectation value . we can now diagonalize * * , and determine the set of its eigenvectors / pcs , @xmath62 , and eigenvalues , @xmath63 . following the standard convention , we enumerate @xmath64 from the largest to the smallest associated eigenvalue . our ability to determine the form of each pc is given by @xmath65 . the set @xmath62 forms a complete base of uncorrelated vectors . this allows us to use a subspace of @xmath62 , @xmath66 , to rewrite @xmath58 as a linear combination of all the elements in @xmath66 , @xmath67 . the data is then used to find the values of the linear expansion coefficients , @xmath68 ( for a detailed discussion see i2011 ) . the question of how many pcs should be used in the final reconstruction , or how to choose the dimensionality of @xmath66 , depends on the particular data set analyzed and our expectation towards them . to provide an idea of how much of the initial information ( variance ) is included in our plots , we shall order them following their _ cumulative percentage of total variance_. a reconstruction with the first @xmath69 pcs encloses a percentage of this value @xmath70 it is important to emphasize that each added pc brings its associated uncertainty ( @xmath71 ) into the reconstruction . so , although the best - fitted reconstruction converges to the real " function as m increases , the uncertainty associated also raises . as a consequence , the question of how many pcs turns into a matter of what percentage of total variance we are willing to enclose . to specify our method of sfh reconstruction from grb data , let us consider a data set formed by @xmath72 measurements of the cumulative number of grb up to redshift @xmath73 , @xmath74 , and its corresponding uncertainty ( @xmath75 ) . the likelihood is given by @xmath76.\end{aligned}\ ] ] using equation ( [ eq : fmdefinicao ] ) , the fisher matrix components are @xmath77 the fisher matrix determination is now a matter of calculating the derivatives of @xmath78 , for which we use the theoretical prescriptions of sec . [ sec_grb_rate ] . aiming at model independence and simplicity , we model the sfh as @xmath79 where @xmath59 are constants , @xmath80 is the total number of redshift bins , and @xmath81 is a window function which returns 1 if @xmath82 and 0 otherwise . using this description , we may write any functional form with resolution limited by our computational power . the derivatives of @xmath45 can be computed analytically : @xmath83 , \label{eq : nderiv}\end{aligned}\ ] ] where @xmath84 is a step function which returns 0 if @xmath85 and 1 otherwise , @xmath86 corresponds to the number of integer bins up to redshift @xmath4 , @xmath87 is the kronecker delta function , @xmath88 is the lower bound of the @xmath89-th redshift bin , and @xmath90 from these relations the fisher matrix can be computed and the functional form of the sfh reconstructed through pca . . the black - thick line is the final reconstruction for each case and the red - dashed - thick lines corresponds to @xmath91 confidence levels . the inset shows the cumulative percentage of total variance , @xmath92 . ] [ [ sec : mock ] ] mock data + + + + + + + + + the mock sample is composed of data pairs @xmath93 , distributed in redshift bins of width @xmath94 , where @xmath95 represents the middle of each bin . the fiducial model used for the sfh is a simple double - exponential function , @xmath96 , fitted to numerical results by @xcite , @xmath97 with @xmath98 for @xmath99 0.993 ; @xmath100 for @xmath101 and @xmath102 for @xmath103 . we generated 500 simulations , each realization with uncertainty in the determination of @xmath45 set to unit and containing 65 redshift bins . after we generated the mock sample , the information about @xmath104 is discarded . our goal from now on is to re - obtain the functional form of eq . ( [ rho_star ] ) from pca . the fisher matrix is obtained as described above , assuming an observing time @xmath105 yr . as the mock sample purpose is to test the procedure under an ideal scenario , we did not include uncertainties in the parameters of eq . ( [ psigrb ] ) . having obtained the pcs , we can rewrite the sfh as @xmath106 where @xmath107 and @xmath108 are constants to be determined and @xmath69 is the number of pcs we choose to use in the reconstruction . the simulated data points are then used to find the appropriate values for the parameters @xmath107 and @xmath108 as those that minimize the expression @xmath109 where @xmath110 for all redshift bins . the reconstructions obtained using 1 to 6 pcs are shown in fig . [ fig : rec_sim_err10_all ] . the uncertainty in the final reconstruction was calculated by a quadrature sum that includes the parameters @xmath111 and the uncertainty in the determination of parameters @xmath112 and @xmath113 . from fig . [ fig : rec_sim_err10_all ] we can appreciate the success of the procedure in reconstructing the underlying unknown sfh in an ideal scenario , with increasing agreement as the number of pcs raises . confidence levels also become wider as @xmath69 increases , with the only exception of the reconstruction with 1 pc . since @xmath114 dominates the errors due to the limited freedom to fit the second peak of the fiducial model with only 1 pc . with 2 pcs fitting the second peak becomes easier , and as a consequence , the magnitude of @xmath114 decreases to levels below those of @xmath111 . [ [ sec : swift ] ] _ swift _ data + + + + + + + + + + + + after validating pca reconstruction under ideal conditions , we turn to the use of currently available _ swift _ data , and compare these results with independent measurements of sfh from the literature . first , we need to properly choose our data set . since only grbs with measured redshifts can be used in our analysis , the question of how the redshift measurements were obtained must be examined carefully . grbs redshifts are generally obtained from optical afterglow spectra using absorption lines or photometry , or from the spectrum of the host galaxy using emission lines . as pointed by @xcite , different methods yield different redshift distributions : a visual inspection of fig . [ fig : swift_hist ] illustrates this point . most noticeably , the grb redshift distribution determined from their hosts lacks very high-@xmath4 events . moreover , emission ( and to a lesser extent , absorption ) lines are susceptible to a selection effect known as the redshift desert " in the range @xmath115 @xcite . additional bias sources are preliminary discussed by @xcite . to avoid systematic errors affecting the overall redshift distribution , our data sample is composed by 120 _ swift _ grbs with redshift determined from absorption lines and photometry ( gray region in fig . [ fig : swift_hist ] , top panel ) . the next step is to choose the appropriate redshift bin width . in principle , the quality of the reconstruction should increase with the number of bins . however , as grb are discrete events , if we pick a bin width based on the available data ( for example , in such a way that each bin has at least one grb ) , the bins will be too wide ( @xmath116 ) . in this case , the assumption that the sfh is constant inside the bin will not hold , leading to reconstructions with bad resolution . to overcome this limitation we performed a gaussian kernel fit for a kernel @xmath117 , bin width @xmath118 and a total of @xmath119 bins . ] to the data ( black line in fig . [ fig : swift_hist ] , top panel ) . now we have a continuous probability distribution function ( pdf ) for @xmath120 , which follows the real data distribution and allows us to set the bin width as small as required . we kept @xmath94 and use the pdf to calculate the cumulative number of observed grbs in each bin . the comparison between the real data cumulative distribution and the one calculated via the fitted pdf are shown in fig . [ fig : swift_hist ] , bottom panel . the fisher matrix is calculated using @xmath121 yr of observation time . the parameters @xmath75 were obtained by summing in quadrature the uncertainties in the quantities involved in eq . ( [ psigrb ] ) . [ fig : rec_data_err10_all ] shows the first 2 pcs and the corresponding reconstruction using both of them , which already encloses more than @xmath122 of total variance . in the lower panel , the points correspond to completely independent measurements from the literature . these data points are shown only for comparison purposes and have not been used in our calculations . confidence levels . the inset shows the cumulative percentage of total variance , @xmath92 . ] we have proposed the use of pca as a powerful tool to reconstruct the cosmic star formation history exploiting the measured gamma - ray burst redshift distribution . the procedure was successfully validated using synthetic data and next applied to actual _ swift _ data ( fig . 3 ) . it is important to highlight that the approach is completely independent of the initial choice of the theoretical model parameter vector , @xmath57 . this has the obvious advantage of avoiding any a priori hypothesis on the sfh , @xmath123 . however , the degeneracy between @xmath123 and any other factor we are failing to take into account can not be removed , i.e. the reconstructed sfh contains also the behavior of all the agents influencing the determination of @xmath45 and not included in the model . for example , @xcite have argued that grb progenitors will have a low metallicity . such an effect would be a consequence of the mass and angular momentum loss caused by winds in high - metallicity stars . this would prevent such stars of becoming grbs and consequently , change their expected redshift distribution @xcite . we implicitly considered that this and other such effects will span within the error bars in our analysis . in spite of the remaining uncertainties , which are probably less severe than those affecting other methods aimed at recovering the high-@xmath4 tail of the sfh , there are robust indications that we can gather from the analysis of our results . the first is that the combination of grb data and pca suggest that the level of star formation activity at @xmath124 could have been already as high as the present - day one ( @xmath1 yr@xmath2 mpc@xmath3 ) . this is a factor 3 - 5 times higher than deduced from high-@xmath4 galaxy searches through drop - out techniques , similarly to the trend found by @xcite . if true , this might alleviate the long - standing problem of a photon - starving reionization ; it might also indicate that galaxies accounting for most of the star formation activity at high redshift go undetected by even the most deep searches . finally it is worth noticing that a sustained high-@xmath4 star formation activity is consistent with predictions of reionization models that simultaneously match a number of observable experimental constraints as the gunn - peterson effect , thomson free - electron optical depth , lyman limit systems statistics etc . ( @xcite , @xcite ) . given the expected large growth of grb detections from the next generation of instruments , the method proposed here promises to become one of the most suitable and reliable tools to constrain the star formation activity in the young universe . we thank k. ioka , r. salvaterra and n. yoshida for useful comments . e.e.o.i . thanks capes ( 1313 - 10 - 0 ) for financial support . r.s.s . thanks cnpq ( 200297/2010 - 4 ) for financial support . af acknowledges support from ipmu where this research started . this work was supported by wpi initiative , mext , japan .

we propose a novel approach , based on principal components analysis , to the use of gamma - ray bursts ( grbs ) as probes of cosmic star formation history ( sfh ) up to very high redshifts . the main advantage of such approach is to avoid the necessity of assuming an _ ad hoc _ parameterization of the sfh . we first validate the method by reconstructing a known sfh from monte carlo - generated mock data . we then apply the method to the most recent _ swift _ data of grbs with known redshift and compare it against the sfh obtained by independent methods . the main conclusion is that the level of star formation activity at @xmath0 could have been already as high as the present - day one ( @xmath1 yr@xmath2 mpc@xmath3 ) . this is a factor 3 - 5 times higher than deduced from high-@xmath4 galaxy searches through drop - out techniques . if true , this might alleviate the long - standing problem of a photon - starving reionization ; it might also indicate that galaxies accounting for most of the star formation activity at high redshift go undetected by even the most deep searches . [ firstpage ] methods : statistical , gamma - ray burst , star formation

You are an expert at summarizing long articles. Proceed to summarize the following text: the tendency toward for the formation of the charge ordering is a common characteristic of the transition metal oxides with a perovskite structure including high - temperature superconducting cuprates and colossal magnetoresistive manganites . @xcite in particular , various kind of the manganites with the doping level @xmath4 of 1/2 show the charge - exchange ( ce)-type charge ordering , @xcite in which mn@xmath5 and mn@xmath6 ions by the ratio of 1:1 regularly distribute in the underlying lattice below the charge and orbital ordering temperature @xmath7 [ fig . [ fig1](a ) ] . such a real - space charge ordering is characterized by a single - particle excitation across the charge gap @xmath8 . using the light of frequency @xmath9 , the single - particle excitation spectrum of the charge - ordered ( co ) manganites have been extensively investigated in recent years ; @xcite for example , okimoto _ et al_. identified @xmath10 mev of pr@xmath11ca@xmath12mno@xmath2 by using polarized reflectivity measurements and clarified that the electronic structure is dramatically reconstructed in the order of ev by varying temperature @xmath13 and magnetic field @xmath14 ( refs . and ) . using transmission measurements , calvani _ et al_. found that @xmath15 of la@xmath16ca@xmath17mno@xmath2 with @xmath18 and 0.67 can be well described by the bardeen - cooper - schrieffer ( bcs ) relation . another important viewpoint is the collective nature of @xmath19 carriers including the orbital degree of freedom , which is thought to have a great impact on extraordinary properties of the manganites , i.e. , colossal magnetoresistance ( cmr ) . @xcite the conspicuous examples are recent findings of the `` orbiton '' , a new elementary excitation of the orbital - ordered state observed in lamno@xmath2 @xcite and of the `` charge stripes '' , a new periodic form composed of a pairing of diagonal mn@xmath5 with mn@xmath6 observed in la@xmath16ca@xmath17mno@xmath2 @xmath20 . @xcite therefore , the understandings of the role of @xmath19 carriers with variation of @xmath4thus the collective phenomena of @xmath19 carriers in connection with the specific kind of the charge ordering depending on @xmath4have attracted interests and may be of central topics in co manganites . more recently , asaka _ et al_. proposed the model of the ce - type structure in pr@xmath21ca@xmath22mno@xmath2 by using the transmission electron spectroscopy as schematically shown in fig . [ fig1](b ) . @xcite it may be viewed as a quasi - one dimensional electronic structure with the reduced dimensionality as compared to the well - known ce - type structure [ fig . [ fig1](a ) ] : @xmath23 or @xmath24 orbital ordering of mn@xmath5 occurs along @xmath25-axis . in another angle , @xmath26 orbital of mn@xmath5 was cliped by three mn@xmath6 ions . such a regular pattern of the distinct charges modifies the uniformity of the charge density , leading to the development of the charge - density - wave ( cdw ) condensate . @xcite this situation is depicted in fig . [ fig1](c ) ; the charge density ( shared area ) periodically varies with the position and also time ( dashed line ) , so the new modulation occurs in the underlying lattice ( the period of this modulation is larger than the lattice constant ) . such a cdw formation is recently deduced from the one - dimensional fermi - surface ( fs ) topology in a layered manganite la@xmath27sr@xmath28mn@xmath29o@xmath30 by chuang _ et al_. ; @xcite they determined the detailed map of the fs by using the angle - resolved photoemission spectroscopy and concluded that fs is subject to cdw instabilities even in the ferromagnetic metallic state . in the sight of these considerations , the charge ordering is also characterized by a collective excitation well below @xmath8 . however , no clear evidence for the collective excitation of the co state was reported so far ( see , _ note added _ ) . the collective excitation is associated with the mode depending on both the position and the time ; the modes arising from the former and the later are referred to as amplitudons and phasons , respectively . @xcite according to lee _ et al_. , @xcite the dispersion relation of the phasons is given by @xmath31 , where @xmath32 is the wavenumber . by the illumination of light , the collective excitation can make a coupled oscillation , which yields the change of the dielectric constant at @xmath33 , being in proportional to @xmath34 , where @xmath35 and @xmath36 are the longitudinal and transverse optical frequencies , respectively . therefore , this kind of the coupled mode comes to the surface as a collective excitation mode in the complex optical spectrum . it is also important that the impurity and/or the lattice imperfection has striking influence on the cdw dynamics , because the cdw condensate is pinned by the impurity and therefore the peak position of the collective excitation mode shifts to a finite frequency . @xcite in this article we provide the spectroscopic evidence for the cdw condensate in the typical co manganite pr@xmath0ca@xmath1mno@xmath2 . the basis of this conclusion is derived from the first observation of the collective excitation mode in the optical conductivity spectrum . as well - known cdw system like a quasi - one dimensional k@xmath1moo@xmath2 , the collective excitation mode frequently appears in the millimeter @xmath9 range @xmath37 mev , @xcite where it is difficult to get the probing light by the conventional optical spectroscopy . we overcome this limitation by using thz time - domain spectroscopy ( tds ) , @xcite which is a powerful tool to unveil the low - energy charge dynamics of the cmr manganites . @xcite in addition to this advantage , thz - tds is especially suited to capture the signal of the fluctuation phenomenon as the high @xmath9 thz pulse is sensitive and matches on the typical time scale of the charge fluctuation in correlated electron system , which is instantiated by thz - tds experiments on underdoped bi@xmath29sr@xmath29cacu@xmath29o@xmath38 . @xcite _ material_material we used in this work is the typical co manganite ; pr@xmath0ca@xmath1mno@xmath2 with a orthorhombically disordered three dimensional structure . this material exhibits the insulating behavior and undergoes the charge @xcite and orbital @xcite ordering below @xmath39 k and the antiferromagnetic spin ordering below @xmath40 k. due to the deviation of @xmath4 from 1/2 , at which the charge ordering is most stable , the extra electrons @xmath41 occupy at the mn@xmath6 site with maintaining the charge ordering [ fig . [ fig1](b ) ] . the remarkable characteristics compared to other co manganites are such an admissibility of the charge ordering and the cmr effect , in which the dramatic variance of the resistance by more than 10 orders of magnitude was attained @xcite therefore , various kind of the experimental results are accumulated to date . @xcite it should be mentioned that there are variant interpretaions about the low - temperature phase below 110 k : one is due to the coexistence of the ferromagnetic and antiferromagnetic phases proposed by jirk _ et al_. ; @xcite another is the canted antiferromagnetic phase ( or the spin - glass phase ) reported by yoshizawa _ et al_. @xcite however , the intimate identification is beyond the scope of this study , but we can say that magnetic properties of our samples are consistent with the results by deac _ et al . _ @xcite dependence of the magnetization @xmath42 in a unit of bohr magnetron @xmath43 at the mn - site of the sample a. circles and squares represent @xmath44 taken from the field cooling ( fc ) and zero field cooling ( zfc ) scans under the magnetic field of 500 oe or 10 koe , respectively.,width=264 ] [ cols="^,^,^,^,^,^,^ " , ] dependence of the scattering rate @xmath45 of the sample a deduced from eq . ( [ eqn1 ] ) . the data are taken from the @xmath13-warming run . the solid line denotes a fitting result to the data ( open circles ) using eq . ( [ eqn3 ] ) . the dashed line indicates the charge and orbital ordering temperature @xmath7.,width=245 ] one may consider the phase separation ( ps ) picture @xcite the dynamical coexistence of the ferromagnetic metallic and co insulating states @xcite , which is thought to be a key feature of the cmr phenomena and indeed experimentally discussed in pr@xmath0ca@xmath1mno@xmath2 . @xcite such ps as well as the presence of the disorder commonly seen in thin films using mgo substrates , produces a finite peak structure as a result of the shifting of the drude peak centered at @xmath46 . our samples show the ferromagnetic order at low - temperature ; there is a significant hysteresis loop in @xmath47 curve at 30 k with a coercive force of @xmath48 oe ( figs . [ fig3 ] ) , which may imply the presence of the ferromagnetic domain in the co insulating phase . therefore , we compare our experimental results to the typical characteristics of the ps in co manganites . @xcite best example in the ps picture is the slow relaxation effect observed in pr@xmath49ca@xmath50mno@xmath2 by anane _ et al_. ( ref . ) and cr - doped manganites by kimura _ et al_. , @xcite in which the ferromagnetic fraction can be artificially controlled by @xmath14-annealing process and also the cr ( impurity)-doping at the mn - site . in these materials , the physical quantities [ i.e. , @xmath51 and @xmath42 ] strongly depend on the history of the external perturbations . for example , anane _ et al_. measured the time dependence of @xmath51 in the ferromagnetic phase of pr@xmath49ca@xmath50mno@xmath2 after @xmath14 of 100 koe is applied at low - temperature . @xcite they found that the co insulating phase is restored in the time range of @xmath53@xmath54 second ( the slow relaxation effect ) . very recently , mathieu _ et al_. reported that @xmath55 under 20 oe in nd@xmath56ca@xmath56mno@xmath2 exhibits the different relaxation effect in the typical characteristic time of @xmath57 second by changing only the cooling rate . @xcite moreover , the difference of @xmath55 in the history of the cooling rate increases with time . based on these studies , we have done following thz - tds experiments : we rapidly cooled the another sample used in sec . [ thz ] ( the growth condition is same as samples a and b ) from room temperature to 4 k with a cooling rate of @xmath58 k per minute and measured the time dependence of @xmath59 while keeping @xmath60 k constant . we observed no slow relaxation effect in the measured @xmath9 range ; the spectral shape of the peak structure and the peak frequency @xmath61 mev exhibit no variation per @xmath62 second , which is a contrarious evidence for the ps picture . as described in sec . [ intro ] , we infer the possibility of the cdw condensate due to the quasi - one dimensional charge and orbital ordering in pr@xmath0ca@xmath1mno@xmath2 . we discuss here that the observed peak structure arises from the cdw condensate as compared to the low - energy optical properties of the well - known cdw system . the typical characteristics of the collective excitation mode observed in low - dimensional cdw materials are as follows : @xcite ( i ) as mentioned earlier , the pinning frequency is in general of the order of mev . @xcite ( ii ) the spectral weight of the collective excitation mode is typically 2 orders of magnitude smaller than that of the single - particle excitation . @xcite this is due to the relatively large @xmath63 mev , which is in strong contrast to the small spin gap @xmath64 mev observed in the spin - density - wave ( sdw ) system . in the sdw system , the spectral weight of the collective excitation mode should be comparable to that of the single - particle excitation due to the electron - electron interaction.@xcite ( iii ) kim _ et al_. found that the pinning frequency in ( ta@xmath65nb@xmath66se@xmath67)@xmath29i alloys linearly shifts to higher @xmath9 and @xmath59 at the pinning frequency does not change with increasing the impurity concentration @xmath68 . @xcite same tendencies are also observed in k@xmath1mo@xmath65w@xmath66o@xmath2 alloys . @xcite ( iv ) mihly _ et al_. identified the swelling structure in the low - energy side of the collective excitation mode in k@xmath1moo@xmath2 as the internal deformation on the basis of the generalized debye analysis . @xcite our present findings ( i)-(iv ) described in sec . [ thz ] and various works ( i)-(iv ) described above in the cdw system , both of which have apparent similar properties and have no contrariety . moreover , @xmath69 dependence of @xmath45 ( fig . [ fig7 ] ) has been also found in the cdw system @xcite and eqs . ( [ eqn1 ] ) and ( [ eqn2 ] ) are same expressions derived from the equation of motion for the cdw dynamics . @xcite therefore , we arrive at the conclusion that the observed structure in pr@xmath0ca@xmath1mno@xmath2 is assigned to the collective excitation mode arising from the cdw condensate . to the best of our knowledge , this is a first spectroscopic evidence for the cdw condensate in co manganites . the description of the cdw condensate in pr@xmath0ca@xmath1mno@xmath2 is consistent and may explain following previous studies . the cdw easily couples with the lattice and affects phonon modes . therefore , new infrared modes are expected to be active below @xmath7 . et al_. found that in addition to phonon modes due to the orthorhombic distortion , many new modes emerge as sharp excitations in pr@xmath11ca@xmath12mno@xmath2 at 10 k. @xcite asamitsu _ et al_. have measured current - voltage characteristics of pr@xmath0ca@xmath1mno@xmath2 and found the beginning of the nonlinear conduction above the threshold electric field , @xcite which reminds us of the sudden motion of the cdw ( cdw depinning ) as observed in cdw materials . @xcite we should note the previous work about the impurity effect on @xmath59 in a copper oxide superconductor . et al_. reported that the zn ( impurity)-doping of the cu - site in yba@xmath29cu@xmath67o@xmath38 distracts the superconductivity and induced the finite peak structure around 10 mev ( ref . ) . they concluded that this structure arises from the localization of the charge carriers produced by the randam distribution of the zn on the cu - site ( disorder ) on the basis that the integrated area of @xmath59 up to 1 ev conserves as far as 3.5% zn - doping . however , same authors also pointed out that if the drude - like response in yba@xmath29cu@xmath67o@xmath38 appears as a result of the collective excitation which is prospective by theory , the peak structure in yba@xmath29(cu@xmath65zn@xmath66)@xmath67o@xmath38 can be regard as the collective excitation mode as in both cases of the cdw and sdw . @xcite it is well known that the collective excitation mode originates in two different states ; one is a pinned " state due to the pinning of the cdw condensate by the impurity and/or the lattice imperfection and another is the bound " state , which is created by the coupling of the pinned state with the optical phonon or the impurity near the pinned state . @xcite despite the fact that the swelling of @xmath59 below the low - energy side of the finite structure , which is usually ascribed to a internal deformation of the pinned collective excitation mode , @xcite can be seen in figs . [ fig4 ] , we can not clearly claim whether the observed mode is assigned to the pinned or the bound collective excitation mode of the cdw . one pregnant result is that @xmath70 below the peak structure crosses zero around 8 mev and reaches the minus value with decreasing @xmath9 [ fig . [ fig4](a ) ] . this implies the presence of another @xmath71 and of the another peak structure below the measured @xmath9 range . further experiments on @xmath72 in the ghz @xmath9 range are necessary to perform the detailed discussion and are now in progress using the cavity perturbation technique . as shown in fig . [ fig5](d ) , the collective excitation mode can be seen even above @xmath7 , while its spectral shape is blurred due to the large proportion of the background contribution above @xmath7 . @xmath45 continues to follow @xmath69 dependence above @xmath73 as shown in fig . [ fig7 ] [ the fitting procedure by eq . ( [ eqn3 ] ) was performed using the data below @xmath73 ] . this excludes the thermal fluctuation as the origin of this broadening , leading us to conclude that the dynamical short - range co ( charge fluctuation ) is still subsistent above @xmath7 instead of the long - range co below @xmath7 . such a short - range co has been already reported by other experiments : radaeli _ et al_. and shimomura _ et al_. have revealed that the short - range co of pr@xmath16ca@xmath17mno@xmath2 with @xmath75 , @xcite 0.35 , 0.4 , and 0.5 , @xcite exists even above @xmath7 by neutron and x - ray scattering experiments , respectively . as in the case of the cdw system such as k@xmath1mo@xmath65w@xmath66o@xmath2 and ( tase@xmath67)@xmath29i , the collective excitation modes are also observed above the cdw transition temperature @xmath76 due to the presence of the fluctuating cdw segments , which are systematically investigated by schwartz _ et al_. @xcite for example , the collective excitation mode of k@xmath1moo@xmath2 is visible even at 300 k ( @xmath77 k ) . finally , we comment on thz radiation from pr@xmath0ca@xmath1mno@xmath2 thin films excited by the femtosecond optical pulses , which we found recently ; @xcite the radiated spectrum decreases rapidly in intensity with increasing @xmath9 and seems to show the depletion around 2.4 mev . as compared to figs . [ fig4 ] , such a thz response is due to the absorption of the collective excitation mode during the propagation of the generated thz pulse inside the material . summarizing , by using thz - tds , for the first time , we reported the presence of the finite peak structure around 23 mev in pr@xmath0ca@xmath1mno@xmath2 and assigned it to the collective excitation mode arising from the cdw condensate . the measurements on the polarization dependence of @xmath59 with the grazing incidence of light using the single - domain single crystal will provide the direct evidence and more detailed information for the collective excitation mode . in especially , the measurements of @xmath59 in the ghz @xmath9 range are needed to perform further quantitative discussions . further interests are to study the melting process under external perturbations and @xmath4 dependence of the collective excitation mode ; for example , @xmath8 decreases linearly with @xmath4 from 0.3 to 0.5 , @xcite whereas @xmath7 gradually increases and behaves the less-@xmath4 dependent . @xcite this indicates the breaking of the bcs relation given by @xmath78 . so , it is indispensable to clarify how the collective excitation mode manifesting well below @xmath8 changes with @xmath4 using thz - tds . _ note added._after the submission of this article , we became an aware of some reports related in the present study . et al_. [ europhys . lett . * 56 * , 434 ( 2001 ) ] and gorshunov _ et al_. [ cond - mat/0201413 ( unpublished ) ] found peak structures around 0.11 mev in spin - ladder materials sr@xmath79ca@xmath17cu@xmath80o@xmath81 by means of optical spectroscopy and assigned them to the collective excitation modes due to the cdw formation . about manganites , campbell _ et al_. [ phys . b * 65 * , 014427 ( 2001 ) ] interpreted that cdw fluctuations inhere in a layered ferromagnetic manganite la@xmath27sr@xmath28mn@xmath29o@xmath30 based on their diffuse x - ray scattering data . recently , et al_. [ phys . rev . b * 65 * , 060405(r ) ( 2002 ) ] reported that the electron microscopic data of a layered co manganite nd@xmath16sr@xmath82mno@xmath67 are consistent with assumptive images of the cdw formation . very recently , the finite peak structure around 4 mev was also found in 1/8 hole - doped la@xmath83nd@xmath11sr@xmath84cuo@xmath67 by dumm _ et al_. [ phys . lett . * 88 * , 147003 ( 2002 ) ] as in the case of yba@xmath29(cu@xmath65zn@xmath66)@xmath67o@xmath38 . they also ascribe this structure to the localization of the charge carriers in the static stripe phase due to the lack of @xmath8 in this material . however , fujita _ et al_. reported the evidence for cdw and sdw formations in 1/8 hole - doped la@xmath85ba@xmath86sr@xmath17cuo@xmath67 by elastic neutron scattering experiments [ phys . lett . * 88 * , 167008 ( 2002 ) ] . we would like to thank in particular y. okimoto and also k. miyano , y. ogimoto , and y. tokura for giving fruitful comments and discussions . we are also grateful to t. kawai for giving an opportunity of using squid apparatus , t. kanki , i. kawayama , k. takahashi , and h. tanaka for their help to squid measurements , and m. misra for reading the manuscript . it is worth noticing that the appearance of the finite peak structure is a common feature in the disordered system but only in _ conductors _ near the compositional insulator - metal phase boundary . in general , the characteristic energy scale of the localized state in the impurity - doped _ conducting _ manganites , i.e. , al ( impurity)-doping at the mn - site of la@xmath87sr@xmath88mno@xmath2 ( ref . ) is about 100 mev , which is considerably different from the peak position of the observed structure .

thz time - domain spectroscopy was used to directly probe the low - energy ( 0.55 mev ) electrodynamics of the charge - ordered manganite pr@xmath0ca@xmath1mno@xmath2 . we revealed the existence of a finite peak structure around 23 mev well below the charge gap @xmath3 mev . in analogy to the low - energy optical properties of the well - studied low - dimensional materials , we attributed this observed structure to the collective excitation mode arising from the charge - density - wave condensate . this finding provides the importance role of the quasi - one dimensional nature of the charge and orbital ordering in pr@xmath0ca@xmath1mno@xmath2 .

You are an expert at summarizing long articles. Proceed to summarize the following text: quantum entanglement is a feature of quantum mechanics that has captured much recent interest due to its essential role in quantum information processing @xcite . it may be characterized and manipulated independently of its physical realization , and it obeys a set of conservation laws ; as such , it is regarded and treated much like a physical resource . it proves useful in making quantitative predictions to quantify entanglement.when one has complete information about a bipartite system subsystems @xmath0 and @xmath1the state of the system is pure and there exists a well established measure of entanglement the _ entropy of entanglement _ , evaluated as the von neumann entropy of the reduced density matrix , @xmath2 with @xmath3 . this measure is unity for the bell states and is conserved under local operations and classical communication . unfortunately , however , quantum systems in nature interact with their environment ; states of practical concern are therefore mixed , in which case the quantification of entanglement becomes less clear . given an ensemble of pure states , @xmath4 with probabilities @xmath5 , a natural generalization of @xmath6 is its weighted average @xmath7 . a difficulty arises , though , when one considers that a given density operator may be decomposed in infinitely many ways , leading to infinitely many values for this average entanglement . the density operator for an equal mixture of bell states @xmath8 , for example , is identical to that for a mixture of @xmath9 and @xmath10 , yet by the above measure the two decompositions have entanglement one and zero , respectively . various measures have been proposed to circumvent this problem , most of which evaluate a lower bound . one such measure , the _ entanglement of formation _ , @xmath11 @xcite , is defined as the minimal amount of entanglement required to form the density operator @xmath12 , while the _ entanglement of distillation _ , @xmath13 @xcite , is the guaranteed amount of entanglement that can be extracted from @xmath12 . these measures satisfy the requirements for a physical entanglement measure set out by horodecki _ et al_. @xcite . they give the value zero for @xmath14 , which might be thought somewhat counterintuitive , since this state can be viewed as representing a sequence of random `` choices '' between two bell states , both of which are maximally entangled . this is unavoidable , however , because assigning @xmath15 a non - zero value of entanglement would imply that entanglement can be generated by local operations . the problem is fundamental , steming from the inherent uncertainty surrounding a mixed state : the state provides an incomplete description of the physical system , and in view of the lack of knowledge a definitive measure of entanglement can not be given . an interacting system and environment inevitably become entangled . the problem of bipartite entanglement for an open system is therefore one of tripartite entanglement for the system and environment . complicating the situation , the state of the environment is complex and unknown . conventionally , the partial trace with respect to the environment is taken , yielding a mixed state for the bipartite system . if one wishes for a more complete characterization of the entanglement than provided by the above measures , somehow the inherent uncertainty of the mixed state description must be removed . to this end , nha and carmichael @xcite recently introduced a measure of entanglement for open systems based upon quantum trajectory unravelings of the open system dynamics @xcite . central to their approach is a consideration of the way in which information about the system is read , by making measurements , from the environment . the evolution of the system conditioned on the measurement record is followed , and the entanglement measure is then contextual dependent upon the kind of measurements made . suppose , for example , that at some time @xmath16 the system and environment are in the entangled state @xmath17 a partial trace with respect to @xmath18 yields a mixed state for @xmath19 . if , on the other hand , an observer makes a measurement on the environment with respect to the basis @xmath20 , obtaining the `` result '' @xmath21 , the reduced state of the system and environment is @xmath22 with conditional system state @xmath23 where @xmath24 is the probability of the particular measurement result . thus , the system and environment are disentangled , so the system state is pure and its bipartite entanglement is defined by the von neumann entropy , eq . ( [ eq : von - neumann ] ) . nha and carmichael @xcite apply this idea to the continuous measurement limit , where @xmath25 executes a conditional evolution over time . in this paper we follow the lead of nha and carmichael , also carvalho _ et al . _ @xcite , not to compute their entanglement measure _ per se _ , but to examine the entanglement _ dynamics _ of a cascaded qubit system coupled through the oneway exchange of photons . the system considered has been shown to produce unconditional entangled states generally a superposition of bell states as the steady - state solution to a master equation @xcite . for a special choice of parameters ( resonance ) , a maximally entangled bell state is achieved @xmath26 except that the approach to the steady state takes place over an infinite amount of time . here we analyze the conditional evolution of the qubit system to illuminate the dynamical creation of entanglement in the general case , and to explain , in particular , the infinitely slow approach to steady - state in the special case . we demonstrate that in the special case the conditional dynamics exhibit a distinct bimodality , where the approach to the bell state is only one of two possibilities for the asymptotic evolution : the second we call an _ entangled - state cycle _ , where the qubits execute a sustained stochastic switching between two bell states . though involving just two qubits and elementary quantum transitions , the situation is similar to that of a bimodal system in classical statistical physics in the limit of a vanishing transition rate between attractors . the physical model of the cascaded qubit system is presented in sec . [ subsec2_1 ] and the quantum trajectory unraveling of its conditional dynamics in sec . [ subsec3_1 ] . in sec . [ subsec4_1 ] we analyze the quantum trajectory equations to demonstrate bimodality and the existence of entangled - state cycles . finally , a discussion and conclusions are presented in sec . [ subsec5_1 ] . in this section we briefly outline the physical model for the cascaded qubit system to be analyzed . a more detailed description , together with the techniques and assumptions used to derive the model master equation presented here , is available in @xcite . the system considered consists of two high - finesse optical cavities , each containing a single tightly - confined atom , the cavities arranged in a cascaded configuration with unidirectional coupling from cavity 1 to cavity 2 ( fig . [ fig : fig1 ] ) . for simplicity , we consider the cavity modes to be identical , with resonance frequency @xmath27 and field decay rate @xmath28 . inefficiencies and losses in the coupling between the cavities are modeled by a real parameter @xmath29 , @xmath30 , with perfect coupling corresponding to @xmath31 . the atoms are assumed to have five relevant electronic levels , of which two ground states , @xmath32 and @xmath33 , represent an effective two - state system , or qubit . and @xmath33 , to three excited states , @xmath34 , @xmath35 , and @xmath36.,scaledwidth=40.0% ] for each atom , the cavity field in combination with auxiliary laser fields ( incident from the side of the cavity ) drives two separate resonant raman transitions between states @xmath32 and @xmath33 . an additional laser field coupled to the @xmath37 transition provides a tunable light shift of the energy of state @xmath32 . all fields are assumed far detuned from the atomic excited states , so these states may be adiabatically eliminated and atomic spontaneous emission ignored . under the further assumption that the cavity field decay rate is much larger than the transition rates between @xmath32 and @xmath33 , the cavity fields may also be adiabatically eliminated to yield a master equation for the reduced two - atom density matrix @xmath12 , @xmath38}+{\left[\hat r_2,\rho\hat r_1^\dag\right]}\right)},\end{aligned}\ ] ] with @xmath39 where @xmath40 , and @xmath41 and @xmath42 are the rates of @xmath43 and @xmath44 transitions , respectively . by virtue of the cavity output , the system is an open system and solutions to master equation ( [ eq : me ] ) generally describe mixed states . under appropriate conditions , however , the system evolves to a pure and entangled steady state . if the coupling between cavities is perfect ( @xmath31 ) and the parameters of the subsystems are the same ( @xmath45 , @xmath46 ) then the steady state is the pure state @xmath47 where we use the abbreviated notation @xmath48 and @xmath49 . then when @xmath50 , which we shall refer to as the _ resonance _ condition , the steady state is a maximally - entangled bell state . this may seem to be ideal , but a problem arises when we consider the eigenvalues of the operator @xmath51 . specifically , the characteristic time for the system to reach steady state , @xmath52 , where @xmath53 denotes the eigenvalue of @xmath51 with smallest ( in magnitude ) non - zero real part , approaches infinity as the resonance condition is approached . this is shown by the plot in fig . [ fig : fig2 ] . thus the master equation itself , in particular its steady state , offers limited insight into the behavior of the system at resonance . we wish to learn more about this special case ; in particular , how does the entanglement develop dynamically . also , if additional information is factored into the description , by making measurements on the environment , can we better characterize the long term behavior , or possibly find perfect entanglement after a finite time ? we demonstrate that quantum trajectory theory can provide answers to these questions . the relaxation time @xmath54 plotted as a function of @xmath55 . note the singularity at resonance , @xmath56.,scaledwidth=40.0% ] as with any open system , the first step in unraveling the master equation is to identify the points of coupling to the environment . the first is obvious the output from cavity 2 . to measure this output , let us assume the existence of an ideal photon detector in the path of the output from cavity 2 ; we call it _ detector 1_. the second point of coupling to the environment is more subtle . our model does not assume the inter - cavity coupling to be perfect ; only a fraction @xmath29 of the output photon flux from cavity 1 makes it into cavity 2 . physically , this loss may be caused , for example , by non - ideal transmissivity of the faraday isolators or by absorption in the cavity mirrors . these imperfections cause photons to be scattered into the environment in some uncontrollable fashion . formally , though , this is equivalent to assuming that the apparatus is ideal , except that there exists a beamsplitter between the cavities , as drawn schematically in fig . [ fig : fig3 ] . we therefore further assume the existence of a second photon detector to collect photons reflected by this beamsplitter ; we call it _ detector 2_. we now proceed to develop the quantum trajectory formalism for the cascaded qubit system . in this approach the system is described by a pure state which is dependent on ( conditioned on ) the counting histories , or records , of detectors 1 and 2 . firstly , we rewrite the master equation in a form suitable for translation into the quantum trajectory language . we reexpress eq . ( [ eq : me ] ) in the form @xmath57 with @xmath58-\frac{1}{2}\sum_{i=1,2 } \left(\hat c_i^{\dag}\hat c_i \rho+\rho\hat c_i^{\dag}\hat c_i\right),\\ \mathcal{s}\rho&=\sum_{i=1,2}\hat c_i \rho\hat c_i^{\dag},\end{aligned}\ ] ] where @xmath59 then , within quantum trajectory theory , the evolution of the system is described by a pure state @xmath60 which evolves under the non - hermitian effective hamiltonian @xmath61 the continuous evolution interrupted at random times by quantum jumps , @xmath62 , where the jumps occur with probability @xmath63 in time interval @xmath64 . physically , the jump operators @xmath65 and @xmath66 account for the reduction of the state of the system , given a photon count is recorded by detector 1 or detector 2 , respectively . thus , within the quantum trajectory description of the coupled cavity system , we consider an experiment in which ideal detectors are employed , such that every scattered photon is detected and recorded . given the history of detector ` clicks ' , one has complete information about the system state , in the sense that that state is always pure ; hence , although the solution to the master equation is generally mixed , one is able to characterize the entanglement in an unambiguous ( conditional ) fashion @xcite . consider the special case where the coupling between the cavities is optimal ( @xmath31 ) . in this case there is only one output from the system , that from cavity 2 , recorded by detector 1 . standard numerical algorithms @xcite have been used to simulate typical quantum trajectories for various values of @xmath55 . specifically , we consider the evolution of the conditional expectation of the operator product @xmath67 , where @xmath68 is the pauli operator diagonal in the @xmath69representation , @xmath70 this expectation has a number of convenient properties ; for example , the steady - state value @xmath71 regardless of the value of @xmath55 , which makes it easy to compare rates of convergence to the steady state for different system parameters . -0.5 cm figure [ fig : fig4 ] contrasts the solution to the master equation and a single quantum trajectory . the solution to the master equation exhibits a completely smooth evolution that tends asymptotically towards the steady state . the quantum trajectory , on the other hand , undergoes a sequence of switches between two extreme values of @xmath72 , which occur at each photon detection . provided the parameters are chosen away from resonance , the photon detections eventually stop and the trajectory settles into the steady state ( [ eq : ss ] ) , with @xmath73 ; the steady state is clearly a dark state . at resonance , however , the photon detections may continue indefinitely . physically , this seems plausible , since it simply implies that the atoms continue to switch between states @xmath32 and @xmath33 , scattering one photon with each transition . at resonance , apparently , a unique equilibrium dark state can not be established . the cyclic behavior that replaces it is completely invisible if we consider only the ensemble average a vivid demonstration of how single quantum trajectories can provide additional insight into the evolution of an open quantum system . the oscillatory behavior featured in fig . [ fig : fig4 ] hints at a simple cyclic process . in fact , it is simple enough that we can understand why it occurs without resorting to numerics . in this section we formulate a graphical description of individual trajectories . figure [ fig : fig4 ] demonstrates that the conditional expectation @xmath72 is conserved during the periods of evolution between quantum jumps . the positively and negatively correlated subspaces @xmath74 are coupled only through quantum jumps . noting that @xmath75 are each @xmath76-dimensional ( assuming real amplitudes without loss of generality ) , we manage to break up a @xmath77-dimensional space into two @xmath76-dimensional planes , linked to one another by the quantum jumps . we refer to this representation as the _ cascaded system phase space_. trajectories within it can be viewed as lines moving continuosly within either plane and jumping discontinuously between the planes . we use phase space portraits within @xmath78 and @xmath79 to characterize the behavior of the system , where for the sake of simplicity , and without loss of generality , we are assuming @xmath80 and @xmath81 to be real . we define @xmath82 and scale time by setting @xmath83 . the master equation then takes the form ( @xmath31 ) @xmath84 where @xmath85 the resonance condition is now @xmath86 . it is useful to convert to a matrix notation , such that a pure state @xmath60 of the system is represented by a @xmath77-vector , @xmath87 and system operators are written as @xmath88 matrices , e.g. , @xmath89 and @xmath90 the evolution of @xmath60under @xmath91 is written as a linear differential equation in four variables , @xmath92\ ! { |\phi\rangle}\nonumber\\ & = \left(\begin{array}{cccc } -2&0&0&2r\\ 0&-(1+r^2)&2r^2&0\\ 0&2&-(1+r^2)&0\\ 2r&0&0&-2r^2\\\end{array}\right ) \label{eqn : basicde}\!{|\phi\rangle}.\end{aligned}\ ] ] as noted above , this evolution is constrained within either @xmath78 or @xmath79 . thus we can write @xmath60 as a vector sum of two orthogonal components @xmath93 and @xmath94 , @xmath95 , to obtain the decoupled dynamics @xmath96 eigenvectors of the two dynamical matrices correspond to states of the system that are preserved under the evolution between quantum jumps . note , however , that it does not necessarily follow that such a state is a steady state of the quantum trajectory evolution as a whole ; it must eventually experience a quantum jump if its norm decays i.e . , the corresponding eigenvalue is not zero . recall from quantum trajectory theory that the probability for a state not to jump prior to time @xmath16 is given by its norm @xcite . for the systems of equations given above we find the following ( unnormalised ) eigenstates and eigenvalues : 1 . @xmath97 , @xmath98 ; this is the steady state of the system for @xmath99 . 2 . @xmath100 , @xmath101 ; this state in @xmath78 is orthogonal to @xmath102 and must eventually jump to a state in @xmath79 . 3 . @xmath103 , @xmath104 ; this state in @xmath79 must eventually jump to a state in @xmath78 unless @xmath86 ; in the latter case it plays no role once an entangled - state cycle is initiated ( see below ) . 4 . @xmath105 , @xmath106 ; this state in @xmath79 must eventually jump to a state in @xmath78 . in the special case of resonance , @xmath86 , there are two independent steady states , @xmath102 and @xmath107 , which helps to explain the failure of the master equation evolution to approach a unique steady state . it also suggests a fundamental feature of the indefinite switching , the cyclic behavior , revealed by individual quantum trajectories : during such an _ entangled - state cycle _ , the system state must remain orthogonal to @xmath102 and @xmath107 . we verify this shortly , after examining the trajectory evolution away from resonance , where the steady state @xmath102 is always reached for perfect inter - cavity coupling . typical quantum trajectories for @xmath108 are shown in figs . [ fig : fig5 ] and [ fig : fig6 ] , where the @xmath78 and @xmath79 subspaces are drawn as circular planes . normalized states are located on the circumferences of the circles . the bell states @xmath109 lie at intersections of the circumference with the dotted lines as shown . between quantum jumps , under the influence of the non - hermitian hamiltonian @xmath91 , the norm of the state decays and the point representing it within the phase space moves to the interior of one of the circles . quantum jumps cause a switch from @xmath78 to @xmath79 or vice - versa . they are represented by the lines connecting the two planes , where for illustrative purposes , the system state is renormalized after each quantum jump ; thus jumps terminate at points on the circumference of the circles . we restrict ourselves to separable initial states located in one or other of the two subspaces ; for example , the states @xmath110 and @xmath111 , respectively , are considered in figs . [ fig : fig5 ] and [ fig : fig6 ] . the action of the jump operator @xmath65 on states located in @xmath78 ( with renormalization ) is @xmath112 while the action of @xmath65 on states in @xmath79 is @xmath113 thus , when a quantum jump occurs , any state within @xmath78 collapses onto the bell state @xmath114 in @xmath79 , while any state within @xmath79 collapses onto the state @xmath115 in @xmath78 . consider an initial normalized state in @xmath78 , @xmath116 , for some ( real ) @xmath117 . given that @xmath102 is a steady state of the evolution between quantum jumps , the probability of an eventual quantum jump to @xmath79 is @xmath118 while with probability @xmath119 the system evolves to the steady state @xmath102 without any photon emissions . if a jump from @xmath79 to @xmath120 has just occurred , then by the same argument one shows that the probability of a future quantum jump to @xmath79 is @xmath121 , or , alternatively , the probability of reaching the steady state after such a jump is @xmath122 ^ 2 $ ] . consider now an initial state in @xmath79 , @xmath123 , for some ( real ) @xmath124 . owing to the instability of both @xmath107 and @xmath125 for @xmath99 , an eventual quantum jump is guaranteed ; thus , @xmath126 armed with this information , we move to an explanation of the quantum trajectories displayed in figs . [ fig : fig5 ] and [ fig : fig6 ] . in fig . [ fig : fig5 ] we plot three typical phase - space trajectories for @xmath108 and @xmath127 . [ fig : fig5](a ) illustrates the case where the system evolves directly to the steady state @xmath102 . the probability of this event is @xmath128 , so it is the most likely occurrence for the chosen parameters . if a first quantum jump does occur , then typical trajectories are shown in figs . [ fig : fig5](b ) and ( c ) . following the jump to @xmath129 in @xmath79 , a second jump returning the state to @xmath78 is guaranteed . for @xmath108 , this leaves the system in the state @xmath130 , from which the probability of a further cycle of jumps is @xmath131 . thus , after a first quantum jump cycle , it is most likely that further cycles will follow , as seen in figs . [ fig : fig5](b ) and ( c ) , where in both cases a total of five cycles ( ten photon detections ) occur before the system finally reaches the steady state . in fig . [ fig : fig6 ] we plot three typical phase - space trajectories for @xmath108 and @xmath132 . in this case , at least one quantum jump is certain to occur , following which the probability of further jumps is @xmath131 , as above . so for this initial condition , the most likely outcome is a sequence of quantum jump cycles following a first guaranteed photon detection . in fig . [ fig : fig6 ] ( a ) only the first detection occurs , while in figs . [ fig : fig6](b ) and ( c ) this detection is followed by a sequence of cycles before the steady state is eventually achieved . the case @xmath86 is of particular interest . the normalized eigenstates of the evolution between quantum jumps are the bell states @xmath133 , @xmath134 , @xmath135 , and @xmath136 . the eigenvalues are @xmath137 and @xmath138 . the action of the jump operator @xmath65 on states within @xmath78 simplifies to @xmath139 and its action on states within @xmath79 to @xmath140 for @xmath86 , photon detections , if they occur , are associated with collapses onto one of two maximally - entangled bell states . for initial states @xmath141 and @xmath142 in @xmath78 and @xmath79 , respectively , the system evolves continuously , without the emission of any photons , to @xmath133 and @xmath143 , with probabilities @xmath144 and @xmath145 in @xmath79 or @xmath146 in @xmath78 . in this case , as both terminal states are unstable under the between - jump evolution , a second detection and quantum jump must follow . according to eqs . ( [ eq : plusjump ] ) and ( [ eq : minusjump ] ) this simply exchanges @xmath147 for @xmath146 and vice - versa . hence , a perpetual switching between bell states @xmath147 and @xmath146 occurs . we designate this behavior an _ entangled - state cycle_. thus , at resonance we find a distinctly bimodal behavior . the system either evolves into a maximally - entangled bell state without emitting photons , or an entangled - state cycle is initiated under which the system switches indefinitely between orthogonal bell states while emitting a continual stream of photons . as an aside , such behavior can be regarded as a quantum measurement that distinguishes the bell states @xmath148 from @xmath149 . the two alternative outcomes of the quantum trajectory evolution are illustrated in figs . [ fig : fig7 ] and [ fig : fig8 ] for the initial states @xmath150 in @xmath78 and @xmath132 in @xmath79 , respectively . with this choice of initial states there are equal probabilities for reaching the steady states , @xmath151 [ fig . [ fig : fig7](a ) ] and @xmath152 [ fig . [ fig : fig8](a ) ] , and for commencing an entangled - state cycle [ figs . [ fig : fig7](b ) and [ fig : fig8](b ) ] . note that once an entangled - state cycle is initiated , the trajectory remains in a plane orthogonal to the lines defining @xmath151 and @xmath152 ; the cycle continues indefinitely our original model allowed for the possibility of imperfect intercavity coupling , through the parameter @xmath29 and the jump operator @xmath66 which describe the effects of photon loss in propagation between the two cavities . focusing on the resonant case ( @xmath86 ) , we now consider the situation in which @xmath153 . typical trajectories for @xmath154 are shown in figs . [ fig : fig9](a ) and [ fig : fig10](a ) , with the two photon count records shown in frames ( b ) and ( c ) of the figures . remarkably , entangled - state cycles persist , but now the system settles into one or other of two distinct cycles , involving either the symmetric or antisymmetric bell states . to understand the behavior , consider the forms of the operators involved ; in particular , for @xmath86 , we have effective hamiltonian @xmath155 and jump operators @xmath156 and @xmath157 significantly , these operators commute with one another , @xmath158=[\hat c_1,\hat h_{\rm eff}]=[\hat c_2,\hat h_{\rm eff}]=0.\end{aligned}\ ] ] their operation upon the bell states is given by @xmath159 and @xmath160 thus , the bell states are eigenstates of @xmath91 , and the jump operators interchange bell states in @xmath78 and @xmath79 : each jump operator converts the symmetric ( antisymmetric ) bell state in @xmath78 to the symmetric ( antisymmetric ) bell state in @xmath79 and vice - versa . now , let us consider a particular quantum trajectory for which a total of @xmath161 jumps occur , separated by the time intervals @xmath162 . for an initial state @xmath163 , the ( unnormalized ) state at the conclusion of the @xmath161 jumps is written as @xmath164 where each @xmath165 is either @xmath65 or @xmath66 . since all operators in the string acting on @xmath166 commute , this expression can be rewritten in a variety of forms , two of which prove to be especially useful in explaining the distinct behaviors illustrated by figs . [ fig : fig9 ] and [ fig : fig10 ] . in the first case , we may write @xmath167 passing all @xmath168 ocurrences of @xmath65 to the right and all @xmath169 occurrences of @xmath66 to the left ( @xmath170 ) ; in the second we write @xmath171 where all jump operators are passed to the left . the arbitrary ( pure ) initial state can be expressed as a superposition of bell states , @xmath172 where @xmath173 , @xmath174 , @xmath175 , and @xmath176 are expansion coefficients , generally complex . substituting this expansion into eqs . ( [ phit1 ] ) and ( [ phit2 ] ) , and using eqs . ( [ heff1])([c2psi])assuming for simplicity that @xmath168 and @xmath169 are even the two forms for the state @xmath177 are @xmath178 where @xmath179 observe now that the ratio of the eigenvalues satisfies @xmath180 it follows that @xmath181 and @xmath182 allow us to predict quite distinct asymptotic behaviors for the system state . for sufficiently large @xmath168 , the contribution to @xmath181 from the symmetric bell states is negligible compared with the contribution from the antisymmetric bell states , in which case , using eqs . ( [ eq : cycle1 ] ) and ( [ eq : cycle1prime ] ) , @xmath183 the system is locked into a cycle between the two antisymmetric bell states , the situation illustrated in fig . [ fig : fig9 ] ( for @xmath184 , @xmath185 ) . in contrast , for sufficiently large @xmath16 , the contribution to @xmath182 from the antisymmetric bell states is negligible compared with that from the symmetric bell states , and using eqs . ( [ eq : cycle2 ] ) and ( [ eq : cycle2prime ] ) , @xmath186 the system is locked into a cycle between the two symmetric bell states , as shown in fig . [ fig : fig10 ] . which of the two cycles is chosen in a particular realization of the photon counting record is random , as is the time taken to settle into the cycle . effectively , the decision is the outcome of a competition between the periods of evolution between quantum jumps and the jumps themselves specifically , those associated with photon counts at detector 1 . considering eqs . ( [ eq : cycle1prime ] ) and ( [ eq : ratio ] ) , we see that every count at detector 1 results in an increased probability to find the system in one of the antisymmetric bell states . on the other hand , from eqs . ( [ eq : cycle2prime ] ) and ( [ eq : ratio ] ) , the periods of evolution between counts have the reverse effect they increase the probability for the system to be found in a symmetric bell state . the critical factor that decides which tendency wins is the number of photon counts occuring at detector 1 over a given ( substantial ) interval of time . if there are many , as in fig . [ fig : fig9](b ) , the entangled - state cycle between antisymmetric bell states wins out ; if there are few , fig . [ fig : fig10](b ) , the cycle between symmetric bell states occurs . the same decision mechanism is observed in other examples @xcite . note that counts at detector 2 are not involved not directly at least . they do figure indirectly as a mechanism reducing the average number of counts at detector 1 ; indeed , they are the ultimate source of the asymmetry reflected in the ratio @xmath187 . as the system approaches a particular cycle the quantum trajectory evolution tends to reinforce the establishment of the cycle . close to the antisymmetric cycle , the evolution between jumps is dominantly governed by @xmath188 and is therefore relatively fast . this leads to frequent photon counts at detector 1 [ fig . [ fig : fig9](b ) ] . close to the symmetric cycle , the between - jump evolution is dominantly governed by @xmath189 , hence is relatively slow . photon counts at detector 1 become much less frequent [ fig . [ fig : fig10](b ) ] . from the dramatic difference in count rates at detector 1 for the two cycles , it is clear that one can determine which entanglement cycle the system evolves to for a particular realization . however , without knowledge of the record of photon counts at detector 2 , which by definition we do not have , one can not know where on the cycle the system is , i.e. , whether the state is in @xmath78 or @xmath79 . thus , the ensemble average state of the system is mixed , described by one of the density operators @xmath190 consider a thought experiment where the cascaded qubit system , set to resonance , evolves freely and its entire output is collected and stored inside a black box . at some time the lasers driving the raman transitions are turned off , so the evolution ceases . the box and qubits are separated and moved to causally disconnected regions of space time . let alice and bob be standard observers of the qubits , and give eve jurisdiction over the box . we can now ask , how much entanglement exists between the qubits of alice and bob ? while this is simply a roundabout way of asking how entanglement evolves , it helps elucidate some of the key concepts behind the quantum trajectory measure of entanglement . conventional entanglement measures are based upon an analysis of the density matrix at this time . they throw away the box and look at the system of qubits alone they disregard eve and view the system from the perspective of alice and bob . yet in general every interaction between two objects entangles them , and as the qubit system and box interacted in the past , their states are intertwined . neither possess an independent reality , and neither , considered alone , can be completely described . eve s box contains information , which , if discarded , adds entropy to the qubit system of alice and bob . this entropy is the source of ambiguity in the quantification of entanglement . from this point of view , as noted in the introduction , the problem of bi - partite entanglement in an open system relates to that of tri - partite entanglement in a closed one . to completely characterize the entanglement of the present example , in addition to the entanglement between alice and bob , we must consider their entanglement with eve . a quantum description of the box is impractical , but it is feasible to extract classical information about what it contains , through measurement . quantum trajectories facilitate this , and allow us not to discard the box completely . in turn , the system state retains its purity , conditional on the classical information extracted from the box . with this extra information , we can extract more entanglement from the cascaded qubit system . working from the master equation for the cascaded system @xcite , previously it was assumed that the system evolved gradually into a pure state , whereby entanglement was generated . the behaviour at resonance , however , was unclear , since there the master equation had two zero eigenvalues and no well - defined steady state . by considering the conditional evolution we have shown that , at resonance , asymptotically the system is either in the bell state @xmath151 or oscillating ( stochastically switching ) between two bell states , @xmath146 and @xmath147 . from the density matrix point of view , the latter is an equal mixture of bell states and would yield no entanglement under any mixed state measure ; physically , alice and bob , without collaboration from eve , can not extract any entanglement from their qubits . suppose , however , that eve opens her box to count the number of photons inside . seeing whether the count is even or odd , she is able to deduce exactly which bell state alice and bob s system is in . thus , her measurement unravels the density operator , creating entanglement , despite the fact that the measurement is not causally connected to alice and bob s qubits . it is tempting to say that the entanglement was always there , as a matter of fact , until one realizes that there are many other ways in which eve could choose to measure her state , each producing a different unravelling of the qubit system and yielding a different value of entanglement . the entanglement facilitated by eve s measurements is _ contextual _ in this sense . this thought experiment demonstrates why any attempt to quantify the entanglement of an open system from the density operator alone can not be considered complete . the density operator should not be treated as a fundamental object , as it does not provide a complete description of the physical state . we have presented a simple example where oscillations between maximally entangled states are hidden within a separable density operator . the fact that the density operator contains entropy , implies that information about its entanglement with an external system was discarded at some time . in studying such a mixed state , there is benefit from considering , not only the mixed state itself , but the process through which it was generated , and the access this potentially gives to a conditional dynamics . the results of this paper could be extended by employing quantum trajectories in a broader sense . in cases where the results of environmental interactions can not be measured , such as coupling loss , wiseman and vaccaro @xcite have shown that only certain unravelings can be physically realized . a conceivable measure of entanglement would take the minimum of all physically realizable unravelings . alternatively , one might take the maximum of all physically realizable unravellings , which would measure the maximum distillable entanglement when local measurements on the environment are taken into account . h. j. carmichael , p. kochan , and l. tian , `` coherent states and open quantum systems : a comment on the stern - gerlach experiment and schrdinger cats , '' in _ coherent states : past , present , and future _ , eds . d. h. feng , j. r. klauder , and m. r. strayer ( world scientific , singapore , 1994 ) , pp . 75 - 91 .

a system of cascaded qubits interacting via the oneway exchange of photons is studied . while for general operating conditions the system evolves to a superposition of bell states ( a dark state ) in the long - time limit , under a particular _ resonance _ condition no steady state is reached within a finite time . we analyze the conditional quantum evolution ( quantum trajectories ) to characterize the asymptotic behavior under this resonance condition . a distinct bimodality is observed : for perfect qubit coupling , the system either evolves to a maximally entangled bell state without emitting photons ( the dark state ) , or executes a sustained entangled - state cycle random switching between a pair of bell states while emitting a continuous photon stream ; for imperfect coupling , two entangled - state cycles coexist , between which a random selection is made from one quantum trajectory to another .

You are an expert at summarizing long articles. Proceed to summarize the following text: a _ spherical curve _ is a smooth immersion of the circle into the sphere where the self - intersections are transverse and double points ( we call the double point _ crossing _ ) . in this paper we assume every spherical curve is oriented , and has at least one crossing . we represent , if necessary , the orientation of a spherical curve by an arrow as depicted in the left - hand side of fig . [ ex - reductivity ] . a spherical curve @xmath0 is _ reducible _ and has a _ reducible crossing _ @xmath1 if @xmath0 has a crossing @xmath1 as shown in fig . [ red ] , where @xmath2 and @xmath3 imply parts of the spherical curve . @xmath0 is _ reduced _ if @xmath0 is not reducible such as the spherical curves in fig . [ ex - reductivity ] . note that around a reducible ( resp . non - reducible ) crossing , there are exactly three ( resp . four ) disjoint regions , where a _ region _ of a spherical curve is a part of the sphere divided by the spherical curve . + a _ half - twisted splice _ is the local transformation on spherical curves as depicted in fig . [ halftwisted ] ( @xcite ) . then the inverse is the transformation depicted in fig . [ halftwisted - i ] . in this paper we call the inverse of the half - twisted splice _ inverse - half - twisted splice _ , and denote by @xmath4 . we remark that the half - twisted splice and the inverse - half - twisted splice do not preserve the orientation of spherical curves . then we give an orientation again to the spherical curve we obtain . we also remark that the half - twisted splice and the inverse - half - twisted splice do not depend on the orientations of spherical curves , but depend only on the relative orientations . now we define the reductivity . the _ reductivity _ @xmath5 of a spherical curve @xmath0 is the minimal number of @xmath4 which are needed to obtain a reducible spherical curve from @xmath0 . for example , a reducible spherical curve has the reductivity 0 , and the spherical curves @xmath0 , @xmath6 and @xmath7 in fig . [ ex - reductivity ] have the reductivity 1 , 2 and 3 , respectively ( see fig . [ eight - two ] for @xmath6 . note that we can not obtain a reducible curve by any single @xmath4 from @xmath6 . ) is 2.,width=377 ] in this paper , we show the following theorem : every spherical curve has the reductivity four or less . [ main ] this implies that we can obtain a reducible spherical curve from any spherical curve by four or less @xmath4 . we have the following question . is it true that every spherical curve has the reductivity three or less ? [ main - q ] in other words , is it true that there are no spherical curve with reductivity four ? the rest of this paper is organized as follows : in section 2 , we discuss the properties of reductivity by considering chord diagrams , and prove theorem [ main ] . in section 3 , we study the unavoidable sets of tangles for spherical curves as an approach to question [ main - q ] . in this section we show theorem [ main ] by using chord diagrams . we consider a spherical curve @xmath0 as an immersion @xmath8 of the circle into the sphere with some double points ( crossings ) . a _ chord diagram _ for a spherical curve is an oriented circle considered as the preimage of the immersed circle with chords connecting the preimages of each crossing ( @xcite ) . a chord diagram is for a reducible spherical curve if and only if the chord diagram could have a chord without crossings . for example , the chord diagram in fig . [ chord - ex ] has the chord labeled 4 without crossings . the move @xmath4 at a crossing @xmath1 on a spherical curve corresponds to the move shown in fig . [ a - on - chord ] on the chord diagram ; cut the chord diagram along the chord labeled @xmath1 ( then @xmath1 vanishes ) , turn over the semicircle , and join the two semicircles again . on a chord diagram.,width=491 ] for example , @xmath4 at the crossing labeled @xmath9 on the spherical curve in fig . [ chord - ex ] is represented on the chord diagram as shown in fig . [ a - on - chord - ex ] . at 2 on a chord diagram.,width=302 ] a region of a spherical curve is _ coherent _ ( resp . _ incoherent _ ) if the boundary of the region is coherent ( resp . incoherent ) ( see , for example , fig . [ bigons ] ) . a coherent bigon and an incoherent bigon are represented by chord diagrams as shown in fig . [ chord - bigon ] . for coherent and incoherent bigons , we have the following lemmas : if a spherical curve @xmath0 has an incoherent bigon , then @xmath10 . by applying @xmath4 at one of the crossing on an incoherent bigon , we obtain a reducible spherical curve as shown in fig . [ c - non - c-2 ] . [ non - c-2 ] if a spherical curve @xmath0 has a coherent bigon , then @xmath11 . if @xmath0 is reducible , the reductivity is zero . if @xmath0 is reduced , there is a crossing @xmath1 as shown in the chord diagram in fig . [ c - c-2 ] . by applying @xmath4 at @xmath1 , we obtain a spherical curve which has an incoherent bigon . hence @xmath0 has the reductivity two or less . [ c-2 ] a trigon of a spherical curve is one of the types a , b , c and d in fig . [ abcd ] with respect to the outer connections . these trigons are represented in chord diagrams as shown in fig . [ chord - trigon ] . we have the following lemmas for trigons of type a and b. if a spherical curve @xmath0 has a trigon of type a , then @xmath11 . by applying @xmath4 at @xmath1 in fig . [ c-3a ] , we have a spherical curve which has an incoherent bigon . [ 3a ] if a spherical curve @xmath0 has a trigon of type b , then @xmath12 . by applying @xmath4 at @xmath1 in fig . [ c-3b ] , we have a spherical curve which has a coherent bigon . [ 3b ] a _ connected sum _ @xmath13 of two spherical curves @xmath0 and @xmath6 is a spherical curve as depicted in fig . [ conn ] . for trigons of type c , we have the following lemma : if a spherical curve @xmath0 has a trigon of type c , then @xmath12 . if the arcs @xmath14 , @xmath15 , and @xmath16 in fig . [ lmn - c ] around a trigon of type c have no mutual crossings , then @xmath0 is a connected sum of a trefoil ( _ trefoil _ is the spherical curve depicted in the left - hand side of fig . [ ex - reductivity ] ) and some ( trivial or non - trivial ) spherical curves as depicted in fig . [ conn - trefoil ] , whose reductivity is one or zero . if @xmath0 has a mutual crossing @xmath1 of @xmath14 , @xmath15 and @xmath16 , we obtain a spherical curve which has a trigon of type a by applying @xmath4 at @xmath1 ( see fig . [ c-3c ] ) . [ 3c ] for type d , we have the following lemma : if a spherical curve @xmath0 has a trigon of type d , then @xmath17 . if @xmath0 is reducible , the reductivity is zero . if @xmath0 is reduced , @xmath0 has a crossing @xmath1 as shown in the chord diagram in fig . apply @xmath4 at @xmath1 , and we obtain a trigon of type b. hence the reductivity of @xmath0 is four or less . [ c-3d ] adams , shinjo and tanaka showed the following lemma @xcite : every reduced spherical curve has a bigon or trigon . [ ast - lem ] they proved this lemma by considering the equality @xmath18 which is obtained by the euler characteristic of @xmath19 , where @xmath20 is the number of @xmath16-gons . we prove theorem [ main ] . _ proof of theorem [ main ] . _ if @xmath0 is reducible , then @xmath21 . if @xmath0 is reduced , then @xmath17 because @xmath0 has a bigon or trigon . in this section , we discuss the unavoidable sets of tangles for reduced spherical curves . a _ tangle _ of a spherical curve is a part of the spherical curve . in this section we do not distinguish the mirror image of tangles . a set @xmath23 of tangles is an _ unavoidable set for spherical curves _ if any spherical curve has at least one tangle in @xmath23 . for example , the set @xmath24 in fig . [ s1 ] is an unavoidable set for _ reduced _ spherical curves because of lemma [ ast - lem ] . is an unavoidable set.,width=226 ] we have the following proposition : the sets @xmath25 , @xmath26 and @xmath27 in fig . [ s11 ] are unavoidable sets for reduced spherical curves . , @xmath26 and @xmath27 are unavoidable sets.,width=491 ] we use the discharging method which is used in graph coloring theory ( see , for example , @xcite ) . we assume there exists a reduced spherical curve @xmath0 which has none of @xmath25 . then @xmath0 has trigons by lemma [ ast - lem ] , and all the regions around each trigon are 5-gons or more . we assign a charge of @xmath28 to each @xmath29-gon . then each trigon receives 1 , 4-gon receives 0 , 5-gon receives -1 , @xmath30 . the total charge @xmath31 on @xmath0 is @xmath32 where @xmath20 is the number of @xmath16-gons . by the equality @xmath33 for lemma [ ast - lem ] and that @xmath34 , we have @xmath35 . now we do discharging ; at each trigon , we take off the charge 1 , and distribute it to the 6 regions around the trigon by @xmath36 ( see fig . [ discharging ] ) . after the discharging , each trigon has the charge 0 because it has been taken the initial charge , and receive no charge from other trigons because they are not adjacent . each 4-gon has also 0 because the initial charge was 0 and receive no charge from trigons . next we consider 5-gons . if a 5-gon @xmath37 has @xmath15 trigons around it , then @xmath37 has the charge @xmath38 after the discharging . remark @xmath39 because if @xmath40 , some of the trigons around @xmath37 are adjacent . hence each 5-gon has a negative charge after the discharging . for @xmath16-gons ( @xmath41 ) , they have @xmath42 after the discharging , where @xmath15 is the number of trigons around them . we have @xmath43 because if @xmath44 , then @xmath45 for @xmath46 , and this means all the @xmath47 regions around the @xmath16-gon are trigons , and they are adjacent . therefore , after discharging , the total charge turns a negative number despite that the discharging preserves the total charge 8 . hence @xmath25 is an unavoidable set . it is similarly shown for @xmath26 and @xmath27 by considering the discharging @xmath48 and @xmath49 in fig . [ discharging2 ] , respectively . and @xmath49.,width=491 ] we have the following question : is the set @xmath50 in fig . [ s2 ] an unavoidable set for reduced spherical curves ? if the answer to this question is yes , then the answer to question [ main - q ] is also yes . a set @xmath2 of tangles is an _ avoidable set _ for spherical curves if @xmath2 is not an unavoidable set , i.e. , there is a spherical curve which has no tangles in @xmath2 . since we have the spherical curve in fig . [ counter ] , the sets @xmath51 and @xmath52 in fig . [ t1 ] are avoidable sets for reduced spherical curves . remark that @xmath52 is the set consisting of the tangles of the bigon and the _ incoherent _ trigons . the author is deeply grateful to professors tetsuya abe , reiko shinjo and kokoro tanaka for valuable discussions and suggestions . she is also grateful to the members of the friday seminar on knot theory at osaka city university and kobe topology seminar for helpful advice and encouragements . she is supported by grant - in - aid for research activity start - up ( 24840030 ) . 99 c. c. adams , r. shinjo and k. tanaka , complementary regions of knot and link diagrams , _ ann . _ * 15 * ( 2011 ) , 549563 . j. a. calvo , knot enumeration through flypes and twisted splices , _ j. knot theory ramifications _ * 6 * ( 1997 ) , 785798 . m. goussarov , m. polyak and o. viro , finite type invariants of classical and virtual knots , _ topology _ * 39 * ( 2000 ) , 10451068 . n. ito and a. shimizu , the half - twisted splice operation on reduced knot projections , _ j. knot theory ramifications _ * 21 * , 1250112 ( 2012 ) [ 10 pages ] . r. wilson , _ four colors suffice _ , princeton science library ( 2002 ) .

we show that we can obtain a reducible spherical curve from any non - trivial spherical curve by four or less inverse - half - twisted splices , i.e. , the reductivity , which represents how reduced a spherical curve is , is four or less . we also discuss unavoidable sets of tangles for spherical curves .

You are an expert at summarizing long articles. Proceed to summarize the following text: a celebrated invariant of a knot @xmath3 in 3-space is its _ function _ @xmath4 defined for complex numbers of absolute value @xmath5 , and taking values in the set of integers . the signature function of a knot is a concordance invariant , and plays a key role in the study of knots via surgery theory , @xcite . it turns out that the signature function is a locally constant function away from the ( possibly empty ) set @xmath6 of roots of the alexander polynomial on the unit circle . in view of this , the interesting part of the signature function is its _ jumping behavior _ on the set @xmath7 . in other words , we may consider the associated _ jump function _ @xmath8 defined by @xmath9 . we may identify the jump function with a _ jump divisor _ @xmath10 $ ] in @xmath11 . since @xmath12 and @xmath13 , it follows that the jump function uniquely determines the signature function away from the set @xmath7 . since @xmath14 , it follows in particular that @xmath15 determines the _ _ of the knot @xmath16 . the signature of a knot may be defined using a seifert surface of a knot ( see section [ sub.symmetries ] below ) . an intrinsic definition of the jump function of a knot was given by milnor @xcite , using the blanchfield pairing of the universal abelian cover of a knot . this definition , among other things , makes evident the role played by the roots of the alexander polynomial on the unit circle ( as opposed to the rest of the roots of the alexander polynomial , which are ignored ) . from the point of view of gauge theory and mathematical physics , the signature function of a knot may be identified with the spectral flow of a 1-parameter family of the signature operator , twisted along abelian ( that is , @xmath0-valued ) representations of the knot complement . the moduli space of @xmath0 representations of the knot complement is well understood ; it may be identified with the unit circle . on the other hand , the moduli space of @xmath1 representations is less understood , and carries nontrivial topological information about the knot and its dehn fillings , as was originally discovered by casson ( see @xcite ) and also by x - s . lin ; see @xcite . one may ask to identify the @xmath0 representations which _ deform to _ irreducible @xmath1 representations . using a linearization argument , klassen and frohman showed that a necessary condition for a @xmath0 representation @xmath17 to deform is that @xmath18 . this brings us to the ( square of the ) set @xmath7 . conversely , frohman - klassen proved sufficiency provided that the alexander polynomial has _ simple roots _ on the unit circle ; see @xcite . herald proved sufficiency under the ( more relaxed condition that ) the jump function vanishes nowhere ; see @xcite . it is unknown at present whether sufficiency holds without any further assumptions . let us summarize the two key properties of the jump divisor @xmath19 , in the spirit of mazur ( see @xcite ) : * the jump divisor controls the signature function of a knot . * the jump divisor controls ( infinitesimally ) deformations of @xmath0 representations of the knot complement to irreducible @xmath1 representations . it is a long standing problem to find a formula for the function of a knot in terms of its _ colored jones function_. the latter is a sequence of jones polynomials associated to a knot . recall that given a knot @xmath3 and a positive integer @xmath20 ( which corresponds to an @xmath20-dimensional irreducible representation of @xmath21 ) , one can define a laurrent polynomial @xmath22 $ ] . in @xcite , rozansky considered a repackaging of the sequence @xmath23 . namely , he defined a sequence of rational functions @xmath24 for @xmath25 with the following properties : * @xmath26 for some polynomials @xmath27 $ ] with @xmath28 and such that @xmath29 . * for every @xmath20 we have : @xmath30\ ] ] where @xmath31 $ ] is the ring of formal power series in @xmath32 with rational coefficients equation is often called the _ euler expansion _ of the colored jones function . in physical terms , the above expansion is an asymptotic expansion of the chern - simons path integral of the knot complement , expanded around a backround @xmath0 flat connection . thus , philosophically , it should not be a surprise to discover that this expansion has something to do with the signature of the knot . for the curious reader , let us point out that rozansky conjectured such an expansion for the full kontsevich integral of a knot , graded by the negative euler characteristic of graphs ( thus the name , euler expansion ) . this conjecture was proven by kricker and the author ; @xcite . furthermore , a close relation was discovered between residues of the rational functions @xmath33 at roots of unity and the lmo invariant of cyclic branched coverings of the knot ; @xcite . in an attempt to understand the euler expansion , a theory of finite type invariants of knots ( different from the usual theory of vassiliev invariants ) was proposed in @xcite . according to that theory , two knots are @xmath34-equivalent iff they are @xmath35-equivalent ; @xcite . moreover , @xmath33 is a finite type invariant of type @xmath36 . technically , the euler expansion of the colored jones function is an integrality statement . namely , it is easy to see that there exist unique sequence of power series @xmath37 $ ] for @xmath25 that satisfies equation . the hard part is to show that these power series are taylor series expansions of rational functions with integer coefficients and prescribed denominators . the statement @xmath28 in the leading term of the euler expansion is nothing but the melvin - morton - rozansky conjecture , proven by bar - natan and the author in @xcite . thus , the leading order term in the euler expansion is a well - understood topological invariant of knots . ever since the euler expansion was established , it has been a question to establish a topological understanding of the lower order terms . consider @xmath38 where @xmath39 . we will think of @xmath40 as a function ( with singularities ) defined on the unit circle . if @xmath41 is a root of the alexander polynomial on @xmath11 , we may expand @xmath42 around @xmath43 . the result is a power series with lowest term @xmath44 , for some integer @xmath45 and some nonzero real number @xmath46 . let us define the _ jones jump function _ of a knot @xmath3 @xmath47 by @xmath48 where @xmath49 is the imaginary part of a complex number @xmath50 and @xmath51 is the _ sign _ of a real number @xmath52 is defined by @xmath53 or @xmath54 according to @xmath55 or @xmath56 respectively . we say that a knot @xmath3 is _ simple _ if its alexander polynomial @xmath57 has simple roots on the unit circle . if @xmath3 is simple , then @xmath58 . a modest corollary is : if @xmath3 is simple , conjecture [ conj.1 ] implies that the colored jones function of @xmath3 determines the signature @xmath16 . notice that @xmath59 and @xmath60 . thus , it suffices to check the conjecture on the upper semicircle . the conjecture is false if @xmath57 has multiple roots ( of odd or even multiplicity ) . for example , consider the connected sum @xmath61 of @xmath20 right trefoils . then , @xmath62 and @xmath63 . we present the following evidence for the conjecture : @xmath64 conjecture [ conj.1 ] is true for torus knots , and for knots with at most 8 crossings . + @xmath65 the conjecture is compatible with the operations of mirror image , connected sum ( assuming the resulting knot is simple ) and @xmath66 parallels of knots . en route to establish our results , we give a skein formula that uniquely characterizes the jump function of simple knots ; see theorem [ thm.unique ] . let us compare conjecture [ conj.1 ] with existing conjectures about the structure of the colored jones function . at the time of the writing , there are two conjectures that relate the colored jones function to hyperbolic geometry . namely , * the _ hyperbolic volume conjecture _ , after kashaev and j&j.murakami , which states that for a hyperbolic knot @xmath3 , @xmath67 where @xmath68 . * the _ characteristic equals deformation variety conjecture _ , due to the author , which compares the deformation curve of @xmath69 representations of a knot complement ( viewed from the boundary ) with a complex curve which is defined using the recursion relations ( with respect to @xmath20 ) of the sequence @xmath23 ; see @xcite and @xcite . the hyperbolic volume conjecture is an analytic statement , which involves the existence and identification of a sequence of real numbers . on the other hand , the characteristic equals deformation variety conjecture is an algebraic statement , since it is equivalent to the equality of two polynomials with integer coefficients , one of which is obtained by noncommutative elimination , and the other obtained by commutative elimination . conjecture [ conj.1 ] appears to be an analytic conjecture , since its basic ingredients are signs of real numbers . in the field of quantum topology , analytic conjectures have held the longest . let us end the introduction with the following understand the underlying geometry and perturbative quantum field theory behind the taylor expansion of the @xmath70 function ( and more generally , euler expansion of the colored jones function ) . in particular , use the higher order terms @xmath33 in the expansion to formulate a conjecture for the jump function of all knots . the author wishes to thank s. orevkov , l. rozansky and a. stoimenov , and especially j. levine for helpful conversations . given a seifert matrix @xmath71 of a knot @xmath3 , consider the _ hermitian matrix _ @xmath72 , for @xmath73 . the eigenvalues of @xmath74 are real , and we define @xmath75 , where @xmath76 denotes the _ signature _ of a hermitian matrix @xmath77 . it turns out that @xmath78 is independent of the seifert surface @xmath71 chosen . since @xmath79 , where @xmath80 , and @xmath81 is the _ symmetrized alexander polynomial _ of @xmath3 , it follows that @xmath78 is a locally constant function with possible jumps along the set @xmath7 . the next lemma , which follows from the proof of ( * ? ? ? * corollary 2 ) , summarizes the symmetries of the jump function . if @xmath17 is a root of the alexander polynomial on @xmath11 , then @xmath82 , where * @xmath83 is an integer * @xmath84 , where @xmath85 is the multiplicity of @xmath17 in @xmath57 , and * @xmath86 . moreover , @xmath87 . in particular , if @xmath3 is simple , @xmath15 takes values in the set @xmath88 . for a precise formula for the jump function in that case , see theorem [ thm.jump ] . let us begin with a useful definition . a triple of links @xmath89 is called _ bordered _ if there is an embedded ball @xmath90 in @xmath91 that locally intersects them as in figure [ crossing ] . @xmath92 if we choose planar projection and a crossing , then a bordered triple corresponds to replacing the crossing by a positive , negative or smoothening . notice that if @xmath93 is a link with @xmath5 component , then @xmath94 and @xmath95 are links with @xmath5 and @xmath96 components respectively . the next lemma computes the change of the signature function with respect to the change of a crossing , in terms of the sign of the alexander polynomials . if @xmath3 is a knot , @xmath97 such that @xmath98 , then @xmath99 we can choose seifert surfaces @xmath100 for @xmath101 such that @xmath102 where @xmath103 and @xmath104 are some row vectors . hermitianizing , we get : @xmath105 let us call a triple of hermitian matrices @xmath106 _ @xmath17-bordered _ if @xmath107 for @xmath108 and some row vector @xmath109 . using lemma [ lem.tbordered ] the result follows . if @xmath106 is a @xmath17-bordered triple , and @xmath110 , then @xmath111 this is well - known for @xmath112 ; @xcite and also ( * ? ? ? * lemma 3.1 ) . we give a proof here for all @xmath17 . by a similarity transformation ( that is a replacement of @xmath113 by @xmath114 where @xmath115 is an invertible matrix , and @xmath116 is the conjugate transpose of @xmath115 ) , we can assume that @xmath117^r \oplus d , \ ] ] where @xmath118 is a nonsingular diagonal matrix , @xmath119^r$ ] is the zero @xmath120 matrix , @xmath109 is a @xmath121 vector and @xmath122 a real number . since the nullity ( that is , the dimension of the kernel ) and the signature of the matrix @xmath123 are given by : [ cols="<,^,^,^",options="header " , ] the result follows by a case - by - case argument . the next theorem computes the jump function of a simple knot in terms of a relative sign of alexander polynomials . first , a preliminary definition . if @xmath124 is a real - valued analytic function of @xmath52 in a neighborhood of @xmath122 , we define _ the sign of @xmath125 at @xmath122 _ @xmath126 to be the sign of the first nonvanishing taylor series coefficient ( around @xmath122 ) , if there is such , and zero otherwise . in other words , if @xmath127 , we have : @xmath128 where @xmath129 for @xmath130 and @xmath131 . notice that if @xmath132 , then @xmath133 , and that if @xmath122 is a simple root , then @xmath134 where @xmath135 is sufficiently small and positive . fix a simple knot @xmath3 and a complex number @xmath136 . choose a planar projection of @xmath3 and a crossing ( positive or negative ) . then , @xmath137 , where @xmath138 is the sign of the chosen crossing . suppose that @xmath139 . such a projection and choice of crossing will be called @xmath140-_good_. fix @xmath140 as above . for every @xmath140-good projection , we have @xmath141 without loss of generality , let us assume @xmath142 , that is @xmath143 . we will apply lemma [ lem.skeins ] twice to @xmath144 and @xmath145 for sufficiently small positive @xmath135 . under these assumptions , we have that @xmath146 ( since @xmath17 is an isolated root of a polynomial ) and @xmath147 ( since @xmath148 by assumption ) , and similarly for @xmath149 . thus , the hypothesis of lemma [ lem.skeins ] are satisfied . applying lemma [ lem.skeins ] twice , we get @xmath150 and @xmath151 now , subtract and remember that @xmath152 is continuous at @xmath17 since @xmath153 . we get @xmath154 since @xmath3 is simple , it follows that @xmath155 , thus the cases @xmath156 or @xmath157 do not occur above . thus , @xmath158 the result follows using remark [ rem.signf ] . indeed , @xmath159 and @xmath160 . there is a unique invariant @xmath161 defined for a simple knot @xmath3 and @xmath162 such that for every @xmath140-good projection we have : @xmath163 in view of theorem [ thm.jump ] , we need to prove that there is at most one such invariant . fix a simple knot @xmath3 and a complex number @xmath136 . we need to prove that there exists a @xmath140-good projection . start with any planar projection of @xmath3 and a crossing . if it is not good , apply reidemaster moves ii , which frohman - klassen call _ threading _ and improve it to be good , using the proof of ( * ? ? ? * theorem 6.2 ) . thus , conjecture [ conj.1 ] is equivalent to the following : @xmath64 for every simple knot @xmath3 , and every @xmath164 , we have @xmath165 . + @xmath65 moreover , for every @xmath140-good projection we have : @xmath166 in this section we will prove conjecture [ conj.1 ] for torus knots . let @xmath167 denote the @xmath168 _ torus knot _ , where @xmath169 are coprime natural numbers . for example , @xmath170 is the right - hand trefoil . the alexander polynomial of torus knots is given by : @xmath171 the roots of @xmath172 on the unit circle are @xmath173 complex roots of unity which are not @xmath122 or @xmath174 order roots of unity . they are all simple . using a useful parametrization of them , following kearton ( * ? ? ? * sec.13 ) , we obtain that latexmath:[\[\mathrm{roots}_{\d(t_{a , b})}=\{t(m , n):=e^{2 \pi i ( m / a+n / b ) } \ , @xmath59 , we need only compute the jump at the points @xmath176 where @xmath177 , @xmath178 and @xmath179 . in @xcite kearton computes the jump function of torus knots by @xmath180 in other words , we have : @xmath181 now we discuss the @xmath70 function of torus knots , which was originally computed by rozansky ( see ( * ? ? ? * eqn.(2.2 ) ) ) , and most recently , it has been recomputed by march and ohtsuki ; see @xcite . we understand that bar - natan has unpublished computations of the euler expansion of the kontsevich integral of torus knots . according to ( * eqn.(2.2 ) ) , the @xmath70 function of torus knots is given by : @xmath182 given an analytic function @xmath183 let us define @xmath184 we have that @xmath185 when we expand @xmath186 around a root @xmath41 , only the last two terms of the numerator contribute to the coefficient of @xmath187 . that is , @xmath188 now , suppose that @xmath183 is a laurrent polynomial with real coefficients that satisfies @xmath189 . then , @xmath190 . thus , @xmath191 and if we substitute @xmath192 , we get @xmath193 if @xmath194 is a simple root of @xmath195 on the unit circle ( as is the case for the alexander polynomial of torus knots ) , then the above real number is negative . this proves that @xmath196 and confirms conjecture [ conj.1 ] for torus knots . let @xmath125 denote either the @xmath70 function or the function of a knot . the following list describes some well - known properties of @xmath125 . * if @xmath197 denote the knot @xmath3 with opposite orientation , then @xmath198 . * if @xmath199 denote the mirror image of @xmath3 , then @xmath200 . * if @xmath201 denotes the connected sum of knots , then @xmath202 . * if @xmath203 denote the @xmath66 parallel of a knot @xmath3 with zero framing , then @xmath204 . the stated behavior of the signature function under @xmath66 parallel was proven by kearton @xcite , and for the @xmath70 function was proven by ohtsuki ( * ? ? ? 3.1 ) . from this , it follows that if conjecture [ conj.1 ] is true for a simple knot @xmath3 , then it is true for @xmath197 , @xmath199 , @xmath203 ( for all @xmath20 ) . furthermore , if @xmath205 is simple , and conjecture [ conj.1 ] is true for @xmath206 and @xmath207 , then it is also true for @xmath205 . in this section we will verify conjecture [ conj.1 ] by computer calculations . rozansky has written a maple program that computes the @xmath70 function of a knot ; see @xcite . we will use a minor modification qfunction.mws of rozansky s program , adopted for our needs . in qfunction.mws , the knot is described by a braid word . for example , @xmath208 $ ] represents the braid @xmath209 whose closure is the @xmath210 knot in classical notation . the command @xmath211)$ ] gives a list whose first , second and third entries are the braid word , the polynomials @xmath212 and @xmath57 , where @xmath213 . a sample output of the program is : .... > # the right trefoil 3_1 > br1([1,1,1 ] ) ; > 2 2 4 [ [ 1 , 1 , 1 ] , 1 + z , 2 z + z ] > # the 4_1 knot > br1([1,-2,1,-2 ] ) ; > 2 [ [ 1 , -2 , 1 , -2 ] , 1 - z , 0 ] > # the 7_2 knot > br1([-1,3,3,3,2,1,1,-3,2 ] ) ; > 2 2 4 [ [ -1 , 3 , 3 , 3 , 2 , 1 , 1 , -3 , 2 ] , 1 + 3 z , 12 z + 14 z ] > # 7_3 > br1([1,1,2,-1,2,2,2,2 ] ) ; > 2 4 [ [ 1 , 1 , 2 , -1 , 2 , 2 , 2 , 2 ] , 1 + 5 z + 2 z , 2 4 6 8 22 z + 65 z + 46 z + 9 z ] .... for example , for the right hand trefoil , we have : @xmath214 the mathematica program jjump.m computes the @xmath215 function . for example , we may launch the jjump.m program from a mathematica session . .... ( math100)/home / stavros : math mathematica 5.0 for sun solaris ( ultrasparc ) copyright 1988 - 2003 wolfram research , inc . -- motif graphics initialized -- in[1]:= < < jjump.m in[2]:= poles[1+z^2,2z^2+z^4 ] solve::ifun : inverse functions are being used by solve , so some solutions may not be found ; use reduce for complete solution information . out[2]= { { 0.16666666666666666667 , -0.00844343197019481429 } } .... we learn that the coefficient of @xmath216 of @xmath217 ( where @xmath218 is the right trefoil ) around the root @xmath219 , is @xmath220 . this computes that @xmath221 , as needed . similarly , .... in[4]:= poles[1 + 5z^2 + 2z^4,22z^2 + 65z^4 + 46z^6 + 9z^8 ] solve::ifun : inverse functions are being used by solve , so some solutions may not be found ; use reduce for complete solution information . out[4]= { { 0.075216475230034463796 , -0.00388836700144941422 } , > { 0.27241752919082620707 , -0.00542424178920663095 } } .... we learn that the coefficient of @xmath222 of @xmath223 around the roots @xmath224 and @xmath225 are @xmath226 and @xmath227 respectively . this computes the jump function @xmath228 for @xmath229 . now , let us compute the jump function of a knot . in @xcite orevkov gives a mathematica program sm.mat which takes as input a braid presentation of a knot , and gives as output a seifert surface of a knot . launching the jump.m version of it in a mathematica session produces .... ( math100)/home / stavros : math mathematica 5.0 for sun solaris ( ultrasparc ) copyright 1988 - 2003 wolfram research , inc . -- motif graphics initialized -- in[1]:= < < jump.m in[2]:= jump[{1,1,1 } ] inversefunction::ifun : inverse functions are being used . values may be lost for multivalued inverses . solve::ifun : inverse functions are being used by solve , so some solutions may not be found ; use reduce for complete solution information . out[2]= { -2 } .... which computes the jump function on the upper semicircle for the right trefoil @xmath218 . .... in[3]:= jump[{1,1,2,-1,2,2,2,2 } ] inversefunction::ifun : inverse functions are being used . values may be lost for multivalued inverses . solve::ifun : inverse functions are being used by solve , so some solutions may not be found ; use reduce for complete solution information . out[3]= { -2 , -2 } .... which computes the jump function on the upper semicircle for the @xmath230 knot . this confirms the conjecture for the @xmath218 and @xmath230 knots . in the appendix , we give the source code of two mathematica programs , jump.m and jjump.m which compute the @xmath231 and the @xmath215 function of knots . .... ( * poles[p , ap ] computes the poles of the rational functions p / ap^2 * ) ( * at the roots of ap=0 on the unit circle . p , ap are polynomials in z * ) ( * poles2[p , ap ] lists the coefficients of the taylor expansion at * ) ( * ( t - a)^{-2}. * ) ( * poles[p , ap ] lists { a , coefficient of taylor expansion at ( t - a)^{-2 } } * ) ff[x_]:=x[[2 ] ] ; poles[ap_,p_]:=module [ { quotient , apt , roots , poles , k } , quotient = simplify[p / ap^2 /. ( z->z^{1/2 } ) /. ( z->2 cos[2*pi*t]-2 ) ] ; apt= simplify[ap /. ( z->z^{1/2 } ) /. ( z->2 cos[2*pi*t]-2 ) ] ; roots = select[map[ff , flatten [ nsolve[apt = = 0 , t , 20 ] ] ] , 1/2 > # > 0 & ] ; poles= { } ; table[flatten[{roots[[k ] ] , coefficient[series[quotient,{t , roots[[k]],0 } ] , t - roots[[k]],-2 ] } ] , { k , length[roots ] } ] ] ( * for the 3_1 knot : poles[1+z^2,2z^2+z^4 ] * ) ( * for the 4_1 knot : poles[1-z^2,0 ] * ) ( * for the 7_2 knot : poles[1 + 3z^2,12z^2 + 14z^4 ] * ) ( * for the 7_3 knot : poles[1 + 5z^2 + 2z^4,22z^2 + 65z^4 + 46z^6 + 9z^8 ] * ) .... .... ( * computing the signature and jump function of knots presented as * ) ( * closures of braids . * ) ( * the signature of the right trefoil is signaturebraid[{1,1,1}]=-2 * ) ( * signaturem[a ] of a matrix a is the signature of a+a^ * * ) ( * jump[{1,1,1 } ] is the jumps of the signature of the right trefoil * ) jump[brd_]:=module [ { m , v , aps , hermitian , roots , k } , m = max[abs @ brd]+1 ; v = n[seifertmatrix[m , brd ] ] ; hermitian=(1-exp[2*pi*i*s])v+(1-exp[-2*pi*i*s ] ) transpose[v ] ; aps = n[det[(cos[2*pi*s/2]+i sin[2*pi*s/2])v-(cos[2*pi*s/2]-i sin[2*pi*s/2 ] ) transpose[v]],20 ] ; roots = select[map[ff , flatten [ nsolve[{aps = = 0 , im[s]==0 } , s , 15 ] ] ] , 1/2 > # > 0 & ] ; if[length[roots]==0 , { } , flatten[table[signaturem[hermitian /. s->(roots[[k]]+1/1000 ) ] -signaturem[hermitian /. s->(roots[[k]]-1/1000 ) ] , { k , length[roots ] } ] ] ] ] ( * 7_3 knot signaturebraid[{1,1,2,-1,2,2,2,2 } ] * ) ( * 7_5 knot signaturebraid[{1,1,1,1,2,-1,2,2 } ] * ) ( * 8_2 knot signaturebraid[{-1,2,2,2,2,2,-1,2 } ] * ) ( * 8_5 knot signaturebraid[{1,1,1,-2,1,1,1,-2 } ] * ) ( * 8_15 knot signaturebraid[{1,1,-2,1,3,3,2,2,3 } ] * ) ( * 7_3 , 7_5 , 8_2 , 8_5 , 8_15 have signature -4 * ) ....

the signature function of a knot is a locally constant integer valued function with domain the unit circle . the jumps ( i.e. , the discontinuities ) of the signature function can occur only at the roots of the alexander polynomial on the unit circle . the latter are important in deforming @xmath0 representations of knot groups to irreducible @xmath1 representations . under the assumption that these roots are simple , we formulate a conjecture that explicitly computes the jumps of the signature function in terms of the jones polynomial of a knot and its parallels . as evidence , we prove our conjecture for torus knots , and also ( using computer calculations ) for knots with at most @xmath2 crossings . we also give a formula for the jump function at simple roots in terms of relative signs of alexander polynomials .

You are an expert at summarizing long articles. Proceed to summarize the following text: one of the intriguing issues is not only to describe the late - time accelerated expansion of our universe but also to explain the smooth transition from decelerating phase to accelerating one . in the context of einstein theory of general relativity , the accelerating universe means that the parameter @xmath0 of equation of state @xcite is negative , where @xmath1 and @xmath2 are the energy density and the pressure , respectively . so , in the ordinary friedmann equation , the energy density is assumed to be positive while the pressure is negative . even more @xmath3 can be required to compensate the effect of ordinary matters in our universe . in some sense , it implies that the state parameter may depend on time and make it possible to explain the transition from decelerating phase to accelerating one . in the quintessence model based on supergravity or m / string theory , the transition has been studied in terms of the numerical simulation @xcite . on the other hand , a two - dimensional dilaton gravity may be useful in studying the transition from the decelerated phase and the accelerated phase because there are fewer degrees of freedom rather than the four - dimensional counterpart . furthermore , there exist exactly soluble models semiclassically @xcite , whose quantum back reactions of the geometry are easily treated so that various cosmological problems have been studied in refs . however , in even this semiclassically soluble gravity , it is difficult to realize the smooth phase transition because the solution shows the only decelerating or accelerating behavior . recently , it has been shown that it is possible to obtain the transition from decelerating phase to accelerating one by assuming the modified poisson brackets @xcite corresponding to noncommutativity of fields @xcite . unfortunately , the future singularity appears at finite time in this model , and the decelerated geometry has been patched by hand for the regularity . so , in this paper , we would like to study the smooth phase transition from the decelerated expansion to accelerated expansion without any curvature singularity in the bose - parker - peleg ( bpp ) model @xcite , which is one of the exactly soluble model semiclassically . in particular , even though the classical cosmological constant is not assumed , the initial state is asymptotically anti - de sitter ( ads ) and the late time behavior of our universe is asymptotically de sitter ( ds ) . this interesting feature is due to the noncommutativity in the modified poisson algebra . in sec . [ sec : bpp ] , we find the semiclassical hamiltonian in the bpp model and also define semiclassical energy - momentum tensors , and obtain the energy density and the pressure in view of a perfect fluid . in sec . [ sec : pb ] , solving the semiclassical hamiltonian equations of motion with the ordinary poisson brackets in the bpp model , we obtain the accelerated expansion solution . in sec . [ sec : mpb ] , we will take the modified poisson brackets instead of the conventional poisson algebra . under some conditions for integration constants , the solution shows that the smooth transition from ads ( decelerating ) phase at the past infinity to ds ( accelerating ) phase is possible . finally , some discussions are given in sec . [ sec : dis ] . in the low - energy string theory , the two - dimensional dilaton gravity are described by @xmath4 , \label{action : dg}\ ] ] and the conformal matter fields is given as @xmath5 , \label{action : cl}\ ] ] where @xmath6 and @xmath7 s are the dilaton and the conformal matter fields , respectively . we set the vanishing cosmological constant @xmath8 for simplicity in what follows . the quantum effective action for the conformal matter ( [ action : cl ] ) is written as @xmath9,\ ] ] where @xmath10 . the first term in eq . ( [ action : qt ] ) comes from the polyakov effective action of the classical matter fields @xcite and the other two local terms have been introduced in order to solve the semiclassical equations of motion exactly @xcite . the higher order of quantum correction beyond the one - loop is negligible in the large @xmath11 approximation where @xmath12 and @xmath13 , so that @xmath14 is assumed to be positive finite constant . in order to study consider the quantum back reaction semiclassically , we take the total action as @xmath15 in the conformal gauge , @xmath16 , the total action and the constraint equations are written as @xmath17 \label{action : conf}\end{aligned}\ ] ] and @xmath18 + \frac12\sum_{i=1}^n\left(\partial_\pm f_i \right)^2 + \kappa\left [ \partial_\pm^2\rho - \left(\partial_\pm\rho\right)^2\right ] \nonumber \\ & & \qquad\qquad\qquad - \kappa\left ( \partial_\pm^2\phi - 2\partial_\pm\rho\partial_\pm\phi\right ) - \kappa\left(\partial_\pm\phi\right)^2 - \kappa t_\pm = 0 , \label{constr : conf}\end{aligned}\ ] ] where @xmath19 reflects the nonlocality of the induced gravity of the conformal anomaly . then , we take the vanishing classical matter , @xmath20 in order to take into account only the quantum - mechanically induced source . defining new fields as @xcite @xmath21 the gauge fixed action is obtained in the simplest form of @xmath22\ ] ] and the constraints are given by @xmath23 in the homogeneous space , using the relations of @xmath24 , the lagrangian and the constraints are obtained , @xmath25 where the action is redefined by @xmath26 with @xmath27 , and the overdot denotes the derivative with respect to the conformal time @xmath28 . then , the hamiltonian becomes @xmath29 in terms of the canonical momenta @xmath30 , @xmath31 . since the semiclassical energy - momentum tensors are defined by @xmath32 , they can be written as @xmath33 ^ 2 \nonumber \\ & = & -\kappa t_\pm + \frac14 ( \ddot\chi - \ddot\omega ) - \frac{1}{4\kappa } ( \dot\chi - \dot\omega)^2 , \label{t++ } \\ t_{+-}^{\rm qt } & = & -\kappa \partial_+ \partial_- ( \chi - \omega ) \nonumber \\ & = & -\frac14 ( \ddot\chi - \ddot\omega ) . \label{t+-}\end{aligned}\ ] ] they can be regarded as a perfect fluid written in the form of @xmath34 where @xmath1 and @xmath2 are the energy density and the pressure , respectively , and @xmath35 is the 4-velocity vector field of flow . in the comoving coordinate , @xmath36 , the 4-velocity is given by @xmath37 , and then we can obtain the distributions of the energy density and the pressure . note that the comoving time are related to the conformal time , @xmath38 dt$ ] . then , the energy density and pressure are written as @xmath39 note that the state parameter @xmath40 has been defined as the equation of state @xmath41 . in this section , we would like to recapitulate the evolution of the two - dimensional universe by solving the semiclassical equations of motion in the bpp model . even if the solutions can be obtained directly from the lagrangian equations of motion , we will solve them in terms of the hamiltonian formulation since the latter case is more convenient to modify the original equations of motion . let us now define the conventional poisson brackets , @xmath42 and then the hamiltonian equations of motion in ref . @xcite are given by @xmath43 where @xmath44 represents fields and corresponding momenta . then they are explicitly written as @xmath45 since the momenta @xmath46 and @xmath47 are constants of motion , we can easily obtain the solutions , @xmath48 where @xmath49 , @xmath50 , @xmath51 , and @xmath52 are arbitrary constants . from the definition ( [ def : omega ] ) , the solution @xmath53 in eq . ( [ sol : omega_com ] ) must be positive . this leads to three cases of conformal time @xmath28 : one is @xmath54 with @xmath55 , another is @xmath56 with @xmath57 , and the other is @xmath58 with @xmath59 and @xmath60 . next , the dynamical solutions ( [ sol : omega_com ] ) and ( [ sol : chi_com ] ) should by satisfied with constraint ( [ con ] ) , which results in @xmath61 note that the integration functions @xmath62 determined by the matter state are time - independent . on the other hand , by using eqs . ( [ sol : omega_com ] ) and ( [ sol : chi_com ] ) , the curvature scalar is calculated as @xmath63 where the equality corresponds to the case of @xmath64 and @xmath65 , in other words , which means flat spacetime . plugging the constraint ( [ constr : com ] ) into eqs . ( [ t++ ] ) and ( [ t+- ] ) , the induced energy - momentum tensors are explicitly written as @xmath66 which yields from eqs . ( [ def : energy ] ) and ( [ def : pressure ] ) , @xmath67 . \label{energy : com}\ ] ] note that the state parameter is simply @xmath68 in this semiclassical case , and the curvature scalar which is proportional to the acceleration is always positive under the condition of @xmath69 . so , there is no phase transition from the deceleration to the acceleration , and we can not obtain the ads - ds phase transition . in this section , we now study whether the phase change of the universe is possible or not in the context of the modified semiclassical equations of motion . the similar analysis to the previous section will be done along with the noncommutative algebra @xcite , @xmath70 where @xmath71 is a positive constant . note that our starting semiclassical action seems to be quantized one more , however , this is not the case since these modified poisson brackets are simply the counterpart of the conventional poisson brackets which are not quantum commutators . if the fields had been taken as operators by decomposing the positive and the negative frequency modes along with the normal ordering , then it would be the quantization of a quantization . but our modified poisson brackets just modify the conventional ( semiclassical ) hamiltonian equations of motion , which still result in the semiclassical solutions , of course , they are @xmath71-dependent due to the modification of the poisson brackets . using the hamiltonian ( [ h ] ) , the previous equations of motion are promoted to the followings , @xmath72 note that the momenta are no more constants of motion because of nonvanishing @xmath71 , hereby , a new set of equations of motion from eqs . ( [ 1st : x_non ] ) and ( [ 1st : p_non ] ) are obtained , @xmath73 of course , the parameter @xmath71 is independent of the quantization where the modified semiclassical equations of motion ( [ 1st : p_non ] ) is reduced to eq . ( [ 1st : p_com ] ) for @xmath74 . from the coupled equations of motion ( [ eom : non ] ) , we obtained the solutions as @xmath75 where @xmath14 has been assumed to be a positive constant , and @xmath76 , @xmath77 , @xmath78 , and @xmath79 are constants of integration . since @xmath53 should be positive in eq . ( [ nc : omega ] ) , the constants @xmath76 , @xmath77 , and @xmath78 are appropriately restricted . then , the scale factor and the expanding velocity are given as @xmath80}{\sqrt{\alpha e^{-\kappa\theta t } + \beta e^{\kappa\theta t } + a } } , \label{nc : a } \\ \frac{da}{d\tau } & = & \dot\rho = \frac12 \theta \frac{\kappa(\alpha e^{-\kappa\theta t } - \beta e^{\kappa\theta t } ) - ( 2\beta e^{\kappa\theta t } + a)^2 + a^2 - 4 \alpha\beta}{\alpha e^{-\kappa\theta t } + \beta e^{\kappa\theta t } + a } , \label{nc : vel } \end{aligned}\ ] ] respectively , where we used @xmath81 and @xmath82 . the overdot denotes the derivative with respect to @xmath28 and comoving time @xmath83 is related to conformal time @xmath28 by @xmath84 , which can be explicitly calculated from the scale factor ( [ nc : a ] ) . subsequently , the acceleration and the curvature scalar are calculated as @xmath85}{\sqrt{\alpha e^{-\kappa\theta t } + \beta e^{\kappa\theta t } + a } } \bigg[\kappa \frac{a^2 - 4\alpha\beta}{\alpha e^{-\kappa\theta t } + \beta e^{\kappa\theta t } + a } \nonumber \\ & & \quad\qquad\ - ( 2\beta e^{\kappa\theta t } + a)^2 - 4\alpha\beta + a^2 - \kappa a \bigg ] , \label{nc : accel}\\ r & = & \frac{2}{a}\frac{d^2a}{d\tau^2 } = \kappa \theta^2 \exp\left [ \frac{2}{\kappa } ( a - b ) + \frac{4}{\kappa } \beta e^{\kappa\theta t}\right ] \bigg[\kappa \frac{a^2 - 4\alpha\beta}{\alpha e^{-\kappa\theta t } + \beta e^{\kappa\theta t } + a } \nonumber \\ & & \qquad\qquad - ( 2\beta e^{\kappa\theta t } + a)^2 - 4\alpha\beta + a^2 - \kappa a \bigg ] , \label{nc : r}\end{aligned}\ ] ] respectively . the solid line and the dashed line denote the curvature scalar and the acceleration of the scale factor , respectively . the dotted line is an asymptotic value of the curvature scalar as @xmath83 goes to infinity . note that the comoving time is defined by @xmath86 . the curvature scalar comes to be a negative constant around @xmath87 and a positive constant as @xmath83 goes to infinity . this fact indicates that there is the phase transition from anti - de sitter universe to de sitter universe . this figure is plotted in the case of @xmath88 , @xmath89 , @xmath90 , @xmath91 , @xmath92 , and @xmath93 in this bpp model . ] in order to describe the smooth transition from the decelerated phase to the accelerated universe , eventually , from the ads to the ds phase , we will consider the special case of @xmath94 , @xmath95 , and @xmath96 with the condition @xmath97 in what follows . these constants tells us that the range of the conformal time is @xmath98 as seen from eq . ( [ nc : omega ] ) , and then the range of the comoving time should be @xmath86 . under this restriction , the expanding velocity @xmath99 is always positive and the scale factor increases from zero to infinity . note that @xmath100 is a monotonic increasing function with respect to @xmath28 . on the other hand , the acceleration @xmath101 is zero at the initial time @xmath87 and is negative before @xmath102 , where @xmath103 where @xmath104 $ ] . after @xmath105 , the acceleration becomes positive , which shows the smooth phase transition . although the acceleration diverges as @xmath83 goes to infinity , but there exists no curvature singularity as shown in fig . [ fig : r ] due to the infinite scale factor . in fact , the curvature scalar is almost negative constant , @xmath106 < 0 $ ] , around @xmath107 and it becomes zero at @xmath105 , and then approaches the positive constant , @xmath108 , at @xmath109 . this fact shows that the phase transition from ads universe to ds appears . the solid , the dashed , and the dotted lines denote the energy density , the pressure , and the state parameter of perfect fluid . note that the pressure is always negative , so that the state parameter can be exotic . this figure is plotted with the same constants used in fig . [ fig : r ] . ] now , the solutions ( [ nc : omega ] ) and ( [ nc : chi ] ) should be satisfied with the constraint ( [ con ] ) , which determines the integration function @xmath110 , @xmath111 . \label{nc : t}\ ] ] then , the induced energy - momentum tensors ( [ t++ ] ) , and ( [ t+- ] ) are obtained as @xmath112 , \label{nc : t++ } \\ t_{+-}^{\rm qt } & = & \frac12 \beta \kappa^2\theta^2 e^{\kappa\theta t}. \label{nc : t+- } \end{aligned}\ ] ] using eqs . ( [ def : energy ] ) and ( [ def : pressure ] ) , the energy density , the pressure are explicitly given as @xmath113 , \label{nc : p } \end{aligned}\ ] ] so that the state parameter @xmath114 reads @xmath115 where its profile is plotted in fig . [ fig : energy ] for the special case giving the ads - ds transition . the energy density and the pressure are the same value of @xmath116 approximately at the initial time @xmath87 corresponding to @xmath117 , and then the state parameter becomes @xmath118 . the energy density becomes zero at the comoving time @xmath119 , where @xmath120 where @xmath121 \ln [ \kappa/(4\beta)]$ ] . it changes from negative value to positive around @xmath119 , but the pressure is always negative . the state parameter diverges at @xmath122 since the energy density vanishes faster than the pressure . the decelerated expansion of the early universe is due to the negative energy density with the negative pressure induced by quantum back reaction @xmath123 , and the accelerated late - time universe comes from the positive energy and the negative pressure which behave like dark energy source @xmath124 . we have shown that the phase changing transition from the ads to the ds phase is possible by assuming the modified poisson brackets to the semiclassical equations of motion in the bpp model . the usual bpp model does not generate this kind of transition since the integration function @xmath62 related to the vacuum state is trivially constant , and the equation of state parameter is simply one which is independent of the time . so , we have taken the nontrivial poisson brackets at the semiclassical level to overcome this triviality . the modified poisson brackets are not the quantum commutators so that it does not mean the quantization of the quantization since the fields @xmath53 and @xmath125 are not the operators . in fact , the modified poisson brackets can be applied to any stage of quantization in order to modify the original equations of motion . for example , if one considers the modified poisson brackets at the classical dilaton gravity , then the corresponding solution can be obtained , however , it is difficult to obtain the meaningful solution in spite of its complexity . the other heuristic example may be a two - dimensional simple harmonic oscillator with the mass @xmath126 and the spring constant @xmath127 , where its hamiltonian is like @xmath128 . the conventional poisson brackets generate the two independent set of hamiltonian equations of motion and then the well - known harmonic solutions are obtained . on the other hand , at this classical level , if we assume the modified poisson brackets , @xmath129 , @xmath130 , then the hamiltonian equations of motion are modified and the equations of motion can be written in the second order form of @xmath131 , @xmath132 , where @xmath133 and @xmath134 . the first order of hamiltonian equations of motion have been written in the form of the second order euler - lagrange equations of motion in order to show the explicit difference between the noncommutative case and the commutative case . then , the solutions are @xmath135 , @xmath136 , where @xmath137 and @xmath138 , @xmath139 , @xmath140 , and @xmath141 are constants of integration . note that these are just modified classical solutions rather than the quantum - mechanical ones . the equation of state parameter is singular at a certain time as seen in fig . [ fig : energy ] . in order for the phase transition from the ads ( @xmath142 ) to the ds universe @xmath143 , the state parameter also changes its signature at a certain time , in our case at @xmath144 . in fact , there are two options satisfying this condition . if the energy density is always positive then the pressure should change its sign , however , in this model , the pressure is always negative , so that the energy density should change its sign . the latter case gives the singular behavior . of course , the quantum - mechanically induced energy density allows the negative value . one might wonder how to derive the nontrivial poisson brackets which are similar to the noncommutativity in string theory @xcite . in the string theory , the noncommutative brackets between the coordinates are derived in the d - brane system applied in the constant external tensor field . this is a higher dimensional realization of the slowly moving point particle on the constant magnetic field . all of these systems can be interpreted as constraint systems @xcite , so we can expect our model may be a similar constraint system , however , it remains unsolved . this work was supported by the science research center program of the korea science and engineering foundation through the center for quantum spacetime * ( cquest ) * of sogang university with grant number r11 - 2005 - 021 .

it can be shown that in the bpp model the smooth phase transition from the asymptotically decelerated ads universe to the asymptotically accelerated ds universe is possible by solving the modified semiclassical equations of motion . this transition comes from noncommutative poisson algebra , which gives the constant curvature scalars asymptotically . the decelerated expansion of the early universe is due to the negative energy density with the negative pressure induced by quantum back reaction , and the accelerated late - time universe comes from the positive energy and the negative pressure which behave like dark energy source in recent cosmological models .

You are an expert at summarizing long articles. Proceed to summarize the following text: sodium nitrite is a ferroelectric at room temperature . it has the orthorhombic structure , space group @xmath2 , with the dipole vector of the v - shaped nitrite anions aligned parallel to the crystallographic @xmath3 direction , as shown in fig . [ fig : nano2 ] . crystal structure of nano@xmath0 in the ferroelectric phase . ] the ferroelectric - paraelectric phase transition takes place at about @xmath4 k , where the high temperature phase is orthorhombic , space group @xmath5 , with the dipoles disordered with respect to the @xmath3 axis . in a narrow temperature range from @xmath6 k to @xmath4 k , there exists an incommensurate antiferroelectric phase . the melting temperature is @xmath7 k. distinguished from displacive ferroelectrics in which the ferroelectric transition is driven by soft phonon modes , nano@xmath0 offers a model system for research of the order - disorder structural phase transition and any associated ferroelectric instability . @xcite extensive experimental work on nano@xmath0 has been devoted to probing the mechanism of the no@xmath8 polarization reversal that triggers the order - disorder transition . the majority of studies support the @xmath1-axis rotation model , but there were also results favoring the @xmath9-axis rotation model.@xcite recently , refined x - ray studies over a wide temperature range reinforced the @xmath1-axis rotation model.@xcite on the theoretical side , the microscopic model calculations done by ehrhardt and michel supported the @xmath1-axis rotation mechanism,@xcite whereas mixed double rotations around the @xmath9-axis and the @xmath1-axis was suggested by kinase and takahashi.@xcite it has long been desirable to apply computer molecular dynamics ( md ) simulations to nano@xmath0 in order to achieve unambiguous understanding of the polarization reversal mechanism . earlier md simulations with empirical born - mayer pair potentials detected the @xmath1-axis rotation in above - room - temperature nano@xmath0.@xcite unfortunately , the low - temperature structure produced by those simulations was antiferroelectric and apparently disagreed with the experimental observations . lu and hardy pointed out that the overall phase behavior of nano@xmath0 could be simulated by using an _ a priori _ approach to construct the crystal potential surface ( pes).@xcite the lu - hardy ( lh ) approach was originally designed to deal with molecular crystals such as k@xmath0seo@xmath10 , where exists a mix of bonding types , that is , the intermolecular interactions are mostly ionic , but the constituent atoms in a molecule ( seo@xmath11 in k@xmath0seo@xmath10 ) bond covalently . in the lh approach , the intra - molecule interactions were treated by applying the _ ab initio _ self - consistent field method to the gas - phase molecules , while the intermolecular pair potentials were computed within the gordon - kim ( gk ) electron gas theory.@xcite the crux of their application of the gk theory is how to partition the _ ab initio _ molecular charge density between the constituent atoms . since there is no unique way to separate the charge density of a highly covalently bonded molecule , lu and hardy suggested equal separation in a spirit similar to the mulliken population analysis ( mpa ) . by using this atomic - level method , we could successfully describe the phase transitions in fluoroperovskites,@xcite and ionic crystals with polyatomic molecules including seo@xmath11,@xcite clo@xmath12,@xcite so@xmath11,@xcite sio@xmath13,@xcite and no@xmath14.@xcite note that the mpa happens to preserve the ( zero ) dipole moment of these molecules . however , several problems appear when we moved on to deal with nano@xmath0 where the no@xmath15 radical has nonzero dipole moment and stronger chemical bonding . first , it is well known that the mpa , while certainly the most widely employed , is also somewhat arbitrary and the most criticized.@xcite in particular , the mpa overestimates the dipole moment of the free no@xmath8 ion by about @xmath16 . other difficulties involved the free - ion approximation . unlike in monatomic ionic crystals , there may exist considerable _ internal _ charge - transfer effects in molecular ionic crystals . electronic band structure calculations @xcite indicated that within a nitrite entity , the nitrogen atom and two oxygen atoms bond covalently , leading to high charge transferability between these constituent atoms . therefore , in solid nano@xmath0 the no@xmath8 group will feel different crystal - field environments as it rotates and responds by redistributing the charge density among its three constituent atoms . our goals in this paper are twofold . first , we show that our atomistic level simulation methods involving pair potentials with the rigid - ion model is capable of correctly describing the phase behavior of nano@xmath0 . second , we systematically examine the lh approach to understand the reason why it works so well in molecular ionic crystal systems by the following steps : ( i ) we develop another population analysis method that preserves the molecular dipole moment by directly fitting the _ ab initio _ charge density of a molecule ; ( ii ) we carry out _ ab initio _ hartree - fock ( hf ) calculations of the intermolecular interactions and find that the pair potentials from the rigid - ion model can correctly reproduce the _ ab initio _ results ; ( iii ) we investigate the crystal - field effects on the no@xmath8 ion by embedding the ion ( and its first shell of neighbors ) in a lattice of point charges and find a remarkable internal charge - transfer effect . @xcite several md simulations based on these modifications of the lh approach are also performed . the ferroelectric - paraelectric transition triggered by the @xmath1-axis rotation of the nitrite ions is observed in all versions of the lh approach . however , the transition temperatures predicted by these simulations are lower than the experimental values . furthermore , the transition temperatures obtained from the original version are higher than those predicted by modified versions and closer to the experimental values . after careful examination , we notice that in the lh approach , the no@xmath17 dipole moments were generally overestimated by about 120% , which reinforces the ground state . this implies that the crystal structure of nano@xmath0 is stabilized by the anion polarization effects . thus , we conclude polarization effects are particularly important for the quantitative study of nano@xmath0 . this paper is organized as follows . section ii describes the method we used to obtain the intermolecular interactions . section iii analyzes the resulting intermolecular potentials in comparison with those obtained from _ ab initio _ calculations . section iv presents the results of our md simulations . the crystal - field effects on no@xmath17 are discussed in section v. concluding remarks are made in section vi . our md simulation technique method originates from the gk model for simple ionic crystals such as alkali halides , assuming that ( molecular ) ions in a crystal environment may be described as free ions . @xcite then it was extended to deal with molecular ionic crystals like k@xmath0seo@xmath10 , in which strong intramolecular covalency exists . @xcite the main idea is that the molecular ion ( seo@xmath18 in k@xmath0seo@xmath10 ) is treated as a single entity , and intramolecular and intermolecular interactions are treated separately : first we perform _ ab initio _ quantum chemistry calculations for the whole molecular ion , to obtain the optimized structure , the force constants , and the whole electron density @xmath19 . the intramolecular interactions are described by force constants within the harmonic approximation . as for the intermolecular interactions , we have to carry out electron population analysis to separate @xmath19 onto each individual atom in the molecular ion , then use the gordon - kim electron gas model to calculate the intermolecular pair potentials . this approach provides a parameter - free description for the crystal potential - energy surfaces , which allow structural relaxation , md simulation , and lattice dynamics calculations . in calculating the intermolecular forces , there are three major approximations as discussed in the following : \(1 ) we assume that the geometries and electronic densities of the separate ions remain unchanged once they have been obtained under given circumstance , such as in the equilibrium state of the gas or crystal phases . this approximation is the fundamental basis for the gk electron gas theory . general speaking , we found that in an ionic crystal there is no strong chemical bond between ions , hence this approximation is reasonable . \(2 ) when dealing with the intermolecular interaction , we assume that the charge density of a rigid ion can be separated into its atomic constituents . \(3 ) we assume that the crystal potential energy is composed of the intermolecular and intramolecular interaction , where the intramolecular interaction is expressed in terms of force fields and the intermolecular interaction is a sum of interatomic pair potentials . atomistic level simulations utilizing pair potentials and the rigid - ion model have great success in describing many ionic systems . @xcite we showed that this scheme can correctly describe the phase transition behaviors of alkali halide fluoperovskites , @xcite and molecular crystals with tetrahedral @xcite and equilateral triangular @xcite radicals . however , for nano@xmath0 in which no@xmath17 has only a two fold symmetrical axis , the results were less satisfactory . @xcite note that the mean hf polarizability of no@xmath8 , 14.156 ( atomic units ) , calculated with the d95 * basis , @xcite is much higher than that of na@xmath20 , 0.343 ( atomic units ) , calculated with the 6 - 31 * basis . therefore , in solid nano@xmath0 , the rotation of no@xmath8 in the crystal field will induce charge redistribution within the molecule . hence this dynamic effect may invalidate the rigid ion approximation . in this paper we shall perform hf calculations for various geometries to verify this scenario . in the gk model , we evaluate the interaction between two molecules based on the electron density , @xcite which is approximated as the sum of component densities taken from hf calculations . that is , if @xmath21 and @xmath22 are the component densities , then the total density is @xmath23 . whereas , interaction potential is computed as the sum of four terms : coulombic , kinetic , exchange , and correlation energies which are expressed in terms of the charge densities . therefore , suppose the @xmath24 and @xmath25 molecules are made up of @xmath26 and @xmath27 atoms , respectively , then the coulombic interaction between them is @xmath28 where @xmath29 , @xmath30 , @xmath31 and @xmath32 are the nuclear charges and coordinations of the @xmath33-th atom in the @xmath24 molecule and @xmath34-th atom in the @xmath25 molecule , respectively this potential energy can be split into two parts : first the long - range part , @xmath35[z_{b , j}-\int \rho _ b^{}(\mathbf{r}_2)d\mathbf{r}_2]}{|\mathbf{r}_{a , i}-\mathbf{r}_{b , j}| } , \label{cl}\ ] ] and the short - range part @xmath36 eq . ( [ cl ] ) is essentially the electrostatic interaction energy when the charge densities of the molecules are distributed as point charges on the constituent atoms , which is known as the madelung potential energy . the non - coulombic energy terms are expressed in the uniform electron gas formula , @xmath37 , \label{kxc}\ ] ] where @xmath38 is one of the energy functionals for the kinetic , exchange , and correlation interactions . @xcite note that eq . ( [ kxc ] ) is not composed of pair potentials . in order to obtain the effective pairwise potentials , we approximate eq . ( [ kxc ] ) using @xmath39 , \label{pair}\ ] ] where @xmath40 . @xmath41 and @xmath42 are the charge densities of individual atoms in the @xmath24 and @xmath25 molecules , respectively , which are obtained by a population analysis as described in the next subsection . even though the non - coulombic forces as determined by eq . ( [ kxc ] ) are not strictly additive , the above approximation appears to be adequate except at very short distances . as pointed out by waldman and gordon , @xcite the main reason as to why this approximation is valid is because the coulombic force , the largest contribution to the potentials , is additive . based on our calculations , we find additivity of @xmath43 holds only to within about 50% ; however , the overlap contribution to the electrostatic energy dominates @xmath43 and renders additivity to within 10% . one final remark is in order , for the sake of simplifying the two - electron integral in eq . ( [ coul ] ) , the charge densities , @xmath41 and @xmath42 , are taken as its spherical average . as a result , the coulombic interaction is not exactly evaluated . nevertheless , as we shall show in figs . [ fig : na - no2 ] and [ fig : no2-no2 ] , this error is compensated by those due to the pairwise additive approximation . to summarize this subsection , we demonstrated that it is possible to analytically express the intermolecular potentials @xmath44 using eqs . ( [ cl ] ) , ( [ cs ] ) and ( [ pair ] ) once the charge density of each individual atom is obtained by an electronic population analysis . in the next subsection , we shall present further analysis on the charge density . in this subsection , we discuss the ways to separate the electron density @xmath19 of a molecule into its atomic constituents . suppose the molecule consists of @xmath26 atoms , then the wave function of the molecule @xmath45 can be written as a linear superposition of atomic wave functions @xmath46 , @xmath47 , centered at each atom , @xmath48 in turn , the atomic wave functions @xmath46 can be written as a linear superposition of the basis functions @xmath49 @xmath50 where \{@xmath51 } are usually the gaussian basis functions , and the coefficients @xmath52 can be obtained from the variational method . then the electronic density of the molecule is , @xmath53 where @xmath54 , which can be divided into two parts , namely the _ net _ ( @xmath55 ) and _ overlap _ ( @xmath56 ) populations . the latter can not be ignored in the presence of strong intramolecular covalency . therefore , separating @xmath19 into its atomic constituents is to split the overlap population . however , the way to achieve that is not unique . for example , we can introduce a set of weights @xmath57 due to different criteria such that @xmath58 then we can rewrite eq . ( [ density ] ) as following @xmath59 where @xmath60 is the atomic density of atom @xmath33 . .[table : multipole ] electronic multipole moments of molecule ( @xmath61 ) calculated from the mulliken population analysis . the _ ab initio _ values are shown in parentheses . all quantities are in atomic units . [ cols="<,^,^,^,^",options="header " , ] before we proceed with the molecular dynamics simulations , we perform lattice relaxation for the ferroelectric structure of nano@xmath0 both with and without the @xmath62 space group symmetry constraints . this relaxation procedure provides the crystal structure with zero force on each atom , that is an energy extremum ; it also produces a test to the pes because the resultant structures have to agree reasonably with the experimental data for further simulations to be reliable . we perform both static and dynamic relaxations : the static one is an application of the newton - raphson algorithm and usually results in finding a local minimum of the energy , and the dynamic one is a simulated annealing calculation for overcoming that limitation . we start the static lattice relaxation with the experimental parameters . in table [ table : par ] we present the lattice and basis parameters deduced from the experiments and static relaxation . in all cases , the static relaxation produced essentially the same structure that strongly resembles the experimental structure . most of the lattice constants in the relaxed structure are shorter than the experimental values ( by 3.7% , 1.5% , and 8.5% for @xmath9 , @xmath3 , and @xmath1 , respectively , in model i , and by 0.5% , -2.4% , and 10% for @xmath9 , @xmath3 , and @xmath1 , respectively , in model ii ) . hence the calculated volume is smaller than the experimental one by 13% for model i and 10% for model ii , a common feature for simulations using the gk model , which will be addressed in more detail in the next subsection . next , we go on to relax the statically relaxed crystal structure to zero temperature using a simulated annealing algorithm , in which the amount of kinetic energy in the molecules slowly decreases over the course of the simulation . we find that the ( zero temperature ) ground states in models i and ii are close to the statically relaxed structures , whereas there are substantial changes taking place in model iii . by monitoring the orientations of the nitrite ions , we find that the ground structure in model iii , still orthorhombic with @xmath63 , @xmath64 , and @xmath65 , is ferroelectric with the dipole moments of no@xmath8 aligned along the @xmath9 axis rather than the experimental @xmath3 axis . so we conclude that the pes given by models iii did not reflect reality . this concurs with the previous discussion ( section iii ) on the intermolecular potentials . in the following we use only models i and ii to simulate the phase transition in nano@xmath0 . temperature variation of lattice constants @xmath9 , @xmath3 , @xmath1 ( solid , dashed , dotted lines , respectively ; left scale ) and volume of the unit cell ( open circles ; right scale ) for ( a ) model i and ( b ) model ii . ] mean dipole moment @xmath66 and quadrupole moment @xmath67 of the whole nano@xmath0 crystal as a function of temperature for the md runs for ( a ) model i and ( b ) model ii . ] diagonal elements of the mean - square atomic displacements @xmath68 vs temperature . ( a ) na , ( b ) n , and ( c ) o atom . ] atomic positions of nano@xmath0 viewed from the @xmath9 direction obtained from the md simulation for model i at ( a ) @xmath69 k , ( b ) @xmath70 k , ( c ) @xmath71 k , and ( d ) @xmath72 k. ] using the isoenthalpic , isobaric ensemble , our md simulation is started with a zero - temperature zero - pressure orthorhombic cell ( @xmath73 ) , consisting of 512 atoms . periodic boundary conditions are imposed to simulate an infinite crystal . the md calculations are carried out in the parrinello - rahman scheme @xcite which allows both the volume and the shape of the md cell to vary with time . the calculation of the energies and forces was handled by the ewald method . a time step of 0.002 ps was used to integrate the equations of motion . in our heating runs , we raise the temperature of the sample in stages , 20 k each time , up to 1000 k. at each stage , the first 2000 time steps were employed to equilibrate the system , then 10000 time steps were collected for subsequent statistical analysis . since our simulations employ periodic boundary conditions , we can not distinguish the incommensurate structure ( i.e. , phase ii of solid nano@xmath0 ) . in figs . [ fig : abc ] through [ fig : ellipsoid ] , we demonstrate that as the md cell is heated , it undergoes two phase transitions : in the first one , the system retains its orthorhombic structure with a change of space group from @xmath62 to @xmath74 , in agreement with the experiments . the critical temperature @xmath75 is around @xmath76 k for model i and @xmath77 k for model ii , respectively . in the second transition , the crystal structure violently changes from orthorhombic to tetragonal at temperature ( @xmath78 ) which is around @xmath79 k for model i and @xmath80 k for model ii , as shown in fig . [ fig : abc ] . however , we argue that the crystal has already melted before this type of transition could be observed in reality . to investigate the mechanism of the polarization reversal of no@xmath8 , we monitor the crystal polarization and display the results in fig . [ fig : fe ] . let the dipole moment of anion @xmath33 be @xmath81 and the quadrupole moment be @xmath82 calculated by using the point charges on the @xmath27 and @xmath83 atoms . then the mean dipole moment per anion at temperature @xmath84 is @xmath85 where @xmath86 is the number of no@xmath8 in the md cell and the brackets denote an average over the md run . in addition , we define the antiferroelectric polarization as @xmath87 where @xmath88 and @xmath89 is the lattice vectors associated with the @xmath33-th ion . within our statistical uncertainty we find over all temperature range @xmath90 , while @xmath91 and @xmath92 . this fact confirms that the transition taking place at @xmath75 is the ferroelectric - paraelectric phase transition . furthermore , we calculated the mean quadrupole moment @xmath93 . when the dipole vector of a no@xmath8 is aligned along the @xmath3 axis , @xmath94 , @xmath95 , @xmath96 for model i and @xmath97 , @xmath98 , @xmath99 for model ii ; thus @xmath100 . this relation holds as the no@xmath8 ion rotates around the @xmath1 axis ; nevertheless , one would expect @xmath101 when the no@xmath8 ion rotates without directional preference . the fact that @xmath102 for @xmath103 ( fig . [ fig : fe ] ) reveals that the no@xmath8 anions rotate primarily about the @xmath1 axis . when @xmath104 , @xmath105 , i.e. , nano@xmath0 becomes an orientational liquid . further , in fig . [ fig : uu ] we show the mean - square atomic displacements @xmath106 where @xmath107 denotes the displacements along the @xmath108 axes , respectively . different models of no@xmath8 reversal are expected to satisfy the following conditions : ( 1 ) rotation around the @xmath1 axis : @xmath109 and @xmath110 ; ( 2 ) rotation around the @xmath9 axis : @xmath111 and @xmath112 . this figure relates to recent x - ray experiments which used the same quantities to investigate the polarization reversal mechanism . @xcite the experiments confirmed that the first condition holds for both ferroelectric and paraelectric phases . another important feature revealed by the experiments is that @xmath113 in the ferroelectric phase , whereas @xmath114 in the paraelectric phase . that is , @xmath115 and @xmath116 are reversed across @xmath75 . these features are reproduced in fig . [ fig : uu ] with exception of @xmath112 in the paraelectric phase . this means the no@xmath8 motions in our simulations are more mobile than those in the real crystal , rendering the simulated transition temperatures lower than the experimental values of @xmath117 k and the melting temperature @xmath7 k. in other words , the barriers to no@xmath8 rotation in our models are too small . in addition , in fig . [ fig : ellipsoid ] , we show the average crystal structures of nano@xmath0 at different temperatures . the ellipsoids in these pictures represent the root - mean - square deviations of the atoms from their average positions and thus indicate the thermal motions of these atoms . the @xmath1-axis rotation mode can be clearly seen , particularly in fig . [ fig : ellipsoid](c ) . according to the calculations described in the previous section , model ii generally gives a better description of free no@xmath8 and the intermolecular potential energies than model i. however , the simulation based on model i matches closer the experimental results than that based on model ii , that is , @xmath118 and @xmath78 predicted by model i are closer to experiment , this indicates that the crystal fields and polarization effects are particularly important for quantitatively studying the nano@xmath0 system , where the ferroelectric structure is considerably stabilized by anion polarization effects . actually , the mpa employed by model i does not preserve the _ ab initio _ dipole moment of free no@xmath8 . rather , it overestimates the dipole moment by 120% , thus leading to higher no@xmath17 rotational barriers than those predicted by model ii , which in turn raises the transition temperature and provides a better simulation in comparison with the experiments . it is worth mentioning the less desirable agreement between theoretical and experimental volumes , namely , the 13% discrepancy for model i and 10% for model ii . to address this we make one simple change : by following waldman and gordon,@xcite we increase the kinetic energy term in the gordon - kim potentials by 5% , this reduces the discrepancy to 9% for model i and 6% for model ii . having done this we rerun the md to obtain values of @xmath119 of 360 k for model i and 303 k for model ii . while this change worsens the value for model i , the value for model ii is virtually unchanged . and in both cases the transition mechanism is unaltered . thus the slight hardening of the short - range potentials removes most of the volume discrepancies . however , there is no material change in the mechanism of the phase transition . this robustness of the results with respect to minor variations in the potential demonstrates that our basic conclusion remain valid . based on the previous simulation results , the order - disorder phase transition in nano@xmath0 involves the rotation of the nitrite ions . we devise a scheme to calculate the three rotational barriers for no@xmath8 around the crystallographic @xmath9 , @xmath3 , and @xmath1 axes with its center of mass fixed : we start from the experimental ferroelectric structure @xcite as the ground state and calculate its energy difference with one of the two nitrite ions in the unit cell being rotated about the respective axis . the results are shown in figs . [ fig : barrier](a ) and [ fig : barrier](b ) . rotational barriers of one of the two nitrite ions in the unit cell around the @xmath9 , @xmath3 , and @xmath1 axes with its center of mass fixed . ( a)(b ) for the gk model , ( c)(d ) for the point charge model . left and right panels are for model i and ii , respectively . ] the bottom of each barrier , zero angle , is in the ferroelectric structure . for both models i and ii , the rotation around the @xmath1-axis has an energy barrier 5 - 10 times smaller than those of the other rotations , which is a characteristic of nitrites . @xcite hence , our calculations unambiguously reveal that the reorientation of no@xmath8 in the paraelectric phase occur essentially by rotations around the @xmath1 axis . in addition , the barriers calculated in model i are higher than that in model ii , confirming that the transition temperature in model i will be higher than in model ii . in figs . [ fig : barrier](c ) and [ fig : barrier](d ) , we also plot the contribution to the rotational barriers purely from the electrostatic interaction , that is , in the point - charge model with the short range forces deleted . comparing fig . [ fig : barrier](a ) with fig . [ fig : barrier](c ) , or fig . [ fig : barrier](b ) with fig . [ fig : barrier](d ) , we notice that the point - charge model gives rise to a quite different rotation barrier about the @xmath9 axis : it bottoms at about @xmath120 and is lower than the ground state energy due to the omission of short - range interactions , which comes from overlap of the charge cloud of an atom with those of its neighbors . in this section , we investigate the crystal - field effects on the no@xmath8 ion , which include electrostatic interaction from the background lattice , overlap compression of the no@xmath8 charge cloud through interaction with its neighbors , and charge - transfer between molecules which is usually small in ionic crystals . in the studies of monatomic ions @xcite and cyanides @xcite , fowler _ et al . _ showed that these effects could be successfully described by embedding the ion of interest , or a cluster consisting of the ion and its first shell of neighbors , into a lattice of point charges . we therefore perform hf calculations based on the following two models . @xcite first , the crystal field of ferroelectric nano@xmath0 is simulated by placing the nitrite ion at the center of a @xmath121 orthorhombic point charge lattice with spacings equal to the experimental lattice parameters . charges in the faces of the lattice are scaled to maintain overall neutrality . all anions except the central no@xmath8 are represented by single point charges . hence , there are 174 point charges surrounding the no@xmath8 ion . calculations of this type are referred to as cryst . obviously , in the cryst calculation we take into account only the crystal - field effect arising purely from electrostatic interaction . at the next level of sophistication , we replace the six nearest positive charges of the central no@xmath8 in the above lattice by the na cations . calculations of this type are referred to as clust . in both cryst and clust calculations , the geometrical structure of the no@xmath17 is fixed at its experimental values . we employ the same basis set , d95 * , for the in - crystal no@xmath17 ion as for the free no@xmath8 ion . in order to keep the clust calculations to a manageable size , we use the minimal basis set , sto-3 g , for the na@xmath20 ions unless specified . the cations , however , are relatively insensitive to the crystal environment and they are included here only to account for their compressing effect on the no@xmath8 wave functions . we find that adding extra basis functions to na@xmath20 will not change the results significantly . the in - crystal no@xmath8 initially points in the @xmath3 direction as in the ferroelectric phase of nano@xmath0 @xcite . its dipole moment is 0.636 ( cryst ) and 0.661 ( clust ) debye , close to that in the free ion model , 0.661 debye . thus it appears that the crystal - field effect is small for this configuration . subsequently , we rotate the no@xmath8 about the @xmath9 , @xmath3 , and @xmath1 axes and calculate the dipole moment of the rotated no@xmath8 . the rotation center is taken to be the center of a ( na@xmath20)@xmath122 cage formed by the 6 neighboring sodium ions of the central no@xmath15,@xcite ( 0,0.279 , 0 ) in the coordinate convention of fig . [ fig : nano2 ] . the clust and cryst results are depicted in figs . [ fig : polarizable](a ) and [ fig : polarizable](b ) , respectively . dipole moment of the central no@xmath8 in a @xmath121 lattice as it rotates around the @xmath108 axes through the center of its ( na@xmath20)@xmath122 cage . ( a ) clust and ( b ) cryst . ] clearly , the dipole moment of the no@xmath8 changes considerably as it rotates , indicating strong crystal field effects on the reorientation of the no@xmath8 . since the electron density of the no@xmath8 is compressed by its 6 neighboring sodium cations , the variation in magnitude of its dipole moment is smaller in the clust model than that in the cryst model . the electron cloud of the no@xmath8 is most and least variable when it rotates around the @xmath9 and @xmath3 axes , respectively ; for the @xmath1-axis rotation , which has the lowest rotational barrier , the dipole moment goes down from 0.661 debye at @xmath123 to 0.534 debye at @xmath124 in the clust model , as shown fig . [ fig : polarizable](a ) . in the context of population analysis , increase of the dipole moment of no@xmath15 implies that more electrons are distributed on the o atom , i.e. , electrons are flowing from the nitrogen atom to the oxygen atoms . conversely , decrease of the dipole moment indicates a reversal in electron transfer . therefore , we have demonstrated considerable _ intramolecular _ charge - transfer , although the intermolecular charge - transfer is usually small in ionic crystals . this intramolecular charge - transfer could be further elucidated based on the language of molecular orbitals ( mos ) : in the minimal basis set of atomic orbitals ( aos ) on nitrogen and oxygen , each atom of the no@xmath17 molecule contributes one @xmath125-orbital perpendicular to the molecular plane . thus their linear combinations , which are determined by the crystal field , form three different @xmath126 mos extended over the entire molecule , thus leading to the above - mentioned intramolecular charge - transfer . note that the mean hf polarizability of free no@xmath8 , @xmath127 in the atomic units , is much higher than that of free na@xmath20 , @xmath128 , calculated by using the 6 - 31 * basis . therefore , it is reasonable to expect that no@xmath8 in solid nano@xmath0 encountering different crystal - field environments as it rotates , redistributes its charge among the three constituent atoms to lower its energy . in fig . [ fig : barrier444 ] _ ab initio _ barriers to rotation of the central no@xmath8 in a @xmath121 lattice around the @xmath9 , @xmath3 , @xmath1 axes through the center of its ( na@xmath20)@xmath122 cage . ( a ) clust and ( b ) cryst . ] we show that the rotational barriers of the central no@xmath8 . on the whole , the clust results are similar to those in figs . [ fig : barrier](a ) and [ fig : barrier](b ) , while the cryst results are similar to those in figs . [ fig : barrier](c ) and [ fig : barrier](d ) . the reason is that in the cryst calculations we consider only the crystal - field effects originating purely from electrostatic interactions with the background point charges , similar to the point charge model used to obtain figs . [ fig : barrier](c ) and [ fig : barrier](d ) . whereas , in the clust calculations , overlap compression is also taken into account , which is reflected in figs . [ fig : barrier](a ) and [ fig : barrier](b ) as inclusion of short range repulsion . in fig . [ fig : barrier444](a ) , the barrier to the @xmath1-axis rotation is the lowest , but comparable with that to the @xmath3-axis rotation . this feature is caused by the fact that all background anions are represented by single point charges in our clust and cryst calculations ; however , we anticipate that restoring multipole moments of these background anions would increase the barrier difference among rotations about the @xmath9 , @xmath3 , and @xmath1 axes . to further elaborate the electronic polarization effect on no@xmath15 arising from the crystal environment , we change the rotation center to the center of mass of the no@xmath15 which was assumed in the model by kremer and siems . @xcite in this case , the dipole moment of no@xmath15 and the rotational barriers are presented in figs . [ fig : crystal](a ) and [ fig : crystal](b ) , respectively . in the clust model , no@xmath8 rotates around the @xmath108 axes through its center of mass . ( a ) dipole moment of no@xmath8 , and ( b ) rotational barriers . ] the dipole moment changes in a different way from that shown in fig . [ fig : polarizable](a ) . in particular , the dipole moment in the ferroelectric phase ( @xmath123-rotation ) is larger than at @xmath129-rotation in fig . [ fig : polarizable](a ) , while it is smaller in fig . [ fig : crystal](a ) . on the other hand , there is no qualitative discrepancy between fig . [ fig : crystal](b ) and fig . [ fig : barrier444](a ) ; the main differences are : the rotational barrier about the @xmath9 axis has risen by 135% while the barrier about the @xmath1 axis is depressed by 26% . this means that the order of rotational barriers is enhanced by the change of the rotation center . although strong crystal field effects have been revealed by these _ ab initio _ calculations , the rotational barriers obtained from the polarizable - ion models are in qualitative agreement with those from the rigid - ion models , confirming that the rigid - ion model is capable of describing the phase behavior in nano@xmath0 . we have presented md simulations of nano@xmath0 using a hybrid _ a priori _ method consisting of _ ab initio _ calculations and gordon - kim electron gas theory to analytically calculate the crystal potential surface . this method has been carefully examined by using different population analysis methods . we have carried out _ ab initio _ hartree - fock calculations of the intermolecular interactions for no@xmath8:na@xmath130 and no@xmath8:no@xmath8 dimers and concluded that the pair potentials of the rigid - ion model can correctly reproduce the _ ab initio _ results . we demonstrated that a rigid - ion model is capable of describing phase behavior in solid nano@xmath0 . the crystal - field effects on the no@xmath8 ion are also addressed in two polarizable - ion models for which the ferroelectric phase of nano@xmath0 was found to have a larger dipole moment of no@xmath8 than the @xmath129-rotation phase . remarkable internal charge - transfer effect is found to be stabilizing the crystal structure of nano@xmath0 . in our md simulations , two rigid - ion models using mpa and fpa , respectively , have been studied . the one using mpa , which overestimates the dipole moment of no@xmath8 , gives rise to the more comparable results with the experiments , since such overestimation also stabilizes the crystal structure , thus mimics the anion polarization effect . to quantitatively simulate nano@xmath0 , a more elaborate polarizable - ion model is needed . helpful discussions with dr . l. l. boyer are gratefully acknowledged . this work was supposed by the nebraska research initiative , the nebraska epscor - nsf grant eps-9720643 , and department of the army grants daag 55 - 98 - 1 - 0273 and daag 55 - 99 - 1 - 0106 . w. n. m. is grateful for the support from the office of naval research .

we present molecular dynamics simulations of solid nano@xmath0 using pair potentials with the rigid - ion model . the crystal potential surface is calculated by using an _ a priori _ method which integrates the _ ab initio _ calculations with the gordon - kim electron gas theory . this approach is carefully examined by using different population analysis methods and comparing the intermolecular interactions resulting from this approach with those from the _ ab initio _ hartree - fock calculations . our numerics shows that the ferroelectric - paraelectric phase transition in solid nano@xmath0 is triggered by rotation of the nitrite ions around the crystallographical @xmath1 axis , in agreement with recent x - ray experiments [ gohda _ et al . _ , phys . rev.b * 63 * , 14101 ( 2000 ) ] . the crystal - field effects on the nitrite ion are also addressed . remarkable internal charge - transfer effect is found .

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Proceed to summarize the following text: finite element analysis ( fea ) is the most widely used analysing tool in computer aided engineering ( cae ) . one key factor to achieve an accurate fea is the layout of the finite element mesh , including both mesh density and element shape @xcite . regions containing complex boundaries , rapid transitions between geometric features or singularities require finer discretisation @xcite . this leads to the development of adaptive meshing techniques that assure the solution accuracy without sacrificing the computational efficiency @xcite . the construction of a high quality mesh , in general , takes the most of the analysis time @xcite . the recent rapid development of the isogeometric analysis @xcite , which suppressed the meshing process , has emphasised the significance of mesh automation in engineering design and analysis . quadtree in fea is a kind of hierarchical tree - based techniques for adaptive meshing of a 2d geometry @xcite . it discretises the geometry into a number of square cells of different size . the process is illustrated in fig.[fig : qtreerep1 ] using a circular domain . the geometry is first covered with a single square cell , also known as the root cell of the quadtree ( fig.[fig : qtreerep1]a ) . as shown in fig.[fig : qtreerep1]b , the root cell is subdivided into 4 equal - sized square cells and each of the cells is recursively subdivided to refine the mesh until certain termination criteria are reached . in this example , a cell is subdivided to better represent the boundary of the circle and the subdivision stops when the predefined maximum number of division is reached . the final mesh is obtained after deleting all the cells outside the domain ( fig.[fig : qtreerep1]c ) . the cell information is stored in a tree - type data structure , in which the root cell is at the highest level . it is common practice to limit the maximum difference of the division levels between two adjacent cells to one @xcite . this is referred to the @xmath0 rule and the resulting mesh is called a balanced @xcite or restricted quadtree mesh @xcite . a balanced quadtree mesh not only ensures there is no large size difference between adjacent cells , but also reduces the types of quadtree cells in a mesh to the 6 shown in fig.[fig : qtreecell ] . owing to its simplicity and large degree of flexibility , the quadtree mesh is also recognised in large - scale flood / tsunami simulations @xcite and image processing @xcite . generation of quadtree mesh on a circular domain . ( a ) cover the domain with a square root cell ( b ) subdivide the square cells ( c ) select the cells based on the domain boundary ] 6 main types of master quadtree cells with @xmath0 rule enforced ] it is , however , not straightforward to integrate a quadtree mesh in a fea . the two major issues are illustrated by fig.[fig : qtreerep ] , which shows the quadtree mesh of the top - right quadrant of the circular domain in fig.[fig : qtreerep1 ] . 1 . _ hanging nodes _ middle nodes , shown as solid dots in fig.[fig : qtreerep ] , exist at the common edges between the adjacent cells with different division levels . when conventional quadrilateral finite elements are used , a middle node is connected to the two smaller elements ( lower level ) but not to the larger element ( higher level ) . this leads to incompatible displacement along the edges and the middle nodes are called the hanging nodes @xcite . fitting of curved boundary _ quadtree cells are composed of horizontal and vertical lines only . as shown in fig.[fig : qtreerep ] , the quadtree cells intersected with the curved boundary have to be further divided into smaller ones to improve the fitting of the boundary . generally , the mesh has to be refined in the area surrounding the boundary . despite this , the boundary may still not be smooth ( fig.[fig : qtreerep1]c ) and may result in unrealistically high stresses . additional procedure is required to conform the mesh to the boundary . there exist a number of different approaches to ensure displacement compatibility when hanging nodes are present @xcite . three typical approaches among all are briefly discussed here . the first one is to subdivide the higher level quadtree cells next to a hanging node into smaller triangular elements @xcite as shown in fig[fig : qtreerep ] . additional nodes may be added to improve the mesh quality and/or reduce the number of element types . these techniques lead to a final mesh that only contains conforming triangular elements . a similar approach was adpoted by @xcite , in which the quadtree mesh was subdivided into a conforming mesh dominated by quadrilateral elements . the second approach introduces special conforming shape functions @xcite to ensure the displacement compatibility . an early work by gupta @xcite reported the development of a transition element that had additional node along its side . a conforming set of shape functions was derived based on the shape functions of the bilinear quadrilateral elements . owing to its simplicity and applicability , gupta s work was further extended by @xcite and @xcite to hexahedral elements . fries et al . @xcite investigated two approaches to handle the hanging nodes within the framework of the extended finite element method ( xfem ) . they were different in whether the enriched degrees - of - freedom ( dofs ) were assigned to the hanging node . a similar work was reported by legrain et al . @xcite , in which the selected dofs are enriched and properly constrained to ensure the continuity of the field . quadtree mesh of the top - right quadrant of a circular domain . demonstration of subdivision ( dashed lines ) is given in two quadtree cells with hanging nodes on their sides . ] the third approach is to model the quadtree cells as _ _ n-__sided polygon elements by treating hanging nodes as vertices of the polygon . this approach generally requires a set of polygonal basis function . special techniques are usually required to integrate the resulting equations over arbitrary polygon domain @xcite . this development was initiated by @xcite who showed the use of rational basis functions for elements with arbitrary number of sides . tabarraei and sukumar @xcite in their work adapted their polygon element @xcite to quadtree mesh . the set of polygonal basis functions was derived using laplace interpolant . by using an affine map on the reference polygon , the conforming shape functions of a quadtree cell with the same number of vertices ( including the hanging nodes ) were obtained . they also reported a fast technique for computing the global stiffness matrix , making use of the quadtree structure by defining parent elements @xcite . in this way , the elemental stiffness matrix has to be computed only 15 times ( 4-node cell not included ) for a balanced quadtree mesh ( when @xmath0 rule is enforced ) . further development of their work with xfem was also reported in @xcite . as mentioned in a recent paper @xcite , the development of high - order polygon elements received relatively less attention . mibradt and pick @xcite devised high order basis functions for polygons based on the natural element coordinates . those basis functions , however , are not complete polynomials . rand et al . @xcite developed a quadratic serendipity element for arbitrary convex polygons based on generalised barycentric coordinates . the potential of using their approach for higher order serendipity elements on convex polygons was also reported . based on the same approach , sukumar @xcite recently developed the quadratic serendipity shape functions that were applicable for convex and nonconvex polygons and were complete quadratic polynomials . the shape functions were obtained through solving an optimisation problem , which was derived from the maximum - entropy principle . besides dealing with the hanging nodes in a quadtree mesh , fitting complex boundaries is another challenging part in the mesh generation . @xcite proposed trimming the quadtree cells , intersected with the boundary , into polygons before a further subdivision process into triangles or quadrilaterals . alternatively those cells were first subdivided and some of the vertices were repositioned based on their projections onto the boundary @xcite . in @xcite and @xcite , after the subdivision , a buffer zone was introduced between the boundary and the internal quadtree cells . a compatible mesh was then constructed to fill up this zone . all these techniques require an additional optimisation step to ensure the final mesh quality . within the framework of xfem , quadtree cells intersected with the boundary were not modified in pre - processing stage @xcite . however , when constructing the stiffness matrix , the domain boundary is still required to identify the portion of the cell within the domain for numerical integration . in the integration process , that portion of the cell within the domain is either subdivided into geometric sub - cells @xcite or treated as a polygon @xcite . the scaled boundary finite element method ( sbfem ) provides an attractive alternate technique to construct polygon elements ( scaled boundary polygon ) @xcite ( fig.[fig : sbfempolygon ] ) . it is a semi - analytical procedure developed by song and wolf to solve boundary value problems @xcite . the only requirement for a scaled boundary polygon is that its entire boundary is visible from the _ scaling centre _ only the edges of the polygon are discretised into line elements . the number of line elements on an edge can be as many as required . any type of displacement - based line elements , including high - order spectral elements , can be used . the domain of the scaled boundary polygon is constructed by scaling from its scaling centre to its boundary , and the solution within the polygon is expressed semi - analytically @xcite . a salient feature of the scaled boundary polygons is that stress singularities occurring at crack and notch tips , formed by one or several materials , can be accurately modelled without resorting to asymptotic enrichment and local mesh refinement . its high accuracy and flexibility in mesh generation lead to simple remeshing procedures when modelling crack propagation @xcite . scaled boundary representation of a polygon ] scaled boundary representation of quadtree cells ] this paper presents a technique for the stress and fracture analysis by integrating the scaled boundary finite element method ( sbfem ) with quadtree mesh of high - order elements . this integrated technique possesses the following features : 1 . hanging nodes are treated without cell subdivision . each quadtree cell is modelled as a scaled boundary polygon as shown in fig.[fig : sbfequadtreerep ] . the edges of a quadtree cell can be divided into more than one line element to ensure displacement compatibility with the adjacent smaller cells . hanging - nodes are thus treated the same as other nodes . owing to the sbfe formulations @xcite , no additional procedure is required to compute the shape functions for the quadtree cells . high - order elements can also be used within each quadtree cell directly . the entire quadtree meshing process is simple and automatic . the boundary of the problem domain is defined using signed distance functions @xcite . only seed points @xcite are required to be predefined to control the mesh density . owing to the ability of the sbfem in constructing polygon elements of , practically , arbitrary shape and order , the quadtree cells trimmed by curved boundaries are simply treated as a non - square scaled boundary polygon . high - order elements can be used to fit curved boundaries closely . the resulting mesh conforms to the boundary without excessive mesh refinement ( see fig.[fig : sbfequadtreerep ] ) . no local mesh refinement or asymptotic enrichment is required for a quadtree cell containing a crack tip to accurately model the stress singularity . the present paper is organised as follows . the summary of the sbfem and its application to quadtree cells are first presented in the next section . it is followed by the developed algorithm of quadtree mesh generation in section[sec : quadtree - mesh - generation ] . five examples are given in section[sec : numerical - examples ] with detailed discussion on accuracy and convergence . finally , conclusions of the present work are stated in section[sec : conclusion ] . this section summarises the scaled boundary finite element method for 2d stress and fracture analysis . only the key equations that are related to its use with a quadtree mesh are listed . a detailed derivation of the method based on a virtual work approach is given in @xcite . the sbfem can be formulated on quadtree cells by treating each cell as a polygon with arbitrary number of sides ( fig . [ fig : sbfequadtreerep ] ) . in each cell , a local coordinate system @xmath1 is defined at a point called the scaling centre from which the entire boundary is visible . @xmath2 is the radial coordinate with @xmath3 at the scaling centre and @xmath4 at the cell boundary . the edges of each cell are discretised using one - dimensional finite elements with a local coordinate @xmath5 having an interval of @xmath6 . it is noted that the hanging nodes appearing in the quadtree structure do not require any special treatment in the sbfem formulation . they are simply used as end nodes of the 1d elements . the coordinate transformation between the cartesian @xmath7 and the local @xmath1 coordinate systems are given by the scaled boundary transformation equations @xcite : @xmath8 where @xmath9^{\mathrm{t}}$ ] is the cartesian coordinates of a point in the cell , @xmath10 is the shape function matrix and @xmath11^{\mathrm{t}}$ ] is the vector of nodal coordinates of a cell with @xmath12 nodes . the displacement field in each cell @xmath13 is interpolated as @xmath14 where @xmath15 are radial displacement functions and are obtained by solving the scaled boundary finite element equation in displacement @xcite : @xmath16 with coefficient matrices @xmath17 where @xmath18 is the material constitutive matrix , @xmath19 and @xmath20 are the sbfem strain - displacement matrices and @xmath21 is the jacobian on the boundary required for coordinate transformation . . is solved by introducing the variable @xmath22 @xcite @xmath23^{\mathrm{t}}\label{eq : xksi}\end{aligned}\ ] ] where @xmath24 so that eq . is transformed into a first order ordinary differential equation with twice the number of unknowns : @xmath25 with the hamiltonian matrix @xmath26 @xcite @xmath27\label{eq : hamilton}\end{aligned}\ ] ] an eigenvalue decomposition of the @xmath26 results in @xmath28= & \left[\begin{array}{cc } \boldsymbol{\phi}_{\mathrm{u}}^{\mathrm{(n ) } } & \boldsymbol{\phi}_{\mathrm{u}}^{\mathrm{(p)}}\\ \boldsymbol{\phi}_{\mathrm{q}}^{\mathrm{(n ) } } & \boldsymbol{\phi}_{\mathrm{q}}^{\mathrm{(p ) } } \end{array}\right]\left[\begin{array}{cc } \boldsymbol{\lambda}^{\mathrm{(n ) } } & 0\\ 0 & \boldsymbol{\lambda}^{\mathrm{(p ) } } \end{array}\right]\label{eq : eigendecomp}\end{aligned}\ ] ] where @xmath29 and @xmath30 are the eigenvalue matrices with real parts satisfying @xmath31 and @xmath32 , respectively . @xmath33 and @xmath34 are the corresponding eigenvectors of @xmath29 whereas @xmath35 and @xmath36 are the eigenvectors corresponding to @xmath30 . for bounded domains such as those considered in this paper , only the eigenvalues satisfying @xmath31 lead to finite displacements at the scaling centre . using eq . and , the solutions for @xmath15 and @xmath37 are @xmath38 the integration constants @xmath39 in eq . and eq . are obtained from the nodal displacements at the cell boundary @xmath40 as @xmath41 the stiffness matrix of each quadtree cell is formulated as @xcite @xmath42 substituting eq . into eq . , the displacement field in a cell is @xmath43 using the hooke s law and the strain - displacement relationship , the stress at a point in a cell is @xcite @xmath44 where @xmath45^{\mathrm{t}}$ ] is the stress mode @xmath46 fig.[fig : crackrep ] shows how a crack is modelled with a quadtree cell . the crack tip is chosen as the scaling centre . the crack surfaces are not discretised . the line elements discretising the cell boundary do not form a closed loop . modelling of a crack with the scaled boundary finite element method . ] when a crack is modelled by the sbfem , two eigenvalues , @xmath47 , @xmath48 satisfying @xmath49 appear in @xmath29 . from eq.[eq : stresfield ] , it can be discovered that these eigenvalues lead to a stress singularity as @xmath50 . using the two modes corresponding to these two eigenvalues , the singular stresses are expressed as @xmath51 where @xmath52\label{eq : singeigenval}\end{aligned}\ ] ] and @xmath53 are the integration constants corresponding to @xmath54 . the singular singular stress modes @xmath55^{\mathrm{t}}$ ] is written as @xmath56 where @xmath57 are the modal displacements in @xmath33 corresponding to @xmath54 . the stress intensity factors can be computed directly from their definitions . for a crack that is aligned with the cartesian coordinate system as shown in fig.[fig : crackrep ] , the stress intensity factors are defined as @xmath58 substituting the stress components in eq . into eq . and using the relation @xmath59 ( @xmath60 is the distance from the scaling centre to the boundary along the direction of the crack , see fig.[fig : crackrep ] ) at @xmath61 leads to @xmath62 this section presents the developed algorithm for quadtree mesh generation . fig.[flowchart ] shows the flow chart of the overall process . the entire generation process is automatic with minimal number of inputs required from the user , which include * maximum allowed number of seed points in a cell @xmath63 , * seed points on each boundary @xmath64 and region of interest @xmath65 , * maximum difference between the division levels of adjacent cells @xmath66 , which is equal to 1 for a balanced quadtree mesh . flow chart of the quadtree mesh generation . ] this section is organised based on fig[flowchart ] . it first presents defining geometry using signed distance function , and assigning seed points on the boundary and the regions of interest . detailed explanations of the meshing steps , which include generating the initial quadtree grid , trimming the boundary quadtree cells into polygons and merging cells surrounding a crack tip , are then followed . to facilitate the description of the meshing steps , fig.[qtreedes0 ] shows a square plate with a circular hole and two local refinement features to be used as an example throughout this section . an efficient computation of the global stiffness matrix , by taking advantage on the quadtree mesh , is described at the end of this section . example to illustrate the quadtree mesh generation process : a square plate with a circular hole . an additional circle and an inclined line ( dashed lines ) are included to control local mesh density . ] the geometry is defined by using the signed distance function @xcite . it provides all the essential information of a geometry and can be operated with simple boolean operations to build up more complex geometries @xcite . the signed distance function of a point @xmath67 associated with a domain @xmath68 , which is a subset of @xmath69 , is given as @xmath70 where @xmath71 represents the boundary of the domain and @xmath72 is the _ euclidean norm _ in @xmath69 with @xmath73 . the sign function @xmath74 is equal @xmath75 when @xmath76 lies inside the domain and is equal 1 otherwise . this definition of the signed distance function is visualised in fig.[dispfunc ] . a number of distance functions in matlab for simple geometries are given in @xcite , including their boolean operations . signed distance function of the points inside the domain ( @xmath77 ) , on the boundary ( @xmath78 ) and outside the domain ( @xmath79 and @xmath80 ) ] for each boundary and region of interest , a set of pre - defined seed points @xcite is introduced to control the quadtree mesh density . there require four sets of predefined seed points for the example in fig.[qtreedes0 ] . two sets are for the square and the circular hole representing the actual domain boundary . the number of seed points directly controls the local density of the quadtree cells and the quality of fitting the boundary . this is further discussed in section[sub : polygon - boundary - cells ] . the other two sets are for the large circle and inclined line controlling local mesh density only . the meshing process starts with covering the problem domain with a single square cell ( the root cell ) . the dimension of the root cell is based on the larger one between the maximum vertical and maximum horizontal dimension of the geometry . the developed algorithm will check the number of seed points in the cell . if the number is larger than the predefined maximum allowed number , the cell will be divided into 4 equal - sized cells . this generation process is applied recursively until all the cells have seed points no more than the predefined value . for each recursive loop , the maximum difference between the division levels of adjacent cells @xmath66 is enforced . for cells that have division level difference with the adjacent cells larger than @xmath81 , the higher level cell is subdivided into 4 equal - sized cells . fig.[qtreedes1 ] shows the initial quadtree grid of the example in fig.[qtreedes0 ] . initial quadtree grid of the example in fig . [ qtreedes0 ] . vertices with solid square markers are on the boundary , with square box markers are inside the domain , and without any markers are outside the domain . ] the initial quadtree grid shown in fig.[qtreedes1 ] does not conform to the boundary . those cells that have edges intersected with the boundary need to be identified and trimmed . by using the signed distance function , the locations of the vertices ( inside the domain , on the domain boundary or outside the domain as shown in fig.[qtreedes1 ] ) are identified based on the sign and value of the function . for edges containing two vertices with opposite signs , they are identified as the edges intersected with the boundary . for each of those edges , the intersection point with the boundary is computed . some quadtree cells could have vertices very close to the boundary in comparison with the lengths of their edges . after trimming , poorly shaped polygon cells with some edges much shorter than the others could be generated and may adversely affect the mesh quality . to avoid this situation , the vertices that are within a threshold distance away from the boundary are identified and then moved to their closest points on the boundary . in the present work , @xmath82 of the length of the cell edge ( based on the smallest cell attaching to the vertex ) is used as the threshold value . the edges connecting to these vertices will no longer be cut by the boundary . the trade - off of this process is the presence of additional non - square cells that lead to additional computation of the stiffness matrix . this is discussed in section[sub : an - efficient - assembly ] . model curved boundary by quadtree refinement or using high - order elements . nodes are represented with small circles along the cell edges . ] at the end of the trimming process , the edges of a cell cut by the boundary are updated with the intersection points and the enclosed segment of boundary is added to the cell . this will result in polygon cells . after trimming the quadtree in fig.[qtreedes1 ] , the polygon cells around the hole of the example problem is shown in fig.[qtreedes2 ] . it is clear from fig.[qtreedes2 ] that the circular boundary is not represented accurately if a single linear element is used on the edge of the cell . in order to represent the curved boundary more accurately , two alternates are available in the developed algorithm . the first is to reduce the element size ( @xmath83-refinement ) . this is achieved by increasing the number of seed points on the curved boundary . fig.[qtreedes3 ] shows the initial quadtree layout of the example problem after increasing the seed points around the hole by 4 times . it can be seen by comparing fig.[qtreedes1 ] with fig . [ qtreedes3 ] that the refinement is limited to a small region around the hole . the refined quadtree ( fig[qtreedes2 ] ) demonstrates the improvement of capturing the circular boundary . quadtree mesh after refinement ] the second option to improve the modelling of curved boundaries is to utilise high - order elements ( @xmath84-refinement ) . fig.[qtreedes2 ] shows the example problem with each line segment on the circular boundary modelled with a 4th order element . with this approach , curved boundaries can be captured more accurately using fewer elements . both options to improve the modelling of the boundaries can be applied simultaneously without conflicts . the numerical accuracy of both approaches is discussed through numerical examples given in section[sec : numerical - examples ] . owing to the capability of the sbfem for fracture analysis @xcite , the domain containing a crack tip is modelled with a single cell . in the stress solution , the variation along the radial direction , including the stress singularity , is given analytically and the variation along the circumference of the cell is represented numerically by the line elements on boundary . to obtain accurate results , sufficient nodes have to present on the boundary of the cell to cover the angular variation of the solution . in the developed algorithm , the size of a cell containing a crack tip is controlled , as shown in fig.[qtreedes8 ] with an inclined crack , by a predefined set of seed points on a circle . quadtree mesh for a crack problem before and after merging cells . the two crack tips are marked with a cross . the two circles are to control the size of quadtree cells covering the crack tips . ] for problems with cracks , only one additional step is required after the initial mesh is generated . the cells surrounding the crack tip are refined to the same division level and then merged into a single cell as shown in fig.[qtreedes8 ] . this step avoids having a crack tip too close to the edges of the cell , which could affect the mesh quality and the solution accuracy @xcite . after the cells are merged , the intersection point between the edge of the resulting cell and the crack is computed to define the two crack mouth points . the other cells on the crack path are split by the crack into two cells . the splitting process is similar to the trimming of cells by the boundary , but two vertices are created at every intersection point between the cell edge and the crack to split the original cells . the global stiffness matrix is simply the assembly of the stiffness matrices of each master quadtree and polygon cell . when the @xmath0 rule is enforced to the mesh , only 6 main types of master quadtree cells are present as given in fig.[fig : qtreecell ] . by rotating the geometry of the master cells orthogonally , the maximum number of types of these master quadtree cells are 24 . for isotropic homogeneous materials , rotation does not have effect on 4-node or 8-node cells and only two 2 rotations are required for the first type of 6-node cell ( the top one in fig.[fig : qtreecell ] ) . the maximum number of master quadtree cells that require stiffness matrix calculation reduces to 16 ( only 15 in @xcite as 4-node cell is excluded ) . after the mesh generation , the algorithm will check which master cells out of the 16 appear in the mesh . their stiffness matrices are then computed and stored . during the stiffness assembling process , the stiffness matrix of each regular quadtree cell is directly extracted from those computed stiffness matrices . for the polygon cells and those irregular quadtree cells ( with their vertices moved to fit the boundaries ) , individual stiffness matrix calculation is required . this approach clearly improves the computational efficiency of constructing the global stiffness matrix , especially for large scale problems that contain a significant number of cells . with the use of high - order elements in the quadtree mesh , this assembling approach becomes even more economical . this section presents five numerical examples to highlight the capability and the performance of the proposed technique . in the first example , an infinite plate with a circular hole is modelled and the results are compared with the analytical solution . the proposed technique is then used to analyse a square plate with multiple holes to highlight the automatic meshing capability in handling transition between geometric features . in the third example , a square plate with a central hole and multiple cracks is studied to demonstrate the performance of the proposed technique in handling complicated geometries with singularities . thereafter , a square plate with two cracks cross each other is analysed . it is aimed to emphasise the automation and simplicity of the mesh generation in the proposed technique . in the first four examples , the same material properties , with young s modulus @xmath85 and poisson s ratio @xmath86 , are used . the final example is a cracked nuclear reactor under internal pressure . it is aimed to show the simplicity of the present technique in modelling practical non - regular structures . the computation time reported in this section is based on a desktop pc with intel(r ) core(tm ) i7 3.40ghz cpu and 16 gb of memory . the proposed technique is implemented in matlab and the computation time is extracted in interactive mode of matlab . an infinite plate containing a circular hole with radius @xmath87 at its centre is considered in this example . the plate is subject to a uniaxial tensile load as shown in fig.[openhole ] . the analytical solution of the stresses in polar coordinates @xmath88 is given by @xcite : @xmath89 the displacement solutions are : @xmath90\nonumber \\ u_{2}(r,\theta ) & = \frac{a}{8\mu}\left[\frac{r}{a}(\kappa-3)\sin\theta+\frac{2a}{r}\left((1-\kappa)\sin\theta+\sin3\theta\right)-\frac{2a^{3}}{r^{3}}\sin3\theta\right],\label{eq : exohdisp}\end{aligned}\ ] ] where @xmath91 is the shear modulus and @xmath92 is the kolosov constant for plane stress condition . the problem is solved by analysing a finite dimension of the plate with a dimension of @xmath93 ( see fig.[openhole ] ) . analytical traction ( eq.[eq : exohstr ] ) is applied at the four edges of this finite plate . fig.[ohm1 ] shows the quadtree mesh of the plate for @xmath94 . each edge on a quadtree cell is discretised with 1st order line elements . the @xmath0 rule is enforced . based on the proposed technique , the curved boundary is handled as shown in fig.[ohm2 ] with polygon cells . convergence study is conducted based on the @xmath95refinement . three different element orders @xmath96 are investigated . fig.[openholecon ] shows the present results of the relative error in the displacement norm @xmath97 , with @xmath98 the analytical solution given in eq . and @xmath99 the solution computed by the proposed technique . the results show that all three types of elements have monotonic convergence . for higher order elements , more accurate results with similar number of dof are obtained and the convergence rate is also faster . is the element order and @xmath100 is the slope of the fitted line ] there are 37 out of 100 cells calculated for the stiffness matrices . among those 37 cells , 9 are regular quadtree cells and 28 are polygon cells surrounding the hole . for the remaining cells , their stiffness matrices are simply extracted from those 9 regular quadtree cells . to further demonstrate the accuracy of the proposed technique , @xmath101 along @xmath102 ( see fig.[openhole ] ) is plotted in fig.[openholesigmatheta ] using the mesh given in fig.[ohm1 ] with 4th order elements . it can be seen that the results of the proposed technique agree well with the analytical solution , which has @xmath103 at @xmath104 ( @xmath105 ) . for points away from @xmath104 , @xmath101 approaches 1 . the same infinite plate can be approximated by increasing the @xmath106 ratio . the application of quadtree mesh facilitates such a study . only the left and right sides of the plate are subjected to uniaxial in - plane tension stress @xmath107 . the element order used in this study is @xmath108 . the same mesh given in fig.[ohm1 ] is used for @xmath94 . the adaptive capability of quadtree mesh leads to the same mesh pattern for all @xmath106 ratios . fig.[ohm3 ] shows the cells around the hole for @xmath109 and it is exactly the same as the one shown in fig.[ohm2 ] . for @xmath109 , although there are 316 cells in total , only 37 cells are calculated for the stiffness matrices , which is the same as the previous study . the results of @xmath101 at @xmath104 with varying @xmath106 ratio are given in table[openholesigmathetavsratio ] . it is seen that the analytical solution ( @xmath103 ) is quickly approached when increasing the @xmath106 ratio . ] .normalised stress ( @xmath101 ) at @xmath104 of the thin square plate with a circular hole [ cols="^,^,^",options="header " , ] [ nreacttab1 ] this paper has presented a numerical technique to automate stress and fracture analysis using the sbfem and quadtree mesh of high - order elements . owing to the nature of the sbfem , the proposed technique has no specific requirement , such as deriving conforming shape functions or sub - triangulation , to handle quadtree cells with hanging nodes . high - order elements are used within each quadtree cell directly . the quadtree mesh generation is fully automatic and involves minimal number of user inputs and operation steps . boundaries are modelled with scaled boundary polygons and this allows the proposed technique to conform the boundary without excessive mesh refinement . the meshing algorithm is also applicable for problems with singularities . the use of quadtree mesh leads to an efficient approach to compute the global stiffness matrix . this facilitates the analysis that requires a significant number of cells using high - order elements . five numerical examples are presented to highlight the functionality and performance of the proposed technique . the present results show excellent agreement with analytical solutions and those computed by the fem . [ [ section ] ]

this paper presents a technique for stress and fracture analysis by using the scaled boundary finite element method ( sbfem ) with quadtree mesh of high - order elements . the cells of the quadtree mesh are modelled as scaled boundary polygons that can have any number of edges , be of any high orders and represent the stress singularity around a crack tip accurately without asymptotic enrichment or other special techniques . owing to these features , a simple and automatic meshing algorithm is devised . no special treatment is required for the hanging nodes and no displacement incompatibility occurs . curved boundaries and cracks are modelled without excessive local refinement . five numerical examples are presented to demonstrate the simplicity and applicability of the proposed technique . scaled boundary finite - element method ; quadtree mesh ; high order elements ; polygon elements

You are an expert at summarizing long articles. Proceed to summarize the following text: models of cyclic dominance are traditionally employed to study biodiversity in biologically inspired settings @xcite . the simplest such model is the rock - paper - scissors game @xcite , where rock crashes scissors , scissors cut paper , and paper wraps rock to close the loop of dominance . the game has no obvious winner and is very simple , yet still , it is an adequate model that captures the essence of many realistic biological systems . examples include the mating strategy of side - blotched lizards @xcite , the overgrowth of marine sessile organisms @xcite , genetic regulation in the repressilator @xcite , parasitic plant on host plant communities @xcite , and competition in microbial populations @xcite . cyclical interactions may also emerge spontaneously in the public goods game with correlated reward and punishment @xcite , in the ultimatum game @xcite , and in evolutionary social dilemmas with jokers @xcite or coevolution @xcite . an important result of research involving the rock - paper - scissors game is that the introduction of randomness into the interaction network results in global oscillations @xcite , which often leads to the extinction of one species , and thus to the destruction of the closed loop of dominance that sustains biodiversity . more precisely , in a structured population where the interactions among players are determined by a translation invariant lattice , the frequency of every species is practically time - independent because oscillations that emerge locally can not synchronize and come together to form global , population - wide oscillations . however , if shortcuts or long - range interactions are introduced to the lattice , or if the original lattice is simply replaced by a small - world network @xcite , then initially locally occurring oscillations do synchronize , leading to global oscillations and to the accidental extinction of one species in the loop , and thus to loss of biodiversity @xcite . if the degree distribution of interaction graph is seriously heterogeneous , however , then such kind of heterogeneity can facilitate stable coexistence of competing species @xcite . interestingly , other type of randomness , namely the introduction of mobility of players , also promotes the emergence of global oscillations that jeopardize biodiversity @xcite . interestingly , however , although long - range interactions and small - world networks abound in nature , and although mobility is an inherent part to virtually all animal groups , global oscillations are rarely observed in actual biological systems . it is thus warranted to search for universal features in models of cyclic dominance that work in the opposite way of the aforementioned types of randomness . the questions is , what is the missing ingredient that would prevent local oscillations to synchronize across the population to form global oscillations ? preceding research has already provided some possible answers . for instance peltomki and alava observed that global oscillations do not occur if the total number of players is conserved @xcite . mobility , for example , then has no particular impact on biodiversity because oscillations are damped by the conservation law . however , the consequence of the conservation law does not work anymore if a tiny fraction of links forming the regular lattice is randomly rewired @xcite . zealots , on the other hand , have been identified as a viable means to suppress global oscillations in the rock - paper - scissors game in the presence of both mobility and interaction randomness @xcite . in addition to these examples , especially in the realm of statistical physics , there is a wealth of studies on the preservation and destruction of biodiversity in models of cyclic dominance @xcite . here we wish to extend the scope of this research by considering a partly overlooked property , namely the consideration of site - specific heterogeneous invasion rates . importantly , we wish to emphasize an important distinction to species - specific heterogeneous invasion rates , which have been considered intensively before . in the latter case , different pairs of species are characterized by different invasion rates , but these differences are then applied uniformly across the population . in case of spatially variable invasion rates , these could be site - specific , and hence particular pairs of species may have different invasion rates even though they are of the same type . such a setup has many analogies in real life , ranging from differing resources , quality or quantity wise , to variations in the environment , all of which can significantly influence the local success rate of the governing microscopic dynamics . notably , this kind of heterogeneity was already studied in a two - species lotka - volterra - like system @xcite , and in a three - species cyclic dominant system where a lattice has been used as the interaction network @xcite . the latter work concluded that the invasion heterogeneity in spatial rock - paper - scissors models has very little effect on the long - time properties of the coexistence state . in this paper , we go beyond the lattice interaction topology , exploring the consequences of quenched and annealed randomness being present in the interaction network . in the latter case , as we will show , it could be a decisive how heterogeneity is introduced into the invasion rate because annealed randomness does not change the oscillation but quenched heterogeneity can mitigate the global oscillation effectively . in what follows , we first present the main results and discuss the implications of our research , while details concerning the model and the methodology are described in the methods section . we first consider results obtained with species - specific invasion rates . indeed , it is possible to argue that it is too idealistic to assume homogenous invasion rates between different species , and that it would be more realistic to assume that these invasion rates are heterogeneous . but as results presented in fig . [ suppressed ] show , this kind of generalization does not bring about a mechanism that would suppress global oscillations . these oscillations clearly emerge for homogeneous species - specific invasion rates , as soon as the fraction of rewired links of the square lattice @xmath0 exceeds a threshold . if we then assume that species - specific invasion rates are heterogeneous , say @xmath1 , @xmath2 , and @xmath3 ( here @xmath4 denotes the invasion rates of @xmath5 transition where @xmath6 runs from @xmath7 to @xmath8 in a cyclic manner ) , it can be observed that nothing really changes . in fact , the threshold in @xmath0 remains much the same , and the order parameter @xmath9 ( the area of the limit cycle in the ternary diagram ) reaches the same close to @xmath10 plateau it does when these invasion rates are homogenous . further along this line , we can even adopt invasion rates that are chosen uniformly at random from the unit interval at each particular instance of the games . more precisely , we still keep the original @xmath5 direction of invasion , but the strength of the invasion rate @xmath4 is chosen randomly in each particular case . but no matter the fact that this rather drastically modifies the microscopic dynamics , the presence of shortcuts will still trigger global oscillations ( marked random in fig . [ suppressed ] ) . we thus arrive at the same conclusion that was already pointed out in @xcite , which is that heterogeneous invasion reaction rates have very little effect on the dynamics and the long - time properties of the coexistence state . having established the ineffectiveness of heterogeneous species - specific invasion rates to prevent local oscillations to synchronize across the population to form global oscillations , we next consider site - specific heterogeneous interaction rates , denoted as @xmath11 and applied to each site @xmath12 . here @xmath11 determines the probability that a neighbor will be successful when trying to invade player @xmath12 according to the original @xmath5 rule . as such , different values of @xmath11 influence the success of microscopic dynamics locally . moreover , these invasion rates are determined once at the start of the game and can be drawn from different distributions . the simplest case is thus to consider values drawn uniformly at random from the unit interval . as results in fig . [ suppressed ] show ( see quenched random ) , this modification of the rock - paper - scissors game clearly blocks the emergence of global oscillations regardless of the value of @xmath0 . indeed , even if the square lattice is , through rewiring , transformed into a regular random graph , the order parameter @xmath9 still remains zero . even if the uniform distribution is replaced by a simple discrete double - peaked distribution ( practically it means that half of the players has @xmath13 for example , while the other half retains @xmath14 ) , the global oscillations never emerge ( see quenched double in fig . [ suppressed ] ) . the coordination effect leading up to global oscillations is thus very effectively disrupted by heterogeneous site - specific invasion rates , and this regardless of the distribution from which these rates are drawn . to illustrate the dramatically contrasting consequences of different types of randomness , we show in fig . [ ternary ] representative time evolutions for both cases . the comparison reveals that , as deduced from the values of the order parameter @xmath9 displayed in fig . [ suppressed ] , time - varying invasion rates fail to suppress global oscillations , the emergence of which is supported by the small - world properties of the interaction network ( ternary diagram and the time course on the left ) . the limit cycle denoted black in the ternary diagram and the large - amplitude oscillations of the densities of species in the corresponding bottom panel clearly attest to this fact . this stationary state is robust and is reached independently of the initial mixture of competing strategies . conversely , quenched heterogeneous interaction rates drawn from a uniform distribution clearly suppress global oscillations ( ternary diagram and the time course on the right ) . here , the system will always evolve into the @xmath15 state , central point of the diagram , even if we launched the evolution from a biased initial state . thus , if heterogeneities are fixed in space , just like in several realistic biological systems , then this effectively prohibits global oscillation by disrupting the organization of a coordinated state , i.e. , synchronization of locally occurring oscillations across the population . as demonstrated previously @xcite , the type of randomness in the interaction network responsible for the emergence of global oscillations plays a negligible role . be it quenched through the one - time rewiring of a fraction @xmath0 of links forming the original translation invariant lattice , or be it annealed through the random selection of far - away players to replace nearest neighbors as targets of invasion with probability @xmath16 , there exist a critical threshold in both where global oscillations emerge if invasion rates are homogeneous . accordingly , it makes sense to test whether heterogeneous site - specific invasions rates are able to suppress such oscillations regardless of the type of randomness that supports them . to that effect , we make use of the discrete double - peaked distribution , where the fraction of sites @xmath17 having a lower invasion rate @xmath13 then the rest of the population at @xmath14 can be a free parameter determining the level of heterogeneity . evidently , at @xmath18 we retain the traditional rock - paper - scissors game with homogeneous invasion rates ( all sites have @xmath14 ) , while for @xmath19 the fraction of sites having @xmath13 , and thus the level of heterogeneity , increases . at the other extreme , for @xmath20 , we of course again obtain a homogeneous population where everybody has @xmath13 , but we do not explore this option since it is practically identical , albeit the evolutionary process is much slower . by introducing heterogeneity into the system gradually , we can monitor how it influences the stationary state . in fig . [ nu ] , we present representative results for both quenched and annealed randomness of interaction graph ( see legend ) . the first observation is that only a minute fraction of suppressed nodes ( less than @xmath21 ) suffices to fully suppress global oscillations , and this regardless of the applied high @xmath0 and @xmath16 values that practically ensure an optimal support for local oscillations to synchronize across the population into global oscillations . moreover , it can be observed that both transitions to the oscillation - free state are continuous . in other words , there does not exist a sharp drop in the value of @xmath9 at a particular value of @xmath17 . instead , the suppression of global oscillations is gradual as the level of site - specific invasion heterogeneity in the population increases . similar in spirit , another way to introduce invasion heterogeneity gradually into the population is to use a fixed fraction of nodes with a lower invasion rate , but vary the difference to @xmath14 . accordingly , we have a fraction @xmath22 of nodes , which instead of @xmath14 have the invasion rate @xmath23 . here @xmath24 becomes the free parameter , which for zero returns the traditional rock - paper - scissors game with homogeneous invasion rates , while for @xmath25 the distance in the peaks of the discrete double - peaked distribution , and thus the level of heterogeneity in the population , increases . representative results obtained with this approach are shown in fig . [ dif ] for both quenched and annealed randomness of interaction graph ( see legend ) . in comparison with results presented in fig . [ nu ] , it can be observed that increasing @xmath24 has somewhat different consequences than increasing @xmath17 . in the former case , when @xmath24 is small , the slight heterogeneity has no particular influence on the stationary state and global oscillations persist well beyond @xmath26 for annealed randomness and @xmath27 for quenched randomness . but if the difference reaches a sufficiently large value , global oscillations disappear in much the same gradual way as observed before in fig . [ nu ] , although the transition for annealed randomness is more sudden . to sum up our observations thus far , different versions of the same concept reveal that spatially quenched heterogeneity in site - specific invasion rates is capable to effectively suppress global oscillations that would otherwise be brought about by either annealed or quenched randomness in the interaction network . however , there is yet another possible source or large - amplitude global oscillations in the population , namely mobility . as is well - known , mobility can give rise to global oscillation that jeopardizes biodiversity @xcite . although subsequent research revealed that global oscillations due to mobility do not emerge if the total number of competing players is conserved @xcite , more recently it was shown that , if in addition to a conservation law also either quenched or annealed randomness is present in the interaction network , then mobility still induces global oscillations @xcite . in particular , if the site exchange is intensive then only a tiny level of randomness in the host lattice suffices to evoke global oscillations . lastly , we thus verify if heterogeneity in site - specific invasion rates is able to suppress global oscillations brought about by mobility . as results in fig . [ mob ] show , the impact of quenched invasion heterogeneity is very similar to the above - discussed cases . it is worth noting that conceptually similar behavior can be observed when biological species are hosted in a turbulent flow of fluid environment @xcite . in fact , as a general conclusion , neither randomness in the interaction network nor the mobility of players can compensate for the detrimental impact of spatial invasion heterogeneity on global oscillations , thus establishing the latter as a very potent proponent of biodiversity in models of cyclic dominance . we have studied the impact of site - specific heterogeneous invasion rates on the emergence of global oscillations in the spatial rock - paper - scissors game . we have first confirmed that species - specific heterogeneous invasion rates , either fixed or varying over time , fail to disrupt the synchronization of locally emerging oscillations into a global oscillatory state on a regular small - world network . on the contrary , we have then demonstrated that site - specific heterogeneous invasion rates , determined once at the start of the game , successfully hinder the emergence of global oscillations and thus preserves biodiversity . we have shown this conclusion to be valid independently of the properties of the distribution that determines the invasion heterogeneity , specifically demonstrating the failure of coordination for uniformly and double - peak distributed site - specific invasion rates . moreover , our research has revealed that quenched site - specific heterogeneous invasion rates preserve biodiversity regardless of the type of randomness that would be responsible for the emerge of global oscillations . in particular , we have considered quenched and annealed randomness in the interaction network , as well as mobility . regardless of the type of randomness that would promote local oscillations to synchronize across the population to form global oscillations , site - specific heterogeneous invasion rates were always found to be extremely effective in suppressing the emergence of global oscillations . drawing from the colloquial expression used to refer to alcohol that is consumed with the aim of lessening the effects of previous alcohol consumption , the introduction of randomness in the form of site - specific heterogeneous invasion rates lessens , in fact fully suppresses , the effects of other types of randomness hence the `` hair of the dog '' phenomenon . our setup takes into account heterogeneities that are inherently present in virtually all uncontrolled environments , ranging from bacterial films to plant communities . examples include qualitative and quantitative variations in the availability of nutrients , local differences in the habitat , or any other factors that are likely to influence the local success rate of the governing dynamics @xcite . the consideration of site - specific heterogeneous invasion rates and their ability to suppress global oscillations joins the line of recent research on the subject , showing for example that the preservation of biodiversity is promoted if a conservation law is in place for the total number of competing players @xcite , or if zealots are introduced to the population @xcite . notably , previously it was shown that zealotry can have a significant impact on the segregation in a two - state voter model @xcite , and research in the realm of the rock - paper - scissors game confirmed such an important role of this rather special uncompromising behavior . in general , since global , population - wide oscillations are rarely observed in nature , it is of significance to determine key mechanisms that may explain this , especially since factors that promote such oscillations , like small - world properties , long - range interactions , or mobility , are very common . in this sense , site - specific heterogeneous invasion rates fill an important gap in our understanding of the missing ingredient that would prevent local oscillations to synchronize across the population to form global oscillations . there is certainly no perfect spatial system where microscopic processes would unfold identically across the whole population . these imperfections are elegantly modeled by the heterogeneous host matrix that stores the individual invasion rates of each player . as we have shown , the coordination of species evolution is highly sensitive on such kind of heterogeneities when they are fixed in space . ultimately , this prevents the synchronization of locally emerging oscillations , and gives rise to a `` hair of the dog''-like phenomenon , where one type of randomness is used to mitigate the adverse effects of other types of randomness . we hope that these theoretical explorations will help us to better understand the rare emergence of global oscillations in nature , as well as inspire further research , both experimental and theoretical , along similar lines . the spatial rock - paper - scissors game evolves on a @xmath28 square lattice with periodic boundary conditions , where each site @xmath12 is initially randomly populated by one of the three competing species . for convenience , we introduce the @xmath5 notation , where @xmath6 runs from @xmath7 to @xmath8 in a cyclic manner . hence , species @xmath29 ( for example paper ) invades species @xmath30 ( rock ) , while species @xmath30 invades species @xmath31 ( scissors ) , which in turn invades species @xmath29 to close the loop of dominance . the evolution of species proceeds in agreement with a random sequential update , where during a full monte carlo step ( @xmath32 ) we have chosen every site once on average and a neighbor randomly . in case of different players the invasion was executed according to the rock - scissors - paper rule with probability @xmath33 . in the simplest , traditional version of the game , all invasion rates between species are equal to @xmath34 . species - specific heterogeneous invasion rates can be introduced through the parameter @xmath35 , which is simply the probability for the @xmath5 invasion to occur when given a chance . the values @xmath36 , @xmath37 and @xmath38 can be determined once at the start of the game , or they can be chosen uniformly at random from the unit interval at each particular instance of the game . on the other hand , site - specific heterogeneous invasion rates , which we denote as @xmath11 , apply to each site @xmath12 in particular , and determine the probability that a neighbor will be successful when trying to invade player @xmath12 according to the @xmath5 rule . this rule can be considered as `` prey - dependent '' because the @xmath39 value at the prey s position determines the probability of invasion . as an alternative rule , we can consider the @xmath39 value of predator s position that determines the invasion probability . lastly , we can assume that the @xmath39 values of both the predator and prey s positions influence the invasion rate via their @xmath40 product . while the time dependence of the evolution will be different in the mentioned three cases but the qualitative behavior is robust . therefore we restrict ourself to the first mentioned `` prey - dependent '' rule . these invasion rates are determined once at the start of the game and can be drawn uniformly at random from the unit interval , or from any other distribution . here , in addition to uniformly distributed @xmath11 , we also consider site - specific heterogeneous invasion rates drawn from a discrete double - peaked distribution , where a fraction @xmath17 of sites have @xmath23 , while the remaining @xmath41 have @xmath14 . to test the impact of site - specific heterogeneous invasion rates under different circumstances , we consider interaction randomness in the form of both quenched and annealed randomness . quenched randomness is introduced by randomly rewiring a fraction @xmath0 of the links that form the square lattice whilst preserving the degree @xmath42 of each site . this is done only once at the start of the game . this procedure returns regular small - world networks for small values of @xmath0 and a regular random network in the @xmath43 limit @xcite . annealed randomness , on the other hand , is introduced so that at each instance of the game a potential target for an invasion is selected randomly from the whole population with probability @xmath16 , while with probability @xmath44 the invasion is restricted to a randomly selected nearest neighbor @xcite . this procedure returns well - mixed conditions for @xmath45 , while for @xmath46 only short - range invasions as allowed by the original square lattice are possible . we also consider the impact of site - specific heterogeneous invasion rates in the presence of mobility . the latter is implemented so that during each instance of the game we choose a nearest - neighbor pair randomly where players exchange their positions with probability @xmath47 . oppositely , with probability @xmath48 , the dominant species in the pair invades the other in agreement with the rules or the rock - paper - scissors game . the parameter @xmath47 hence determines the intensity of mobility while the number of players is conserved . technically , however , the strategy exchange between neighboring players is determined not only by the level of mobility @xmath47 , but it also depends on the individual @xmath11 and @xmath49 values characterizing the neighboring sites @xmath12 and @xmath50 . in this way , we can consider the fact that different sites may be differently sensitive to the change of strategy , and the success of mutual change is then practically determined by the site that is more reluctant to change its state . accordingly , when the strategy exchange is supposed to be executed , then this happens only with the probability @xmath51 that is equal to the smaller of @xmath11 and @xmath49 values ( all the other details of the model remain the same as above ) . global oscillations are characterized with the order parameter @xmath9 , which is defined as the area of the limit cycle in the ternary diagram @xcite . this order parameter is zero when each species occupies one third of the population , and becomes one when the system terminates into an absorbing , single - species state . we have used lattices with up to @xmath52 sites , which was large enough to avoid accidental fixations when the amplitude of oscillations was large , and which allowed an accurate determination of strategy concentrations that are valid in the large population size limit . naturally , the relaxation time depends sensitively on the model parameters and the system size , but @xmath53 mcs was long enough even for the slowest evolution that we have encountered during this study .

global , population - wide oscillations in models of cyclic dominance may result in the collapse of biodiversity due to the accidental extinction of one species in the loop . previous research has shown that such oscillations can emerge if the interaction network has small - world properties , and more generally , because of long - range interactions among individuals or because of mobility . but although these features are all common in nature , global oscillations are rarely observed in actual biological systems . this begets the question what is the missing ingredient that would prevent local oscillations to synchronize across the population to form global oscillations . here we show that , although heterogeneous species - specific invasion rates fail to have a noticeable impact on species coexistence , randomness in site - specific invasion rates successfully hinders the emergence of global oscillations and thus preserves biodiversity . our model takes into account that the environment is often not uniform but rather spatially heterogeneous , which may influence the success of microscopic dynamics locally . this prevents the synchronization of locally emerging oscillations , and ultimately results in a phenomenon where one type of randomness is used to mitigate the adverse effects of other types of randomness in the system .

You are an expert at summarizing long articles. Proceed to summarize the following text: one of the challenging problems in pulsar astrophysics is to find a consistent and observationally verified explanation for the high peculiar velocities of pulsars . these velocities can be as high as 1000 km / s and have a mean value of 450 km / s , much greater than the random velocities of ordinary stars ( harrison et al . @xcite ; lyne & lorimer @xcite ) . several mechanisms have been put forward in the past to explain the origin of the large proper motions . since it is believed that pulsars are born during the first stages of some type ii or core - collapsing supernovae , an asymmetric core collapse or explosion could give the pulsars the observed velocities ( shklovskii @xcite ; woosley @xcite ; woosley & weaver @xcite ; janka & mller @xcite ; burrows & hayes @xcite ) . the evolution of close binary systems could also be responsible for the large pulsar velocities ( gott et al . @xcite ) . alternatively , the emission of electromagnetic radiation during the first months after the supernova explosion , stemming from an off - centered rotating magnetic dipole in the newborn pulsar , could give the pulsar a substantial kick ( harrison & tademaru ) . another approach is based on the assumption that most of the energy and momentum released during a type ii supernova explosion ( @xmath0 erg ) are carried off by the relativistic neutrinos , as was observationally confirmed by the detection of a neutrino burst associated with sn1987a ( hirata et al . @xcite ; bionta et al . therefore , an asymmetric neutrino emission , caused for example by convection ( woosley @xcite ; woosley & weaver @xcite ; janka & mller @xcite ) or strong magnetic fields ( chuga@xmath1 @xcite ; dorofeev et al . @xcite ; vilenkin @xcite ; horowitz & piekarewicz @xcite ) , may play an important role when trying to understand the origin of pulsar velocities . not all these mechanisms , however , seem to be able to produce the required high velocities and many of them suffer from the fact that they need magnetic fields of the order of @xmath2 g , pulsar periods of @xmath3 ms or that they are not sustained by other observed pulsar properties ( duncan & thompson @xcite ) . in a recent paper kusenko & segr ( @xcite ) , hereafter ks , have proposed a new mechanism of asymmetric neutrino emission based on resonant neutrino oscillations . the three types of neutrinos , @xmath4 , @xmath5 and @xmath6 , are abundantly produced in the core of a collapsing star which later on may become a pulsar . the matter density is so high in the core that the neutrinos do not escape but get trapped . they undergo a diffusion process until they reach higher radii , where the density has decreased and they can freely stream away . the emission surface , the so - called neutrino sphere , is not the same for the three types of neutrinos . since electron neutrinos can interact via both charged and neutral currents they interact more strongly in the protoneutron star than muon and tau neutrinos hence , the electron neutrino sphere is at a larger radius than the muon and tau neutrino spheres . the authors in ks showed that under these conditions neutrinos @xmath6 can resonantly turn into @xmath4 , by means of the adiabatic msw effect ( smirnov @xcite ) , in the region between the tauonic and the electronic neutrino spheres and @xmath6 have been discussed with @xmath7 and small mixing . ] . the emerging electron neutrino , however , will be absorbed by the medium and therefore the resonant surface becomes the effective surface of emission of the @xmath6 . neutrinos propagating in media with a longitudinal magnetic field @xmath8 have different electromagnetic properties than in the vacuum case . they acquire an effective electromagnetic vertex which is induced by weak interactions with the charged particles in the background and generates a contribution @xmath9 to the effective self - energy of the neutrino , @xmath10 being the neutrino momentum ( esposito & capone @xcite ; dolivo & nieves @xcite ; elmfors et al . the induced vertex modifies the flavor transformation whilst preserving chirality and , as a result , the location at which the resonance occurs is affected , leading to the spherical symmetry of the effective emission surface being replaced by a dipolar asymmetry . the condition for resonant oscillations to take place is accordingly given by @xmath11 where @xmath12 is the neutrino vacuum mixing angle , @xmath13 the fermi constant , @xmath14 the charge density of the degenerate electron gas in which the neutrino propagates and @xmath15 the radial coordinate . neutrinos emitted from the two magnetic poles of the resonant surface then have slightly different temperatures because the two poles are at slightly different radii . the outcome is an asymmetric emission of momentum carried by neutrinos which gives the neutron star a kick in the direction of the magnetic field and thus leads to a recoil velocity in agreement with observational data . quantitatively the kick is described by the asymmetry in the third component of momentum and estimated by @xmath16 since the total momentum carried away by neutrinos emitted by the protoneutron star is times the momentum of the proper motion of the pulsar , an asymmetry of 1% would give a kick in agreement with observation . assuming an average energy for the tau neutrinos leaving the protoneutron star of @xmath17 mev , which corresponds to @xmath18 mev , the authors in ks obtain the desired asymmetry of 1% for values of @xmath19 g. as an advantage over other neutrino emission mechanisms the one discussed here works for smaller magnitudes of the magnetic field and does not demand for any constraints on the pulsar period . if the resonant neutrino conversion turned out to be the origin of pulsar velocities , one could use pulsar observations to obtain information on neutrino masses . the implications for particle physics models would be of great importance . a possible weak point of the model , however , lies in the fact that the mass of the tau neutrino is required to be @xmath20 ev in order to have a resonant conversion in the protoneutron star between the electron and the tau neutrino spheres . although such a mass is not excluded by laboratory experiments , it is difficult to accommodate it in the standard big bang cosmological model . indeed , the present age of the universe can be used to set an upper bound on the mass of any light , stable neutrino species ( gerstein & zeldovich @xcite ; cowsik & mcclelland @xcite ) : @xmath21 ev , where @xmath22 is the hubble constant in units of 100 km s@xmath23/mpc . nevertheless , this problematic point might be overruled by unstable tau neutrinos or by a better understanding of the evolution of the universe . another comment can be made in the context of the supernova physics . from treatments of neutrino transport it was found that the average energy for tau neutrinos leaving the supernova can be as high as 27 mev ( janka @xcite ) . such a value , however , would demand for a higher magnetic field in eq . ( [ kick ] ) in order to still explain the observed pulsar velocities thus weakening one of the advantages of the proposed model . on the other hand , it is interesting to note that neutrino oscillations between @xmath4 and a tau neutrino with a mass of @xmath24 mev could , for a large range of mixing angles , help to explode supernovae while leaving r - process nucleosynthesis unscathed ( raffelt @xcite ) . a better understanding of supernova explosions and data from future galactic supernovae could be used to test whether the asymmetric resonant neutrino emission model is actually realistic . since none of the above mentioned points represent a serious danger to the discussed mechanism , but its implications for astrophysics and physics beyond the standard model of the fundamental interactions would be significant , it is worthwhile to examine further related observational consequences with the aim of arriving at independent constraints . in addition , the derived constraints can be used to test the viability of some of the other proposed models for the origin of pulsar velocities . in this paper we derive , using the mechanism introduced in ks , a new expression relating the transverse velocity , the magnetic field and geometric parameters of fast spinning pulsars . we analyze observational data on pulsars to see whether this relation is fulfilled . our work , however , seems to indicate that pulsars do not satisfy this relation , and therefore present data do not support the neutrino oscillation mechanism . according to the authors of ks the asymmetric neutrino emission from a magnetically distorted neutrino sphere gives a net momentum to a pulsar along the magnetic axis . it is believed that , in general , the spin and magnetic axes of pulsars are not aligned , and thus the magnetic field rotates around the spin axis with the period @xmath25 of the pulsar . from this it can be seen that a net velocity along the magnetic axis is only obtained if the pulsar period is much larger than the characteristic time scale @xmath26 on which the neutrino flux from the cooling protoneutron star remains constant . for such a case , the length of the velocity vector @xmath27 and magnetic field vector @xmath8 in the core of the pulsar should be proportional : @xmath28 in a different paper kusenko & segr ( @xcite ) apply the model to pulsars which are slow spinning at birth , @xmath29 s and @xmath30 s , and argue that this @xmath31 correlation should become increasingly more significant as the period of rotation approaches a few seconds . they claim that observational data show such an increase in the correlation for the examined periods , which would support the investigated model . in the present paper we consider fast spinning pulsars , @xmath32 s , instead . the characteristic time @xmath26 is of the order @xmath33 s ( suzuki @xcite ) . therefore , a reliable test of the mentioned correlation between @xmath34 and @xmath35 for slow spinning pulsars should only be possible for periods @xmath36 s which are not observed . we believe that pulsars with @xmath37 s form a better set to test the model . a period @xmath37 s is sufficiently smaller than the time interval in which neutrinos are emitted and therefore it can be safely assumed that the magnetic field and thus the net momentum vector rotate around the spin axis several times during the neutrino emission . the momentum component on the plane orthogonal to the spin axis averages out and only the component parallel to the spin axis survives . thus , for fast spinning pulsars , the proper motion should be in the direction along the pulsar spin axis . as an important consequence of this alignment of the velocity and spin axis for fast spinning pulsars one can easily prove that the relation @xmath38 must apply where @xmath39 is the projected pulsar velocity in the sky , @xmath40 the angle between the magnetic and spin axes , and @xmath41 is the angle between the magnetic axis and the line of sight ( impact parameter ) . we will see that observational data allow a test of this relation . pulsar proper motions are determined by radio interferometry ( harrison et al . @xcite ) and by interstellar scintillation observations ( cordes @xcite ) . in order to determine the emission geometry parameters @xmath40 and @xmath41 one has to rely on some emission model for pulsars . the so - called magnetic - pole model assumes that the electromagnetic radiation originates in the vicinity of a magnetic pole . the radiation is emitted in a narrow cone - shaped beam centered around the magnetic axis . the nonvanishing angle @xmath40 between the spin axis and the magnetic axis produces the characteristic pulses every time the beam sweeps the line of sight ( lyne & graham - smith ) . two different groups ( lyne & manchester ; rankin @xcite , @xcite , @xcite ) have determined @xmath40 and @xmath41 for more than one hundred pulsars . the method used by both groups is based on the apparent increase of pulse width as @xmath42 . the classification scheme of pulsars , however , differs between the two groups and , for calculating the angles , they find different empirical laws relating the pulse width and the pulsar period : while lyne & manchester ( ) use width @xmath43 , rankin ( @xcite , @xcite , @xcite ) employs width @xmath44 . the results of both groups show , in general , a remarkable agreement for angles @xmath45 but they can widely differ for larger angles ( miller & hamilton ) . our aim has been to find out whether the relation given by eq . ( [ vbangles ] ) is corroborated by observational data . bearing in mind the lack of complete agreement between the different measurements of the emission angles , we have studied three different sets of data : ( a ) angles from lyne & manchester ( ) , ( b ) angles from rankin ( @xcite ) and ( c ) the average of the angles from sets ( a ) and ( b ) for those pulsars which are common in both of these sets and which have similar @xmath40 values , namely @xmath46 . furthermore , it has to be taken into account that eq . ( [ vbangles ] ) , true at the pulsar birth , may be spoiled by the temporal evolution of pulsar parameters ( angles and magnetic field ) . it is believed that the magnetic field of a pulsar decays with a characteristic time @xmath47 years ( harrison et al . there is also observational evidence supporting that the magnetic and spin axes become aligned when the pulsar ages ( lyne & manchester ) . to avoid these problems associated with time evolution we have selected from the data the pulsars younger than @xmath48 years and , to be even safer , younger than @xmath49 years ( lorimer et al . @xcite ) thus creating two subsets for each of the sets ( a ) , ( b ) and ( c ) . we disregard older pulsars since their present properties may be rather different from those at birth . we do not consider pulsars in binary systems either since for them eq . ( [ vbangles ] ) could be affected by the evolution of close binary systems . the selected pulsars for the three cases ( a ) , ( b ) and ( c ) are listed in tables [ tab1 ] and [ tab2 ] . lll psr b0136 + 57@xmath50 & psr b0329 + 54 & psr b0450@xmath5118@xmath50 + psr b0458 + 46@xmath50 & psr b0559@xmath5105 & psr b0656 + 14@xmath50 + psr b0736@xmath5140 & psr b0823 + 26 & psr b1508 + 55@xmath50 + psr b1642@xmath5103 & psr b1706@xmath5116@xmath50 & psr b1929 + 10 + psr b1933 + 16@xmath50 & psr b1946 + 35@xmath50 & psr b2020 + 28@xmath50 + [ tab1 ] lll psr b0136 + 57@xmath50 & psr b0329 + 54 & psr b0355 + 54@xmath50 + psr b0450@xmath5118@xmath50 & psr b0450 + 55@xmath50 & psr b0458 + 46@xmath50 + psr b0736@xmath5140 & psr b0740@xmath5128@xmath50 & psr b0823 + 26 + psr b0919 + 06@xmath50 & psr b1449@xmath5164@xmath50 & psr b1508 + 55@xmath50 + psr b1642@xmath5103 & psr b1706@xmath5116@xmath50 & psr b1818@xmath5104@xmath50 + psr b1822@xmath5109@xmath50 & psr b1842 + 14 & psr b1933 + 16@xmath50 + psr b1946 + 35@xmath50 & psr b2020 + 28@xmath50 & psr b2021 + 51@xmath50 + psr b2217 + 47 & psr b2224 + 65@xmath50 & + [ tab2 ] we plot @xmath52 $ ] versus @xmath53 for young , fast spinning pulsars in figs . [ figu1 ] , [ figu2 ] and [ figu3 ] , for the cases ( a ) , ( b ) and ( c ) , respectively . the transverse velocity @xmath39 is calculated from the measured angular velocity @xmath54 using the pulsar distance @xmath55 derived from its dispersion measure . the pulsar velocities have been corrected for galactic rotation by removing a flat rotation curve of rotational velocity 225 km / s , with the sun at a distance of 8.5 kpc from the galactic center . they have also been corrected for the peculiar solar motion of the sun which was assumed to to have a velocity of 15.6 km / s in the direction @xmath56 and @xmath57 ( murray @xcite ) . the parameters @xmath55 , @xmath34 , @xmath25 and the pulsar ages as well as the parameters needed to compute @xmath54 have been taken from taylor et al . ( @xcite , @xcite ) , an updated catalog containing 706 pulsars . the scatter of points in the plots we depict suggests that there is hardly any correlation . to evaluate rigorously their significance , we use the spearman rank - order correlation coefficient @xmath58 and its probability @xmath59 . the coefficient @xmath58 ranges from @xmath60 ( perfect correlation ) to @xmath61 ( perfect anticorrelation ) . further information is given by @xmath59 , the probability that @xmath58 for two uncorrelated data sets would be larger than the spearman coefficient found . hence , to have a significant correlation between two sets , one must obtain @xmath62 and @xmath63 . the calculated @xmath58 and @xmath59 corresponding to our three cases are shown in table [ tab3 ] . we calculate the two numbers using pulsars younger than @xmath49 yr and pulsars younger than @xmath48 yr . we are unable to find any correlation of the sort given by eq . ( [ vbangles ] ) ( except possibly for the case ( b ) with pulsars younger than @xmath49 yr ) . ccccc case & age[@xmath48 yr]@xmath64 & number & @xmath58 & @xmath65 + ( a ) & 1 & 15 & @xmath66 & 0.83 + ( a ) & 0.3 & 9 & 0.25 & 0.52 + ( b ) & 1 & 23 & 0.31 & 0.15 + ( b ) & 0.3 & 17 & 0.68 & @xmath67 + ( c ) & 1 & 10 & @xmath68 & 0.78 + ( c ) & 0.3 & 7 & @xmath69 & 0.88 + [ tab3 ] it is interesting to point out the existence of fast spinning pulsars like psr b0833 - 45 which have large peculiar velocities whilst the magnetic and spin axes are nearly perpendicular . clearly , the mechanism presented in ks can not account for the proper motion of these pulsars . we have also examined the case of young , fast spinning pulsars with small angles @xmath70 . in the framework of the studied mechanism those pulsars should have small transverse velocities @xmath39 ( since @xmath41 is small for most pulsars ) . for the mean transverse velocity of the pulsars fulfilling the selection criteria we obtain 344 km / s . this has to be compared with the mean average transverse velocity for pulsars without any restrictions on @xmath40 or the pulsar period . such mean values were computed to be @xmath71 km / s for 29 pulsars which are younger than 3 myr , and @xmath72 km / s for 99 pulsars without the age constraint ( lyne & lorimer @xcite ) . the fact that there is no significant difference between the mean transverse velocity we obtain and the latter ones above does not support the model examined here . there is yet another important consequence of the alignment of the spin axis and the velocity vector for fast spinning pulsars , which has been explored some time ago ( tademaru @xcite ; anderson & lyne @xcite ) in the context of the asymmetric emission of electromagnetic radiation mentioned in the introduction of this paper . if these two vectors are aligned , the difference between the projected pulsar velocity angle in the sky @xmath73 and the position angle of the linear polarization vector in the center of a pulse ( corrected for faraday rotation ) @xmath74 should be either @xmath75 or @xmath76 depending on the emission mechanism and the inclination of the spin and magnetic axes to the line of sight . while tademaru ( @xcite ) found that the histogram of @xmath77 showed two peaks at @xmath75 and @xmath76 , a subsequent study performed by anderson & lyne ( @xcite ) concluded that such peaks were not present . the latter result was argued to be in disagreement with the mechanism introduced by harrison & tademaru ( ) . it would not support the resonant neutrino emission model discussed here either . given the significant increase in the amount and quality of available pulsar data over the last years it would be of interest to carry out the @xmath78 test again . this could not only help to resolve the contradicting conclusions of the previous studies but also yield an additional test of the neutrino emission model if only fast spinning pulsars are selected from the updated and enlarged sets of observational data . moreover , both tests have different sources of errors . while the @xmath78 test does not depend on the angles @xmath40 and @xmath41 the derivation of which is model dependent , it relies on the measurement of the rotation measure rm of a pulsar . the contribution from faraday rotation , rm@xmath79 , has to be subtracted from the measured polarization angle to obtain the intrinsic polarization angle @xmath74 . our test relies on the measurement of @xmath40 and @xmath41 but does not need the value of rm for each pulsar which can be a significant source of errors . in this sense , both tests constitute complementary ways of checking whether the model given in ks and related models are realistic or not . we have looked into possible ways of testing the new mechanism suggested by ks to explain the large proper motion of pulsars . it was argued that for fast spinning pulsars this mechanism forces the vector velocity of a pulsar to be aligned with its spin axis . making use of this alignment we have found eq . ( [ vbangles ] ) , which is a relation between the magnetic field of a pulsar , its proper motion , the angle @xmath40 between the magnetic and spin axes and the impact parameter @xmath41 . this relation should be true as long as pulsar velocities are produced by asymmetric emission from a magnetically distorted resonant neutrino - sphere . we have studied whether pulsar data actually show this correlation . to carry out this study we have had to face the fact that there is some controversy about the true values of @xmath40 and @xmath41 in the sense that the angles from two different groups show discrepancies for @xmath80 . therefore , we have in a first step considered separately the data from each group . in addition , we have also selected the pulsars studied by both groups for which there is a reasonable agreement between the angles measured by either group . we believe that the results we derive in the latter case are the most reliable . in this case , observational data do not show any significant correlation and therefore , the mechanism introduced in ks is unlikely to explain the proper motion of pulsars . we have indicated that the temporal evolution of pulsars may spoil any initial correlation . to diminish this effect we have only considered young pulsars . another possible caveat is the fact that eq . ( [ vbangles ] ) involves the magnetic field in the core of the pulsar whilst what is measured is the surface magnetic field . it is , however , plausible and accepted to assume that both fields are proportional ( manchester & taylor @xcite ; ruderman @xcite ) . looking at table [ tab3 ] one can see that the number of pulsars with available data which satisfy all the requirements regarding period , age and availability of emission angles as well as proper motions is small in all the cases considered . although our results cast doubt on the resonant neutrino emission as the mechanism responsible for the proper motion of pulsars , clearly one needs more data , and clarify the uncertainties related to the true emission angles , to rule out definitely or support this mechanism . as a final remark we would like to point out that our conclusions also apply to other mechanisms proposed to explain the large velocities of pulsars this is the case for fast spinning pulsars in all models where the momentum kick is collinear to the magnetic axis and the resulting velocity proportional to @xmath34 . under these conditions velocities along the spin axis are predicted . hence , the work by chuga@xmath1 ( @xcite ) and dorofeev et al . ( @xcite ) is affected . they study how the polarization of electrons and positrons by a high magnetic field in a newborn neutron star will generate an anisotropic neutrino emission . also the mechanism suggested by vilenkin ( @xcite ) is disfavored for a uniform magnetic field , as is the mechanism in horowitz & piekarewicz ( @xcite ) . pulsar velocities from resonant spin - flavor precession of neutrinos are studied by akhmedov et al . ( @xcite ) . again , this mechanism is put under pressure by our work as well as the model suggested by kusenko & segr ( @xcite ) where a resonant conversion to sterile neutrinos induced by neutral currents is considered . we would like to thank dr . p. podsiadlowski and dr . s. sarkar for helpful discussions as well as prof . manchester , prof . rankin and dr . k. xilouri for useful information . m.b . gratefully acknowledges financial support from the fellowship hsp ii / aufe of the german academic exchange service ( daad ) . r.t . would like to thank dr . s. sarkar for his invitation to oxford university and the european theoretical astroparticle network for financial support under the eec contract no . chrx - ct93 - 0120 ( direction generale 12 coma ) .

it has been recently suggested that magnetically affected neutrino oscillations inside a cooling protoneutron star , created in a supernova explosion , could explain the large proper motion of pulsars . we investigate whether this hypothesis is in agreement with the observed properties of pulsars and find that present data disfavor the suggested mechanism . the relevance of our results for other models proposed to understand the origin of pulsar velocities is also discussed .

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Proceed to summarize the following text: the study of nuclear properties at high temperature , spin and isospin has gained much of interest in recent times . thanks to the recent developments in experimental facilities , these highly - excited nuclei are becoming more accessible and provide theorists with a challenging task . apart from these extremes , there are still some unexplored regimes of hot nuclei . the properties of nuclei at very low temperatures and the phase transitions associated with that belong to such area where conclusive experimental results are scarce . at such low temperatures , the shell ( quantal ) and pairing effects are quite active though being modified by thermal effects . among the famous and open questions in this regime are the existence of pairing phase transition , the order of it if it exists , the role of fluctuations , etc . in hot nuclei , thermal fluctuations are expected to be large since the nucleus is a tiny finite system . thermal shape fluctuations and fluctuations in the pairing field are the dominating fluctuations and they have been so far studied separately within different models @xcite . both of these fluctuations are expected to be present at low temperatures . however , the interplay between them has not been investigated so far . the present work addresses this subject and we study the influence of this interplay on the experimental observables , namely the width of giant dipole resonance ( gdr ) . gdr is a fundamental mode of excitation of nuclei caused by the out of phase oscillations between the proton and neutron fluids under the influence of the electromagnetic field induced by the emitted / absorbed photon . in general , the resonance parameters of any resonating object are related to the geometry of the object . in this way , the gdr width and its cross - section could yield direct information about the shape of the nucleus . this is only a macroscopic description of gdr and there are microscopic approaches , which couple the gdr to particle - hole , particle - particle and hole - hole excitations @xcite . for hot nuclei , which are not accessible by the discrete @xmath3-ray spectroscopy and other conventional techniques , the measurement of gdr is considered to be a major probe to obtain the details of nuclear structure . the importance of measuring the gdr width at low temperatures was insisted by one among the authors @xcite and the first results were reported in ref . @xcite , where it was found that the gdr width in @xmath1sn at @xmath4 1 mev is nearly the same as that in the ground state . this data point was successfully explained only after treating properly the pairing correlations with the phonon damping model ( pdm ) @xcite . similarly , it was found in ref . @xcite that the gdr width in @xmath5au extracted at @xmath4 0.7 mev is almost the same as its ground state value . however , it was misattributed to the shell effects as in the case of @xmath2pb . this was clarified later @xcite where the proper inclusion of shell effect was found to act in the opposite direction to raise the width and the pairing fluctuations were speculated to explain this anomaly . preliminary results in this regard were reported in refs . @xcite where the importance of considering pairing in the thermal shape fluctuation model has been emphasized . the recent low temperature gdr measurements done at variable energy cyclotron centre , kolkata @xcite highlight the interesting features of the gdr width in heavy nuclei observed at low temperatures . the thermal shape fluctuation model ( tsfm ) @xcite , which is often used by experimentalists , describes the increase of the gdr width with temperature by averaging the gdr cross section over all the quadrupole shapes . however this model is known to largely overestimate the gdr width in open - shell nuclei at low temperatures . the success of a proper treatment of pairing within the pdm @xcite suggests the necessity of including pairing correlations to cure this shortcoming of the tsfm . the pdm is a microscopic model , whose mechanism is different from that of the tsfm . the pairing has never been taken into account within the tsfm so far because of the ( incorrect ) assumption that the pairing gap disappears at @xmath6 1 mev . given the popularity of the tsfm the inclusion of pairing in the tsfm is quite important . this is done for the first time in the present work . apart from the tsfm , two phenomenological parameterizations have been reported @xcite which are very successful in approximating global behaviour of the gdr widths . in recent literature @xcite these parameterizations referred to as , phenomenological tsfm ( ptsfm ) @xcite and its modification to take into account the quenching of width at low @xmath7 has been referred to as a critical temperature included fluctuation model ( ctfm ) @xcite . it has to be noted that these empirical formulae constructed to mimic the results of tsfm should not be confused with the tsfm itself . the ptsfm and ctfm are merely a phenomenological parameterizations based on empirical data , which has neither microscopic nor macroscopic foundation . it is interesting to note , although both the pdm and ctfm give results consistent with the measurements of the gdr width in @xmath0tc @xcite , these two models are quite different from the tsfm . it is indeed mentioned in ref . @xcite that it would be interesting to compare the data with tsfm by including the effect of thermal pairing . here we employ the thermal fluctuation model built on nilsson - strutinsky calculations with a macroscopic approach to gdr and examine the inclusion of the fluctuations in the pairing field . in refs . @xcite the same hamiltonian was used to generate the free energy as well as the gdr observables in a consistent way , which led to a slow thermal damping of gdr width when compared to the experimental results . the formalism adopted in the present work is well tested to reproduced several gdr observations at higher temperatures @xcite . this formalism is extended to include thermal pairing and our results are discussed in the forthcoming sections . the present theoretical approach is explained below in threefold within the models for a ) deformation energy calculations , b ) relating the shapes to gdr observables and c ) considering the fluctuations due to thermal effects in finite systems . here we follow the finite temperature nilsson - strutinsky method @xcite . the total free energy ( @xmath8 ) at a fixed deformation is calculated using the expression @xmath9the liquid - drop energy ( @xmath10 ) is calculated by summing up the coulomb and surface energies corresponding to a triaxially deformed shape defined by the deformation parameters @xmath11 and @xmath12 . the shell correction ( @xmath13 ) is obtained with exact temperature dependence @xcite using the single - particle energies given by the triaxial nilsson model . while considering the pairing fluctuations , the nucleus is assumed to behave as a grand canonical ensemble ( gce ) , which allows fluctuations in particle number @xcite . the corresponding free energy is determined as @xmath14where @xmath15 is the nuclear hamiltonian which is independent of temperature , @xmath16 is the chemical potential , @xmath17 is the particle number , @xmath7 is the temperature and @xmath18 is the entropy . the above expression can be expanded to@xmath19+\frac{% \delta ^{2}}{g}\ ] ] where @xmath20 are the single - particle energies obtained by diagonalizing @xmath15 with a harmonic oscillator basis comprising the first 12 major shells , @xmath21 are the quasiparticle energies , @xmath22 is the pairing gap obtained by solving the temperature dependent bcs equations @xcite by assuming a constant pairing strength given by @xcite @xmath23/a^2 $ ] . the smoothed free energy , in the strutinsky way , can be written as @xmath24 with the third term included to give better plateau conditions @xcite . here @xmath25 is the averaging function given by@xmath26@xmath27 if @xmath28 is even and @xmath29 if @xmath28 is odd ; @xmath30 , @xmath31 is the smearing parameter satisfying the plateau condition @xmath32 ; @xmath33 is the order of smearing and @xmath34 are the hermite polynomials . the averaged occupation numbers and single - particle entropies are given by @xmath35 and @xmath36 , respectively . the quasiparticle occupation numbers at finite temperature @xmath7 is given by @xmath37^{-1}\;$ ] , so that the total entropy can be written as@xmath38.\end{aligned}\ ] ] for calculations without pairing ( @xmath39 ) , we consider the canonical ensemble ( ce ) for which the expression for free energy reduces to those given in ref . @xcite . the nuclear shapes are related to the gdr observables using a model @xcite comprising an anisotropic harmonic oscillator potential with separable dipole - dipole interaction . in this formalism the gdr hamiltonian can be written as @xmath40 where @xmath41 stands for the anisotropic harmonic oscillator hamiltonian , the parameter @xmath42 characterizes the isovector component of the neutron and proton average fields and @xmath43 denotes the strength of the pairing interaction . the pairing interaction changes the oscillator frequencies [ @xmath44(@xmath45 ) ] , resulting in the new set of frequencies @xmath46 , where @xmath47 with @xmath48 having the units of mev@xmath49 . alternatively , the role of pairing is to renormalize the dipole - dipole interaction strength such that , @xmath50 , with @xmath51 having the units of mev@xmath52 . including the dipole - dipole and pairing interactions , the gdr frequencies in the laboratory frame are obtained as @xmath53 @xmath54 with the pairing field , the gdr hamiltonian has to be redefined , which will affect the oscillator frequencies . the gdr cross section is constructed as a sum of lorentzians given by @xmath55 where @xmath56 , @xmath57 and @xmath58 are the resonance energy , peak cross - section and full width at half maximum , respectively . here @xmath59 represents the number of components of the gdr and is determined from the shape of the nucleus @xcite . @xmath60 is assumed to depend on the centroid energy through the relation @xcite @xmath61 the peak cross section @xmath57 is given by @xcite @xmath62 the parameter @xmath63 , which takes care of the sum rule is fixed at 0.3 for all the nuclei . in most of the cases we normalize the peak with the experimental data and hence the choice of @xmath64 has a negligible effect on the results . the other parameters @xmath65 ( or @xmath42 ) and @xmath51 ( or @xmath48 ) vary with nuclei so that the experimental width of the gdr built on the ground state is reproduced . the choice for @xmath1sn is @xmath66 , @xmath67 mev@xmath52 and for @xmath0tc it is @xmath68 , @xmath69 mev@xmath52 . when the nucleus is observed at finite temperature , the effective gdr cross - sections carry information on the relative time scales for shape rearrangements @xcite , which lead to shape fluctuations . the general expression for the expectation value of an observable @xmath70 incorporating such thermal shape fluctuations is given by @xcite@xmath71\exp[-f(t;\beta , \gamma ) /t]\mathcal{o}}{\int \mathcal{d}[\alpha ] \exp[-f(t;\beta , \gamma ) /t]}\;\]]with @xmath72=\beta ^{4}|\sin 3\gamma |\,d\beta \,d\gamma $ ] . with the inclusion of pairing fluctuations , we have @xmath73\exp[-f(t;\beta , \gamma , \delta _ { p},\delta _ { n})/t]% \mathcal{o}}{\int \mathcal{d}[\alpha ] \exp[-f(t;\beta , \gamma , \delta _ { p},\delta _ { n})/t]}\;\]]with @xmath72=\beta ^{4}|\sin 3\gamma |\,d\beta \,d\gamma \ \delta _ { p}\,\delta _ { n}\,d\delta _ { p}\,d\delta _ { n}$ ] . we perform the tsf calculations exactly by numerically computing the integrations in eq . ( [ average_2 ] ) with the free energy and the observables calculated at every mesh point ( deformations and pairing gaps ) , utilizing the microscopic - macroscopic approach outlined in sec . [ sec_defene ] . sn with @xmath74 calculated from the simple bcs approach and with the pairing fluctuations ( without shape fluctuations ) . full results for both protons and neutrons are also shown : ( c ) average pairing gap , ( d ) gdr width in the case of @xmath1sn , as a function of temperature . the calculations done without pairing utilize a ce approach ( ce ) , whereas the calculations with simple bcs pairing ( bcs ) and with pairing fluctuations ( pf ) utilize a gce approach ( gce ) . the results obtained using the liquid drop model ( ldm ) are also presented . experimental values for @xmath1sn are taken from refs . @xcite are shown by solid squares . for comparison , data for @xmath75sb , taken from ref . @xcite , are also shown with open circles . ] sn at @xmath76 mev plotted without pairing and with pairing where the calculations are with ce and gce , respectively . the contour line spacing is 0.5 mev , the most probable shape is marked by a solid ( red ) circle and the first two minima are shaded . ] ( c ) and [ fig1](d ) but for the nucleus @xmath0tc . experimental values are taken from ref . @xcite are shown with solid squares . ] ( d ) but for the nucleus @xmath2pb . experimental values are taken from ref . @xcite are shown with solid squares . ] we categorize our different approaches into the following cases : 1 . * ldm * - the free energies correspond to that of the liquid drop model and hence no pairing is included . 2 . * without pairing ( ce ) * - the free energies are calculated assuming a ce and the pairing correlations are neglected . 3 . * bcs ( gce ) * - the free energies are calculated assuming a gce and the pairing has been taken care of within the simple bcs approach . * pf ( gce ) * - the free energies are calculated assuming a gce and along with the pairing and its fluctuations are also has been taken care of ( [ average_2 ] ) . the first three cases are with tsf only ( [ average_1 ] ) . it is convenient to consider the following important factors which can affect the gdr width ( @xmath77 ) in our calculations : 1 . the shell effects through the modification of free energy surfaces . 2 . the pairing effects : 1 . modification of free energy surfaces . 2 . damping of the gdr through the term @xmath78 we start with the study of temperature dependence of gdr width in the case of @xmath1sn where the measured values @xcite have been used as a benchmark for several theories . the results from our calculations are presented in fig . [ fig1 ] . in fig . [ fig1](a ) we see that with the inclusion of fluctuations in the pairing field , even for a closed shell , the average pairing gap is finite and also sustained at high @xmath7 . the trend for @xmath79 ( fig . [ fig1](b ) ) is in accordance with earlier works @xcite with the average value converging to the bcs value as @xmath80 . once we include the tsf along with the pf , the average pairing gaps acquire a larger value as seen in fig . [ fig1](c ) . the gdr widths obtained within various approaches are presented in fig . [ fig1](d ) . we observe that the shell effects have a very little role on @xmath77 because the results with ldm and those without pairing are almost the same . shell effects can contribute to change in @xmath77 only by modifying the free energy surfaces . the modification in free energy surface can be of two types viz . change in the minimum with @xmath77 proportional to the quadrupole deformation ( @xmath81 ) corresponding to the minimum . 2 . change in surface around the minimum : a deeper or crisper minimum will lead to sampling over a smaller region of deformations ( lesser tsf ) and a shallow or well - spread minimum will lead to contributions from larger deformations ( stronger tsf ) . with @xmath82 , lesser tsf leads to a quenched @xmath77 . the free energy surfaces for @xmath1sn at @xmath76 mev are presented in fig . [ fig2 ] where we can see that the one without pairing resembles that of a ldm and hence the shell effects are subtle . with pairing , the surface around minimum energy becomes much crisper and leads to a quenched @xmath77 . we can notice from fig . [ fig1](c ) that , while treating pairing within the bcs approach there is no proton pairing and the averaged neutron pairing gap vanishes at @xmath83 mev . with the inclusion of pf , the proton pairing develops , the neutron pairing strengthens and it is sustained even at @xmath84 mev . subsequently the quenching of @xmath77 is also sustained at higher @xmath7 and the results agree with the experimental observations . very recently , experimental data at low @xmath7 for the nuclei @xmath85tl @xcite and @xmath0tc @xcite are reported and our results for @xmath0tc nucleus is presented here . in @xmath0tc , with bcs calculations , the proton pairing gap and neutron pairing gap vanish at @xmath86 mev and @xmath87 mev respectively as shown in fig . [ fig3](a ) . while considering the pf along with tsf , the average pairing gaps continue to be strong even @xmath84 mev . at low @xmath7 , the discrepancy between ldm " and without pairing " cases are due to the fact that the shell effects drive the shape of this nucleus to a deformed one ( and hence a larger @xmath77 ) . the inclusion of pairing leads to a deeper minimum and hence quenched back @xmath77 as shown by the dashed line in fig . [ fig3](b ) . however , this quenching in @xmath77 owing to bcs pairing is not sufficient to explain the experimental data . the latter can be explained only after the pf along with tsf are included as shown by the solid line in fig . [ fig3](b ) . here we have demonstrated that with the inclusion of not just pairing but its fluctuations also is crucial to explain the observed quenching . at low @xmath7 , in @xmath1sn we have seen that the shell effects have no role on @xmath77 and they increase @xmath77 in the case of @xmath0tc at very low @xmath7 . in @xmath2pb , it is known @xcite that the shell effects quench the gdr width as a consequence of a deeper minimum in the free energy surface . our results in this case are shown in fig . [ fig4 ] , where we see that the quenching in @xmath77 occurs once we include the shell effects . even for such a doubly magic nucleus , we can see a significant pairing gap with the inclusion of pf . however , the pf has no contribution to @xmath77 , which is dominated by the shell effects only . more experimental data at lower @xmath7 could be useful to validate this argument . however , our results for @xmath0tc , @xmath1sn and @xmath2pb clearly demonstrate that the tsfm can be quite successful , if the shell effects ( with explicit temperature dependence ) and the pairing effects are properly incorporated in the free energy . to strengthen our arguments , it would be nice to have more experimental data for a single nucleus but at several temperatures along with a precise data at @xmath88 . we have constructed a theoretical framework to study the gdr with proper treatment of pairing and its fluctuations along with the thermal shape fluctuations . our study reveals that the observed quenching of gdr width at low temperature in the open - shell nuclei @xmath0tc and @xmath1sn can be understood in terms of simple shape effects caused by the pairing correlations . for a precise match with the experimental data , the consideration of pairing fluctuations is crucial . our results for @xmath0tc , @xmath1sn and @xmath2pb clearly demonstrate that the tsfm can be quite successful if the shell effects ( with explicit temperature dependence ) and the pairing ones are properly incorporated in the free energy . it has to be noted that often the empirical formula in ref . @xcite ( constructed to mimic the results of tsfm at higher temperatures ) is quoted as the phenomenological tsfm . the subsequent failure of this empirical formula as well as its modifications in describing gdr width at low temperature should not be confused with that of the tsfm ( also called the adiabatic tsfm ) because of the absence of thermal pairing in the latter . this work is supported by the department of science and technology , government of india . project no . sr / ftp / ps-086/2011 . a part of this work was completed at riken and the numerical calculations were carried out using riken integrated cluster of clusters ( ricc ) .

we present an approach that includes temperature - dependent shell effects and fluctuations of the pairing field in the thermal shape fluctuation model ( tsfm ) . we apply this approach to study the width of giant dipole resonance ( gdr ) in @xmath0tc , @xmath1sn and @xmath2pb . our results demonstrate that the tsfm that includes pairing fluctuations can explain the recently observed quenching in the gdr width . we also show that to validate pairing prescriptions and the parameters involved , we require more and precise data .

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Proceed to summarize the following text: knowledge of earth s internal structure and composition has primarily been obtained through complimentary approaches of geophysical and geochemical investigation . however , debate still persists about the mechanisms responsible for generating the chemical and physical characteristics of earth s mantle observed today , as well as their evolution through time . the large - scale composition of the mantle may be inferred in part from mantle - derived melts erupted at or near the earth s surface , including mid - ocean ridge basalts ( morb ) and ocean island basalts ( oib ) . therefore , analyses of oceanic basalts remain a key component in understanding the geochemical complexities of the mantle . trace element ratios , radiogenic isotope variations ( e.g. , sr , nd , hf , os , pb ) , and noble gas systematics have traditionally been studied in mantle - derived samples to better understand the chemical composition and physical processes occurring within the mantle throughout earth history ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? ocean island basalts exhibit systematic differences in isotope compositions when compared with morb ; this has generally been attributed to mantle heterogeneities arising from the recycling of various materials into oib mantle source regions ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? various different mantle endmembers have been proposed to explain the variable sr , nd , and pb isotope ratios , in that each plume samples chemically distinct mantle sources @xcite ; this has been recently extended to hf ( e.g. , * ? ? ? * ) and os isotopes ( e.g. , * ? ? ? the main compositional mantle endmembers include dmm ( depleted morb mantle ) , em-1 ( enriched mantle 1 ) , em-2 ( enriched mantle 2 ) , and himu ( `` high-@xmath4 '' , where @xmath4 is defined as the time - integrated ^238^u/^204^pb ) . each endmember is defined in terms of their relative radiogenic sr and pb isotope characteristics : dmm displays low ^87^sr/^86^sr and relatively unradiogenic pb , em-1 has intermediate ^87^sr/^86^sr and low ^206^pb/^204^pb , em-2 shows high ^87^sr/^86^sr and intermediate ^206^pb/^204^pb , and himu is defined by low ^87^sr/^86^sr and high ^206^pb/^204^pb @xcite . these geochemical characteristics are thought to be a reflection of specific lithologies present as a component of the source : em-1 may incorporate recycled lower continental crust material or pelagic sediments @xcite or delaminated subcontinental lithosphere @xcite , em-2 may arise from the recycling of continental - derived sediments @xcite , and himu may involve recycled oceanic crust and lithospheric mantle @xcite . it has even been proposed that an ancient marine carbonate component may exist in the himu mantle source @xcite . the traditional mantle endmembers represent extremes in terms of isotopic composition , with each oib location potentially representing a unique mixture of mantle components . however , linking observations of chemical heterogeneity of oib with the mechanisms responsible for generating such variations is currently a major challenge in mantle geochemistry . in contrast to radiogenic isotope systems , the mantle end - members are less well defined by stable isotope variations ; although o , mg , and ca show possible recycled signatures ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , generally it is assumed that the mantle is homogeneous with respect to most stable isotope systems . to date , this has been the case for silicon isotopes . for example , samples of and from the shallow mantle ( peridotites and morb ) are assumed to be representative of the si isotope composition of bulk silicate earth ( bse ) , with any si isotope heterogeneity being obliterated through convective mantle mixing @xcite . however , a systematic study of the si isotope composition of oib has not previously been performed ; only a limited number of high - precision analyses are present in the literature ( see * ? ? ? * for a recent review ) . given the implications of possible mantle heterogeneities on a wide variety of mantle processes and the importance of an accurate characterization of the isotopic composition of bse for geo- and cosmo - chemical comparison purposes , a dedicated investigation of potential si isotope variations in oib is warranted . the si isotope system is a potentially useful tool with which to investigate recycling of terrestrial surface materials within the mantle . silicon is a key element in silicate minerals and is composed of three stable isotopes : ^28^si ( 92.23% ) , ^29^si ( 4.68% ) , and ^30^si ( 3.09% ) . relatively large si isotope fractionations have been observed as the result of low - temperature bio- and geo - chemical surficial / supergene processes , with the light si isotopes becoming preferentially enriched in precipitated silica phases compared to dissolved si in the fluid phase @xcite . previous studies of the si isotope systematics in low - temperature environments , including chert formation @xcite , diatom formation @xcite , and silicate weathering @xcite , have shown that si isotope variations can range over several per mil in surface environments . in contrast , high - temperature igneous processes typically result in relatively small fractionations ; terrestrial igneous materials display only a limited range ( @xmath00.2 per mil on the ^30^si/^28^si ratio ) of si isotope compositions @xcite , with the bulk silicate earth being relatively homogeneous ( @xmath1 ^ 30^si = @xmath20.29 @xmath3 0.07 , 2 sd ; * ? ? ? partial mantle melting to form basaltic melts does not appear to generate large si isotope fractionations @xcite compared to low - temperature processes , so incorporation of recycled surface materials in the mantle could produce si isotope variations that may be observed in oib . here we present high - precision si isotope data obtained by multi - collector inductively - coupled - plasma mass - spectrometry ( mc - icp - ms ) for a suite of oib , including samples representative of the em-1 , em-2 , and himu mantle endmembers , to quantify potential si isotope variations in the mantle and assess the processes that may generate si isotope variability . a set of 46 oib representing locations in the pacific , atlantic , and indian oceans were selected for si isotope analysis . in particular , samples were selected to represent the proposed mantle endmembers , including eight samples from pitcairn ( em-1 ) , eleven samples from samoa ( em-2 ) , and four samples from mangaia ( himu ) . additional samples represent geographic locations with unique geochemical features , such as primitive noble gas compositions , or areas that display less extreme mantle endmember compositions . for example , three samples were selected from so nicolau , one of the northern islands of cape verde that has been identified as himu - like based on radiogenic pb isotope compositions and low ^87^sr/^86^sr , but that also exhibits high ^3^he/^4^he ( @xmath5 r / r@xmath6 ) , possibly indicative of a primitive lower mantle component @xcite . similarly , two samples were selected from st . helena , which has been identified as himu - like based on radiogenic isotope compositions but displays low ^3^he/^4^he @xcite . six samples were selected from the canary islands , particularly the islands of el hierro and la palma , whose mantle sources have been interpreted to contain distinct proportions of recycled oceanic crust and lithospheric mantle based on he o sr nd os pb isotope systematics @xcite . four icelandic samples from volcanic complexes in the northern rift zone ( nrz ) that display depleted tholeiitic compositions were selected . iceland is one of the most well - studied oib - type localities due to its association with both a spreading center ( the mid - atlantic ridge ) and a deep - seated mantle plume @xcite . the relatively unradiogenic he and ne isotope ratios measured previously in these samples compared to global oib is possible evidence for a deep , undegassed mantle source @xcite . ^29^si versus @xmath1 ^ 30^si for oib . all data are within error ( @xmath3 2 se ) of the calculated mass - dependent equilibrium ( solid line , slope 0.5178 ) and kinetic ( dashed line , slope 0.5092 ) fractionation lines . ] lllrrrrrrrr bhvo-2 & hawaii & pacific & @xmath20.15 & 0.07 & 0.01 & @xmath20.30 & 0.12 & 0.01 & 76 & 49.90 + ahanemo2d20-b & galapagos & pacific & @xmath20.14 & 0.06 & 0.02 & @xmath20.30 & 0.13 & 0.05 & 6 & 48.89 + isi09 - 9 & iceland & atlantic & @xmath20.22 & 0.08 & 0.03 & @xmath20.41 & 0.15 & 0.06 & 6 & 44.34 + kbd408702 & iceland - kambsfell & atlantic & @xmath20.23 & 0.14 & 0.05 & @xmath20.44 & 0.19 & 0.07 & 7 & 47.76 + kbd408709 & iceland - kistufell & atlantic & @xmath20.20 & 0.09 & 0.04 & @xmath20.40 & 0.13 & 0.06 & 5 & 48.26 + kbd408729 & iceland - blafjall & atlantic & @xmath20.17 & 0.08 & 0.04 & @xmath20.36 & 0.15 & 0.08 & 4 & 51.19 + _ * average - iceland * _ & & & * @xmath20.21 * & * 0.06 * & * 0.03 * & * @xmath20.40 * & * 0.06 * & * 0.03 * & * 4 * & - + mg-1001 # 1 & mangaia & pacific & @xmath20.23 & 0.08 & 0.02 & @xmath20.46 & 0.16 & 0.05 & 10 & 43.57 + mg-1001 # 2 & mangaia & pacific & @xmath20.20 & 0.07 & 0.02 & @xmath20.38 & 0.14 & 0.04 & 10 & 43.57 + _ average - mg-1001 _ & & & @xmath20.21 & 0.04 & 0.03 & @xmath20.42 & 0.11 & 0.08 & 2 & 43.57 + mg-1002 # 1 & mangaia & pacific & @xmath20.20 & 0.05 & 0.02 & @xmath20.39 & 0.12 & 0.04 & 10 & 43.35 + mg-1002 # 2 & mangaia & pacific & @xmath20.19 & 0.12 & 0.05 & @xmath20.36 & 0.17 & 0.08 & 5 & 43.35 + _ average - mg-1002 _ & & & @xmath20.20 & 0.02 & 0.02 & @xmath20.37 & 0.05 & 0.04 & 2 & 43.35 + mg-1006 # 1 & mangaia & pacific & @xmath20.15 & 0.04 & 0.02 & @xmath20.30 & 0.09 & 0.04 & 6 & 43.23 + mg-1006 # 2 & mangaia & pacific & @xmath20.14 & 0.04 & 0.02 & @xmath20.26 & 0.07 & 0.03 & 6 & 43.23 + _ average - mg-1006 _ & & & @xmath20.15 & 0.01 & 0.01 & @xmath20.28 & 0.05 & 0.04 & 2 & 43.23 + mg-1008 & mangaia & pacific & @xmath20.20 & 0.05 & 0.02 & @xmath20.38 & 0.07 & 0.03 & 6 & 43.00 + _ * average - mangaia * _ & & & * @xmath20.19 * & * 0.06 * & * 0.03 * & * @xmath20.36 * & * 0.12 * & * 0.06 * & * 4 * & - + fe0903 & canary - fuerteventura & atlantic & @xmath20.12 & 0.04 & 0.02 & @xmath20.24 & 0.05 & 0.02 & 6 & 47.05 + eh03 & canary - el hierro & atlantic & @xmath20.13 & 0.05 & 0.02 & @xmath20.24 & 0.10 & 0.04 & 6 & 42.74 + eh12 & canary - el hierro & atlantic & @xmath20.14 & 0.01 & 0.01 & @xmath20.24 & 0.06 & 0.03 & 6 & 42.08 + eh15 & canary - el hierro & atlantic & @xmath20.17 & 0.02 & 0.01 & @xmath20.34 & 0.11 & 0.05 & 6 & 42.27 + _ * average - el hierro * _ & & & * @xmath20.15 * & * 0.04 * & * 0.02 * & * @xmath20.27 * & * 0.12 * & * 0.07 * & * 3 * & - + lp03 & canary - la palma & atlantic & @xmath20.16 & 0.05 & 0.02 & @xmath20.36 & 0.05 & 0.02 & 6 & 39.04 + lp15 & canary - la palma & atlantic & @xmath20.16 & 0.05 & 0.02 & @xmath20.36 & 0.11 & 0.04 & 6 & 43.23 + _ * average - la palma * _ & & & * @xmath20.16 * & * 0.00 * & * 0.00 * & * @xmath20.36 * & * 0.00 * & * 0.00 * & * 2 * & - + cv sn 98 - 01 & cape verde - nicolau & atlantic & @xmath20.21 & 0.09 & 0.04 & @xmath20.40 & 0.19 & 0.09 & 4 & 41.01 + cv sn 98 - 15 & cape verde - nicolau & atlantic & @xmath20.17 & 0.06 & 0.03 & @xmath20.36 & 0.08 & 0.04 & 4 & 39.50 + cv sn 98 - 18 & cape verde - nicolau & atlantic & @xmath20.22 & 0.08 & 0.04 & @xmath20.40 & 0.15 & 0.09 & 3 & 44.30 + _ * average - cape verde * _ & & & * @xmath20.20 * & * 0.05 * & * 0.03 * & * @xmath20.39 * & * 0.04 * & * 0.02 * & * 3 * & - + sh25 & st helena & atlantic & @xmath20.15 & 0.12 & 0.05 & @xmath20.29 & 0.10 & 0.05 & 5 & 44.95 + bm.1911,1626(1 ) & st helena & atlantic & @xmath20.15 & 0.04 & 0.02 & @xmath20.30 & 0.13 & 0.06 & 5 & 45.56 + _ * average - st helena * _ & & & @xmath20.15 & 0.01 & 0.01 & @xmath20.30 & 0.01 & 0.01 & 2 & - + pit 1 & pitcairn & pacific & @xmath20.13 & 0.04 & 0.02 & @xmath20.26 & 0.10 & 0.04 & 6 & 48.10 + pit 3 & pitcairn & pacific & @xmath20.14 & 0.07 & 0.03 & @xmath20.29 & 0.08 & 0.04 & 5 & 48.73 + pit 4a & pitcairn & pacific & @xmath20.12 & 0.06 & 0.02 & @xmath20.25 & 0.08 & 0.03 & 6 & 48.91 + pit 6 & pitcairn & pacific & @xmath20.17 & 0.06 & 0.03 & @xmath20.31 & 0.08 & 0.03 & 6 & 49.54 + pit 7 & pitcairn & pacific & @xmath20.15 & 0.12 & 0.05 & @xmath20.30 & 0.20 & 0.08 & 6 & 48.15 + pit 8 & pitcairn & pacific & @xmath20.15 & 0.08 & 0.03 & @xmath20.31 & 0.13 & 0.05 & 6 & 46.32 + pit 12 & pitcairn & pacific & @xmath20.16 & 0.10 & 0.04 & @xmath20.30 & 0.22 & 0.09 & 6 & 49.22 + pit 16 & pitcairn & pacific & @xmath20.11 & 0.06 & 0.03 & @xmath20.22 & 0.04 & 0.02 & 6 & 48.71 + _ * average - pitcairn * _ & & & * @xmath20.14 * & * 0.04 * & * 0.01 * & * @xmath20.28 * & * 0.07 * & * 0.02 * & * 8 * & - + ag132 & kerguelen & indian & @xmath20.19 & 0.06 & 0.04 & @xmath20.31 & 0.10 & 0.06 & 3 & 45.08 + bm1962 128 ( 114 ) & tristan da cunha & atlantic & @xmath20.17 & 0.09 & 0.04 & @xmath20.31 & 0.19 & 0.07 & 7 & 43.18 + alr 40 g & gough & atlantic & @xmath20.21 & 0.09 & 0.04 & @xmath20.35 & 0.17 & 0.08 & 4 & 48.54 + bv2 & bouvet & atlantic & @xmath20.17 & 0.07 & 0.03 & @xmath20.35 & 0.11 & 0.05 & 6 & 55.16 + ofu-04 - 14 & samoa - ofu & pacific & @xmath20.17 & 0.08 & 0.03 & @xmath20.31 & 0.10 & 0.03 & 9 & 45.36 + alia-115 - 03 & samoa - savaii & pacific & @xmath20.15 & 0.12 & 0.05 & @xmath20.26 & 0.19 & 0.08 & 6 & 50.31 + alia-115 - 18 & samoa - savaii & pacific & @xmath20.16 & 0.06 & 0.02 & @xmath20.28 & 0.06 & 0.02 & 6 & 54.87 + t16 & samoa - tau & pacific & @xmath20.18 & 0.09 & 0.04 & @xmath20.34 & 0.10 & 0.04 & 6 & 46.71 + t21 & samoa - tau & pacific & @xmath20.16 & 0.08 & 0.03 & @xmath20.32 & 0.19 & 0.08 & 6 & 48.61 + t38 & samoa - tau & pacific & @xmath20.17 & 0.03 & 0.01 & @xmath20.32 & 0.06 & 0.03 & 5 & 45.34 + 74 - 4 & samoa - tau & pacific & @xmath20.18 & 0.04 & 0.02 & @xmath20.33 & 0.05 & 0.02 & 5 & 45.52 + 63 - 2 & samoa - vailuluu & pacific & @xmath20.19 & 0.04 & 0.02 & @xmath20.34 & 0.08 & 0.03 & 6 & 46.14 + 76 - 9 & samoa - malumalu & pacific & @xmath20.17 & 0.05 & 0.02 & @xmath20.36 & 0.07 & 0.03 & 6 & 45.91 + 78 - 1 & samoa - malumalu & pacific & @xmath20.15 & 0.07 & 0.03 & @xmath20.28 & 0.05 & 0.02 & 6 & 45.54 + 128 - 21 & samoa - taumatau & pacific & @xmath20.12 & 0.05 & 0.02 & @xmath20.27 & 0.07 & 0.03 & 6 & 49.82 + _ * average - samoa * _ & & & * @xmath20.16 * & * 0.04 * & * 0.01 * & * @xmath20.31 * & * 0.07 * & * 0.02 * & * 11 * & - + aco95 - 3 ( 1562 ad ) & azores - so miguel & atlantic & @xmath20.17 & 0.06 & 0.03 & @xmath20.33 & 0.15 & 0.07 & 5 & 47.81 + aco2000 - 51 & azores - so jorge & atlantic & @xmath20.14 & 0.15 & 0.08 & @xmath20.28 & 0.21 & 0.10 & 4 & 45.21 + _ * average - azores * _ & & & * @xmath20.16 * & * 0.04 * & * 0.02 * & * @xmath20.30 * & * 0.07 * & * 0.05 * & * 2 * & - + eruption 1931 & la reunion & indian & @xmath20.14 & 0.14 & 0.06 & @xmath20.30 & 0.12 & 0.05 & 5 & 43.91 + * mineral separates * & & & & & & & & & & + mg-1001 & zeolite & & @xmath20.07 & 0.07 & 0.04 & 0.00 & 0.15 & 0.08 & 4 & 7.40 + mg-1001 & pyroxene & & @xmath20.22 & 0.10 & 0.04 & @xmath20.42 & 0.21 & 0.09 & 5 & 52.70 + when possible , unaltered samples ( based on visual inspection and absence of secondary minerals ) were chosen for analysis in order to minimize the potential effects of post - eruptive weathering , which is a process that could alter the si isotope composition of basalts . in many cases samples were fresh submarine or subglacial basaltic glasses . major- and trace - element abundances and radiogenic isotopic ratios for the present sample set have been previously reported elsewhere @xcite . the alkali fusion sample dissolution and chemical purification methods used here have been described recently @xcite , and are based on the methods first described by @xcite . samples were ground to a fine powder using an agate mortar ; at least 1 g of each whole rock sample was processed to ensure representative sampling . samples were then digested by an alkali fusion technique . this sample digestion method avoids the use of hf , which can lead to the loss of si in the form of volatile sif complexes following conventional acid digestion procedures . approximately 10 mg of powdered sample was combined with @xmath0200 mg naoh flux in an ag crucible and heated for 12 minutes in a muffle furnace at 720 c. the resulting fused sample was dissolved using milli - q h@xmath7o and weakly acidified to 1% hno@xmath8 to obtain a si stock solution sufficiently dilute to avoid silicic acid polymerization . the samples were then purified for si isotope analysis through ion - exchange chromatography using 1.8 ml biorad ag50 x12 ( 200 - 400 mesh ) cation exchange resin loaded in polyprep columns . although the speciation of dissolved si in water is ph dependent @xcite , it occurs as neutral or anionic species at low to neutral phs and thus is not retained by the resin whereas cationic matrix elements are efficiently separated . quantitative recovery of si was achieved through elution by 5 ml milli - q h@xmath7o . column loads were calculated to yield final solution si concentrations of 2 ppm for isotope analyses . 2 se ) . the dashed line and shaded box represent the estimate ( @xmath3 2 sd ) for the si isotope composition of the bulk silicate earth ( bse ) as given by @xcite . most samples fall in the range of the estimate for bse , but some himu and iceland localities exhibit systematically lower @xmath1 ^ 30^si values . ] silicon isotope ratios were measured on thermo scientific neptune plus mc - icp - ms instruments at washington university in st . louis and at the institut de physique du globe de paris . each measurement represents 30 cycles with an 8.389 s integration time . sample solutions were introduced using a wet plasma system and measurements were made in medium resolution on peak shoulder to avoid the large ^14^n^16^o interference on mass 30 . sample bracketing was used to correct for instrumental mass bias ; the quartz sand standard nbs28 ( nist rm8546 ) and the well - characterized basalt geostandard bhvo-2 were each used as bracketing standards to obtain si isotope measurements during separate analytical sessions and were subjected to the same dissolution and si purification procedure as the samples . silicon isotope compositions are expressed relative to a bracketing standard ( nbs28 or bhvo-2 ) using the following delta notation : @xmath9 where x = 29 or 30 . reported isotope ratios are averages of repeated measurements of each sample ( `` n '' in table 1 ) , and errors are determined from these repeated measurements . in a @xmath1 ^ 29^si versus @xmath1 ^ 30^si plot ( figure 1 ) , all of the samples measured in this study fall within error of the calculated mass - dependent equilibrium ( slope 0.5178 ) and kinetic ( slope 0.5092 ) fractionation lines , and no si isotope anomalies have been observed in terrestrial samples @xcite . therefore , the remainder of the text discusses the si isotope data in terms of ^30^si . silicon isotope data relative to nbs28 as the bracketing standard are reported in table 1 and figure 2 . the errors discussed throughout the text are 2 standard error ( 2 se ) unless otherwise specified . to assess the reproducibility of the data , the si isotope composition of the well - characterized basalt geostandard bhvo-2 was measured during each analytical session . the bhvo-2 value measured here ( @xmath1 ^ 30^si = @xmath20.30 @xmath3 0.01 ) represents 76 measurements during 13 separate analytical sessions and is in good agreement with previously published literature data ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? on average , @xmath1 ^ 30^si values for oib ( @xmath20.32 @xmath3 0.09 , 2 sd ) , including the em-1 endmember pitcairn ( @xmath20.28 @xmath3 0.07 , 2 sd ) and the em-2 endmember samoa ( @xmath20.31 @xmath3 0.07 , 2 sd ) , are in general agreement with previous estimates for the @xmath1 ^ 30^si value of bse ( @xmath1 ^ 30^si = @xmath20.29 @xmath3 0.07 , 2 sd ; * ? ? ? * ) . in addition , no variation in si isotope composition with ocean basin is observable . however , some locations exhibit light si isotope enrichments relative to morb ; on average , oib from iceland nrz ( @xmath1 ^ 30^si = @xmath20.40 @xmath3 0.06 , 2 sd ) and the himu localities mangaia ( cook islands ; @xmath1 ^ 30^si = @xmath20.36 @xmath3 0.12 , 2 sd ) , la palma ( canary islands ; @xmath1 ^ 30^si = @xmath20.36 @xmath3 0.00 , 2 sd ) , and cape verde ( @xmath1 ^ 30^si = @xmath20.39 @xmath3 0.04 , 2 sd ) are lighter than the other oib measured in this study ( on average , @xmath1 ^ 30^si = @xmath20.30 @xmath3 0.06 , 2 sd ) . there are two notable exceptions within these localities with low @xmath1 ^ 30^si . one sample from mangaia ( mg-1006 ) has anomalously high @xmath1 ^ 30^si , but of the mangaia group it has the lowest cao / al@xmath7o@xmath8 value . additionally , one el hierro sample ( eh15 ) has high @xmath1 ^ 30^si relative to the other el hierro samples ; this sample has the lowest @xmath1 ^ 18^o of the group . llrrrrrrr nbs28 & & 0.12 & 0.09 & 0.04 & 0.24 & 0.13 & 0.05 & 6 + isi09 - 9 & iceland & @xmath20.05 & 0.05 & 0.02 & @xmath20.10 & 0.10 & 0.04 & 6 + kbd408702 & iceland & @xmath20.03 & 0.07 & 0.03 & @xmath20.09 & 0.12 & 0.05 & 6 + _ * average - iceland * _ & & * @xmath20.04 * & * 0.02 * & - & * @xmath20.10 * & * 0.01 * & - & * 2 * + mg-1001 # 1 & mangaia & @xmath20.04 & 0.10 & 0.04 & @xmath20.07 & 0.21 & 0.08 & 6 + mg-1001 # 2 & mangaia & @xmath20.05 & 0.14 & 0.05 & @xmath20.10 & 0.20 & 0.07 & 6 + _ average - mg-1001 _ & & @xmath20.05 & 0.01 & 0.01 & @xmath20.08 & 0.04 & 0.03 & 2 + mg-1002 # 1 & mangaia & @xmath20.02 & 0.06 & 0.02 & @xmath20.01 & 0.08 & 0.03 & 6 + _ * average - mangaia * _ & & * @xmath20.03 * & * 0.03 * & - & * @xmath20.05 * & * 0.10 * & - & * 2 * + sh25 & st helena & 0.02 & 0.12 & 0.05 & @xmath20.01 & 0.16 & 0.06 & 6 + pit 1 & pitcairn & 0.01 & 0.10 & 0.04 & 0.00 & 0.10 & 0.04 & 6 + ofu-04 - 14 & samoa - ofu & @xmath20.02 & 0.07 & 0.03 & @xmath20.03 & 0.12 & 0.05 & 6 + aco2000 - 51 & azores - so jorge & @xmath20.01 & 0.06 & 0.02 & @xmath20.01 & 0.21 & 0.08 & 6 + 2 se ) . the dashed line and shaded box represent the si isotope composition of bhvo-2 ( @xmath3 2 se ) , which is used as a proxy for bse . these additional data directly compare selected oib samples to further distinguish si isotope compositions and confirm the variations observed in figure 2 . ] in order to directly measure the variations in si isotope composition among oib , the basalt geostandard bhvo-2 was used as the bracketing standard in a separate measurement session ; these data are denoted @xmath1 ^ 30^si@xmath10 and are presented in table 2 and figure 3 . bhvo-2 was chosen as a bracketing standard because its si isotope composition is well characterized in the literature and it has previously been found to have the same si isotope composition as the estimated value for bse @xcite . bhvo-2 is an oib from kilauea volcano , taken from a surface layer of the pahoehoe lava that overflowed from the halemaumau crater in fall 1919 and has 7.23 wt.% mgo . this direct comparison between oib makes non - zero isotopic offsets easier to resolve since it eliminates a source of analytical error ; when comparing data bracketed using the traditional nbs28 international si isotope standard , the propagation of error compounds the uncertainty . when measured against bhvo-2 , samples from iceland ( @xmath1 ^ 30^si@xmath10 = @xmath20.10 @xmath3 0.01 ) and mangaia ( @xmath1 ^ 30^si@xmath10 = @xmath20.05 @xmath3 0.10 ) are relatively lighter in si isotopes , while samples from st . helena , pitcairn , samoa , and the azores are indistinguishable from bhvo-2 ( on average , @xmath1 ^ 30^si@xmath10 = @xmath20.01 @xmath3 0.02 ) . these results are consistent with the data obtained through bracketing with the traditional si isotope standard nbs28 and confirm the observed si isotope variations in oib . at first order , oib exhibit generally uniform si isotope compositions that are in good agreement with previous estimates for the @xmath1 ^ 30^si value of bse . however , small variations in the data are present , with some localities enriched in light si isotopes by @xmath00.1 on @xmath1 ^ 30^si compared to the estimate for bse . although this difference is small , it could have significant implications for mantle processes responsible for creating and preserving such heterogeneity . below , we discuss the possible causes of the si isotope variability observed in oib , including post - eruptive alteration , magmatic differentiation , sampling of a primitive mantle reservoir , crustal assimilation , and recycling of crust and lithospheric mantle . where possible , unaltered samples were chosen to minimize the effects of post - eruptive alteration . however , mangaia volcanic rocks have crystallization ages of @xmath020 myrs and therefore have undergone surface weathering processes @xcite . although minor chemical weathering should not significantly affect the si isotope composition of the bulk rock @xcite , @xcite found that basalt weathering may result in the formation of secondary minerals that are isotopically light relative to the dissolved si in the fluid phase . to assess the possible contribution of isotopically fractionated secondary minerals on the bulk si isotope composition of mangaia samples , zeolite and pyroxene mineral separates from one mangaia sample ( mg-1001 ) were selected for si isotope analysis by mc - icp - ms . the si isotope compositions of the mg-1001 mineral separates are reported in table 1 . the pyroxene fraction ( @xmath1 ^ 30^si = @xmath20.42 @xmath3 0.09 ) is indistinguishable from the mg-1001 whole - rock ( @xmath1 ^ 30^si = @xmath20.42 @xmath3 0.08 ) in si isotope composition , suggesting that secondary minerals in this sample do not measurably affect the isotopic composition of the bulk rock . in contrast , the zeolite separate ( @xmath1 ^ 30^si = 0.00 @xmath3 0.08 ) is significantly heavier in si isotope composition than the mg-1001 whole - rock , so the zeolites are not the carrier of the light si isotope enrichment in the mangaia oib . however , this is unexpected considering that fluid alteration generally leads to secondary phases enriched in light si isotopes . the heavier si isotope composition of the zeolite can be explained as a precipitate that formed from si - bearing surface fluids that were unassociated with dissolution of the host rock ( i.e. , fluids that do not have @xmath1 ^ 30^si values matching those of basalts ) . assuming a typical surface fluid si isotope composition of 1.3 @xcite and considering that the si isotope fractionation factor between precipitated and dissolved si is around @xmath21 to @xmath21.5 for a variety of precipitates ( e.g. , * ? ? ? * ; * ? ? ? * ) , the formation of secondary minerals with an isotopic composition around 0 is possible if the zeolite si source is dissolved si in such a fluid . isotopically - heavy silica coatings have been observed in weathered hawaiian basalts and were interpreted as the result of weathering @xcite . since the zeolite fraction in mg-1001 has a relatively low modal abundance ( < 10% ) and low concentration of si ( < 4 wt.% ) , based on a simple mass balance the effect of the zeolite fraction on the bulk @xmath1 ^ 30^si value is less than 0.04 . this would result in a positive @xmath1 ^ 30^si shift , suggesting that formation of secondary minerals due to weathering does not account for the light si isotope enrichment in oib lavas from mangaia . since equilibrium isotope fractionation decreases with increasing temperature ( e.g. , * ? ? ? * ) , only a limited range of si isotope fractionation ( on the sub permil level ) is expected for high - temperature igneous processes . in general , evolved , silica - rich lithologies ( e.g. , rhyolites , dacites , granites ) have heavier si isotope compositions than less - evolved ( ultramafic and basaltic ) material , suggesting a small isotope fractionation during magmatic differentiation , with a @xmath00.2 offset in ^30^si/^28^si from 40 to 75 wt.% sio@xmath7 @xcite . first - principles calculations predict equilibrium si isotope fractionation between co - existing mineral phases , with the more polymerized ( i.e. , more si - rich ) phase generally enriched in the heavier isotopes @xcite ; this suggests that si isotope fractionation may result from fractional crystallization or partial melting . however , no significant si isotope fractionation has been observed as the result of basalt extraction by mantle partial melting @xcite . ^30^si as a function of sio@xmath7 content for the oib measured in this study . no correlation between si isotope composition and silica content is observed , and widely variable @xmath1 ^ 30^si values are observed for a given sio@xmath7 value . a plot of @xmath1 ^ 30^si versus mgo or mg # shows similar results . ] the si isotope compositions of oib span a similar range ( i.e. , from the lightest oib at @xmath1 ^ 30^si = @xmath20.44 to the heaviest oib at @xmath1 ^ 30^si = @xmath20.22 ) as that defined by ultramafic to rhyolitic material in the literature ( e.g. , * ? ? ? * ) , but since the sample set presented here consists of basaltic rocks , there is a much smaller range of sio@xmath7 contents ( table 1 ) . furthermore , samples with similar sio@xmath7 contents ( e.g. , @xmath048 wt.% sio@xmath7 ) can span a range of si isotope compositions up to 0.2 ( figure 4 ) . a plot of @xmath1 ^ 30^si versus mgo or mg # shows a similar lack of correlation with @xmath1 ^ 30^si ; this further argues against accumulation of isotopically light olivine and/or pyroxene as the carrier of low @xmath1 ^ 30^si values in oib . the lack of correlation between @xmath1 ^ 30^si and sio@xmath7 content suggests that magmatic differentiation is not the primary cause of si isotope variability observed in oib . recent theoretical calculations of equilibrium fractionation factors show that si isotopes may be fractionated between silicate minerals at the elevated pressure conditions in the lower mantle @xcite . due to the large pressure gradients within the deep mantle , high - pressure silicates have a different structure than those at lower pressure ; low - pressure olivine polymorphs have four - fold coordination of si while high - pressure bridgmanite ( mg - perovskite ) has six - fold coordination of si . this difference in si coordination changes the vibrational frequency associated with the si - o bonds , and isotopic substitution is sensitive to this effect . although isotopic fractionations are generally small at high - temperature , the work of @xcite predicts significant si isotope variations among silicate minerals with four - fold coordination of si ( e.g. , olivine , pyroxene , garnet ) and six - fold coordination of si ( bridgmanite ) . as a result , based on the calculated fractionation factors and estimates of the size and composition of mantle reservoirs , @xcite estimated the bridgmanite dominated lower mantle to be @xmath00.1 lighter on the ^30^si/^28^si ratio compared to the upper mantle . if si isotope heterogeneity was created by bridgmanite crystallization at the base of the magma ocean during earth s silicate differentiation and subsequently preserved through mantle evolution , then the lower mantle may still be enriched in lighter si isotopes compared to the upper mantle . although there is much debate about the internal structure of the mantle and the mechanisms responsible for the long - term preservation of primitive mantle reservoirs , layered convection models suggest that geochemical heterogeneities may be preserved for long periods ( e.g. , * ? ? ? * ; * ? ? ? the degree of variation in si isotope composition between oib measured in this study and previously measured morb derived from the shallow mantle ( e.g. , * ? ? ? * ) is in agreement with the @xmath00.1 difference in si isotope composition between the upper and lower mantle predicted by @xcite , suggesting that preservation of si isotope heterogeneity within the mantle is a possible explanation for the @xmath1 ^ 30^si variability in oib . the variable si isotope compositions of oib could potentially reflect selective sampling of preserved primordial components with light si isotope enrichment within the lower mantle . in the case of icelandic nrz oib , their light si isotope enrichments may be consistent with sampling of a deep , primitive , undegassed mantle source based on the he and ne isotope systematics in these particular samples @xcite . similarly , the so nicolau island of cape verde displays both relatively primitive ^3^he/^4^he ( 10 - 15 r / r@xmath6 ; * ? ? ? * ) and a light si isotope enrichment compared to morb . arguments against light si isotope compositions reflecting a lower mantle signature include the observation that samoan ofu samples have primitive he @xcite and ne @xcite but morb - like si isotope composition ( @xmath1 ^ 30^si = @xmath20.31 @xmath3 0.03 ) . conversely , canary island samples have ^3^he/^4^he values within the range of morb @xcite , but have variable si isotope compositions . therefore , sampling of a primitive mantle reservoir with a light si isotope composition can not solely explain the si isotope variability in oib but remains a possibility . experimental work to constrain si isotope fractionation between silicates with different si coordination are required to ground - truth the theoretical calculations of @xcite . although only a limited number of samples in this study have previously been analyzed for o isotope compositions , it is noteworthy that most of the locations that exhibit light si isotope enrichments also have been identified as areas with low @xmath1 ^ 18^o signatures , including himu - type oib and iceland @xcite . this raises the question of whether there is a common mechanism that generates the isotopically - light si and o isotope compositions of oib . like si isotopes , o isotopes display large variations in surface environments relative to mantle materials ; therefore , o isotopes can be a sensitive measure of low - temperature processes , particularly water - rock interactions . altered oceanic crust can be enriched in the heavier isotopes of o through low - temperature alteration by surface fluids ( characteristic of the upper oceanic crust ) or depleted in the heavier isotopes through high - temperature hydrothermal alteration ( characteristic of the lower oceanic crust ) , so variations in @xmath1 ^ 18^o may indicate the presence of oceanic crustal material in oib magmas . the depletions in the heavy isotopes of o found in some himu locations and iceland are consistent with either recycling or assimilation of hydrothermally altered lower oceanic crust @xcite . it is difficult at this point to distinguish between recycling and assimilation of oceanic crust based solely on si isotopes . however , due to the much lower concentration of dissolved si in fluids compared to the high o concentration , and the similar si isotope composition of si saturated hydrothermal fluids to their rock setting @xcite , the effect of hydrothermal alteration is likely to be much smaller for si isotopes than for o isotopes . large amounts of hydrothermally altered material would likely be needed to create resolvable si isotope variations in oib . based on the comparison of @xmath1 ^ 18^o and ^187^os/^188^os in oib , @xcite suggested that himu - type lavas compositions are inconsistent with the incorporation of recycled oceanic crust , and @xmath1 ^ 18^o depletions in oib reflect assimilation of lower oceanic crust . however , @xcite argued that radiogenic ^187^os/^188^os coupled with high os concentrations in some canary island oib can not be explained by crustal assimilation , but rather indicates a contribution of recycled oceanic lithosphere . this suggests that crustal assimilation is not the main cause of the light si enrichments in this oib suite , but processes of crustal recycling may play an important role in generating si isotope variability in the mantle . subduction introduces a wide variety of materials into the mantle , including oceanic and continental crust , oceanic lithospheric mantle , and sediments , all of which may have undergone variable degrees of alteration on the seafloor and/or during subduction ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? subduction of geochemically diverse material that is incorporated into hotspot sources may give rise to observed heterogeneities in the products of these magmas , and it has been proposed that the type of material subducted generates the isotopic signatures of the different mantle components sampled by some oib @xcite . furthermore , subducted sediment is a significant source of incompatible elements to the mantle , and sediments may aid in creating chemical and isotopic heterogeneity within the mantle @xcite . the presence in the mantle of unaltered continent - derived material ( in the form of lower continental crust or sediments derived from the upper continental crust that has not been altered in si isotope composition due to metamorphic dehydration and fluid - rock interaction during subduction ) is unlikely to significantly affect oib si isotope compositions . estimates of the si isotope composition of both the lower and upper continental crust ( @xmath1 ^ 30^si = @xmath20.29 and @xmath1 ^ 30^si = @xmath20.25 , respectively ) are similar to morb and estimates for the bse @xcite . although specific continental inputs may have si isotope compositions that differ from these values for the lower and upper continental reservoirs ( e.g. , shale , which exhibits @xmath1 ^ 30^si values from zero to @xmath20.82 ; * ? ? ? * ) , we assume that on a large scale a single rock type would not dominate the subducted material and such sub - permil @xmath1 ^ 30^si variations would be averaged out . similarly , the presence of unaltered oceanic crust ( with morb - like si isotope composition ) would not create si isotope variations in oib . however , isotopic variations in himu - like oib and the iceland nrz oib in this study have most often previously been explained by the incorporation of recycled oceanic crust into the plume source ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? this suggests that if recycling of continental or oceanic crust is the source of the light si isotope enrichment observed in the himu and iceland nrz oib , the material must be geochemically modified either prior to or during subduction . there are a number of processes acting on oceanic crust that could modify its si isotope composition , including high - temperature hydrothermal alteration at seafloor spreading centers , low - temperature seafloor alteration , and metasomatism and metamorphic dehydration during subduction @xcite . the preferential loss of heavy si isotopes to the interacting fluid would result in isotopically light crustal material , which could lead to heterogeneities in the mantle as observed in oib . although a high - precision investigation of the si isotope composition of altered and subducted lithologies ( e.g. , ophiolite sequences ) does not currently exist in the literature , there is some evidence for preservation of initial si isotope compositions during high - temperature metamorphism and metasomatism @xcite . additionally , island arc basalts ( iab ) display si isotope compositions similar to morb @xcite , so processing during subduction may preserve primary si isotope compositions to the extent that the signature of interacting fluids is not communicated to the island arc . altered oceanic crust displays enrichments in light si isotopes relative to morb @xcite . it is therefore possible that oceanic crust has experienced si isotope fractionations due to seafloor and hydrothermal alteration , and this composition is preserved during subduction . however , a systematic study of the behavior of si isotopes during metamorphic dehydration is necessary to ascertain the effects of subduction on mantle inputs . in contrast to oceanic and continental crust , oceanic sediments have widely variable compositions and can be significantly enriched in light si isotopes compared to typical igneous rocks . this suggests that the incorporation of oceanic sediments with isotopically light si in the oib plume sources is one viable mechanism for creating the light si isotope enrichments observed in oib . previous measurements of si isotopes in silicified and precipitated materials such as cherts show a large range in si isotope composition , with @xmath1 ^ 30^si values down to @xmath24 or more @xcite , but sedimentary material with extremely fractionated si isotope compositions ( potentially including clays and cherts ) would be diluted by other subducting material . additionally , much of the sedimentary material that enters subduction zones may be removed before entering the convecting mantle by forearc scraping or through arc volcanism @xcite ; in general , oib sources are believed to incorporate at most a few percent of sediments @xcite . for these reasons it is difficult to quantify the si isotope composition of subducting material that enters the mantle . the potential contribution of subducted material on the si isotope signature of oib can be calculated from a simple mass balance using estimates for the si isotope composition and fraction of recycled material in the oib source . here we consider a subducted package with a bulk @xmath1 ^ 30^si of @xmath21.0 , @xmath20.75 , and @xmath20.5 . these values are consistent with plausible compositions of altered oceanic crust @xcite with or without a small portion of sediment ( we consider a few percent of sediments with @xmath1 ^ 30^si values down to @xmath24 corresponding to compositions of silicified and precipitated materials ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? results of the mass balance calculations are shown in figure 5 . these calculations suggest that the relatively light si isotope enrichment observed in the himu and iceland nrz basalts can be explained by the incorporation 25% of recycled altered oceanic crust or 10 - 25% altered oceanic crust plus a few percent of sediments in oib magmas . using a similar mass balance shows that 1 - 2% sediment added to peridotite mantle is unlikely to measurably shift the @xmath1 ^ 30^si of the mixture , which can explain the morb - like @xmath1 ^ 30^si values of em-1 and em-2 type oib despite their hosting some recycled marine sediment . the addition of more than a few percent of sediments in the sources of oib , particularly himu - type , is inconsistent with pb and sr isotopic constraints and trace - element ratios unless sediments are significantly modified during subduction zone processing @xcite . sediments constitute a thin veneer that overlies oceanic crust , so that the subduction package is dominated by basaltic oceanic crust and lithospheric mantle . for this reason , it is more likely that the recycled material that can reproduce the @xmath1 ^ 30^si signatures of himu and iceland nrz oib is primarily altered ` basaltic ' oceanic crust with a bulk composition of @xmath1 ^ 30^si < @xmath20.5 that contributes melt to generate @xmath025% of the mass of the lavas . furthermore , it should be noted that the mass fraction of recycled material in the mantle source is unlikely to be the same as the mass fraction present in the plume source melts because recycled lithosphere in the form of eclogite preferentially melts compared to ambient peridotitic mantle , potentially leading to pyroxenite - peridotite mixtures in oib mantle sources ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? ^30^si as a function of the fraction of recycled component present in the melt derived from the mantle source . the dashed line and shaded box represent the average si isotope composition ( @xmath3 2 se ) of the light - isotope enriched oib measured in this study ( iceland , mangaia , cape verde ) . the bse reservoir is considered to have a si concentration [ si ] = 0.21 wt.% and @xmath1 ^ 30^si = @xmath20.29 @xcite . the recycled component is modeled with a nearly identical bulk [ si ] = 0.23 wt.% but variable @xmath1 ^ 30^si values . here @xmath20.50 , @xmath20.75 , and @xmath21.00 are used as illustrative @xmath1 ^ 30^si values for the bulk recycled component ; these values are consistent with altered oceanic crust @xcite or a mixture of altered oceanic crust with a few percent of sediments ( @xmath1 ^ 30^si values between @xmath22 and @xmath24 ; see text for details ) . the si isotope compositions of the anomalous oib localities are consistent with the incorporation of 10 - 30% recycled material in the plume source melt . ] a key indication that the si isotope variability that potentially arises from recycling of subducted oceanic crust and lithospheric mantle lies in the difference in si isotope composition between lavas from the canary islands of la palma and el hierro . these two islands have previously been found to have contrasting o isotope compositions , with relatively lower @xmath1 ^ 18^o values in la palma samples compared to el hierro samples @xcite . furthermore , the coupled o and os isotope systematics of la palma and el hierro reflect different proportions of recycled oceanic crust and lithospheric mantle in the plume sources @xcite . specifically , the relatively lower @xmath1 ^ 18^o values in la palma are generated by a contribution of recycled oceanic gabbroic crust @xcite . this suggests that a single process may be responsible for generating the low @xmath1 ^ 18^o and low @xmath1 ^ 30^si values observed in oceanic basalts , although further work is necessary to confirm this relationship . these observations indicate that the si isotope system may be an important tool for investigating the processes responsible for generating stable isotopic variability present in oib . to a first order , oib exhibit homogeneous si isotope compositions generally in agreement with estimates for the @xmath1 ^ 30^si value of bse ( @xmath20.29 @xmath3 0.07 , 2 sd ; * ? ? ? however , some systematic variations are present ; some himu - type and nrz basalts exhibit light si isotope enrichments relative to other oib , morb , and the estimate for the bse . in contrast , em - type oib do not show any systematic si isotope variations . one possible cause of the light si isotope enrichments in oib is primitive si isotope heterogeneity created by bridgmanite crystallization at the base of the magma ocean during the silicate differentiation of the earth as suggested by @xcite , although the potential to preserve such a reservoir over the history of the earth is debated . an alternative explanation is that subduction of material that is incorporated into hotspot sources may give rise to observed heterogeneities . the si isotope variability in the himu and iceland nrz can be explained by the presence of @xmath025% recycled altered oceanic crust and lithospheric mantle in the plume source . a reservoir of isotopically light si in the earth s mantle would have implications for the calculated amount of si in the core . silicon has long been proposed as a potential light element in the core to account for the density deficit relative to pure feni , and an estimate of the si content of the earth s core was one of the early applications of the si isotope system to high temperature geochemistry @xcite . debate still persists about the degree of offset between the si isotope composition of the earth s mantle and that of chondrites ( assumed to represent the bulk earth composition ) , resulting in widely varying estimates of the si content of the earth s core from @xmath01 wt.% to more than 10 wt.% , although high amounts of si in the terrestrial core are likely irreconcilable with other geochemical and geophysical constraints ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? however , a reservoir of isotopically light si in the mantle may be indicated by these new measurements of oib . this would lower the amount of si in the core required by mass balance , driving the calculated si content of the core based on si isotopes closer to that estimated by geophysical means . we thank the three anonymous reviewers as well as the associate editor fangzhen teng for their thorough reviews and comments , which have greatly improved this paper . ep thanks the chateaubriand stem fellowship program for funding . fm thanks the european research council under the european community s h2020 framework program / erc grant agreement # 637503 ( pristine ) and the agence nationale de la recherche for a chaire dexcellence sorbonne paris cit ( idex13c445 ) and for the univearths labex program ( anr-10-labx-0023 and anr-11-idex-0005 - 02 ) . ps thanks the support of the marie curie fp7-iof fellowship `` isovolc '' . abraham k. , opfergelt s. , fripiat f. , cavagna a. , de jong j.t.m . , foley s.f . , andr l. , cardinal d. , 2008 . @xmath1 ^ 30^si and @xmath1 ^ 29^si determinations on usgs bhvo-1 and bhvo-2 reference materials with a new configuration on a nu plasma multi - collector icp - ms . geostandards geoanalytical res . 32 , 193202 . badro j. , fiquet g. , guyot f. , gregoryanz e. , occelli f. , antonangeli d. , dastuto m. , 2007 . effect of light elements on the sound velocities in solid iron : implications for the composition of earth s core . earth planet . 254 , 233238 . brandenburg j.p . , hauri e.h . , van keken p.e . , ballentine c.j . , 2008 . a multiple system study of the geochemical evolution of the mantle with force - balanced plates and thermochemical effects . earth planet . 276 , 113 . cabral r.a . , jackson m.g . , rose - koga e.f . , koga k.t . , whitehouse m. j. , antonelli m.a . , farquhar j. , day j.m.d . , hauri e.h . , 2013 . anomalous sulphur isotopes in plume lavas reveal deep mantle storage of archaean crust . nature 496 , 490493 . chemtob s.m . , rossman g.r . , young e.d . , ziegler k. , moynier f. , eiler j.m . , hurowitz j.a . , 2015 . silicon isotope systematics of acidic weathering of fresh basalts , kilauea volcano , hawaii . . cosmochim . acta 169 , 6381 . claude - 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scale mantle heterogeneities . earth planet . 94 , 257273 . workman r.k . , hart s.r . , jackson m. , regelous m. , farley k.a . , blusztajn j. , kurz m. , staudigel h. , 2004 . recycled metasomatized lithosphere as the origin of the enriched mantle ii ( em2 ) end - member : evidence from the samoan volcanic chain . geochem . geophys . geosys . 5 , q04008 . zambardi t. , poitrasson f. , corgne a. , mheut m. , quitt g. , anand m. , 2013 . silicon isotope variations in the inner solar system : implications for planetary formation , differentiation and composition . geochim . cosmochim . acta 121 , 6783 .

the study of silicon ( si ) isotopes in ocean island basalts ( oib ) has the potential to discern between different models for the origins of geochemical heterogeneities in the mantle . relatively large ( @xmath0several per mil per atomic mass unit ) si isotope fractionation occurs in low - temperature environments during biochemical and geochemical precipitation of dissolved si , where the precipitate is preferentially enriched in the lighter isotopes relative to the dissolved si . in contrast , only a limited range ( @xmath0tenths of a per mil ) of si isotope fractionation has been observed from high - temperature igneous processes . therefore , si isotopes may be useful as tracers for the presence of crustal material within oib mantle source regions that experienced relatively low - temperature surface processes in a manner similar to other stable isotope systems , such as oxygen . characterizing the isotopic composition of the mantle is also of central importance to the use of the si isotope system as a basis for comparisons with other planetary bodies ( e.g. , moon , mars , asteroids ) . here we present the first comprehensive suite of high - precision si isotope data obtained by mc - icp - ms for a diverse suite of oib . samples originate from ocean islands in the pacific , atlantic , and indian ocean basins and include representative end - members for the em-1 , em-2 , and himu mantle components . on average , @xmath1 ^ 30^si values for oib ( @xmath20.32 @xmath3 0.09 , 2 sd ) are in general agreement with previous estimates for the @xmath1 ^ 30^si value of bulk silicate earth ( @xmath20.29 @xmath3 0.07 , 2 sd ; * ? ? ? * ) . nonetheless , some small systematic variations are present ; specifically , most himu - type ( mangaia ; cape verde ; la palma , canary islands ) and iceland oib are enriched in the lighter isotopes of si ( @xmath1 ^ 30^si values lower than morb ) , consistent with recycled altered oceanic crust and lithospheric mantle in their mantle sources .

You are an expert at summarizing long articles. Proceed to summarize the following text: quantum transport in low - dimensional systems has been a topic of interest within the past few decades due to its potential applicability in the field of nanoscience and nanotechnology . exploitation of the spin degree of freedom adds a possibility of integrating memory and logic into a single device , leading to remarkable development in the fields on magnetic data storage application , device processing technique , quantum computation @xcite , etc . naturally a lot of attention has been paid to study spin transport in low - dimensional systems both from experimental @xcite and theoretical @xcite points of view . the understanding of electronic localization in low - dimensional model quantum systems is always an interesting issue . whereas , it is a well established fact that in an infinite one - dimensional ( @xmath0d ) system with random site potentials all energy eigenstates are exponentially localized irrespective of the strength of randomness due to anderson localization @xcite , there exists another kind of localization , known as wannier - stark localization , which results from a static bias applied to a regular @xmath0d lattice , even in absence of any disorder @xcite . till date a large number of works have been done to explore the understanding of anderson localization and scaling hypothesis in one- and two - dimensional systems @xcite . similarly , wannier - stark localization has also drawn the attention of many theorists @xcite as well as experimentalists @xcite . for both these two cases , viz , infinite @xmath0d materials with random site energies and @xmath0d systems subjected to an external electric field , one never encounters any _ mobility edge _ i.e. , energy eigenvalues separating localized states from the extended ones , since all the eigenstates are localized . but there exist some special types of @xmath0d materials , like quasi - periodic aubry - andre model and correlated disordered systems where mobility edge phenomenon at some particular energies is obtained @xcite . although the studies involving mobility edge phenomenon in low - dimensional systems have already generated a wealth of literature @xcite there is still need to look deeper into the problem to address several interesting issues those have not yet been explored . for example , whether the mobility edges can be observed in some other simple @xmath0d materials or the number of mobility edges separating the extended and localized regions in the full energy band of an @xmath0d material can be regulated , are still to be investigated . to address these issues in the present article we investigate two - terminal spin dependent transport in a @xmath0d mesoscopic chain composed of magnetic and non - magnetic atomic sites in presence of external electric field . to the best of our knowledge , no rigorous effort has been made so far to explore the effect of an external electric field on electron transport in such a @xmath0d magnetic - non - magnetic superlattice geometry . here we show that , depending on the unit cell configuration , a @xmath0d superlattice structure subjected to an external electric field exhibits multiple mobility edges at different values of the carrier energy . we use a simple tight - binding ( tb ) framework to illustrate the model quantum system and numerically evaluate two - terminal spin dependent transmission probabilities through the superlattice geometry based on the green s function formalism . from our exact numerical analysis we establish that a sharp crossover from a completely opaque to a fully or partly transmitting zone takes place which leads to a possibility of tuning the electron transport by gating the transmission zone . in addition to this behavior we also show that the magnetic - non - magnetic superlattice structure can be used as a pure spin filter for a wide range of energy . these phenomena enhance the prospect of such simple superlattice structures as switching devices at multiple energies as well as spin filter devices , the design of which has significant impact in the present age of nanotechnology . with an introduction in section i , we organize the paper as follows . in section ii , first we present the model , then describe the theoretical formulation which include the hamiltonian and the formulation for transmission probabilities through the model quantum system . the numerical results are illustrated in section iii and finally , in section iv , we draw our conclusions . let us start with fig . [ chain ] where a @xmath0d mesoscopic chain composed of magnetic and non - magnetic atomic sites is attached to two semi - infinite @xmath0d non - magnetic electrodes , namely , source and drain . the chain consists of @xmath1 ( @xmath1 being an integer ) number of unit cells in which each unit cell contains @xmath2 and @xmath3 numbers of magnetic and non - magnetic atoms , respectively . both the chain and side - attached electrodes are described by simple tb framework within nearest - neighbor hopping approximation . the hamiltonian for the entire system can be written as a sum of three terms as , @xmath4 the first term represents the hamiltonian for the chain and it reads @xmath5 \label{eqn2}\ ] ] where , @xmath6 @xmath7 @xmath8 ; + @xmath9 + @xmath10 = @xmath11 . + + here , @xmath12 refers to the on - site energy of an electron at the site @xmath13 with spin @xmath14 ( @xmath15 ) , @xmath16 is the nearest - neighbor hopping strength , @xmath17 ( @xmath18 ) is the creation ( annihilation ) operator of an electron at the @xmath13th site with spin @xmath14 and @xmath19 is the strength of local magnetic moment where @xmath20 for non - magnetic sites . the term @xmath10 corresponds to the interaction of the spin of the injected electron with the local magnetic moment placed at the site @xmath13 . the direction of magnetization in each magnetic site is chosen to be arbitrary and specified by angles @xmath21 and @xmath22 in spherical polar co - ordinate system for the @xmath13th atomic site . here , @xmath21 represents the angle between the direction of magnetization and the chosen @xmath23 axis , and @xmath22 represents the azimuthal angle made by the projection of the local moment on @xmath24-@xmath25 plane with the @xmath24 axis . in presence of bias voltage @xmath26 between the source and drain an electric field is developed , and therefore , the site energies of the chain becomes voltage dependent . mathematically we can express it as @xmath27 , where @xmath28 is the voltage independent term . the voltage dependence of @xmath29 reflects the bare electric field in the bias junction as well as screening due to longer range electron - electron interaction . in the absence of such screening the electric field varies uniformly along the chain and it reads @xmath30 , where @xmath31 corresponds to the total number of atomic sites in the chain . in our present work , we consider both the linear and screened electric field profiles . as illustrative example , in fig . [ potprofile ] we show the variation of voltage dependent site energies for three different electrostatic potential profiles for a chain considering @xmath32 atomic sites and describe the nature of electronic localization for these profiles in the forthcoming section . the second and third terms of eq . [ eqn1 ] describe the tb hamiltonians for the @xmath0d semi - infinite non - magnetic electrodes and the chain - to - electrode coupling . these hamiltonians are written as follows . @xmath33 \right ] \label{eqn3}\ ] ] and , @xmath34 + \tau_d[\mbox { \boldmath $ c$}_n^{\dag } \mbox { \boldmath $ c$}_{n+1 } + h.c . ] . \label{eqn4}\end{aligned}\ ] ] the summation over s and d in eq . [ eqn3 ] implies the incorporation of both the two electrodes , viz , source and drain . @xmath35 and @xmath36 stand for the site energy and nearest - neighbor coupling , respectively . the electrodes are directly coupled to the chain through the lattice sites @xmath0 and @xmath31 , and the coupling strengths between these electrodes with the chain are described by @xmath37 and @xmath38 , respectively . to obtain spin resolved transmission probabilities of an electron through the source - chain - drain bridge system , we use green s function formalism . the single particle green s function operator representing the entire system for an electron with energy @xmath39 is defined as , @xmath40 where , @xmath41 . following the matrix form of and the problem of finding in the full hilbert space can be mapped exactly to a green s function @xmath42 corresponding to an effective hamiltonian in the reduced hilbert space of the chain itself and we have , @xmath43 where , @xmath44 these and are the self - energies introduced to incorporate the effect of coupling of the chain to the source and drain , respectively . using dyson equation the analytic form of the self energies can be evaluated as follows , @xmath45 where , @xmath46 . following fisher - lee relation , the transmission probability of an electron from the source to drain is given by the expression , @xmath47 . \label{eqn9}\ ] ] where , @xmath48@xmath49 s are the coupling matrices representing the coupling between the chain and the electrodes and they are defined as , @xmath50 . \label{eqn10}\ ] ] here , @xmath51 and @xmath52 are the retarded and advanced self - energies associated with the @xmath53-th ( @xmath54 ) electrode , respectively . finally , we determine the average density of states ( ados ) , @xmath55 , from the following relation , @xmath56\right ] . \label{equ11}\ ] ] in what follows we limit ourselves to absolute zero temperature and use the units where @xmath57 . for the numerical calculations we set @xmath58 , @xmath59 , @xmath60 for the magnetic sites , @xmath61 , @xmath62 , @xmath63 and @xmath64 . the energy scale is measured in unit of @xmath16 . throughout our numerical calculations we assume that the magnetic moments are aligned along @xmath65 direction ( @xmath66 ) , which yields vanishing spin flip transmission probability , viz , @xmath67 , across the bridge system . the net transmission probability is therefore a sum @xmath68 , and the origin of this zero spin flipping can be explained from the following arguments . the operators @xmath69 @xmath70 and @xmath71 @xmath72 associated with the term in the tb hamiltonian eq . [ eqn2 ] are responsible for the spin flipping , where being the pauli spin vector with components @xmath73 , @xmath74 and @xmath75 for the injecting electron . in our present model since we consider that all the magnetic moments are aligned along @xmath65 direction , the term @xmath76 gets the form @xmath77 , and accordingly , the hamiltonian does not contain @xmath73 and @xmath74 and so @xmath69 and @xmath71 do not appear , which leads to the vanishing spin flip transmission probability across the @xmath0d chain . below , we address the central results of our study i.e , the possibility of getting multiple mobility edges in @xmath0d magnetic - non - magnetic superlattice geometries and how such a simple model quantum system can be used as a perfect spin filter for a wide range of energy . in fig . [ localization1 ] we show the variation of total transmission probability @xmath78 along with the average density of states for a @xmath0d magnetic - non - magnetic superlattice geometry considering a linear bias drop . here we consider a @xmath79-site chain in which each unit cell contains one magnetic and four non - magnetic sites and the results are shown for three different bias voltages . for the particular case when the chain is free from external electric field i.e. , @xmath80 electronic conduction through the bridge takes place for the entire energy band as shown in fig . [ localization1](a ) which predicts that all the energy eigenstates are extended in nature . the situation becomes really very interesting when the superlattice geometry is subjected to an external electric field . it is illustrated in figs . [ localization1](b ) and [ localization1](c ) . from these spectra we notice that there are some energy regions for which the transmission probability completely drops to zero which reveals that the eigenstates associated with these energies are localized , and they are separated from the extended energy eigenstates . thus , sharp mobility edges are obtained in the spectrum , and , the total number of such mobility edges separating the extended and localized regions in a superlattice geometry in presence electric field strongly depends on the unit cell configuration and it can be regulated by adjusting the number of magnetic and non - magnetic sites . this phenomenon describes the existence of multiple mobility edges in a superlattice geometry under finite bias condition . now if the fermi energy is fixed at a suitable energy zone where @xmath78 drops to zero an insulating phase will appear , while for the other case , where @xmath78 is finite , a metallic phase is observed and it leads to the possibility of controlling the electronic transmission by gating the transmission zone . the width of the localized regions between the band of extended regions increases with the strength of the electric field as clearly shown by comparing the spectra given in figs . [ localization1](b ) and [ localization1](c ) , and , for strong enough field strength almost all energy eigenstates are localized . in that particular limit metal - to - insulator transition will no longer be observed . the above results are analyzed for a particular ( linear ) variation of electric field along the chain . to explore the sensitivity of getting metal - to - insulator transition on the distribution of electric field , in figs . [ localization2 ] and [ localization3 ] we present the results for two different screened electric field profiles taking the identical chain length . from the spectra we clearly observe that the width of the localized region gradually disappears with the flatness of the electric field profile in the interior of the bridge system . if the potential drop takes place only at the chain - to - electrode interfaces , i.e. , when the potential profile becomes almost flat along the chain the width of the localized region almost vanishes and the metal - to - insulator transition is not observed , as is the case for the zero bias limit . finally , we illustrate how such a simple magnetic - non - magnetic superlattice geometry can be utilized as a perfect spin filter for a wide range of energy in absence of any external electric field . as illustrative example , in fig . [ filter1 ] we present the transmission probabilities for up and down spin electrons together with the average density of states as a function of energy for a @xmath0d magnetic - non - magnetic superlattice geometry . from the spectra we observe that the up and down spin electrons follow two different channels while traversing through the superlattice geometry , since the spin flipping is completely blocked for this configuration . this splitting of up and down spin conduction channels is responsible for spin filtering action and the total number of these channels strongly depends on the unit cell configuration . from figs . [ filter1](a ) and ( b ) we clearly see that for a wide range of energy for which the transmission probability of up spin electrons drops to zero value , shows non - zero transmission probability of down spin electrons . therefore , setting the fermi energy to a suitable energy region we can control the transmission characteristics of up and down spin electrons , and , a spin selective transmission is thus obtained through the bridge system . before we end , we would like to point out that since the overlap between the up and down spin conduction channels depends on the magnitudes of the local magnetic moments , we can regulate the spin degree of polarization ( dop ) simply by tuning the strength of these magnetic moments and for a wide range of energies it ( dop ) almost reaches to @xmath81 . thus , our proposed magnetic - non - magnetic superlattice geometry is a very good example for designing a spin filter . to conclude , in the present work we investigate in detail the spin dependent transport under finite bias condition through a @xmath0d magnetic - non - magnetic superlattice geometry using green s function formalism . we use a simple tb framework to describe the model quantum system where all the calculations are done numerically . from our exact numerical analysis we predict that in such a simple @xmath0d magnetic - non - magnetic superlattice geometry multiple mobility edges separating the localized and extended regions are obtained in presence of external electric field and the total number of mobility edges in the full energy spectrum can be controlled by arranging the unit cell configuration . this phenomenon reveals that the superlattice geometry can be used as a switching device for multiple values of fermi energy . the sensitivity of metal - to - insulator transition and vice versa on the electrostatic potential profile is thoroughly discussed . finally , we analyze how such a superlattice geometry can be utilized in designing a tailor made spin filter device for wide range of energies . setting the fermi energy at a suitable energy zone , a spin selective transmission is obtained through the bridge system . all these predicted results may be utilized in fabricating spin based nano electronic devices . the results presented in this communication are worked out for absolute zero temperature . however , they should remain valid even in a certain range of finite temperatures ( @xmath82k ) . this is because the broadening of the energy levels of the chain due to the chain - to - electrode coupling is , in general , much larger than that of the thermal broadening @xcite . s. aubry and g. andr , in _ group theoretical methods in physics _ , annals of the israel physical society vol . 3 , edited by l. horwitz and y. neeman ( american institute of physics , new york , 1980 ) , p. 133 .

based on green s function formalism , the existence of multiple mobility edges in a one - dimensional magnetic - non - magnetic superlattice geometry in presence of external electric field is predicted , and , it leads to the possibility of getting a metal - insulator transition at multiple values of fermi energy . the role of electric field on electron localization is discussed for different arrangements of magnetic and non - magnetic atomic sites in the chain . we also analyze that the model quantum system can be used as a perfect spin filter for a wide range of energy .

You are an expert at summarizing long articles. Proceed to summarize the following text: the process of phase - separation occurring in binary systems quenched below the coexistence line has been matter of intensive investigations since many years . the overall phenomenology is nowadays reasonably well understood . in an early stage , whose properties depend on the system specific details , domains of the two phases form ; the relative concentration of the two species well inside the domains quickly saturates to a value very similar to the equilibrium composition . then a crossover leads to the late stage dynamics which is characterized by the growth of ordered regions of typical size @xmath4 and by a statistically invariant morphology . this phenomenology is at the basis of the well known scaling property according to which a single length @xmath4 is asymptotically relevant : system configurations at different times are simply related by a scale transformation with a scale factor @xmath4 . in particular , for the equal time density - density correlation function one has g(r , t)=g , [ sca ] meaning precisely that @xmath5 is invariant as a function of scaled lengths @xmath6 . the growth of @xmath4 is of the power law type @xmath7 . the dynamic exponent @xmath8 depends on the specific microscopic mechanisms responsible for the segregation process ; in general different dynamical regimes can be observed , characterized by different values of @xmath8 , by changing the experimental time window @xcite . when the system enters the late stage the memory of the initial state is lost and , due to the presence of one single growing length , its properties become universal . then the dynamical exponents , the scaling function @xmath9 and other observables become independent on the system specific details and are only determined by a few relevant parameters . besides the spatial dimensionality , the presence of conservation laws is a relevant feature . for binary fluids , to which we restrict our attention , the concentration of the species is conserved as the phase - separation proceeds . the initial stage of the scaling regime , before hydrodynamic effects become relevant , is the subject of this paper . for viscous fluid this time window can be sufficiently wide to be detected experimentally . in this condition the phase - separation process is ruled by the evaporation , diffusion and subsequent recondensation of monomers @xcite and the exponent @xmath8 is known to be @xmath10 , independent on the space dimension @xmath11 . the spatial properties are encoded into the scaling function @xmath12 or , equivalently , into its fourier transform @xmath13 . for large @xmath14 , @xmath15 has a power - law tail of the form @xmath16 : this is the famous porod s law @xcite , long recognized as arising from sharp domains walls . another relevant parameter affecting the scaling properties is the number of components @xmath17 of the order parameter @xmath18 . for binary fluids @xmath19 is a scalar field . most of the analytical techniques developed to study the kinetics , among which the bray - humayun scheme that is considered throughout this article , are however suited for vectorial systems with large-@xmath17 . let us briefly discuss the main features of vectorial systems that will be recalled in this paper . for systems with a continuous symmetry , rotations of the order parameter on the @xmath20 symmetric manifold of the local ground state of the hamiltonian , instead of evaporation and condensation , is the important growth mechanism . this changes @xmath8 to @xmath21 , regardless both of @xmath11 and @xmath17 ( apart from exceptional cases @xcite ) . concerning the large-@xmath14 behavior of @xmath15 a generalization of the porod s law , @xmath22 is obeyed @xcite for @xmath23 . this form relies on the existence of stable localized topological defects . much less is known about the tails of @xmath15 when these defects are not present , for @xmath24 . numerical simulations @xcite found a stretched - exponential decay @xmath25 $ ] . despite this rather simple phenomenology the problem of deriving the scaling properties from first principles is still open . on the analytical side , the only soluble model @xcite for conserved order parameter is the one with @xmath26 . when considering this unphysical limit one hopes that the gross features of the kinetic process are retained for @xmath26 leaving further refinements to @xmath27 perturbation theories . however , although the global behavior of the @xmath26 model is qualitatively resemblant @xcite of real systems , the scaling symmetry ( [ sca ] ) is violated already at a qualitative level . the explicit solution of the model @xcite shows in fact that instead of the standard scaling ( [ sca ] ) one has a _ multiscaling _ symmetry produced by the existence of two logarithmically different lengths . actually the ordering process with @xmath26 is more a _ condensation _ in momentum space @xcite , than a genuine phase - separation process . these results shadow the possibility to develop a successful theory of the segregation process by systematic @xmath27 expansions around the large-@xmath17 limit . in this scenario a prominent role is played by a class of approximate theories known as gaussian auxiliary field theories , originally introduced by mazenko @xcite . the essence of these is a non - linear mapping between the order parameter @xmath28 and an auxiliary field @xmath29 . the actual form of the relationship @xmath30 is largely arbitrary and , depending on different physically motivated choices , different schemes have been developed @xcite . the basic idea is that with a proper choice of @xmath30 the equation obeyed by the auxiliary field can be treated to lowest order in a mean - field approximation while the basic non - linearities of the evolution are still retained through the non - linear mapping @xmath30 . ad hoc _ assumption is generally uncontrolled and its use is justified _ a posteriori _ by the satisfactory quality of the predictions . starting from the method of mazenko , bray and humayun ( bh ) have derived @xcite a non - linear closed equation of motion for the correlation function @xmath31 within the framework of the @xmath27 perturbative expansion . letting @xmath32 in the bh equation one recovers the large-@xmath17 model previously discussed ; the bh approach , however , does not amount to a straightforward @xmath27-expansion because all orders in @xmath27 are reintroduced through the gaussian auxiliary field approximation . on one side , the uncontrolled character is the limit of the bh scheme , since it is not clear how to improve the approximation , but on the other side , represents a powerful tool to overcome the shortcomings of standard perturbation theory previously discussed . as shown by bh their equation reinstates standard scaling ( [ sca ] ) as a truly asymptotic symmetry , leaving to multiscaling a merely pre - asymptotic role . an additional feature at variance with the @xmath26 limit is the form of the scaling function @xmath13 , which , in the bh approximation falls exponentially at large momenta , as opposed to the quartic exponential decay of the @xmath33 model . in this paper we are interested in the diffusive phase separation process of a binary fluid in the presence of an applied shear flow , namely an imposed fluid velocity along @xmath34 with a constant gradient along @xmath35 . the presence of the flow radically changes the behavior and even the phenomenology is much less understood . a first trivial effect is the anisotropic deformation of the growing pattern that appears greatly elongated along @xmath34 @xcite . then , when dealing with growth laws one has to specify the length of the equilibrated regions for each direction , @xmath36 , and in general @xmath37 . a more subtle point , that may have important consequences , is the following : stretching of the domains causes ruptures of the network @xcite that may render the segregation incomplete . actually , the characteristic lengths in the different directions could keep on growing indefinitely , as without shear , or due to domains break - up the system may eventually enter a stationary state characterized by domains with a finite thickness . in general there is no agreement on this point both from the experimental @xcite and theoretical point of view . on the theoretical side large-@xmath17 calculations @xcite show the existence of a multiscaling regime with ever growing lengths in all directions . however numerical simulations @xcite can not still establish a clear evidence due to finite size and discretization effects . another intriguing feature which is probably related to the stretching and break - up of the domains is the observation of an oscillatory pattern in the observables . let us consider the excess viscosity @xmath38 as an example . stretching of domains require work against surface tension leading to an increase of @xmath38 . however when a domain breaks up the stress is released since the fragments are more isotropic ; this produces a decrease of @xmath38 . if repeated in time this mechanism produces an oscillatory pattern that is observed in simulations @xcite and possibly in experiments @xcite . in the large-@xmath17 limit it can be shown that this process has a periodicity in @xmath39 . similar patterns are also reported in apparently different contexts as fractures in heterogeneous solids @xcite or in stock market indices @xcite , suggesting a possible common underlying interpretation . however we do not have nowadays analytical tools to enlighten this feature . the spatial properties of the mixture are encoded into the correlation function @xmath31 or , equivalently , into its fourier transform @xmath40 , the structure factor . the latter is directly measured with scattering techniques and is more suited for comparison with the experiments . the form and the evolution of @xmath41 are themselves modified by the presence of the flow . when the shear becomes effective , after an initial isotropic regime where @xmath41 has the shape of a circular volcano , the tip of @xmath41 is deformed into an ellipses . this is expected as a consequence of the anisotropy , and is generally observed in experiments . in some cases the presence of four pronounced peaks on the edge of @xmath41 can be detected @xcite . the existence of these multiple maxima has been also observed in numerical studies @xcite of phase separation and in different physical systems such as stationary microemulsion in shear flow @xcite , suggesting that this is a rather general feature of sheared fluids . a peak in the structure factor is generally interpreted as the signature of a characteristic length proportional to the inverse of the wavevector where the peak is located . in this anisotropic case to each peak one associates one typical length for each space direction . taking into account the symmetry @xmath42 this suggests that a couple of characteristic lengths for each direction is present : it is not completely clear , however , to which extent this interpretation can be pushed . in this partly unclear scenario the development of theoretical tools for reliable predictions to be tested in experiments is important . in the case without shear the bh scheme turns out to be a reference theory capable to describe the main ingredients of the phenomenology . therefore in this paper we address a numerical solution of the bh equation in the presence of shear flow . we will compare its behavior with previous results pertaining to the large-@xmath17 limit or to numerical simulations of the fully non - linear scalar model . our results confirm the occurrence of a scaling regime with growing lengths in every direction . the dynamical exponents are found to coincide with those of the @xmath26 model . the four peak pattern for the structure factor is exhibited , similarly to what observed both for @xmath33 and in the scalar model , the actual form of @xmath41 being , however , rather different in the three cases . concerning the oscillating behavior of the typical lengths and of the excess viscosity , on the other hand , the bh model behaves quite differently , in that these are strongly inhibited or even absent , at variance with @xmath26 and @xmath43 . this paper is divided in 4 sections : in sec . [ model ] we introduce the bh equation and discuss some numerical details about its integration ; in sec . [ results ] the results of the numerical solution are presented and a comparison with different approaches is proposed . in sec . [ conclusions ] we draw the conclusions of this work . we consider a system described by an order parameter with @xmath17 components . the evolution is described by the convection - diffusion equation @xcite that generalizes the cahn - hilliard - cook equation in the case of an applied velocity field @xmath44 + = ^2 + * * ( x , t ) [ chc ] where @xmath45 is the vector order parameter that in the scalar case represents the concentration difference between the two species of the fluid , @xmath46 is the space coordinate , @xmath47 is a transport coefficient and @xmath48 is a gaussian stochastic field with expectations @xmath49 describing thermal fluctuations . the symbol @xmath50 means an ensemble average . @xmath51 is the equilibrium ginzburg - landau free energy ( \ { * * } ) = dx [ freen ] where @xmath52 . the parameter @xmath53 distinguishes between the mixed state with @xmath54 ( @xmath55 ) and the phase separated states with @xmath56 . for a plane shear flow the velocity term is given by v = y e_x [ shear ] where @xmath57 is the unit vector in the flow direction @xmath34 and @xmath58 is the shear rate . in the approximation framework developed by bh @xcite the following equation of motion can be derived for the equal time correlation function @xmath59 + y = -2 ^ 2\ { ^2 g(r , t)-r(t ) } [ bh ] where we have dropped the component indices due to internal symmetry . in deriving eq . ( [ bh ] ) from eq . ( [ chc ] ) we have let @xmath60 and @xmath61 since this simply amounts to a redefinition of time and space scales and we have neglected the thermal disturbance @xmath62 because it can be shown @xcite that the temperature is an irrelevant parameter below the coexistence line . the quantity @xmath63 is a function of time that is asymptotically determined by the requirement @xmath64 . here we use the self - consistent determination of @xmath63 r(t)=g[g(0,t)-s_eq ] [ r ] that is usually adopted @xcite . for @xmath26 , namely dropping the cubic term on the r.h.s . , eq . ( [ bh ] ) becomes the @xmath26 model . we have considered the evolution of eq . ( [ bh ] ) with @xmath43 . some comments about the role of @xmath17 are in order . ( [ bh ] ) is derived in an @xmath27 perturbative framework . then , in principle , the quality of the approximation is expected to be better for large @xmath17 and , in any case , this whole approach is meaningful for vectorial systems with continuous symmetry . given that the nature of the symmetry is relevant for the scaling properties , as discussed in sec . [ intro ] , one can not pretend to push this scheme down to the physically relevant case with @xmath43 in a completely successful way . however , besides numerical simulation that are quite difficult and not particularly instructive in this case , most of the understanding of phase separation with shear comes nowadays from the @xmath26 model , which surely suffers of much more profound shortcomings than the present approach , if used to infer the properties of physical systems . the bh scheme represents a first step beyond @xmath26 , despite of course being not conclusive . it must be recalled , moreover , that at least without shear the bh equation substantially reproduces the behavior of the system with @xmath26 , including multiscaling , in a preasymptotic time domain @xmath65 after which the cubic correction in eq . ( [ bh ] ) becomes effective @xcite . the choice @xmath43 , besides being appropriate for physical systems , is also the most efficient in order to amplify as much as possible the role of finite-@xmath17 corrections , given the limited timescale numerically accessible . ( [ bh ] ) have been solved on a two dimensional lattice of size @xmath66 , the lattice constant being @xmath67 , via a finite difference first order euler scheme with @xmath68 . periodic boundary conditions have been adopted . the initial condition @xmath69 , @xmath70 being a constant , corresponds to a quench from an equilibrium state at infinite temperature . the parameter of the free energy we use are @xmath71 ; different choices correspond to a redefinition of the order parameter amplitude . we will present data relative to the case @xmath72 and @xmath73 ; we have tested that other choices produce similar results . the structure factor @xmath74 obeys the equation -k_x = -2k^2 c(k , t)-2r(t)d(k , t ) [ ck ] where @xmath75 is the fourier transform of @xmath76 . due to the structure of this equation , however , it is more efficient to compute @xmath5 from eq . ( [ bh ] ) and to obtain @xmath41 by fourier transform . from the knowledge of @xmath41 one can extract the characteristic lengths along @xmath34 and @xmath35 as @xmath77 and rheological indicators such as the shear stress _ xy ( t)= k_x k_y c(k , t ) [ stress ] for a steady flow the excess viscosity is defined in terms of the shear stress as @xmath78 . the anisotropic character of the evolution , with the ordered regions aligning along the flow direction , is reflected in the behavior of the correlation function @xmath5 , shown in fig . ( [ figgr ] ) . @xmath5 has a peak at @xmath79 and is continuously stretched along the @xmath34-direction , assumining increasingly eccentric elliptical patterns . snapshots of the structure factor at the same times are shown in the @xmath80 plane in fig . ( [ figck ] ) . initially , not shown here , @xmath41 develops the form of a circular volcano , as in the case without shear . a later observation , at @xmath81 , shows a deep in the volcano profile along the direction @xmath82 . this deep becomes progressively more pronounced until @xmath41 appears separated into two symmetric foils . at times of order @xmath83 in each foil a couple of maxima are built up . we denote by the letter a the peak that in the figure at time @xmath83 is higher . with respect to the other maximum , denoted by b , a is characterized by having @xmath84 , whereas in b @xmath85 . the relative height of these maxima changes in time . this is observable in fig . ( [ figck ] ) at time @xmath86 : now b is prevailing . up to this time the qualitative evolution of the structure factor is comparable to what reported for @xmath33 @xcite and for the numerical solution @xcite of the full model equation with @xmath43 . computer limitations prevent the observation of the system on longer times . only in the @xmath33 case the dynamics can be followed by numerical integration of the equations on larger timescales and it can be shown that the repeated prevalence of either a or b maxima continues cyclically in time . it turns also out that the recurrent dominance of a and b is periodic in the logarithm of the strain @xmath87 . a physical interpretation of this pattern in terms of stretching and break - up of domains is proposed in @xcite . the spherical average c_sf(k , t)=n^-1dc(k , t ) , [ sf ] where @xmath88 and @xmath89 is the number of lattice points contained in a circular shell of width @xmath90 centered around @xmath91 , and the averages along @xmath92 or @xmath93 c_x(k_y , t)=l^-1dk_x c(k , t ) [ cx ] and c_y(k_x , t)=l^-1dk_y c(k , t ) [ cy ] are plotted in fig . ( [ figmedie ] ) . this figure shows that in the large-@xmath94 tail the structure factor decays exponentially , as already observed without imposed flow @xcite , at variance with the faster decay observed for @xmath26 @xcite . as already mentioned , the absence of power - law tails is related to the absence of stable localized topological defects in the bray - humayun approximation . next we consider the growth laws . for @xmath33 , in the large time region @xcite @xmath95 and @xmath96 keep increasing with ( logarithmically corrected ) power laws @xmath97 , @xmath98 modulated by a log - periodic oscillation . the short time part of this pattern is observed in fig . ( [ figradii]b ) . the same growth exponents are expected for the bh model ( but without logarithmic corrections ) ; this is because the model amounts to an @xmath27 perturbative expansion and , therefore , it is accurate for vectorial systems whose growth exponents belong to the same universality class of @xmath33 . actually the value of the exponents can be easily determined by means of a scaling analysis . if standard scaling holds the structure factor can be cast as c(k , t)=l_x(t)l_y(t)f(x , y ) [ scalc ] where @xmath99 and @xmath100 and @xmath15 is a scaling function . from ( [ scalc ] ) one also has d(k , t ) = b l_x(t)l_y(t)h(x , y ) [ scald ] where @xmath101 is a constant and @xmath102 another scaling function . inserting expressions ( [ scalc],[scald ] ) into the equation of motion of @xmath41 ( [ ck ] ) and assuming that asymptotically @xmath95 prevails over @xmath96 one obtains @xmath103y^2 + \frac{l_y(t)^5}{l_x(t)}\gamma x\frac{\partial f(x , y)}{\partial y } \nonumber \\ & - & \frac{1}{4}\frac{dl_y(t)^4}{dt } \left \ { 1+\frac{\frac{d \left [ \ln l_x(t ) \right ] } { dt } } { \frac{d\left [ \ln l_y(t)\right ] } { dt } } \right \ } \label{inferscal}\end{aligned}\ ] ] the l.h.s . of eq . ( [ inferscal ] ) does not depend explicitly either on time and @xmath58 . then in order to fulfill this equation for each @xmath104 both @xmath105 and @xmath58 must drop out from the r.h.s . then one has \ { ll l_x(t)~t^ + l_y(t)~t^ + r(t)~t^- . [ exponents ] as already anticipated the exponents are those pertaining to systems with continuous symmetry . furthermore one can observe that the shear rate enters linearly in @xmath95 but the growth in the shear direction @xmath35 is unaffected by @xmath58 . for completeness we recall that in the full scalar model one expects different exponents , namely @xmath106 , @xmath107 from scaling @xcite or renormalization group @xcite arguments , but these are not yet clearly observed in simulations . moreover the very existence of an asymptotic scaling dynamics is debated in this case . ( [ figradii]a ) shows that initially both @xmath95 and @xmath96 start growing with an exponent which is consistent with the value @xmath108 as in absence of shear . in this time regime the fluid essentially does not feel the presence of the flow , as already discussed for the structure factor . then , starting from @xmath109 , @xmath96 keeps growing unexpectedly larger than @xmath95 untill at @xmath110 it decreases . up to this point the bh model roughly resembles the @xmath33 case , despite the fact that the overshoot of @xmath96 is much more pronounced in the latter case . fig . ( [ figradii]c ) , instead , shows that the full scalar model behaves quite differently , with a pronounced decrease of @xmath96 in the range of times where in the other models a fast growth was observed . for longer times the asymptotic stage is entered . the numerical integration of the bh model presented insofar shows that eq . ( [ exponents ] ) is obeyed but , at variance with the other two cases , the oscillatory pattern is absent or , at least , strongly depressed . now we turn to the discussion of fig . ( [ figexc ] ) where the behavior of the excess viscosity is reported . in all the cases considered @xmath111 initially grows to a maximum , then falls negative and raises to a second maximum . this pattern has been observed also in experiments @xcite and is referred to as _ double overshoot_. after the second maximum the asymptotic stage is entered . on the basis of scaling arguments one expects that @xmath111 scales as @xmath112 times the inverse of the typical volume @xmath113 of the domains of the two species . then @xmath114 for vector order parameter and @xmath115 for the scalar model . the former exponent is observed in fig . ( [ figexc]a ) for the bh model , although in the very last part of the simulation . for @xmath33 the same exponent ( with logarithmic corrections ) has been exactly computed @xcite and observed numerically on larger timescales , but modulated by log - time periodic oscillations @xcite . in the scalar case a definite exponent can not be extracted from simulations , but the oscillations are observed . in this article we have considered the behavior of a binary fluid quenched below the mixing temperature in the presence of an applied shear flow . the description of this system has been carried out in the context of the continuum convection - diffusion equation based on the ginzburg - landau equilibrium free energy . the nature of our approach , which neglects hydrodynamic effects , limits the domain of applicability of the model to the diffusive regime which takes place at short times . we have studied this model in the approximation scheme introduced by bray and humayun . this amounts to an expansion in @xmath27 on top of the auxiliary field theory approximation originally introduced by mazenko . a comparison is also presented with two different approaches : the @xmath33 limit and the full scalar model , which is the most appropriate for fluids . the first has the great advantage to be fully soluble analytically , al least in the large time domain , providing an unambiguous description . furthermore it can be easily studied numerically from the instant of the quench onwards . however , besides the fact that the case @xmath33 is far from physical , it has also proved to violate the qualitative symmetry of standard scaling in favor of multiscaling , rendering this model insecure as an approximation to real systems . on the other hand the full scalar model is well suited for describing fluids but it can not be studied analytically . numerical simulations , although important , have not yet provided clear evidences of the basic phenomenology , due to technical limitations . the bh model is ideally located in between these two : it shares with the large-@xmath17 limit the property of being a closed equation for an observable , the correlation function , that is already an ensemble average . then , even if one resorts in the end to a numerical solution , one still has the great advantage to avoid the average over a large number of realizations . secondly , due to its perturbative character , it is believed to represent a step forward with respect to the case @xmath33 . this statement has been proven to be correct at least without shear . given the essence of the bh scheme , one does not expect to obtain a fully reliable approximation to real systems with @xmath43 . the spirit of this work is more to establish if and to which extent its phenomenology compares better to the numerical simulations of the physical model than to the case @xmath33 . from the analysis of the behavior of the characteristic lengths and of the excess viscosity we can conclude that the bh model behaves more like the @xmath33 case with damped oscillations than like the full scalar model , despite that , with the choice @xmath43 , the corrections to the case @xmath33 contained in eq . ( [ bh ] ) are effective almost from the beginning , as it is clear from figs . ( [ figradii],[figexc ] ) . this is true not only for the value of the dynamic exponents , but also for the qualitative behavior of the typical lengths . on the other hand , regarding the oscillating character of most observables , the bh model behaves rather differently from both @xmath33 and @xmath43 in that strong oscillations are observed in the latter two while they are practically absent in the former . under this respect the bh model is atypical and can not be regarded as a bridge between @xmath33 and @xmath43 . it is possible that the bh scheme resembles more the behavior of vectorial system with a finite @xmath17 under shear , but this is a totally unexplored field . regarding the important issue of the existence of an asymptotic scaling regime characterized by ever growing lengths @xmath116 , in the time range accessed in this work we do not see any saturation effect and both @xmath95 and @xmath96 keep firmly growing with the expected power laws . in the bh scheme , then , the presence of an asymptotic scaling regime is confirmed while a stationary state with domains of a finite thickness is not observed . bray _ adv . phys . _ * 43 * , 357 , ( 1994 ) ; j.d . gunton , m. san miguel and p.s . sahni , in _ phase transitions and critical phenomena _ , edited by c. domb and j.l . lebowitz ( academic press , london 1983 ) , vol.8 . a. coniglio , p. ruggiero and m. zannetti _ phys . * e 50 * , 1046 ( 1994 ) ; u. marini bettolo marconi and f. corberi , _ europhys . _ * 30 * , 349 , ( 1995 ) ; s. glotzer and a. coniglio , _ phys . rev . _ * e 50 * , 4241 , ( 1994 ) ; f. corberi and c. castellano , _ phys . rev . _ * e 58 * , 4658 ( 1998 ) ; c. castellano and f. corberi , _ phys . rev . _ * e 57 * , 672 ( 1998 ) . c. yeung , y. oono and a.shinozaki , _ phys . * e 49 * , 2693 , ( 1994 ) ; s. de siena and m. zannetti , _ phys . * e 50 * , 2621 , ( 1994 ) ; t. otha , d. jasnow and k. kawasaky , _ phys . lett . _ * 49 * , 1223 , ( 1982 ) ; a.j . bray and k. humayun , _ phys . * e 48 * , r1609 ( 1993 ) . t. hashimoto , k. matsuzaka , e. moses and a. onuki , _ phys . lett . _ * 74 * , 126 , ( 1995 ) ; t. takebe , f. fujioka , r. sawaoka and t. hashimoto , j. chem . phys . * 98 * , 717 , ( 1993 ) ; j. luger , c. laubner and w. gronsky , _ phys . lett . _ * 75 * , 3576 , ( 1995 ) ; c.k . chan , f. perrot and d. beysens , _ phys . rev . _ * a 43 * , 1826 , ( 1991 ) . z. laufer , h.l . jalink and a.j . staverman , _ journal of polymer science _ * 11 * , 3005 , ( 1973 ) ; s. mani , m.f . malone , h.h . winter , j.l . halary and l. monnerie , _ macromolecules _ * 24 * , 5451 , ( 1991 ) ; s. mani , m.f . malone and h.h . winter , _ macromolecules _ * 25 * , 5671 , ( 1992 ) .

the phase separation process which follows a sudden quench inside the coexistence region is considered for a binary fluid subjected to an applied shear flow . this issue is studied in the framework of the convection - diffusion equation based on a ginzburg - landau free energy functional in the approximation scheme introduced by bray and humayun [ _ phys.rev.lett . _ * 68 * , 1559 , ( 1992 ) ] . after an early stage where domains form and shear effects become effective the system enters a scaling regime where the typical domains sizes @xmath0 , @xmath1 along the flow and perpendicular to it grow as @xmath2 and @xmath3 . the structure factor is characterized by the existence of four peaks , similarly to previous theoretical and experimental observations , and by exponential tails at large wavevectors . # 1#2 # # 1##2#2##1

You are an expert at summarizing long articles. Proceed to summarize the following text: in recent years , the experimental realization of landau - zener - stckelberg ( lzs ) interferometry@xcite in several systems has emerged as a tool to study quantum coherence under strong driving . lzs interferometry is realized in two - level systems ( tls ) which are driven with a time periodic force . the periodic sweeps through an avoided crossing in the energy level spectrum result in successive landau - zener transitions . the accumulated phase between these repeated tunneling events gives place to constructive or destructive interferences , depending on the driving amplitude and the detuning from the avoided crossing . these lzs interferences have been observed in a variety of quantum systems , such as rydberg atoms,@xcite superconducting qubits,@xcite ultracold molecular gases,@xcite quantum dot devices,@xcite single spins in nitrogen vacancy centers in diamond,@xcite nanomechanical circuits,@xcite and ultracold atoms in accelerated optical lattices.@xcite several other related experimental and theoretical works have studied lzs interferometry in systems under different shapes of periodic driving,@xcite in two coupled qubits,@xcite in optomechanical systems,@xcite and the effect of a geometric phase.@xcite furthermore , experiments in superconducting flux qubits under strong driving have allowed to extend lzs interferometry beyond two levels.@xcite in this later case , the multi - level structure of the flux qubit , with several different avoided crossings in the energy spectrum , exhibited a series of diamond - like interference patterns as a function of dc flux detuning and microwave amplitude.@xcite driven two - level systems have been extensively studied theoretically in the past . under strong time - periodic driving fields , phenomena such as coherent destruction of tunneling ( cdt ) @xcite and multiphoton resonances @xcite have been analyzed . the influence of the environment has been studied within the driven spin - boson model,@xcite applying various techniques like the path - integral formalism,@xcite or the solution of the time dependent equations for the populations of the density matrix,@xcite either as an integro - differential kinetic equation,@xcite or considering for weak coupling the underlying bloch - redfield equations,@xcite or using the decomposition of the quantum master equation into floquet states,@xcite or performing a rotating - wave approximation.@xcite however , the theoretical results for driven tls in contact with a quantum bath @xcite present some discrepancies with the experiments of lzs interferometry , particularly in the flux qubit.@xcite in fact , these works typically predict population inversion,@xcite which has not been observed in the experiments of ref . . in a recent work we showed,@xcite that for the case of a non - resonant detuning , population inversion arises for very long driving times and it is mediated by a slow mechanism of interactions with the bath . this long - time asymptotic regime was not reached in the experiment . here , we will study in full detail the lzs interference patterns of the flux qubit , by considering the dependence with dc flux detuning in addition to the dependence with microwave amplitude . to this end , we calculate numerically the finite time and the asymptotic stationary population of the qubit states using the floquet - markov approach,@xcite solving a realistic model of the flux qubit in contact with an ohmic quantum bath . for the time scale of the experiments we find a very good agreement with the diamond patterns of ref .. for longer time scales , we find a dynamic transition within the first diamond ( which corresponds to the two level regime ) , manifested by a symmetry change in the structure of the lzs interference pattern . we also consider the case of an structured quantum bath , which can be due to a squid detector with a resonant plasma frequency @xmath0 . different types of lzs interference patterns can arise in this case , depending on the magnitude of @xmath0 . the paper is organized as follows . in sec.ii we present the model for the flux qubit and we review the basics of lzs interferometry . in sec . iii , we show our results for the emergence of a dynamic transition in the lzs interference pattern at long times . in sec . iv we give our conclusions and an appendix is included to describe in detail the floquet - markov formalism used in this paper . superconducting circuits with mesoscopic josephson junctions are used as quantum bits@xcite and can behave as artificial atoms.@xcite . a well studied circuit is the flux qubit ( fq ) @xcite which , for millikelvin temperatures , exhibits quantized energies levels that are sensitive to an external magnetic field . the fq consists on a superconducting ring with three josephson junctions@xcite enclosing a magnetic flux @xmath1 , with @xmath2 . the circuits that are used for the fq have two of the junctions with the same josephson coupling energy , @xmath3 , and capacitance , @xmath4 , while the third junction has smaller coupling @xmath5 and capacitance @xmath6 , with @xmath7 . the junctions have gauge invariant phase differences defined as @xmath8 , @xmath9 and @xmath10 , respectively . typically the circuit inductance can be neglected and the phase difference of the third junction is : @xmath11 . therefore , the system can be described with two independent dynamical variables . a convenient choice is @xmath12 ( longitudinal phase ) @xmath13 ( transverse phase ) . in terms of this variables , the hamiltonian of the fq ( in units of @xmath14 ) is:@xcite @xmath15 with @xmath16 and @xmath17 . the kinetic term of the hamiltonian corresponds to the electrostatic energy of the system and the potential term corresponds to the josephson energy of the junctions , given by @xmath18 typical flux qubit experiments have values of @xmath19 in the range @xmath20 and @xmath21 in the range @xmath22.@xcite in quantum computation implementations @xcite the fq is operated at magnetic fields near the half - flux quantum , @xmath23 , with @xmath24 . for @xmath25 , the potential @xmath26 has two minima at @xmath27 separated by a maximum at @xmath28 . each minima corresponds to macroscopic persistent currents of opposite sign , and for @xmath29 ( @xmath30 ) a ground state @xmath31 ( @xmath32 ) with positive ( negative ) loop current is favored . , for the qubit parameters @xmath33 and @xmath34 considered throughout this work . energy is measured in units of @xmath14 and flux in units of @xmath35 . the gaps at the avoided crossings are indicated as @xmath36 . black lines correspond to `` longitudinal '' modes and red lines to `` transverse '' modes ( see text for description ) . the blue arrows at the bottom of the plot illustrate the relation between ac driving amplitudes and level crossing positions for a particular dc flux detunning @xmath37 . the indicated values correspond to the edges of the spectroscopic diamonds of fig.[las ] for the given @xmath37 . ] in fig.[fig - espec ] we plot the lowest energy levels @xmath38 as a function of @xmath39 , obtained by numerical diagonalization of @xmath40.@xcite in this case we set @xmath41 and @xmath42 , close to the experimental values employed in flux qubits experiments.@xcite negative ( positive ) slopes in fig.[fig - espec ] , correspond to eigenstates with positive ( negative ) loop current . a gap @xmath36 opens at the avoided crossings of the @xmath43-th level of positive slope with the @xmath44-th level of negative slope at @xmath45 . for our choice of device parameters , the avoided crossing of the two lowest levels at @xmath46 has a gap @xmath47 ( in units of @xmath14 ) . at larger @xmath48 , we find the avoided crossing with a third level at @xmath49 with gap @xmath50 , and next the avoided crossing at @xmath51 with gap @xmath52 . ( there are also energy levels that correspond to excited transverse modes,@xcite plotted with red lines in fig.[fig - espec ] , but they have a negligible contribution to the dynamics considered here ) . the two - level regime , involving only the lowest eigenstates at @xmath53 and @xmath54 , corresponds to @xmath55 , such that the avoided crossings with the third energy level are not reached . in this case , the hamiltonian of eq . ( [ ham - sys ] ) can be reduced to a two - level system@xcite @xmath56 in the basis defined by the persistent current states @xmath57 , with @xmath58 and @xmath59 the ground and excited states at @xmath60 . the parameters of @xmath61 are the gap @xmath62 and the detuning energy @xmath63 . here @xmath64 is the magnitude of the loop current , which for our case with @xmath42 and @xmath33 is @xmath65 ( in units of @xmath66 ) . landau - zener - stckelberg ( lzs ) interferometry is performed by applying an harmonic field on top of the static field with @xmath67 if the ac driving amplitude is such that @xmath68 , which is fulfilled for @xmath69 , the quantum dynamics can be described within the two - level regime . in this case , we use @xmath70 in the hamiltonian of eq.([htls ] ) , with @xmath71 and @xmath72 . when @xmath73 the central avoided crossing at @xmath60 is reached within the range of the driving amplitude , @xmath74 . in this case the periodically repeated landau - zener transitions at @xmath60 give place to lzs interference patterns as a function of @xmath75 and @xmath76 , which are characterized by _ multiphoton resonances_@xcite and _ coherent destruction of tunneling_,@xcite as we describe below . there are _ multiphoton resonances _ when@xcite @xmath77 ( where @xmath78 is written in units of @xmath79 and energies in units of @xmath14 ) . if @xmath80 , these resonances are at @xmath81.@xcite calling @xmath82 , the @xmath83-resonance condition can also be written as @xmath84 . an example of the dynamic behavior in a @xmath85 resonance is shown in fig.[fo1](a ) , calculated as described in ref . ( see also the appendix for details of the calculation ) . in this case the fq is driven with frequency @xmath86 ( for @xmath87ghz it corresponds to @xmath88mhz ) . the dc detuning is @xmath89 , corresponding to @xmath90 ( since @xmath91 ) . the fq is started at @xmath92 in the ground state @xmath93 , and the probability of having a positive loop current is calculated , @xmath94 . since for @xmath95 we have @xmath96 , the initial probability is @xmath97 . the red - dashed line in fig [ fo1](a ) shows the time dependence of @xmath98 . the resonant dynamics is clearly seen : the fq oscillates coherently between positive and negative current states . therefore , the probability @xmath98 oscillates between @xmath99 and @xmath100 , having a time averaged value of @xmath101 . in contrast , fig.[fo1](b ) shows the time dependence of @xmath98 in an off - resonant case , for @xmath102 . we see that for off - resonance the fq stays in the positive loop current state , since @xmath98 fluctuates around @xmath103 . state as a function of time , for a flux qubit driven with frequency @xmath86 ( in units of @xmath79 ) . ( a ) @xmath104 ( second resonance ) and @xmath105 . ( b ) @xmath106 ( off resonance ) and @xmath107 . ( c ) same as ( b ) but for a long time scale . red dashed line shows the results in the non dissipative case and the black line corresponds to the dissipative case with an ohmic bath at @xmath108 . ] for @xmath109 , it has been shown , in a rotating wave approximation , that the average probability @xmath110 near a @xmath111 photon resonance is@xcite @xmath112 at the resonance , @xmath113 , eq.([l2ls ] ) gives @xmath114 and away of the resonance @xmath115 . the width of the resonance is @xmath116 , with @xmath117 the bessel function of the first kind . this gives a quasiperiodic dependence as a function of @xmath76 for @xmath75 fixed near the resonance . in particular , at the zeros of @xmath117 , the resonance is destroyed , giving @xmath118 instead of @xmath101 , a phenomenon known as _ coherent destruction of tunneling_.@xcite plots of @xmath119 as a function of flux detuning @xmath75 and ac amplitude @xmath76 give the typical lzs interference patterns , which have been measured experimentally by oliver et al in flux qubits,@xcite and have also been observed in other systems . @xcite the eq.([l2ls ] ) corresponds to ideally isolated flux qubits , neglecting the coupling of the qubit with the environment . the dynamics of the fq as an open quantum system is usually characterized by the energy relaxation time @xmath120 and the decoherence time @xmath121 . several phenomenological approaches have taken into account relaxation and decoherence in lzs interferometry , obtaining a broadening of the lorentzian - shape @xmath111 photon resonances of eq.([l2ls ] ) . for example , in ref . a bloch equation approach is used , obtaining for zero temperature : @xmath122 with @xmath123 and @xmath124 . similar results were obtained by berns et al,@xcite considering a pauli rate equation with an effective transition rate that adds to @xmath125 a driving induced rate @xmath126 . the latter was obtained assuming that decoherence is due to a classical white noise in the magnetic flux , an approach that is valid for time scales smaller than the relaxation time and larger than the decoherence time , for @xmath127 . the case of low frequency @xmath128 noise has been considered in a similar approach,@xcite finding a gaussian line shape for the @xmath83-resonances instead of the lorentzian line shape . the description of the lzs interference patterns with phenomenological approaches like eq.([l2lsr ] ) gives a good agreement with most of the experimental results.@xcite however , they rely on approximations valid either for large frequencies , or for low ac amplitudes , or for time scales smaller than the relaxation time . here we will study lzs interferometry using the floquet formalism for time - periodic hamiltonians,@xcite which allows for an exact treatment of driving forces of arbitrary strength and frequency . the time - dependent schrdinger equation is transformed to an equivalent eigenvalue equation for the floquet states @xmath129 and quasi - energies @xmath130 , which can be solved numerically . ( see the appendix for details ) . to describe relaxation and decoherence processes , we consider that the qubit is weakly coupled to a bath of harmonic oscillators . the markov approximation of the bath correlations is performed after writing the reduced density matrix @xmath131 of the qubit in the floquet basis , @xmath132.@xcite by following this procedure , the quantum master equation obtained for @xmath133 is valid for periodic driving forces of arbitrary strength . ( in contrast , the standard born - markov approach is valid for small driving forces ) . the obtained floquet - markov quantum master equation is @xcite : @xmath134 the coefficients @xmath135 depend on the spectral density @xmath136 of the bath , the temperature @xmath137 , and on the qubit - bath coupling . using the numerically obtained floquet states @xmath129 , we can calculate the coefficients @xmath135 , and then we compute the solution of @xmath138 as described in the appendix . here , we will consider the dynamics of a fq coupled to a bath with an ohmic spectral density@xcite @xmath139 with @xmath140 a cutoff frequency . the ohmic bath mimics an unstructured electromagnetic environment that in the classical limit leads to white noise . we consider @xmath141 , corresponding to weak dissipation,@xcite and a large cutoff @xmath142 . the bath temperature is taken as @xmath143 ( @xmath144 for @xmath145 ) . experimentally , the probability of having a state of positive or negative persistent current in the flux qubit is measured.@xcite the probability of a positive loop current measurement can be calculated in general as@xcite @xmath146\ ] ] with @xmath147 the operator that projects wave functions on the @xmath148 subspace , as described in the appendix . in fig [ fo1 ] the black lines show the population @xmath149 as a function of time obtained from the numerical solution of eq.([drho ] ) , taking the ground state @xmath58 as initial condition , and compared with the isolated fq ( red dashed lines ) discussed in the previous section . in fig [ fo1](a ) , for the @xmath150 resonance , we see that the population @xmath98 has damped oscillations that tend to the asymptotic average value of @xmath101 for large times . on the other hand , for the off - resonant case of fig [ fo1](b ) and ( c ) we see that there is a clear difference between the short time behavior of @xmath98 , shown in fig [ fo1](b ) , and the large time behavior shown in fig [ fo1](c ) . moreover , we observe that @xmath98 tends to asymptotic values that are very different than in the isolated system . for the particular case shown in the plot , we find population inversion , i.e @xmath151 , in the asymptotic long time limit , a phenomenon we discussed in ref . for off - resonant cases . in the following , we will see how this change of behavior in the long time is reflected in the full lzs interference pattern as a function of @xmath75 and@xmath76 . as a function of the driving amplitude @xmath76 and dc detuning @xmath75 for ( a ) @xmath152 , time scale of the experiments , and ( b ) asymptotic regime for @xmath153 . in ( a ) we mark the ac amplitudes defined in the text that correspond to the edges of the spectroscopic diamonds for a given @xmath37 . ( c ) @xmath149 for @xmath106 as a function of the driving amplitude @xmath76 , at @xmath152 ( blue dashed line ) and asymptotic average population @xmath154 ( black line ) . the calculations were performed for @xmath155 and ohmic bath at @xmath156mk . vertical lines separate diamond regimes d1 , d2 , d3 , described in the text . , title="fig : " ] as a function of the driving amplitude @xmath76 and dc detuning @xmath75 for ( a ) @xmath152 , time scale of the experiments , and ( b ) asymptotic regime for @xmath153 . in ( a ) we mark the ac amplitudes defined in the text that correspond to the edges of the spectroscopic diamonds for a given @xmath37 . ( c ) @xmath149 for @xmath106 as a function of the driving amplitude @xmath76 , at @xmath152 ( blue dashed line ) and asymptotic average population @xmath154 ( black line ) . the calculations were performed for @xmath155 and ohmic bath at @xmath156mk . vertical lines separate diamond regimes d1 , d2 , d3 , described in the text . , title="fig : " ] as a function of the driving amplitude @xmath76 and dc detuning @xmath75 for ( a ) @xmath152 , time scale of the experiments , and ( b ) asymptotic regime for @xmath153 . in ( a ) we mark the ac amplitudes defined in the text that correspond to the edges of the spectroscopic diamonds for a given @xmath37 . ( c ) @xmath149 for @xmath106 as a function of the driving amplitude @xmath76 , at @xmath152 ( blue dashed line ) and asymptotic average population @xmath154 ( black line ) . the calculations were performed for @xmath155 and ohmic bath at @xmath156mk . vertical lines separate diamond regimes d1 , d2 , d3 , described in the text . , title="fig : " ] in figs.[las](a ) and [ las](b ) we show an intensity plot of @xmath149 as a function of @xmath75 and @xmath76 , calculated at a time near the experimental time scale of refs . , @xmath157 , with @xmath158 the period of the ac driving . for @xmath69 we observe an lzs pattern modulated by multiphoton resonances and coherence destruction of tunneling , which can be described within a good approximation with eq.([l2lsr ] ) . this lzs interference plot is very similar to the experimental results of ref .. furthermore , for higher ac amplitudes , we find a pattern of `` spectroscopic diamonds '' , also in good agreement with experiments.@xcite the diamond structure obtained for high ac amplitudes can be related to the energy level spectrum of fig.[fig - espec ] , for a fixed dc flux detuning @xmath159 , as follows.@xcite the first diamond , d1 , starts when the @xmath160 avoided crossing at @xmath161 is reached by the amplitude of the ac drive , _ i.e. _ when @xmath162 . this defines an onset ac amplitude @xmath163 . the first diamond ends when the nearest second avoided crossing is reached ( with gap @xmath164 ) at the ac amplitude @xmath165 . then the second diamond , d2 , starts when the other second avoided crossing is reached , at @xmath166 , and it ends when the next avoided crossing is reached at @xmath167 . similarly , the third diamond , d3 , starts at @xmath168 , and so on . the analysis of the positions of the resonances as a function of @xmath76 and @xmath75 was the route followed in refs . to obtain the parameters characterizing the different avoided crossings of the flux qubit . the fq of refs . have short decoherence times ( @xmath169 ) and large relaxation times ( @xmath170 ) . the typical duration of the driving pulses in these experimental measurements ( @xmath171 ) is in between this two time scales , @xmath172 . due to this time scale separation , a model with classical noise,@xcite valid for @xmath173 , can qualitatively explain the experimentally observed behavior of @xmath149 within the first diamond , through eq.([l2lsr]).@xcite moreover , a multilevel extension of the model of ref . can also describe the higher order diamonds ( provided one gives as an input parameter the positions @xmath174 and the gaps @xmath36 of the avoided crossings).@xcite for large times @xmath175 one would expect naively , that after full relaxation with the environment , a blurred picture of the lzs interference pattern of figs.[las](a ) with broadened resonance lobes should be observed . the asymptotic @xmath176 , averaged over one period @xmath177 , is shown in fig.[las](b ) . [ @xmath178 can be calculated exactly after obtaining numerically the right eigenvector of @xmath179 with zero eigenvalue . see the appendix for details . ] surprisingly , we see in fig.[las](b ) that the asymptotic behavior of @xmath149 gives a qualitatively different lzs interference pattern within the first diamond , and not a mere blurred version of fig.[las](a ) . in particular , a cut at constant @xmath75 is shown in fig.[las](c ) . there we see clearly that within the first diamond regime , @xmath180 , the value of @xmath149 at the experimental time scale is very different than the asymptotic value @xmath178 . on the other hand , beyond the first diamond , for @xmath181 the results of @xmath182 and @xmath178 are nearly coincident . as a function of the @xmath183 flux detuning @xmath75 for the flux qubit driven with @xmath107 and @xmath155 , and coupled to an ohmic bath at @xmath184mk ( for @xmath185 ) . red line with squares : @xmath152 . black line with circles : asymptotic ( @xmath153 ) average population @xmath154 . blue line with triangles : time averaged population in the isolated system . horizontal orange line : indicates the @xmath186 level to help to identify when there is population inversion . the flux detuning @xmath48 is normalized by @xmath187 , such that the @xmath83-photon resonances are at @xmath188 . ( b ) enlarged view around the @xmath85 resonance . the green dashed line is a plot of the best fit with eq.([l2lsr ] ) , while a plot of eq.([l2ls ] ) ( not shown ) falls exactly over the blue dotted line . ] in fig.[fpf0 ] we show @xmath149 vs. @xmath75 , for @xmath107 , within the first diamond . the blue line with triangles corresponds to the solution of the isolated system , which shows dips where @xmath189 that correspond to the @xmath83-photon resonances at @xmath190 . the red line with squares corresponds to @xmath182 , which shows values of @xmath149 smaller than in the isolated case ( due to effect of decoherence and relaxation ) and in agreement with the experiments . the asymptotic @xmath178 is also shown in fig.[fpf0 ] ( black line with circles ) , which is lower than @xmath182 . fig.[fpf0](b ) shows in detail the behavior near the @xmath85 resonance . the isolated case shows a dip with a lorentzian shape accurately described by eq.([l2ls ] ) . the open system at the experimental time scale shows a broadened peak for @xmath182 , partially consistent with a description like eq.([l2lsr ] ) , except that near the resonance the behavior of @xmath149 vs @xmath75 becomes antisymmetric around @xmath190 . in the asymptotic steady regime , @xmath178 vs @xmath75 is antisymmetric in a wide region around @xmath190 , showing population inversion ( @xmath191 ) below the resonance , for @xmath192 . + the temporal evolution from symmetric to antisymmetric resonances can be seen in detail in fig.[figtemp ] , where we show an intensity plot of @xmath149 as a function of @xmath75 and the duration time @xmath193 , for @xmath107 . two different temperatures are considered , a low temperature @xmath194 in figs.[figtemp](a ) and ( c ) , and @xmath195 in figs.[figtemp](b ) and ( d ) . the relaxation time @xmath196 for the driving amplitude considered in fig.[figtemp ] is @xmath197 ( as we will see in sec.iiic , @xmath196 depends on @xmath76 ) . figs.[figtemp](a ) and ( b ) are for duration times up to the time scale @xmath198 of the experiments , for the two different temperatures . we see that within this time scale @xmath199 , the resonances are symmetric . in figs.[figtemp](c ) and ( d ) the duration time is plotted in logarithmic scale and the evolution at very large times is shown . we see that at a time @xmath200 the resonances start to change form , becoming asymmetric in the long time limit . the asymmetry is stronger for low temperatures , in particular for @xmath201 there is full inversion of population on one side of the resonances . , to ( b ) triangular checkerboard pattern for asymptotic long times . enlarged views of fig.[las](a ) and ( b ) , respectively , showing a sector of the first diamond including the @xmath202 multiphoton resonances . the flux detuning @xmath48 is normalized by @xmath187 , such that the @xmath83-photon resonances are at @xmath188 . , title="fig : " ] , to ( b ) triangular checkerboard pattern for asymptotic long times . enlarged views of fig.[las](a ) and ( b ) , respectively , showing a sector of the first diamond including the @xmath202 multiphoton resonances . the flux detuning @xmath48 is normalized by @xmath187 , such that the @xmath83-photon resonances are at @xmath188 . , title="fig : " ] fig.[figtemp ] shows that , as a function of time , _ there is a dynamic transition manifested by a symmetry change in the structure of the lzs interference pattern_. the dynamic transition is from nearly symmetric resonances at short times ( @xmath173 ) to _ antisymmetric resonances _ at long times ( @xmath175 ) . this is clearly illustrated in fig.[las2 ] where an enlarged view of a part of the first diamond is shown . fig.[las2](a ) corresponds to @xmath182 and shows the characteristic features observed in experiments : ( i ) @xmath203 away from the resonances , ( ii ) there are resonance lobes where @xmath204 , and ( iii ) the @xmath83-resonance lobes are limited by the points where there is coherent destruction of tunneling ( cdt ) , given by @xmath205 . on the other hand , in the asymptotic regime , [ ( fig.[las2](b ) ] , the pattern of symmetric @xmath83-resonance lobes is replaced by a pattern of antisymmetric resonances , which form a triangular checkerboard picture defined by triangles with @xmath206 and @xmath207 alternatively , with their vertices located at the cdt points . we name , in short , the former pattern of lzs interferometry as symmetric resonances " ( sr ) and the latter as antisymmetric resonances " ( ar ) . that contribute to the relaxation rate . as a function of the driving amplitude @xmath76 . the calculations were performed for @xmath106 , @xmath155 and an ohmic bath at @xmath208 ( for @xmath185 ) . ( b ) asymptotic average population @xmath154 for the same case . the dashed vertical lines highlight one of the regions of @xmath76 where there is population inversion . ] the asymptotic ar interference patterns show population inversion ( pi ) on one side of a multiphoton resonance , as seen in fig.[fpf0](b ) and figs.[figtemp](c ) and ( d ) . this makes @xmath209 antisymmetric around the resonance . if we fix @xmath75 at a value in between two resonances , the probability oscillates around @xmath210 as function of @xmath76 , showing pi whenever a `` triangle '' with @xmath211 is traversed . this is shown in fig.[fo4](a ) . the underlying mechanism of this population inversion can be understood by analyzing the contribution to relaxation of virtual photon exchange processes with the bath.@xcite as we show in the appendix , within the first diamond regime , the total relaxation rate @xmath212 can be decomposed as @xmath213 with @xmath214 the relaxation rates due to virtual @xmath83-photon transitions to bath oscillator states.@xcite in fig.[fo4](b ) we plot @xmath214 as a function of @xmath76 for the same @xmath75 considered in fig.[fo4](a ) . we show the cases with @xmath215 , where @xmath216 describes the relaxation without exchange of virtual photons , corresponding to the `` conventional '' dc relaxation mechanism , while @xmath217 correspond to the the ac contribution due to the exchange of one virtual photon with energy @xmath218 . we show in fig.[fo4 ] that , whenever there is population inversion , the dc relaxation terms vanish ( @xmath219 ) , while the @xmath220 term is the largest one . this indicates that the relevant mechanism leading to pi is a transition to a virtual level at energy @xmath221 ( one photon absorption , @xmath222 ) , followed by a relaxation to the level @xmath223.@xcite and decoherence rates @xmath224 and @xmath225 as a function of the driving amplitude @xmath76 . the calculations were performed for @xmath106 , @xmath155 and an ohmic bath at @xmath208 ( for @xmath185 ) . vertical lines separate diamond regimes d1 and d2 described in the text . ] to better understand why the ar patterns have not been observed yet in current experiments , we now analyze the time scales of relaxation and decoherence . to this end , we calculate numerically the full relaxation rate @xmath212 and the decoherence rates @xmath226 , from the eigenvalues of the @xmath179 matrix , as discussed in the appendix . in fig.[fo4c ] we show the relaxation rate @xmath212 and two decoherence rates @xmath226 , as a function of the driving amplitude @xmath76 for @xmath106 ( away from a @xmath83-resonance ) . for @xmath227 the relaxation rate @xmath212 corresponds to the @xmath228 measured experimentally , _ i.e. _ , @xmath229 when @xmath230 . since to a good approximation the density matrix becomes diagonal in the floquet basis in the asymptotic regime,@xcite the decoherence rate @xmath224 shown in fig.[fo4c ] corresponds to the decoherence between the @xmath231 and the @xmath232 floquet states . when @xmath230 , @xmath233 , since in this limit it corresponds to the decoherence between the two lowest energy levels . we also plot in fig.[fo4c ] the rate @xmath225 , which is the decoherence rate between the @xmath234 and the @xmath235 floquet states . at @xmath227 it corresponds to the decoherence between the third and the fourth energy level . for small @xmath76 , below the onset of the first diamond ( @xmath236 ) , the relaxation and decoherence rates stay in values similar to the undriven case . however , when @xmath237 we see in fig.[fo4c ] that both rates depend strongly on @xmath76 . above the onset of the first diamond , the overall behavior is that the decoherence rate @xmath224 increases and the relaxation rate @xmath212 decreases as a function of @xmath76 . therefore , within the first spectroscopic diamond , the difference between decoherence and relaxation is much larger than in the undriven case ( @xmath238 ) . this explains the important difference between @xmath182 and @xmath178 , within the first diamond in fig.[las ] , due to the large time window where ar patterns can be observed for @xmath239 . beyond the first diamond , for @xmath181 , the relaxation rate @xmath212 increases strongly with @xmath76 , becoming nearly of the same order of the decoherence rates within the second diamond and above . this behavior is a consequence of the fact that when more than two - levels are involved in the dynamics , there are several possible decay transitions between energy levels that contribute to a faster relaxation of the system . therefore , in the second diamond and beyond decoherence and relaxation rates become comparable . for this reason , @xmath240 in this case , since the relaxation time is significantly reduced and thus @xmath241 . although previous works have found pi for the asymptotic regime of two level systems,@xcite the time - dependent dynamics with different time scales has been overlooked . in fact , the relevant point from our findings is that the asymptotic regime is difficult to reach in the experiment , since pi needs long times ( @xmath242 ) to emerge when mediated by the bath . and the two virtual levels @xmath243 as a function of the flux detuning @xmath48 , for the flux qubit coupled with an structured bath with a resonant mode at frequency @xmath0 . the location of the `` virtual crossings '' at @xmath244 are also indicated . the inset shows a plot of the spectral density @xmath136 of the structured bath . ] the measurement of the state of the fq is performed with a read - out dc squid , which is inductively coupled to the qubit.@xcite this modifies the bath spectral density by adding a resonant mode at the plasma frequency @xmath0 of the squid detector . the so - called `` structured bath '' spectral density is given by@xcite @xmath245 with @xmath246 the width of the resonant peak at @xmath247 ( see the inset of fig.[majo ] for a schematic plot of @xmath136 ) . as a function of the driving amplitude @xmath76 and dc detuning @xmath75 for ( a ) @xmath248 and ( b ) @xmath249 . we indicate in the plot the regions corresponding to the regimes ( i ) , ( ii ) and ( iii ) described in the text . , title="fig : " ] as a function of the driving amplitude @xmath76 and dc detuning @xmath75 for ( a ) @xmath248 and ( b ) @xmath249 . we indicate in the plot the regions corresponding to the regimes ( i ) , ( ii ) and ( iii ) described in the text . , title="fig : " ] here , we study the effect of this resonant mode at @xmath0 on lzs interferometry , in the case @xmath250 . to this end , we calculate the asymptotic @xmath178 using the spectral density of eq.([jsb ] ) , considering different values of @xmath0 , with @xmath251 fixed . in fig.[lasp ] we show the intensity plots of @xmath178 as a function of @xmath252 for @xmath248 and @xmath249 . as it is evident , the diamond patterns are strongly affected by the structured bath . in the case of @xmath248 , we observe in fig.[lasp](a ) that the @xmath252 region formerly occupied by the first diamond in the ohmic case [ shown in fig.[las](b ) ] is now divided in three parts : two new sub - diamonds indicated as regimes ( i ) and ( iii ) , and the region in between them , indicated as regime ( ii ) . when lowering @xmath0 the regime ( iii ) becomes more predominant , as can be seen in fig.[lasp](b ) for @xmath249 . from now on , to describe the above mentioned changes of the first diamond , we restrict to the two lowest levels of the fq . it has been shown that a two - level system coupled to an structured bath that has a localized mode at @xmath0 is equivalent to a two level system weakly coupled to a single mode quantum oscillator with frequency @xmath0 , and both coupled to an ohmic bath.@xcite in fact , most of the results discussed in this section can be qualitatively interpreted by assuming that at low energies there are two virtual levels at @xmath253 and @xmath254 , as sketched in fig.[majo ] . these virtual levels are not stable and decay fast to their `` underlying '' energy level , _ i.e. _ , @xmath255 and @xmath256 . the three regimes found in fig.[lasp ] can be understood by considering that there are two `` virtual crossings '' when @xmath257 , at the field detunings @xmath258 as shown schematically in fig.[majo ] . the boundaries of these sub - diamonds can be defined in a similar way as in sec.iii.b , replacing @xmath259 by @xmath260 in the argument ( when @xmath261 ) . this gives that the sub - diamond of regime ( i ) is within the limits @xmath262 , and the sub - diamond of regime ( iii ) is within the limits @xmath263 ( assuming @xmath264 ) . however , since the virtual levels @xmath265 and @xmath266 are not truly stable , the regimes ( ii ) and ( iii ) show interference patterns that are different than the analyzed in the previous section , as we describe below . regime ( i ) : _ antisymmetric resonances_. if the nearest ( in energy scale ) virtual bath mode at @xmath265 is never reached within the driving interval @xmath267 , the behavior is the same as the one analyzed previously for the ohmic bath in figs.[las](b ) and [ las2](b ) . the only difference is that the ar interference pattern is now within a smaller sub - diamond , since it is limited by the condition @xmath268 within the driving interval . regime ( ii ) : _ symmetric resonances_. in this case , one of the virtual crossings of @xmath269 with @xmath265 is reached within the driving interval @xmath267 . therefore , transitions from the @xmath269 level to the virtual level at @xmath265 are possible . since this later virtual level is unstable , the system decays to the ground state at @xmath270 . by going repeatedly through this process during the periodic driving , population is pumped from the @xmath269 level to the ground state at @xmath270 , providing a ` cooling ' mechanism . the transitions via the unstable mode at @xmath265 lead to a fast relaxation of the system , before the slow mechanisms of bath mediated population inversion , discussed in sec.iiic , can take place . this impedes the dynamic transition to antisymmetric resonances . in this way , the interference pattern with symmetric resonant lobes remains _ in the asymptotic regime_. this is shown in fig.[lasp2](a ) , which corresponds to an enlarged section of the regime ( ii ) of fig.[lasp](a ) . for the ohmic bath . ( a ) for @xmath248 . ( b ) for @xmath249 , title="fig : " ] for the ohmic bath . ( a ) for @xmath248 . ( b ) for @xmath249 , title="fig : " ] regime ( iii ) : _ sideband resonances_. in this case , the two virtual crossings of @xmath223 with @xmath265 are reached within the driving interval @xmath267 . this makes possible to access also the @xmath266 level through landau - zener transitions at @xmath60 , which allows for new `` sideband '' resonances involving the @xmath265 and @xmath266 levels , in addition to the direct resonances at @xmath271 . when @xmath272 , called a blue sideband resonance,@xcite the qubit resonates between the @xmath270 and the @xmath266 levels . however , the @xmath266 virtual level is unstable and it decays to the @xmath269 level . in this way , the ac drive is continuously pumping population from the ground state to the excited state at @xmath269 , leading to full inversion of the qubit population : @xmath273 . when @xmath274 , called a red sideband resonance,@xcite the qubit resonates between the @xmath269 and the @xmath265 levels . in this case , the @xmath265 virtual level decays fast to the @xmath270 level , and thus the ac drive is continuously pumping population from the @xmath269 level to the ground state , leading to @xmath275 . in fig.[lasp2](b ) we can see in detail the sideband resonance patterns that characterize this regime , with alternating @xmath276 and @xmath277 . this type of resonances have been analyzed in ref . , within a perturbative approach for @xmath278 . in that case , only the regime ( iii ) is realized . on the other hand , when @xmath279 the three regimes described above are possible . to observe the subdivision of the first diamond in the three regimes , the crossing of @xmath269 with @xmath265 should occur before the avoided crossing of @xmath269 with @xmath280 . this means that the condition @xmath261 is required , or @xmath281 . for the typical flux qubit parameters considered in this work , this later condition corresponds to @xmath282 . the squid detectors used in the measurements of refs . have typically @xmath283 , which is @xmath284 in our normalized units . therefore in the case of these lzs interferometry experiments , the effects of the resonant mode at @xmath0 are negligible , and everything is within the regime ( i ) of asymptotic asymmetric resonances . on the opposite side , the flux qubits studied in ref . have @xmath285 , which situates them deeply in the case of the regime ( iii ) . in fact , the @xmath286 blue and red sideband resonances have been observed in ref .. however , in these devices the oscillator mode of the squid detector is strongly coupled to the qubit , while eq . ( [ jsb ] ) is valid for weak coupling . a full analysis in this case has to consider a quantum oscillator with frequency @xmath0 coupled to the qubit within the system hamiltonian , see refs .. in any case , our results show that it will be interesting to perform experiments on lzs interferometry using squid detectors with low @xmath0 and weakly coupled to the qubit , to observe the three regimes shown in figs.[lasp ] and [ lasp2 ] . to summarize , by performing a realistic modeling of the flux qubit we were able to analyze the time dependence of the lzs interference patterns ( as a function of ac amplitude and dc detuning ) taking into account decoherence and relaxation . we found an important difference between the lzs patterns observed for the time scale of current experiments and those for the asymptotic long time limit : a symmetry change in their structure . this is a dynamic transition as a function of time from a lzs pattern with nearly symmetric multiphoton resonance lobes to antisymmetric multiphoton resonances . this transition is observable only when driving the system for very long times , after full relaxation with the bath degrees of freedom ( @xmath175 ) . the large time scale separation , @xmath287 , present in the device of ref . explains why in their case the asymptotic lzs pattern is beyond the experimental time window . it will be interesting if experiments could be carried out for longer driving times in this device . for example , measurements of curves of @xmath149 at growing time scales near a multiphoton resonance , could show the transition from symmetric to antisymmetric behavior . another interesting finding is the dependence of the lzs interference patterns on the frequency @xmath0 of the squid detector . different types of lzs interference patterns can arise , depending on the magnitude of @xmath0 . in particular , we showed that the resonant mode at @xmath0 can impede the dynamic transition when @xmath0 is of the order of the qubit gap , in the regime ( ii ) discussed in sec.iiid . in principle , the frequency @xmath0 can be varied ( in a small range ) by varying the driving current of the squid detector @xcite or with a variable shunt capacitor . we acknowledge discussions with w. oliver and s. kohler . we also acknowledge financial support from cnea , uncuyo ( p 06/c455 ) conicet ( pip11220080101821,pip11220090100051,pip 11220110100981,11220150100218 ) and anpcyt ( pict2011 - 1537 , pict2014 - 1382 ) . since the fq of eq.([ham - sys ] ) is driven with a magnetic flux @xmath288 , the hamiltonian is time periodic @xmath289 , with @xmath290 . in this case , it is convenient to use the floquet formalism , that allows to treat periodic forces of arbitrary strength and frequency . @xcite according to the floquet theorem,@xcite the solutions of the schrdinger equation are of the form @xmath291 where @xmath292 are time independent coefficients and @xmath130 are the so - called quasienergies . the floquet states @xmath129 are time - periodic , @xmath293 and satisfy the eigenvalue equation @xmath294 where @xmath295 is defined as the floquet hamiltonian . the time periodicity of @xmath296 and @xmath129 makes convenient to use the fourier representation , @xmath297 |\alpha_{q}\rangle= \varepsilon_\alpha |\alpha_{k } \rangle \label{eqfloq1}\ ] ] with @xmath298 the identity . for the driven fq , we write @xmath299 with @xmath300 the time independent part of the hamiltonian of eq.([ham - sys ] ) , and @xmath301 \sin(2\pi f_{dc } + 2\varphi_l)+\nonumber\\ & & \!\!\!2\alpha e_j\sin^2[\pi f_{ac}\sin{(\omega_{0 } t)}]\cos(2\pi f_{dc } + 2\varphi_l)\;.\nonumber\\ \label{vt1}\end{aligned}\ ] ] in the energy eigenbasis of @xmath302 , given by @xmath303 , the eq.([eqfloq1 ] ) can be written as @xmath304\langle m|\alpha_q\rangle= \varepsilon_\alpha \langle n|\alpha_k\rangle \label{floq - eig}\ ] ] the static eigenvalues @xmath38 and eigenstates @xmath306 are obtained by numerical diagonalization of @xmath302 , using @xmath307-periodic boundary conditions on @xmath308 and a discretization grid of @xmath309 ( with @xmath310).@xcite then , the @xmath311 are evaluated from eqs . ( [ flo9 ] ) and ( [ vt1 ] ) , where the matrix elements @xmath312 and @xmath313 have been calculated using the obtained eigenstates @xmath306 . in order to solve the floquet eigenvalue problem numerically we have to truncate the eq . ( [ floq - eig ] ) both in the fourier indices @xmath314 and the in the number of energy levels of @xmath315 considered.@xcite the truncated eigenproblem is of dimension @xmath316 where @xmath317 is defined by the maximum value of the fourier index and @xmath318 by the number of levels considered in the diagonalization of @xmath315 . the obtained floquet states @xmath319 and quasienergies @xmath130 contain all the information to study the quantum dynamics of the system described above . an alternative method , which is more efficient for large @xmath318 , is to consider the time evolution operator @xmath320 , which in the floquet representation can be expanded as @xmath321 since @xmath322 , the floquet state @xmath129 is an eigenvector of @xmath323 with eigenvalue @xmath324 . therefore , it is possible to calculate the floquet states and quasienergies from the diagonalization of @xmath323 . numerically , one needs to compute the evolution operator @xmath325 within a period , for @xmath326 . taking discretized time steps of length @xmath327 , we use the second - order trotter - suzuki approximation @xmath328 for times @xmath329 , and we compute the product @xmath330 for @xmath331 , starting with @xmath332 . the floquet states are then obtained as eigenvectors of @xmath333 . we diagonalize numerically the hermitian matrix @xmath334 , solving @xmath335 , where @xmath336 and @xmath337 , the floquet states at any time are then calculated as @xmath338 , and their fourier components @xmath339 can be obtained using a fast fourier transform routine . we find that for @xmath340 this numerical procedure is more efficient than the direct diagonalization of eq.([floq - eig ] ) . experimentally , the probability of having a state of positive or negative persistent current in the flux qubit is measured.@xcite the probability of a positive current measurement ( `` right '' side of the double - well potential ) can be calculated , for @xmath24 , integrating the probability @xmath341 within the subspace with @xmath342:@xcite @xmath343 where @xmath344 and @xmath345 . for later generalizations , it is better to define the projector corresponding to a positive current measurement : @xmath346 in terms of this operator , we have @xmath347 . for an initial condition @xmath348 at @xmath349 , we can express @xmath98 in the floquet basis as @xmath350 with @xmath351 . in experiments the initial time , or equivalently the initial phase of the driving field seen by the system in repeated realizations of the measurement , is not well defined . then , the quantities of interest are the probabilities averaged over initial times , @xmath349.@xcite using the properties of the floquet functions , the average over the initial phase time @xmath349 gives @xmath352 and we obtain , @xmath353}\ ] ] it is worth noticing that averaging over the initial phase time of the driving field is equivalent to defining a density matrix @xmath354 which , due to the average over @xmath349 , is diagonal in the floquet basis , @xmath355 . finally , we average within a period over the observation time @xmath193 , to obtain a time independent `` stationary '' probability @xmath356 . after defining the time - averaged projector in the floquet basis as @xmath357 we obtain the simple result @xmath358 with @xmath359 . numerically , after diagonalization of @xmath315 we compute the matrix elements of the projector @xmath360 , with @xmath361.@xcite then , for each @xmath78 and @xmath76 the floquet states and quasienergies are obtained , and the coefficients @xmath362 are evaluated ( with @xmath363 ) . of a state with positive loop current as a function of @xmath75 for the undriven qubit ( red squares ) and for the driven qubit ( black circles ) with @xmath364 and @xmath365 . the flux detuning @xmath75 is normalized by @xmath187 , such that the @xmath83-photon resonances are at @xmath188 . ( b ) floquet quasienergies ( in units of @xmath14 ) as a function of @xmath75 for the same case as in ( a ) . ( c ) floquet quasienergies ( in units of @xmath14 ) as a function of @xmath76 for the @xmath286 resonant state at @xmath366 [ corresponding to black square in panel ( a ) ] . ( d ) floquet quasienergies ( in units of @xmath14 ) as a function of @xmath76 for an out of resonance state at @xmath367 [ corresponding to blue diamond in panel ( a ) ] . device parameters of the flux qubit are @xmath42 and @xmath33 . ] as an example , we calculate @xmath368 for the driven fq in the two - level regime , as described by the hamiltonian @xmath369 of eq.([htls ] ) . at zero temperature and in the absence of driving ( @xmath227 ) the isolated fq is in the ground state . in this case , the probability @xmath149 is simply the projection of the ground state on the subspace of positive persistent current , and we have @xmath370 . for @xmath371 we have @xmath372 except near @xmath373 where @xmath374 , as can be seen in fig.[qef0](a ) . ( while for @xmath375 , we have @xmath376 , since the ground state has the loop current in the opposite direction in this case ) . in the presence of an ac drive , @xmath377 , we calculate the time averaged @xmath378 , following the procedure discussed above . the time averaged @xmath378 vs. @xmath379 shows dips corresponding to @xmath111photon resonances . as shown in fig.[qef0](a ) . these @xmath83-resonances are at @xmath380 , which is equivalent to @xmath84 , after defining @xmath82 . the @xmath83-resonances are at @xmath84 . in the floquet picture , these resonances correspond to avoided crossings of the floquet quasienergies@xcite as a function of @xmath75 as we illustrate in fig.[qef0](b ) . when increasing @xmath76 , we see that for a @xmath83-resonance the quasienergies have a small gap , @xmath381 ( with the difference @xmath382 defined modulo @xmath78 ) . on the other hand , for a value of @xmath75 away of a resonance the quasienergies maintain a finite gap ( compared to @xmath78 ) as a function of @xmath76 , see fig.[qef0](c ) and ( d ) . experimentally , the system is affected by the electromagnetic environment that introduces decoherence and relaxation processes . a standard theoretical model to study enviromental effects is to couple the system bilinearly to a bath of non - interacting harmonic oscillators with masses @xmath383 , frequencies @xmath384 , momenta @xmath385 , and coordinates @xmath386 , with the coupling strength @xmath387.@xcite the total hamiltonian of system and bath is then given by @xmath388 where @xmath389 is the time - periodic hamiltonian of the system , @xmath390 is the hamiltonian that describes a bath of harmonic oscillators and @xmath391 its system - bath coupling hamiltonian term , with @xmath394 the operator of the system that couples to the bath . the bath degrees of freedom are characterized by the spectral density @xmath395 it is further assumed that at time @xmath92 the bath is in thermal equilibrium and uncorrelated to the system . then , the full density matrix @xmath396 has at initial time the form @xmath397 , where @xmath398 is the density matrix of the system and @xmath399 is the bath temperature . after expanding the density matrix of the system in the time - periodic floquet states @xmath400 the born ( weak coupling ) and markov ( fast relaxation ) approximations for the time evolution of @xmath138 are performed . in this way , the floquet - markov master equation is obtained@xcite @xmath401 the first term in eq.([fbr1 ] ) represents the nondissipative dynamics and the influence of the bath is described by the time - dependent rate coefficients @xmath402 with @xmath403 the nature of the bath is encoded in the coefficients @xmath404 with @xmath405 and @xmath406 , and defining @xmath407 for @xmath408 . the system - bath interaction is encoded in the transition matrix elements @xmath409 in the floquet basis in the case of the driven fq , the system hamiltonian is @xmath411 and the bath degrees of freedom couple with the system variable @xmath412 since the dominating source of decoherence is flux noise ( see ref . ) . thus , after taking @xmath413 and in terms of the eigenbasis of @xmath414 , we have to compute @xmath415 considering that the time scale @xmath196 for full relaxation satisfies @xmath416 , the transition rates @xmath417 can be approximated by their average over one period @xmath177 , @xcite @xmath418 , obtaining @xmath419 the rates @xmath420 can be interpreted as sums of @xmath421-photon exchange terms . this formalism has been extensively employed to study relaxation and decoherence for time dependent periodic evolutions in double - well potentials and in two level systems.@xcite here we use it to model the ac driven fq , considering the full multilevel hamiltonian of eq . ( [ ham - sys]).@xcite to calculate the time dependence of @xmath138 it is convenient to work in the superoperator formalism of the so - called liouville space.@xcite we write , @xmath425 with @xmath426 . then we change notation rewriting the @xmath427 matrix @xmath428 as an @xmath429 vector represented as the ket @xmath430 , and the @xmath431 `` supermatrix '' @xmath179 as the operator @xmath432 acting on this linear space , where the inner product is defined as @xmath433 . in particular , for the identity matrix @xmath298 we have @xmath434 , the later equality corresponding to the normalization of @xmath428 , which is a conserved quantity . on the other hand , the norm of the vector @xmath430 is @xmath435 . in this notation , we can rewrite the floquet - markov equation as @xmath436 the superoperator @xmath437 is non - hermitian and has left and right eigenvectors with complex eigenvalues @xmath438 , @xmath439 which are mutually orthogonal , @xmath440 . in general , the number of independent eigenvectors of @xmath432 can be less than the dimensionality of @xmath432 . a formal solution of eq.([eqfm ] ) for @xmath441 can be obtained using a similarity transformation to the jordan normal form of @xmath432.@xcite in the cases considered in this work we found numerically that it was always possible to diagonalize @xmath432 , in which case the solution of eq.([eqfm ] ) can be expressed as @xmath442 from where we can calculate numerically @xmath443 . the probability of a positive current measurement is then obtained combining eq.([p+t ] ) with eq.([rhot1 ] ) . the asymptotic state @xmath444 satisfies , @xmath445 therefore , the asymptotic state can be constructed from the right - eigenvectors @xmath446 of @xmath432 with eigenvalue @xmath447 ( _ i.e. _ , the kernel of @xmath432 ) . if @xmath448 is non - degenerate , then the asymptotic state is unique and independent of the initial condition @xmath449 . in the cases considered in this work we found , within the numerical accuracy , that the eigenvalue @xmath447 was non - degenerate , and so the asymptotic state was given by the eigenvector @xmath450 , and then @xmath451 . the time independence of the asymptotic @xmath452 , implies that quantities like @xmath149 in eq([p+t ] ) , become time - periodic with period @xmath177 in the asymptotic state . therefore , it is convenient to calculate the asymptotic @xmath98 averaged over one period as @xmath453 it is also clear from eq.([rhot1 ] ) that information about the relaxation and decoherence rates is contained in the non - zero eigenvalues @xmath438 of @xmath432.@xcite on one hand , the relaxation rates are given by the negative real eigenvalues of @xmath432 , where the long time relaxation rate @xmath454 is given by the minimum of @xmath455 ( excluding @xmath447 ) . on the other hand , the decoherence rates are given by the negative real parts of the complex conjugate pairs of eigenvalues of @xmath432 . it is well known that the density matrix becomes diagonal in the energy basis for times larger than the relaxation time in undriven systems.@xcite in this case , the decay time of the offdiagonal @xmath456 defines the decoherence time @xmath457 between the eigenstates @xmath38 and @xmath458 . for time scales @xmath459 , relaxation is described by the pauli rate equation for the populations of the energy levels @xmath460 . in the case of a system with a time periodic drive , it is usually assumed that for large times the density matrix becomes approximately diagonal in the floquet basis.@xcite more precisely , this approximation can be justified when @xmath461 , which is fulfilled for very weak coupling with the environment and away from resonances , see ref .. from fig.[qef0](c ) and ( d ) it is clear that this condition will be easily satisfied in the offresonant case considered here , where the floquet gap @xmath462 is large . on the other hand at an near a resonance where @xmath463 this condition can not be fulfilled , unless the system - bath coupling is extremely small.@xcite as a function of time . black lines : off - diagonal matrix element @xmath464 in the basis of eigenstates of the undriven hamiltonian . red lines : off - diagonal element @xmath465 in the floquet basis . ( a ) for a resonant state at @xmath466 driven with @xmath364 and @xmath365 . ( b ) for an off - resonant state at @xmath367 driven with @xmath107 and @xmath365 . insets in ( a ) and ( b ) show @xmath465 at large times . ] in fig.[foff ] we show the time evolution of matrix elements of @xmath428 calculated in the eigenbasis of @xmath414 and in the floquet basis , both for an offresonant and for a resonant case . to clarify notation , for the @xmath414 eigenbasis we use latin indices @xmath467 with @xmath468 ordered for increasing eigenenergy @xmath469 . for the floquet basis we use greek indices @xmath470 with @xmath471 , and ordered such that @xmath472 corresponds to the state that in the limit @xmath230 maps to the ground state ( with index @xmath473 ) ; @xmath474 corresponds to the state that in the limit @xmath230 maps to the first excited state ( with index @xmath475 ) , and so on . the time dependence of the offdiagonal elements @xmath476 in the the @xmath414 eigenbasis , are shown in fig.[foff](a ) and fig.[foff](b ) for the resonant and offresonant cases , respectively . ( here we consider the flux qubit in contact with an ohmic bath with @xmath477 ) . we see that for long times @xmath478 goes to a finite non - zero value that can not be neglected , showing explicitly that the driven system density matrix is not diagonal in the eigenenergy basis . therefore approaches based on the use of the pauli rate equation in the eigenenergy basis will not be correct for the analysis of the asymptotic long time behavior . in the case of the floquet basis , we see in fig.[foff](a ) and ( b ) that the offdiagonal @xmath465 after having oscillations at short time scales , decreases exponentially to very low values for long times . we find that @xmath479 for the offresonant case ( while diagonal elements @xmath480 ) . this confirms that it is a good approximation to neglect the @xmath465 at long times in this case . on the other hand , in the resonant case we find @xmath481 . thus , in this case neglecting @xmath465 is not a good approximation as in the offresonant case . the results reported in the main body of the paper correspond to the solution of the full floquet - markov equation , eq . [ eqfm ] . however , we have verified that most of our results ( including the dynamic transition discussed in sec.iii ) are accurately reproduced by the approximation that assumes an asymptotic density matrix diagonal in the floquet basis . assuming that the density matrix becomes diagonal in the floquet basis , one can separate the dynamics of the diagonal and the off - diagonal density matrix . the off - diagonal part is dominated by the dependence @xmath482 \,\rho_{\alpha\beta}\;\;\;\ ; \alpha\not=\beta\ ] ] in this approximation , the decoherence rate between the @xmath483 and the @xmath484 floquet state , is given by @xmath485 . the dynamics for the diagonal part of the density matrix gives a rate equation for the population of the floquet states @xmath486 : @xmath487 where @xmath488 , after eq . 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w. thomas pollard , anthony k. felts , and richard a. friesner , advances in chemical physics , volume xciii , edited by i. prigogine and stuart a. rice . 1996 john wiley & sons , inc .

we study landau - zener - stuckelberg ( lzs ) interferometry in multilevel systems coupled to an ohmic quantum bath . we consider the case of superconducting flux qubits driven by a dc+ac magnetic fields , but our results can apply to other similar systems . we find a dynamic transition manifested by a symmetry change in the structure of the lzs interference pattern , plotted as a function of ac amplitude and dc detuning . the dynamic transition is from a lzs pattern with nearly symmetric multiphoton resonances to antisymmetric multiphoton resonances at long times ( above the relaxation time ) . we also show that the presence of a resonant mode in the quantum bath can impede the dynamic transition when the resonant frequency is of the order of the qubit gap . our results are obtained by a numerical calculation of the finite time and the asymptotic stationary population of the qubit states , using the floquet - markov approach to solve a realistic model of the flux qubit considering up to 10 energy levels .

You are an expert at summarizing long articles. Proceed to summarize the following text: different cosmological probes seem to point toward a consensus scenario characterized by a universe which is experiencing an accelerated expansion . standard candles ( snia , e.g. @xcite ) , standard rulers ( clusters , cmb and bao , e.g. @xcite ) and the angular power spectrum of the cmb ( e.g. @xcite ) are used to probe the hubble expansion flow up to redshifts 1 - 2 ( by means of snia or clusters ) or up to the epoch of recombination ( through the cmb anisotropies ) . gamma ray bursts are observed up to very high redshifts : the farthest is grb080913c at z=6.7 @xcite . among grbs with known redshifts , 45% are at @xmath1 and 8% at @xmath2 . the high grb luminosity ( 10@xmath3 erg ) and their detection in the @xmath4ray band makes them attractive as a potential and complementary cosmological tool to constrain the cosmological models at @xmath1 . however , the problem is that grbs are all but standard candles @xcite : their isotropic equivalent energetics and luminosities span 3 - 4 orders of magnitudes . similarly to snia , it has been proposed to use correlations between various properties of the prompt emission @xcite and also of the afterglow emission @xcite to standardize grb energetics . * isotropic * energies ( luminosities ) can be computed for grbs with measured redshifts and well constrained spectral properties . the spectrum gives the bolometric fluence @xmath5 ( peak flux @xmath6 ) and then @xmath7 ( @xmath8 ) . these two quantities are strongly correlated with the rest frame peak energy of the @xmath9 spectrum @xcite . we have updated the sample of bursts with known @xmath10 and spectral parameters to jan 2009 ( the last being grb 090102 at z=1.547 ) . these are 97 grbs . the fit of both correlations with a powerlaw gives @xmath11@xmath12 and @xmath11@xmath13 . however , due to the large scatter of the data points @xmath14 is extremely large ( 492 and 612 for 95 degrees of freedom for the and correlations , respectively ) . the scatter of the data points is defined by their distance from the best fit line . by modeling the scatter distribution with a gaussian we find a logarithmic dispersion of @xmath150.23 dex and @xmath16=0.28 dex for the and correlation , respectively . these scatters are much larger than the statistical errors associated with the observables @xmath17 , @xmath18 , @xmath19 . there is then the possibility that a third variable is responsible for this large scatter . dispersion ( modeled as a gaussian ) of the data points around the best fit lines . red open symbols are the grbs with a jet break time measurement reported in the literature . ] the large dispersion of the and correlations prevents their use to standardize grb energetics . however , grbs are thought to be collimated sources . in the standard grb model the jetted outflow should produce a break in the afterglow light curve decay @xcite . the measure of the time of occurrence of this break @xmath20 allows to infer the jet opening angle @xmath21 ( under the standard afterglow model assumptions ) and to recover the * collimation corrected * energy @xmath22 . by collecting available estimates of jet break times from the literature we found @xcite and later confirmed @xcite that the collimation corrected energy is strongly correlated with . in fig.[fg1 ] the correlation is shown through the 29 bursts having a jet break in their optical afterglow light curves . we find @xmath23 with @xmath14=37.8 for 27 degrees of freedom assuming a wind like profile of the circumburst medium . this correlation has a dispersion @xmath24 which is consistent with the average statistical uncertainties on and and it is much smaller than the dispersion of the and correlations . this result also suggests that the dispersion of the correlation is due to the jet opening angle : by correcting for @xmath21 ( for each burst ) , the scatter of the correlation is reduced . due to its tightness the correlation can be used to constrain the cosmological parameters @xcite . the correlation , shown in fig.[fg1 ] , was found by assuming a cosmological model ( i.e. @xmath25 , @xmath26 ) . it would be a circular argument to use this particular correlation to constrain the cosmological parameters . we originally solved this problem by properly accounting for the dependence of the correlation from the cosmological parameters @xcite or by adopting a bayesian fitting method @xcite . fig.[fg2 ] shows the cosmological constraints obtained through the correlation with the most updated sample of 29 grb . these are compared with the constraints obtained with the sample of 19 grbs @xcite . also in this case we have applied the bayesian method that overcomes the circularity problem . the correlation is derived in the standard uniform jet scenario assuming a constant radiative efficiency and either a uniform @xcite or a wind @xcite circumburst density profile . @xcite discovered a completely empirical correlation @xmath27 between the three observables which are combined in the correlation . also the empirical correlation , due to its low scatter , can be used to derive constraints on the cosmological parameters @xcite . the mostly debated issue of the spectral energy correlations in general is that they are due to selection effects @xcite . it has also been claimed that the slope and normalization of the correlation evolve with redshift @xcite selection effects can be studied in the observational plane corresponding to these correlations : the and the planes , where @xmath5 and @xmath6 are the bolometic fluence and peak flux . we have studied @xcite two instrumental selection effects : the minimum flux required to trigger a burst ( trigger threshold ) and the minimum flux to properly analyze its prompt emission spectrum ( spectral threshold ) . the former has been claimed to bias the correlation @xcite . the sample of 76 bursts ( updated to sept . 2007 ) with measured redshifts is composed by grbs detected by different instruments . for this reason we modeled the trigger threshold and the spectral analysis threshold of the different detectors . we can exclude that the correlation is biased by the trigger threshold . also we can exclude that the spectral threshold is biasing the pre - swift sample . instead , the swift grb sample ( 27 events ) is biased by the spectral threshold . this is also due to the limited spectral energy range of the bat instrument on - board swift which limits the measure of in the 15 - 150 kev energy range . by considering sub - samples of bursts at different redshifts , we also exclude that the correlation slope or normalization change . recently @xcite added grbs without redshift in the and planes to compare the distribution of bursts with respect to the two selection effects . to this purpose a sample of 100 faint batse bursts , representative of a larger population of 1000 objects , was analyzed . by means of this complete , fluence - limited , grb sample , it was found that the fainter batse bursts have smaller than those of bright events . as a consequence , the of these bursts is correlated with the fluence , though with a slope flatter than that defined by bursts with z. selection effects , which are present , are not responsible for the existence of such a correlation . about six per cent of these bursts are surely outliers of the correlation , since they are inconsistent with it for any redshift . also correlates with the peak flux , with a slope similar to the correlation . in this case , there is only one sure outlier . the scatter of the correlation defined by the batse bursts of this sample is significantly smaller than the correlation of the same bursts , while for the bursts with known redshift the correlation is tighter than the one . once a very large number of bursts with and redshift will be available , we expect that the correlation will be similar to that currently found , whereas it is very likely that the correlation will become flatter and with a larger scatter . one of the main drawback of the correlation for cosmological use is that it still has few points : the original sample of 15 events @xcite has only doubled since 2004 . with the launch of swift @xcite several jet breaks were expected to be measured , especially in the x ray band . jet breaks should be achromatic because they are produced by a geometric effect . all the jet breaks used to compute in the pre - swift era were , instead , obtained from the optical light curves . swift revealed a complex x ray afterglow light curve : the early afterglow is often characterized by a steep decay followed by a shallow phase lasting thousands of seconds @xcite . a characteristic break time is that ending the x ray shallow phase . @xcite showed that this time is not a jet break . a possible interpretation is that the shallow phase is produced by a long lasting central engine activity @xcite as also supported by the presence of strong precursors , post - cursors , and x - ray flares in a sizable fraction of bursts . often the x - ray and the optical afterglow light curves do not track one another , suggesting that they are two different emission components . we selected a sample of 33 gamma ray bursts ( grbs ) detected by swift , with known redshift and optical extinction at the host frame @xcite . the de absorbed and k corrected x ray and optical rest frame light curves are modelled as the sum of two components : emission from the forward shock due to the interaction of a fireball with the circum - burst medium and an additional component , treated in a completely phenomenological way . the latter can be identified , among other possibilities , as `` late prompt '' emission produced by a long lived central engine with mechanisms similar to those responsible for the production of the `` standard '' early prompt radiation . we find a good agreement with the data , despite of their complexity and diversity . our approach allows us to interpret the behaviour of the optical and x - ray afterglows in a coherent way , by a relatively simple scenario . within this context it is possible to explain why sometimes no jet break is observed ; why , even if a jet break is observed , it is often chromatic ; why the steepening after the jet break time is often shallower than predicted . the use of the correlation for cosmology requires to measure the redshift , the prompt emission and @xmath20 . the latter is the most critical observable : the jet break is typically observed at 1 - 2 days after the trigger and the light curve needs to be sampled at much later epochs in order to infer @xmath20 when the afterglow can be very dim , also because swift detects higher redshift bursts than before @xcite . therefore , it is interesting to explore if other correlations can be employed to standardize grb energetics . recent attempts @xcite tried to use the and correlations . in this cases , however , one has to take into account that these correlations are affected by a dispersion which is much larger than the statistical uncertainty on the data points . the scatter of these correlations is due to three terms . one is the statistical uncertainty in the measurements of the parameters @xmath28 . a second contribution to the scatter comes from systematic errors @xmath29 which could also have a physical origin but is difficult to model . a third contribution to the scatter could finally be due to the cosmological model @xmath30 : in the real cosmology its contribution should be minimized . in the and correlations the last two terms are dominating the scatter . vs @xmath31 are normalized to the average value of the statistical error associate to the variable they are assigned to . top panel : the extra - scatter term is assigned to and the results similar to those of @xcite are found . a minimum ( though weak ) is found for @xmath32 . the solid curve is found with the likelihood function of eq.1 , the dotted and dashed curved are found with the symmetric likelihood function of @xcite . the dot - dashed line is found with the @xmath33 fitting method . bottom panel : the extra - scatter term is assigned to . this is also the most obvious assumption as it is to depend on the cosmological parameters . no minimum is found in this case of the extra - scatter term for any value of @xmath31.,scaledwidth=80.0% ] @xcite ( @xcite ) proposed the use of the ( ) correlation to constrain the cosmological parameters . what is appealing is the possibility to use a correlation defined by a large grb sample ( much larger than that defining the correlation ) and to use only on prompt emission observables ( and or ) . however , in both correlations the non statistical scatter @xmath34 need to be modeled . these terms ( combined ) are treated as a free parameter , i.e. the extra scatter @xmath35 which is assumed to have a gaussian distribution equal for all the data points . to test the possibility of using the correlation for cosmology , @xcite fit the correlation in different `` cosmologies '' ( a flat universe is assumed ) and derive , in function of @xmath31 , the best fit values of the free parameters , i.e. the slope @xmath36 and the normalization @xmath37 of the correlation and the extra scatter term @xmath35 . the correlation is fitted with the likelihood function : @xmath38=\frac{1}{2}\sum_{i } \log[{1\over 2\pi(m^2\sigma_{x , i}^2+\sigma_{y , i}^2+m^2\sigma_{x}^2+\sigma_{y}^2 } ] + & \\ - \frac{(y_{i}-mx_{i}-q)^2}{m^2\sigma_{x , i}^2+\sigma_{y , i}^2+m^2\sigma_{x}^2+\sigma_{y}^2 } & \end{aligned}\ ] ] @xcite assume y= , x = and set the extra scatter term @xmath41 , i.e. they give the ( free ) extra scatter only to . they find that the extra scatter @xmath42 shows a minimum corresponding @xmath43 . therefore , they apply the standard procedure to derive constraints on the cosmological models through the correlation . however , in the ( ) correlations it is ( ) that depends on the cosmological parameters ( through the luminosity distance @xmath44 ) . therefore , the extra - scatter term should be assigned to x= , i.e. @xmath45 . for this reason we repeated the same test on the and correlations and we do not find any minimum of the extra - scatter @xmath46 when it is assigned to which is actually the variable that depends on the cosmological parameters . we verified our results also by ( a ) adopting the symmetric likelihood function of @xcite ( having the term @xmath47 in the numerator of the first term of eq.1 ) ; ( b ) fitting with the least square method ; ( c ) inverting the order of the fitting variables ( i.e. setting y = and x= ) . similar results are also found for the correlation . our results are shown in fig.[fg3 ] . in the correlation ( fig.[fg1 ] ) the scatter of the data points is already consistent with the statistical errors associated with and . the only residual scatter is due to the cosmological model . this is why , without assumptions on the nature and `` normality '' of the unknown extra scatter term , the correlation is preferable to standardize the grb energetics . the correlation also proves that most of the scatter of the correlation is due to the jet opening angle , which is different from burst to burst . this shows ( as also recently demonstrated by @xcite ) that the and correlations can not be used straightforwardly to constrain the cosmological parameters due to the unknown nature of the extra scatter they are affected by . 9 astier , p. , guy , j. , regnault , n. , et al . , _ a&a _ , 2006 , 447 , 31 percival , w. j. , cole , s. , eisenstein , d. j. , et al . , _ mnras _ , 2007 , 381 , 1053 lewis , a. , _ phrvd _ , 2008 , 78 , 3002 greiner , j. , kruehler , t. , fynbo , j. p. u. , et al . , _ apj subm . 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several interesting correlations among gamma ray bursts ( grb ) prompt and afterglow properties have been found in the recent years . some of these correlations have been proposed also to standardize grb energetics to use them as standard candles in constraining the expansion history of the universe up to @xmath0 . however , given the still unexplained nature of most of these correlations , only the less scattered correlations can be used for constraining the cosmological parameters . the updated correlation is presented . caveats of alternative methods of standardizing grb energetics are discussed . address = inaf - osservatorio astronomico di brera . via e. bianchi 46 , i-23807 merate ( lc ) , italy

You are an expert at summarizing long articles. Proceed to summarize the following text: about half of the nuclei with @xmath0 observed in nature are formed by the so - called rapid neutron - capture process ( or r - process ) of nucleosynthesis , occurring in explosive stellar events . the r - process is believed to take place in environments characterized by high neutron densities ( @xmath1 ) , so that successive neutron captures proceed into neutron - rich regions well off the @xmath2-stability valley forming exotic nuclei that can not be produced and therefore studied in the laboratory . if the temperatures or the neutron densities characterizing the r - process are low enough to break the @xmath3 equilibrium , the r - abundance distribution depends directly on the neutron capture rates by the so - produced exotic neutron - rich nuclei @xcite . the neutron capture rates are commonly evaluated within the framework of the statistical model of hauser - feshbach ( although the direct capture contribution can play an important role for such exotic nuclei ) . this model makes the fundamental assumption that the capture process takes place with the intermediary formation of a compound nucleus in thermodynamic equilibrium . in this approach , the maxwellian - averaged @xmath4 rate at temperatures of relevance in r - process environments strongly depends on the electromagnetic interaction , i.e the photon de - excitation probability . the well known challenge of understanding the r - process abundances thus requires that one be able to make reliable extrapolations of the e1-strength function out towards the neutron - drip line . to put the description of the r - process on safer grounds , a great effort must therefore be made to improve the reliability of the nuclear model . generally speaking , the more microscopic the underlying theory , the greater will be one s confidence in the extrapolations out towards the neutron - drip line , provided , of course , the available experimental data are also well fitted . large scale prediction of e1-strength functions are usually performed using phenomenological lorentzian models of the giant dipole resonance ( gdr ) @xcite . several refinements can be made , such as the energy dependence of the width and its temperature dependence @xcite to describe all available experimental data . the lorentzian gdr approach suffers , however , from shortcomings of various sorts . on the one hand , it is unable to predict the enhancement of the e1 strength at energies around the neutron separation energy demonstrated by various experiments , such as the nuclear resonance fluorescence . on the other hand , even if a lorentzian - like function provides a suitable representation of the e1 strength for stable nuclei , the location of its maximum and its width remain to be predicted from some systematics or underlying model for each nucleus . for astrophysical applications , these properties have often been obtained from a droplet - type model @xcite . this approach clearly lacks reliability when dealing with exotic nuclei , as already demonstrated by @xcite . recently an attempt was made to derive microscopically the e1 strength for the whole nuclear chart @xcite . the dipole response was calculated with the quasiparticle random phase approximation ( qrpa ) on top of hartree - fock+bcs ( hfbcs ) description @xcite . the only input of this approach was the skyrme effective interaction injected in the hfbcs model . these microscopic calculations predicted the presence of a systematic low - lying component in the e1 strength for very neutron - rich nuclei . this low - lying component influences the neutron capture rate , especially if located in the vicinity of the neutron separation energy @xmath5 . in our previous hfbcs and qrpa microscopic approach @xcite , the pairing correlation in the bcs model was determined assuming a simple constant - gap pairing interaction . in addition , in the case of the highly neutron - rich nuclei that are of particular interest in the context of the r - process , the validity of the bcs approach to pairing is questionable , essentially because of the role played by the continuum of single - particle neutron states ( see @xcite , and references therein ) . therefore the impact of the newly - derived e1-strength functions on the cross section prediction could only be evaluated qualitatively . it was found that the radiative neutron capture cross sections by neutron - rich nuclei were systematically increased by the hfbcs+qrpa calculations @xcite with respect to the one obtained using lorentzian - like strength functions . predictions with different forces have been compared , but no conclusions could be drawn regarding their intrinsic quality to predict the e1 strength . the final large - scale hfbcs+qrpa calculations performed in @xcite were obtained on the basis of the skyrme force denoted sly4 @xcite . in the present paper we calculate the dipole strength with one of the most accurate and reliable microscopic model available to date , namely the hartree - fock - bogoliubov ( hfb ) and qrpa models @xcite . as recalled in sect . 2.1 , the ground state is described within the hfb model . effective interactions of the skyrme type characterized by different values of the nucleon effective mass and prescriptions for the pairing interaction are considered . the collective gdr mode is obtained by qrpa calculations on top of the hfb calculations , as described in sect . the residual interaction is derived self - consistently from the nucleon - nucleon effective interaction , which is the only input of the hfb calculation . to describe the damping of the collective motions on microscopic grounds , the second - rpa ( srpa ) described by @xcite is adopted ( sect . this approach strongly improves the reliability of the predictions by eliminating the phenomenological spreading of the qrpa strength determined in our previous hfbcs+qrpa calculations @xcite . this new approach allows us to determine on a more quantitative and reliable ground the photoabsorption cross section and consequently to judge the ability of the forces to reproduce experimental data . in order to select the most adequate interaction for the e1-strength calculation , the hfb+qrpa prediction are compared with photoabsorption data for 48 spherical nuclei @xcite ( sect . 3 ) . the hfb+qrpa predictions are further compared , in sect . 3 , with low - energy experimental data and generalized to include deformation effects in a phenomenological way . all these drastic improvements compared to the previous hfbcs and qrpa models allow to provide quantitative predictions of the e1-strength function , also for exotic neutron - rich nuclei ( sect . 4 ) . the predicted gdr strengths are used to estimate all the radiative neutron capture rates of relevance for nucleosynthesis applications ( sect . the long - term goal of microscopic models is to describe on the same ground a wide variety of nuclear structure properties ( in particular , magicity and pairing correlations in open - shell nuclei ) for both stable and exotic nuclei . the hfb and qrpa models allows to treat , in a self - consistent way , pairing effects on the ground state as well as collective excitations for nuclei ranging from the valley of stability to the drip - line . the qrpa considers nuclear excitation as a collective superposition of two quasiparticle ( qp ) states built on top of the hfb ground state @xcite . this collective aspect of the excitation makes the qrpa an accurate tool to investigate the e1-strength function , in both closed and open shell nuclei . the hfb calculations considered in the present work are fully detailed in @xcite . they are based on the conventional skyrme force of the form @xmath6 the pairing force acting between like nucleons is treated in the full bogoliubov framework with a @xmath7-function pairing force of the form @xcite @xmath8~ \delta(\mbox{\boldmath$r$}_{ij } ) \quad , \label{eq_surf}\ ] ] where @xmath9 is the density and @xmath10 its equilibrium value in symmetric nuclear matter . two types of pairing forces are considered here , a volume density - independent force characterized by @xmath11 and a volume plus surface ( i.e density - dependent ) force with @xmath12 and @xmath13 . this latter prescription originates from the calculations of the pairing gap in infinite nuclear matter at different densities performed by garrido et al . @xcite using a bare " or realistic " nucleon - nucleon interaction . this density - dependent pairing has also been found to be compatible with experimental nuclear masses by @xcite , provided the space of single - particle states over which such a pairing force is allowed to act is truncated to about @xmath14 mev around the fermi energy . note finally that in the present approach the strength parameter @xmath15 is allowed to be different for neutrons and protons , and also to be slightly stronger for an odd number of nucleons ( @xmath16 ) than for an even number ( @xmath17 ) , i.e. , the pairing force between neutrons , for example , depends on whether @xmath18 is even or odd . for odd-@xmath19 and odd - odd nuclei , the blocking approximation is used , as detailed in @xcite . based on this skyrme - hfb approach , a number of effective forces have been determined recently @xcite , the parameters of the underlying forces being fitted _ exclusively _ to all the 2135 available experimental masses @xcite , with some additional constraints regarding the stability of neutron matter and the incompressibility of nuclear matter . the parameters corresponding to these six forces named bsk2-bsk7 are summarized in table [ tab_sky ] . are also included in table [ tab_sky ] the effective isoscalar ( @xmath20 ) , isovector ( @xmath21 ) nucleon mass and the root - mean - square ( rms ) deviations @xmath22 between the measured and estimated masses for the 2135 nuclei with @xmath23 . more details about these forces can be found in @xcite . the major differences between these six forces are found in the density - dependence of the pairing force and the adopted isoscalar effective nucleon mass . while bsk2 and bsk3 have been built without constraining the effective mass ( leading to a value of @xmath24 ) , bsk4 and bsk5 are constrained by @xmath25 , as inferred from the extended brckner - hartree - fock calculations of asymmetric nuclear matter @xcite and bsk6 and bsk7 by @xmath26 , as inspired from the more traditional symmetric nuclear - matter calculations ( e.g @xcite ) . the mass - data fits with the bsk47 interaction are almost as good as those obtained with bsk23 , in which @xmath27 is unconstrained , so that such a mass fit can not be used to discriminate between the different skyrme forces and the corresponding optimal choice for the nucleon effective mass or the pairing interaction . for comparison purposes , we also consider the sly4 skyrme force @xcite used in our previous hfbcs+qrpa calculation @xcite . the sly4 parameters are given in table [ tab_sky ] , the pairing interaction corresponding to the one determined in @xcite ( the mass rms deviation is however not available for the sly4 force ) . .some properties of the skyrme forces bsk2-bsk7 and sly4 ( see text for more details ) . the last line corresponds to the rms deviation between predicted and experimental masses for the full set of 2135 spherical and deformed nuclei . [ cols="^,^,^,^,^,^,^,^",options="header " , ] as illustrated in figs . [ fig_gdr1]-[fig_gdr3 ] and table [ tab_cst ] , the prediction of the gdr parameters is force - dependent . as far as the position of the peak energy is concerned , most of the forces overpredict the peak energy of light nuclei and underpredict it for the heavier species . the best agreement is found for the bsk6 - 7 forces which still overestimate @xmath28 for the lightest elements , but give an excellent agreement for @xmath29 elements . it can be concluded that , within the present hfb+qrpa model , skyrme forces need to have a low effective nucleon mass @xmath30 to correctly predict the gdr characteristics . an effective mass as low as the one used in sly4 requires a particularly low value of the interference factor ( see table [ tab_cst ] ) which simultaneously give rise to fine structure effects not observed experimentally ( see fig . [ fig_gdr3 ] ) . interestingly , the density dependence of the pairing interaction has a minor impact on the prediction of the e1-strength function . almost no differences are found among the 2 couples of forces bsk4 vs bsk5 or bsk6 vs bsk7 ( see in particular fig . [ fig_gdr3 ] ) . finally , note that the srpa effect to shift the energy peak was not taken into account by the phenomenological lorentzian damping adopted in our previous hfbcs+qrpa work @xcite , and obviously modifies the conclusion drawn regarding the ability of the sly4 force to predict the location of the gdr . the agreement found here for the sly4 interaction ( fig . [ fig_gdr3 ] ) is worse than it used to be in @xcite where an rms deviation of 0.457 mev was obtained on the centroid energy for the same sample of spherical nuclei . finally , regarding the amplitude of the e1-strength function , the qrpa equations are solved so as to exhaust the thomas - reiche - kuhn sum rule . this adopted ewsr corresponds to @xmath31\ ] ] and is found to reproduce well the peak cross section measured experimentally , as illustrated in fig . [ fig_sigmax ] . the resulting deviation can be characterized by an rms deviation factor @xmath32 defined as @xmath33^{1/2 } \label{eq_rms}\ ] ] where @xmath34(@xmath35 ) is the theoretical ( experimental ) peak cross section and @xmath36 the number of nuclei in the experimental sample . all these results show that among the six skyrme forces studied here , both the bsk6 and bsk7 forces not only reproduce extremely well the experimental masses ( with a rms deviation as low as 0.676 mev on the 2135 known masses ) , but also is well adapted to describe the e1 collective excitations . for this reason , all further calculations are performed with bsk7 as our standard force . it should be stressed that for more than thirty years , phenomenological effective interactions were developed using exclusively ground state properties of nuclei , such as binding energies , radii or spectroscopic quantities . this was initiated through the skyrme hartree - fock model by @xcite . nuclear forces are traditionally determined by fitting such ground state properties for less than ten or so nuclei . recently , progress has been achieved in determining the skyrme force by fitting essentially all the mass data @xcite . the only excited feature taken into account so far was the giant monopolar resonance energy @xcite in order to predict the infinite matter compressibility modulus . the present hfb+qrpa model ( with the srpa corrections ) allows to consider nuclear excitations such as gdr in the development of phenomenological effective interactions . in the case of deformed spheroidal nuclei , the gdr splits into two major resonances as a result of the different resonance conditions characterizing the oscillations of protons against neutrons along the axis of rotational symmetry and an arbitrary axis perpendicular to it . in the phenomenological approach , the lorentzian - type formula is generalized to a sum of two lorentzian - type functions of energies @xmath37 and width @xmath38 @xcite , such that @xmath39 where @xmath40 is the ratio of the diameter along the axis of symmetry to the diameter along an axis perpendicular to it . in turn , the width @xmath38 of each peak is given by the same deformation dependence as the respective energy @xmath37 @xcite . a similar splitting of the resonance strength for deformed nuclei is applied within the srpa procedure given by eq . ( [ loren ] ) , the lorentzian function at a given energy @xmath41 being split with an equal strength into two lorentzian functions centered according to eq . ( [ eq_def ] ) and characterized by a width @xmath42 ( see eq . [ width ] ) obtained from the same relations ( eq . [ eq_def ] ) . as already found in @xcite , distributing the strength equally between the two resonance peaks gives optimal location and relative strength of both gdr centroid energies as observed experimentally . we illustrate in fig . [ fig_u235 ] how the photoabsorption cross section in @xmath43u peaked around 12 mev in the spherical approximation is split into the two observed peaks . the same deformation effects are applied to all nuclei predicted to be deformed by the hfb calculation based on the bsk7 force . for practical astrophysics applications , it is of first importance to describe the tail of the gdr at low energies , i.e around the neutron separation energy , as reliably as possible @xcite . experimental e1 strengths at low energies are available through average resonance capture ( arc ) data @xcite or recent measurements of @xmath44-ray spectra in light - ion reactions @xcite . however , such data are related to the so - called `` downwards '' e1-strength function which determines the average width of the @xmath44-decay , while the photoexcitation data considered so far depend on the `` upwards '' e1-strength function associated with @xmath44-absorption . when dealing with @xmath44-decay data , a temperature - dependent correction factor is traditionally introduced in the expression of the gdr width to take the collision of quasiparticles into account @xcite . in order to guarantee the compatibility with photoabsorption data , we introduce in the srpa procedure such a collision term by adding to the width @xmath42 ( see eq . [ width ] ) a temperature - dependent correction term as @xmath45 \label{tdep}\ ] ] where @xmath46 refers to the temperature of the absorbing state , @xmath28 is the peak energy of the gdr and @xmath47 a normalization constant . in all calculations performed in the present work , the temperature is derived from the microscopic statistical model of nuclear level densities @xcite . as shown below , adopting @xmath48 gives excellent agreement with most of the available data . [ fig_nd144 ] illustrates in the specific case of the spherical @xmath49nd nucleus , that the e1-strength data derived from primary photon spectra in the ( n,@xmath44 ) reaction around 68 mev @xcite or ( n,@xmath50 ) reaction around @xmath51 mev @xcite are correctly reproduced at low energies with the @xmath46-dependent correction given by eq . ( [ tdep ] ) with @xmath48 . the energy dependence of the collision term introduced in eq . ( [ tdep ] ) and already suggested in @xcite is of particular importance , since it is responsible for the @xmath52 behavior of the e1-strength function observed experimentally @xcite . it is also found to affect the e1 strength around the neutron binding energy , as seen in figs . [ fig_nd144][fig_fe1 ] . in addition , we compare in fig . [ fig_fe1 ] the qrpa predictions with the compilation of experimental e1-strength functions at low energies ranging from 4 to 8 mev @xcite for nuclei from @xmath53 up to @xmath54 . the data set includes resolved - resonance measurements , thermal - captures measurements and photonuclear data . in a certain number of cases the original experimental values need to be corrected , typically for non - statistical effects , so that only values recommended by @xcite are considered in fig . [ fig_fe1 ] . qrpa predictions are globally in good agreement with experimental data at low energies in the whole nuclear chart . the average and rms deviations , as defined in eq . ( [ eq_rms ] ) , on the 62 experimental data have been estimated . the @xmath46-independent predictions underpredict the e1 strength by an average factor of 1.6 , while on average the @xmath46-dependent formula ( assuming @xmath48 in eq . [ tdep ] ) is in perfect agreement with the data . the respective rms deviation factors are @xmath55 and 2.1 for the @xmath46-independent and @xmath46-dependent results . these results show that including a @xmath46-dependence in the e1 strength to describe the @xmath44-decay data globally improves the agreement . a qualitative agreement is also obtained with the e1-strength function derived at low energy from primary photon spectra in light - ion reactions @xcite , although the fine structure pattern are not reproduced . large - scale qrpa calculations based on the bsk7 skyrme force have been performed for all @xmath56 nuclei lying between the proton and the neutron driplines , i.e some 8300 nuclei . the srpa is applied to all distributions . in the neutron - deficient region , as well as along the valley of @xmath2-stability , the resulting e1-strength functions are very similar to the empirical lorentzian - like approximation . when dealing with neutron - rich nuclei , the qrpa predictions start deviating from a simple lorentzian shape and results quantitatively similar to @xcite are obtained . in particular , some extra strength is found to be located at an energy lower than the gdr energy . the more exotic the nucleus , the stronger this low - energy component . this is illustrated in fig . [ fig_sn ] for the e1-strength function in the sn isotopic chain . all nuclei shown in fig . [ fig_sn ] are predicted to be spherical in the hfb calculations based on the bsk7 force @xcite . for the @xmath57 neutron - rich isotopes , an important part of the strength is concentrated at low energies ( @xmath58 mev ) . phenomenological models are unable to predict such low energy components . in particular for @xmath59sn , all phenomenological systematics ( as used for cross section calculation ) predict a @xmath44-ray strength peaked around 15 mev with a full width at half maximum of about 4.5 mev @xcite which is obviously very different from the microscopic estimate ( fig . [ fig_sn ] ) . more generally , the present hfb+qrpa calculation confirms that the neutron excess affects the spreading of the isovector dipole strength , as well as the centroid of the strength function . the energy shift is larger than predicted by the usual @xmath60 or @xmath61 dependence given by the phenomenological liquid drop approximations @xcite . the above - described feature of the qrpa e1-strength function for nuclei with a large neutron excess is qualitatively independent of the adopted effective interaction . the radiative neutron capture cross section is estimated within the statistical model of hauser - feshbach making use of the most code @xcite . it should be noted that this version makes use of the nuclear ground state properties derived coherently from the same microscopic hfb method with the bsk7 skyrme force @xcite . it also benefits from the improved nuclear level density prescription based on the microscopic statistical model , also used to estimate the nuclear temperature in eq . ( [ tdep ] ) @xcite . the direct capture contribution as well as the possible overestimate of the statistical predictions for resonance - deficient nuclei are effects that could have an important impact on the radiative neutron captures by exotic nuclei @xcite , but are not included in the present study . the maxwellian - averaged radiative neutron capture rate at a temperature @xmath62 k , typical of the r - process nucleosynthesis , obtained with the qrpa e1-strength are compared in fig . [ fig_rate ] with those based on the hybrid lorentzian - type formula @xcite . these rates are sensitive to the neutron capture cross section at incident energies around 130 kev , and therefore depend on the e1 strength in a narrow range of a few hundred kev around @xmath5 . the temperature - dependent hybrid formula corresponds to a generalization of the energy- and temperature - dependent lorentzian formula including an improved description of the e1-strength function at energies below @xmath5 as derived from @xcite . the hybrid e1 strength differs from the qrpa estimate not only in the location of the centroid energy , but also in the low - energy tail . no extra low - lying strength is included in the phenomenological hybrid formula , but its temperature dependence increases the e1 strength at low energies and is responsible for its non - zero @xmath63 limit . the newly - derived strength gives an increase of the rate by a factor up to 6 close to the neutron drip line . r - process nuclei characterized by @xmath64 mev are seen to have a neutron capture rate about at least twice faster than the one predicted with the phenomenological hybrid formula . this is due to the shift of the gdr to lower energies compared with the usually adopted liquid - drop @xmath61 rule , as well as to the appearance of some extra strength at low energies as explained above . both effects tend to increase the e1 strength at energies below the gdr , i.e in the energy window of relevance in the neutron capture process . for less exotic nuclei , the qrpa impact is relatively small , differences being mainly due to the exact position of the gdr energy and the resulting low - energy tail . when compared to our previous hfbcs+qrpa predictions @xcite , the hfb+qrpa model gives larger neutron capture rates close to the neutron drip line , but lower rates for many of the @xmath65 \lsimeq 2 $ ] nuclei , as seen in fig . [ fig_rate ] ( lower panel ) . these differences justify the use of the hfb approach for exotic neutron - rich nuclei . the e1-strength function is estimated with one of the most accurate and reliable microscopic model available to date , namely the hartree - fock - bogoliubov ( hfb ) and qrpa models . the spreading width of the gdr is taken into account by analogy with the srpa method . the analysis of hfb+qrpa model based on various skyrme forces with different pairing prescriptions and parameterizations shows that the effective nucleon - nucleon interaction can be constrained with the gdr data . in particular , it is found that the skyrme force characterized with a low effective mass @xmath66 is a necessary condition to reproduce the location and width of the gdr , at least within the present hfb+qrpa model to which the srpa is applied . in contrast , gdr data can not be used to discriminate between the surface or volume property of the pairing interaction . in addition to its reliability , it is shown that the hfb+qrpa model also gives accurate predictions and that globally it agrees fairly well with experimental data . the present hfb+qrpa model brings important improvement with respect to our previous hfbcs+qrpa model and can provide quantitative predictions of the dipole strength . large - scale calculations of the e1-strength function are performed and used to estimate the radiative neutron capture rates of relevance for the r - process nucleosynthesis . a systematic increase of the reaction rates for exotic neutron - rich nuclei is found . further improvements may be useful . a proper treatment of the continuum states and its impact on the dipole strength is an important issue . it is expected to be significant for drip - line nuclei . continuum - qrpa models are available @xcite and study along these lines are in progress . the particle - vibration coupling also affects the low - energy strength and could contribute to an extra increase of the radiative neutron capture rate by exotic nuclei . * acknowledgments * m.s . and s.g . are fnrs research fellow and associate , respectively . this work has been performed within the scientific collaboration ( tournesol ) between the wallonie bruxelles community and france . + 100 s. goriely , phys . lett . b436 ( 1998 ) 10 . mccullagh , m.l . stelts , r.e . chrien , phys . c23 ( 1981 ) 1394 . kadmenskii , v.p . markushev , v.i . furman , sov . 37 ( 1983 ) 165 . j. kopecky , r. e. chrien , nucl . phys . a468 ( 1987 ) 285 . myers , w.j . swiatecki , et al . c15 ( 1977 ) 2032 . f. catara , e.g. lanza , m.a nagarajan , a. vitturi , nucl . a624 ( 1997 ) , 449 . s. goriely , e. khan , nucl . a706 ( 2002 ) 10 . e. khan , nguyen van giai , phys . b472 ( 2000 ) 253 . j. dobaczewski , w. nazarewicz , t.r . werner , j.f . berger , c.r . chinn , j. decharge , phys . c53 ( 1996 ) 2809 . e. chabanat , p. bonche , p. haensel , j. meyer , r. schaeffer , nucl . a635 ( 1998 ) 231 . m. grasso , n. sandulescu , nguyen van giai , r. j. liotta , phys . c64 ( 2001 ) 064321 . e. khan , n. sandulescu , nguyen van giai , m. grasso , phys . c66 ( 2002 ) 024309 . s. drod , s. nishizaki , j. speth , j. wambach , phys . 197 ( 1990 ) 1 . dietrich , b. l. berman , at . data nucl . data tables 38 ( 1989 ) 199 . photonuclear data for applications ; cross sections and spectra , iaea - tecdoc-1178 ( 2000 ) . p. ring , p. schuck , _ the nuclear many - body problem , springer - verlag _ ( 1980 ) . m. samyn , s. goriely , p .- h . heenen , j.m . pearson , f. tondeur , nucl . a700 ( 2002 ) 142 . s. goriely , m. samyn , p .- h . heenen , j.m . pearson , f. tondeur , phys . c 66 ( 2002 ) 024326 . m. samyn , s. goriely , j.m . pearson , nucl . phys . a ( 2003 ) submitted . s. goriely , m. samyn , m. bender , j.m . pearson , nucl . a ( 2003 ) in preparation . bckman , a.d . jackson , j. speth , phys . b56 ( 1975 ) 209 . g. col , p.f . bortignon , nucl . a696 ( 2001 ) 427 . p. schuck , s. ayik , nucl . a687 ( 2001 ) 220c . smith , j. wambach , phys . c38 ( 1988 ) 100 . baker , l. bimbot , c. djalali , et al . 289 ( 1997 ) 235 . d. vautherin , j.d.m . brink , phys . c5 ( 1972 ) 626 . nguyen van giai , h. sagawa , nucl . a371 ( 1981 ) 1 . j. kopecky , m. uhl , phys . rev . c41 ( 1990 ) 1941 . a. voinov , m. guttormsen , e. melby , j. rekstad , a. schiller , s. siem , phys . c * 63 * , 044313 ( 2001 ) . s. siem , m. guttormsen , k. ingeberg , e. melby , j. rekstad , a. schiller , a. voinov , phys . c * 65 * , 044318 ( 2002 ) . p.f . bortignon , nucl . a687 ( 2001 ) 329c . p. demetriou , s. goriely , nucl . a695 ( 2001 ) 95 . popov , in proc . conf . , physics and application , vol . p. oblozinsky ) , p. 121 reference input parameter library , iaea - tecdoc , in press ( 2003 ) .

large - scale qrpa calculations of the e1 strength are performed on top of hfb calculations in order to derive the radiative neutron capture cross sections for the whole nuclear chart . the spreading width of the gdr is taken into account by analogy with the second - rpa ( srpa ) method . the accuracy of hfb+qrpa model based on various skyrme forces with different pairing prescription and parameterization is analyzed . it is shown that the present model allows to constrain the effective nucleon - nucleon interaction with the gdr data and to provide quantitative predictions of dipole strengths . 16.2 cm -.54 cm -.54 cm nuclear reactions : qrpa , e1-strength , nuclear forces

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Proceed to summarize the following text: un rsultat classique d brouwer nonce que tout homomorphisme du plan @xmath3 qui prserve lorientation et possde une orbite priodique , possde galement un point fixe . dans le mme ordre dides , on peut montrer @xcite quun tel homomorphisme @xmath0 possde un point fixe li cette orbite priodique , en ce sens quil nexiste pas de courbe de jordan @xmath4 , bordant un disque @xmath5 contenant lorbite priodique mais ne contenant pas le point fixe , et telle que @xmath6 soit homotope @xmath4 dans le complmentaire de lorbite priodique et du point fixe . une question pose par john franks dans @xcite demeure toujours sans rponse : tant donn un homomorphisme @xmath7 prservant lorientation , existe - t - il pour toute orbite priodique de @xmath0 un point fixe ayant un nombre denlacement non nul avec cette orbite priodique ? on sait que la rponse cette question est positive pour les orbites de priode @xmath8 ( voir @xcite ) ou de priode @xmath9 @xcite . dautre part , dans une prpublication rcente , franks @xcite utilise la rponse affirmative cette question comme tant un thorme de handel ( sans rfrence ) : on peut donc supposer que cette question est , soit rsolue , soit en passe de ltre . nous montrons ici quun raisonnement trs simple et trs rapide permet de rpondre par laffirmative la question de franks pour les orbites de toutes les priodes des homomorphismes @xmath0 de @xmath3 pour lesquels @xmath1 vrifie un condition de lipschitz . plus prcisment : soit @xmath7 un homomorphisme qui prserve lorientation et tel que @xmath1 soit lipschitzienne de rapport @xmath10 $ ] . alors , pour toute orbite priodique @xmath11 de @xmath0 , il existe un point fixe de @xmath0 , @xmath12 , ayant un nombre denlacement non nul avec @xmath2 . outre lintrt du rsultat , la simplicit de la preuve met en valeur lavantage quil y a tester sur cette classe ( pas trop petite ) dhomomorphismes , les conjectures concernant les homomorphismes des surfaces . soit @xmath7 un homomorphisme . on note @xmath13 lensemble de ses points fixes . un point @xmath14 est priodique de priode @xmath15 si @xmath16 mais @xmath17 pour @xmath18 . on note @xmath19 lorbite de @xmath20 sous @xmath0 . soit @xmath21 , @xmath20 un point priodique de priode @xmath15 et @xmath22 un arc joignant @xmath20 et @xmath23 dans @xmath24 , on note @xmath25 la courbe ferme obtenue en joignant bout bout les arcs , @xmath26 . on note @xmath27 le nombre denroulement de @xmath25 autour de @xmath12 cest dire le nombre dintersection algbrique dune demi - droite gnrique issue de @xmath12 avec @xmath25 ( ce nombre est souvent appel indice de @xmath12 par rapport @xmath25 ) . [ lem1 ] soient @xmath22 et @xmath28 deux arcs quelconques joignant @xmath20 et @xmath23 dans @xmath29 . si @xmath0 prserve lorientation , alors @xmath30 . on a @xmath31 or si @xmath0 prserve lorientation @xmath32 do @xmath33 . on peut montrer galement que la valeur de @xmath34 ( mod @xmath15 ) ne dpend pas du choix du point @xmath20 de @xmath2 choisi pour le construire . avec les notations ci - dessus , on note @xmath35 lunique entier @xmath36 tel que @xmath37 pour un choix quelconque de @xmath22 et on lappelle le _ nombre denlacement _ ( ou _ linking number _ ) du point fixe @xmath12 avec lorbite priodique @xmath2 . lappellation nombre denlacement est justifie par la remarque suivante : @xmath38 est aussi le nombre denlacement des deux orbites fermes @xmath39 et @xmath40 du champ de vecteurs canonique induit dans la suspension @xmath41 de @xmath0 . le point fixe de la rotation dangle @xmath42 a pour nombre denlacement @xmath43 avec lune quelconque de ses orbites priodiques . soit @xmath44 une application continue . si lensemble @xmath45 est non vide , on note @xmath46 sa borne infrieure . sinon , on pose @xmath47 . soient @xmath20 et @xmath48 deux points de @xmath3 , on note @xmath49 $ ] le segment de droite joignant @xmath20 et @xmath48 . [ lem2 ] soit @xmath7 un homomorphisme tel que @xmath50 et @xmath51 . alors @xmath52=\emptyset$ ] . par labsurde , supposons quil existe @xmath53 . on a alors : @xmath54 dautre part @xmath55 car @xmath0 est injective , ce qui conclut . [ lem3 ] soit @xmath56 un homomorphisme tel que @xmath57 et @xmath51 . alors , @xmath58)$ ] et @xmath59 $ ] sont homotopes relativement @xmath23 , @xmath60 dans @xmath61 . soit @xmath62^{2}\rightarrow { \mathbb r}^{2}$ ] dfinie par : @xmath63 limage de @xmath64 est la runion des segments de droite @xmath65(y\in[x , f(x)])$ ] . daprs le lemme [ lem2 ] @xmath66 $ ] nest pas un point fixe de @xmath0 , donc @xmath65\subset { \mathbb r}^{2}\setminus fix(f)$ ] . do @xmath67 . dautre part , @xmath68 . on obtient donc une application du disque : @xmath69^{2}/(s , 0)\sim(0 , s)-{\mathbb r}^{2}\setminus fix(f)\ ] ] telle que @xmath70\cup f([x , f(x)])$ ] . on a ainsi ralis lhomotopie . soient @xmath71 , @xmath72 deux vecteurs de @xmath3 . on note @xmath73 $ ] langle non orient des vecteurs @xmath71 et @xmath72 . [ lem4 ] soit @xmath7 un homomorphisme tel que @xmath50 et @xmath51 . alors pour tout @xmath66 $ ] : @xmath74 on a : @xmath75 et par suite : @xmath76 . si langle est @xmath77 , cela implique @xmath78 ce qui est impossible en vertu du lemme [ lem2 ] . soit @xmath79 lorbite dun point priodique de priode @xmath15 dun homomorphisme de @xmath3 qui prserve lorientation et tel que @xmath50 . soit @xmath80 la courbe polygonale obtenue en joignant bout bout les segments @xmath81 $ ] , @xmath59 $ ] , @xmath82 , @xmath83 $ ] ( voir figure [ fig1 ] ) et soit @xmath84 $ ] . lassertion ( 1 ) rsulte du lemme [ lem2 ] . en raisonnant par rcurrence et en utilisant le lemme [ lem3 ] , on tablit que @xmath80 est homotope @xmath25 dans @xmath24 , do lgalit @xmath87 . par ailleurs , le nombre denroulement de @xmath80 par rapport @xmath12 est aussi le nombre algbrique de croisements dune demi - droite issue de @xmath12 avec @xmath80 . ce nombre est donc ncessairement infrieur @xmath88 en valeur absolue . soient @xmath89 les composantes connexes bornes de @xmath90 ( remarquer quil en existe au moins une , sinon @xmath80 serait rduit un segment de droite ce qui est exclu en vertu du lemme [ lem4 ] ) et @xmath91 la composante connexe non borne . si @xmath92 on dfinit lindice de @xmath93 en posant : @xmath94 o @xmath95 dsigne une dtermination continue de langle du vecteur @xmath96 avec une direction fixe et o @xmath97 est le bord orient de @xmath93 . dans chaque composante dindice non nul il existe au moins un point fixe de @xmath0 . soient @xmath98 les sommets de @xmath97 et @xmath99 ses arrtes munies de lorientation induite par celle de @xmath80 . en un sommet @xmath100 , il y a deux configurations possibles quant lorientation des artes adjacentes @xmath100 : ou bien ces deux arrtes ont des orientations compatibles , ou bien il y a un changement dorientation ( voir figure [ fig2 ] ) . on a : @xmath103 o @xmath104 et @xmath105 sont les valeurs respectives en @xmath100 et @xmath106 ( @xmath107 ) dune dtermination continue @xmath108 de langle @xmath109 ( @xmath110 ) avec la tangente oriente @xmath111 . il rsulte alors du lemme [ lem4 ] que @xmath112 $ ] ( autrement dit le vecteur @xmath109 ne dcrit pas de tour complet lorsque @xmath48 parcourt @xmath111 ) . en dsignant alors par @xmath113 $ ] langle intrieur @xmath93 en @xmath100 , on a ( voir figure [ fig3 ] ) : @xmath114 par suite : @xmath115\\ & = \frac{1}{2\pi}\left[\sum_{i=0}^{m-1}(\varphi_{i-1}^{1}-\varphi_{i}^{0})\right]\\ & = \frac{1}{2\pi}\left[\sum_{i=0}^{m-1}(\pi-\beta_{i})-2p\pi\right]\\ & = 1-p.\end{aligned}\ ] ] nous pouvons maintenant tablir le thorme . le nombre denroulement @xmath116 dun point @xmath117 ne dpendant que de la composante @xmath93 laquelle il appartient , on notera @xmath118 cette valeur commune . il nous reste donc tablir lexistence dune composante @xmath119 de @xmath90 telle que : @xmath120 remarquons dabord que @xmath124 et que @xmath125 pour toute composante @xmath119 adjacente @xmath91 . il existe donc bien @xmath126 tel que : @xmath127 par labsurde , supposons que @xmath128 . daprs le lemme [ lem6 ] , il existe alors au moins un changement dorientation en un des sommets @xmath129 de @xmath130 . alors lune des composantes @xmath131 adjacente @xmath132 en @xmath129 vrifie @xmath133 et lautre @xmath134 vrifie @xmath135 ( voir figure [ fig4 ] ) . en effet , les valeurs de @xmath136 et @xmath137 ne dpendent que de lorientation avec laquelle on franchit @xmath80 pour passer de @xmath132 @xmath131 et @xmath134 . il existe donc @xmath138 ( @xmath139 ) tel que : @xmath140 ce qui contredit lhypothse faite sur @xmath141 .

soit @xmath0 un homomorphisme du plan qui prserve lorientation et tel que @xmath1 soit une contraction . sous ces hypothses , on tablit lexistence , pour toute orbite priodique @xmath2 , dun point fixe ayant un nombre denlacement non nul avec @xmath2 .

You are an expert at summarizing long articles. Proceed to summarize the following text: over the past decade , michelson stellar interferometry has seen some tremendous advances in applicability . it has evolved from a primarily experimental technique towards a general astrophysical observational mode extensively used by the community for galactic and extragalactic science ( see the various contributions in these proceedings ) . observations with milli arcseconds angular resolution are now routinely performed in the near infrared . strategic planning of next generation michelson interferometers should aim for higher resolution , higher sensitivity and image reconstruction capabilities ( synthesis or direct imaging ) . an alternative technique to michelson interferometry that has the potential of delivering similar scientific products is the technique of intensity interferometry ( ii ) . it is able to attain @xmath0-arcseconds resolution and provide image synthesis capacity and can be implemented in a cost effective and straightforward instrument . the principle of ii is based on the partial correlation of intensity fluctuations of coherent light beams measured at different points in space or in time . the fluctuations ( also called wave - noise , although `` noise '' is a bit of a misnomer ) was a well - known phenomenon at radio wavelengths , but its nature and even reality at optical wavelengths were seriously debated in the 1950s . hanbury brown & twiss @xcite conclusively demonstrated in a series of papers how the phenomenon is firmly rooted in both theory and experiment . the intensity fluctuations can be interpreted in a semi - classical sense as the superposition of light waves at different frequencies producing beats . the fluctuations can also be interpreted in a quantum mechanical sense as an effect related to so - called photon bunching @xcite . at optical wavelengths , the resulting fluctuations are much smaller than the shot - noise component of a ( recorded ) light beam . given that shot - noise is random , any coherence of different light beams is solely determined by the intensity fluctuation effect . hanbury brown et al . @xcite measured the stellar diameter for 32 stars using a dedicated stellar intensity interferometer stationed at narrabri , australia . their pioneering experiments constituted the first successful stellar diameter measurements after the michelson & pease experiments . we will discuss in this contribution the potential of future cherenkov telescope arrays for a revival of intensity interferometry as a mainstream high angular resolution imaging technique in astronomy . all modern interferometers in operation are based on the principle of michelson stellar interferometry . it provides a measure of the spatial coherence of two light beams by using the fringe contrast ( visibility ) of the beams interference pattern . the technique requires highly accurate metrology to correct for path differences between the beams , which is on the order of the wavelength of light itself . active correction of the optics ( adaptive optics , tip - tilt ) are required in order to make the otherwise corrugated wave front planar . light beams are made to interfere , and allow in principle the determination of the modulus and phase of the complex visibility . in practise however , atmospheric limitations corrupt the phase ( the fringes wander over the detector ) , and in broad - band interferometric observation a quadratic visibility estimator ( @xmath2 ) is generally employed eliminating the corrupted phase , but at the same time introducing a bias in the estimator that needs to be taken care of , e.g. , @xcite . in spectrally resolved observations a linear visibility estimator can be employed , thanks to the achromatic nature of the atmospheric disturbances , and relative phase can be retrieved . absolute phase information with ground based interferometers can be obtained by simultaneously observing a phase reference star @xcite , or by exploiting the principle of closure phase when working with more than two telescopes , see @xcite . intensity interferometry provides a measure of the square of the michelson fringe visibility . this fundamental difference is the basis for some of its technical advantages . ii is virtually insensitive to atmospheric and instrumental instabilities as no actual waves are interfered . the technique can thus make do with relatively coarse light collectors and long baselines . the signals from each receiver can be electronically recorded and correlated after detection . the correlation of the multiple signals measured by a telescope array can thus all be determined for all possible pairs , and maybe even higher order correlations can be exploited . the coherence length is set by the length of the frequency beats ( wave group ) which is of order cm rather than nm ( depending on bandwith ) , alleviating the strong constraints on accurate delay tracking . ii is however less sensitive in measuring coherence than michelson interferometry , as the sought fluctuations are very small relative to the shot - noise of the photon stream . however , without stringent requirements for the light collectors , the decreased sensitivity can be offset by large coarse mirrors in order to maximise photon collection . in addition @xcite have shown that under certain conditions higher order correlations will increase the sensitivity ( see @xcite ) . a second drawback is no phase information is contained within the measurement . although this does not impede the measurement of centro - symmetric objects ( indeed as was done with the narrabri stellar intensity interferometer ) , it has been shown that the phase can be recovered using correlations between more than two signals @xcite . also superimposing a coherent beam from a known reference source on the light beam of the target source would allow the recovery of the phase @xcite , see also @xcite . imaging air cherenkov telescope ( iact ) arrays are multi - telescope arrays designed to image air showers that are produced by high energy particles and @xmath3-ray photons ( @xmath4 gev ) that impinge on the earth s atmosphere . the air showers consist of secondary particles some of which carry electrical charge and produce cherenkov light . as the cherenkov radiation is faint ( depending on the energy of the incident particle / photon ) , large collecting areas are required to obtain decent signal - to - noise ratios . the flash of cherenkov radiation produced is also very brief ( a few nanoseconds ) and fast photon counting detectors are therefore mandatory . the advantage of employing telescope arrays in air - shower observations is the ability to reconstruct the spatial geometry of the shower in the earth s atmosphere . this allows one to make a distinction between cosmic ray showers and @xmath3-ray showers , and also to find the spatial direction of the incident particle / photon , thus obtaining angular information on the galactic or extragalactic source . in short , iacts aim to determine the energy and the astrophysical source of @xmath3-ray photons by observing the optical light of induced cherenkov radiation . it should be clear that the basic technical specifications of an iact array as touched upon here are very similar to the ones of an intensity interferometer as briefly described in the previous section . a more detailed description of the components of iact arrays relevant for ii ( fast photon detection , signal communication , correlators ) can be found in @xcite . currently , there are two major and successful iact arrays operational . the h.e.s.s . array @xcite consists of 4 telescopes of 108m@xmath5 each . they are arranged in a square with 120 m side length . the veritas @xcite array is similar and also consists of 4 telescope with 110m@xmath5 light collecting area . in contrast to hess , it has variable baselines ranging from 34 to 109 metres . hess will be upgraded to hess ii with one extra 30 m dish telescope placed at the centre of the configuration , its completion foreseen for the 4th quarter of 2009 . future iact projects are currently under study , such as agis[multiblock footnote omitted ] and cta[multiblock footnote omitted ] , and aim at increasing the number of telescopes up to 100 , each with a @xmath6m@xmath5 light collectors arranged over an area covering 1km@xmath5 or more . these envisioned projects ( expected to be operational around 2015 ) would not only benefit the resolving power and sensitivity for the high - energy science purposes of iacts , but they are in harmony with the requirements of a more sensitive intensity interferometer with a higher angular resolution and denser ( u , v)-coverage @xcite . future iacts will offer thousands of baselines from 50 m to at least 1 km and when used as an intensity interferometer may attain angular resolution down to a few tens of @xmath0-arcseconds . _ left : _ proposed lay - out for the future cta . small red dots are the 85 @xmath7 dishes , large blue dots are the four @xmath8 dishes ( adapted from @xcite ) . _ sensitivity estimate as function of the 48 non - redundant baselines for a 5@xmath9 detection , in a 5 hr integration on a centro - symmetric object with 50% visibility . final values depend on signal bandwith and cta design details , see @xcite.,title="fig:",width=226,height=226 ] _ left : _ proposed lay - out for the future cta . small red dots are the 85 @xmath7 dishes , large blue dots are the four @xmath8 dishes ( adapted from @xcite ) . _ sensitivity estimate as function of the 48 non - redundant baselines for a 5@xmath9 detection , in a 5 hr integration on a centro - symmetric object with 50% visibility . final values depend on signal bandwith and cta design details , see @xcite.,title="fig:",width=302,height=226 ] limiting @xmath10 of the cta concept is illustrated in the right panel of fig.1 with simulated performance that meet the goal sensitivity for @xmath3-ray astronomy @xcite . targets are limited to a @xmath11 for a s / n = 5 , and a 5 hours integration in case of 50% visibility ( see @xcite ) . these specifications allow important interferometry studies regarding binary stars , stellar radii and pulsating stars with unprecedented resolution on @xmath0-arcsecond scales . we highlight three potential science cases below . [ [ star - formation ] ] * star formation * + + + + + + + + + + + + + + + + key questions relating to the physics of mass accretion and pre - main sequence ( pms ) evolution can be addressed by means of very high resolution imaging as provided by the next generation iacts and ii . they involve the absolute calibration of pms tracks , the mass accretion process , continuum emission variability , and stellar magnetic activity . the technique may allow us to the resolve spot features on the stellar surface . hot spots deliver direct information regarding the accretion of material onto the stellar surface . cool spots , on the other hand , may cover 50% of the stellar surface , and they are the product of the slowly decaying rapid rotation of young stars . imaging them will constrain ideas regarding the interplay of rotation , convection , and chromospheric activity as traced by cool spots . it may provide direct practical application as the explanation for the anomalous photometry observed in young stars @xcite . in practise , about 50 young stars with @xmath12 are within reach of future iacts . in the last decade several young coeval stellar groups have been discovered in close proximity ( @xmath1350pc ) to the sun . famous examples are the tw hydra and @xmath14 pic comoving groups . the majority of the spectral types within reach range between a and g - type . their ages lie within the range 8 to 50 myr ( see for an overview ) . the age intervals ensures that a substantial fraction of the low - mass members are still in their pms contraction phase . measurement of their angular size can be used in the calibration of evolutionary tracks , fundamental in deriving the properties of star forming regions and young stellar clusters . the proximity of the comoving groups ensures that their members are bright . their proximity renders the comoving group also relatively sparse making them very suitable , unconfused targets despite the large optical psf of a few arcminutes . the sparseness is also the reason for incomplete group memberships , making it likely that the number of young stars close to the sun will increase with the years to come . [ [ distance - scale - and - pulsating - stars ] ] * distance scale and pulsating stars * + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + measuring diameters of cepheids is a basic method with far reaching implications . a radius estimate of a cepheid can be obtained using the baade - wesselink method . the baade - wesselink method relies on the measurement of the ratio of the star size at times @xmath15 and @xmath16 , based on the luminosity and colour . combined with a simultaneous measurement of the radial velocity , this method delivers the difference in the radius between @xmath15 and @xmath16 . with the known difference and ratio of the radius at two times , one can derive the radius of the cepheid . combining ii angular size measurement with the radius estimate one obtains the distance to the cepheid ( see @xcite ) . this makes possible the calibration of the all important cepheid period - luminosity relation using local cepheids . a count of cepheids observed with _ hipparcos _ shows that at least 60 cepheids with @xmath17 are in reach with future iacts . [ [ rapidly - rotating - stars ] ] * rapidly rotating stars * + + + + + + + + + + + + + + + + + + + + + + + + as a group , classical be stars are particularly well - known for their close to break - up rotational velocities as deduced from photospheric absorption lines . in addition the stars show balmer line emission firmly established as due to gaseous circumstellar disks , that appear and disappear on timescales of months to years . these two properties are somehow related , but many open questions regarding the detailed physical processes at play exist . the be - phenomenon is an important phenomenon given the number of stars and stellar physics involved ( fraction of be stars to normal b - type peaks at nearly 50% for b0 stars , ) . absorption lines will however never provide the final answer to their actual rotational velocity due to strong gravity darkening at the equator and brightening at the pole areas . direct measurement of the shape of the rotating star is not hampered by gravity darkening , and provides a direct indication of the rotational speed ( see , e.g. , @xmath18 eri with the vlti , ) . the be star disk formation and dissolution activity is little understood . photometric observations of be star disks seem to indicate that they may actually evolve into ring structures before disappearing into the interstellar medium ( e.g. ) . the disk s bremsstrahlung can constitute @xmath19 of the total light in @xmath20-band . there are about 300 be stars[multiblock footnote omitted ] brighter than @xmath21 , roughly corresponding to a distance limit of 700pc . signifying that be star phenomena can be probed in depth with iact based ii . technical requirements for a relatively sensitive and @xmath0-arcsecond resolution synthesis intensity interferometer go hand in hand with the designs for next generation air cherenkov telescope arrays . this prompts close study of the possibilities in designing and building a iact instrument that incorporates the ii capability . in principle there is no competition between the two modes , as @xmath3-ray observations need to be executed when the moon is less than half full . this leaves half the available night time of an iact for interferometric observations . the renewed interest in ii has resulted in the formation of an iau working group on intensity interferometry . laboratory experiments are performed to test and demonstrate the various aspects of ii integrated in a iact . a pair of 3-m telescopes is now available in utah , usa ( star base utah 2008[multiblock footnote omitted ] ) for testing the techniques in a realistic astronomical environment . in conclusion , the time is now to assess the applicability of ii as a future high resolution imaging mode that allows science to enter the realm of astrophysics on @xmath0-arcsecond angular scales . this contribution is partly based on two presentations @xcite given at the meeting `` the universe at sub - second timescales '' held in edinburgh , september 2007 . 10 url # 1#1urlprefix[2][]#2 r and twiss r q 1956 _ nature _ * 178 * 1046 s , daniel m , de wit w j , hinton j a , jose e , holder j a , smith j and white r j 2008 _ high time resolution astrophysics : the universe at sub - second timescales _ ( _ american institute of physics conference series _ vol 984 ) p 205 w j , le bohec s , hinton j a , white r j , daniel m k and holder j 2008 _ high time resolution astrophysics : the universe at sub - second timescales _ ( _ american institute of physics conference series _ vol 984 ) p 268

in this poster contribution we highlight the equivalence between an imaging air cherenkov telescope ( iact ) array and an intensity interferometer for a range of technical requirements . we touch on the differences between a michelson and an intensity interferometer and give a brief overview of the current iact arrays , their upgrades and next generation concepts ( cta , agis , completion 2015 ) . the latter are foreseen to include 30 - 90 telescopes that will provide 400 - 4000 different baselines that range in length between 50 m and a kilometre . intensity interferometry with such arrays of telescopes attains 50@xmath0-arcseconds resolution for a limiting @xmath1 . this technique opens the possibility of a wide range of studies , amongst others , probing the stellar surface activity and the dynamic au scale circumstellar environment of stars in various crucial evolutionary stages . here we discuss possibilities for using iact arrays as optical intensity interferometers .

You are an expert at summarizing long articles. Proceed to summarize the following text: ultraluminous and luminous infrared galaxies ( ulirgs / lirgs ) have strong infrared ( ir ) luminosities of @xmath5 and @xmath6 , respectively @xcite . their intense ir luminosities are mainly due to starburst activities , and the contribution of active galactic nuclei ( agn ) to ir luminosity seems to be rather limited , i.e. , @xmath7 @xcite . the starburst activities in ( u)lirgs are likely to be triggered by mergings of galaxies , in particular gas - rich galaxies , since observations suggest that ( u)lirgs have complex and disturbed morphologies @xcite . in the @xmath8-band ( f814w filter ) images of 100 sampled ( u)lirgs at @xmath9 taken with _ hubble space telescope _ ( _ hst _ ) , about @xmath10 of ulirgs have multiple ( @xmath0 ) nuclei @xcite . the fraction increases to more than @xmath11 , if all probable galaxies are included @xcite . the origin of the multiple nuclei has been thought to be multiple major merger events . in order to explain multiple mergers , @xcite argued that the progenitors of ( u)lirgs , were compact groups of galaxies . however , the evolution timescale of compact groups is very long ( @xmath12 ) @xcite . therefore , the probability that two galaxies merge in a compact group and the merging galaxy has double cores is @xmath13 @xmath14 @xmath15 , where @xmath16 is the merging timescale of the galactic cores . the probability that another galaxy merges with such merging galaxy in the compact group and the merging galaxy has triple cores is @xmath17 , where @xmath18 ( @xmath19 ) is the typical number of galaxies in a compact group @xcite . then , the number density of ulirgs with multiple cores is @xmath20 @xmath21 , where @xmath22 ( @xmath23 ) is the number density of galaixes in a compact group @xcite . since the number density of ulirgs is @xmath24 @xcite , the fraction of multiple merger to ulirgs is @xmath25 . thus , it is unliky that multiple merger is a major cause of the observed multiple nuclei in ( u)lirgs . an alternative explanation for the origin of multiple nuclei in ( u)lirgs seems necessary . previous numerical simulations of merging galaxies @xcite have shown that starbursts occurred during the merging process . however , these simulations failed to reproduce formation of star clusters as observed in many interacting and merging galaxies @xcite , since their spatial and mass resolutions ( _ e.g. _ , @xmath26 and @xmath27 ) were not sufficiently high to distinguish star forming region and formation of star clusters whose sizes are less than @xmath26 . in addition , previous simulations did not allow the interstellar medium ( ism ) to cool below @xmath28 . this is another reason why formation of star clusters was not reproduced in the previous simulations . in order to reproduce formation of star clusters , high resolution simulations have been attempted . @xcite have performed the simulation of a merging galaxy using tree+sph code gadget . in this simulation , mass and spatial resolutions were @xmath29 and @xmath30 , respectively . they assumed an isothermal ism and used sink particles , that absorb their surrounding gas , to represent clusters . they have shown that a number of massive clusters form in the merging process . in this simulation , the most massive cluster has mass of @xmath31 . @xcite have performed the simulations of merging galaxies using sticky particles , that collide inelastically , instead of solving hydrodynamics . in these simulations , mass and spatial resolutions were @xmath32 and @xmath33 , respectively . they have also shown formations of massive star clusters with masses of @xmath34 . these simulations are , however , unrealistic in a sense that dynamical evolution of star clusters themselves can not be properly followed due to the limitation of the sticky and sink particle methods . @xcite improved the spatial and mass resolutions ( @xmath35-@xmath36 and @xmath37 ) and the ism model that is allowed to cool to @xmath38 . these simulations showed that the behavior of the multiphase ism in the merging galaxies is considerably altered and the formation of shock - induced star clusters is naturally reproduced @xcite . there are several high resolution simulations of merging galaxies , which resolve the low temperature gas ( @xmath39-@xmath40 ) using adaptive mesh refinement ( amr ) methods @xcite . these simulations also showed the difference in the behavior of the multiphase ism from that of ism used in the previous simulations . @xcite considered the merger of low - mass galaxies ( @xmath41 ) . several spiky peaks of star formation rate were seen in thier simulation although the most prominet starburst was found at the beginning the simualtions and the significant fraction of the ism was blown away by energetic wind before the merging event . thus , it would be hard to investigate detailed evolution of the ism in merging galaxies . in @xcite , the polytropic equation of state was used instead of solving the energy equation , which is essentially different from @xcite and present paper . the validity of this approximation is unclear for understanding the evolution of the ism in the merging galaxies . high - resolution images of the local ( u)lirgs obtained by the integral field spectroscopy using _ william herschel telescope _ @xcite and _ vlt - vimos _ @xcite showed that ( u)lirgs generally have very complex structures , such as h@xmath42 bright knots , rings , and tidal tails . as in the case of formation of star clusters , the reason why these structures have not been reproduced in numerical simulations might be just the inadequate treatment of ism and limited resolution . thus , high resolution simulations resolving multiphase ism are essential to comprehend the complex structures in ( u)lirgs . we have performed high - resolution simulations of merging galaxies with sufficiently high spatial resolution and cooling function of ism that covers a wide range of temperature ( @xmath43 ) . in this paper , we focus on the origin of the observed ulirgs with multiple nuclei . the detailed evolution of merging galaxies will be described in forthcoming papers . the structure of this paper is as follows . we first describe the numerical methods and models in [ sec : methods ] . numerical results are shown in [ sec : results ] . in [ sec : discussion ] , we compare our results with observations . conclusions and discussion are presented in [ sec : conclusions ] . the model parameters of the initial disk galaxy are the same as those used in @xcite . the masses of the dark matter halo and old stellar and gas disks are @xmath44 , @xmath45 , and @xmath46 , respectively . the gas fraction in the disk is @xmath47% of the total disk mass . the gas fraction is set to be slightly higher than that of local spiral galaxies since the gas fraction decreases due to star formations during the isolated phase for @xmath48 before interactions ( see below ) . we also assume the halo gas component of which total mass is @xmath49 . it has the same distribution as the dark matter halo . the initial gas temperatures are @xmath50 in the disk and @xmath51 in the halo . the scale radii of the stellar and gas disks are @xmath52 and @xmath53 , respectively . the dark - matter halo and the stellar disk are expressed by @xmath54-body particles , whereas the gas disk is expressed by smoothed particle hydrodynamics ( sph ) particles . unlike @xcite , before starting the simulations , we let an isolated disk galaxy evolve for @xmath48 in order to stabilize the gas component . after this evolution , the amount of disk and halo gas in each galaxy is @xmath55 . the sfr during this period is @xmath56 . during the first @xmath48 , the disk galaxy holds a quasi - steady state and is free from a global instability due to gravity . the evolution of the isolated disk is different from that of a gas rich disk at high-@xmath57 . simulations of the gas - rich disk , in which gas mass fraction of a disk is @xmath58 % , have shown that massive clumps form from gravitational instabillity of the gas rich disk @xcite . in contrast to the simulations of merging of such clumpy disks at high - z by @xcite , we focus on the merging process of two local disk galaxies . we let the pre - evolved two galaxies collide in a parabolic orbit with the pericentric distance of @xmath59 . the simulations start with the initial separation of @xmath60 , where the separation distance is measured between the mass centers of the two galaxies . we have performed nine runs . three of them are high - resolution runs ( @xmath61 in tab . [ param_tab ] ) and the rests are low - resolultion runs ( @xmath62 ) . the initial masses of sph , star , and dark matter particles are the same . they are @xmath63 and @xmath64 , for @xmath61 and @xmath62 runs , respectively . subscripts p , t , and r denote prograde , tilted , and retrograde spin axes of two galaxies . we performed six possible combinations of the spin orientations in low resolution and only pp and tt cases in high resolution . the gravitational softening length , @xmath65 , is set to be @xmath36 for eight runs except for @xmath66 run in which the softening length is @xmath67 . the orientation of the disk axis is specified by two angles , @xmath68 and @xmath69 . here , @xmath68 is the inclination and @xmath69 is the argument of pericenter . they are defined as follows . we use the coordinate system in which the orbital plane of two galaxies is in the @xmath70-@xmath71 plane . the laplace - runge - lenz vector of the orbit is @xmath72 . the spin axis of a galaxy is then given by @xmath73 . these angles are defined relative to the orbit of each galaxy . thus , in the case of `` tt '' runs , spin axes of two galaxies are parallel to each other . lccccccccc run & @xmath74}$ ] & @xmath75}$ ] & @xmath76}$ ] & @xmath77}$ ] & @xmath78 $ ] & @xmath79 & @xmath80 & @xmath81 & @xmath82 $ ] + @xmath83 & @xmath84 & - & @xmath84 & - & @xmath85 & @xmath86 & @xmath87 & @xmath88 & 20 + @xmath89 & @xmath90 & @xmath91 & @xmath92 & @xmath91 & @xmath85 & @xmath86 & @xmath87 & @xmath88 & 20 + @xmath93 & @xmath84 & - & @xmath84 & - & @xmath94 & @xmath95 & @xmath96 & @xmath97 & 20 + @xmath98 & @xmath84 & - & @xmath99 & - & @xmath94 & @xmath95 & @xmath96 & @xmath97 & 20 + @xmath100 & @xmath99 & - & @xmath99 & - & @xmath94 & @xmath95 & @xmath96 & @xmath97 & 20 + @xmath101 & @xmath84 & - & @xmath92 & @xmath102 & @xmath94 & @xmath95 & @xmath96 & @xmath97 & 20 + @xmath103 & @xmath90 & @xmath102 & @xmath99 & - & @xmath94 & @xmath95 & @xmath96 & @xmath97 & 20 + @xmath104 & @xmath90 & @xmath91 & @xmath92 & @xmath91 & @xmath94 & @xmath95 & @xmath96 & @xmath97 & 20 + @xmath66 & @xmath90 & @xmath91 & @xmath92 & @xmath91 & @xmath85 & @xmath86 & @xmath87 & @xmath88 & 5 + @xmath105 the disk inclination for the galaxy @xmath106 . @xmath107 the argument of the pericenter for the galaxy @xmath106 . @xmath108 the disk inclination for the galaxy @xmath109 . @xmath110 the argument of the pericenter for the galaxy @xmath109 . @xmath111 the mass of sph , star , and dark matter particles . @xmath112 the number of sph particles . @xmath113 the number of star particles . @xmath114 the number of dark matter particles . @xmath115 the gravitational softening length . numerical simulations were performed by an @xmath54-body / sph code asura @xcite that can optionally utilize grape @xcite in order to accelerate calculation of gravitational force . we adopted a time - step limiter @xcite for the time - integration of sph particles with individual time - steps , which keeps the differences in time - steps of neighboring particles to be small enough to handle the strong shocks correctly . we also used the fast scheme @xcite that allows self - gravitating sph particles to use different time - steps for integrations of hydrodynamics and gravity . this method accelerates simulations without losing the accuracy of the time - integration . the leapfrog method is used for the time integrations of both gravity and hydrodynamics . these improvements allow us to follow the formation of cold and dense gas clumps and their expansion by supernovae ( sne ) feedback without numerical problems . for the radiative cooling and heating , we follow the treatment of @xcite , in which the gas is allowed to cool down to @xmath38 . the treatment of star formation and sn feedback are also the same as those in @xcite . when an sph particle satisfied ( 1 ) high number density ( @xmath116 ) , ( 2 ) a low temperature ( @xmath117 ) , and ( 3 ) a collapsing region ( @xmath118 ) , the sph particle spawns a star particle of which mass is one - third of the original sph particle mass . we assume simple stellar polulation approximation for newly formed star particles . salpeter s initial mass function with mass range of @xmath119 is adopted @xcite . stars with mass @xmath120 in each star particle explode as type ii sne and release @xmath121 of thermal energy per one sn into the surrounding ism . initially , the two galaxies move on the parabolic orbits and approach to each other . at around @xmath122 , they reach the pericenter ( first encounter ) . since orbital angular momenta of their main bodies are converted into the disk internal spin , they lose their orbital angular momenta @xcite and their trajectories deviate from the parabolic orbits . the main bodies approach to each other again ( second encounter ) . finally , after several encounters , their cores completely merge . figures [ snapshot_hpp ] and [ snapshot_htt ] show the evolution of distribution of gas and stellar particles for runs @xmath83 and @xmath89 . gas disks are strongly disturbed at the first encounter ( the panels at @xmath123 , @xmath124 , and @xmath125 ) and at the second encounter ( the panels at @xmath126 and @xmath127 ) in run @xmath83 and at the second encounter ( the panels at @xmath128 and @xmath129 ) in run @xmath89 . because of the strong perturbation , gas is compressed by large - scale shocks . the dense gas radiatively cools and becomes gravitationally unstable . as a result , massive star clusters form ( see also * ? ? ? * ; * ? ? ? figure [ image ] shows the synthesized @xmath8-band images from our merger simulations after the massive star clusters formed . here , the @xmath8-band luminosity emitted from newly formed stars was calculated using population synthesis code pgase @xcite , and effects of dust absorption and re - emission were neglected . the two top panels show the high resolution runs , and the six lower panels show the low resolution runs . in all images , multiple compact sources appear in the central regions of galaxies . two of them with arrows are the progenitor galactic cores , and the others are newly - formed massive star clusters . the absolute magnitudes of these clusters in @xmath8-band are @xmath130 , which are comparable to those of the galactic central cores . the effect of dust extinction is discussed in 4.2 . since the cores form through the gravitational instability of strongly perturbed gas disks , they do not contain dark matter and consist only of young stars . on the other hand , the galactic luminous nuclei contain dark matter and a large amount of old stars that have formed before the merger event . the properties of newly formed cores are , therefore , different from those of the galactic nuclei . in addition , such cores are more massive than usual star clusters with @xmath131 , by an order of magnitude . we , therefore , call the newly formed cores `` hypermassive star clusters '' . since the hypermassive star clusters form in the galactic central region ( @xmath3 ) , they are different from tidal dwarves that are formed in tidal arms ( e.g. , * ? ? ? * ; * ? ? ? the reason why they are extremely massive can be explained as follows . the total mass of the gas which becomes gravitationally unstable is very large ( @xmath132 ) because of the strong large - scale gravitational and hydrodynamical disturbances from another galaxy at the encounter phase . the strong disturbances deform the gravitational potential into strongly non - axisymmentric shape . therefore , the rotation of the disk becomes ineffective in stabilizing long - wavelength perturbations . even though the gas disk fragments to a number of small clumps with @xmath133 , these small clumps are still gravitationally bound to each other and eventually merge to form larger clusters . figure [ snapshot ] shows the snapshots during the formation of the most massive cluster from @xmath134 to @xmath135 in run @xmath89 . multiple gas clumps formed from the disturbed gas disk merge with each other and become one massive cluster . in other words , the merging of smaller clumps within the short timescale causes quick growth of the mass of star clusters . as a result , hypermassive star clusters form . in fig . [ mass_evolution ] , we show the evolution of the masses of these hypermassive star clusters ( upper panels ) , and their distances from the center of the galaxy ( lower panels ) . here , the definition of the distance is the smaller of the two distances from the two galactic centers . the clusters first grow quickly , reaching about a half of the final mass in around @xmath136 . after that , the growth of the mass slows down , and the clusters fall to the center of the galaxy due to the dynamical friction . as a result , the merger remnant has a very compact and luminous core similar to those observed in a large number of elliptical galaxies ( e.g. , * ? ? ? figure [ sfr ] shows the evolution of the sfr and bolometric luminosity coming from the stars in the merging event . the effects of dust absorption and re - emission are not taken into account . if we take them into account , the ir luminosity would become higher since ultraviolet and optical photons are absorbed by dust and re - emitted in the ir regime . therefore , we expect that the bolometric luminosity of our runs is a good indicator of the ir luminosity . in all runs , the maximum sfr reaches @xmath137 , and the luminosity reaches @xmath138 , which is about an order of magnitude higher than that before the encounter . the figure shows that the evolution of bolometric luminosity reflects the sfr . the peak luminosity is comparable to those of lirgs @xcite . if the progenitor galaxies are more massive and/or the initial gas fraction is much higher than those used in this simulation , the peak of sfr would become higher and the merger galaxy would become ulirgs with infrared emission @xmath139 . agns would also provide an additional ir luminosity , although our simulations do not take into account the effect of agns . the influence of agns to the total ir luminosity is typically @xmath47 % of the total one @xcite . the peak mass - to - light ratio of our run , @xmath140 , is less than @xmath141 and is comparable to that of ulirgs @xcite . the period of the high luminosity phase is @xmath142 . the hypermassive star clusters after their formation contain a large amount of dusty gas and young stars as shown in fig . [ snapshot ] . therefore , a strong infrared emission from the hypermassive star clusters is likely to be observable . in order to study the dependence on the numerical resolution of the formation of hypermassive star clusters , we have performed simulations with different mass resolutions . in both high and low resolution runs , hypermassive star clusters with @xmath143 are formed as shown in fig . [ image ] . their formation process is the same as that of the high resolution runs . in addition , the time evolution of sfr is also similar to that of high resolution runs . in both runs , the peak of sfr is @xmath144 , and the duration time of the active star formation is about @xmath145 . here , we compare our numerical results with observations of ulirgs with nuclei . @xcite observed nuclei with absolute magnitude from @xmath146 to @xmath147 in @xmath8-band . the @xmath8-band luminosity of the identified putative nuclei is typically around a few percent of far - infrared luminosity of their host galaxy @xcite . we estimated the absolute @xmath8-band magnitude of hypermassive star clusters formed in our simulations using pgase . some hypermassive star clusters have luminosity comparable to the observed putative nuclei . in the upper panels of fig . [ image ] , there is one hypermassive star cluster with @xmath148 in run @xmath83 , and there are three hypermassive star clusters with @xmath149 , @xmath150 , and @xmath150 in run @xmath89 , respectively . these @xmath8-band absolute magnitudes are comparable to those of the galactic cores , which are @xmath151 and @xmath152 in run @xmath83 and @xmath153 and @xmath154 in run @xmath89 . these results indicate that the galactic nuclei and hypermassive star clusters are not distinguishable from their luminosities . the @xmath8-band ( bolometric ) luminosity of hypermassive star clusters is a few percent ( @xmath155% ) of the bolometric luminosity of the host galaxy with @xmath156 . these characteristics are in good agreement with the observations . we also compare the spatial distribution of the observed putative nuclei with that of simulated hypermassive star clusters . if we select relatively compact ulirgs , the average separation between nuclei is about @xmath157 @xcite . in our simulations , the separation between hypermassive star clusters and galactic cores depends on the evolution phase . however , the typical separation is a few @xmath158 in the most luminous phase of the galaxies . thus , the spatial distribution also agrees with the observations . in this subsection , we investigate the effect of the dust extinction in @xmath8-band . here , we use the high spatial resolution run , @xmath66 , in which the softening length is @xmath67 for mock observatations , since the detailed structure of the ism is important to estimate the extinction . in this run , similarly to run @xmath89 , hypermassive star clusters with @xmath159 form after the second encounter . the masses of the first , second , third most massive star clusters are @xmath160 , @xmath161 , and @xmath162 , respectively . the flux , taking dust extinction into account , from a star is estimated by @xmath163 . here , @xmath164 is the flux expected without dust extinction , and @xmath165 is optical depth for each sph particle and @xmath166 @xcite , where @xmath167 , @xmath168 , @xmath169 , and @xmath170 represent the clumpiness parameter , the dust cross - section per h atom at @xmath8-band for the milkyway dust model , column number density of atomic h of a sph particle , and the metallicity of a sph particle , respectively . the sph particles between the star and the observer are used for calculating the optical depth , @xmath171 . the clumpiness parameter is introduced in order to represent the clumpiness of the dust distribution in the ism . it is assumed to be @xmath172 , @xmath173 , and @xmath174 . the value of @xmath175 is used in ly@xmath42 emitters @xcite , whereas that of @xmath176 is the case without the dust clumpiness under the spatial resolution . here , we do not estimate the `` clumpiness '' of sph particles along the line of sight directly , since the `` real '' clumpiness comes from much smaller scale in the ism . we adopt @xmath177 @xcite . the column density of an sph particle is given by @xmath178 , where @xmath179 and @xmath180 are the mass and 2-dimensional kernel function of the sph particle , respectively , where @xmath181 is the 2-dimensional distance between the sph and star particles in the projected plane , and @xmath182 is the kernel size of the sph particle . figure [ extinction_map ] illustrates the effect of dust extinctions . the upper row shows the time evolution of the @xmath8-band map without extinction . the middle row represents the expected observation taking dust extinction with @xmath183 into account for run @xmath66 . here , the map is projected to the orbital plane of the galaxies . after formation of hypermassive star clusters , strong dust extinctions take place in the clusters since there is a large amount of dusty gas . the dust extinction reduces the flux from the formed hypermassive star clusters . the extinction of the clusters in @xmath8-band is @xmath184 magnitude in the @xmath185 case as shown in the bottom panels of fig . [ extinction_map ] . in the models of @xmath186 and @xmath187 , the extinctions are @xmath188 and @xmath189 , respectively . the strong extinction continues till @xmath190 ( the left and the middle panels ) , which corresponds to the epoch about @xmath191 after the formation of the clusters . in this dusty phase , multiple core structures are buried by dust . after @xmath192 ( the right panels ) , the clusters become optically thin because of consumption of dusty gas by star formation and escape of dusty gas from the cluster by sn feedback . as a result , the multiple cores appear . the extinction becomes @xmath193 magnitude . on the other hand , the galactic central region is still buried by dust due to the continuous gas accretion . in this phase , the extinction does not depend strongly on the clumpiness parameter since dusty gas is restricted to the central region of the cluster rather than distributed in the broad region of the cluster . the timescale in which hypermassive star clusters become optically thin in @xmath8-band is less than the ulirg lifetime ( @xmath142 as shown in fig . [ sfr ] ) . therefore , it is possible for the clusters to be observed as ulirgs . note , however , that the estimate of the dust extinction depends strongly on the distribution of the dust and that resolutions of current simulations are insufficient to resolve the `` real '' structure of the ism . futher high resolution simulations are necessary to involve the clumpiness of the dust distribution in the ism . the fraction of ulirgs with multiple nuclei strongly depends on the wavelength used for observation . in the analysis of _ hst _ @xmath8-band data of @xmath194 ulirgs samples by @xcite and @xcite , the fraction of the ulirgs with multiple nuclei is @xmath195 . in addition , @xcite have analyzed the nine samples of @xcite in @xmath8-band . two ulirgs are found to have multiple nuclei , i.e. , the fraction is consistent with the results of @xcite and @xcite . on the other hand , @xcite and @xcite have claimed that the fraction of ulirgs with multiple nuclei is less than @xmath196 , based on their analysis of the data observed in near ir bands . using the _ hst _ @xmath197-band data , @xcite reanalyzed 27 samples randomly selected from the 123 @xmath8-band samples , which include a part of borne s and cui s samples with multiple nuclei . in their analysis , only one sample is classified as a ulirg with multiple nuclei . the other sample includes some ulirgs that have been already classified as ulirgs with multiple nuclei by @xmath8-band image analysis @xcite . the reasons for excluding them from ulirgs with multiple nuclei are as follows : ( 1 ) the nucleus seems to locate on the tidal arm and is thought to be the tidal arm or ( 2 ) multiple nuclei morphology do not appear in the @xmath197-band images . furthermore , @xcite analyzed the @xmath198-band data of 118 ulirgs taken by the keck observatory . their samples also include candidates of ulirgs with multiple nuclei in hst @xmath8-band data . some of them are excluded since multiple cores are not apparent in @xmath198-band in spite of their appearance in @xmath8-band . as a result , only 5 ulirgs ( @xmath199 ) are classified as ulirgs with multiple nuclei . in order to understand the difference between the results obtained using images taken in different bands , we made @xmath8- , @xmath197- , and @xmath200-band images of our simulation data using pgase . we assumed that galaxies were at @xmath201 ( @xmath202 ) . the point spread function was assumed to be gaussian , and its dispersion was calculated by @xmath203 , where @xmath204 is the spatial resolution . the spatial resolutions of @xmath8- and @xmath197-band images were given by the diffraction limit since we assumed the observations by _ hst_. the spatial resolutions of @xmath8- and @xmath197-bands are @xmath205 and @xmath206 , respectively . for @xmath200-band images , the seeing determines the spatial resolution since we assumed ground - based observations by the university of hawaii @xmath207 telescope or the keck spectroscopy . the spatial resolution , therefore , is set to be @xmath208 @xcite . for comparison , we also made the emulated images with @xmath200-band by _ james web space telescope _ ( _ jwst _ ) or ground - based @xmath209 m telescopes with adaptive optics ( ao ) . we assume that the angular resolution of these future instruments is @xmath210 . figure [ multibands ] shows @xmath8- , @xmath197- , and @xmath200-band images , in which dust extinction are not taken into account , of runs @xmath83 and @xmath89 . in @xmath8-band images , multiple core - like structures are clearly visible , while such cores are blurred in other bands due to the limited resolution and the contribution of the luminosity of old stars . in @xmath197-band images , cores are connected one another and form arm - like structures . as a result , the cores seem to be in arms . in @xmath200-band images , multiple core structures are not resolved at all . as a result , ulirgs seem to have only single or double cores . thus , multiple cores were identified as a single or double cores in @xmath197- and @xmath198-bands . our analysis indicates that ulirgs with multiple cores can be misclassified as ones with single or double cores . the apparent discrepancy in the observational fractions of ulirgs with multiple nuclei might have been caused by this difference in the spatial resolution . in the rightmost column in fig . [ multibands ] , we show the emulated @xmath200-band images of our simulations with the angular resolution of @xmath211 . we can clearly see the multiple bright sources in these @xmath200-band images that were not visible in grand - base observations , since the resolution of the image is remarkably improved . we expect that , in the near future , observations of ( u)lirgs by _ jwst _ or ground - based @xmath209 m telescopes with ao will settle the problem of the band - to - band difference in the fraction of ulirgs with multiple nuclei . figure [ multibands_dust ] shows the observed @xmath8- , @xmath197- , and @xmath200-band images taking dust extinction into account . here , the cross - sections per h atom at @xmath197- and @xmath200-bands are assumed to be @xmath212 and @xmath213 , respectively @xcite . we adopt @xmath183 as a fiducial value . in the @xmath8-band image , the hypermassive star clusters and the galactic cores are obscured because of dust extinction although the resolution is enough to observe them . on the other hand , in the @xmath197- and ground - based @xmath200-band images , dust extinction becomes weaker although the clusters and the galactic cores are not resolved . in the @xmath200-band image by _ jwst _ or ground - based @xmath209 m telescopes with ao , the clusters and the galactic cores are clearly observable since not only dust extinction is small but also they are resolved . when dust extinction is significant in @xmath8-band , we need the observations by _ jwst _ or ground - based @xmath209 m telescopes with ao . we have performed high resolution @xmath54-body / sph simulations of merging galaxies . we found that hypermassive star clusters with @xmath143 form from disturbed gas disks in the central region ( @xmath3 ) . in these clusters , active star formations take place , so that some bright core structures appear in the merging galaxy . the features of formed hypermassive star clusters agree with the observations of ulirgs with multiple nuclei . although dust extinction may obscure the clusters after their formation , the clusters become optically thin within the ulirg lifetime . these results indicate that one major merger of two spiral galaxies can explain complex structures of multiple nuclei in ulirgs . rare multiple major merger events on a short timescale are not necessary . in the high - redshift universe , gas - rich major mergers must occur frequently . our result suggests that these merging galaxies have multiple hypermassive star clusters that would be observed as bright sources in either near or far infrared bands . future observations using e.g. , _ jwst _ or ground - based @xmath209 m telescopes with ao are expected to find a number of such merging galaxies with multiple bright sources . in our simulations , we assume that main feedback source is type ii sne , and radiative feedback from newly formed stars is not taken into account . if the radiative feedback is effective , gas in hypermassive star clusters is heated and is ejected from the cluster @xcite . this effect may reduce the cluster mass . in order to study the effect , we need radiation - hydrodynamic simulations as performed by @xcite . since the hypermassive star clusters are compact ( @xmath214 within @xmath2 ) , it is possible that star - star collisions and mergings occur in these clusters @xcite . if the stellar mass loss in the main sequence phase is not very large , such merging of stars might result in the runaway growth of supermassive stars and the formation of intermediate - mass black holes ( imbhs ) @xcite . these imbhs are carried to the center of the galaxy together with the parent hypermassive star cluster by dynamical friction . if there is already a supermassive black hole at the center of the progenitor galaxies , the imbh merges with the smbh through dynamical friction and eccentricity evolution @xcite . if there was no supermassive black hole , multiple imbhs conveyed to the center by the parent cluster finally merge each other to form the seed of a smbh . the authours thank the anonymous referee for many useful comments . the authors also thank junichi baba , nozomu kawakatu , masakazu a. r. kobayashi , and tohru nagao for helpful discussions . numerical computations were carried out on cray xt4 and grape system at center for computational astrophysics ( cfca ) of national astronomical observatory of japan . this project is supported by nec corporation , molecular - based new computational science program of nins , grant - in - aid for scientific research ( 17340059 ) of jsps , and hpci strategic program field 5 `` the origin of matter and the universe '' . is financially supported by a research fellowship from the japan society for the promotion of science for young scientists . is financially supported by grant - in - aid for scientific research ( s ) by jsps ( 20224002 ) and by grant - in - aid for young scientists ( 21840015 ) . , t. r. , daisaka , h. , kokubo , e. , makino , j. , oakmoto , t. , tomisaka , k. , wada , k. , & yoshida , n. 2010 , in astronomical society of the pacific conference series , vol . 423 , astronomical society of the pacific conference series , ed . b. smith , j. higdon , s. higdon , & n. bastian , 185+

ultraluminous infrared galaxies ( ulirgs ) with multiple ( @xmath0 ) nuclei are frequently observed . it has been suggested that these nuclei are produced by multiple major mergers of galaxies . the expected rate of such mergers is , however , too low to reproduce the observed number of ulirgs with multiple nuclei . we have performed high - resolution simulations of the merging of two gas - rich disk galaxies . we found that extremely massive and compact star clusters form from the strongly disturbed gas disks after the first or second encounter between the galaxies . the mass of such clusters reaches @xmath1 , and their half - mass radii are @xmath2 . since these clusters consist of young stars , they appear to be several bright cores in the galactic central region ( @xmath3 ) . the peak luminosity of these clusters reaches @xmath4 of the total luminosity of the merging galaxy . these massive and compact clusters are consistent with the characteristics of the observed multiple nuclei in ulirgs . multiple mergers are not necessary to explain multiple nuclei in ulirgs .

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Proceed to summarize the following text: axisymmetric spreading of an insoluble surfactant on a thin layer of incompressible fluid has been the subject of numerous experimental and mathematical studies @xcite . motivated by the biomedical application of aerosol medications delivered to the thin fluid lining the lung , gaver and grotberg derived a mathematical model , based on lubrication theory , that couples the height profile of the fluid surface @xmath2 to the local surfactant concentration @xmath3 . this model captures the driving force associated with the marangoni surface stress induced by spatial variations in surfactant concentration , which in turn depends on an equation of state that specifies the dependence of surface tension @xmath4 on @xmath5 . while the model was developed for monolayer applications of surfactant , it has come to be applied both above @xcite and near @xcite the critical monolayer concentration @xmath6 , the concentration above which a single layer of surfactant molecules can no longer form . similar models have been used to study thin films in bronchial systems @xcite , ocular systems including blinking dynamics @xcite , bulk solute transport @xcite , drying of latex paint @xcite , ink - jet printing @xcite , and secondary oil recovery @xcite . numerical simulations have been used to confirm several predictions of the model @xcite based on analysis of self - similar solutions . the simulations have also yielded detailed information about the spatiotemporal evolution of the free surface height and surfactant concentration profiles . one key observation is that the motion of the leading edge of the surfactant - covered region sets its spreading rate . the decrease in concentration at the leading edge induces a surface stress ( marangoni stress ) , which drives a capillary ridge ( local maximum ) in the fluid free surface that propagates along with the leading edge . in spite of these advances in understanding , experimental confirmation of the model has been more difficult to obtain due to the difficulty of measuring the surfactant concentration and its dynamics . consequently , attention has primarily focused on the evolution of the surface height profile @xcite , with the location of the leading edge @xmath7 of the surfactant layer inferred from other dynamics @xcite . within the near - monolayer regimes for which the model was developed , two experiments have observed @xmath8 spreading behavior on millimetric glycerin films , for both oleic acid @xcite and fluorescently - tagged phosphocholine ( nbd - pc ) @xcite . in experiments with initial concentrations @xmath9 of surfactant exceeding @xmath6 , spreading behavior consistent with the predicted @xmath10 were observed by dussaud et al . for oleic acid on a sub - millimetric water - glycerin mixture , and by fallest et al . for nbd - pc on millimeter - thick glycerin . the latter experiments were able to simultaneously measure both the capillary ridge and the spatiotemporal dynamics of @xmath3 , to be analyzed in more detail below . sec . ] in order to verify the model prediction for the time - dependent distribution of surfactant concentration , we explore stricter tests of the model than have been performed previously . using data from fallest et al . ( for which @xmath5 is well above @xmath6 ) , we make a detailed comparison between the model predictions and the measured surface height profiles @xmath2 ( from a laser line ) and measured surfactant concentration profiles @xmath3 ( from azimuthal averages of fluorescent intensity at each point @xmath11 ) . figure [ f : exp - image ] provides a schematic of the apparatus and a sample image . the well - specified physical parameters additionally allow us to evaluate the accuracy of the characteristic timescale predicted for the spreading rate , rather than just the exponent . while we find approximate agreement in the spreading exponent and the coincidence of the surfactant leading edge with the capillary ridge , we also find two significant inconsistencies . first , there is a mismatch between the characteristic timescale between model and experiment . second , the spatial distribution of the surfactant differs markedly from what is predicted in simulations . to account for the extent to which these discrepancies might be due to amounts of surfactant well beyond the monolayer regime , we perform new experiments , modified to allow for the detection of monolayer concentrations of surfactant ( @xmath12 ) . in these experiments , we observe a distribution of surfactant which differs from the model predictions . in addition , we find that there is no spreading capillary ridge , and that the spreading exponent for the leading edge of the surfactant is less than @xmath1 . this value is well below predictions of the theory . the outline of the paper is as follows . in [ s : experiment ] , we provide details about the experimental setup and the materials used . in [ s : model ] , we review the gaver - grotberg model @xcite , including a discussion of the choice of equation of state relating surfactant concentration to surface tension , and outline the finite difference method used for numerical simulation . in [ s : highgamma ] , we describe comparisons between numerical simulations and experimental results at initial concentrations well above @xmath6 . as described above , we find partial agreement but also two inconsistencies when comparing experimental observations to numerical simulations . in an attempt to address the latter problem , in [ s : hybrid ] we describe a hybrid model that takes the experimentally - measured @xmath3 and uses that quantity ( instead of the model for surfactant evolution ) to generate the evolution of the surface height profile . we find that this provides reasonable agreement with the experimental data for @xmath2 . in particular , the timescale in the simulations is set by the experimental surfactant distribution , and the height profile dynamics therefore evolve according to the experimentally - observed timescale . in [ s : lowgamma ] , we report new experiments with @xmath12 and find significant disagreement with the model predictions , as summarized above . we conclude in [ s : discussion ] with a discussion of the results and their significance . in our experiments , we simultaneously record the surface height profile @xmath2 of the underlying glycerin fluid layer , as well as the local fluorescence intensity , which corresponds to the local concentration @xmath13 of insoluble lipids ( surfactant ) spreading on the surface . the basic apparatus , described in and shown in figure [ f : exp - image ] , consists of an aluminum well , a black light for exciting the nbd fluorophores , an oblique red laser line to illuminate the profile of the fluid surface , and a digital camera positioned directly above the experiment to capture the laser line and fluorescence ; this apparatus is used for the data presented in [ s : highgamma ] . new experiments , presented in [ s : lowgamma ] , are optimized to permit visualization of monolayer concentrations of surfactant . the bottom of the aluminum well is covered with a plasma - cleaned silicon wafer for improved reflectivity , and the fluorescent excitation is provided by 467 nm ( blue ) leds which coincide with the 464 nm absorption peak of the nbd fluorophore . a green laser line illuminates the profile of the fluid surface , so that both it and the fluorescently - emitted light pass through a band - pass filter centered at the emission peak ( 531 nm ) . these improvements to the optics permit us to collect images of the spreading dynamics at a framerate of 3 hz and an integration time of @xmath14 second , using an andor luca r camera optimized for fluorescence measurements . the signal - to - noise ratio now sets a lower limit of @xmath15 for the detection of surfactant . for all experiments , we deposit 1-palmitoyl-2-\{12-[(7-nitro-2 - 1 , 3-benzoxadiazol-4-yl)amino]lauroyl } -sn - glycero-3-phosphocholine ( abbreviated nbd - pc , from avanti polar lipids ) within a retaining ring which is lifted to begin the spreading process . this lipid molecule has one 12-carbon chain and one 16-carbon chain ; the nbd fluorophore is attached to the 12-carbon chain . experiments are conducted on a layer of 99.5% anhydrous glycerin , at an initial depth of @xmath16 mm , and held at room temperature ( @xmath17 @xmath18c ) . the lipids are initially deposited while dissolved in chloroform , which is allowed to evaporate for at least @xmath19 min before the retaining ring is lifted by a motor at @xmath20 mm / min . this allows sufficient time for the meniscus to drain before it detaches from the ring . .key dimensional parameters . [ cols="^,^,^",options="header " , ] the key material parameters for nbd - pc and glycerin are summarized in table [ t : ndval ] . the initial conditions ( ic ) for the experiment are distinguished by the initial concentration @xmath21 of surfactant deposited within the ring , defined by @xmath22 where @xmath23 is the mass of nbd - pc deposited and @xmath24 is the radius of the ring . although a small amount of surfactant remains on the ring after it has been lifted , we nonetheless use the nominal concentration @xmath21 to describe the different initial conditions in table [ t : initcond ] . the experiments presented in [ s : highgamma ] all begin from ic6 , with @xmath21 above the critical monolayer concentration @xmath6 . the experiments of [ s : lowgamma ] employ initial conditions that probe the monolayer regime ( below @xmath6 ) . sec , for experiments with initial conditions ( a ) ic6 and ( b ) ic4 . the dashed circles ( of radii 3.4 cm and 1.1 cm , respectively ) , highlight the corrugations in the leading edge . ] figure [ f : exp - image]b shows a sample image of both the laser line ( measures the height profile @xmath25 and the location @xmath26 of its maximum ) , and the fluorescence intensity ( measures the surfactant distribution @xmath27 after azimuthal averaging and the location @xmath7 of its leading edge ) . figure [ f : idist ] shows sample images of the surfactant distribution alone , for a representative @xmath28 case and a @xmath29 case . in each image , a sharp interface between the surfactant - covered and bare glycerin is readily visible ; the location @xmath7 of this interface is determined by identifying the annulus of maximum fluorescence intensity gradient . while the surfactant distribution is uniform for @xmath30 , several heterogeneities are present when @xmath31 . first , the central region contains a greater concentration of surfactant than the regions closer to the leading edge , an effect that we will explore in more detail below . second , there are filamentary patches of high concentration which also propagate out from the central region , becoming more dilute during the spreading dynamics . we also note that the outer edge of surfactant in both cases has corrugations ; in the figure , white circles are imposed to emphasize that the edges are not quite circular . although the surfactant distributions are never precisely axisymmetric , we nonetheless record the distribution by averaging azimuthally . moreover , in the model and simulations of the following sections , we assume that the surfactant distributions are axisymmetric . we consider the model derived by gaver and grotberg for a single layer of surfactant molecules spreading on a thin liquid film . the model is a coupled system of partial differential equations for the height @xmath2 of the fluid free surface and the concentration @xmath3 ( mass per unit area ) of surfactant . we assume axisymmetric spreading , and the variables are nondimensionalized : @xmath32 , @xmath33 , @xmath34 , @xmath35 , where @xmath36 indicates the dimensional variable . [ rfulleqn1 ] @xmath37 the nondimensional parameter groups @xmath38 , @xmath39 , and @xmath40 result from nondimensionalization , using values of physical parameters listed in table [ t : ndval ] . the parameter @xmath41 balances gravity and marangoni forces , @xmath42 is the ratio of the capillary driving forces to the forces from the surface tension gradient , and @xmath43 represents the surface diffusion of the surfactant molecules where @xmath44 is the pclet number . the function @xmath45 expresses the dependence of surface tension @xmath4 on surfactant concentration @xmath5 . it is specified by an equation of state , as discussed in the next subsection . the timescale @xmath46 sec achieves a balance between the terms on the left side of ( [ rfulleqn1 ] ) . in [ modelparams ] , we test this predicted timescale directly . with @xmath47 , the leading edge of the surfactant distribution is not precisely defined , since @xmath48 for all @xmath49 and for all @xmath50 . nonetheless , although @xmath40 is very small , it is retained in numerical simulations , apart from one case , in which @xmath40 is set to zero in order to track the leading edge of the surfactant distribution . since the boundary at @xmath51 is not a physical boundary , boundary conditions there are natural ; for large @xmath49 , the free surface is undisturbed on the time scale of the experiment , and the surfactant concentration is expected to be identically zero : @xmath52 a finite difference method is used to simulate ( [ rfulleqn1 ] ) and is summarized in the appendix . the initial condition @xmath53 is chosen to reflect the initial height profile in the experiment , as the fluid meniscus detaches from the ring , thereby releasing the surfactant to spread across the fluid surface . the initial distribution @xmath54 of surfactant within the ring is unknown , and is varied in the simulations to test its effect on the spreading . @xmath55 1,&\frac{11\pi}{12}<r < r_{max } , \end{cases}\nonumber\\ \gamma(r,0)&=&\begin{cases}\gamma_0(r),&\mbox 0\leq r<\frac{5\pi}{12}\\[6pt ] 0,&\frac{5\pi}{12}<r < r_{max}. \end{cases}\end{aligned}\ ] ] in the simulations , we set @xmath56 in order provide a match between the initial condition and the measured height profiles at later times . for the simulations , the location @xmath57 of the edge of the domain is taken large enough that the influence of the boundary conditions at @xmath58 is negligible over the time of the experiment . we take @xmath59 , but display graphs of @xmath60 and @xmath5 over the smaller domain , @xmath61 . ) , g : langmuir ( [ langeos ] ) , e : measured ( [ bulleos ] ) , m : multilayer ( [ meos ] ) . ] in order to compare the model ( [ rfulleqn1 ] ) to the results of the experiments described in , we need to choose an appropriate equation of state relating the surfactant concentration @xmath5 to the surface tension @xmath4 . however , the model is valid only for a single layer of surfactant molecules ( @xmath62 ) but the experiments are conducted with initial surfactant concentrations up to @xmath63 . in an attempt to extend the model to the regime of the experiment , we consider different equations of state @xmath64 that have been proposed in the literature . in figure [ f : eos ] , we show the graphs of four such functions ; we argue below that only the graph labeled * m * is suitable for modeling the full range of surfactant concentrations we wish to consider . the equation of state we seek should have the following properties : @xmath65 , expressing the effect that increasing surfactant concentration decreases surface tension ; @xmath66 , since this is the range of values of surface tension in our nondimensionalization , with @xmath67 . as can be seen in the figure , only graph * m * has these properties . the linear equation of state @xmath68 and has been used widely @xcite . this equation is generally chosen for simplicity ; it is also a reasonable linear approximation to nonlinear equations of state at low concentration . note that @xmath69 is a negative constant ( @xmath70 in the figure ) . in the case of more than a monolayer of surfactant , this equation suggests that the surface tension decreases endlessly which is not physical , as surfactant concentration beyond a monolayer has little further effect in decreasing surface tension . bull et al . determined an equation of state for nbd - pc on glycerin by fitting the data obtained using a tensiometer . using the nondimensional parameters in table [ t : ndval ] ( @xmath71 @xmath72g/@xmath73 ) , the corresponding formula for @xmath45 is @xmath74 however , this formula is applicable only for surfactant concentrations below approximately @xmath75 ( @xmath76 in figure [ f : eos ] ) , as @xmath77 decreases sharply for larger values of @xmath5 . the langmuir equation of state , used in and , is @xmath78 where @xmath79 and @xmath80 . when only a small amount of surfactant is introduced , a large change in the surface tension occurs and as the surfactant comes close to saturation ( a monolayer ) then adding more surfactant does not alter the surface tension very much . however , the range of @xmath45 is @xmath81 $ ] rather than @xmath82 $ ] . the multiple layer equation of state used by borgas and grotberg is related to the langmuir equation of state : @xmath83 this formulation is based on properties of surface tension discussed by sheludko and by an experimental fit by foda and cox who worked with an oil layer on water . in addition , @xmath45 remains positive at large @xmath5 , allowing us to simulate much higher concentrations of surfactant . therefore , we use eq . ( [ meos ] ) for the simulations . the @xmath84 spreading behavior predicted by ( [ rfulleqn1 ] ) has already been observed in prior experiments with oleic acid on a water - glycerin mixture @xcite and nbd - pc on glycerin @xcite . in this section , we make a more detailed comparison between the results from experiments and numerical simulations , using data from fallest et al . . while we are able to obtain reasonable agreement in the height profile shapes , a comparison of the dynamics ( [ modelparams ] ) requires that we adjust the timescale . in [ compprofile ] , we show a significant discrepancy between the observed distribution of surfactant and the prediction from simulations , even though the location and time evolution of the leading edge of the surfactant layer agree well , as detailed in [ spreadingexp ] . we also describe attempts to capture the experimentally observed surfactant distribution by varying the initial distribution @xmath54 in the simulations . measured at @xmath85 sec in the experiment ( short dashed lines , red ) , compared to numerical solutions at nondimensional times . long dashed lines : @xmath86 ( uses calculated @xmath87 sec ) ; solid line : @xmath88 ( uses better - fitting @xmath89 sec ) . data are from fallest et al . , with initial condition ic6 . ] in order to compare the model and experiment , we convert the simulation results from dimensionless time @xmath90 to dimensional time @xmath91 using the relation @xmath92 , where @xmath93 sec is calculated from the dimensional parameters listed in table [ t : ndval ] . previous simulations @xcite have treated the lengthscale @xmath24 as a free parameter , effectively adjusting the timescale to agree with the experimental observations . however , for the experiments analyzed here , the ring radius @xmath94 cm is known and consequently the time scale @xmath95 is determined , with no free parameters . in figure [ f : thalf ] , we observe that the simulated height profile and the experimental data are inconsistent at @xmath96 sec if the determined value @xmath87 sec is used : neither the peak location nor its width are in agreement with the model . however , if we use @xmath97 sec instead of @xmath87 sec , thereby comparing the simulation at the later time @xmath98 to the same experimental data at @xmath96 sec , then both the position and width of the ridge are in approximate agreement between the model and experiment . this agreement between simulation and experiment using the timescale @xmath97 sec is observed to hold for all times beyond an initial transient . although an explanation for the discrepancy between times scales awaits further investigation , we use the empirically - determined @xmath97 sec as the timescale for the remaining comparisons in the paper . measured at @xmath96 seconds in the experiment ( dashed lines ) and @xmath98 in numerical solution ( solid line ) ; the location @xmath99 of the peak is marked with a @xmath100 . ( b ) the corresponding surfactant concentration profiles @xmath101 in the experiment ( noisy ) and numerical solution ( smooth ) ; the location @xmath102 of the leading edge of the surfactant is marked with a @xmath100 . data are from fallest et al . 2010 , with initial condition ic6 . ] in figure [ f : profile1 ] we compare the simulated and measured surface profiles and surfactant distributions , using the model parameters described in [ modelparams ] and experimental data from fallest et al . . as can be seen in figure [ f : profile1 ] , the height profiles @xmath103 are in approximate agreement : locations of the maximum and minimum are in approximate agreement between simulation and experiment , and the overall shapes are similar . in contrast , the measured surfactant distribution has quite a different shape from the distribution predicted by the simulations . while the model predicts a smooth decrease in @xmath104 away from the central peak , experiments instead show an extended plateau over which the surfactant concentration is nearly constant , and which appears to be drawn out of a reservoir , near the peak concentration at @xmath51 . for longer times , the plateau extends and decreases in height . these features do not appear in the numerical simulations . and ( g - i ) surface height profiles @xmath105 from numerical solutions at @xmath98 . the experimental height profiles at @xmath96 seconds are the same in each case , and are shown as dashed lines . ] in the experiment , the total surfactant mass is known , but its initial spatial distribution within the retaining ring is not measured . consequently , there is some uncertainty about the appropriate initial condition @xmath106 . we explore whether the choice of @xmath54 could change the simulations enough to replicate the expanding plateau in the experimentally observed surfactant distributions . we tested three different functions @xmath107 , shown in figure [ f : surfic ] : ( a ) uniform distribution ; ( b ) surfactant more concentrated near the retaining ring ; ( c ) step distribution . the initial free surface height profile @xmath108 is the same in each case , given by ( [ numic ] ) . the results of these simulations at time @xmath96 sec ( after short - time transients have died out ) are shown in figure [ f : surfic ] . in the middle column , we observe that the distribution @xmath5 of surfactant does not exhibit a plateau for any of the initial conditions , while in the final column we see that the height profiles show broad agreement with the experiment in each case . and ( b ) the location of the leading edge of surfactant in both experiment and simulation . colored lines are from experiments with ic6 adapted from fallest et al . 2010 , black dots are from simulations . dashed lines are ( a ) comparison to best - fit @xmath109 with @xmath110 and ( b ) comparison to @xmath0 . ] in spite of the disagreements above , we observe that the spreading dynamics of the model and experiment are in good agreement when the artificial choice of @xmath111 sec is used . the surface diffusion term @xmath112 in ( [ rfulleqn1h ] ) smooths the surfactant profile , and guarantees @xmath48 for all @xmath113 and @xmath50 . this means the surfactant distribution has no clearly defined leading edge . since @xmath114 has a very small effect , we take @xmath115 in these simulations . the surfactant distribution is then supported at each @xmath50 on a bounded interval @xmath116 , and the leading edge @xmath117 can be tracked using the numerical scheme described in with @xmath118 . in figure [ f : spread](b ) , @xmath119 is shown as a dotted line ; on the log - log plot the numerical solution is compared to the experimental results and to the analytic form @xmath120 derived from the similarity solution of , in which @xmath121 . we also tracked the capillary ridge @xmath122 , where the height profile @xmath2 has a maximum . in figure [ f : spread](a ) we show the numerical solution as a dotted line , and compare to experimental results , with an approximate slope shown with a dashed line . the data for the leading edge of the surfactant agrees with the @xmath123 prediction of the model . the capillary ridge moves faster as it catches up to the surfactant leading edge ; it is best fit by @xmath124 over the duration of the experiment . note that for @xmath121 in the model , the fluid surface experiences a discontinuity at the leading edge @xmath117 of the surfactant , and the fluid is undisturbed ahead of this front . it is worth noting that with @xmath115 , the surfactant distribution still has a finite extent when @xmath38 or @xmath39 is non - zero , but the disturbance of the film does not . in fact , the surface tension gradient induced at the leading edge @xmath117 of the surfactant generates fluid motion ahead of @xmath7 . due to the failure of the model equations ( [ rfulleqn1 ] ) to capture the spatial distribution of the surfactant , as discussed in [ s : highgamma ] , we consider whether there is a fundamental modeling problem with the equation ( [ rfulleqn1 g ] ) for the evolution of @xmath3 . given that the height profile @xmath2 evolves in a quantitatively reasonable way , we leave ( [ rfulleqn1h ] ) intact , but take @xmath3 from the experiment and use it to determine @xmath2 numerically . the experimental data for @xmath5 are noisy , and occur at discrete times ; we must smooth and interpolate the data for use in the numerical scheme to determine @xmath60 . the first step is to smooth the experimental data at each recording time . this is achieved using the matlab _ smooth _ function , which performs a moving spatial averages over a specified span of data points ; we found a low - pass filter with a span of 21 points to be effective . the experimental data are recorded at one second intervals ; however , the numerical method requires the time step @xmath125 to be on the order of @xmath126 for stability . to remedy this inconsistency , we interpolate the smoothed surfactant profiles to obtain functions that can be used to represent the surfactant concentration @xmath3 at all values of @xmath49 and @xmath90 . from the numerical simulations ( solid blue ) and experimental data ( dashed red ) . ( e - h ) the corresponding surfactant profiles from the experiment , smoothed data ( smooth blue ) and raw data ( noisy green ) . the profiles are shown at @xmath127 sec . ] first , we use the nonlinear fit function , _ nlinfit _ in matlab to fit the smoothed surfactant data at each experimental recording time to a function with a graph consistent with the two step structure of the experimental surfactant distributions . as observed in figure [ f : exp4 ] , the experimental surfactant distribution @xmath5 is roughly constant in each of the two steps . with this in mind , we use the function @xmath128 to fit the surfactant profile at each time @xmath91 for which we have data , where @xmath129 sec . this procedure generates coefficients @xmath130 . next we create polynomial functions , @xmath131 from the discrete values @xmath132 using _ polyfit _ in matlab , but with a restriction that the mass of the surfactant must be conserved . since the numerical code requires nondimensional time @xmath133 , we define @xmath134 . these functions define the surfactant concentration profiles that approximate the experimental data : @xmath135 we update the height profile using the finite difference scheme used in [ s : highgamma ] with boundary conditions ( [ bcsym ] ) and the surfactant concentration profile using ( [ gfit ] ) : @xmath136 the flux functions @xmath137 are described in the appendix . in figure [ f : exp4 ] , we show simulation results using initial condition ( [ numic ] ) and parameters @xmath138 . note that this value of @xmath39 is larger than the value suggested by the nondimensional grouping . this is in order to smooth the height profile , which otherwise would develop multiple persistent ridges , due to the steep gradient in the surfactant concentration . multiple ridges are observed at very early time in the simulations discussed in [ compprofile ] but then the gradient in the surfactant is quickly smoothed . in this case , the ridge in the height profile from the surfactant gradient and the one from the initial condition ( due to the lifting of the ring ) combine at very early time and then propagate as one . by contrast , in the experiment the surfactant distribution does not experience this smoothing and consequently when the experimental values for the surfactant concentration are input into the height equation a second ridge develops and persists . by increasing the capillarity parameter @xmath39 , the height profile is smoothed , and the numerical height profile becomes more similar to the experimental height profile . the surface height profile simulated using this hybrid model exhibits an evolution and shape that is comparable to the experiment . we conclude that the equation modeling surfactant molecule motion through passive transport by the surface fluid is fundamentally flawed , and is missing some physical or chemical properties that would generate the surfactant distributions observed in the experiment . because the original model ( [ rfulleqn1 ] ) was developed for use with monolayer concentrations of surfactants ( @xmath30 ) , we conduct new experiments in this regime . in addition , these experiments help elucidate the discrepancies between model and experiment at the higher concentrations . performing experiments at lower surfactant concentrations requires improvements of our earlier techniques ( see [ s : experiment ] ) in order to visualize lower surfactant concentrations . we perform experiments starting from four different initial concentrations , ic1-ic4 in table [ t : initcond ] , all of which result in similar spreading dynamics , described below . in no case do we find that the agreement with the model is improved over the @xmath31 case : height profiles , surfactant distribution , and the spreading dynamics all significantly disagree with the model predictions . from the center of the surfactant region , shown at representative times , taken from an experiment with ic4 ( @xmath30 ) . ] as illustrated in figure [ f : idist ] , and shown quantitatively in figure [ f : fluoro ] , the spreading region has an approximately uniform surfactant distribution throughout the lipid - covered area . the leading edge , located at @xmath7 , exhibits a sharp interface ( approximately @xmath139 mm wide , neglecting azimuthal corrugations ) that does not broaden as the surfactant spreads outward . instead , the overall concentration decreases throughout the lipid - covered region . these observations are in disagreement with the model , which predicts a monotonically decreasing profile ( see figure [ f : profile1 ] ) with a gradual and broadening transition to @xmath140 at @xmath7 . ) and ic2 ( f - j ) ( @xmath141 ) at five different times . the uppermost bright line in each image is the reflection from the air glycerin interface , and is used to measure the free surface height profile @xmath142 . ] figure [ f : mridge ] provides a comparison between the behavior of the fluid ridge in two regimes . for @xmath143 , the glycerin is pushed out by the surface tension gradient , and forms a capillary ridge at the surfactant edge ( panels f - j ) . as already discussed in [ compprofile ] , these two features travel outward together . this is consistent with simulations at high or low surfactant concentrations , that show a fluid ridge propagating outwards due to marangoni forces . in contrast , the experimental data in the @xmath144 regime ( panels a - e ) show an initial ridge ( due to the meniscus at the ring ) collapsing rapidly in place , and no discernible ridge propagating outwards . ) , scaled by the ring radius @xmath24 . the dashed lines corresponding to @xmath0 and @xmath145 spreading behavior are shown for comparison . ] given the significant differences between the observed and modeled @xmath2 and @xmath3 , it is unsurprising that we observe the spreading dynamics to be quite different as well . in figure [ f : rvst ] , we plot the position of the leading edge @xmath7 as a function of time , for experiments in both the monolayer regime of ic1-ic4 and for higher initial concentrations ( ic5-ic6 ) . for @xmath30 ( ic1-ic4 ) the dynamics all follow a form @xmath146 , with @xmath147 . remarkably , this is reminiscent of tanner s law for fluid spreading on a solid @xcite . for slightly larger values of @xmath21 ( ic5 ) , we observe faster surfactant spreading dynamics initially , but @xmath146 with @xmath148 at later times . this slow - spreading regime was not reached in the runs with much larger values of @xmath21 ( ic6 ) , for which @xmath149 remains close to @xmath150 , as predicted in @xcite . the decrease in @xmath149 for low surfactant concentrations suggests a transition in the dynamics which is not covered by the model equations . the differing prefactors for the various runs is probably a result of variations in @xmath21 due to some lipid molecules remaining on the ring after it lifts off , an effect that is more significant at lower concentrations . current mathematical models that describe the dynamics of the free surface of a thin fluid layer subject to forces induced by variations in surface tension contain two key assumptions : ( 1 ) lubrication theory is valid , and ( 2 ) surfactant molecules are primarily passively - transported by the fluid motion along the surface , along with negligible molecular diffusion . however , the dependence of surface forces on local variations in surfactant concentration is not completely settled , especially for larger concentrations . due to the coupling between the motion of the underlying fluid and the spreading of surfactant molecules , it is crucial to compare results from simulations and experiments for both fluid motion ( through surface deformations ) and the dynamics of surfactant distribution . in this paper , we compare model predictions to experiments that include the simultaneous visualization of the fluid height profile and the distribution of surfactant , both above and below the critical monolayer concentration @xmath6 . in both cases , we find serious inconsistencies between the model and the experiments . the aspect ratio for the experiments is @xmath151 , which justifies the use of the lubrication approximation ; we have not verified the magnitude of the vertical velocity profile . at all initial concentrations , both above and below @xmath6 , the spatial distribution of surfactants does not follow the smooth , monotonically - decreasing profiles predicted by the model . at low surfactant concentrations ( @xmath30 , for which the models were originally developed ) , the distribution is highly - uniform with a sharp interface at the leading edge . second , spreading occurs much more slowly than is predicted by the model . for all experiments with @xmath30 , the spreading dynamics of the leading edge approximately follow a power law @xmath152 , with @xmath153 . this is significantly smaller than the @xmath154 predicted by the natural scaling in the model . interestingly , this exponent is also markedly different from the exponent of @xmath155 to @xmath156 observed by gaver and grotberg for oleic acid and by bull et al . @xcite for nbd - pc . in the former case , the measurement technique relies on the model for interpretation of experimental results , while in the later the @xmath157 mm fluid layer thickness may be large enough that deviations from the lubrication approximation are significant . for @xmath158 , even though @xmath159 , there is a mismatch by a factor of two between the timescales predicted in the model and observed in experiment . these inconsistencies are not resolved by changing assumptions concerning the initial distribution of surfactant . moreover , if a measured @xmath3 is incorporated directly into the lubrication model , then the timescale issue is no longer present , as the spreading ridge is simply driven by the spreading surfactant . one untested assumption is the functional form of the equations of state which have been considered to date . the lack of spreading ( @xmath153 for very low concentrations ) might indicate that the assumed @xmath45 equation of state is inadequate . if there were a value of @xmath5 below which there were no longer a significant surface tension gradient , a lack of spreading would be expected . in fact , in static surface - 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( 8) , 11071117 . levy , r. , m. shearer , and t. p. witelski : 2007 , ` gravity - driven thin liquid films with insoluble surfactant : smooth traveling waves ' . , 679708 . matar , o. k. and s. m. troian : 1999 , ` spreading of a surfactant monolayer on a thin liquid film : onset and evolution of digitated structures ' . ( 1 ) , 141153 . peterson , e. r. and m. shearer : 2010 , ` radial spreading of surfactant on a thin liquid film ' . . peterson , e. r. and m. shearer : 2012 , ` simulation of spreading surfactant on a thin liquid film ' . ( 9 ) , 5157 5167 . sakata , e. k. and j. c. berg : 1969 , ` surface diffusion in monolayers ' . , 570575 . segur , j. b. and h. e. oberstar : 1951 , ` viscosity of glycerol and its aqueous solutions ' . ( 9 ) , 21172120 . sheludko , a. : 1966 , _ colloid chemistry_. elsevier . sinz , d. k. n. , m. hanyak , j. c. h. zeegers , and a. darhuber : 2011 , ` insoluble surfactant spreading along thin liquid films confined by chemical surface patterns . ' . ( 20 ) , 976877 . tanner , l. h. : 1979 , ` the spreading of silicone oil drops on horizontal surfaces ' . , 14731484 . warner , m. r. e. , r. v. craster , and o. k. matar : 2004 , ` fingering phenomena created by a soluble surfactant deposition on a thin liquid film ' . ( 8) , 29332951 . wulf , m. , s. michel , k. grundke , o. i. del rio , d. y. kwok , and a. w. neumann : 1999 , ` simultaneous determination of surface tension and density of polymer melts using axisymmetric drop shape analysis ' . ( 1 ) , 172 181 . we summarize the finite difference method used to generate numerical results for system ( [ rfulleqn1 ] ) . simulations are conducted on a large interval @xmath160 . for simplicity , we consider uniformly distributed grid points @xmath161 , where @xmath162 . at each time @xmath163 , let @xmath164 and @xmath165 . we use the standard notation for spatial averages of @xmath166 , @xmath167 the numerical method is an implicit finite difference scheme in conservative form : in the hybrid model of [ s : hybrid ] , the values of @xmath172 are derived from the experimental data , as described in that section . the height profile evolution can then be computed from equation and the companion equation is discarded .

the spreading dynamics of surfactant molecules on a thin fluid layer is of both fundamental and practical interest . a mathematical model formulated by gaver and grotberg @xcite describing the spreading of a single layer of insoluble surfactant has become widely accepted , and several experiments on axisymmetric spreading have confirmed its predictions for both the height profile of the free surface and the spreading exponent ( the radius of the circular area covered by surfactant grows as @xmath0 ) . however , these prior experiments have primarily utilized surfactant quantities exceeding ( sometimes far exceeding ) a monolayer . in this paper , we report that this regime is characterized by a mismatch between the timescales of the experiment and model , and additionally find that the spatial distribution of surfactant molecules differs substantially from the model prediction . for experiments performed in the monolayer regime for which the model was developed , the surfactant layer is observed to have a spreading exponent of less than @xmath1 , far below the predicted value , and the surfactant distribution is also in disagreement . these findings suggest that the model is inadequate for describing the spreading of insoluble surfactants on thin fluid layers .

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Proceed to summarize the following text: on february 28 , 1997 , the first x - ray afterglow of a gamma - ray burst ( grb ) was detected , leading to the identification of its progenitor at cosmological distances @xcite . in a few days , the afterglow faded away with time as a power law . this behavior is satisfactorily explained in the spherical ( isotropic ) fireball model involving relativistic ejecta decelerated by circumburst medium @xcite . the introduction of collimated jets relaxes the energy requirement of grbs by a factor of several hundred , as well as explains the steeper temporal decay of afterglows @xcite . grbs are extra - galactic sources of gev and probably higher energy photons . evidences of distinct high - energy component have been accumulated by egret onboard the compton gamma - ray observatory : ( 1 ) @xcite reported the detection of long - duration mev gev emission of grb 940217 lasting up to 1.5 hour after the kev burst , including an @xmath218 gev photon . this burst is the longest and the most energetic among those grbs with detected high - energy emission so far ; ( 2 ) @xcite revealed a high - energy component of grb 941017 temporally and spectrally different from the low - energy component . in the fireball model , synchrotron emission of shock - accelerated electrons is commonly thought to produce prompt @xmath3-ray emission as well as afterglow emission at lower energies ( e.g. , * ? ? ? it is natural to expect that these photons are inverse - compton up - scattered by electrons , giving rise to a higher energy component peaking at sub - gev to tev energies @xcite . when electrons scatter the self - emitting synchrotron photons , synchrotron self - compton ( ssc ) emission is resulted . in the external shock scenario , the temporal profile of the ssc emission from forward shock electrons is similar to that of the low energy afterglow emission and no significant time lag is expected . the _ swift _ satellite , thanks to its rapid response time and accurate localization , has started a new era of research on grbs . different modifications to the standard afterglow model are put forward to explain the peculiar behaviors exhibited in the x - ray light curves , in particular the shallow declining phase @xcite . recently , the ssc emission of the modified forward shock has been extensively discussed in the literature @xcite and applied to the case of grb 940217 @xcite . the agile ( astro - rivelatore gamma a immagini leggero ) satellite , launched on april 23 , 2007 , is dedicated to high - energy @xmath3-ray astronomy . the fermi gamma - ray space telescope ( fgst ) was launched on june 11 , 2008 . the large area telescope ( lat ) onboard covers the energy range from 20 mev to 300 gev and its effective area is about 5 times larger than that of egret at gev energies . the first grb observations with lat have resulted in detection of photons with energies larger than @xmath4 gev from several grbs @xcite . @xcite , @xcite , and @xcite predicted promising and detectable ssc emission from the forward shock with fgst out to z@xmath21 . most of the discussions in the literature have focused on the afterglow emission from tens of mev to gev . lat can also detect very - high - energy ( vhe ; @xmath5100 gev ) afterglow emission . however , with a small effective area @xmath6 , it is very hard to have a significant detection at such high energy . imaging atmospheric cherenkov telescopes such as h.e.s.s . , magic , and veritas may serve better at energies above @xmath2100 gev because of their much larger effective area ( @xmath7 ) and a high rejection rate of hadronic background . the effective collecting area of cherenkov telescopes increases with energy @xcite . some of these large area cherenkov detectors have been used to set constraints on the possible vhe afterglow component of grbs @xcite . it is thus desirable to see whether these results are consistent with the predictions of the fireball model . our aim of this paper is also to investigate the prospect of significant detections in the future . to have a reliable estimate of the afterglow emission at energies above 100 gev , one needs to calculate the forward shock emission ( both synchrotron and ssc emission of the shocked electrons ) carefully . the attenuation of vhe photons by the cosmic infrared background is also taken into account.since the attenuation effect for photons with an energy @xmath5100 gev is more severe for high - redshift grbs , we limit our grb sample to nearby events . this paper is organized as follows : in [ sect_afterglow_modeling ] , we describe the grb afterglow model , introduce the code that is used in the afterglow modeling and the calculation of the ssc emission from grb forward shock . in [ sect_model_prediction ] , we present the expected results of the ssc model using reasonable parameter values for grbs . in [ sect_vheafterglow_fromnearbygrbs ] , we describe the grb sample which includes six nearby grbs with sufficient multi - wavelength afterglow data and predict their corrected energy flux after the attenuation by the cosmic background during the afterglow phase , which is then compared with the available observational data . we summarize our results and discuss their implications in [ sect_discussion ] . we conclude in [ sect_conclusions ] . while synchrotron emission is widely considered to be responsible for the radio , optical , and x - ray afterglows ( e.g. * ? ? ? * ) , inverse compton scattering of forward shock photons is considered in details by @xcite and @xcite . inverse compton scattering may considerably change the temporal and spectral behavior of grb afterglows , and its cooling effect on electrons accelerated in external shocks will contribute to the photon spectra at sub - gev to tev energy range @xcite . in the afterglow model , both synchrotron emission and inverse compton emission are taken into account . it is assumed that : ( 1 ) the external medium is homogenous with a density @xmath8 or a wind profile @xmath9 ; ( 2 ) the relativistic jet is uniform , i.e. energy per solid angle is independent of direction within the jet ; ( 3 ) the shock parameters ( @xmath10 and @xmath11 , fractions of the shock energy given to the electrons and the magnetic field , respectively ) are constant ; ( 4 ) the energy distribution of electrons accelerated in shocks follows @xmath12 ; ( 5 ) the possible achromatic flattening in the afterglow light curve is due to energy injection in the form @xmath13 @xcite or @xmath14^{-1}$ ] with @xmath15 being the initial spin - down time scale @xcite . the parameters involved in this afterglow model include : @xmath16 ( the initial isotropic outflow energy ) , @xmath17 ( the initial half - angle of the jet ) , @xmath8 ( the density of the homogeneous external medium ) or @xmath18 ( the wind parameter ) , @xmath19 ( the power - law index of energy distribution of shock - accelerated electrons ) , @xmath20 , and @xmath21 ( shock parameters ) . in the case where energy injection is necessary , three additional parameters : @xmath22 ( the injected luminosity in the rest frame ) , the timescale of energy injection and @xmath23 , are included . the code used in our afterglow modeling and the prediction of the ssc emission is that developed by @xcite , who carry out numerical calculations of synchrotron and ssc emission of the external forward shock in the afterglow phase . the reverse shock emission , predicted in the fireball model but not detected in most events @xcite , is not taken into account . the key treatments ( see 3 of * ? ? ? * for details ) are as follows : ( i ) the dynamical evolution of the outflow is followed using the formulae in @xcite , which describes the hydrodynamics in both relativistic and non - relativistic phases . ( ii ) the arbitrary assumption that the distribution of shocked electrons is always in a quasi - stationary state is considered to be unsatisfactory and the energy distribution of electrons is calculated by solving the continuity equation with the power - law source function q = k@xmath24 , normalized by a local injection rate @xcite . ( iii ) the observed flux is integrated over the equal - arrival surface " . ( iv ) the klein - nishina correction of the inverse compton emission has been included . ( v ) energy injection into the outflow is considered necessary in reproducing some multi - frequency afterglow data of some grbs . this may change the dynamics significantly . by fitting the low - energy multi - waveband afterglows , from radio to x - ray band , parameters involved in the afterglow model are gotten . _ simultaneous afterglow data in at least two well - separated wavebands are needed to get a relatively well - constrained set of parameters . _ a rough estimate of the energy - integrated vhe afterglow flux ( without correction by the cosmic background ) is given by @xmath25 where @xmath26 is the total luminosity of the ssc emission , @xmath27 and @xmath28 are the typical ssc emission frequency and the ssc cooling frequency of the forward shock electrons ( see eq.(33 - 34 ) in * ? ? ? * the case of @xmath29 , for the expressions ) and @xmath30 is the luminosity distance of the event . high energy photons , especially those in the tev range , will be attenuated by the cosmic background light . various models of the spectral energy distribution of the cosmic infrared background are proposed @xcite , but all these models give comparable opacities for low redshifts . in this work , a level consistent with a study of two distant blazars and galaxy counts is used ( p0.45 , * ? ? ? we adopt reasonable values of parameters for nearby grbs and predict the spectra in high - energy to vhe range . after corrected for the attenuation by extragalactic background , we compare them with the sensitivity levels of @xmath3-ray instruments . parameters assumed and the time - averaged spectra , including both synchrotron and ssc components from the forward shocks , are shown in figure [ fig : spectrum ] . for this fictitious burst , current imaging atmospheric cherenkov telescopes would be more likely than space satellites such as egret , agile / grid and _ fgst_/lat to detect the modeled emission , as seen in figure [ fig : spectrum ] . = @xmath31@xmath32 , @xmath33 , @xmath34 , @xmath35 , @xmath36 , @xmath37 and @xmath38 . the solid and dashed line are calculated with the same parameters represented above during the first epoch but occurring at larger redshift , @xmath39 and @xmath40 , respectively . the sensitivity curves of egret @xcite , agile / grid @xcite , fgst / lat @xcite , veritas @xcite and hess ( assuming @xmath41 , * ? ? ? * ) for an integration time of 0.5 hour are plotted as labeled . magic 2-sigma upper limits derived from 30-min observations of grb 060206 are also plotted , taken from albert et al . ( 2007 ) . ] for photons with energy higher than @xmath42 gev , the attenuation due to interaction with background photons is significant if the source has a high redshift . therefore nearby bursts ( those with z@xmath430.25 ) are chosen in this study . to predict the vhe afterglow emission of nearby grbs and compare the calculated results with sensitivity of different @xmath3-ray telescopes , grbs in our sample must satisfy : nearby events to alleviate the attenuation effect ; at least two independent waveband low - energy afterglows are recorded to get relatively constrained parameters ; the low - energy afterglow can be reproduced according to the afterglow model . in this work , we consider nearby grbs ( @xmath44 ) with relatively high luminosity and multi - wavelength afterglow data sufficient to meaningfully constrain the properties of the grbs ( i.e. the model parameter values as described in [ subsect_grb_afterglow_model ] ) up to march 2007 . five grbs meet such criteria : grb 030329 , grb 050509b , grb 050709 , grb 060505 , and grb 060614 . though having a relatively large redshift of z@xmath20.55 , grb 051221a is also considered in this work because it is one of the brightest short grbs detected so far . grb 030329 triggered the high energy transient explorer , hete-2 @xcite . very detailed bvri afterglow light curves , spanning from @xmath20.05 to @xmath2 80 days , were compiled by @xcite . @xcite reported xmm - newton and rossi - xte late - time observations of this burst . based on the emission and absorption lines in the optical afterglow , a redshift of z=0.1685 has been identified @xcite . the x - ray telescope ( xrt ) onboard _ swift _ began observations of grb 050509b 62s after the trigger of the burst alert telescope ( bat ) @xcite . optical and infrared data were reported in @xcite . @xcite and @xcite reported a redshift of z@xmath20.22 based on numerous absorption features and a putative host galaxy , respectively . grb 050709 was discovered by hete-2 its prompt emission lasted 70 ms in the 3 - 400 kev energy band , followed by a weaker , soft bump of @xmath2100 s duration . the optical counterpart of this burst was observed with the danish 1.5-m telescope at the la silla observatory . the observations started 33 hours after the burst and spanned over the following 18 days @xcite . observations with the chandra x - ray observatory revealed a faint , uncatalogued x - ray source inside the hete-2 error circle @xcite , which was coincident with a pointlike object embedded in a bright galaxy @xcite at z = 0.16 @xcite . grb 051221a was localized by bat @xcite and also simultaneously observed by the konus - wind instrument . the x - ray ( @xmath45s ; * ? ? ? * ) and the optical ( @xmath46s ; * ? ? ? * ) afterglow light curves of grb 051221a were well detected , while in the radio band only one detection followed by several upper limits are available @xcite . @xcite detected several bright emission lines , indicating a redshift of @xmath47 . grb 060505 was detected by bat in the 15 - 150 kev band @xcite . @xcite reported the detection of the optical transient , later confirmed by vlt fors2 observations @xcite . xrt detected a source which was located about 4@xmath48 from a galaxy with z=0.0894 @xcite . grb 060614 triggered both _ swift_-bat @xcite and konus - wind @xcite . xrt found a very bright ( @xmath21300 counts s@xmath49 ) un - catalogued source inside the bat error circle . ground - based optical and infrared follow - up observations were performed using several instruments ( e.g. , * ? ? ? * ; * ? ? ? based on the detection of the host galaxy emission lines , a redshift of z = 0.125 was proposed by @xcite and confirmed by @xcite . noted that the classification of grb 060614 is ambiguous in the commonly - used long / short burst scheme , since it has a long duration but no accompanying sn @xcite . the available multi - frequency afterglow data are then used to obtain the model parameters . in this work , we have reproduced the multi - frequency afterglow data of grb 030329 and grb 060614 . the well - sampled distinguishing afterglow behavior of grb 030329 has gained much attention . some authors concentrated on the rebrightening occurring at 1.6 days after the trigger and considered different mechanisms to explain the rebrightening features seen in the optical light curves @xcite . we concentrate on the multi - waveband emission , from radio @xcite , optical @xcite to x - ray band @xcite for the purpose in this paper . we show in figure [ fig : fits ] that , with a certain set of parameters , the numerical results can describe the observed data in all three wavebands . fluctuations were captured in @xmath50 band afterglow light curves after @xmath51s from the burst trigger , which may imply the multiple energy injection into the outflow @xcite or a two component jet @xcite . we ignore these details and focus on the general trend of the optical emission ( particularly , in our calculation the energy of the relativistic ejecta is a constant ) . as shown in figure [ fig : fits ] , the difference is that in the time range @xmath52 sec our approach gives a ( little bit ) brighter optical emission , so will be the high energy emission . the modeled and observed afterglow light curves of grb 060614 are shown in figure [ fig : fit1 ] . energy injection , starting around 30 minutes after the grb onset , is needed in the afterglow modeling to reproduce the increase in flux ( instead of simple power - law decay seen in other grbs ) . the early x - ray flux before 500s from the burst trigger , which is much brighter than the modeled flux , results from the dominating contribution from the prompt emission . table [ sample ] lists the physical parameters derived from the afterglow modeling for grb 030329 and grb 060614 . parameters of grb 050509b , grb 050709 , grb 051221a and grb 060505 are taken from the literature , also listed in table 1 . we are interested in vhe observations during the afterglow phase when the ssc is likely to dominate ( see [ sect_discussion ] ) . vhe @xmath3-ray afterglow data of three of the grbs in the sample ( i.e. grb 030329 , grb 050509b , and grb 060505 ) are available . @xcite reported a total of 4 hours of observations , which spanned five nights , using the whipple 10-m telescope . no evidence for vhe @xmath3-ray signal was found during any of the observation periods . when combining all data , a flux upper limit of @xmath53 was derived . the first observation , lasting for about an hour , was started 64.6 hours after the burst . the 99.7% c.l . flux upper limit above an energy of @xmath54 gev derived from this observation is shown in table [ flux ] , as well as in figure [ fig : highenergy ] . the 28-minute h.e.s.s . observation of grb 030329 were taken 11.5 days after the burst @xcite . since the burst position was located above the northern hemisphere , the zenith angle of the grb observation was relatively large , i.e. 60@xmath55 , resulting in an energy threshold of 1.36 tev . no evidence for vhe @xmath3-ray signal was found . the 99% c.l . flux upper limit ( @xmath56 tev ) is @xmath57 , assuming a photon index of @xmath58 . the observations of this burst using the stacee detector employ an on - off observation mode and contain two 28-minute on / off pairs . the first on - source observation started 20 minutes after the burst and the second 80 minutes after the burst . after data quality cuts , about 18 minutes of useful on - source data remain in each observation . no evidence for vhe @xmath3-ray signal above the energy threshold of 150 gev was reported by @xcite . the 95% c.l flux upper limits ( above 150 gev , assuming a photon spectrum of @xmath59 ) were @xmath60 and @xmath61 for the first and second on - source observation , respectively ( a. jarvis private communication ) . the h.e.s.s . observations began 19.4 hours after the burst and lasted for 2 hours @xcite . no evidence for vhe @xmath3-ray signal was found . the 99% c.l . flux upper limit ( @xmath62 tev ) is @xmath63 , assuming a photon index of @xmath58 . based on the parameters obtained in 4.2 , the gev - tev emission is obtained using the code described in 2.2 . we depict the calculated high energy afterglow spectrum in figure [ fig : highenergy ] , which shows the time - integrated high energy afterglow spectrum of these six events . the solid and dashed lines represent the intrinsic ssc spectra and corrected spectra for each grb , respectively . the absorption is based on the cosmic infrared background model `` p0.45 '' @xcite . such model is constrained by the upper limits provided by two unexpectedly hard spectra of blazars at optical / nir wavelengths and is close to the lower limit from integrated light of resolved galaxies . in order to compare with the vhe observational data which are usually given in integrated photon fluxes , we integrate the spectra above the energy threshold . we consider first the grbs with vhe data . these include grb 030329 , grb 050509b , and grb 060505 . in table 2 we list the modeled integrated energy fluxes after correction due to interaction with photons from cosmic infrared background , as well as the vhe @xmath3-ray observations and the derived upper limits . all predicted fluxes are below the upper limits derived from the vhe observations . we then investigate whether a sensitive vhe instrument is expected to detect the predicted vhe signal from nearby grbs during the late afterglow phase . we use h.e.s.s . sensitivity as an example of an array of sensitive atmospheric cherenkov telescopes . the sensitivity level of h.e.s.s . detector is shown in figure [ fig : sensitivity ] , assuming a @xmath64=2.6 spectrum @xcite . assuming softer spectra , the level is higher , and the difference is about 50% between @xmath64=2.0 and @xmath64=3.0 ( c.f . we show the temporal evolution of energy fluxes ( @xmath5200gev ) of six grbs in our sample in figure [ fig : sensitivity ] , indicating only the vhe signal from grb 030329 may be above the h.e.s.s . sensitivity . for grb 030329 which is a bright burst with low redshift , the expected energy flux would be high enough to be detected with a delayed observation time of @xmath65 hours if grb position was favorable , i.e. with zenith angle @xmath66 ( and thus an energy threshold of @xmath67 gev is attained ) . in this paper , we have calculated the ssc emission from the forward shock electrons following @xcite . we shall discuss here the importance of other radiation processes in the late afterglow phase . possible vhe @xmath3-ray emission initiated from protons has been suggested @xcite . however , the proton - synchrotron component , as well as the hadron - related photo - meson electromagnetic components , is in most cases overshadowed by the ssc component of electrons in the afterglow phase . this is especially the case for the parameter values of @xmath10 and @xmath11 used here in the modeling of these six grbs ( e.g. , * ? ? ? additional ic component will play a role and will enhance any vhe emission . ssc is , in general , only a lower limit . another possible contribution to vhe emission is related to the central engine afterglow " emission , like the flares ( e.g. , * ? ? ? * ) or the plateaus followed by a sudden drop ( e.g. , * ? ? ? the ssc radiation of the late internal shocks or the external inverse compton radiation in the external forward shock front may be able to give rise to some vhe emission signals ( see fan & piran 2008 for a review ) . however , no central engine afterglow " emission has been reported in the late afterglow phase for the six bursts we studied . so it is hard for us to estimate its ability to enhance the detectability of the vhe emission . as shown in figure [ fig : sensitivity ] , we only expect detectable signal using a sensitive ground - based @xmath3-ray detector for a bright and nearby grb like grb 030329 . the rate of such nearby and energetic grbs is very uncertain . grb 940217 might have been another event of this kind @xcite . several factors that reduce the chance of detecting vhe photons can be summarized as follows : firstly , as a result of large zenith angles ( e.g. , 60@xmath55 for grb 030329 ) , the energy thresholds of some observations are relatively high ( @xmath21.4 tev ) . any vhe photon is severely attenuated by the cosmic infrared background , unless the level is very low . secondly , the observations were taken at very late epochs , e.g. 11.5 days after the burst for h.e.s.s . observations of grb 030329 , when expected vhe flux had largely decayed . thirdly , the fraction of low - redshift grbs is small , e.g. see the catalogue in @xcite . for grb 051221a ( at @xmath68 ) studied here , the attenuation is severe at energies @xmath69200 gev . despite these practical limitations , detection of vhe afterglow emission of grbs is probable . those grbs close enough ( z@xmath430.5 ) and with an intrinsic high luminosity ( like grb 030329 ) , can be detected above @xmath2200 gev when the observation is taken within @xmath210 hours after the burst . from eq . 1 , grbs with low @xmath70 , large @xmath71 , large @xmath10 , and small @xmath11 are more likely to be detected in the vhe band . besides , in general the expected vhe afterglow flux decays as @xmath72 where @xmath73 is at least one . thus , observations with shorter delay are more likely to probe the predicted vhe emission . _ the main purpose of this work is to provide a relatively reliable prediction of the detectability of the vhe emission from nearby grbs_. it is also interesting to explore the role of the vhe afterglow emission detection in revealing the grb physics . perhaps the most robust conclusion is that the vhe emission in late afterglow phase can not be attributed to the synchrotron radiation of the forward shock , for which the maximal photon energy is @xmath74 @xcite , where @xmath64 is the bulk lorentz factor of the decelerating outflow . in general , for @xmath75 , the inverse compton emission is expected to be very weak . so the detection of vhe afterglow emission from grbs requires that @xmath76 , in support of the current radio / optical / x - ray afterglow modeling ( e.g. , see tab.1 ) . together with _ , agile , fgst and some ground - based optical telescopes , ground - based @xmath3-ray detectors can provide us continuous spectra in the optical to tev energy band during the afterglow phase . a self - consistent modeling of these data in a very wide energy band , in principle , will impose very tight constraint on the physical parameters and on the environment . however , given the small number of the vhe photons expected from a single burst , we do not expect that the vhe emission can help us a lot to achieve such goals . in this work , we discuss the prospect of detecting vhe @xmath3-rays with current ground - based detectors in the late afterglow phase . during this phase , the dominant radiation process in the vhe @xmath3-ray regime is the ssc emission from the forward shock electrons . klein - nishina effects and attenuation by the cosmic infrared background , both known to suppress the vhe @xmath3-ray spectra , were taken into account . to minimize the attenuation effect , we chose a sample of six nearby grbs in this study . we have calculated the detailed ssc emission numerically using the model developed by @xcite , with a set of parameters which are able to reproduce the available multi - wavelength afterglow light curves . the results are consistent with the upper limits obtained using vhe observations of grb 030329 , grb 050509b , and grb 060505 . moreover , assuming observations taken 10 hours after the burst , the vhe signal predicted from five grbs is below the sensitivity level of a current sensitive atmospheric cherenkov detectors mainly due to the low fluence of these outflows . for those bright and nearby bursts like grb 030329 , a vhe detection is possible even with a delayed observation time of @xmath210 hours . we thank the anonymous referee for helpful comments and d. xu for providing us data in figure 3 . this work is supported by the national science foundation ( grants 10673034 and 10621303 ) and national basic research program ( 973 programs 2007cb815404 and 2009cb824800 ) of china . yzf is also supported by a grant from danish national research foundation . pht acknowledges support from imprs - hd . aharonian , f. , et al . 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the synchrotron self - compton ( ssc ) emission from gamma - ray burst ( grb ) forward shock can extend to the very - high - energy ( vhe ; @xmath0100 gev ) range . such high energy photons are rare and are attenuated by the cosmic infrared background before reaching us . in this work , we discuss the prospect to detect these vhe photons using the current ground - based cherenkov detectors . our calculated results are consistent with the upper limits obtained with several cherenkov detectors for grb 030329 , grb 050509b , and grb 060505 during the afterglow phase . for 5 bursts in our nearby grb sample ( except for grb 030329 ) , current ground - based cherenkov detectors would not be expected to detect the modeled vhe signal . only for those very bright and nearby bursts like grb 030329 , detection of vhe photons is possible under favorable observing conditions and a delayed observation time of @xmath110 hours .

You are an expert at summarizing long articles. Proceed to summarize the following text: frustrated quantum magnets can exhibit highly unconventional ground states , in which local moments are highly correlated but nevertheless evade a conventional ordering transition and remain strongly fluctuating down to zero temperature . these unusual states are commonly referred to as quantum spin liquids @xcite despite their rather diverse physical properties ranging from gapped states with an emergent topological order to gapless states with an emergent spinon fermi surface . a common motif in the search for quantum spin liquids has been to look for quantum antiferromagnets on geometrically frustrated lattices , i.e. lattices where the elementary building blocks prohibit the formation of a conventional nel state . paradigmatic examples of geometric frustration include lattices formed by corner - sharing tetraedra such as the pyrochlore lattice , or by corner - sharing triangles such as the kagome lattice in two spatial dimensions and the hyperkagome lattice in three spatial dimensions . an alternative route to induce frustration in a quantum magnet is to look for systems , in which competing interactions can not be simultaneously satisfied . archetypal examples of such exchange frustration are given by the quantum compass models @xcite , in which the easy - axis of an anisotropic spin exchange strongly depends on the spatial orientation of the exchange path a scenario which can prohibit even a ferromagnet on a bipartite lattice from undergoing a finite - temperature ordering transition . the best known example in this class of compass models is the kitaev model @xcite on the honeycomb lattice , in which the easy - axis of an ising - like spin exchange points along the @xmath5 , @xmath6 , and @xmath7 directions for the three different bond types of the hexagonal lattice , which is captured by the hamiltonian @xmath8 where su(2 ) spins @xmath9 on sites @xmath10 and @xmath11 are connected via a bond in the @xmath12 direction . the kitaev model is quintessential in that it harbors three different types of quantum spin liquids a gapped , @xmath13 topological spin liquid if one of the three exchange couplings is significantly larger than the couplings associated with the two other bond directions ( i.e. @xmath14 ) , and a gapless spin liquid in the vicinity of equal - strength exchange couplings ( @xmath15 ) . if applying an external magnetic field along the @xmath16-direction , the latter can be gapped out into a topological spin liquid with non - abelian vortex excitations . the kitaev model not only stands out for the unusual richness of its ground states , but the fact that it is one of the very few examples of an interacting spin model that can be rigorously solved . it should , however , be pointed out that the kitaev model has not only attracted the imagination of phenomenologically inclined theorists , but has also stirred some excitement in the materials oriented community after it has been pointed out that the significantly enhanced spin - orbit coupling in 5d transition metal oxides and in particular certain iridates can give rise to unconventional mott insulators where the local moment is a spin - orbit entangled @xmath3 moment @xcite . the orbital contribution to these moments results in a highly anisotropic , spatially oriented exchange @xcite , which can in fact mimic those of the kitaev model . in terms of actual materials the layered iridates na@xmath1iro@xmath2 and li@xmath1iro@xmath2 have attracted much recent interest and are intensely discussed @xcite as possible candidate materials realizing the two - dimensional honeycomb kitaev model . in this manuscript , we turn to generalizations of the kitaev model on three - dimensional lattices a move that is prompted by the recent synthesis of @xmath0-li@xmath1iro@xmath2 @xcite , which forms a truly three - dimensional lattice structure of the ir@xmath4 ions . this structure , which has quickly been dubbed hyperhoneycomb lattice @xcite , keeps the trivalent vertex structure of the hexagonal lattice and thereby the essential feature allowing for an analytical solution of the kitaev model . in fact , the kitaev model on the hyperhoneycomb lattice had been identified and studied before by mandal and surendran @xcite who reported the occurrence of a gapless spin liquid with an emergent spinon fermi surface on a line in momentum space for approximately equal - strength interactions ( @xmath15 ) as well as the occurrence of a gapped topological spin liquid for anisotropic exchange strength @xcite . more recently , extensions to a heisenberg - kitaev model @xcite have established the stability of this gapless phase in the presence of weak isotropic spin exchange @xcite . this motivated us to ponder alternative three - dimensional lattices that keep the trivalent vertex structure and led us to consider what we call the hyperoctagon lattice @xcite illustrated in fig . [ fig : hyperoctagonlattice ] . the hyperoctagon lattice is closely related to the hyperkagome lattice the hyperoctagon lattice is the premedial lattice of the hyperkagome lattice obtained by shrinking each triangle of the hyperkagome lattice to a single vertex and the new bonds indicating the original connectivity of the triangles , schematically summarized in fig . [ fig : latticesummary ] . the hyperoctagon lattice is a chiral lattice , which contains two elementary motifs a spiraling octagonal helix and a counter - spiraling square helix as illustrated in fig . [ fig : hyperoctagonlattice ] . its space group i4@xmath1732 ( no . 214 ) indicates the presence of 4- , 3- and 2-fold ( skew ) symmetries ( the details of which we will provide below ) that will turn out to play an essential role in stabilizing the gapless modes of the quantum spin liquid emerging for the kitaev model on this lattice . the presence of these symmetries is also the key distinction to the hyperhoneycomb lattice , another somewhat less symmetric three - dimensional trivalent lattice structure which has been revealed in the recent synthesis of @xmath0-li@xmath1iro@xmath2 . our main result is the observation of a gapless quantum spin liquid with an extended two - dimensional majorana fermi surface around the point of isotropic couplings for the kitaev model on the hyperoctagon lattice . this result is rigorously established by an exact analytical solution of the spin model , which can be cast into a free fermion system by majorana fermionization , thus employing the same powerful techniques that have already allowed the solution of the kitaev model on other trivalent lattices @xcite . our discussion in the remainder of the paper is structured as follows . in section [ sec : lattice ] we will discuss trivalent lattice structures in two and three spatial dimensions and in particular provide a detailed introduction of the hyperoctagon lattice . the kitaev model on the hyperoctagon lattice is subsequently introduced and exactly solved in section [ sec : kitaev_model ] where we also provide a detailed discussion of its ground state phase diagram , in particular the gapless spin liquid with a majorana fermi surface emerging for a broad range of parameters . possible instabilities of the majorana fermi surface are discussed in section [ sec : fermisurface ] . we conclude with an outlook in section [ sec : outlook ] . before we dive into the physics of the kitaev model we start our discussion with a short review of the underlying lattice structure . in its original form the kitaev model has been discussed for the honeycomb lattice , a two - dimensional lattice with a trivalent coordination of all vertices as depicted in fig . [ fig : honeycombs ] a ) . keeping this motif of a trivalent lattice structure the model can readily be associated with a broader class of lattices a move that not only allows a @xmath18 assignment of the three different exchange types to the bonds around the vertices , but is also key to keep the analytical tractability of the model , which we will review in the following section @xcite . in two spatial dimensions one such generalization is the square - octagon lattice of ref . . in three spatial dimensions such trivalent lattices are considerably less common . one example is the so - called hyperhoneycomb lattice , which is depicted in fig . [ fig : honeycombs ] b ) . the elementary building blocks of the hyperhoneycomb lattice are zig - zag chains running along the crystallographic @xmath19 and @xmath20 axis , respectively , as depicted in fig . [ fig : honeycombs ] b ) . these zig - zag chains are coupled by bonds along the @xmath21 axis , readily implying that there is no general symmetry possibly interchanging the three crystallographic axises . another example of a trivalent lattice in three dimensions is the so - called hyperoctagon lattice , which we describe in detail in the following . the hyperoctagon lattice is a body - centered cubic lattice without inversion symmetry . its symmetries correspond to space group i4@xmath1732 ( no . 214 ) . in cartesian coordinates , the atomic positions can be constructed starting from the point @xmath22 in the unit cell and applying all symmetry transformations of the space group on it . in particular , the symmetries of space group i4@xmath1732 include the following : i ) a four - fold symmetry which is obtained by 90 degree _ screw_-rotations around the ( 1,0,0 ) , ( 0,1,0 ) , or ( 0,0,1 ) directions . ii ) a three - fold symmetry which leave the lattice invariant under 120 degree rotations around the ( 1,1,1 ) , ( -1,1,1 ) , ( 1,-1,1 ) , or ( 1,1,-1 ) directions , and iii ) a two - fold symmetry corresponding to 180 degree rotations around the directions @xmath23 , @xmath24 , and @xmath25 . as a guide to the eye , [ fig : rotations ] shows the projection of the lattice onto the planes normal to the rotation axis for three examples of the above ( screw-)rotations . note that the ( projected ) square - octagon structure in fig . [ fig : rotations ] a ) is not planar , which is why the additional translation is needed . @xmath26 ) . a ) view along ( 0,1,0 ) as an example of a 90 degree screw - rotation symmetry . b ) view along ( 1,1,-1 ) as an example of a 120 degree rotation symmetry . c ) view along ( 0,1,1 ) as an example of a 180 degree rotation symmetry . ] while the hyperoctagon lattice arises quite naturally as the premedial lattice of the hyperkagome lattice , which is realized for instance in the spin liquid material na@xmath27ir@xmath2o@xmath28 @xcite , there are so far no known realizations of the hyperoctagon lattice in the diverse family of recently synthesized iridates . to provide some abstract guidance as to which chemical compositions might possibly realize magnetic hyperoctagon systems , we have made an attempt at designing possible materials candidates . with the iridium atoms assumed to occupy the sites of the hyperoctagon lattice , our further thinking is guided by the microscopic prerequisites that allow a dominant anisotropic kitaev - like interaction to emerge the occurrence of double ir - o - ir exchange paths that suppress the isotropic spin exchange @xcite . the latter can be achieved by placing the iridium atoms in bond - sharing iro@xmath29 cages . in fact , the symmetries of the hyperoctagon lattice allow to embed each iridium atom into a perfectly undistorted iro@xmath29 octahedron , when placing the oxygen atoms at position @xmath30 in the unit cell . the resulting iro@xmath2 structure is illustrated in fig . [ fig : o6cages ] . such a sparse octahedron structure has indeed been observed for the subhalides , e.g. la@xmath2br@xmath2si @xcite , where the silicium atoms form the hyperoctagon lattice and the lanthanum atoms form octahedra around them . finally , one might want to attempt to fill the remaining interstitial sites of the octahedron structure . taking into account the space group symmetries there are several distinct ways of doing so , as described in some detail in appendix [ sec : appspacegroupmaterials ] . in particular , one might start to add atoms to a single interstitial site ( and its space group related siblings ) as illustrated in fig . [ fig : space_group ] a ) of the appendix . this would result in the chemical composition of the alkaline - earth iridates airo@xmath2 where a is one of the alkaline - earth elements ca , sr , or ba . the alkaline - earth iridates are known to exhibit quite distinct electronic properties for the different a - site materials , including an @xmath31 antiferromagnetic mott insulator for cairo@xmath2 @xcite , a weak ferromagnetic semiconductor for bairo@xmath2 @xcite and a non - fermi liquid metal in sriro@xmath2 @xcite . while various crystal structures have been reported for the different airo@xmath2 compounds , no crystals in space group i@xmath32 have so far been synthesized for any of the alkaline - earth iridates . an alternative possibility to fill the interstitial sites is to add two additional atoms resulting in a chemical composition of the form a@xmath1iro@xmath2 with a being one of the alkali metals na or li . the resulting crystal structure is illustrated in fig . [ fig : space_group ] b ) of the appendix . this is a particularly interesting idea to entertain as it would point to a possible existence of a third crystallization pattern for a@xmath1iro@xmath2 beyond the already known examples of quasi two - dimensional honeycomb layers and the recently synthesized three - dimensional hyperhoneycomb structure . octahedra around the central ir atoms . a ) view along the ( 1,0,0 ) direction . b ) view along the ( 1,1,1 ) direction . ] not only motivated by a possible relevance to future materials , but also driven by a curiosity to explore unusual spin liquid states we now turn to a three - dimensional variant of the kitaev model on the hyperoctagon lattice . we proceed with an introduction and precise definition of the model and discuss some of its general properties before presenting an analytical solution of the model in terms of an exact majorana fermionization of the spin degrees of freedom . finally , we present our main result of identifying a gapless spin liquid ground state with a majorana fermi surface . kitaev originally introduced his elementary spin model as a system of su(2 ) spin-1/2 degrees of freedom interacting on the two - dimensional honeycomb lattice . its fundamental beauty not only arises from its exact analytical solution , but the fact that the spin model harbors a number of paradigmatic ground states besides an abelian topological phase , it exhibits an extended gapless spin liquid ground state , which can be gapped out into a non - abelian topological phase by an external magnetic field . this variety of different ground states arises from highly frustrated spin interactions which favor the alignment ( or anti - alignment ) of different spin components along the three principle directions of the honeycomb lattice . here we generalize this idea to the three - dimensional hyperoctagon lattice , which because of its trivalent vertices allows the definition of an analogous spin model . to this end , we cover the lattice with bonds that favor spin alignments along the @xmath5 , @xmath6 , and @xmath7 directions and which we call @xmath33 , @xmath34 , and @xmath35-bonds , respectively . while there are many different ways to realize such coverings on a given lattice , there is only a single covering that is compatible with all the lattice translation symmetries of the hyperoctagon lattice . this unique covering is illustrated in fig . [ fig : kitaevmodel ] and will serve as our primary definition of the kitaev model on this lattice . we will briefly discuss alternative models based on other coverings in section [ sec : fermisurface ] . -couplings , red bonds to @xmath34-couplings , and blue bonds to @xmath35-couplings , respectively . ] in order to provide a self - contained description of the model , we start by introducing the four - site unit cell compatible with the unique covering of exchange bonds such that all lattice translation symmetries are kept . the atomic positions in this unit cell are given by @xmath36 where @xmath37 is the unit cell position . the offset in @xmath38 is chosen to be consistent with the conventions in section [ sec : lattice_symmetries ] and can be mostly ignored in the following discussion . the corresponding lattice translation vectors are then given as @xmath39 which are also illustrated in fig . [ fig : kitaevmodel ] . with these definitions in place we can now define the kitaev hamiltonian on the hyperoctagon lattice as @xmath40 for the following discussion , it is beneficial to introduce a ` bond operator ' @xmath41 for a bond @xmath42 @xmath43 in terms of these bond operators , the hamiltonian then reduces to the compact form introduced earlier @xmath44 our first step in analyzing hamiltonian is to identify conserved quantities , which we will find to be intimately connected to closed paths ( or loops ) on the lattice . the elementary loops of the hyperoctagon lattice have length ten . for each unit cell there are six distinct such loops , which are visualized in fig . [ fig : loops_hyperoctagon ] . all other elementary loops can be obtained by lattice translations . for each loop @xmath45 we can define a corresponding loop operator @xmath46 , which measures the ` magnetic flux ' through the plaquette that is enclosed by @xmath45 . we can define the loop operator by the product of bond operators of all the bonds contained in the loop @xmath47 because of the even length of the loops these loop operators square to the identity , thus they have eigenvalues @xmath48 . it can further be verified that the loop operators commute with the hamiltonian as well as with each other . each loop operator thus defines an ` integral of motion ' and a corresponding conserved quantity the extensive number of which greatly simplifies the problem . for one , we can divide the hilbert space into distinct sectors that are each labeled by the eigenvalues of all the loop operators @xmath46 and restrict the hamiltonian to a particular sector . before proceeding we note that an alternative definition of the loop operators can be formulated as a product over all sites contained in the loop : @xmath49 where @xmath50 corresponds to the spin component at each vertex , which is not included along the loop . for instance , the loop operator of the loop in fig . [ fig : loops_numbered ] a ) is given by @xmath51 in both the honeycomb and hyperhoneycomb model , the two definitions of the loop operator , @xmath46 in and @xmath52 in , are identical . however , in our particular case there is a relative minus sign between the two , i.e. @xmath53 , which leaves some freedom in how to define the magnetic flux . in the following , we define magnetic flux by the eigenvalue of : if @xmath46 has eigenvalue @xmath54 we say that there is a magnetic flux ( vortex ) penetrating the plaquette enclosed by @xmath45 , while an eigenvalue @xmath55 corresponds to no flux @xcite . a further important difference to the two - dimensional case is that the loop operators are not all linearly independent . as an example , we consider the two loops depicted in fig . [ fig : loops_numbered ] . the loop operator for the loop in fig . [ fig : loops_numbered ] a ) is given in eq . , while the one for the loop in fig . [ fig : loops_numbered ] b ) is given by @xmath56 note that we can define a third loop of length ten by combining the bonds that are contained in @xmath57 or @xmath58 , but not in both of them ( see the loop illustrated in panel e ) of fig . [ fig : loops_hyperoctagon ] ) @xmath59 the product of the three loops is the identity operator @xmath60 which implies that the eigenvalue of @xmath61 is uniquely determined by the ones of @xmath57 and @xmath58 . a direct consequence of this linear dependence of three loops is that there is no full - flux sector in this model for which all loop operators have eigenvalue -1 . note that if we would have chosen the alternative definition of the loop operator in eq . then we would have concluded that there is no zero - flux sector in this model . this constraint can also be understood graphically . each loop defines an enclosed surface as illustrated in fig . [ fig : volumes_hyperoctagon ] . in general , a product of loop operators is constrained if their respective surfaces form a closed ` volume ' . for the hyperoctagon lattice , note that the three surfaces corresponding to the respective loops indeed form a closed object though the encompassed volume actually vanishes . this situation should be contrasted to the three - dimensional hyperhoneycomb lattice , which for a complete and self - consistent presentation we discuss in appendix [ app : hyperhoneycomb ] . in the hyperhoneycomb lattice , four loops of length ten encompass a closed volume as depicted in fig . [ fig : loops_hyperhoneycomb ] in the appendix and are thus linearly dependent . however , with an even number of linear dependent loops , the corresponding spin model allows for both a zero - flux and a full - flux sector , while in our case only one of the two sectors can exist . in order to gain a better insight into the physics underlying the different magnetic flux sectors , let us note that the elementary loops of length ten can be uniquely labeled by their midpoint @xmath62 . these midpoints form a ( deformed ) hyperkagome lattice . the constraint on the loop operator eigenvalues is then enforced on each of the triangles of the hyperkagome lattice : there are either zero or two loops per triangle that carry flux . we can thus count the number of independent configurations by noting that there are six midpoints ( = loops ) and four triangles ( = constraints ) per unit cell . thus , there are only @xmath63 loop eigenvalues per unit cell that can be chosen freely . as a result there are in total @xmath64 distinct flux sectors , where @xmath65 is the number of unit cells . in order to determine in which magnetic flux sector the ground state resides , we can not follow the same route as taken in two spatial dimensions and resort to lieb s theorem stating that the ground state always resides in the flux - free sector @xcite , but instead have to carefully consider the energetics of the different flux sectors / loops . for the great majority of points in parameter space , we numerically observe that creating or enlarging loops costs energy and as such the ground state lies in the flux - free sector also in three spatial dimensions . we therefore restrict our following discussion to the zero - flux sector . let us briefly consider the effective theory for the magnetic flux excitations arising from flipping a loop operator eigenvalue from @xmath55 to @xmath54 . we note that the midpoints of loops with loop operator eigenvalue @xmath54 form themselves closed loop configurations , which live again on the links of a hyperoctagon lattice albeit with opposite chirality to the original one . due to the constraint , only closed loop configurations are allowed , i.e. there are no magnetic monopoles in this @xmath66 theory . we now proceed to discuss the exact analytic solution of hamiltonian . in analogy to the two - dimensional kitaev model , such an analytical solution is possible by recasting the original spin degrees of freedom in terms of majorana fermions a step that effectively reduces the interacting spin system to a free fermion problem , which is given by majorana fermions hopping in a static gauge field @xcite . the fermion system can thus be diagonalized in a straight - forward way thereby also revealing the physics of the interacting spin model . as first step , we rewrite the original spin degrees of freedom by introducing four majorana fermion degrees of freedom @xmath67 and @xmath20 per spin @xmath9 , such that @xmath68 where @xmath69 denotes the spin component . the four - dimensional hilbert space of the four majorana fermions can be projected back to the two - dimensional physical hilbert space of the original spin degrees of freedom by requiring @xmath70 as we need to introduce four majorana fermions per site , we have to introduce additional labels to indicate the unit cell index @xmath11 as well as the unit cell position @xmath38 , i.e. @xmath71 . the majorana fermions obey the usual anti - commutation relations @xmath72 of a clifford - algebra . in terms of the majorana fermions , the hamiltonian becomes @xmath73 where we introduced the link operators @xmath74 with @xmath75 being the label of the bond @xmath42 . the link operators commute among themselves as well as with the hamiltonian , which implies that we can fix the eigenvalues of all the link operators i.e. choose a specific ` reference configuration ' and compute the spectrum of the resulting quadratic hamiltonian for any given @xmath76 sector . when doing this , one needs to define a direction on the bonds , as @xmath77 . we choose the convention that the @xmath33-bonds are directed along the @xmath78-direction , the @xmath34-bonds along the @xmath79-direction , and the @xmath35-bonds along the @xmath80-direction . this convention ensures that the following discussion remains symmetric in permutations of @xmath81 , @xmath82 , and @xmath83 . one may think of the link degrees of freedom as a static @xmath66 gauge field . the gauge transformations are generated by the @xmath84 operators and eq . is equivalent to demanding the physical states to be gauge invariant . in fact , the gauge invariant objects are precisely the loop operators introduced earlier . choosing a reference configuration is equivalent to choosing a specific gauge . the physical properties , such as the majorana excitation spectrum , are independent of the specific gauge choice , as was already pointed out in kitaev s original solution @xcite of the honeycomb model . as we restrict our discussion to the flux - free sector , we may choose all link operators to have eigenvalues @xmath55 . using the fourier transformation @xmath85 with @xmath65 being the number of unit cells , we can compute the majorana hamiltonian in momentum space @xmath86 where @xmath87 is defined as the coefficient of the reciprocal lattice vectors @xmath88 with @xmath89 arriving at hamiltonian has thus reduced the original problem to a four - band hamiltonian that can be easily diagonalized @xcite . from the diagonal form of the hamiltonian we can readily read off the elementary structure of the phase diagram in @xmath90-parameter space by carefully analyzing the excitation spectrum of the majorana sector . in particular , we observe that the hamiltonian allows for zero - energy solutions indicative of a gapless phase in a range of parameters , while both the excitations of the majorana sector and the magnetic flux sector remain gapped in other parts of the phase diagram . ( indicated by a point ) . around the three corners of the phase diagram where one of the couplings dominates extends a gapped spin liquid phase , which is separated from the gapless phase via a line of ( continuous ) phase transitions indicated by the yellow line . ] the occurrence of zero - energy solutions is equivalent to requiring that @xmath91 , which becomes @xmath92&=\frac 1{16 } ( j_x^4+j_y^4+j_z^4 + 2j_y^2 j_z^2 \cos(k_x ) \nonumber\\ & + \ , 2j_x^2j_z^2\cos(k_y)+2j_x^2 j_y^2 \cos(k_z ) ) \nonumber\\ & \equiv 0\end{aligned}\ ] ] in cartesian coordinates . in order to determine whether or not the above equation has solutions , let us analyze the limiting behavior of the determinant . expression is bounded from above by @xmath93 & \leq \frac 1{16 } ( j_x^2+j_y^2+j_z^2)^2,\end{aligned}\ ] ] when setting @xmath94 and bounded from below by @xmath95 & \geq - \frac 1{16 } ( j_x+j_y -j_z)(j_x+j_z - j_y ) \nonumber\\ & \times ( j_y+j_z - j_x)(j_x+j_y+j_z ) , \end{aligned}\ ] ] when setting @xmath96 . as the upper bound is always strictly positive , there is a zero - energy solution iff the lower bound is negative ( or zero ) . the latter is equivalent to requiring the triangular inequality @xmath97 . in the region around the isotropic point @xmath98 , there are gapless modes in the majorana sector and the ground state is a gapless spin liquid . if one of the three coupling dominates the majorana spectrum remains gapped and the ground state is a gapped spin liquid . the gapped and gapless phases are connected via lines of phase transitions which are parametrized by the equalities in the triangular inequality . it should be noted that the fundamental shape of this phase diagram is precisely the same one as the ones found for the two - dimensional honeycomb lattice @xcite and the three - dimensional hyperhoneycomb lattice @xcite . what sets the phase diagrams apart is the actual nature of the two principle gapped and gapless phases for the respective lattices as we will discuss in the following section . the fundamental distinction in the phase diagram of the hyperoctagon model is the nature of the gapless phase in the vicinity of the isotropic coupling point ( @xmath98 ) . its main feature is an extended two - dimensional majorana fermi surface of gapless modes . to see the emergence of such a fermi surface in the majorana spectrum around the isotropic coupling point one needs to invert eq . , which gives @xmath99 \ , . \label{eq : surface}\end{gathered}\ ] ] in combination with the requirement that the lower bound in eq . becomes negative or zero this parametrizes an entire manifold of @xmath100-points , or more precisely two distinct , non - intersecting continuous surfaces in momentum space centered around the corners of the brillouine zone at @xmath101 , respectively , as illustrated in fig . [ fig : majoranafermisurface ] . it is important to note that while the two surfaces are symmetry related they can not be mapped onto each other by a reciprocal lattice vector . as a direct consequence momentum conservation ensures that the zero - energy modes can not gap out in a pairwise fashion and the surfaces have to remain stable throughout the gapless region . indeed varying the coupling constants away from the point of isotropic coupling only deforms the surfaces , but does not destroy them . this is illustrated in the sequence of panels of fig . [ fig : majoranafermisurface ] where we plot the evolution of the surfaces along a line in parameter space defined by @xmath81 and @xmath102 . starting from the isotropic point and increasing @xmath81 elongates the surface along the @xmath80-direction and contracts it in the orthogonal @xmath78- and @xmath79-directions as illustrated in the upper panel of fig . [ fig : majoranafermisurface ] . upon further increasing @xmath81 the surface contracts towards the corner of the brillouine zone . as one approaches the phase transition to the gapped spin liquid at @xmath103 , the surfaces have reduced to the points @xmath104 at the corners of the brillouine zone . on the other hand , decreasing @xmath81 from the isotropic point flattens the surface in the @xmath80-direction as illustrated in the lower panel of fig . [ fig : majoranafermisurface ] . as @xmath81 goes to zero the two opposite sides of the surface approach each other and touch precisely at the decoupling point @xmath105 ( and at which the notion of a two - dimensional surface ultimately breaks down as well ) . to reveal the nature of the zero - energy gapless modes we plot the dispersion of the four principal bands of hamiltonian along certain high - symmetry lines in fig . [ fig : dispersion ] . for the entire gapless phase the dispersion of the bands crossing zero energy is always linear along the direction normal to the surface reminiscent of the energy spectrum of a fermi liquid in the vicinity of the fermi energy . one should however keep in mind that the four bands in our hamiltonian are not spanned by conventional fermionic degrees of freedom , but by majorana fermions . as such the zero - energy surfaces revealing themselves in the energy spectrum should in fact be thought of as majorana fermi surfaces . as the phase transition to the gapped phase at @xmath106 is approached , the relevant majorana band moves up in energy and at @xmath103 no longer crosses the zero - energy level , but merely touches @xmath107 in a single point with a quadratic dispersion . this scenario is in complete analogy to the phase transitions from the gapless to gapped majorana phases in both the honeycomb and hyperhoneycomb models . for conventional fermi liquids it is well appreciated that the fermi surface is susceptible to a variety of instabilities , the most notable of which is the formation of superconductivity . as such two questions immediately arise with regard to the majorana fermi surface in our hyperoctagon model why is the majorana fermi surface stable in the first place and what are its possible instabilities ? we will address the first question the stability of the majorana fermi surface in the following by showing that it can be tracked to the underlying lattice symmetries . we will then devote the remainder of this section to a discussion of possible instabilities of the majorana fermi surface focusing on instabilities arising from a reduction in lattice symmetries . to discuss the stability of the majorana fermi surface let us first recall some basic facts about majorana fermions . an immediate consequence of the majorana condition @xmath108 in real space is that the ` creation operator ' @xmath109 in momentum space is defined by @xmath110 . the latter implies that for every energy state @xmath111 there is a ` particle - hole - conjugate ' partner at @xmath112 for which @xmath113 quite generally additional energy relations might exist , which depend on the underlying lattice geometry . of particular importance in our case is the bipartite nature of the lattice . for a majorana hamiltonian on a bipartite lattice with vanishing intra - sublattice hopping amplitudes , one can verify that @xmath114 where @xmath115 is the reciprocal lattice vector of the translation relating the two sublattices . note that for the honeycomb and hyperhoneycomb lattice we have two - site and four - site unit cells , which allow @xmath116 and , combining the two energy relations found above , @xmath117 . this implies that zero - energy modes always occur in pairs at a given momentum . on the other hand , it is important to note that for the hyperoctagon lattice @xmath115 generically does not vanish , because its elementary four - site unit cell is not consistent with a bipartite coloring of the lattice , see also fig . [ fig : kitaevmodel ] . for the specific example discussed in the previous section , @xmath118 . as a consequence , the zero - modes are in general all separated in momentum space . in order to see what effect this has on the stability of the gapless modes , we start by revising the situation for the honeycomb model following kitaev s original arguments and afterwards extend this discussion to the three - dimensional generalizations . in the honeycomb model , the majorana hamiltonian in momentum space is a @xmath119 matrix of the form @xmath120 where @xmath121 is a complex - valued function . the vanishing diagonal elements are in fact protected by time - reversal symmetry @xcite . the eigenvalues of the hamiltonian are given by @xmath122 and zero - energy modes occur when @xmath123 . let us start by noting that the conditions @xmath124=0 $ ] and @xmath125=0 $ ] define ( several ) closed lines in momentum space , denoted in the following by @xmath126 and @xmath127 . the zeroes of @xmath121 are then , in general , given by the intersections of these lines . as a consequence , each pair of zero - modes at momentum @xmath128 comes with a partner , which in fact is located at @xmath112 . changing parameters deforms the line @xmath129 , which in turn moves the zeroes . the only way to gap out the system is by moving the lines @xmath129 sufficiently , such that they do not intersect any longer . the phase transition corresponds to a situation , where @xmath126 and @xmath127 merely touch . this structure of the eigenenergies readily confirms that _ separated pairs _ of zero - modes are stable in a finite parameter regime . on the other hand , a similar line of reasoning shows that _ lines _ of zero - modes are not stable and can generically be gapped out completely by even an infinitesimal change in parameters . there is , however , a generic way to stabilize lines of gapless modes in two - dimensional generalizations of the kitaev model , which we will comment on below . we now extend the above discussion to the three - dimensional models , first considering the hyperhoneycomb model . in fact , the arguments of this section are valid for any three - dimensional kitaev - type model on a bipartite lattice , which is time - reversal invariant and where the unit cell is compatible with a bipartite coloring of the lattice . these assumptions are sufficient to determine the majorana hamiltonian to be a block matrix , where only the off - diagonal matrices are non - vanishing @xmath130 where @xmath131 is a complex matrix . the eigenvalues of the hamiltonian are given by @xmath132 , where @xmath133 are the eigenvalues of @xmath131 . analogously to the two - dimensional case , there are zero - energy solutions for @xmath134 . however , as the model is three dimensional the conditions re@xmath135=0 $ ] and im@xmath134 now define ( several ) surfaces in momentum space , denoted again by @xmath129 . zero - energy modes occur at the intersection of @xmath126 with @xmath127 and thus in general form closed lines in momentum space . changing parameters in the model deforms the surfaces @xmath129 and , thus , the corresponding line of gapless modes , but can not induce a gap in the system . the latter can only be done by changing parameters sufficiently , such that the surfaces @xmath126 with @xmath127 do not intersect any longer , during which the line shrinks to a single point and vanishes . this shows that extended lines of gapless modes are topologically stable in these types of models . a similar line of reasoning substantiates that this does not apply to separated points as well as surfaces of gapless modes both of which are not stable objects for these kinds of models and can only be accidental . the important distinction of the hyperoctagon model to the ones discussed above , is that the unit cell is not compatible with a bipartite coloring of the lattice . time - reversal invariance thus only requires the diagonal elements to vanish . the hamiltonian can then be written as @xmath136 which is a band hamiltonian with the additional property @xmath137 due to the majorana condition @xcite . zero - energy modes occur , when bands cross the @xmath107 line , which results in _ surfaces _ of gapless modes . in general , the incompatibility of the unit cell and the bipartite coloring of the lattice implies that @xmath138 . as a result , there is generically only a _ single _ majorana zero - mode at a given momentum . the surfaces are thus trivially stable changing parameters deforms the energy bands and thus the surfaces , but gapping a surface can only be done by either shrinking it to a point or by superimposing two such surfaces , in which case there are two gapless majorana modes at the same momentum . the latter is not stable and can always be gapped out by an infinitesimal change in parameters . likewise , one can verify that lines or separated points of gapless modes are not stable objects for these types of majorana hamiltonians . this line of reasoning also sheds light on how to obtain two - dimensional models with a stable fermi surface . in analogy to the case above , one needs to consider two - dimensional lattices , where the unit cell is not compatible with the bipartite coloring of the lattice . an example would be the square - octagon lattice studied in @xcite . considering various flux sectors , which do not enlarge the unit cell , indeed demonstrates that separated points of gapless modes are not stable , while closed lines are . following the line of reasoning in the previous subsection points to a natural way to destabilize the fermi surface by enlarging the unit cell such that the two majorana fermi surfaces are mapped onto each other . this requirement is identical to identifying an enlarged unit cell that allows for a bipartite coloring of the lattice within that unit cell ( and thus @xmath115 vanishes ) and the disappearance of a lattice symmetry that prohibits the zero - energy modes from gapping out pairwise . below we will see that the majorana fermi surface is indeed no longer stable when enlarging the unit cell . however , the surface does not gap out completely , but instead reduces to a line of gapless modes similar to the situation of the kitaev model on the hyperhoneycomb lattice . a closed line of gapless modes is , on the other hand , a topologically stable object for three - dimensional hamiltonians such as . changing parameters can only deform the line , but not gap it out . in order to elucidate this , let us consider an alternative covering of the hyperoctagon lattice with @xmath33 , @xmath34 , and @xmath35 bonds , which is no longer invariant under @xmath139 translations . the enlarged unit cell thus allows for a bipartite coloring of the lattice . a specific realization of this is shown in figure [ fig : kitaev2 ] a ) . compared to the original model discussed in section [ sec : the_model ] , the @xmath33 and @xmath34 bonds are switched in every second unit cell along the @xmath140 direction . this enlarges the unit cell with the new translation vectors becoming @xmath141 , @xmath142 and @xmath143 . note that these are the simple cubic translations , i.e. we have moved from a body - centered cubic structure to a simple cubic one by doubling the unit cell along the @xmath140 direction . and b ) @xmath144 direction . the arrow marks the broken translation vector . the unit cell of the model in a ) is compatible with a bipartite coloring of the lattice while the one in b ) is not . ] the model still fulfills that no vertex is connected to the same bond type twice and is , therefore , still exactly solvable with the methods used above . in contrast to the original hamiltonian , the new hamiltonian is no longer isotropic in the coupling constants , as can already be deduced from fig . [ fig : kitaev2 ] a ) the resulting kitaev model is only symmetric in @xmath145 , while the @xmath35 type bond stands out . thus , we expect qualitatively different behavior when @xmath146 compared to @xmath147 . let us first comment on the flux sectors on the model . it can be shown that the constraints on the loop operators are independent on the choice of the covering of the hyperoctagon lattice in @xmath33 , @xmath34 , and @xmath35 bonds , even though the loop operators themselves change . in analogy to our original discussion in section [ sec : the_model ] we restrict ourselves to the vortex - free sector in the following and focus on the changes in the majorana sector induced by the alternation of the kitaev interactions . in particular , we are interested in the gapless modes in the majorana sector . the parameter region , in which gapless modes exist , is identical to the original model , i.e. determined by the triangular inequality . the transition lines to the gapped phases are parametrized by the equalities of eq . . in addition , the behavior of the model along the @xmath146 line in parameter space is identical to the original model . the latter can be understood by noting that the modified majorana hamiltonian at this line ` looks ' like the original one , albeit with an enlarged unit cell . in particular , on the line @xmath146 we still find a surface of gapless majorana modes when @xmath146 . however , the surface is not stable against departing from that parametrization condition . even an infinitesimal discrepancy in the coupling constants @xmath147 immediately opens up a gap on most of the surface and the two - dimensional majorana fermi surface collapses onto a line of gapless excitations , with linear dispersion in the normal directions . this line of gapless modes then remains stable throughout the rest of the gapless phase and contracts to a point on the transition lines to the gapped phases . the phase diagram for this model is visualized in fig . [ fig : phasediagramline ] . a ) . along the line @xmath146 , there is a majorana fermi surface , which is identical to the original model defined in section [ sec : the_model ] . away from this line , the majorana surface is partly gapped out and reduced to a closed line of dirac cones . ] , the model exhibits a majorana fermi surface centered around @xmath148 as shown in a ) . away from @xmath146 , the surface reduces to a line , which lies in the @xmath149 plane . panels b ) and c ) show the behavior of the gapless line , when b ) increasing @xmath81 and c ) decreasing @xmath81 , and setting @xmath102 . ] this behavior should be contrasted to what happens when increasing the unit cell in the @xmath144 direction , such that the enlarged unit cell is still incompatible with a bipartite coloring of the lattice . one possible way to achieve this is to switch the @xmath33 and @xmath34 bonds in every second unit cell along the @xmath150 direction , shown in fig . [ fig : kitaev2 ] b ) . note that this again breaks the isotropy of the original model in the coupling constants . similarly to the two previously discussed models , the region of parameter space with gapless excitations is again defined by the triangular inequality . for the same reasons as stated above , the model with the enlarged unit cell has the same properties as the original one along the @xmath146 line in parameter space . in contrast to the model with enlarged unit cell along the @xmath151 direction , discussed above , the surface remains stable , even when departing from this parametric condition . the behavior of the surface is similar to the original model . in particular , the dispersion around the gapless surface is linear in the normal direction , except at the phase transition to the gapped phases , where the surface shrinks to a single point with quadratic dispersion . the reason for the stability of the surfaces can again be tracked to their relative displacement in momentum space a direct consequence that the unit cell is not compatible with the bipartite coloring of the lattice . to a certain extent one can take the perspective that our results for the kitaev model on the hyperoctagon lattice complete a family of analytically tractable spin liquids of growing complexity starting from a two - dimensional dirac spin liquid on the honeycomb lattice over the intermediate step of a three - dimensional spin liquid with a line of gapless modes on the hyperhoneycomb lattice to finally a spin liquid with a full , two - dimensional surface of gapless modes for the hyperoctagon lattice . despite this ascent in complexity , we want to point out that all three instances share certain features such as rapidly decaying dynamical spin - spin correlation functions while differing in other aspects such as the nature of the phases induced by time - reversal symmetry breaking perturbations such as a magnetic field . starting with the similarities , it is interesting to observe that independent of the nature of the manifold of their gapless modes all three spin liquids exhibit dynamical two - spin correlation functions that are identically zero beyond nearest neighbor separation . this extremely rapid decrease was first observed in the context of the honeycomb model @xcite , but the very same arguments employed there also hold for the hyperhoneycomb and hyperoctagon lattices . another parallel arises when distorting the couplings in the kitaev model such that one of the coupling constants becomes dominant ( i.e. such that the triangular inequality does not hold any longer ) and the majorana sector is gapped out , see the phase diagram in fig . [ fig : phasediagram ] for comparison . in these gapped phases the low - lying excitations are instead given by configurations , where some of the loop operators have negative eigenvalue . while for the two - dimensional honeycomb lattices two types of excitations can be discerned and identified with the point - like ( electric and magnetic ) excitations of a @xmath66 gauge theory , the effective low - energy theory for the three - dimensional lattice is somewhat more elaborate . mandal and surendran argued @xcite that there is only a single loop - like excitation in the gapped phases of the hyperhoneycomb model that exhibits non - trivial ( semionic ) braiding properties . likely , a variant of their argument with similar conclusions can be applied to the hyperoctagon model as well . a clear distinction between the three spin liquid phases arises when considering the effect of time - reversal symmetry breaking perturbations such as a magnetic field . for the honeycomb model such a perturbation is of utmost interest as the gapless majorana modes are protected by time - reversal and even an infinitesimal magnetic field ( applied along the 111-direction ) gaps out the spin liquid into a gapped topological phase with non - abelian vortex excitations . on a more technical level the reason for this drastic change induced by the magnetic field can be seen in terms of the symmetry class classification @xcite of the underlying free ( majorana ) fermion problem @xcite . in the absence of time - reversal symmetry breaking the model is in symmetry class bdi , while in the presence of a magnetic field the symmetry class changes to class d. the latter allows for a @xmath152 classification of topological phases in two spatial dimensions . indeed the two bands split by the magnetic field are characterized by chern numbers @xmath48 indicating the non - abelian nature of the gapped phase . when considering a similar line of arguments for the three - dimensional models a different picture emerges . as noted earlier , the zero - energy modes of the hyperhoneycomb model are protected by time - reversal symmetry , while the ones of the hyperoctagon lattice are not . so while we expect the gapless phase of the hyperhoneycomb model to gap out immediately in the presence of a magnetic field , this is far from obvious for the zero - energy modes of the hyperoctagon model as the spectrum is robust against any two - fermion term that does not break translation symmetry . independent of whether a gap opens in the spectrum , it still holds that the symmetry class of the underlying free ( majorana ) fermion model changes from class bdi to class d in the presence of a time - reversal symmetry breaking term . however , in contrast to its two - dimensional counterpart symmetry class d does not harbor any topological phases in three dimensions @xcite . as a consequence , the three - dimensional systems can not be driven into a ( non - abelian ) topological phase by applying a magnetic field ( or any other time - reversal symmetry breaking perturbation ) . however , one might still be able to employ similar ideas to the ones used by ryu in ref . @xcite to stabilize a non - trivial topological phase by introducing additional ( orbital ) degrees of freedom such that the augmented model can be reformulated as a free fermion model in symmetry class diii . the latter does have a @xmath152 classification in three dimensions and , thus , allows for three dimensional analogs of the topological phase in the honeycomb model . finally , an interesting perspective emerges when recasting our results in the terminology conventionally used to characterize various spin liquid states @xcite . in this language , we have discovered a spin liquid with a _ spinon fermi surface _ that covers an extensive two - dimensional manifold in momentum space . the quest to identify magnetic systems harboring such spinon fermi surfaces has typically inspired theorists to consider a slave - fermion approach where the fermion interacts with a _ fluctuating _ @xmath153 gauge field a situation that is notoriously hard to track analytically and any progress coming at the expense of compromises on the level of various decoupling / mean - field approaches . this situation should be contrasted to the current situation where we have stumbled upon a system with a spinon fermion surface with a much simpler and analytically exact description in terms of majorana fermions interacting with a _ static _ @xmath66 gauge field . however , it is important to note that this difference is not a mere conceptual one , but one that has direct implications for thermodynamic observables such as the specific heat coefficient @xmath154 . for a @xmath153 spin liquid the specific heat diverges as @xmath155 i.e. the specific heat coefficient @xmath156 _ diverges _ logarithmically at low temperatures @xcite . for our case of a spinon fermi surface emerging from majorana fermions interacting with a @xmath66 gauge field we find @xmath157 i.e. the specific heat coefficient @xmath75 goes to a _ constant _ at low temperatures . finally , this situation should be contrasted to the spin liquid with a fermi line , as it was found for the hyperhoneycomb lattice , where the specific heat grows as @xcite @xmath158 i.e. the specific heat coefficient @xmath75 _ vanishes _ in the limit of @xmath159 . remarkably enough , this implies that a simple thermodynamic experiment could immediately distinguish these three seemingly equally exotic spin liquids . returning to the perspective of the free ( majorana ) fermion system underlying our gapless spin liquid , there is one obvious bouquet of questions that we have not addressed in the manuscript at hand namely the various pairing instabilities that the fermi surface of our system might exhibit . one might be particularly interested in asking what instabilities can be induced by additional interactions such as a heisenberg exchange argued to accompany the kitaev interactions in any microscopic description of iridate compounds @xcite . the effective description of the hyperoctagon model in terms of _ spinless _ fermions suggests @xmath160-wave pairing as the natural candidate for opening a gap . interaction terms of this type can indeed arise in a perturbative analysis of the heisenberg exchange . the question of whether or not these terms lead to a collapse of the majorana fermi surface or even gap out all majorana modes in the system is left for future work . + * acknowledgements. * we thank a. akhmerov , a. altland , p. becker - bohaty , s. parameswaran , f. pollmann , and especially a. rosch for insightful discussions . st acknowledges the hospitality of the aspen center for physics where some of the ideas underlying this manuscript were perceived . st is indebted to l. balents for a beginner s guide to 3d printing . we acknowledge partial support from sfb tr 12 of the dfg . in this appendix , we want to expand our discussion of possible magnetic materials candidates in space group i4@xmath1732 ( no . 214 ) . the guiding idea in our analysis is to put the space group symmetries of i4@xmath1732 to work and look for various possible ways to fill the interstitial sites between the network of edge - sharing iro@xmath29 cages as illustrated in fig . [ fig : o6cages ] of the main text . in addition , we have to take into account that the fundamental building blocks of iro@xmath2 have valency @xmath161 and as such are looking for interstitial fillings that allow to chemically compensate this valence . let us first consider the effect of the space group symmetries . there are in total 48 symmetry operations in the space group i4@xmath1732 . a generic point in the cubic cell is , thus , mapped to in total 48 distinct points ; the set of these points will , in the following , often be labeled by a single representative . however , there are several high - symmetry points , respectively lines in the cubic cell , which are mapped to far fewer points . in total , we can distinguish 5 types of lattice point , according to the number of distinct lattice points that can be reached by the symmetry operations . \i ) the lattice point is mapped to 8 distinct lattice points in the unit cell . this is possible in two inequivalent ways . the resulting points form the two chiralities of the hyperoctagon lattice . the representatives of the two possibilities are @xmath162 and @xmath163 . in the main text , we chose the hyperoctagon lattice generated by @xmath164 ; thus , in the following we place the iridium atoms on this set of sites . \ii ) the lattice point is mapped to 12 distinct lattice points in the unit cell . this is again possible in two inequivalent ways . the representatives are given by @xmath165 and @xmath166 . these sites form effective lattices , which are deformations of the two chiral versions of the hyperkagome . in fact , they can be identified as the medial lattices of the two chiral hyperoctagon lattices in i ) . \iii ) the lattice point is mapped to 16 distinct lattice points in the unit cell . there are infinitely many such possibilities , as long as the representative is chosen on the line @xmath167 ( except the high symmetry points already listed in i ) ) . \iv ) the lattice point is mapped to 24 distinct lattice points in the unit cell . there are many high - symmetry lines in the cubic unit cell , which lead to this behavior . the oxygen sites are an example for this type of lattice points represented by @xmath168 . \v ) the lattice point is mapped to 48 distinct lattice points in the unit cell , which applies to all lattice points that do nt lie on one of the above mentioned high - symmetry lines . structures , indicated by the grey octahedra . panel a ) shows the crystal structure by placing atoms of valency @xmath169 on the sites of type i ) ( see text ) . in panel b ) , atoms with valency @xmath55 are placed on sites of type iii ) , which are generated by the representative @xmath170 . ] this structure , imposed by the symmetry group , severely restricts the composition of possible compounds . assuming the presence of edge - sharing iro@xmath29 cages , we note that the above analysis implies that there are 8 iro@xmath2 in a cubic unit cell . thus , the remaining atoms must compensate a total valency of @xmath171 . the latter can , for instance , be achieve by placing 8 atoms with valency @xmath169 on the remaining set of sites of type i ) . the resulting compound is of the form airo@xmath2 , where a is one of the alkaline - earth elements ca , sr or ba . another possibility is to place 16 atoms of valency @xmath55 on the sites of type iii ) . this results in a material of the type a@xmath1iro@xmath2 , where a is one of the alkali atoms na or li . the resulting crystal structures for both possibilities are visualized in fig . [ fig : space_group ] . to complement our discussion of the kitaev model on the hyperoctagon lattice , we will present a brief , self - contained summary of the kitaev model on the hyperhoneycomb lattice , an alternative trivalent 3d lattice , in this appendix . for clarity , we will use similar notations and conventions as in the main text , but emphasize that in doing so we also closely follow the original solution of the kitaev model on the hyperhoneycomb lattice as discussed in some detail in ref . . the hyperhoneycomb lattice is an alternative 3d lattice with a trivalent lattice structure which is illustrated in fig . [ fig : hyperhoneycomb_unitcell ] . its elementary building blocks are zig - zag chains running along the crystallographic @xmath172- and @xmath173-axises , which are coupled along the @xmath174-axis . its crystal structure can be classified as a face centered orthorhombic lattice of space group number 70 . of particular importance for our discussion is the fact that it is not spatially isotropic as the hyperoctagon lattice , but has one ` preferred ' direction the @xmath174 direction in fig . [ fig : hyperhoneycomb_unitcell ] . following our discussion in the main text relating the hyperoctagon and hyperkagome lattices as ( pre)medial lattices of each other , one can establish a similar set of relations for the hyperhoneycomb lattice as well . the medial lattice of the hyperhoneycomb is a lattice of corner - sharing triangles illustrated in fig . [ fig:3dkagome ] , which can be considered another generalization of the kagome lattice to three spatial dimension , albeit one distinct from the hyperkagome and one which we dub _ orthorhombic - kagome _ lattice . its main motif are sheets of two parallel triangle lines , which are staggered in a rotated way as illustrated in fig . [ fig:3dkagome ] . the set of relations between the hyperhoneycomb and orthorhombic - kagome lattices as well as their relation to the pyrochlore and diamond lattices are summarized in fig . [ fig : latticesummary2 ] . in comparison . the hyperhoneycomb lattice is the medial lattice of the so - called orthorhombic - kagome lattice depicted in fig . [ fig:3dkagome ] . like the hyper - kagome lattice the orthorhombic - kagome lattice can be obtained from the pyrochlore lattice via depletion of 1/4 of the triangles , see the inset on the left . the premedial lattice of the pyrochlore is the diamond lattice , which can be depleted by 1/4 of its bonds to obtain the hyperhoneycomb lattice . ] similar to the hyperoctagon lattice we can define a covering of @xmath33 , @xmath34 , and @xmath35-couplings on the hyperhoneycomb lattice which is commensurate with a four - site unit cell . this unit cell and related translation vectors as indicated in fig . [ fig : hyperhoneycomb_unitcell ] . the kitaev hamiltonian then takes the form @xmath175 where @xmath38 denotes the unit cell position . similar to our analysis of the kitaev model on the hyperoctagon lattice we can identify conservative quantities for this model by considering the structure of closed loops , which again have length ten for this model . for each elementary loop one can again identify a conserved quantity via the loop operators @xmath46 , which again have eigenvalues @xmath48 . in contrast to the hyperoctagon lattice , the smallest volume enclosed by these elementary loops is now formed by _ four _ loops as illustrated in fig . [ fig : loops_hyperhoneycomb ] . graphically speaking this constrains each ` tetraeder ' to have an even number of loops with eigenvalue @xmath54 . a counting argument similar to the one presented in our main analysis in section [ sec : flux_sectors ] shows that there are @xmath64 different flux sectors , where @xmath65 is the number of unit cells . numerical simulations @xcite indicate that the ground state indeed resides in the zero flux sector , which is why we restrict the following discussion to this sector . in order to solve the hamiltonian , we proceed as in the main text and introduce four types of majorana fermions per site to write @xmath176 . introducing bond operators @xmath74 and choosing the zero - flux sector as a reference sector , we again obtain a free fermion hamiltonian of majorana fermions hopping in a static @xmath66 gauge field @xmath177 after a fourier transformation , the hamiltonian is straightforward to diagonalize . the principle energy bands in the majorana spectrum are thereby found to be @xmath178}}\,,\\\end{gathered}\ ] ] where @xmath179 the momenta @xmath180 are defined as the coefficients of the reciprocal lattice vectors , i.e. @xmath181 with @xmath182 . . the gapless line is located in the plane @xmath183 , which is indicated in grey . the other panels show the behavior of the gapless line when a ) increasing @xmath81 and b ) decreasing @xmath81 , while setting @xmath102 . the extent of the brillouine zone in the plane @xmath183 is indicated by the hexagon . ] zero - energy solutions are obtained by setting the second term in the root to zero , which implies @xmath184 the first line of eq . ( [ eq : hh_constraint ] ) can be inverted to yield @xmath185 the second line determines the ( unique ) value of @xmath186 given @xmath187 and @xmath188 . the gapless majorana modes thus form a _ line _ in momentum space with linear dispersion along the normal directions . the line of gapless modes and its dependence on the coupling constant @xmath83 setting @xmath189 is shown in fig [ fig : hyperhoneycomb_gapless ] . for this choice of parameters the gapless line always lies in the @xmath183 plane , although this is no longer true when @xmath190 . we note that when approaching the gapped phase at @xmath191 , @xmath192 , the gapless line shrinks to a point at @xmath170 . s. k. choi , r. coldea , a. n. kolmogorov , t. lancaster , i. i. mazin , s. j. blundell , p. g. radaelli , yogesh singh , p. gegenwart , k. r. choi , s .- w . cheong , p. j. baker , c. stock , and j. taylor , phys . lett . * 108 * , 127204 ( 2012 ) . further generalizations of the kitaev models also to lattices with higher vertex coordination number , in particular the kagome , triangular , hyperkagome and pyrochlore lattices have been studied @xcite . it should however be noted that it is precisely the higher coordination number of the vertices in these lattices which prohibits to follow the same analytical route that can be used for the trivalent ones . the relative sign for the two alternative definitions of the loop operator , and subsequent ` freedom ' to define magnetic flux , appears also for other trivalent lattices , for instance the two - dimensional square - octagon lattice . using the operator @xmath46 , eq . , to define the flux ensures that the notion of 0/@xmath193-flux per plaquette is consistent with the one used by lieb @xcite . in particular , the ground state of the square - octagon lattice resides in the _ full - flux sector _ according to lieb s theorem , as the loops have lengths four respectively eight .

motivated by the recent synthesis of @xmath0-li@xmath1iro@xmath2 a spin - orbit entangled @xmath3 mott insulator with a three - dimensional lattice structure of the ir@xmath4 ions we consider generalizations of the kitaev model believed to capture some of the microscopic interactions between the iridium moments on various trivalent lattice structures in three spatial dimensions . of particular interest is the so - called hyperoctagon lattice the premedial lattice of the hyperkagome lattice , for which the ground state is a gapless quantum spin liquid where the gapless majorana modes form an extended two - dimensional majorana fermi surface . we demonstrate that this majorana fermi surface is inherently protected by lattice symmetries and discuss possible instabilities . we thus provide the first example of an analytically tractable microscopic model of interacting su(2 ) spin-1/2 degrees of freedom in three spatial dimensions that harbors a spin liquid with a two - dimensional spinon fermi surface .

You are an expert at summarizing long articles. Proceed to summarize the following text: photometric redshifts is a term that has received recent , wide - spread visibility and interest , though the use of multicolor photometry to estimate the redshifts of galaxies has a history that is relatively long . as of march 1999 , my search of the ads abstracts for the term `` photometric redshifts '' yielded only 9 papers ending in 1994 , with the first in 1982 , i.e. averaging less than one paper per year . a sudden jump then occurs in 1995 with 6 papers , followed by 1996 with 7 , 1997 with 9 , and then another major jump to 35 papers in 1998 . the broad participation and range of topics of this workshop validate this surge in activity . among these papers , the earliest reference to the term `` photometric redshifts '' is found in the abstract of puschell et . al . ( 1982 ) . their attempt to estimate redshifts of faint radio galaxies via broadband photometry was pioneering ( even by today s standards ) in three respects : 1 ) the use of near - infrared bands ( @xmath0 ) along with optical bands ( @xmath1 ) , 2 ) the adoption of @xmath2 fits to spectral energy distributions ( sed ) , and 3 ) the use of different sets of sed templates , ranging from non - evolving local ellipticals , theoretical seds from bruzual , to observed seds from known radio galaxies . the first use of the term in the title was by loh and spillar ( 1986 ) : _ photometric redshifts of galaxies_. this work was pioneering in its use of ccds to reach quite faint limits of @xmath3 , along with the use of 6 medium - band filters and again @xmath2 template fitting , but only to three observed _ local _ sed templates to represent all galaxy types at all redshifts . the use of multicolors to estimate redshifts actually predates the above papers and the true literature on the subject is significantly more extensive , but due to the ambiguity of the term `` photometric redshifts '' as detailed in the next section , many workers in this arena chose not to use this term . probably the first use of multicolor photometry for redshift estimation was by baum ( 1962 ) . he obtained photoelectric photometry through 9 medium - wide filters of galaxies in cluster 3c395 , assumed that the seds were that of ellipticals , and obtained an estimate of redshift @xmath4 , quite close to the eventual spectroscopic value of 0.46 . nearly two decades passed before multicolor photometry was used to estimate redshifts . butchins ( 1981 , 1983 ) used uk schmidt plate @xmath5 photometry that reached @xmath6 . because of the overlap in @xmath5 colors of low - redshift ( @xmath7 ) early - type galaxies with higher redshift ( @xmath8 ) later - type galaxies , butchins applied probablility constraints on luminosities that are qualitatively similar to what has more recently been termed `` bayesian '' . i was able to reach fainter limits ( @xmath9 ) by using kpno 4-m plates and had both superior redshift accuracy @xmath10 and far less degeneracy by exploiting a set of filters ( roughly @xmath11 ) with much longer wavelength coverage ( koo 1981 , 1985 , 1986 ) . since baum is not here and butchins is no longer in astronomy , i qualify as the old - timer in the field at this workshop . another important set of papers that does _ not _ use the term `` photometric redshifts '' , and yet constrains redshifts from broadband photometry , relies on the so - called `` lyman - break technique '' ( see contribution by steidel ) . the basic idea is to discern very high redshift galaxies via a significant drop in the bluest band for galaxies with otherwise very blue colors in two or more redder bands . this situation occurs when the lyman break enters , e.g. , the @xmath12 band , at redshifts @xmath13 . since the apparent drop in the @xmath12 band flux ( or especially in redder bands at higher redshifts ) is also affected by the intergalactic lyman - alpha forest depression , a more accurate term should be `` lyman - drop technique '' . early examples of its use include papers by partridge ( 1974 ) , koo(1986 ) , cowie ( 1988 ) , majewski ( 1988 , 1989 ) , guhathakurta et . 1990 ) , and steidel & hamilton ( 1993 ) . i suggest the following : photometric redshifts are those derived from _ only _ images or photometry with spectral resolution @xmath14 . my choice of 20 is intended to exclude redshifts derived from slit and slitless spectra , narrow band images , ramped - filter images , fabry - perot images , fourier transform spectrometers , etc . this definition still leaves a wide range of approaches to obtain redshifts , examples of which include : 1 . spatial correlations : galaxies are assumed to have , _ statistically _ , the redshifts of their neighbors in apparent close pairs , groups , and especially the cores of rich clusters . photometric redshifts is an unconventional term to apply in these cases and should thus probably be avoided . magnitudes : although using just magnitudes alone to estimate redshifts might be expected to work only for standard candles , such as the brightest cluster galaxies in @xmath15 or strong - flux radio galaxies in @xmath16 , the convolution of accessible volumes with the shape of the luminosity function of galaxies results in a fairly tight correlation between magnitudes and redshift , even for more common field galaxies ( see the redshift - magnitude plot fig . 4 of koo & kron 1992 ) . for any of the following techniques , the addition of magnitudes can thus be expected to provide additional constraints on the redshift probability distribution ( see , e.g. , connolly et . al . 1995 ) . just as in the previous approach , the use of the term `` photometric redshifts '' would be quite unconventional , but technically accurate . 3 . one color : in situations where the sed is known or unique , one color may be sufficient to estimate a redshift . practical applications include redshift estimates for the reddest galaxies in clusters or field and also very high redshift searches , especially when at least one of the two bands is in the near - infrared . a single - color is probably the minimal information to qualify the use of the term `` photometric redshift '' as understood by most astronomers today . two or more colors : the use of three or more filters is necessary to break the degeneracy between instrinsic color and redshift . this approach is probably the most commonly accepted one when referring to photometric redshifts that employ optical bands , and works surprisingly well even with only three bands ( straizys & sviderskiene 1983 ) . this is because most galaxies ( at least locally ) occupy only a small fraction of the possible multicolor volume ( see fig . 2a ) ; galaxy spectra are often composites of old and young stellar populations which result in bowl - shaped or u - shaped sed ( see fig . 1 and note the more bowl - shaped value of the z = 0 locus compared to that of its consituent stars ) ; the pivot point of this curvature lies near the 4000 break and is roughly independent of the galaxy s average color ; and bowl - shaped spectra , when redshifted , result in moving the iso - z loci in a direction perpendicular to these loci ( see fig . 1 ) . obviously one needs longer wavelength bands to sample the pivot point near 4000 as one wishes to discern higher redshifts ( see fig . nature has been kind : if galaxy spectra had instead blackbody shapes of different temperatures or power - laws in shape , we would not be able to separate redshifts using multicolors . surface brightness : if the intrinsic surface brightness is roughly constant or slowly varying with time , as might be expected for large spirals that undergo largely constant star formation rate histories , the @xmath17 surface brightness dimming can be exploited to yield redshifts . one color with light profiles : in principle , the light profile might yield the type of galaxy ( e.g. , @xmath18 profiles might imply a luminous spheroidal or bulge to disk ratios might suggest a probable galaxy type ) , which in turn might be assumed to have a unique sed so that a single color is sufficient for estimating a redshift ( sarajedini et . al . 1999 ) . one color with image structure : this is a generalization of the last two approaches . anderson et . 1996 ) has already been exploring the correlations of galaxies among color , surface brightness , and image concentration . this combination appears to be useful to improve the efficiency of gathering photometric redshifts by not requiring deep @xmath12 photometry , which is very costly in telescope time . etc . whether the term `` photometric redshifts '' should include such a diversity of approaches is debatable , but what should be obvious is that astronomers have yet to exploit the wealth of additional photometric / structural information in refining redshift estimates from image data alone the most popular method for estimating redshifts from a set of photometric measurements is the @xmath2 ( or maximum likelihood ) template - fitting method adopted by puschell et . al ( 1982 ) , largely because the method is simple , does not rely on having any spectroscopic redshifts , and needs only a few templates ( empirical or theoretical ) to yield results . with the availability of multicolor ccd photometry , almost anyone can get into the photometric redshift business today . the achilles heel of the technique is of course the template set . empirical seds are relatively few , almost exclusively confined to local galaxies whose seds may not reflect those of more distant galaxies , and which are not necessarily representative in luminosity , dust , inclination , morphology , etc . to the targets of interest . while theoretical seds avoid some of these problems , they may not be correct and do not yield information on the probability distribution for the estimated redshifts , since this information relies on having the probability distribution of the seds . to show the level of uncertainty between empirical and theoretical seds , see fig . 1 . we note that the empirical spectra ( which have traditionally been based on only a handful of measured integrated spectra ) differ significantly ( 0.05 or more in z ) from the theoretical at all redshifts . in contrast , the empirical fitting method ( e.g. , connolly et . 1995 , wang et . al . 1998 ) uses a large enough pool of spectroscopic redshifts to calibrate the relationship of colors and magnitudes to redshifts . by construction , this method should provide accurate redshifts , but more importantly , it should also yield realistic estimates of the redshift errors . to achieve this in practice , however , requires a sufficiently complete set of known redshifts to the depth of the desired photometry . except for the mean and rms error when compared to spectroscopic redshifts , photometric redshift errors remain poorly characterized . in particular , the position , shape , and asymmetry of the redshift error distribution should be derived ; the sources of the errors and especially whether they are due to random measurement errors , intrinsic dispersions in the seds of galaxies , or unknown systematics should be tracked down ; and , ideally , the dependence of the errors on a variety of other parameters ( color , luminosity , morphology , structure , redshift , environment , etc . ) should also be measured . more importantly , these redshift uncertainties should be incorporatedly explicitly into the analysis , not by using the derived maximum likelihood or minimum @xmath2 value of the photometric redshift for each galaxy , but rather the full probability distribution . this idea has already been incorporated for the c - method of deriving the luminosity function from photometric redshifts ( subbarao et . al . 1996 ) , but this approach should be adopted more universally by others in most statistical analyses . of special concern in the area of redshift errors is the likelihood both that the sed s of distant galaxies are different on average from those today and that the intrinsic dispersion in the seds are greater . if true , we would expect larger random and systematic errors for galaxies at higher redshifts . figures 2 and 3 are telling and sobering . more specifically , figure 2b shows that morphologically peculiar galaxies exhibit a greater spread of values _ perpendicular to the line _ in the @xmath19 two - color plots than that of more normal galaxies shown in figure 2a ( larson and tinsley 1978 ) . interactions and mergers appear to be a primary cause of these peculiarities , so if indeed such galaxies are more common in the past , we should expect greater photometric redshift errors at higher redshifts that result from the greater spread in intrinsic seds alone . larson and tinsley ( 1978 ) explain the wider spread as the result of a greater diversity in the star formation histories of galaxies , namely brief starbursts interrupting an otherwise more quiet or more smoothly varying star formation history . figure 3 shows the possible tracks of galaxies with varying amounts of such bursts in the @xmath19 two - color diagram ; note in particular the regions _ perpendicular _ to the wide , solid line on top of which most local , morphologically - normal galaxies lie . such deviations result in systematic errors in photometric redshifts . clearly what is desired for significant progress in our use and understanding of photometric redshifts is a large pool of spectroscopic redshifts that span the full range of redshifts , depth , photometric bands , etc . in 1985 , we had about 100 spectroscopic redshifts ( @xmath20 ) to @xmath21 from 4 - 5 m class telescopes to study photometric redshifts in four bands @xmath11 ( koo 1985 ) , while roughly the same number has been measured with the keck 10-m to study the hubble deep field to fainter limits @xmath22 and a wider range in redshifts ( @xmath23 ) . our current lack of good calibration is , however , changing dramatically . at the lowest redshifts of @xmath24 , the sloan digital sky survey will yield nearly @xmath25 spectra for a photometric sample that includes 5 filters . at intermediate redshifts between @xmath26 to 0.7 , cnoc2 will have 6000 spectroscopic redshifts for calibrating their 5 band system ( lin et . al . 1999 ) . keck will be providing over 1000 redshifts for @xmath27 to over @xmath28 and @xmath29 to 4 for passbands that include some @xmath16 , while handfuls of redshifts are coming in for the desert at @xmath30 to over 2 and the highest redshifts @xmath31 . with these samples to calibrate the photometric redshift system and especially its errors , photometric redshifts will finally rest on a much more secure foundation . the scientific potential of using photometric redshifts has been recognized for a long time , especially during an earlier era when redshifts for large samples of galaxies fainter than @xmath32 were difficult or impractical , while multicolor photometry easily sampled the distant universe . the following list is meant to be illustrative , rather than representative or exhaustive , of pioneering projects that were based on older multicolor photographic photometry or small - format ccds . the topics of other papers from this workshop provide a more current view of how photometric redshifts are being applied in a new era of hst images , large mosaic ccd cameras , and 8 - 10 m telescopes . the use of photometric redshifts has not only undergone a recent revival , but it is also rapidly becoming a crucial tool of many programs in mainstream observational cosmology . * searches for primeval galaxies at redshifts @xmath33 via the @xmath12 band detecting the lyman break ( cowie 1988 , cowie & lilly 1989 ; guhathakurta et . al . 1990 ; koo 1986 ; majewski 1988 , 1989 ; steidel & hamilton 1993 ) or even via data further to the red to probe higher redshifts ( partridge 1974 ) . * searches for high redshift qso s ( see review by warren & hewett 1990 ) or distant radio galaxies ( puschell et . 1982 ; van der laan 1983 ) . * studies of the evolution of field galaxies ( butchins 1983 ; cowie et . 1988 , 1990 ; guiderdoni 1987 ; koo 1986 ; lilly et . al . 1991 ) or their luminosity function ( subbarao et . al . 1996 ) * discrimination of cluster members and superclusters at moderate redshifts ( connolly et . al . 1996 ; koo 1981 , 1986 , koo et . al . 1988 ) * estimate of the geometry of the universe via the volume test ( loh and spillar 1986 ) i would like to thank ray weymann for letting me be an old - timer , albeit only briefly , and his conference team for organizing such a fun workshop . with an old - timer brain , i have also probably missed a number of important old references , and to their authors , my apologies for any oversights . this work has been partially supported by nsf ast-9529098 and by nasa through grant number ar-07532.01 - 96 from the space telescope science institute , which is operated by aura , inc . , under nasa contract nas 5 - 26555 . anderson , d. r. , et . 1996 , , 188 , 817 baum , w. a. 1962 , iau symp . 15 , 390 butchins , s. a. 1981 , , 97 , 407 butchins , s. a. 1983 , , 203 , 1239 connolly , a. j. , et . 1995 , , 110 , 2655 connolly , a. j. , et . 1996 , , 473 , l67 cowie , l. l. 1988 , _ the post - recombination universe _ , 1 cowie , l. l. , et . 1988 , , 332 , l29 cowie , l. l. , & lilly , s. j. 1989 , , 336 , l41 cowie , l. l. , et . al . 1990 , , 360 , l1 guhathakurta , p. , tyson , j. a. , & majewski , s. r. 1990 , , 357 , l9 guiderdoni , b. 1987 , _ high redshift and primeval galaxies _ , 271 koo , d. c. 1981 , , 252 , l75 koo , d. c. 1985 , , 90 , 418 koo , d. c. 1986 , , 311 , 651 koo , d. c. , & kron , r. g. 1992 , , 30 , 613 koo , d. c. , et . 1988 , , 333 , 586 larson , r. b. , & tinsley , b. m. 1978 , , 219 , 46 lilly , s. j. , cowie , l. l. , & gardner j. p. 1991 , , 369 , 71 lin , h. , et . al . 1999 , , 518 , 533 loh , e. d. , & spillar , e. j. 1986 , , 303 , 154 majewski , s. r. 1988 , _ towards understanding galaxies at high redshifts _ , 203 majewski , s. r. 1989 , _ the epoch of galaxy formation _ , 85 partridge , r. b. 1974 , , 192 , 241 puschell , j. j. , owen , f. n. , & laing , r. a. 1982 , , 275 , l57 sarajedini , v. l. , et . 1999 , , 121 , 417 steidel , c. c. , & hamilton , d. 1993 , , 105 , 2017 straizys , v. , & sviderskiene , z. 1983 , , 94 , 23 subbarao , m. et . 1996 , , 112 , 929 tinsley , b. m. 1980 , fund . cosmic phys . , 5 , 87 van der laan , h. , et . 1983 , iau symp . 104 , 73 wang , y. , bahcall , n. , & turner , e. l 1998 , , 116 , 2081 warren , s. j. , & hewett , p. c. 1990 , rep .

i review the early history of photometric redshifts ; specify a working definition that encompasses a broader range of approaches than commonly adopted ; discuss the pros and cons of template fitting versus empirically - based techniques ; and summarize some past applications . despite its relatively long history , the technique of photometric redshifts remains far from being a mature tool . areas needing development include the use of spatial structure , the incorporation of large redshift samples with multicolor photometry for empirical calibrations of redshift errors , and improved analysis tools that directly include redshift probability distributions rather than singular values . photometric redshifts has not only undergone a recent revival it is also rapidly becoming a crucial tool of mainstream observational cosmology .

You are an expert at summarizing long articles. Proceed to summarize the following text: the proximity effect in a superconductor coupled to a mesoscopic normal conductor leads to a wide range of new quantum phenomena . these include andreev reflection at normal and superconductor interfaces , the formation of andreev bound states ( abs ) in confined geometries , and proximity - induced supercurrent flow through normal conductors.@xcite the recent experimental detection of individual andreev bound states@xcite as well as the efforts towards demonstrating majorana states in superconductor coupled nanostructures devices with strong spin - orbit interaction@xcite received significant experimental and theoretical interest and opened a new area of research . quantum dots ( qds ) coupled to superconducting leads provide an ideal system to test the theoretical predictions on the interplay of abs with the kondo effect.@xcite a 0 to @xmath0 junction transition@xcite has been reported in s - qd - s systems by measuring the sign reversal ( positive to negative ) of the josephson supercurrent for even to odd occupation of the qd.@xcite theoretical calculations suggest that this quantum phase transition in the s - qd - s josephson junction devices is signaled also by the crossing of two andreev levels.@xcite depending on the ratio of the kondo temperature @xmath1 at the center of a coulomb valley and the superconducting gap energy @xmath2 , the abs display a crossing ( @xmath3 ) or a non - crossing ( @xmath4 ) dispersion @xmath5 as a function of gate voltage @xmath6 . these predictions have been confirmed by recent experimental studies using al - contacted semiconductor quantum dot / nanowire devices.@xcite it has also been predicted theoretically that there could be up to four abs for a single level model in the superconducting gap . however the two outer abs may not be visible in the transport spectrum since they can merge with the continuum.@xcite so far most of the experimental studies of abs formation are limited to al as contacts material , which restricts these experiments to very low temperatures and magnetic fields . despite these earlier reports , the study of hybrid nanostructures with larger gap superconducting elements is required to allow for a more complete understanding of abs formation in these devices . here , we report on low temperature tunnelling spectroscopy measurements on an individual carbon nanotube quantum dot device strongly coupled to a nb superconducting loop and weakly coupled to an al tunnel probe . two types of abs are observed in coulomb valleys with different charging energy . in some gate regimes two pairs of abs are found within the superconducting gap . in addition , next to the main abs conductance resonance a weaker conductance peak is present , which is interpreted as quasi - particle tunnelling from the abs to the al tunnel probe . at higher temperatures tunneling from the thermally populated upper abs becomes visible and shows an opposite curvature at the center of the coulomb valleys . calculations based on the superconducting anderson model are used to describe the experimentally observed subgap features . single wall cnts are grown on a si / sio@xmath7 substrate by chemical vapor deposition using a fe / mo based catalyst and methane as precursor gas . the highly doped si substrate with 300 nm sio@xmath7 layer serves as a global back gate . ): a peak in the differential conductance is observed when unoccupied ( occupied ) states of the tunnel probe are aligned to the energy ( @xmath8 ) of the lower ( upper ) state of an abs pair . , width=340 ] a niobium ( @xmath9 pd / @xmath10 nb ) superconducting loop and a tunnel probe ( @xmath11 ti / @xmath10 al ) are patterned using standard electron beam lithography on an individual single wall carbon nanotube ( figure [ fig : device](a ) ) . a @xmath12 pd interlayer is used to improve the coupling between the superconducting fork and the nanotube and thereby increase the superconducting proximity effect . for weak coupling of the tunnel probe to the cnt a @xmath11 thin ti adhesion layer is used . low temperature electrical transport measurements are performed in a @xmath13he/@xmath14he dilution refrigerator with heavily filtered signal lines down to @xmath15 . the differential conductance @xmath16 of the superconducting tunnel probe weakly connected to the nanotube was measured employing conventional lock - in techniques by adding a low frequency ac excitation voltage ( @xmath17 , @xmath18 ) onto the dc bias voltage @xmath19 . variation of the back gate voltage gives access to the abs spectrum in the cnt - qd . the device parameters are extracted by measurements in both the superconducting and the normal conducting state ( by applying a magnetic field ) of the contacts . as function of back gate voltage @xmath20 and bias voltage @xmath21 , for ( a ) @xmath22 and ( b ) @xmath23 . while the superconducting energy gap in the contacts is clearly visible in ( a ) , in ( b ) superconductivity is suppressed and normal - state coulomb blockade behaviour emerges . , width=302 ] corresponding overview plots of the differential conductance are shown in fig . [ fig : overview ] . while in fig . [ fig : overview](a ) at zero applied magnetic field features pertaining to superconductivity in the leads dominate transport at low bias , a magnetic field of @xmath23 restores the typical pattern of coulomb blockade in fig . [ fig : overview](b ) . the superconducting gap of the nb loop is found to be @xmath24 , and for the al tunnel probe @xmath25 and @xmath26 are estimated . typical parameter values of the cnt - qd are the charging energy @xmath27 , the tunnel coupling to the leads @xmath28 , and the coefficient @xmath29 which relates the energies @xmath30 of the quantum dot states to @xmath6 . and back gate voltage @xmath6 for two different gate regimes , ( a ) @xmath31 and ( b ) @xmath32 . the sharp resonances are the signature of the andreev bound states : a main resonance of high conductance ( @xmath33 ) , a weak conductance resonance ( @xmath34 ) running parallel to the main resonance peak ( @xmath33 ) , and a third additional resonance close to the gap edge ( @xmath35 ) . ( c ) , ( d ) abs spectrum calculated using nrg for a two channel superconducting anderson model with parameters @xmath36 and @xmath37 , respectively . inset : schematic representation of the theoretical model used to describe the main multilevel features , see text . two independent transport channels connect the quantum dot with the superconducting reservoir . , width=302 ] figure [ fig : stability ] shows two details of the stability diagram , where the differential conductance @xmath38 is again plotted as a function of bias @xmath21 and gate voltage @xmath20 , at base temperature and zero magnetic field . the two panels ( a ) and ( b ) correspond to two different gate voltage ranges exhibiting different charging energies @xmath39 and @xmath40 , respectively . within the superconducting gap range @xmath41 there are three subgap features : two main resonances of high conductance ( @xmath33 , @xmath35 ) , and a weak conductance resonance ( @xmath34 ) running parallel to the resonance ( @xmath33 ) . strong peaks in the differential conductance measurements are expected when an andreev level at @xmath42 is aligned to the bcs singularity of the density of states of the tunnel probe ( see figure [ fig : device](b ) ) . this results in pronounced conductance peaks at voltages @xmath43.@xcite the weak conductance peak ( @xmath34 ) running parallel to the lower abs ( @xmath33 ) at lower bias voltages is understood as a replica of the abs at low temperatures ( see below ) . as already mentioned the abs ( @xmath33 ) spectrum for odd charge states can show non - crossing ( fig . [ fig : stability](a ) ) or crossing behavior of the pairs of bound states ( fig . [ fig : stability](b ) ) , resulting in a 0-@xmath0 quantum phase transition.@xcite this is controlled by the ratio @xmath44.@xcite in case of non - crossing the system always stays in the 0-phase , i.e. a ( singlet ) ground state whereas for a zero bias crossing of the abs the system changes its ground state from 0 to @xmath0-state ( magnetic doublet ) . [ fig : stability](c ) and ( d ) show the abs spectrum obtained from the superconducting anderson model using the numerical renormalization group ( nrg ) method.@xcite we have found that the presence of two pairs of abs can be explained by assuming the presence of two nearly degenerate levels , e.g. , resulting from the often lifted kk-symmetry that are coupled to two independent channels in the leads , as schematically drawn in the inset of the figure . for the case of fig . [ fig : stability](a ) both channels are in the 0-phase and the abs exhibit a non - crossing behavior as correspondingly shown in fig . [ fig : stability](c ) . on the other hand , in the case of fig . [ fig : stability](b ) the inner abs exhibit a loop indicating the transition to the @xmath0-phase , which is accounted for in the theoretical result of fig . [ fig : stability](d ) by a larger @xmath45 ratio for one of the channels . the @xmath46 values were taken as 8 ( fig . [ fig : stability](c ) ) and 9 ( fig . [ fig : stability](d ) ) close to the experimental estimations while the parameters @xmath47 were chosen to get a qualitative fit of the experimental results . to understand the origin of the weak replicas of the inner pair of abs @xmath48 in our experiment , we investigate their dependence on temperature . figure [ fig : temperature](a d ) shows a detail of the 2d stability diagram in a different coulomb valley at the indicated temperatures . in this gate regime , again a weak conductance resonance ( @xmath34 ) running parallel to a pair of abs ( @xmath33 ) is observed . the apparent replica of the abs can be understood if we assume that the probe dos is finite at the fermi energy @xmath49 and @xmath49 is aligned with the abs : @xmath50 . ) at @xmath51 is observed when the probe dos is finite at the fermi energy @xmath49 and @xmath49 is aligned with the abs . ( b ) at higher temperatures also the upper abs ( @xmath52 ) at @xmath53 is thermally populated . in this case a peak at @xmath54 ) is observed in the differential conductance , where an unoccupied ( occupied ) dos of the tunnel probe is aligned to a thermally populated upper ( lower ) abs . , width=340 ] this level arrangement illustrated in fig . [ fig : sketch](a ) . for a non - superconducting tunnel probe one would expect the disappearance of the replicas together with a shift in the main abs due to the suppression of the probe gap @xmath55 . we have verified this by applying a small magnetic field ( @xmath56 ) that drives the al tunnel probe into the normal state : it suppresses the replica and a clear crossing of the main abs ( @xmath33 ) is then present ( data not shown ) . the main abs peak ( @xmath33 ) remains unchanged up to the maximum temperature @xmath57 investigated in the present study . the satellite peaks ( @xmath34 ) also do not show any significant change up to @xmath58 . at @xmath59 , however , the dispersion of the gate voltage dependence of the satellite peaks flattens at the center of cb valley and an additional conductance resonance starts to emanate from the main abs resonance ( see the arrow in fig . [ fig : temperature](b ) ) . at even higher temperatures @xmath60 , the gate voltage dependence of the satellite peak changes its curvature at the center of the coulomb valley compared to the low temperature ( @xmath61 ) case . as illustrated in the schematic of fig . [ fig : sketch](b ) , we expect a peak in differential conductance when the maximum of the bcs density of states of the tunnel probe is aligned with the thermally populated upper state . the position of the secondary peak ( @xmath52 ) should then be inverted , and vary as @xmath62 ) . at intermediate temperatures ( @xmath63 ) we observe that quasi particle tunnelling via both the lower and the thermally populated upper abs ( fig . [ fig : temperature](b ) ) contributes to the current . the combined contributions lead to a nearly flat dispersion of the secondary peak . at higher temperatures ( fig . [ fig : temperature](c , d ) ) transport of thermally excited quasi - particles via the upper abs dominates , which leads to the opposite curvature of the dispersion in the center of the coulomb valley . the slight shift of both the abs and their replica towards smaller energies results from the reduction of @xmath64 as the temperature is increased . according to our model , the dispersion of all three types of conductance peak originates from the same dispersion relation @xmath5 . hence , it should be possible to collapse the dispersion of the abs and its replicas at low and high temperature on top of each other by suitable inversions and shifts . for ( a ) @xmath65 and ( b ) @xmath66 . right : plots combining the data of the left column graphs to demonstrate that the gate dependence of the peak positions can in each case be reduced to a single dispersion relation @xmath5 ; see the main text for details . , width=302 ] this is illustrated in fig . [ fig : gatedep ] , confirming our interpretation : at low temperature @xmath65 , fig . [ fig : gatedep](a ) left panel , we observe in the differential conductance measurement the main abs at @xmath67 and a satellite peak at @xmath68 . after subtracting the sc gap of the tunnel probe ( @xmath69 ) at @xmath70 , the peak position of the main abs peaks show a good overlap with the satellite peak ( fig . [ fig : gatedep](a ) , right panel ) . at higher temperature @xmath66 , as depicted in fig . [ fig : gatedep](b ) left panel , the dispersion of the satellite peak is inverted . its position follows a relation @xmath71 with gate voltage in the coulomb blockade valley . to verify this gate dependence , we compute and plot @xmath72 = \pm [ 2{\ensuremath{\delta_\text{al}}}-\{{\ensuremath{\delta_\text{al}}}- { \ensuremath{\varepsilon_\text{abs}}}({\ensuremath{v_{\textrm{g}}}})\ } ] = \pm [ { \ensuremath{\delta_\text{al}}}+ { \ensuremath{\varepsilon_\text{abs}}}({\ensuremath{v_{\textrm{g } } } } ) ] = e v_\text{main}({\ensuremath{v_{\textrm{g}}}})$ ] for a reduced @xmath73 , and find a very good match with the observed main peak position @xmath74 , see fig . [ fig : gatedep](b ) right panel . a similar analysis for other temperatures also shows an excellent agreement with the proposed mechanism . in addition to the measurement results , calculated conductance patterns for a quantum dot structure as described here are shown in fig . [ fig : temperature](e)-(h ) . the conductance calculations were performed using a mean field description of the superconducting anderson model coupled to two superconducting leads with two different gap parameters @xmath75 and @xmath76 . as discussed in ref . the magnetic phase can be represented within a mean field description by introducing an exchange parameter @xmath77 which produces a splitting of the dot energy levels for different spin orientations . in these calculations the coupling to the nb superconducting leads is included nonperturbatively ( i.e. to all orders in perturbation theory ) while the coupling to the al probe is introduced to the lowest order in perturbation theory . this approximation is justified by the weak coupling strength to the probe electrode , manifested by the small conductance values experimentally observed ( of the order of @xmath78 ) . more precisely the conductance was calculated using @xmath79 where @xmath80 are the local densities of states at the probe electrode and at the dot coupled to the nb leads , respectively , @xmath81 is the fermi distribution function and @xmath82 is the tunneling rate from the dot to the probe electrode . while @xmath83 is calculated using the mean field approximation for the anderson model coupled to the nb superconducting lead as described in ref . , for @xmath84 we use a standard bcs - like density of states in which we introduce a phenomenological dynes - parameter@xcite of @xmath85 broadening the bcs density of states for the al probe . as can be observed in fig . [ fig : temperature ] , the model calculations give a good description of the evolution of the weak subgap features with temperature , and once more confirm our interpretation in terms of different transport mechanisms . it should be noted that the only change in the model parameters when going from fig . [ fig : temperature](e ) to fig . [ fig : temperature](h ) is due to the reduction of @xmath64 with temperature , while we assume @xmath86 in this temperature range . the comparatively high value of the dynes - parameter is at present not understood . this may be an effect of the electromagnetic environment,@xcite to which superconducting quantum dot devices are known to be extraordinarily sensitive . ( a ) and displaying a multi - loop pattern . ( b ) abs spectrum calculated by nrg assuming a single channel in the leads and using the parameters @xmath87 . inset : schematic of the nrg model , see text . , width=302 ] a peculiar feature of the present experimental results are the multiple loop abs that are observed in a more extended gate range , as depicted in fig . [ fig : multiloop ] . as opposed to the situation considered in fig . [ fig : stability](c ) and ( d ) , a theoretical description of such multi - loop features is obtained when more than one dot level couples to _ the same _ channel in the leads , see the inset of fig . [ fig : multiloop](b ) . a nrg calculation considering such a situation with two hybridized levels on one dot can actually reproduce qualitatively the observed multi - loop patterns , as shown in fig . [ fig : multiloop](b ) . in several of the coulomb valleys investigated in more detail the abs - peaks are accompanied by a pronounced negative differential conductance ( ndc ) , shown in blue color in figs . [ fig : stability ] , [ fig : temperature ] , and [ fig : multiloop ] . ndc features have been predicted to appear due to the presence of the so - called yu - shiba - rusinov ( ysr ) states for a qd with an odd number of electrons@xcite with highly asymmetric coupling to the leads . ysr - states can be regarded as a variant of abs appearing for a magnetic impurity coupled to a superconductor,@xcite and their existence has been experimentally confirmed via scanning tunneling microscopy of magnetic atoms on superconducting surfaces@xcite and qds with superconducting leads.@xcite in conclusion , our transport spectroscopy of an individual carbon nanotube strongly coupled to wide gap nb leads reveals several different types of andreev bound state spectra . weak satellite peaks appear within the smaller probe superconducting gap which are a result of quasiparticle tunneling into a residual density of states within this gap . at higher temperature these satellite peaks change their dispersion as a function of gate voltage due to the thermal population of the upper state of an abs pair . our findings are well reproduced within the superconducting anderson model in terms of combined nrg and mean field calculations . more efforts in this direction could be helpful also to discriminate majorana bound states in similar hybrid nanostructures from other states at zero energy . we gratefully acknowledge financial support from the deutsche forschungsgemeinschaft within grk 1570 , sfb 689 , the e. noether program ( hu 1808 - 1 ) , from spanish mineco through grant fis2011 - 26516 , and from the eu fp7 project se2nd . a. k. thanks the alexander von humboldt foundation for providing financial support during this research . we thank kicheon kang for enlightening discussions .

tunneling spectroscopy of a nb coupled carbon nanotube quantum dot reveals the formation of pairs of andreev bound states ( abs ) within the superconducting gap . a weak replica of the lower abs is found , which is generated by quasi - particle tunnelling from the abs to the al tunnel probe . an inversion of the abs - dispersion is observed at elevated temperatures , which signals the thermal occupation of the upper abs . our experimental findings are well supported by model calculations based on the superconducting anderson model .

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Proceed to summarize the following text: an important step for understanding the dynamics and phenomenology of astrophysical jets is the study of their instabilities . instabilities have a substantial importance , on one hand for the formation and evolution of various observed structures and , on the other hand , for dissipating part of the jet energy and leading to the observed radiation . there are several possible sources of instabilities , like the velocity shear between the jet and the ambient medium , which drives the kelvin - helmholtz instability , the current flowing along magnetic field lines , which drives the current driven instability ( cdi ) and rotation that can drive several kinds of instabilities . since the most promising models for the acceleration and collimation of jets involve the presence of a magnetic field with footpoints anchored to a rotating object ( an accretion disk or a spinning star or black hole ) , the presence of a toroidal field component and of rotation seems to be a natural consequence and both cdi and rotation driven instabilities may play an important role in the jet propagation . cdi have been , for example , suggested as being responsible for the conversion from poynting to kinetic energy flux in the first phases of jet propagation @xcite . khi have been extensively studied in several different configurations both in the newtonian ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and relativistic ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) cases . similarly , cdi have been widely studied in the newtonian limit ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , while the analysis of the relativistic case has been more limited , most of the studies have considered the force - free condition @xcite and only @xcite studied the full mhd case . the study of the effects of rotation have been mainly focused on the accretion disk problems , where the main instability considered is the magnetorotational instability @xcite , however , the combination of magnetic field and rotation can give rise to several other instabilities ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and the interplay between the different modes can become quite complex . our goal is to study these rotation - induced instabilities in the context of magnetized jets . in @xcite ( hereinafter paper i ) we studied the interplay between khi and cdi in a relativistic non - rotating cold jet configuration , characterized by a current distribution concentrated inside the jet and closing at large distances . in this paper , we introduce the effects of rotation which , however , makes the analysis of the unstable modes much more intricate . therefore , before tackling the full relativistic case , in this paper we limit ourselves to a newtonian analysis , neglecting again thermal pressure compared with the magnetic one . moreover , since most of the unstable modes that we will consider are concentrated inside the jet radius and therefore the effect of the jet velocity would be to simply doppler shift their frequencies , in this first step we ignored also the presence of the longitudinal velocity component . the main focus of this paper will then be on the effect of rotation on cdi and on the new modes of instability introduced by rotation . the effect of rotation on cdi was considered by @xcite who analyzed a rigidly rotating jet and found a stabilizing effect for rotation periods shorter than a few alfvn times . an analysis of the unstable modes introduced by rotation in a configuration and parameter range similar to ours has been performed by @xcite . they discuss these modes in the cold plasma limit , however , their study is mainly local , whereas we focus more on global analysis of these instabilities , besides they do not discuss the cdi . another related works are by @xcite and @xcite , who examined the instabilities of axisymmetric perturbations in the presence of rotation and superthermal magnetic fields . the treatment of the first paper is again local and mostly focuses on an equilibrium configuration typical of accretion discs , while the second one analyses the stability a rotating cylindrical plasma dean flow with only axial field both with local and global approach . the plan of the paper is the following : in the next section we present the equilibrium configuration , in section [ sec : linequations ] we derive the linearized equations , in section [ modeclassification ] we discuss the wkbj local dispersion relation and present energetic considerations based on the frieman - rotenberg approach @xcite . the local dispersion relation and the energetic considerations will be useful in understanding the nature of the unstable modes that will be discussed in section [ results ] , where we present our results on global modes . finally in the last section [ summary ] we summarize our findings . we study the linear stability of a cold magnetized cylindrical jet flow . although in the following we will consider only the zero thermal pressure case , we keep here the presentation more general . the relevant equations are the equations of ideal mhd : @xmath0 @xmath1 @xmath2 @xmath3 where @xmath4 is the density , @xmath5 is the pressure , @xmath6 is the sound speed , @xmath7 , @xmath8 , are , respectively , the velocity and magnetic fields . we remark that a factor of @xmath9 is absorbed in the definition of @xmath8 . the first step in the stability analysis is to define an equilibrium state satisfying the stationary form of equations ( [ eq : drho / dt]-[eq : dp / dt ] ) and this will be done in the next subsection . we adopt a cylindrical system of coordinates @xmath10 ( with versors @xmath11 ) and seek for axisymmetric steady - state solutions , i.e. , @xmath12 . the jet propagates in the vertical ( @xmath13 ) direction , the magnetic field and velocity have no radial component and consist of a vertical ( poloidal ) component @xmath14 , and a toroidal component @xmath15 . the magnetic field configuration can be characterized by the pitch parameter @xmath16 the only non - trivial equation is given by the radial component of the momentum equation ( [ eq : dm / dt ] ) which , in the zero pressure case , simplifies to @xmath17 equation ( [ eq : radial_eqmhd ] ) leaves the freedom of choosing the radial profiles of all flow variables but one and then solve for the remaining profile . furthermore , we note that the presence of a longitudinal velocity has no effect on the radial equilibrium . the choice of the radial profiles is somewhat arbitrary since we have no direct information about the magnetic configuration in astrophysical jets . the choice of the @xmath18 distribution is equivalent to a choice of the distribution of the longitudinal component of the current and also determines the behavior of the pitch parameter @xmath19 , that is important for the stability properties . in principle , one can then have several equilibria characterized by different forms of the current distribution , that can be more or less concentrated , can peak on the axis or at the jet boundary , and can close in different ways ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? our choice is to consider a general class of constant density equilibria in which the vertical current density is peaked on the axis and is concentrated in a region of radius @xmath20 . the azimuthal component of magnetic field has therefore to behave linearly with radius close to the origin and decay as @xmath21 at large distances , more precisely we can write it as @xmath22 where the function @xmath23 behaves in the following way at small and large radii : @xmath24 for the rotational frequency @xmath25 we assume the form @xmath26 where @xmath27 is the value of @xmath28 on the axis , i.e. @xmath29 . we make this assumption for simplicity and for avoiding any possible non - monotonic behaviors of @xmath30 . inserting expressions ( [ eq : bphi ] ) and ( [ eq : omega ] ) in the equilibrium condition ( [ eq : radial_eqmhd ] ) , we can get the profile of @xmath30 as @xmath31 where @xmath32 @xmath33 and @xmath34 is a monotonic function with @xmath35 and @xmath36 . depending on the value of @xmath37 , @xmath30 is decreasing ( for @xmath38 ) or increasing ( for @xmath39 ) . from a physical point of view , if we look at equation ( [ eq : radial_eqmhd ] ) , we can see that the equilibrium is given by the balance of three forces : gradient of @xmath40 , the gradient of @xmath41 and the centrifugal force . when there is no rotation , the gradient of @xmath40 , which always points towards the jet axis , is balanced by the gradient of @xmath41 . increasing the rotation rate , the gradient of @xmath41 decreases , until , for @xmath42 , @xmath30 becomes constant . if we still increase rotation beyond this point , the centrifugal term becomes larger than the gradient of @xmath40 , thus @xmath30 has to increase outward for providing an inward force term , needed for having equilibrium . we can now define a radially averaged alfvn velocity via @xmath43 and inserting the expressions for @xmath18 and @xmath30 from equations ( [ eq : bphi ] ) and ( [ eq : bz ] ) in equation ( [ eq : bav ] ) , we get @xmath44 where @xmath45 from which we get @xmath46 where @xmath47 are , respectively , nondimensional measures of the rotation rate and of the value of the pitch on the axis . for a given choice of the function @xmath23 and the values of these two parameters , we can derive the value of @xmath48 from equation ( [ eq : hc ] ) and the equilibrium structure is then fully determined . however not all combinations of @xmath49 and @xmath50 are allowed , because , in order to have a physically meaningful solution , we have to impose the additional constraints that @xmath51 and @xmath52 have to be everywhere positive , which translate as @xmath53 respectively , for equations ( [ eq : hc ] ) and ( [ eq : bz ] ) . in order to exemplify the structure of the equilibrium solution , we now make a specific choice for the function @xmath23 , @xmath54 which gives the same solution used in paper i. this same solution will also be used in the next sections for computing the instability behavior . more specifically , we have the following profiles for @xmath51 , @xmath52 and @xmath55 @xmath56,\ ] ] @xmath57 where @xmath58 is the error function , and @xmath59 in fig . 1 , we present the regions in the @xmath60 plane for which the equilibrium is possible . the red curve represents the condition @xmath61 , while the blue curve represents the condition @xmath62 and the green curve represents the combinations of @xmath49 and @xmath50 for which @xmath30 is constant . therefore the green region represents equilibria for which @xmath63 , while in the red region @xmath64 . we conclude by summarizing the parameters determining the equilibrium : once the magnetization radius @xmath20 and the average alfvn velocity , @xmath65 , are fixed , the equilibrium structure is fully determined by the two parameters @xmath50 and @xmath49 . we also observe that an arbitrary profile of the longitudinal velocity @xmath66 can be superposed to the equilibrium , however in this paper we consider only the case @xmath67 . we consider small perturbations of the form @xmath68 to the equilibrium state described above . by linearizing the ideal mhd equations , we obtain the following system of two first order ordinary differential equations @xmath69 @xmath70 where @xmath71 is the radial component of the lagrangian displacement related to the eulerian perturbation of the velocity field @xmath72 through @xmath73 and @xmath74 is the total pressure perturbation @xmath75 with @xmath76 and @xmath77 being respectively the density and magnetic field perturbations . in equations ( [ eq : lin_system1 ] ) and ( [ eq : lin_system2 ] ) , @xmath78 , @xmath79 , @xmath80 and @xmath81 depend on the equilibrium quantities and on @xmath82 , @xmath83 , @xmath84 and are defined as @xmath85[\tilde{\omega}^2(b_0 ^ 2+\rho_0c_s^2)-c_s^2k_b^2 ] , \end{split}\ ] ] @xmath86 - \frac{2m}{r^2}(k_bb_{0\varphi}+\rho_0v_{0\varphi}\tilde{\omega})[\tilde{\omega}^2(b_0 ^ 2+\rho_0c_s^2)-c_s^2k_b^2 ] , \end{split}\ ] ] @xmath87,\ ] ] @xmath88 -\frac{4[\tilde{\omega}^2(\rho_0c_s^2+b_0 ^ 2)-c_s^2k_b^2](k_bb_{0\varphi } + \rho_0v_{0\varphi}\tilde{\omega})^2}{r^2 } \\ + \frac{\rho_0[\tilde{\omega}^2(b_{0\varphi}^2-\rho_0v_{0\varphi}^2 ) + ( \tilde{\omega}b_{0\varphi}+k_bv_{0\varphi})^2]^2}{r^2 } , \end{split}\ ] ] where quantities with @xmath89 subscript refer to the equilibrium state and @xmath90 we note that the system ( [ eq : lin_system1 ] ) and ( [ eq : lin_system2 ] ) was derived by @xcite and we kept it in its general form even though in the following we will consider only the case with @xmath91 and @xmath92 . this system , supplemented with appropriate boundary conditions at @xmath93 and @xmath94 , poses an eigenvalue problem for @xmath82 . on the axis , at @xmath93 , the equations are singular but the solutions have to be regular , while at infinity the solutions have to decay and no incoming wave is allowed ( sommerfeld condition ) . this asymptotic behaviour of the solutions for small and large radii are used in the numerical integration of the eigenvalue problem . for finding eigenvalues we use a shooting method with a complex secant root finder , as we did in paper i. the numerical integration can not start at @xmath93 because of the singularity , so we start at a small distance from the origin where the solution is obtained through a series expansion of the equations described in the appendix [ ap : small_r ] . similarly , we start a backward integration from a sufficiently large radius , where the asymptotic solution is obtained as described in the appendix [ ap : large_r ] and then we match the two numerical solutions at an intermediate radius . equations ( [ eq : lin_system1 ] ) and ( [ eq : lin_system2 ] ) have singularities whenever @xmath95 which give rise to four distinct continua , two alfvn continua for @xmath96 and two slow continua for @xmath97 in the case of zero pressure and zero longitudinal velocity , that we consider in this paper , the slow continua reduce to the single flow continuum defined by the condition @xmath98 since we are interested only in unstable solutions , our integration path will always avoid the singularities , however the presence and position of the continua is fundamental in shaping the overall mhd spectrum @xcite . to classify the unstable modes present in the jet and understand their physical origin , following @xcite , we employ the wkbj approach and energetic considerations following the frieman - rotenberg formalism @xcite . the combination of these methods allows us to gain insight into the nature of dominant driving forces inducing the instability of each mode and classify them accordingly . equations ( [ eq : lin_system1 ] ) and ( [ eq : lin_system2 ] ) can be combined in a single second - order differential equation only for @xmath99 , @xmath100(r\xi_{1r})=0 . \end{split}\ ] ] assuming the radial wavelength of perturbations small compared to the length scale over which there are significant variations in the equilibrium quantities , we can represent the radial dependence of the displacement as @xmath101 , where the radial wavenumber @xmath102 is assumed to be large , @xmath103 . substituting this into equation ( [ eq : secondorder ] ) and neglecting the radial variations of the equilibrium quantities , we obtain , to leading order in the large parameter @xmath104 , the following local dispersion relation ( see e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) @xmath105 which after using the expressions for @xmath106 and @xmath78 reduces to a sixth - degree polynomial @xmath107 - 4\rho_0\omega \left ( \frac{m}{r } \rho_0(r\omega^2 ) + 2\omega_b kb_{0z } \right ) \tilde{\omega}^3 \\ + \tilde{\omega}^2 \bigg [ k_t^2 k_b^2b_0 ^ 2 + \left(k^2+\frac{m^2}{r^2}\right ) \big ( 4\rho_0\omega^2b_{0\varphi}^2 -\rho_0 ^ 2(r\omega^2)^2 + b_0 ^ 2r \frac{d}{dr } \left ( \rho_0\omega^2 -\omega_b^2 \right ) + 4 \rho_0 b_{0\varphi } \omega_b ( r\omega^2 ) \big ) \\ - 4\frac{m}{r } \rho_0 k_b \omega_b ( r\omega^2 ) - 4 b_0 ^ 2 k^2\omega_b^2 \bigg ] + 4\rho_0b_{0\varphi}k_b\omega(r\omega^2)\left(k^2+\frac{m^2}{r^2}\right)\tilde{\omega } + \rho_0(r\omega^2)^2k_b^2\left(k^2+\frac{m^2}{r^2 } \right)=0 , \end{split}\ ] ] where @xmath108 describes the effect of the curvature of toroidal field lines in cylindrical geometry and @xmath109 is the total wavenumber squared . we intentionally separated out the product @xmath110 , which represents the centrifugal force per unit volume . this general dispersion relation describes , in principle , all the modes in the local approximation , however , the coefficients are complicated and not physically revealing . so , we examine various limiting cases to identify these modes . we start by considering the limits for large and small wavenumbers . at large @xmath83 , when @xmath111 and @xmath112 , to leading order from ( [ eq : fullwkb ] ) we get @xmath113 where @xmath114 is the local alfvn velocity . this dispersion relation resembles that of the parker instability with the driving role of external gravity replaced here by the centrifugal force per unit mass @xmath115 ( see e.g. , * ? ? ? thus , the unstable mode at large @xmath83 , is driven mainly by the centrifugal force and can be identified with a magnetic buoyancy mode @xcite , which in this limit of large vertical wavenumber operates by bending mostly poloidal field lines . the growth rate , @xmath116 , for @xmath117 tends asymptoticallly to its maximum value @xmath118 . at small @xmath119 and for @xmath120 , from equation ( [ eq : fullwkb ] ) we get the following dispersion relation @xmath121 where @xmath122\ ] ] and @xmath123 is the epicyclic frequency , @xmath124 . the solution is given by @xmath125 it is clear that if @xmath126 is positive , which , as we will show below , corresponds to the jet flow being stable against the cold magnetorotational instability ( mri ) , only the last term of equation ( [ eq : tbm ] ) , proportional to the square of the centrifugal acceleration , guarantees the existence of instability . as a result , also in this limit , we have again the centrifugal buoyancy mode , but now , at small @xmath83 , which mainly operates by bending toroidal field lines . in fact , the dispersion relation ( [ eq : tbm ] ) , in the @xmath127-dominated regime , is similar to that of the non - axisymmetric toroidal buoyancy mode derived in @xcite ( see their equation 51 ) . note that this approximation can work only for non - axsymmetric modes , since it is based on the condition @xmath128 , that can not be satisfied for @xmath129 . on the other hand , in the axisymmetric case , it is not possible to bend toroidal field lines . the instability condition for this mode can be derived from equation ( [ eq : tbmsol ] ) as @xmath130 in principle , the cold differentially rotating jet can also support the mri arising from the combined effect of differential rotation and magnetic fields . to capture this instability , in equation ( [ eq : fullwkb ] ) we ignore centrifugal @xmath131 and curvature @xmath132 terms , which are not its main driving factors , but retain rotation @xmath28 ( i.e. , coriolis force ) and shear @xmath133 , which together with azimuthal and vertical magnetic fields cause this instability . as a result , we obtain a more compact dispersion relation describing the mri in cold differentially rotating cylindrical flows ( see also * ? ? ? * ; * ? ? ? * ) @xmath134 \\ + \left(k_r^2+k^2+\frac{m^2}{r^2}\right ) k_b^2b_0 ^ 2 + \left(k^2+\frac{m^2}{r^2}\right ) \left(4 \rho_0\ omega^2 b_{0\varphi}^2 + b_0 ^ 2 r \frac{d}{dr } ( \rho_0 \omega^2 ) \right ) = 0 \end{split}\ ] ] this expression is a quadratic polynomial for @xmath135 from which a condition for the cold mri can be readily deduced . that is , an unstable solution @xmath136 can exist whenever @xmath137 the most favourable condition for the instability is when @xmath138 and from this we can derive the necessary condition @xmath139 . hence , in the cold plasma limit ( @xmath140 ) , the mri vanishes in the case of a purely toroidal field and the presence of a nonzero poloidal / vertical field component is necessary for its operation . in fig . [ fig : figwkbj ] we show a comparison between representative full numerical solutions to equation ( [ eq : fullwkb ] ) , and the analytical approximations given by equations ( [ eq : pbm ] ) , ( [ eq : tbmsol ] ) and ( [ eq : mri ] ) . the left panel is for the axisymmetric mode with @xmath141 and @xmath142 . in this case we have only the poloidal buoyancy mode , the numerical solution is represented by the red curve , while the analytical approximation , given by equation ( [ eq : pbm ] ) , is represented by the green curve and the two curves are indistinguishable . the mid and right panels are for @xmath143 , the same value of @xmath50 and two different values of the rotation rate , @xmath144 for the mid panel and @xmath145 for the right panel . again , the numerical solution is represented by the red curve and , in this case , we have the analytical approximations for the poloidal buoyancy mode ( equation [ eq : pbm ] ) , at large wavenumbers , represented by the green curve , and for the toroidal buoyancy mode ( equation [ eq : tbm ] ) at small wavenumbers , represented by the blue curve . we can furthermore notice that around the value of @xmath83 for which @xmath138 , we have a narrow peak in the growth rate , which corresponds to a very localized mri . the peak is very narrow for @xmath142 and widens with increasing the rotation rate ( right panel with @xmath146 . the black curve , representing the solution to equation ( [ eq : mri ] ) , reproduces the behavior of the numerical solution , although the agreement is not as good as for the other two approximations due to the terms neglected in the derivation of equation ( [ eq : mri]).the solutions are taken at a particular radial position , @xmath147 , however , taking different radial positions , the qualitative behavior of the solution remains the same . alternatively , the classification of modes and related instabilities performed above using the wkbj approach can also be made based on energetic considerations that can be derived following the frieman - rotenberg formalism @xcite . the equation of motion for the lagrangian displacement @xmath148 is @xmath149=0,\ ] ] where the generalized force operator @xmath150 is given by @xmath151 = \rho_0\left(\frac{m}{r}v_{0\varphi}+kv_{0z } \right)^2{\ensuremath{\boldsymbol{\xi}}}_1 - 2i\rho_0\omega\left(\frac{m}{r}v_{0\varphi}+kv_{0z}\right)\left(\xi_{1\varphi}{\hat{\bf r}}-\xi_{1r}\hat{{\ensuremath{\boldsymbol { \varphi}}}}\right)-\\ -2\rho_0r\omega\frac{d\omega}{dr}\xi_{1r}\hat{\bf r}+\rho_1(r\omega^2)\hat{\bf r}+{{\ensuremath{\boldsymbol{j}}}}_0\times { { \ensuremath{\boldsymbol{b}}}}_1+(\nabla\times { { \ensuremath{\boldsymbol{b}}}}_1)\times { { \ensuremath{\boldsymbol{b}}}}_0,\end{gathered}\ ] ] and the @xmath89 subscript indicates again the equilibrium quantities , while the @xmath152 subscript indicates perturbations . substituting @xmath153 in equation ( [ eq : force_op ] ) , we get @xmath154=0.\ ] ] the various terms entering the expression of @xmath150 correspond to different forces acting on the perturbations in the jet flow . the first term comes from a convective derivative and describes the advection of perturbations by the mean flow . in unmagnetized flows , this term contributes to the kh instability . the second term is related to coriolis force due to rotation , the third term is related to shear , or differential rotation of the flow , since it is proportional to the radial derivative of the angular velocity @xmath28 , the fourth term proportional to @xmath76 corresponds to the centrifugal force ( radial buoyancy ) , the fifth and sixth terms are the linearized lorentz force , respectively , due to the equilibrium current @xmath155 and the perturbed magnetic field @xmath156 and due to the perturbed current @xmath157 and the equilibrium magnetic field @xmath158 . one can show that the force operator @xmath150 is self - adjoint @xmath159 d^3 { \ensuremath{\boldsymbol{r}}}= \int { \ensuremath{\boldsymbol{\xi } } } _ 1\cdot { \ensuremath{\boldsymbol{g } } } \left [ { \ensuremath{\boldsymbol{\eta } } } \right ] d^3 { \ensuremath{\boldsymbol{r}}}\ ] ] while the second term in equation ( [ eq : var_disprel ] ) is antisymmetric @xmath160 where @xmath161 is an arbitrary function and integration is performed over an entire fluid volume provided that displacement @xmath148 and @xmath161 vanish at the flow boundaries . if we take @xmath162 , we can write @xmath163 d^3 { \ensuremath{\boldsymbol{r}}}= \int { \ensuremath{\boldsymbol{\xi}}}_1 \cdot { \ensuremath{\boldsymbol{g } } } \left [ { \ensuremath{\boldsymbol{\xi}}}^{\ast}_1\right ] d^3 { \ensuremath{\boldsymbol{r}}}\ ] ] and @xmath164 therefore @xmath165 d^3 { \ensuremath{\boldsymbol{r}}}$ ] is a real quantity and @xmath166 purely imaginary . we will see below that these properties are necessary for establishing stability criteria for the flow . multiplying equation ( [ eq : var_disprel ] ) by @xmath167 , integrating by @xmath168 over the interval @xmath169 $ ] and taking into account that the perturbations vanish for @xmath170 and are regular at @xmath93 , we get @xmath171 where the coefficients @xmath172 and @xmath173 are @xmath174 r d r \nonumber \\ f & = & \int^{\infty}_0 { \ensuremath{\boldsymbol{\xi}}}^{\ast}_1 \cdot { \ensuremath{\boldsymbol{g } } } \left [ { \ensuremath{\boldsymbol{\xi}}}_1 \right ] r d r. \end{aligned}\ ] ] @xmath175 and @xmath176 are real by definition , while @xmath173 is real due to the self - adjointness of the force operator @xmath177 . using the expression of @xmath150 in equation ( [ eq : var1 ] ) , we can write @xmath173 in a symmetric form with respect to @xmath148 and @xmath167 : @xmath178 r dr , \end{gathered}\ ] ] where the various terms are grouped according to driving forces they correspond to , as in @xmath179 . the solution to the quadratic equation ( [ eig3 ] ) is @xmath180 of course this is a formal solution , since the terms @xmath175 , @xmath176 and @xmath173 depend on the eigenfunctions , so they can be computed only after the eigenvalue problem has been solved . if @xmath181 for an eigenmode , this solution comes in complex conjugate pairs that indicates instability of the mode . therefore , in the expression ( [ eq : force_op_sym ] ) for @xmath173 , negative terms are stabilizing and positive ones destabilizing . we distinguish four distinct destabilizing contributions : 1 . the sum of the first two terms @xmath182r dr\ ] ] describe the combined effect of advection by the mean flow and coriolis force . however , these processes also define @xmath176 and only the sign of @xmath183 actually characterizes stabilizing or destabilizing contribution due to these two effects . this term is responsible for the velocity shear , or kh instability . 2 . the third term @xmath184 describes the effect of shear , or differential rotation and is destabilazing when @xmath185 , i.e. , @xmath186 somewhere in the flow field . @xmath187 , together with the combined effect of advection and coriolis force characterized by @xmath188 , determines instability in shear flows . in magnetized shear flows , the condition @xmath186 is necessary for the existence of the mri @xcite . the fourth term @xmath189 which is proportional to the centrifugal acceleration , describes the effect of centrifugal force . if @xmath190 , the centrifugal force can give rise to the magnetic buoyancy instability . this term depends on the density perturbation , which is expressed via @xmath71 and @xmath74 as @xmath191\ ] ] at large @xmath83 , to leading order the density perturbation becomes @xmath192 indicating that it is produced mainly by bending poloidal field lines . by contrast , at small @xmath83 , @xmath193 and the density perturbation is determined primarily by bending the toroidal field ( especially at small pitch , when the growth rates are higher ) . this implies that the density perturbation in this regime arises due to bending of toroidal field lines . the fifth term @xmath194 is proportional to the equilibrium current and corresponds to the lorentz force . if @xmath195 , this term is destabilizing , giving rise to current driven instability . the last term is the magnetic tension force , which is always stabilizing . based on the above analysis , we classify unstable modes according to which of these four contributions prevails over the net effect of other three ones and results in the destabilization of a given mode . we then label the mode according to the type of this dominant destabilizing term . so , for example , if @xmath187 is positive and dominates over the net contribution from all the other terms in the square root in equation ( [ eq : omg_var ] ) , this implies that the instability is caused by differential rotation , which in the case of the considered jet flow threaded by the magnetic field in fact corresponds to the `` cold '' version of mri . if @xmath196 is positive and dominates , the main destabilizing force is the centrifugal force , which via bending magnetic field lines , gives rise to the magnetic buoyancy instability . finally , if @xmath197 term is positive and dominates , the destabilization comes from the lorentz force due to the presence of the equilibrium current and hence the resulting instability is current driven . as discussed in section [ sec : equilibrium ] , the basic equilibrium depends on the two parameters @xmath49 and @xmath50 , wich are defined in equation ( [ eq : param ] ) and represent respectively nondimensional measures of the rotation rate and of the pitch on the vertical axis . instead of @xmath49 , which measures the rotation rate in terms of the average total alfv ' en velocity , we can alternatively make use of the parameter @xmath37 , defined in equation ( [ eq : alfa ] ) , which measures rotation in terms of the alfvn velocity associated only with the azimuthal magnetic field component . reference to parameter @xmath37 can be convenient because its value can be related to the sign of the radial gradient of @xmath30 , i.e. for @xmath38 , @xmath30 decreases outward , for @xmath42 , @xmath30 is constant and , for @xmath39 , @xmath30 increases outward . in the following we will focus our discussion on a number of equilibrium solutions , whose position in the @xmath198 plane is shown in fig . [ fig : figcases ] . the green curve corresponds to solutions with @xmath42 , the red dots represents equilibria with @xmath199 , the blue dots are for @xmath42 and , finally , the black dots are for @xmath200 . we choose these three values of @xmath37 in order to sample the solutions with different gradients of @xmath30 . for which we computed the behavior of unstable modes . the red dots represents equilibria with @xmath199 , the blue dots are for @xmath42 and the black dots are for @xmath200 . the green curve corresponds to solutions with @xmath42.,width=566 ] we start our discussion with the axisymmetric modes , in this case we know that the cdi mode is stable and instabilities can be only due to rotation . in fig . [ fig : complexplanem0 ] we plot in the complex plane the position of unstable modes for a given parameter set @xmath141 , @xmath202 ( @xmath203 ) and @xmath204 . we observe a sequence of modes clustering to @xmath205 , that is the point where the slow continuum collapses in the present conditions ( @xmath206 , @xmath91 and @xmath129 ) . the modes in the sequence differ by the number of radial oscillations which increases as the sequence approaches @xmath207 . , @xmath141 ( corresponding to @xmath208 ) , @xmath129 and @xmath204 . , width=566 ] we can then investigate the physical origin of this sequence of modes by comparing , in fig . [ fig : centrifugalm0 ] , the growth rate of the most unstable one as a function of @xmath49 ( solid curve ; the value of @xmath209 is again 1.5 ) with an approximation obtained by equation ( [ eq : omg_var ] ) in which we consider only the centrifugal term ( dashed curve ) , i.e. we approximate the growth rate by @xmath210 where @xmath196 and @xmath175 are given respectively by equations ( [ eq : fc ] ) and ( [ eq : var1 ] ) and can be computed once we have solved the eigenvalue problem and found the eigenfunctions . we can see that the approximation reproduces very well the behavior of the actual growth rate , so we can regard the centrifugal term as being responsible for the destabilization of these modes and hence identify them as magnetic buoyancy instabilities . we already discussed these instabilities in section [ modeclassification ] , when we considered the wkbj local dispersion relation . they have been also already studied by @xcite and @xcite and , as we mentioned above , are analogous to the parker instability , with the centrifugal force replacing gravity and operate by bending the poloidal field lines . for the most unstable mode of the axisymmetric centrifugal buoyancy mode shown in fig . [ fig : complexplanem0 ] . the dashed curve represents an approximation to the growth rate given by equation ( [ eq : cengr ] ) . , width=566 ] from the local dispersion relation ( [ eq : pbm ] ) appropriate for the poloidal buoyancy mode , we expect their growth rates to be proportional to the square of the rotation rate . in fig . [ fig : m0 ] we then show their growth rates divided by @xmath211 as a function of the wavenumber . the three panels are for three different values of @xmath37 and the different curves in each panel are for different values of @xmath50 . as already discussed , the centrifugal buoyancy modes represent actually a sequence of unstable modes and , in the panels , we show only those branches of this mode with the maximum growth rate . this figure demonstrates that the @xmath211 scaling law is quite good and that , as expected from the local dispersion relation , the growth rate increases with the vertical wavenumber and tends to an asymptotic limit as @xmath212 , in fact the behaviour of the growth rate as a function of the wavenumber is the same as in fig . [ fig : figwkbj ] . these modes appear to be always unstable , the reason , discussed by @xcite , is related to the fact that , since the plasma has no pressure , it is possible to compress it along the field lines and create a density perturbation without performing any work . the centrifugal force can then always overcome the magnetic restoring forces . however , the inclusion of a finite pressure tends to stabilize this mode . , the left panel is for @xmath199 , the mid panel is for @xmath203 and the right panel is for @xmath213 . the different curves refer to different values of @xmath50 and the corresponding values of @xmath50 and @xmath49 are given in the legend.,width=566 ] we start our analysis of non - axisymmetric instabilities by considering first the cases with @xmath42 and @xmath215 and , in order to get a first indication on the number and the kind of modes that we can find , in fig . [ fig : complexplanem1 ] we plot in the complex plane the position of unstable modes for a given parameter set . the figure is for @xmath141 and we consider two values of the wavenumber : squares are for @xmath216 and dots are for @xmath217 . both at large and small wavenumbers , we observe an isolated mode and a series . in fig . [ fig : grvsalfa ] we consider the behavior of the modes represented in fig . [ fig : complexplanem1 ] as a function of the rotation parameter @xmath49 , the colors of the curves are in correspondence with the colors in fig . [ fig : complexplanem1 ] , and , for the series , we have considered only the mode with the largest growth rate . from the figure we can see that the only mode that survives when we let rotation go to zero is the one corresponding to the green dot in fig . [ fig : complexplanem1 ] , all the others become stable . we can then conclude that the mode corresponding to the green dot reduces to the cdi mode in the zero rotation limit , while rotation is at the origin of all the other modes . in this figure , we can further notice a stabilizing effect of rotation on the cdi , however , this is noticeable only at large values of @xmath49 . this can be compared with the results of @xcite who find also a stabilizing effect of rotation , but at smaller values of the rotation rate , in their case however there is a rigid rotation , while in this case the rotation rate decreases radially . , @xmath141 , @xmath143 , the ( green and red ) dots are for @xmath218 , while the ( black and blue ) squares are for @xmath216 . the orange line shows the frequency range of the slow continuum , which , in the zero pressure case , reduces to the flow continuum . as it is discussed in the text , the green dot represents the cdi , the black dot mri , while the red and blue series represent centrifugal modes . , width=566 ] for the modes represented in fig . [ fig : complexplanem1 ] . the colors of the curves correspond to the color in fig . [ fig : complexplanem1 ] . , width=566 ] we can further investigate the physical origin of the different modes by computing the stabilizing and destabilizing terms based on the energetic considerations discussed in the previous section . starting from the cdi mode , we find that the destabilizing terms for this mode is not only @xmath197 , as it should be expected for the current driven mode , but also @xmath187 . in fig . [ fig : cdvar ] we plot the fractional contributions of these two terms as a function of the rotation rate @xmath49 . the fractional contributions for the two terms are defined , respectively , as @xmath219 and @xmath220 we see that at low rotation rates , the dominant term is the current term represented by the green curve , but , as we increase the rotation rate , the contribution by the shear term increases until it becomes dominant for @xmath221 . ( green ) , and of the shear term , @xmath222 ( black ) , as a function of the rotation rate @xmath49 for the cdi mode . the values of other parameters are the same as in fig . [ fig : complexplanem1 ] . , width=566 ] as we did for the axisymmetric modes , we can investigate the physical origin of the two series of modes , represented by blue squares and red dots in fig . [ fig : complexplanem1 ] , by comparing in fig . [ fig : centrifugal ] the growth rate as a function of @xmath49 ( solid curves ) with the approximation given by equation ( [ eq : cengr ] ) ( dashed curves ) . we see that the approximation reproduces fairly well the behavior of the actual growth rate , so we can then regard the centrifugal term as being responsible for the destabilization of these modes and hence identify them as magnetic buoyancy instabilities . as discussed in the previous section and in section [ sec : wkb ] , with the wkbj analysis , we can distinguish them as a toroidal buoyancy mode at low wavenumbers ( red curve ) and a poloidal buoyancy mode at high wavenumbers ( blue curve ) . the two sequences of modes cluster to the edge of the flow continuum ( which is what is left of the slow continua in the zero pressure case ) , whose frequency range is represented in fig . [ fig : complexplanem1 ] by the orange line . consider now the mode represented by the black square in fig . [ fig : complexplanem1 ] . the black curve in fig . [ fig : centrifugal ] traces this mode as @xmath49 varies . similarly to the buoyancy modes , we can aproximate its growth rate by an expression analogous to equation ( [ eq : cengr ] ) in which @xmath196 is replaced by @xmath187 , @xmath223 the dashed black curve in fig . [ fig : centrifugal ] shows the growth rate given by equation ( [ eq : growthmri ] ) , which indeed closely follows an actual one represented by the black curve . so , the main driving force for this mode is related to the shear of the radially decreasing rotation rate and therefore it should be identified with the mri . however , this occurs only in a very limited parameter range , while in other regions it merges with either the buoyancy modes or with the cdi mode , where the driving force become either the centrifugal term or the current term , respectively ( see below ) . so , from now on we mostly concentrate on the cdi and centrifugal buoyancy modes . for the centrifugal buoyancy modes ( blue and red ) and the mri branch ( black ) . the blue and red dashed curves represent an aproximation to the growth rate given by equation ( [ eq : cengr ] ) for the buoyancy modes , whereas the black dashed curve represents an approximation to the growth rate given by equation ( [ eq : growthmri ] ) for the mri . the values of other parameters are the same as in fig . [ fig : complexplanem1].,width=566 ] we can now proceed with a more detailed analysis of the dependence of the growth rates on the wavenumber , the pitch and rotation . in fig . [ fig : gr1 ] , we plot the growth rates as a function of the wavenumber for @xmath224 ( left panel ) and for @xmath225 ( right panel ) for @xmath203 and @xmath143 . in the left panel , we have clearly distinct the cdi and the toroidal and poloidal buoyancy modes . for both values of the pitch , the black part of the curve , which corresponds to the mri , is distinct only in the growing part over a relatively narrow range of wavenumbers and is merged with one of the poloidal buoyancy modes in the constant region . as expected and discussed in paper i , the cdi mode ( green curve ) increases its growth rate and the value of its maximum unstable wavenumber as we decrease the pitch . for @xmath226 , the increase of the maximum unstable wavenumber brings the cdi mode to a complicated interaction with the other modes , merging first with the mri branch and then with one of the poloidal buoyancy modes . for @xmath227 the cdi mode becomes unstable for all wavenumbers , while the driving force , increasing the wavenumber , changes nature , becoming first related to the shear of rotation and then centrifugal . both the toroidal and the poloidal buoyancy modes increase their growth rate as we decrease the pitch , this is because in the equilibrium configuration the rotation rate increases as the pitch decreases . the toroidal buoyancy mode becomes stable at high wavenumbers , while the poloidal buoyancy mode becomes stable at small wavenumbers . decreasing @xmath50 , the stable region between them shows a small increase in width and moves towards high wavenumbers . . the left panel is for @xmath228 while the right panel is for @xmath141 . the other parameters are @xmath42 and @xmath143 . , width=566 ] for discussing in more detail the behavior of the cdi , it can be useful to examine how the properties of the equilibrium structure are modified in the different parameter ranges . figures [ fig : pitch ] and [ fig : jpar ] show the radial profiles of the pitch and the equilibrium current component parallel to the magnetic field , @xmath229 , which is the destabilizing factor for the cdi . in fact , by rearranging expression ( [ eq : fcd ] ) for the current term @xmath197 , one can show that the destabilizing term is due to a contribution proportional to the current component parallel to the magnetic field ( e.g. , see * ? ? ? * ch . 8) . therefore , the latter is a central quantity determining the growth rate of the cdi . the three panels are respectively for @xmath199 ( left panel ) , @xmath42 ( mid panel ) and @xmath200 ( right panel ) and the different curves in each panel refer to different values of @xmath50 , in each panel we also plotted the case with @xmath224 and no rotation for reference ( black curves ) . we remember that a variation of the pitch leads also to a variation of the rotation rate : for lower values of the pitch we have higher values of @xmath49 and these values can be read in the legend . the pitch profile is normalized to the value @xmath50 , on the axis , while the parallel current is multiplied by @xmath50 to bring curves for different values of @xmath50 on the same scale , since for large values of @xmath50 the parallel current scales as @xmath230 . the pitch profile is in general characterized by a flat part up to @xmath231 followed by a steep increase . in the left panel , we see that a decrease of @xmath50 leads to a slower increase of @xmath232 for @xmath233 . correspondingly , in the left panel of fig . [ fig : jpar ] , we observe a slight increase of the parallel current . in the case of @xmath42 ( mid panel ) , when we have @xmath30 constant , the pitch profiles remain essentially unchanged when we decrease @xmath50 , while the parallel current shows a substantial decrease for low values of @xmath50 . for @xmath200 ( right panel ) , @xmath30 increases with radius and consequently the pitch show increasingly steeper profiles as we decrease @xmath50 and the decrease of the parallel current is larger than in the previous case . . the three panels refer to three different values of @xmath37 , the left panel is for @xmath199 , the mid panel is for @xmath203 and the right panel is for @xmath213 . the different curves refer to different values of @xmath50 and the corresponding values of @xmath50 and @xmath49 are given in the legend . , width=566 ] . the three panels refer to three different values of @xmath37 , the left panel is for @xmath199 , the mid panel is for @xmath203 and the right panel is for @xmath213 . the different curves refer to different values of @xmath50 and the corresponding values of @xmath50 and @xmath49 are given in the legend . , width=566 ] in paper i we discussed a scaling law for the growth rate of the cdi of the form @xmath234 and in fig . [ fig : cd ] we can investigate the effect of rotation by considering the deviation from this scaling law . the three panels and the curves in each panel correspond to the same cases shown in figs . [ fig : pitch ] and [ fig : jpar ] . in the left panel we can observe that a first effect of rotation is to move the cutoff wavenumber to smaller values , from @xmath235 without rotation ( paper i ) to @xmath236 . apart from that , the scaling provided by eq . ( [ eq : scaling ] ) is quite good , slight deviations can be observed only for the smallest value of @xmath50 ( blue curve ) , partly due to the interaction with other modes ( mri ) and partly ( as already discussed in paper i ) related to the change of the pitch profile and parallel current observed in the corresponding equilibrium solution . comparing the red curves ( largest values of @xmath50 ) in the three panels we see that the increase of the rotation rate leads the cutoff wavenumber to shift towards increasingly lower values , in parallel , however , we have also a slight increase of the growth rate in the unstable range . from the other curves ( green , blue , orange and purple ) , we see that , for lower values of @xmath50 , at @xmath42 , the cutoff disappears because the cdi mode starts to interact and merge with the centrifugal mode and the growth rate decreases as a result of the decrease of the parallel current in the equilibrium configuration . for @xmath200 , we also observe a decrease of the growth rate for increasingly lower values of @xmath50 , corresponding to the decrease of the parallel current . in summary , rotation has , in general , a stabilizing effect on the cdi mainly because it modifies the equilibrium structure by decreasing the parallel current . this is consistent with @xcite , who also found stabilization of cdi at high rotation rates in the case of rigid rotation . however , especially at large values of @xmath50 , there are situations in which , on the contrary , the growth rate of cdi shows a slight increase with the rotation rate . finally , we recall that the cdi is stable for @xmath237 . for @xmath143 . the three panels refer to three different values of @xmath37 , the left panel is for @xmath199 , the mid panel is for @xmath203 and the right panel is for @xmath213 . the different curves refer to different values of @xmath50 and the corresponding values of @xmath50 and @xmath49 are given in the legend . , width=566 ] we can now turn our attention to the toroidal and poloidal buoyancy modes . from the local dispersion relations , given by equations ( [ eq : pbm ] ) and ( [ eq : tbm ] ) , we expect their growth rates to be proportional to the square of the rotation rate . in figs . [ fig : tb ] and [ fig : pb ] , we then show their growth rates divided by @xmath211 as a function of the wavenumber . the three panels are , as before , for three different values of @xmath37 and the different curves in each panel are for different values of @xmath50 . as already discussed , the toroidal and poloidal buoyancy modes represent actually a series of unstable modes and , in these figures , we show only the modes with the maximum growth rate . the @xmath211 scaling law is quite good for the toroidal buoyancy mode , slightly less valid in the case of the poloidal buoyancy mode . in general , the value of @xmath238 represents a high wavenumber cutoff for the toroidal buoyancy mode and a low wavenumber cutoff for the poloidal buoyancy mode , however , for large values of the rotation rate , as shown in the right panels of figs . [ fig : tb ] and [ fig : pb ] , we observe merging between modes of the two series and deviations from the @xmath238 cutoff ( note that the merging of poloidal and toroidal modes at @xmath213 shown in the right panels of figs . [ fig : tb ] and [ fig : pb ] , may refer to different mode branches in the two series ) . in fig . [ fig : buoym-1 ] we show the results for the case with @xmath237 . overall , the buoyancy modes behave similarly with the wavenumber , although there are some differencies with the @xmath143 case at intermediate @xmath83 . specifically , we see that the buoyancy mode is unstable at all wavenumbers , in fact the narrow stability region around @xmath239 disappears . the growth rate , which is independent of the wavenumber at small values of the latter , shows however a variation around @xmath238 , sometimes a decrease but typically an increase going towards larger values of the wavenumber . of course , this increase of the growth rate eventually asymptotes to a finite value as @xmath117 , as in the @xmath143 case above , but at small @xmath50 , this asymptotic value is reached at higher wavenumbers . so far we concentrated on the values @xmath240 of the azimuthal wavenumber and now we examine different values . figure [ fig : diffm ] shows the toroidal ( red curves ) and poloidal ( blue curves ) buoyancy modes as well as mri ( black curves ) at different @xmath84 and fixed @xmath225 and @xmath203 . first consider the case of positive @xmath84 , when the toroidal and poloidal modes are separated . the growth rate of the toroidal mode increases with @xmath84 and the instability boundary extends to larger @xmath83 . however , it is clear from this figure and also from the local dispersion relation ( [ eq : tbmsol ] ) that the growth rate at small @xmath83 converges to a finite value as @xmath84 becomes large . by contrast , the growth rate of the poloidal mode decreases with @xmath84 and the instability boundary shifts to larger @xmath83 , but the maximum growth achieved in the limit of high @xmath83 is essentially independent of @xmath84 , as it also follows from the local dispersion relation ( [ eq : pbm ] ) . the behaviour of these modes in the case of negative @xmath84 , where , as we discussed above , they are represented by a single curve with respect to @xmath83 ( fig . [ fig : buoym-1 ] ) , are similar to that for positive @xmath84 . at @xmath241 , corresponding to the toridal mode , the growth rate increases with the absolute value of @xmath84 and converges to a constant value at a given @xmath83 , while at @xmath242 , corresponding to the poloidal mode , it decreases with the absolute value of @xmath84 , but tends to the same limiting value . the mri exists only for @xmath243 and 3 in a certain interval of @xmath83 being most unstable at @xmath243 and @xmath244 . this maximum growth rate of the mri is higher than that at @xmath143 , is comparable to that of the toroidal and poloidal modes for the same @xmath243 and it does not merge with the latter as opposed to the case @xmath143 ( see fig . the growth rate of the mri at @xmath245 is decreased , its range in @xmath83 is narrower and shifted to larger values . the mri disappears beyond this azimuthal wavenumber ; we did not find it for larger positive @xmath246 as well as for all negative @xmath84 . for @xmath143 . the three panels refer to three different values of @xmath37 , the left panel is for @xmath199 , the mid panel is for @xmath203 and the right panel is for @xmath213 . the different curves refer to different values of @xmath50 and the corresponding values of @xmath50 and @xmath49 are given in the legend . , width=566 ] for @xmath143 . the three panels refer to three different values of @xmath37 , the left panel is for @xmath199 , the mid panel is for @xmath203 and the right panel is for @xmath213 . the different curves refer to different values of @xmath50 and the corresponding values of @xmath50 and @xmath49 are given in the legend . , width=566 ] for @xmath237 . the three panels refer to three different values of @xmath37 , the left panel is for @xmath199 , the mid panel is for @xmath203 and the right panel is for @xmath213 . the different curves refer to different values of @xmath50 and the corresponding values of @xmath50 and @xmath49 are given in the legend . , width=566 ] ( solid ) , @xmath247 ( dashed ) , @xmath248 ( dash - dot ) . the pitch parameter @xmath141 and the rotation @xmath42 . ] , in the different regions of the parameter plane @xmath198 . in the region marked by a green shading , the instability with the largest growth rate is the cdi , while in the region marked by a blue shading the instability with the largest growth rate is the centrifugal buoyancy instability.,width=566 ] we have examined the stability properties of a rotating magnetized jet flow in the approximation of zero thermal pressure . our study has focused on the effect of rotation on the cdi and on the new modes of instability introduced by rotation . in this spirit , as a first step , we did not consider the presence of a longitudinal flow , whose main effect on the modes concentrated inside the jet radius , as it is for most of the rotationally - induced modes studied in this paper , is only that of doppler shifting the frequency . the instability behaviour depends , of course , on the chosen equilibrium configuration and our results can be considered representative of an equilibrium configuration characterized by a distribution of current concentrated in the jet , with the return current assumed to be mainly found at very large distances . similar stability analyses of rotation - induced modes in magnetized flows in the limit of zero thermal pressure are presented in @xcite and @xcite , they however make use only of a local wkb approach and the last two papers consider only axisymmetric perturbations . our local analysis ( see section [ sec : wkb ] ) generally agrees with the results of these papers in the parameter regimes they consider . however , for our specific jet configuration we found that the mri in the cold limit can be present , but only in a very limited region of the parameter space and is never dominant . we extended the results of these papers to the global domain , where the wkb approach no longer holds , by solving a boundary value problem and revealed new properties of these modes that are summarized below : + 1 . at small and large axial wavenumbers @xmath83 , the growth rates , respectively , for the toroidal and poloidal buoyancy modes obtained from the global calculations ( figs . 6 , 11 and 15 - 18 ) actually exhibit a dependence on @xmath83 similar to what is obtained by the corresponding local dispersion relations ( see fig . + 2 . at intermediate @xmath83 , the behavior can be different from that predicted on the basis of the local dispersion relation , with the presence of stability gaps ( for positive @xmath84 ) and merging of poloidal and toroidal modes ( for negative @xmath84 ) . the properties of the mri in the cold plasma limit , studied in the above papers based on the local dispersion relation , qualitatively agrees with our global calculations . in the nonaxisymmetric case , for positive @xmath84 , the mri is present only in a limited range of wavenumbers , its growth rate and the width of the unstable range reach a maximum for @xmath249 and then ( for larger @xmath84 ) decrease ( fig . 18 ) consistently with @xcite , however , the mri is absent for @xmath246 and for every negative value of @xmath84 . in addition , we did not find the mri for @xmath129 in both local and global cases , likely because of the different equilibrium adopted . in general , the mri has always a growth rate smaller than that of the buoyancy modes . we have shown that two main kinds of instabilities cdi and buoyancy prevail in the considered jet flow . in fig . [ fig : summary ] we represent , in the parameter plane @xmath198 , with different shadings , the regions where each of them has the largest growth rate . the figure refers to non - axisymmetric modes with @xmath250 . we can observe that the cdi is dominant at small rotation rates and that the boundary between the cdi and centrifugal buoyancy instability regions moves towards larger values of @xmath49 as we decrease @xmath50 . for @xmath251 , the cdi is stable for this value of the wavenumber and the only instability is the centrifugal buoyancy , which is , however , obviously stable at zero rotation . it is seen that the buoyancy instability occupies quite a large area in this parameter space in comparison with the cdi and hence should be important in jets with rotation . when we increase the wavenumber , the cdi tends to be stabilized and the centrifugal instability tends to become dominant everywhere in the parameter plane . comparing now the growth rates of axisymmetric and non - axisymmetric centrifugal buoyancy modes , we see that at high wavenumbers , axisymmetric modes have a larger growth rate , which decreases monotonically with decreasing @xmath83 . by contrast , non - axisymmetric modes have a growth rate that is almost independent from the wavenumber and , therefore , become dominant at low values of @xmath83 . summarizing , at low rotation rates , the non - axisymmetric cdi is the instability that grows fastest and has large wavelengths . increasing the rotation rate , the prevailing instability becomes the centrifugal axisymmetric one , which operates at small wavelengths . these results are applicable to magnetically and rotationally dominated jets , since , increasing the importance of thermal pressure , centrifugally driven modes tend to be stabilized and other modes , like pressure driven modes @xcite may appear . at the same time , taking into account the shear of longitudinal velocity can give rise to unstable kh modes in the jet . this first step will be extended by introducing the effects of the longitudinal velocity also in the relativistic regime and these results will be presented in a following paper . the different behaviour in the explored parameter space may be important for understanding the nonlinear stages since distinct types of instability may evolve differently . this study is therefore an essential first step for the interpretation of the results of numerical simulations and for their comparison with astrophysical data . g.m . acknowledges the georg forster postdoctoral research fellowship from the alexander von humboldt foundation . m. , stegun i. a. , 1972 , handbook of mathematical functions , new york : dover s. , 1996 , , 314 , 995 s. , camenzind m. , 1992 , , 256 , 354 s. , lery t. , baty h. , 2000 , , 355 , 818 s. a. , hawley j. f. , 1992 , , 400 , 610 h. , keppens r. , 2002 , , 580 , 800 m. c. , 1998 , , 493 , 291 m. , 1991 , the stability of jets . p. 278 j. w. s. , van der swaluw e. , keppens r. , goedbloed j. p. , 2005 , , 444 , 337 g. , mamatsashvili g. , rossi p. , mignone a. , 2013 , , 434 , 3030 g. , rosner r. , ferrari a. , knobloch e. , 1989 , , 341 , 631 g. , rosner r. , ferrari a. , knobloch e. , 1996 , , 470 , 797 a. , urpin v. , 2006 , , 73 , 066301 a. , urpin v. , 2007 , , 662 , 851 a. , urpin v. , 2008 , , 488 , 1 a. , urpin v. , 2011a , , 84 , 056310 a. , urpin v. , 2011b , , 525 , a100 a. , iacono r. , bhattacharjee a. , 1987 , physics of fluids , 30 , 2167 c. s. , sovinec c. r. , 2009 , , 699 , 362 a. , trussoni e. , zaninetti l. , 1978 , , 64 , 43 j. , 1987 , ideal magnetohydrodynamics , plenum press , new york e. , rotenberg m. , 1960 , reviews of modern physics , 32 , 898 w. , lai d. , 2011 , , 410 , 399 j. p. , 2009 , physics of plasmas , 16 , 122110 goedbloed j. p. , keppens r. , poedts s. , 2010 , advanced magnetohydrodynamics : with applications to laboratory and astrophysical plasmas . cambridge university press , cambridge ; 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we perform a linear stability analysis of magnetized rotating cylindrical jet flows in the approximation of zero thermal pressure . we focus our analysis on the effect of rotation on the current driven mode and on the unstable modes introduced by rotation . we find that rotation has a stabilizing effect on the current driven mode only for rotation velocities of the order of the alfvn velocity . rotation introduces also a new unstable centrifugal buoyancy mode and the `` cold '' magnetorotational instability . the first mode is analogous to the parker instability with the centrifugal force playing the role of effective gravity . the magnetorotational instability can be present , but only in a very limited region of the parameter space and is never dominant . the current driven mode is characterized by large wavelenghts and is dominant at small values of the rotational velocity , while the buoyancy mode becomes dominant as rotation is increased and is characterized by small wavelenghts . [ firstpage ] galaxies : jets , mhd , instabilities

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Proceed to summarize the following text: diffuse reflectance spectroscopy in the visible and near - infrared range ( vis - nir drs ) has proved to be useful to assess various soil properties @xcite . it can be employed to provide more data rapidly and inexpensively compared to classical laboratory analysis . therefore , drs is increasingly used for vast soil surveys in agriculture and environmental research @xcite . recently , several studies have shown the applicability of vis - nir drs _ in situ _ as a proximal soil sensing technique @xcite . to predict soil properties from soil spectra , a model is calibrated , often using partial least squares ( pls ) regression . however , when calibration is based on air - dried spectra collected under laboratory conditions , predictions of soil properties from field spectra tend to be less accurate @xcite . usually , this decrease in accuracy is attributed to varying moisture between air - dried calibration samples and field spectra recorded with a variable moisture content . different remediation techniques have been proposed , ranging from advanced preprocessing of the spectra @xcite to `` spiking '' the calibration set with field spectra @xcite . in our study , we adopt a slightly different view on the calibration problem . it does not only apply to the varying moisture conditions between the calibration data set and the field spectra . indeed , it is also valid if we want to predict soil properties in a range where calibration samples are rare . mining with rarity or learning from imbalanced data is an ongoing research topic in machine learning @xcite . in imbalanced data sets frequent samples outnumber the rare once . therefore , a model will be better at predicting the former and might fail for the latter . two different approaches exist to take care of the data imbalance : we can either adjust the model or `` balance '' the data . the latter approach has the advantage that we can use the usual modelling framework . synthetic minority oversampling technique ( smote ) is one way to balance the data . it was first proposed for classification @xcite and recently for regression @xcite . smote oversamples the rare data by generating synthetic points and thus helps to equalize the distribution . in this study , we propose a strategy to increase the prediction accuracy of soil properties from field spectra when they are rare in calibration . the goal of this study is to build a calibration model to predict soil organic carbon content ( socc ) from field spectra by air - dried samples spiked with synthetic field spectra . the studied soil was sampled at the southern slopes of mt . kilimanjaro , tanzania ( 3@xmath3 4@xmath4 33@xmath5 s , 37@xmath3 21@xmath4 12@xmath5 e ) in coffee plantations . due to favourable soil and climate in this region , extensive coffee plantations constitute a frequent form of land use . we took 31 samples for calibration at 4 different study sites . for validation , we scanned 12 field spectra at a wall of a soil pit and sampled soil material for chemical analysis at the scanned spots . we call these validation field spectra f. after collection , the calibration samples were dried in an oven at 45@xmath3c and sieved @xmath6 2 mm . subsequently , they were scanned with an agrispec portable spectrophotometer equipped with a contact probe ( analytical spectral devices , boulder , colorado ) in the range 3502500 nm with 1 nm intervals . the same spectrometer was used in the field . the instrument was calibrated with a spectralon white tile before scanning the soil samples . for the measurement , a thoroughly mixed aliquot of the sample was placed in a small cup and the surface was smoothed with a spatula . each sample was scanned 30 times and the signal averaged to reduce the noise . in the following , we call this calibration data set l. socc was measured in a cns - analyser by high temperature combustion with conductivity detectors . to generate new data to spike the calibration data set l , we used smote @xcite and its extension for regression @xcite . this algorithm consists of generating new synthetic data using existing data and is summarized below . in our case , we generated new spectra and the related socc using the field spectra f. the new spectra are created by calculating the difference between a field spectrum and one of its nearest neighbours and adding this difference ( weighted by a random number between 0 and 1 ) to the field spectrum . the socc of the synthetic spectrum is then a weighted average between the socc of the field spectrum and the used nearest neighbour . smote has two parameters , namely @xmath7 , the number of points to generate for each existing point ( given in percent of the whole data set ) and @xmath8 , the number of nearest neighbours . to study the influence of these parameters we generated six different synthetic data sets s1 through s6 , varying @xmath9 and @xmath10 . @xmath11 $ ] : original sample @xmath12 @xmath13 $ ] : target value of original sample @xmath14 @xmath15 $ ] : synthetic sample @xmath16 @xmath17 $ ] : target values of synthetic sample @xmath18 @xmath19 @xmath20 : number of synthetic samples to compute for each original sample generate synthetic samples : @xmath21 compute @xmath8 nearest neighbours for @xmath11 $ ] randomly choose @xmath22 @xmath23 $ ] @xmath24 = { \mathit{orig.s}}[i ] + \mathrm{random}(0,1 ) \times { \mathit{diff}}$ ] @xmath25)$ ] @xmath26 @xmath27 we corrected each spectrum ( calibration , validation and synthetic ) for the offset at 1000 and 1830 nm and kept only parts with a high signal - to - noise ratio ( 4502400 nm ) . then , we transformed the spectra to absorbance @xmath28 and smoothed them using the singular spectrum analysis ( ssa ) . ssa is a non - parametric technique to decompose a signal into additive components that can be identified as the signal itself or as noise @xcite . finally , we divided each spectrum by its maximum and calculated the first derivative . in order to assess similarities between the calibration , validation and synthetic data sets , we calculated the principal component analysis ( pca ) of the ( uncorrected original ) spectra l and f and projected the synthetic data into the space spanned by the principal components . we calibrated seven different pls models . for model i we used the data set l , the spectra scanned under laboratory conditions . model ii through vii were calibrated on l spiked with synthetic spectra s1 through s6 . to find the best model i through vii , we varied the number of pls components between 1 and 15 . based on the predictions in the leave - one - out cross - validation ( loocv ) we calculated the corrected akaike information criterion @xcite aic@xmath29 , where @xmath30 is the number of calibration samples , @xmath31 the number of pls components and @xmath32 the root mean - squared error . the latter is defined as @xmath33 , where @xmath34 are the predicted and @xmath35 the measured soccs . we selected the model with the smallest aic@xmath36 as the most plausible . to assess the model quality , we used the @xmath32 , the mean error @xmath37 and the coefficient of determination @xmath38 , where @xmath39 is the mean socc . smote has two random components because it selects spectra randomly ( with replacement ) among the nearest neighbours and weights the difference between spectra by a random number ( between 0 and 1 ) . to study the influence of these random steps , we generated 100 different datasets s1 through s6 . each data set was then used to spike the calibration data set l , to build a new pls model and to predict the data set f. the first principal components ( pcs ) explain 85.4% and 11.2% of variance , respectively . we can clearly identify two distinct groups of samples : the calibration data set l and the field spectra f ( fig . [ fig : pca ] ) . in other words , the data sets l and f differ . the synthetic points that were projected into the space spanned by the pcs resemble the field spectra as expected . the distinct characteristics of the data sets l and f accord well with the difficulties to predict the data set f by using the laboratory spectra l only ( table [ tab : results : calib ] and table [ tab : results : valid ] ) . although the loocv of model i yields a moderate @xmath32 and a large @xmath1 , the validation on the data set f fails . spiking the calibration data set l with synthetic spectra increases the prediction accuracy of the socc in the data set f. actually , the @xmath32 decreases and @xmath1 increases with increasing number of synthetic points both for the loocv and the validation ( table [ tab : results : calib ] and table [ tab : results : valid ] ) . however , the number of model parameters also increases from 2 to 7 . the monte carlo results show only a small variability in the interquartile range . however , some synthetic data sets in model v produced @xmath1 values smaller than @xmath20.53 , the value we obtain in model i on air - dried samples only . this might be due to the combination of neighbours during smoting . in general , models with 5 neighbours were more accurate than those with 3 neighbours . however , the number of neighbours had a smaller influence on the prediction accuracy than the number of synthetic points . it is difficult to decide _ a priory _ how many synthetic points should be included in the calibration . indeed , in a classification problem the goal is to approximate an equal distribution of different classes such that the rare class becomes an ordinary one . in regression , however , we do not know which features of the data make them rare . for our data , the range of socc in the data set l is larger than in the data set f. therefore , we conclude that concentration is not responsible for the difference between these data sets . based on the monte carlo results we chose one synthetic data set from model vi , namely the one with the median number of model parameters and the best @xmath1 in the validation . thus , the calibration data set includes 31 air - dried and 24 synthetic spectra . compared to model i , spiking the air - dried data set l with these synthetic spectra clearly improves the prediction of the data set f ( fig . [ fig : models ] ) . [ cols="<,<,>,>,>,>,>,>",options="header " , ] we propose a framework to predict soil properties from _ in situ _ acquired field spectra by spiking air - dried laboratory calibration data by synthetic ones generated from these field spectra . in general , the prediction accuracy increases when a sufficient number of synthetic points is included in the calibration . however , because it is difficult to determine this number _ a priori _ , we recommend to generate several synthetic data sets to find an appropriate model . this study is part of the project dfg for 1246 `` kilimanjaro ecosystems under global change : linking biodiversity , biotic interactions and biogeochemical ecosystem processes '' and was supported by the deutsche forschungsgemeinschaft . b. stenberg and r. a. viscarra rossel , `` diffuse reflectance spectroscopy for high - resolution soil sensing , '' in _ proximal soil sensing _ , r. a. viscarra rossel , a. b. mcbratney , and b. minasny , eds . , pp . springer , 2010 . k. d. shepherd and m. g. walsh , `` infrared spectroscopy - enabling an evidence - based diagnostic surveillance approach to agricultural and environmental management in developing countries , '' , vol . 1 , pp . 119 , 2007 . vgen , k. d. shepherd , and m. g. walsh , `` sensing landscape level change in soil fertility following deforestation and conversion in the highlands of madagascar using vis - nir spectroscopy , '' , vol . 3 , pp . 281294 , 2006 . r. a. viscarra rossel , s. r. cattle , a. ortega , and y. fouad , `` in situ measurements of soil colour , mineral composition and clay content by vis nir spectroscopy , '' , vol . 3 , pp . 253266 , 2009 . t. h. waiser , c. l. s. morgan , d. j. brown , and c. t. hallmark , `` in situ characterization of soil clay content with visible near - infrared diffuse reflectance spectroscopy , '' , vol . 2 , pp . 389396 , 2007 . b. minasny , a. b. mcbratney , v. bellon - maurel , j .- roger , a. gobrecht , l. ferrand , and s. joalland , `` removing the effect of soil moisture from nir diffuse reflectance spectra for the prediction of soil organic carbon , '' , vol . 118124 , 2011 .

diffuse reflectance spectroscopy is a powerful technique to predict soil properties . it can be used _ in situ _ to provide data inexpensively and rapidly compared to the standard laboratory measurements . because most spectral data bases contain air - dried samples scanned in the laboratory , field spectra acquired _ in situ _ are either absent or rare in calibration data sets . however , when models are calibrated on air - dried spectra , prediction using field spectra are often inaccurate . we propose a framework to calibrate partial least squares models when field spectra are rare using synthetic minority oversampling technique ( smote ) . we calibrated a model to predict soil organic carbon content using air - dried spectra spiked with synthetic field spectra . the root mean - squared error of prediction decreased from 6.18 to 2.12 mg g@xmath0 and @xmath1 increased from @xmath20.53 to 0.82 compared to the model calibrated on air - dried spectra only . = 1 diffuse reflectance spectroscopy , soil , partial least squares , calibration , smote

You are an expert at summarizing long articles. Proceed to summarize the following text: the problem of instantaneous wavefunction collapse ( reduction ) due to measurement @xcite has been a stumbling block for many physicists since the creation of quantum mechanics . the unavoidable `` spookiness '' @xcite of the quantum collapse is related to the impossibility to find a traditional physical mechanism responsible for the collapse . mathematically , the `` spookiness '' can be expressed via violation of the bell inequalities @xcite . even though this violation @xcite is common knowledge nowadays , the mechanism and interpretation of the collapse remain debatable @xcite . a natural approach to understanding the physics of the wavefunction reduction is through analysis of the gradual evolution at a shorter time scale , i.e. , `` inside '' the collapse . a few decades ago there was an idea that such an evolution can be fully described by decoherence . however , nowadays it is becoming common knowledge that gradual collapse of individual quantum systems is governed by a continuous flow of information during the measurement , thus showing essentially the same `` spookiness '' as the textbook collapse . this understanding was significantly influenced by experiments with superconducting qubits in the last decade @xcite , which demonstrated the actual evolution `` inside '' the collapse . there are many approaches to the theoretical description of the evolution `` inside '' the collapse , i.e. , the description of partial or continuous quantum measurement . in spite of very different mathematical treatments , many of these approaches are essentially equivalent . probably the most well - known approach is based on positive operator - valued measure ( povm ) and kraus operators @xcite . let us also mention quantum trajectories @xcite , quantum filtering @xcite , monte carlo wavefunction approach @xcite , quantum state diffusion @xcite , restricted path integral @xcite , quantum bayesian formalism @xcite ( see also @xcite and chap . 2.2 of @xcite ) and many other approaches , e.g. , @xcite . among these approaches , one of the simplest formalisms is the quantum bayesian formalism , which is based only on elementary quantum mechanics and common sense . for solid - state systems , the gradual collapse due to continuous measurement was first described using the quantum bayesian formalism @xcite , and soon after that was also described by the quantum trajectory approach @xcite . from late 1990s to mid-2000s the analysis was mainly focused on the continuous measurement of a charge qubit by a quantum point contact ( qpc ) or a single - electron transistor ( set ) @xcite . the next considered system was based on a partially / continuously measured superconducting phase qubit @xcite ; the first experimental demonstration of a partial collapse @xcite and uncollapsing @xcite was realized with this system . after the development of circuit qed qubit measurement @xcite and the transmon @xcite , much attention was paid to this system since it experimentally allowed truly continuous quantum measurement of qubits @xcite . in this measurement setup the qubit state affects the frequency of a coupled resonator , which in turn is probed by an applied microwave in the homodyne way . for the circuit qed measurement of a qubit the quantum trajectory approach was developed in ref . @xcite and the quantum bayesian approach was introduced in ref . in particular , the quantum bayesian theory was used in several circuit qed experiments on quantum feedback and quantum trajectories @xcite , and several experiments used the quantum trajectory theory @xcite . while the description of the qubit evolution in the process of circuit qed measurement is generally similar to that for measurement by qpc or set , there is one considerable difference . the measurement by a qpc or set is of the broad - band type , while the circuit qed measurement is narrow - band . correspondingly , instead of one output signal @xmath0 in the qpc / set case , there are in general two output signals in the circuit qed case , since a narrow - band signal can be represented as @xmath1 , where @xmath2 is the carrier frequency . the existence of two signals ( two quadratures ) leads to the importance of the question of which amplifier is used in the process of measurement . in the case of a phase - sensitive amplifier , only one quadrature is amplified , and therefore only one signal [ say , @xmath0 ] is available . this makes the phase - sensitive case similar to the measurement by qpc or set ( however , it is still important exactly which quadrature is amplified ) . for the case of a phase - preserving amplifier , both output signals @xmath0 and @xmath4 are available ( and both are noisier than in the phase - sensitive case @xcite ) ; this makes the description of qubit evolution significantly different from that for the qpc / set case . an advantage of the quantum bayesian formalism in comparison with the quantum trajectory formalism is its simplicity , so that it does not require special theoretical training , and can be used by non - experts . this simplicity is due to a transparent physical meaning , which directly relates the quantum back - action to the information acquired during measurement . in ref . @xcite the quantum bayesian formalism for the circuit qed measurement of a qubit was developed for the so - called `` bad cavity '' limit , in which the damping ( bandwidth ) @xmath5 of the measurement resonator is much larger than the rate of qubit collapse ( quantum back - action ) due to measurement . in this limit the qubit is practically unentangled with the resonator and experiences two kinds of back - action due to measurement . the `` spooky '' back - action ( which can also be called `` quantum '' , `` informational '' , or `` non - unitary '' @xcite ) moves the qubit state along the meridians of the bloch sphere and is directly related to the continuous information on the qubit state ( @xmath6 or @xmath7 ) obtained during the measurement . this back - action does not have a physical mechanism , similarly to the einstein - podolsky - rosen - bell example @xcite . the other type , the `` phase '' backaction ( called `` realistic '' , `` classical '' or `` unitary '' back - action in @xcite ) has a physical mechanism : fluctuation of the ( ac stark - shifted ) qubit frequency due to a fluctuating number of photons in the resonator . the phase back - action moves the qubit state along the parallels of the bloch sphere . in spite of the clear physical mechanism , the phase back - action also has some `` spookiness '' ; for example , it is possible to choose the qubit movement along the parallels or meridians afterwards , by choosing the amplified microwave quadrature @xcite ( this prediction has been confirmed experimentally @xcite ) . in the present paper we extend the quantum bayesian formalism to the case when the `` bad cavity '' limit is not applicable . as we will see , in this case the evolution equations remain practically the same as in the `` bad cavity '' regime @xcite ; however , now they should be applied to the entangled qubit - resonator system . in the derivation we will assume that the qubit evolves only due to measurement ; in particular , we assume no rabi oscillations . the rabi oscillations can be added later phenomenologically ; however , such addition is not really correct if the rabi frequency is comparable to or larger than the resonator damping @xmath5 . in this respect the theory discussed here has the same limitation as the `` polaron frame approximation '' usually used in the quantum trajectory approach @xcite ( see also @xcite ) . actually , our theory is equivalent to the quantum trajectory theory with this approximation . however , the evolution equations are formally different and have a simple and intuitive physical meaning . we expect that our approach may have advantage over the quantum trajectory theory in numerical simulations , similar to the qpc / set case . ( in the latter case the reason for the numerical advantage was that the quantum trajectory equation is essentially the lowest - order approximation in the time step , while the quantum bayesian evolution is the exact solution in the absence of rabi oscillations , and this permits using larger time steps even in the presence of rabi oscillations . ) our derivation will be based on elementary quantum mechanics . we will also need some basic facts related to coherent states ; for completeness , they are discussed in appendix a. the paper is mainly addressed to non - experts in continuous quantum measurement and non - experts in quantum optics ; this is why we include brief discussion of facts well - known to experts and focus on simple logic . we hope that our derivation is accessible at the advanced - undergraduate level . while we discuss the circuit qed measurement of one qubit , it is straightforward to extend the discussion to the measurement of several qubits , including entanglement by measurement @xcite . the paper is organized in the following way . in sec . ii we discuss the system and the model . in sec . iii we review the results of ref . @xcite for the `` bad cavity '' regime of circuit qed measurement . the main section of this paper is sec . iv , in which we derive the quantum bayesian formalism for circuit qed measurement with a moderate bandwidth . we first introduce a natural idea of `` history tail '' , which consists of the microwave field emitted by the measurement resonator , and thus carries information about the resonator state at previous time moments ( sec . [ sec : history - tail ] ) . then we develop the bayesian formalism by applying a natural measurement procedure to pieces of the `` history tail '' of short duration @xmath8 ( sec . [ sec : main - idea ] ) . the textbook collapse due to this measurement leads to evolution of the entangled qubit - resonator state . we first derive the results for phase - sensitive measurement ( sec . [ sec : imperfect ] ) and then for phase - preserving measurement ( sec . [ sec : ph - pres ] ) . the obtained evolution equations for short @xmath8 are also converted into the differential form ( sec . [ sec : differential ] ) and integrated for an arbitrary long duration ( sec . [ sec : arb - duration ] ) we conclude in sec . v. appendix a reviews basic facts related to coherent states . in appendix b we derive the formulas for the phase back - action in the `` bad cavity '' regime via a simple language based on vacuum noise . we consider a superconducting qubit ( transmon ) measured in the circuit qed setup ( fig . [ fig : schematic ] ) . the idea of the measurement @xcite is based on the dispersive coupling of the qubit with a microwave resonator , whose frequency slightly changes depending on whether the qubit is in the state @xmath6 or @xmath7 ( both are the eigenstates of qubit energy ) . this frequency shift affects the phase and amplitude of a probing microwave , which is transmitted through or reflected from the resonator ( theoretically , there is no significant difference between the transmission and reflection configurations ; however , in practice it is often better to use reflection ) . the outgoing microwave is amplified , and then the ghz - range signal is downconverted by mixing it with the original microwave tone , so that the low - frequency ( @xmath9 mhz ) output of the iq mixer provides information about the qubit state . the rate of the information acquisition is limited by the output noise , which is mainly determined by the first amplifying stage ( pre - amplifier ) . in recent years nearly quantum - limited superconducting parametric amplifiers @xcite became the standard pre - amplifiers , replacing formerly used cryogenic high - electron - mobility transistors ( hemts ) , which have a much higher noise level . is transmitted through ( or reflected from ) the resonator , whose frequency slightly changes , @xmath10 , depending on the qubit state . after amplification , the microwave is sent to the iq mixer , which produces two quadrature signals : @xmath0 and @xmath4 . in the case of a phase - sensitive amplifier we define @xmath0 as the signal corresponding to the amplified quadrature , while for a phase - preserving amplifier we define @xmath0 as the quadrature carrying information about the qubit state.,width=336 ] the hamiltonian of the qubit interacting with the resonator in the dispersive approximation @xcite is h_q&r/=(_q/2 ) _ z + _ + a^a _ z , [ ham - disp]where @xmath11 is the ( effective ) qubit frequency , @xmath12 is the ( effective ) resonator frequency , @xmath13 is the dispersive coupling , @xmath14 and @xmath15 are the creation and annihilation operators for the resonator ( so that @xmath16 is the number of photons in the resonator ) , and the pauli operator @xmath17 acts on the qubit state in the energy basis @xmath7 and @xmath6 . as we see from this hamiltonian , the resonator frequency increases by @xmath18 when the qubit state changes from @xmath6 to @xmath7 ; conversely , the qubit frequency increases by @xmath18 per each additional photon in the resonator ( ac stark shift ) . the typical value of @xmath19 is crudely 1 mhz , while the qubit and resonator frequencies are typically between 4 and 9 ghz , with the detuning @xmath20 of crudely 1 ghz , where @xmath21 . the microwave drive of the resonator can be described by the standard additional hamiltonian h_d/=(t ) e^-i_dt a^+ ^*(t ) e^i_dt a , [ ham - drive ] where @xmath22 is the drive frequency and @xmath23 is the properly normalized drive amplitude . [ this form is the rotating wave approximation ( rwa ) of the physical hamiltonian @xmath24(a+a^\dagger ) $ ] . ] we do not consider the case when the resonator is driven by a squeezed microwave or a squeezed vacuum . we assume that the field in the resonator decays with the rate @xmath25 ( so that the energy decays with the rate @xmath5 ) due to coupling with transmission lines and possibly due to other mechanisms of decay ( at zero temperature ) . for the ensemble - averaged evolution , the effect of damping with rate @xmath5 can be described via the standard lindblad term in the master equation ; however , we will not use it , since we are interested in evolution of an individual quantum system rather than an ensemble . note that the derivation of the dispersive hamiltonian ( [ ham - disp ] ) for a transmon is somewhat involved ( see , e.g. , ref . @xcite and appendix of ref . @xcite ) because at least 3 transmon levels should be taken into account to find the coupling @xmath13 ( 4 levels are needed for the lowest - order dependence of @xmath13 on @xmath26 ) . the small-@xmath26 value of the coupling @xmath13 can be approximated @xcite as = , where @xmath27 is the coupling in jaynes - cummings hamiltonian and @xmath28 is the transmon anharmonicity ( @xmath29 is the transition frequency between transmon levels @xmath7 and @xmath30 ) . with increasing @xmath26 the value of @xmath13 changes ( as well as @xmath12 ) , and a better description of the evolution should be based on the eigenlevels of the transmon - resonator system , rather than bare levels @xcite . in the present paper we do not take these complications into account and use the simple hamiltonian ( [ ham - disp ] ) ; however , there is a natural way to include these effects into our formalism phenomenologically . one more subtlety is that the resonator damping @xmath5 leads to the qubit energy relaxation @xcite via the purcell effect , which we do not take into account . however , in many present - day experiments this effect is suppressed by a purcell filter @xcite , so description by the simple hamiltonian ( [ ham - disp ] ) again becomes a good approximation . in this paper we will be using the rotating frame , based on the drive frequency @xmath22 for the resonator and the frequency @xmath11 for the qubit . this essentially means that instead of fast - oscillating coefficients in the lab - frame wavefunction @xmath31 ( here @xmath26 is the number of photons in the resonator ) , we implicitly operate with slower - varying coefficients @xmath32 and @xmath33 . equivalently , we can change the hamiltonian ( [ ham - disp ] ) and ( [ ham - drive ] ) to the rotating - frame hamiltonian h_rot/= ( _ r-_d ) a^a + a^a _ z + a^+ ^ * a. [ ham-2]note that in appendix a we use tilde signs for the rotating - frame variables , which are omitted in the main text . our goal in this paper is to find ( in a simple way ) the evolution of the qubit - resonator state in the process of measurement . for that we assume that the qubit evolves only due to measurement , so we explicitly assume the absence of a rabi drive applied to the qubit and absence of qubit energy relaxation . since the hamiltonian ( [ ham - disp ] ) is of the quantum non - demolition ( qnd ) type @xcite , then if the initial qubit state is @xmath6 , it will remain @xmath6 during the whole measurement process . similarly , the initial qubit state @xmath7 will remain @xmath7 . in these two simple cases , evolution of the resonator state is decoupled from the qubit , but the effective resonator frequency @xmath34 depends on the qubit state ( the upper sign is for the qubit state @xmath7 ) . then the classical evolution of the resonator field @xmath35 can be described in the standard rwa way as @xcite _ = -i ( _ r -_d ) _ - _ -i , [ alpha - dot1]where the rotating frame is based on the drive frequency @xmath22 . the quantum evolution is described by exactly the same equation @xcite , with the classical field state replaced by the coherent state @xmath36 ( see appendix a ) . note that @xmath37 is the average number of photons in the resonator . besides the notation @xmath38 , we will interchangeably use a notation that explicitly shows the corresponding qubit state , _ 1 _ + , _ 0 _ - . from eq . ( [ alpha - dot1 ] ) , we see that the resonator field depends on the qubit state . in particular , the steady state is _ , [ alpha - st](it is easy to see that these complex numbers always belong to the circle in the complex plane , which is centered at @xmath39 and passes through the origin . ) the outgoing field @xmath40 in the transmission and reflection configurations ( fig . [ fig : trans - refl ] ) can be described as @xcite f_trans= , f_refl= + , [ f - out]where @xmath41 is the resonator damping due to coupling with the outgoing transmission line ( @xmath42 ) , and in this normalization @xmath43 is the average number of propagating photons per second . ( note that the phase of @xmath40 can be chosen arbitrarily ; in our choice the coefficient between @xmath40 and @xmath44 is real and positive . ) by combining eqs . ( [ alpha - st ] ) and ( [ f - out ] ) it is easy to see that in the case @xmath45 the reflection configuration operates with smaller fields for the same response ( and therefore larger phase response ) than the transmission configuration , and in this sense it is preferable from the practical point of view . however , for our purposes in this paper the two configurations are equivalent ( the well - defined difference @xmath46 can theoretically be simply subtracted ) . we will implicitly assume the transmission configuration ( without loss of generality ) , while all the results are applicable to both the transmission and reflection configurations . the outgoing microwave field @xmath40 is then amplified ( either in a phase - preserving or a phase - sensitive way ) and sent to the iq mixer ( fig . [ fig : schematic ] ) . a phase - sensitive amplifier amplifies only a certain phase ( quadrature ) @xmath47 of the microwave field and de - amplifies the @xmath48-shifted phase ( orthogonal quadrature ) . for the complex number @xmath40 this means amplification of only a certain direction on the complex plane along @xmath49 . for a faster qubit measurement , the obvious choice is to amplify the quadrature that connects the complex numbers @xmath50 and @xmath51 corresponding to qubit states @xmath6 and @xmath7 . we will consider amplification of an arbitrary quadrature , including this optimal case . the iq mixer produces two low - frequency signals , which correspond to the real and imaginary parts of an amplified @xmath40 ; however it is easy to rotate the axes of the complex plane by using the linear combinations of the two outputs . since only one quadrature is amplified by a phase - sensitive detector , there is no information in the mixer output corresponding to the orthogonal quadrature . therefore , the phase - sensitive amplifier essentially produces _ only one output signal _ after the mixer , which we will call @xmath0 . note that the amplified phase @xmath47 can in principle vary in time ; then we also vary the quadrature , corresponding to @xmath0 . a phase - preserving amplifier equally amplifies any quadrature , so both outputs of the iq mixer are important . ( note that usual non - parametric amplifiers , including hemt , are phase - preserving . ) in this case we will call @xmath0 the linear combination of the outputs corresponding to the quadrature connecting @xmath50 and @xmath51 , so that @xmath0 carries information about the measured qubit state @xmath6 or @xmath7 . the output signal for the orthogonal quadrature will be called @xmath4 ; it does not carry information about the qubit state , but will still be important for producing phase back - action . since @xmath52 and @xmath53 evolve before reaching steady values , we will correspondingly vary the quadratures corresponding to @xmath0 and @xmath4 . the main reason why phase - sensitive amplifiers are often preferred for the qubit measurement is that their quantum limitation for the output noise is twice smaller than that for phase - preserving amplifiers @xcite . the output noise of a phase - sensitive amplifier should exceed the `` half quantum '' , which exactly corresponds to the width of the ground state of the oscillator , representing the amplified field ( so that the energy is @xmath54 ) . in other words , this is the amplified vacuum noise of the coherent state of the field , and the ideal phase - sensitive amplifier does not add its own noise ( the output noise can be smaller if a squeezed state is amplified ) . the output noise power of a phase - preserving amplifier is at least two `` half quanta '' : one comes from the amplified vacuum noise , and one more is added by the amplifier @xcite . as discussed above , the dynamics of the system is very simple when the qubit is either in the state @xmath6 or @xmath7 during the whole measurement process . the goal of this paper is to describe the evolution when the initial qubit state is a superposition @xmath55 or , more generally , an arbitrary density matrix @xmath56 . in this section we review results of ref . @xcite for the `` bad cavity '' limit , which assumes @xmath57 , where @xmath58 is the qubit ensemble dephasing due to measurement , discussed below . in this case we can neglect transient evolution of the resonator state , and there is practically no entanglement between the qubit and the measurement resonator , because the two steady states ( [ alpha - st ] ) are very close to each other , @xmath59 . therefore , the evolution of the qubit state can be considered by itself . it is assumed that parameters of the measurement setup ( @xmath5 , @xmath60 , etc . ) do not change in time . we review here the `` bad cavity '' limit mainly for later comparison with the more general case @xmath61 ; the derivation in the next section does not rely on results discussed in this section . ensemble dephasing of the qubit in the `` bad cavity '' regime is @xcite = 2 i m ( _ 1,st^ * _ 0,st ) = |_1,st-_0,st|^2 . [ gamma - bc1]we see that the condition @xmath57 is equivalent to @xmath59 . in the case when @xmath62 , the ensemble dephasing @xcite can be expressed as [ see eq . ( [ alpha - st ] ) ] = , [ gamma - bc2]and the ac stark shift contribution @xmath63 to the effective qubit frequency @xmath64 is @xcite _ q= 2|n . [ stark - bc]if @xmath65 is comparable to @xmath5 ( but still @xmath66 ) , then eqs . ( [ gamma - bc2 ] ) and ( [ stark - bc ] ) should be modified ( see sec . [ sec : ensemble - aver ] ) , but the bayesian formalism reviewed in this section does not change . evolution of the qubit density matrix @xmath67 due to measurement of an _ arbitrary duration _ @xmath68 can be described by simple equations @xcite @xmath69 , \,\,\,\qquad \label{ph - sens - diag - bc}\\ & & \frac{\rho_{10}(t+\tau)}{\rho_{10}(t)}= \frac{\sqrt{\rho_{11}(t+\tau)\ , \rho_{00}(t+\tau ) } } { \sqrt{\rho_{11}(t)\ , \rho_{00}(t)}}\ , \ , \nonumber \\ & & \hspace{1.7 cm } \times \ , \exp [ -i k \tilde{i}_{\rm m}(\tau)\ , \tau ] \ , e^{-\gamma \tau } \ , e^{-i\delta \omega_{\rm q}\tau } , \qquad \label{ph - sens - off - bc } \end{aligned}\ ] ] where i_m ( ) = i_m ( ) - , i_m()= _ t^t+ i(t ) dt , [ ph - sens - im - bc]so that @xmath70 is the measured output signal @xmath0 averaged over the time interval @xmath71 $ ] , while for @xmath72 we also subtract the mean value @xmath73 , with @xmath74 and @xmath75 being the average output signals , corresponding to the qubit states @xmath6 and @xmath7 . the measurement response is @xmath76 where @xmath77 is the phase difference between the amplified quadrature @xmath47 and the optimal quadrature @xmath78 , which gives the largest response @xmath79 . the variance of @xmath80 due to the amplifier noise is d= , [ d - def - bc]where @xmath81 is the single - sided spectral density of the noise [ for different definitions of the spectral density , eq . ( [ d - def - bc ] ) should be changed correspondingly ] . the phase back - action depends on the coefficient @xmath82 , which equals k= _ d= _ d. [ ph - sens - k - bc ] + the dephasing rate @xmath83 is due to non - ideality of the measurement , = - = - , where @xmath58 is the qubit ensemble dephasing [ see eqs . ( [ gamma - bc1 ] ) and ( [ gamma - bc2 ] ) ] . the quantum efficiency of the measurement process can be introduced in two different ways , = 1-==_amp_col , = ^2 _ d , where @xmath84 ( @xmath85 ) takes into account quantum efficiency @xmath86 of the phase - sensitive amplifier and efficiency @xmath87 of the microwave signal collection , while @xmath88 also includes the effect of choosing a non - optimal quadrature for amplification . here @xmath89 is the ratio of the microwave energy reaching amplifier to the total energy loss by the resonator , so that @xmath90 describes the loss in the transmission line before reaching the amplifier . the amplifier efficiency @xmath91 is the ratio between the output noise @xmath92 of an ideal quantum - limited amplification chain to the actual output noise @xmath81 . the last term in eq . ( [ ph - sens - off - bc ] ) is due to the ac stark shift @xmath63 given by eq . ( [ stark - bc ] ) ( note that in ref . @xcite the rotating frame was already accounting for this term and the equation was written for the conjugate variable @xmath93 ) . note that @xmath94 and therefore @xmath95 . equations ( [ ph - sens - diag - bc])([ph - sens - im - bc ] ) can be used to find the qubit evolution in an experiment by using experimental output signal record @xmath0 ; in numerical simulations @xmath80 can be picked randomly from the probability density distribution @xmath96 ^ 2}{2d } \bigg ] , \qquad \label{ph - sens - gauss}\end{aligned}\ ] ] where @xmath97 and @xmath98 is the standard gaussian distribution in the case when the qubit is in the state @xmath99 . for an infinitesimally small averaging time @xmath68 , this is equivalent to using i(t)= + [ _ 11 ( t)-_00(t ) ] + _ i(t ) , s__i = s_i , [ ph - sens - i(t)]where @xmath100 is the white noise with spectral density @xmath81 . the qubit evolution equations ( [ ph - sens - diag - bc ] ) and ( [ ph - sens - off - bc ] ) have a very simple physical meaning . the evolution ( [ ph - sens - diag - bc ] ) of the diagonal matrix elements of the density matrix is the classical bayesian update for the probabilities , _ jj(t+ ) = , where @xmath101 is the likelihood , given by eq . ( [ ph - sens - gauss ] ) . note that another form of eq . ( [ ph - sens - diag - bc ] ) in terms of the non - centered measurement result @xmath70 is = . the evolution ( [ ph - sens - off - bc ] ) of the off - diagonal matrix element contains the natural term due to change of the diagonal elements ( conservation of relative purity ) , the phase back - action term , decoherence due to non - ideality , and contribution from the ac stark shift . the phase back - action has a natural mechanism : when measuring a non - optimal quadrature , @xmath102 , the output signal gives us information about the fluctuating number of photons in the resonator , and therefore the fluctuating ac stark shift . the factor @xmath82 in eq . ( [ ph - sens - off - bc ] ) is the coefficient characterising this linear relation between the ac stark shift and output signal fluctuations . the evolution equations ( [ ph - sens - diag - bc ] ) and ( [ ph - sens - off - bc ] ) have been derived in ref . @xcite in the following way . the bayesian evolution ( [ ph - sens - diag - bc ] ) of the diagonal matrix elements was essentially postulated from the necessary correspondence between the classical and quantum evolution . this follows from common sense as much as the standard collapse postulate in quantum mechanics . for the off - diagonal elements , the logic of the derivation ( sketched below ) was essentially the same as in the first derivation @xcite for measurement by a qpc . using the general inequality @xmath103 and evolution ( [ ph - sens - diag - bc ] ) for the diagonal elements , it is easy to derive inequality for the ensemble dephasing , @xmath104 . in the quantum - limited case [ in this case @xmath105 is resolved with signal - to - noise ratio of 1 after time @xmath106 , and therefore @xmath107/(32\chi^2\bar{n})\,$ ] ] , and for @xmath108 , the lower bound of this inequality for @xmath58 coincides with the actual value ( [ gamma - bc2 ] ) . therefore , in this case the evolution of @xmath109 should be precisely the first term in eq . ( [ ph - sens - off - bc ] ) and possibly a result - independent phase ( which is naturally associated with the qubit frequency shift in the last term ) ; otherwise the ensemble dephasing would be larger than in eq . ( [ gamma - bc2 ] ) . thus , in the ideal case eqs . ( [ ph - sens - diag - bc ] ) and ( [ ph - sens - off - bc ] ) have been derived `` logically '' , by comparing unavoidable evolution due to acquired information with the ensemble dephasing . in the non - optimal case ( @xmath102 ) , the derivation in ref . @xcite took into account the phase back - action by explicitly analyzing the information on the fluctuating photon number in the resonator provided by the measurement result @xmath110 . in this way eq ( [ ph - sens - k - bc ] ) for the correlation factor @xmath82 was obtained , leading to the term with @xmath82 in eq . ( [ ph - sens - off - bc ] ) . finally , the term @xmath111 in eq . ( [ ph - sens - off - bc ] ) was obtained by averaging over the extra noise from a non - ideal amplifier @xcite and averaging over the signal that was lost due to imperfect microwave collection . this is how the qubit evolution equations ( [ ph - sens - diag - bc ] ) and ( [ ph - sens - off - bc ] ) have been obtained in ref . @xcite . actually , the derivation for the phase back - action coefficient @xmath82 in ref . @xcite was presented only for the case of resonant microwave frequency , @xmath112 . in appendix b we show the derivation , which is still valid in the case of a significant detuning , @xmath113 . this derivation is based on an analysis of the effect of vacuum noise entering the resonator from the transmission line . in this analysis the vacuum noise is treated essentially classically , consistent with the poisson statistics @xmath114 for the photon number . note that averaging of the evolution equations ( [ ph - sens - diag - bc ] ) and ( [ ph - sens - off - bc ] ) over random @xmath110 with the probability distribution ( [ ph - sens - p(i ) ] ) produces ensemble - averaged equations @xmath115 in which there is no dependence on the measured phase @xmath77 ( as required by causality ) because + = = - . let us briefly discuss the role of the `` weak response '' condition @xmath116 in the formalism reviewed in this section . in the case of not too small a number of photons in the resonator , @xmath117 , this inequality follows from the `` bad cavity '' condition @xmath59 , and therefore is not needed as an additional condition . however , for @xmath118 it is possible to have @xmath59 even when @xmath119 . in this case the bayesian formalism ( [ ph - sens - diag - bc])([ph - sens - gauss ] ) is still applicable , but the ensemble dephasing @xmath58 and ac stark shift @xmath120 are not necessarily given by eqs . ( [ gamma - bc2 ] ) and ( [ stark - bc ] ) , in particular , because @xmath121 and @xmath122 may be significantly different , @xmath123 . the formulas for @xmath58 and @xmath120 in this case are given in ref . @xcite and also derived in sec . [ sec : ensemble - aver ] [ @xmath58 is given by eq . ( [ gamma - bc1 ] ) , while @xmath63 is given by eq . ( [ delta - omega - sum ] ) ] . note that the bayesian evolution equations ( [ ph - sens - diag - bc ] ) and ( [ ph - sens - off - bc ] ) are exactly the same as for the continuous qubit measurement by qpc or set @xcite . however , the dependence ( [ ph - sens - deltai - bc ] ) and ( [ ph - sens - k - bc ] ) of the response @xmath124 and phase back - action coefficient @xmath82 on the measured quadrature @xmath77 is a specific feature of the circuit qed ( or cavity qed ) setup . as was discussed in sec . ii , in the case of a phase - preserving amplifier we choose @xmath0 to be the output signal , corresponding to the optimal quadrature @xmath78 , while the output @xmath4 corresponds to the orthogonal quadrature @xmath125 . therefore , @xmath108 for @xmath0 and @xmath126 for @xmath4 . the qubit state evolution due to a phase - preserving measurement for an _ arbitrary duration _ @xmath68 is described by equations @xcite @xmath69 , \,\,\,\qquad \label{ph - pres - diag - bc}\\ & & \frac{\rho_{10}(t+\tau)}{\rho_{10}(t)}= \frac{\sqrt{\rho_{11}(t+\tau)\ , \rho_{00}(t+\tau ) } } { \sqrt{\rho_{11}(t)\ , \rho_{00}(t)}}\ , \ , \nonumber \\ & & \hspace{1.7 cm } \times \ , \exp [ -i k \tilde{q}_{\rm m}(\tau)\ , \tau ] \ , e^{-\gamma \tau } \ , e^{-i\delta \omega_{\rm q}\tau } , \qquad \label{ph - pres - off - bc } \end{aligned}\ ] ] which have exactly the same form as eqs . ( [ ph - sens - diag - bc ] ) and ( [ ph - sens - off - bc ] ) , except @xmath127 in eq . ( [ ph - sens - off - bc ] ) is replaced by @xmath128 in eq . ( [ ph - pres - off - bc ] ) . the measurement result @xmath127 is given by eq . ( [ ph - sens - im - bc ] ) , and similarly q_m ( ) = _ t^t+ q(t ) dt - q_0 , [ ph - pres - qm - bc]with equal average values , @xmath129 , for the two qubit states . since @xmath0 and @xmath4 are equally amplified , the variances of @xmath127 and @xmath128 due to amplifier noise are both equal to d= , s_q = s_i . the phase back - action is now caused by @xmath130 , and the coefficient @xmath82 is the same as in eq . ( [ ph - sens - k - bc ] ) for @xmath126 , k== , i = i_1-i_0 . [ ph - pres - k]the dephasing rate @xmath83 due to non - ideality is = - 2 = - , where ensemble dephasing @xmath58 is still given by eqs . ( [ gamma - bc1 ] ) and ( [ gamma - bc2 ] ) ( it can not depend on the detector because of causality ) , and the extra factor of 2 is related to equal contributions due to fluctuations of @xmath0 and @xmath4 . the quantum efficiency can again be defined in two different ways , = 1- = _ amp_col , = = _ amp_col , [ eta - phase - pres - def]where @xmath84 ( @xmath131 ) compares the measurement with the ideal phase - preserving case , while @xmath88 and @xmath132 compare the operation using only @xmath0 channel with the ideal phase - sensitive case ( @xmath133 because of twice larger noise in an ideal phase - preserving amplifier ) . equations ( [ ph - pres - diag - bc ] ) and ( [ ph - pres - off - bc ] ) describe the qubit evolution when the signals @xmath0 and @xmath4 are obtained from an experiment , while in numerical simulations @xmath127 can be generated using eqs . ( [ ph - sens - p(i ) ] ) and ( [ ph - sens - gauss ] ) , while @xmath128 can be picked from the gaussian probability distribution p(_m ) = . for infinitesimally small @xmath68 , the signal @xmath0 can also be generated using eq . ( [ ph - sens - i(t ) ] ) , and for @xmath4 we can use q(t)=q_0 + _ q ( t ) , s__q= s_q= s_i , with equal spectral densities , @xmath134 , of uncorrelated noise in @xmath0 and @xmath4 channels . equations ( [ ph - pres - diag - bc ] ) and ( [ ph - pres - off - bc ] ) have been derived in ref . @xcite in three different ways , leading to the same result . in the first derivation , eq . ( [ ph - pres - diag - bc ] ) has been again postulated from the necessary correspondence with classical evolution of probability , and the phase back - action coefficient @xmath82 in eq . ( [ ph - pres - k ] ) has been calculated from information on fluctuation of photon number , provided by @xmath4 . this gives the inequality @xmath135 , whose lower bound in the ideal case coincides with the actual value ( [ gamma - bc2 ] ) . thus in the ideal case eqs . ( [ ph - pres - diag - bc ] ) and ( [ ph - pres - off - bc ] ) can be derived `` logically '' , while the non - ideal case ( @xmath136 ) can be analyzed by averaging over the extra noise of the amplifier ( @xmath137 ) and over information contained in the lost fraction of the microwave signal ( @xmath138 ) . in the second derivation @xcite , eqs . ( [ ph - pres - diag - bc ] ) and ( [ ph - pres - off - bc ] ) have been obtained from eqs . ( [ ph - sens - diag - bc ] ) and ( [ ph - sens - off - bc ] ) by considering a phase - preserving amplifier as a phase - sensitive amplifier with rapidly rotating amplified phase @xmath47 , so that the difference @xmath77 from the optimal phase is also changing . then averaging the evolution in eqs . ( [ ph - sens - diag - bc ] ) and ( [ ph - sens - off - bc ] ) over the period of phase rotation , we obtain eqs . ( [ ph - pres - diag - bc ] ) and ( [ ph - pres - off - bc ] ) . finally , the third derivation in ref . @xcite has been based on considering a phase - preserving amplifier as two phase - sensitive amplifiers , which amplify orthogonal quadratures in two microwave channels , obtained from the microwave signal by using a symmetric beam splitter . then using eqs . ( [ ph - sens - diag - bc ] ) and ( [ ph - sens - off - bc ] ) for each channel , we again obtain eqs . ( [ ph - pres - diag - bc ] ) and ( [ ph - pres - off - bc ] ) . note that since in the `` bad cavity '' regime the qubit is practically not entangled with the measurement resonator , it is easy to include the qubit evolution due to rabi oscillations , energy relaxation , etc . ( this extra evolution should be much slower than @xmath5 , but can be slower , comparable , or faster than @xmath58 ) . for that we need to take the derivative of the evolution equations ( [ ph - sens - diag - bc ] ) and ( [ ph - sens - off - bc ] ) for the phase - sensitive case or eqs . ( [ ph - pres - diag - bc ] ) and ( [ ph - pres - off - bc ] ) for the phase - preserving case , and simply add the terms due to other mechanisms of evolution ( this is equivalent to interleaving the both types of evolution ) . as always @xcite , in taking the derivative it is important to specify whether the it or stratonovich definition of the derivative is used . now let us discuss the main subject of this paper : the bayesian formalism for continuous qubit measurement in the circuit qed setup ( fig . [ fig : schematic ] ) in the case when the `` bad cavity '' limit is not applicable . therefore , we now assume that the resonator damping rate @xmath5 is comparable to the speed of the qubit evolution due to measurement back - action , which can be characterized by the qubit ensemble dephasing @xmath58 . in this case there is significant entanglement between the qubit and resonator , so we should consider the evolution of the combined qubit - resonator system . also , since the typical measurement time is comparable to @xmath139 , our formalism should focus on the transient evolution . the parameters of the measurement setup ( @xmath60 , @xmath47 , @xmath5 , etc . ) are allowed to change in time ( this change should be much slower than @xmath22 for rwa to be valid , but can be comparable to or even faster than @xmath5 ) . in general , this problem is rather complicated , but we use a simplifying assumption : we assume that the qubit evolution is only due to measurement , i.e. , there are no rabi oscillations , qubit energy relaxation , etc . in practice this means that the frequency of rabi oscillations and rate of energy relaxation should be much smaller than @xmath5 ( then the extra evolution can be added phenomenologically , as discussed in the previous section ) . we will assume that the initial state of the qubit - resonator system is unentangled , and the resonator starts in a coherent state @xmath140 , |(0)= ( c_0 |0+c_1 |1)|_in , [ psi - in]where @xmath141 and , for example , @xmath140 is vacuum . generalization to a mixed initial state , ^q&r ( 0 ) = _ q , in |_in_in| , [ rho - in]or a slightly more general state [ see eq . ( [ rho - qr - def ] ) below ] will be straightforward . in the analysis we will use the rotating frame , corresponding to the hamiltonian ( [ ham-2 ] ) . we will first discuss a simple general point of view , which describes the evolution due to measurement , then derive equations for the ensemble - averaged evolution , and then discuss the evolution during an individual realization of the measurement process . until sec . [ sec : imperfect ] we will assume the ideal case , in particular @xmath142 see fig . [ fig : trans - refl](a ) , in which we need to assume @xmath143 . suppose the initial state of the qubit is @xmath6 . since the measurement is of the qnd type and the qubit does not evolve by itself , it will remain in the state @xmath6 , and since the resonator is initially in a coherent state , its state will remain to be an ( evolving ) coherent state ( see appendix a ) . therefore , the qubit - resonator system will evolve in the rotating frame as |(t)=|0 e^-i_0 ( t ) |_0 ( t ) , where the coherent state amplitude @xmath52 and the overall phase @xmath144 evolve according to eqs . ( [ tilde - alpha - dot ] ) and ( [ phi - dot-2 ] ) with the resonator frequency @xmath145 and drive ( rotating frame ) frequency @xmath22 , @xmath146 here the drive amplitude @xmath60 can be time - dependent and the damping @xmath5 can in general be also time - dependent . note that the evolution ( [ phi0-dot ] ) of the overall phase is often not considered in textbooks , but for us it is very important . derivation of eq . ( [ phi0-dot ] ) from the schrdinger equation with the hamiltonian ( [ ham-2 ] ) is very simple . now let us consider a larger physical system , which includes the field leaking from the resonator to the transmission line ( actually , we also necessarily need to consider the incoming field from the transmission line , but we assume that it is always vacuum ) . this larger system keeps a record of the previous evolution in a form of a `` flying away tail '' ( a propagating microwave ) , which we will call a `` history tail '' ( fig . [ fig : tail ] ) . let us divide this tail into sufficiently short pieces of duration @xmath8 ( @xmath147 ) ; each of them will also be a coherent state , as follows from the property 2.6 discussed in appendix a for a beam splitter ( in our case a leaking `` mirror '' at the end of the resonator ) . the @xmath148th piece of the history tail ( counting back in time ) will be @xmath149 , which is the resonator state at time @xmath150 , passed through the beam splitter with transmission amplitude @xmath151 [ this value follows from the energy conservation , @xmath152 . therefore , the wavefunction , including the history tail is |(t)= e^-i_0 ( t ) |0 |_0 ( t)_m |_0 ( t - m t ) . [ evol-0]if @xmath5 depends on time , then the factor @xmath153 in this equation should be replaced with @xmath154 . note that the coherent states in the tail are unentangled with each other and with the resonator state ( see property 2.6 in appendix a ) . also note that the number of terms in the direct product ( [ evol-0 ] ) increases with time ; this seems unphysical , but it is only a matter of notation ; we can keep the number of terms constant by adding vacuum states of the pieces of field incoming from the transmission line . if the qubit is initially in the state @xmath7 , then it remains in @xmath7 , so that the wavefunction of the system including the history tail is given by eq . ( [ evol-0 ] ) with @xmath6 replaced with @xmath7 , @xmath50 replaced with @xmath51 , and @xmath155 replaced with @xmath156 , where @xmath157 and @xmath158 evolve according to eqs . ( [ tilde - alpha - dot ] ) and ( [ phi - dot-2 ] ) with the resonator frequency @xmath159 , @xmath160 now let us make a simple but very important logical step in the derivation . since the qubit does not evolve by itself , we can consider two evolutions at the same time : for the qubit in the state @xmath6 and in the state @xmath7 , so that the coefficients in the initial superposition ( [ psi - in ] ) do not change in time ( fig . [ fig : tail ] ) . this follows from the general linearity of quantum mechanics and somewhat resembles the `` many worlds '' interpretation . therefore , for the initial state ( [ psi - in ] ) of the qubit - resonator system we obtain the wavefunction evolution ( including the flying away history tail ) @xmath161 where @xmath162 and @xmath163 are constant in time , while @xmath164 , @xmath155 , @xmath51 , and @xmath156 evolve according to eqs . ( [ alpha0-dot ] ) , ( [ phi0-dot ] ) , ( [ alpha1-dot ] ) and ( [ phi1-dot ] ) , starting with @xmath165 and @xmath166 . the approach to qubit measurement via the wavefunction evolution in eq . ( [ evol-01 ] ) is physically transparent and quite powerful . in particular , it will easily allow us to describe evolution of the qubit - resonator system in the process of measurement by applying the textbook collapse postulate to measurement of the tail pieces ( fig . [ fig : tail ] ) . however , let us first discuss the ensemble - averaged evolution . if the result of the tail measurement is not taken into account , we need to average the quantum state over all possible measurement results , which is equivalent to tracing the state ( [ evol-01 ] ) over the tail . this leads to a density operator in the qubit - resonator hilbert space , @xmath167 , in which the parts diagonal in the qubit subspace are @xmath168 while the off - diagonal parts contain the inner product of the tails for the two different evolutions , @xmath169}\ , |1\rangle \langle 0 | \otimes ( t)\rangle \langle \alpha_0 ( t)| \nonumber \\ & & \hspace{0.1 cm } \times \prod_m \langle \alpha_0 ( t - m\ , \delta t ) \sqrt{\kappa \ , \delta t } \ , |\ , \alpha_1 ( t - m\ , \delta t ) \sqrt{\kappa \ , \delta t } \rangle , \qquad \label{rho-10-qr-1}\end{aligned}\ ] ] and similarly for @xmath170 . the inner product for each time piece @xmath8 is given by eq . ( [ inner - product ] ) in appendix a , so that we find @xmath171}\ , |1\rangle \langle 0 | \otimes ( t)\rangle \langle \alpha_0 ( t)| \ , \nonumber \\ & & \hspace{1 cm } \times \exp \left ( - \int_0^t \frac{\kappa}{2}\,|\alpha_1(t')-\alpha_0(t')|^2 \ , dt ' \right ) \nonumber \\ & & \hspace{1 cm } \times \exp \left ( -i \int_0^t \kappa\ , { \rm im}\ , [ \alpha_1^*(t ' ) \ , \alpha_0 ( t')]\ , dt ' \right ) . \qquad \label{rho-10-qr}\end{aligned}\ ] ] in this equation the second line obviously describes dephasing with the rate _ d(t ) = |_1(t)-_0(t)|^2 , [ gamma - d]which is directly related to distinguishability of the field emitted into the transmission line and therefore to the information that can in principle be obtained from measurement . the third line in eq.([rho-10-qr ] ) is the changing phase factor which can be ascribed to the shift of the qubit frequency in the process of measurement , _ q,1(t ) = i m [ _ 1^ * ( t ) _ 0 ( t ) ] . [ delta - omega-1]however , a similar frequency shift comes from the term @xmath172}$ ] in eq . ( [ rho-10-qr ] ) ; using eqs.([phi0-dot ] ) and ( [ phi1-dot ] ) , we obtain the corresponding value _ q,2(t ) = re \ { ^*(t ) [ _ 1 ( t)- _ 0 ( t ) ] } , [ delta - omega-2]so that the total frequency shift of the qubit ( which can be called the ac stark shift ) is _ q , s = _ q,1 + _ q,2 . [ delta - omega-12 ] thus , the ensemble - averaged evolution of the qubit - resonator state can be described ( neglecting the overall phase ) by the wavefunction |(t)= ( t)+ e^-i _ 0^t _ q , s(t ) dt c_1 |1 |_1 ( t ) , subjected to dephasing @xmath173 between the two components . if we also want to trace the state over the resonator , then we have an additional inner product @xmath174 , which changes the dephasing rate ( [ gamma - d ] ) by ( t ) = [ delta - gamma](this change can be positive or negative ) and introduces additional contribution to the qubit frequency shift , _ q,3(t)= i m [ _ 1^*(t)_0(t ) ] , [ delta - omega-3]as follows from eq . ( [ inner - product ] ) . the _ qubit - only _ density matrix elements then become @xmath175 \ , dt'\bigg ] \nonumber \\ & & \hspace{0.9 cm } \times \ , \exp \bigg [ -i\int_0^t [ \delta\omega_{\rm q , s}(t')+\delta\omega_{\rm q,3}(t ' ) ] \ , dt ' \bigg ] . \qquad \end{aligned}\ ] ] [ note that @xmath176 are numbers , while @xmath177 in eqs . ( [ rho-00-qr])([rho-10-qr ] ) are operators . ] using eqs . ( [ alpha0-dot ] ) and ( [ alpha1-dot ] ) , it is easy to show that @xmath178 , \label{gamma - sum}\\ & & \delta\omega_{\rm q , s}(t)+\delta\omega_{\rm q,3}(t)= 2\chi \ , { \rm re } [ \alpha_1^ * ( t ) \ , \alpha_0(t ) ] , \qquad \label{delta - omega - sum}\end{aligned}\ ] ] which coincide with the results of refs . @xcite for the qubit dephasing and ac stark shift . note that the results ( [ gamma - d ] ) and ( [ gamma - sum ] ) for the dephasing rate coincide in the steady state ( because then @xmath179 ) , but they are different during the transient evolution . the rate ( [ gamma - d ] ) reflects the information loss due to emitted field , while the rate ( [ gamma - sum ] ) also includes the effect from changing entanglement between the qubit and the resonator . similarly , the results ( [ delta - omega-12 ] ) and ( [ delta - omega - sum ] ) for the ac stark shift coincide in the steady state ( then @xmath180 ) , but differ during transients . equation ( [ delta - omega-12 ] ) is applicable to the entangled qubit - resonator state , while eq . ( [ delta - omega - sum ] ) assumes tracing over the resonator state . also note that all these results for the dephasing rate and ac stark shift are applicable only in the case of a non - evolving qubit ( i.e. , when the evolution is only due to measurement ) . therefore , they are applicable to the ramsey sequence ( with short pulses ) , but , strictly speaking , not applicable to rabi oscillations , spectroscopic measurement of the ac stark shift , etc . for the echo sequence our results are not applicable directly , but the exact results can still be easily obtained using the same derivation ( assuming sufficiently short pulses applied to the qubit ) . in the `` bad cavity '' limit we can neglect the transients and use the steady - state values @xmath181 and @xmath182 . if additionally @xmath116 , then eqs . ( [ gamma - sum ] ) and ( [ delta - omega - sum ] ) reduce to eqs . ( [ gamma - bc2 ] ) and ( [ stark - bc ] ) . if @xmath183 , then for the qubit dephasing and ac stark shift in sec . iii we need to use steady - state versions of eqs . ( [ gamma - sum ] ) and ( [ delta - omega - sum ] ) or , equivalently , eqs . ( [ gamma - d ] ) and ( [ delta - omega-12 ] ) . to describe an individual measurement realization with a random result and evolution depending on this result , we will measure the pieces of the `` history tail '' in eq . ( [ evol-01 ] ) see fig . [ fig : tail ] . note that each piece can be measured in a different way , so the measurement properties can be changing in time . moreover , in general the sequence of measurement of the pieces can also be arbitrary ( e.g. , in a `` delayed choice '' experiment ) . we are interested in describing homodyne measurement with a phase - sensitive or phase - preserving amplifier . let us start with describing an _ ideal _ ( with perfect quantum efficiency ) _ phase - sensitive _ homodyne measurement . we will use the following physical model to describe such a measurement : 1 . a large coherent - state field @xmath184 ( @xmath185 ) from a pump is added to the piece of the tail . the number of photons @xmath26 is measured in the resulting state . 3 . for a particular random @xmath26 obtained in this measurement , the wavefunction is collapsed in the standard textbook way . this procedure describes well the optical homodyne measurement ( note that the photon number does not actually need to be resolved with single - photon precision since the fluctuations are significant ) . it is also similar to what is done experimentally in a phase - sensitive superconducting parametric amplifier . for example , in refs . @xcite the phase - sensitive parametric amplifier works by adding a pump microwave to the microwave leaked from the resonator using a directional coupler ( a microwave analog of a beam splitter ) . then the resulting microwave is sent to a nonlinear oscillator , whose frequency depends on the oscillation amplitude . this frequency change is then sensed via the corresponding phase change at the mixer . thus we measure the power of the pump with added signal , i.e. , within the time interval @xmath8 we essentially measure the corresponding number of photons @xmath26 ( again , single - photon precision in measuring @xmath26 is not needed ) . in a more complicated case of sideband pumping ( double - pumping ) of the parametric amplifier @xcite , the added resonant pump wave is modulated in amplitude ; however , the general principle remains practically the same . the case of a parametric pumping at the doubled frequency is different , but it is still practically equivalent to the measurement described by our model . note that in order to add the field @xmath184 , we need a beam splitter ( directional coupler ) which almost fully passes the signal , so the applied pump field should be much larger than the already large field @xmath184 . also note that a small part of the signal in this case will be lost , so that perfect quantum efficiency is impossible . however , we will not consider these details , and will also not consider ways to go around these problems ( e.g. , by using a balanced homodyne detection ) . it is rather simple to analyze the measurement using our model . for describing measurement of @xmath148th piece of the tail , let us rewrite eq . ( [ evol-01 ] ) as @xmath186 where @xmath187 is the measured @xmath148th piece of the tail , while remaining terms in eq . ( [ evol-01 ] ) are denoted as the normalized wavefunctions @xmath188 . [ actually , if we measure each piece of the tail immediately as it emerges , then it is sufficient to consider @xmath189 , which contain only the qubit and resonator states and do not contain unmeasured pieces of the tail . however , with eqs . ( [ psi-1])([psi - j ] ) we can in general consider a `` delayed choice '' version of the measurement . ] after the first step in the procedure ( addition of the pump field @xmath184 ) , the tail pieces @xmath190 become @xmath191 $ ] [ see eq.([displacement - comp ] ) for displacement by operator @xmath192 , therefore , the state ( [ psi-1 ] ) becomes @xmath193 at the second step of our procedure we need to measure the number of photons @xmath26 in the pump - plus - piece - of - tail part of the state ( [ psi-2 ] ) . the probability distribution for obtaining a particular @xmath26 is @xmath194 as follows from eqs . ( [ psi-2 ] ) and ( [ prop-2 - 1 ] ) . this distribution is normalized , @xmath195 , because @xmath196 . the third step of the procedure is the orthodox collapse of the state ( [ psi-2 ] ) onto the particular ( random ) measurement result @xmath26 . this means that instead of the states @xmath197 in eq . ( [ psi-2 ] ) , we pick only the amplitude corresponding to @xmath198 , and then renormalize the wavefunction ( [ psi-2 ] ) , so that it becomes [ see eq . ( [ alpha - def-2 ] ) ] @xmath199 where the normalization @xmath200 ensures that @xmath201 . note that the the overall phase of @xmath202 is not important . as we see , the `` quantum back - action '' due to the collapse changes the amplitudes of the pre - measured state ( [ psi-1 ] ) : @xmath203 and @xmath204 . this is the main idea for the description of the evolution due to measurement in the quantum bayesian formalism . the procedure can be applied to measurement of other pieces of the `` history tail '' in the same way . let us transform eqs . ( [ p(n)-1 ] ) and ( [ tilde - c-1 ] ) into a more useful form , using the assumption of a large pump amplitude , @xmath205 and @xmath206 [ see eq . ( [ alpha - def-2 ] ) ] , @xmath207}{\sqrt{2\pi\sigma^2 } } } \nonumber \\ & & \hspace{2.6 cm } \times \exp [ i n\ , { \rm arg } ( \alpha_{\rm p}+\alpha_{j,\rm t } ) ] \ , |n\rangle , \qquad \label{alpha - gauss}\\ & & \hspace{-0.0 cm } \bar{n}_j = |\alpha_{\rm p}|^2 + 2\ , { \rm re } ( \alpha_{\rm p}^*\alpha_{j,\rm t})+ |\alpha_{j,\rm t}|^2 \nonumber \\ & & \hspace{0.5 cm } \approx |\alpha_{\rm p}|^2 + 2\ , { \rm re } ( \alpha_{\rm p}^*\alpha_{j,\rm t } ) , \label{n - bar-01 } \\ & & \sigma=|\alpha_{\rm p}|\gg 1 , \end{aligned}\ ] ] where @xmath208 is the average number of photons for the state @xmath209 ( we can neglect the last term @xmath210 for @xmath208 since @xmath211 ) and @xmath212 is the standard deviation . note that we use the same @xmath213 for both states because @xmath214 . also note that for exact normalization of the gaussian state ( [ alpha - gauss ] ) at finite @xmath215 the denominator @xmath216 should be slightly changed ; however , this is not important for the derivation . the probability distribution ( [ p(n)-1 ] ) for measuring @xmath26 photons in this case becomes p(n)= + , [ p(n)-gauss]and the updated amplitudes @xmath217 and @xmath218 given by eq.([tilde - c-1 ] ) become _ j= . [ tilde - c-2]we see that if the measurement result @xmath26 is closer to @xmath121 than to @xmath122 , then the amplitude @xmath219 increases ( by absolute value ) in this update . this is the expected feature of the quantum bayesian formalism : if the measurement result is more consistent with the qubit state @xmath6 , then the amplitude of this state increases . to simplify the phase factor in eq . ( [ tilde - c-2 ] ) , let us write @xmath220 as @xmath221 , and then expand @xmath222 to the second order , so that e^i n arg ( _ p+_j , t ) = e^i n arg ( _ p ) e^in im(_j , t/_p ) [ 1-re(_j , t/_p ) ] [ phase-1](we need the second order because @xmath223 and we wish to keep the terms linear in @xmath224 ) . the @xmath225-independent phase factor @xmath226 can be ignored as an overall phase . in the remaining phase in eq.([phase-1 ] ) let us represent @xmath26 as @xmath227 with the center point |n_c(|n_0+|n_1)/2 . from eq . ( [ n - bar-01 ] ) , neglecting the term @xmath228 , we find @xmath229 $ ] . if @xmath230 , then the phase in eq . ( [ tilde - c-2 ] ) ( neglecting the overall phase ) is @xmath231 $ ] , which can be written ( neglecting the terms of order @xmath232 ) as @xmath233 $ ] . moving the phase difference to @xmath234 ( i.e. , considering the phase for @xmath235 as an unimportant overall phase ) , we find that the phase evolution in eq . ( [ tilde - c-2 ] ) can be described by multiplying @xmath163 by @xmath236 . this is exactly what we would expect from the phase of the inner product @xmath237 , and it is fully consistent with the result ( [ delta - omega-1 ] ) for the ac stark shift contribution . thus , for @xmath230 the phase shift produced by the collapse is the same as the ensemble - averaged phase shift . when @xmath238 , there is an additional phase factor @xmath239 in eqs . ( [ tilde - c-2 ] ) and ( [ phase-1 ] ) [ we now use @xmath240 , neglecting a phase correction of order @xmath232 ] . moving the phase difference to @xmath234 , we find that @xmath163 should be additionally multiplied by @xmath242}$ ] . thus , the evolution due to measurement [ see eqs . ( [ psi-1 ] ) , ( [ tilde - psi ] ) , and ( [ tilde - c-2 ] ) ] can be described as @xmath243}{\rm norm } , \qquad \label{tilde - c0}\\ & & \tilde{c}_1= \frac{c_1\,\exp [ -(n-\bar{n}_1)^2/4\sigma^2]}{\rm norm } \ , e^{-i\,\delta \varphi } , \label{tilde - c1}\\ & & { \rm norm } = \sqrt{\sum\nolimits_{j=0,1 } |c_j|^2 \exp [ -(n-\bar{n}_j)^2/2\sigma^2 ] } , \\ & & \delta\varphi = -\frac{n-\bar{n}_{\rm c } } { \sigma}\,\sqrt{\kappa \delta t } \,\ , { \rm im}\ { [ \alpha_1(t_m)-\alpha_0(t_m)]\,e^{-i\phi_{\rm a}}\ } \nonumber \\ & & \hspace{0.8 cm } + \kappa \delta t\ , { \rm im}[\alpha_{1 } ^*(t_m)\,\alpha_{0}(t_m ) ] , \,\,\,\ \label{delta - phi } \\ & & \phi_{\rm a } = \phi_{\rm p}={\rm arg } ( \alpha_{\rm p } ) , \end{aligned}\ ] ] where @xmath244 is the time moment when the measured piece of the `` history tail '' leaked from the resonator , and @xmath47 is the phase of the pump , which determines the amplified quadrature . we emphasize that the measured piece becomes unentangled with the rest of the wavefunction [ see eq . ( [ tilde - psi ] ) ] and therefore can be disregarded when the measurement of the next piece is analyzed . note that the pump phase @xmath47 affects the response [ see eq . ( [ n - bar-01 ] ) ] |n_1-|n_0 = 2 re \{[_1(t_m)-_0(t_m ) ] e^-i_a } and also affects the phase shift @xmath245 in eq.([delta - phi ] ) . thus , the choice of the measured quadrature affects evolution of the system ( as in the `` bad cavity '' case @xcite ) . however , it is simple to show that the state update ( [ tilde - c0])([delta - phi ] ) averaged over the measurement result ( [ p(n)-gauss ] ) does not depend on the choice of @xmath47 . the formalism developed in secs . [ sec : main - idea ] and [ sec : gaussian ] allows us to consider measurement of the `` history tail '' pieces in an arbitrary sequence and thus to describe various `` delayed choice '' experiments . however , usually this is not needed , and we can assume measurement of the pieces as soon as they leak from the resonator . in this case it is sufficient to describe the system by an entangled qubit - resonator wavefunction ( we still consider an ideal case ) |(t)= c_0(t ) |0 |_0(t)+ c_1(t ) |1 |_1(t ) , [ psi - cont - meas]with the coefficients @xmath246 and @xmath247 evolving in time due to measurement . then the evolution equations are essentially the same as eqs . ( [ tilde - c0])([delta - phi ] ) , but now there is no delay in measurement , and we have included the phase factors @xmath248 and @xmath249 in eq . ( [ evol-01 ] ) into the coefficients @xmath162 and @xmath163 in eq . ( [ psi - cont - meas ] ) [ actually , we include the phase difference @xmath250 into @xmath163 , neglecting the overall phase @xmath248 ] . also , instead of the measured number of photons @xmath26 , let us introduce the output signal @xmath252 . similarly , @xmath253 and @xmath254 are the corresponding average values , and @xmath255 is the variance of @xmath70 . then eqs . ( [ tilde - c0])([delta - phi ] ) can be rewritten as @xmath256}{\rm norm } , \qquad \label{c0 - 1}\\ & & c_1(t+\delta t)= \frac{c_1 ( t)\,\exp [ -(i_{\rm m}-i_1)^2/4d]}{\rm norm } \ , e^{-i\,\delta \varphi } , \quad \,\,\ , \label{c1 - 1}\\ & & { \rm norm } = \sqrt{\sum\nolimits_{j=0,1 } |c_j(t)|^2 \exp [ -(i_{\rm m}-i_j)^2/2d ] } , \qquad \\ & & \delta\varphi = - \frac{i_{\rm m}-(i_0+i_1)/2}{\sqrt{d}}\,\sqrt{\kappa \delta t } \,\ , { \rm i m } [ ( \alpha_1-\alpha_0)\,e^{-i\phi_{\rm a } } ] \nonumber \\ & & \hspace{1.0 cm } + \ , \delta \omega_{\rm q , s } \ , \delta t , \label{delta - phi-1}\end{aligned}\ ] ] where the qubit ac stark shift @xmath257 due to leaking field is given by eqs . ( [ delta - omega-1])([delta - omega-12 ] ) . note that eqs . ( [ c0 - 1])([delta - phi-1 ] ) do not change if we multiply @xmath70 , @xmath74 , @xmath75 , and @xmath258 by an arbitrary factor . therefore , we can consider them just as experimental output signals ( in arbitrary units ) , so that i_m=_t^t+t i(t ) dt for a continuous measurement output @xmath0 , and the variance @xmath259 is related to the single - sided spectral density @xmath81 of the output signal as d= . [ d - def ] now let us introduce the angle difference @xmath77 between the amplified quadrature along @xmath184 and the `` information - carrying '' quadrature along @xmath260 , _ d = _ a - arg[_1 ( t ) -_0(t ) ] , and also introduce the maximum response @xmath261 , which _ would _ correspond to @xmath108 , so that i = i_1 -i_0 = i_max ( _ d ) . then we can write the factor @xmath262 $ ] in eq . ( [ delta - phi-1 ] ) as @xmath263 . also counting the measurement signal from the central point @xmath73 , i_m = i_m - , we can rewrite evolution equations ( [ c0 - 1])([delta - phi-1 ] ) as @xmath264 where @xmath265 ensures @xmath266 . note that for short @xmath8 the variance @xmath259 of the noisy signal @xmath110 is much larger than @xmath267 , and then @xmath268 , so that the change of @xmath162 and @xmath163 is small . however , from the structure of eqs . ( [ c0 - 2])([delta - phi-2 ] ) it is easy to see that they _ remain valid for an arbitrary long @xmath8 _ if @xmath79 , @xmath77 , @xmath269 and noise @xmath81 do not change with time . therefore , the practical upper limit for the time step @xmath8 ( e.g. , in numerical simulations ) is determined by transients , which change the resonator states @xmath270 and @xmath157 , and by possible changes of the amplified quadrature phase @xmath47 . so far we considered an ideal phase - sensitive measurement , so that the evolution description using a wavefunction was sufficient . to describe a measurement with imperfect efficiency , we need to use the language of density matrices ( we still assume that the qubit evolves only due to measurement ) . then instead of eq . ( [ psi - cont - meas ] ) , the evolution of the entangled qubit - resonator system is described by the density operator ^q & r(t)= _ j , j=0,1 _ jj(t ) |jj| |_j ( t)_j(t)| , [ rho - qr - def]where @xmath270 and @xmath157 are given by eqs . ( [ alpha0-dot ] ) and ( [ alpha1-dot ] ) . we emphasize that the matrix elements @xmath271 describe the entangled qubit - resonator state , not only the qubit state . ( note that using the form ( [ rho - qr - def ] ) for the qubit - resonator state is equivalent to the polaron - frame approximation used in the theory of quantum trajectories @xcite . ) in the ideal case the evolution of the matrix elements @xmath272 can be obtained by converting eqs . ( [ c0 - 2])([delta - phi-2 ] ) into the language of density matrices , @xmath273 , \qquad \label{rho - diag}\\ & & \frac{\rho_{10}(t+\delta t)}{\rho_{10}(t ) } = \frac{\sqrt{\rho_{11}(t+\delta t)\,\rho_{00}(t+\delta t)}}{\sqrt{\rho_{11}(t)\,\rho_{00}(t)}}\ , e^{-i\delta\varphi } , \label{rho - offdiag}\end{aligned}\ ] ] where the phase shift @xmath245 is still given by eq . ( [ delta - phi-2 ] ) . another , more intuitive way to describe the evolution of the diagonal elements is by using the uncentered signal @xmath70 as in eq . ( [ c0 - 1 ] ) : _ jj(t+t)= . [ rho - jj ] note that these evolution equations for @xmath272 are exactly the same as eqs . ( [ ph - sens - diag - bc ] ) and ( [ ph - sens - off - bc ] ) of the quantum bayesian formalism in the `` bad cavity '' limit , except the ac stark shift @xmath120 is now @xmath257 , dephasing is so far absent ( @xmath274 ) , and , most importantly , eqs . ( [ rho - diag ] ) and ( [ rho - offdiag ] ) describe the entangled qubit - resonator state ( [ rho - qr - def ] ) and are capable of describing transient evolution . during transients there is significant time - dependence in @xmath275 and @xmath276 , which also leads to time - dependence in @xmath277 , the quadrature phase difference @xmath278 , the response @xmath279 , and the middle point @xmath280 . therefore , during transients the time step @xmath8 in eqs . ( [ rho - diag ] ) and ( [ rho - offdiag ] ) should be much smaller than @xmath139 , in contrast to arbitrary @xmath68 in eqs . ( [ ph - sens - diag - bc ] ) and ( [ ph - sens - off - bc ] ) . imperfect quantum efficiency @xmath84 of the measurement ( @xmath281 ) , similar to the case discussed in sec . [ sec : ph - sens - bc ] , mainly originates from two mechanisms : imperfect collection efficiency and imperfect amplifier efficiency . first , a fraction of the field leaked from the resonator is lost before reaching the amplifier . second , the amplifier produces more output noise than the quantum limitation . therefore , we can define the total quantum efficiency @xmath84 as = _ col_amp , _ col = , _ amp = , where @xmath282 is the ratio of the `` collected '' microwave energy , which reaches amplifier , to the total energy lost by the resonator ( @xmath283 ) , and @xmath92 is the spectral density of the output noise if a quantum - limited phase - preserving amplifier were used instead of the actual amplifier , which produces a larger noise @xmath81 . let us now discuss the effects produced by imperfect @xmath87 and @xmath86 . an imperfect collection efficiency @xmath87 can be modelled by adding an asymmetric beam splitter on the path of the leaked field , which splits each piece of the `` history tail '' into two pieces , @xmath284 , so that the first piece is measured , while the second one remains unmeasured . since no information can be obtained from the unmeasured piece , we need to trace over it , as in the calculation of the ensemble - averaged evolution , while for the measured piece we use the same procedure as above . the tracing over the unmeasured piece does not change the diagonal matrix elements in eq . ( [ rho - qr - def ] ) ; therefore , the total change of @xmath285 and @xmath286 is still given by eq . ( [ rho - diag ] ) . note , however , that imperfect collection efficiency reduces the response , @xmath287 and changes the central point @xmath73 , while the variance @xmath259 ( determined by the amplifier noise ) remains unchanged . for the off - diagonal matrix element @xmath109 , the tracing over the unmeasured piece @xmath288 produces the factor @xmath289 [ see eqs . ( [ rho-10-qr-1])([delta - omega-12 ] ) in sec . [ sec : ensemble - aver ] ] , while the measured piece gives the evolution described by eq . ( [ rho - offdiag ] ) , with @xmath290 in eq . ( [ delta - phi-2 ] ) multiplied by @xmath87 . therefore , the total evolution of @xmath109 is described by eq . ( [ rho - offdiag ] ) with the extra factor @xmath291 , while the phase @xmath245 is still given by eq . ( [ delta - phi-2 ] ) [ note again that @xmath79 and @xmath73 are affected by @xmath87 , but still correspond to the experimentally measured values ] . thus , the only effect of an imperfect collection efficiency @xmath87 on the system evolution ( [ rho - diag])([rho - offdiag ] ) is the extra factor @xmath291 in eq . ( [ rho - offdiag ] ) , where @xmath292 is given by eq . ( [ gamma - d ] ) . imperfect quantum efficiency @xmath86 of the amplifier produces additional noise at the output , so that the response @xmath79 and the middle point @xmath73 do not change , while the variance @xmath259 given by eq . ( [ d - def ] ) increases because of the increased noise spectral density @xmath81 . to take into account the extra noise , for a given measured output value @xmath72 , we need to guess what was the `` actual '' value @xmath293 ( the probability distribution is given by the classical bayesian analysis ) , then apply the evolution ( [ rho - diag])([rho - offdiag ] ) using the value @xmath293 , and then average over all possible values of @xmath293 . this is exactly what was done in ref . @xcite for a qubit measurement by qpc or set . since the evolution equations ( [ rho - diag ] ) , ( [ rho - offdiag ] ) , and ( [ delta - phi-2 ] ) have exactly the same form as what was considered in ref . @xcite , we can simply use the obtained result : the evolution is still given by eqs . ( [ rho - diag ] ) , ( [ rho - offdiag ] ) , and ( [ delta - phi-2 ] ) with two changes . first , the variance @xmath259 is the actual ( increased ) variance ; second , there is an extra dephasing factor in eq . ( [ rho - offdiag ] ) , which can be found from comparison with ensemble - averaged evolution . even though the formal derivation of this result is rather lengthy @xcite , it is easy to understand it . the evolution of the diagonal elements of the density matrix is the evolution of probabilities , and therefore must obey the classical bayes formula , which directly gives eq . ( [ rho - diag ] ) . the extra dephasing in eq . ( [ rho - offdiag ] ) comes from uncertainty of the phase @xmath294 due to uncertainty of the unknown `` actual '' value @xmath293 . the reduced proportionality factor between @xmath294 and the ( centered ) measurement result @xmath72 in eq . ( [ delta - phi-2 ] ) due to increased value of @xmath259 can be understood from the fact that for uncorrelated gaussian - distributed zero - mean random numbers @xmath295 and @xmath296 , the averaging of @xmath295 for a fixed sum @xmath297 gives the smaller value , @xmath298 $ ] . thus , combining both imperfection mechanisms of the quantum efficiency @xmath84 , we can describe the evolution of the qubit - resonator system ( [ rho - qr - def ] ) , measured using a phase - sensitive amplifier . the resulting equations are very similar to eqs . ( [ ph - sens - diag - bc ] ) and ( [ ph - sens - off - bc ] ) for the `` bad cavity '' case , @xmath299 , \qquad \label{rho - diag-2}\\ & & \hspace{-0.2 cm } \frac{\rho_{10}(t+\delta t)}{\rho_{10}(t ) } = \frac{\sqrt{\rho_{11}(t+\delta t)\,\rho_{00}(t+\delta t)}}{\sqrt{\rho_{11}(t)\,\rho_{00}(t)}}\ , \nonumber \\ & & \hspace{1.5 cm } \times \exp(-ik \tilde{i}_{\rm m } \ , \delta t ) \,\ , e^{-\gamma \delta t } \ , e^{-i\delta\omega_{\rm q , s}\delta t } , \qquad \label{rho - offdiag-2}\end{aligned}\ ] ] where @xmath300 and we repeated several previous formulas here for convenience . note that most parameters in these equations depend on time during transients , and therefore the time step @xmath8 should be sufficiently small . recall that @xmath81 is the single - sided spectral density of the output noise and @xmath301 is the corresponding noise variance of @xmath72 . equations ( [ rho - diag-2])([stark-2 ] ) are the _ main result _ of this paper . if instead of using experimental output signal @xmath0 we want to simulate the process , we can pick @xmath302 from the probability distribution @xmath303 } { \sqrt{2\pi d } } \nonumber \\ & & \hspace{1.0 cm } + \rho_{11}(t)\ , \frac{\exp [ - ( \tilde i_{\rm m}- \delta i/2)^2/2d ] } { \sqrt{2\pi d } } . \qquad \label{tilde - i - distr}\end{aligned}\ ] ] for an infinitesimally small @xmath8 this is equivalent to using i(t)= + [ _ 11 ( t)-_00(t ) ] + _ i(t ) , [ ph - sens - i(t)-2]where @xmath304 is a white noise with spectral density @xmath81 . it is easy to check that averaging of @xmath305 given by eqs . ( [ rho - diag-2 ] ) and ( [ rho - offdiag-2 ] ) over @xmath72 with probability distribution ( [ tilde - i - distr ] ) produces the expected ensemble - averaged equations @xmath306 similar to what was mentioned in sec . [ sec : ph - pres - bc ] , the averaging over the fluctuating phase @xmath307 in eq . ( [ rho - offdiag-2 ] ) produces an ensemble dephasing rate @xmath308 , while the averaging of the first term in eq . ( [ rho - offdiag-2 ] ) produces an ensemble dephasing rate @xmath309 . both dephasing rates depend on the amplified quadrature via the angle @xmath77 , but their sum , @xmath310 , does not depend on @xmath77 . if there is extra ( not measurement - related ) dephasing rate @xmath311 of the qubit - resonator system , e.g. , due to intrinsic pure dephasing of the qubit , then it can be easily included into eq . ( [ rho - offdiag-2 ] ) by adding the factor @xmath312 . alternatively , we can include @xmath311 into the ensemble - averaged dephasing , @xmath313 , so that evolution equations ( [ rho - diag-2 ] ) and ( [ rho - offdiag-2 ] ) remain unchanged , but now @xmath314 . in this case the overall quantum efficiency includes the extra dephasing , @xmath315 , with @xmath316 @xcite . note that for the evolution equations discussed in this section the initial state should not necessarily be pure , so eq . ( [ psi - in ] ) for the initial state can be replaced with eq . ( [ rho - in ] ) . moreover , it is sufficient to have an initial state of the form ( [ rho - qr - def ] ) ; the only necessary condition is that each qubit state @xmath99 corresponds to a certain coherent state @xmath317 . so far we considered the measurement using a phase - sensitive amplifier . in this section we use the results of the previous section to describe the case when a phase - preserving amplifier is used . we will do it in two ways , which give the same result . first , let us model the measurement using a phase - preserving amplifier in the following way . let us pass each piece of the `` history tail '' through a symmetric beam splitter @xmath318 , amplify orthogonal quadratures in these two parts , measure as discussed in sec . [ sec : main - idea ] , and output the results for both quadratures . from the structure of eqs . ( [ rho - diag-2])([tilde - i - distr ] ) it is easy to see that it does not matter how these two orthogonal quadratures are chosen if the amplification conditions in both channels are the same ( the same @xmath259 and @xmath79 , which also means the same quantum efficiency ) . note that for both channels we should simultaneously use either the first or the second gaussian in eq . ( [ tilde - i - distr ] ) , though no correlation is needed in the infinitesimal limit ( [ ph - sens - i(t)-2 ] ) . it is natural to choose one quadrature ( we call it @xmath319 ) along the informational direction @xmath320 , while the other quadrature ( we call it @xmath321 ) is shifted by @xmath48 , so that @xmath108 for the @xmath319-quadrature and @xmath126 for the @xmath321-quadrature . thus , the directions of the @xmath319 and @xmath321 quadratures are changing in time , but they are practically constant during the time step @xmath8 . note that if @xmath259 for both quadratures is kept the same as for a phase - sensitive amplifier , then the response @xmath124 is a factor @xmath322 smaller than @xmath79 for the phase - sensitive case ( because of the beam splitter ) . equivalently , if @xmath124 is kept the same as in the phase - sensitive case ( e.g. , by an additional classical amplification by the factor @xmath322 ) , then @xmath259 for both quadratures is twice larger than for the phase - sensitive case ( in the ideal case this corresponds to the fact that the noise of a phase - preserving amplifier is twice as large as for a phase - sensitive amplifier ) . therefore , for the phase - preserving case we can simply use eqs . ( [ rho - diag-2])([tilde - i - distr ] ) twice , for the optimal quadrature ( @xmath108 ) and for the orthogonal quadrature ( @xmath126 ) , assuming the same output noise , @xmath323 , for both output quadratures @xmath0 and @xmath4 . the @xmath319-quadrature has the maximum response , @xmath324 , while the @xmath321-quadrature has no response , @xmath129 . thus , after the time step @xmath8 the density matrix ( [ rho - qr - def ] ) of the qubit - resonator system changes as @xmath299 , \qquad \label{rho - diag-3}\\ & & \hspace{-0.2 cm } \frac{\rho_{10}(t+\delta t)}{\rho_{10}(t ) } = \frac{\sqrt{\rho_{11}(t+\delta t)\,\rho_{00}(t+\delta t)}}{\sqrt{\rho_{11}(t)\,\rho_{00}(t)}}\ , \nonumber \\ & & \hspace{1.0 cm } \times \exp ( -i\tilde{q}_{\rm m } \delta i/2d ) \,\ , e^{-\gamma \delta t } \ , e^{-i\delta\omega_{\rm q , s}\delta t } , \qquad \label{rho - offdiag-3 } \end{aligned}\ ] ] where @xmath110 is given by eq . ( [ tilde - i - m ] ) , while @xmath325 the ensemble - averaged dephasing @xmath292 is given by eq . ( [ gamma - d ] ) , and the ac stark shift @xmath269 is given by eq . ( [ stark-2 ] ) or eqs . ( [ delta - omega-1])([delta - omega-12 ] ) . an equivalent form for eq . ( [ rho - diag-3 ] ) in terms of the non - centered signal @xmath70 is given by eq . ( [ rho - jj ] ) . the factor of 2 in eq . ( [ gamma-3 ] ) appears because averaging over the result in each channel produces the contribution @xmath309 into the total ensemble dephasing @xmath292 . another way to derived eqs . ( [ rho - diag-3 ] ) and ( [ rho - offdiag-3 ] ) from eqs . ( [ rho - diag-2 ] ) and ( [ rho - offdiag-2 ] ) is to assume a slightly shifted pump frequency for a phase - sensitive amplifier , so that the angle @xmath77 rotates sufficiently fast , and for both quadratures @xmath0 and @xmath4 we collect only the values averaged over @xmath77 . then we have a natural formation of two quadratures in eqs . ( [ rho - diag-2 ] ) and ( [ rho - offdiag-2 ] ) : @xmath326 and @xmath327 , where the superscripts indicate the phase - sensitive ( ps ) or phase - preserving ( pp ) case . the variance of the noise in each quadrature is @xmath328 ( because @xmath329 ) and the response in the information - carrying quadrature is @xmath330 ( because the phase - sensitive response @xmath331 should be multiplied by @xmath332 to project onto the proper quadrature ) . therefore @xmath333 , and eqs . ( [ rho - diag-2 ] ) and ( [ rho - offdiag-2 ] ) directly transform into eqs . ( [ rho - diag-3 ] ) and ( [ rho - offdiag-3 ] ) . for the dephasing we get @xmath334 since @xmath335 and @xmath330 , thus reproducing eq . ( [ gamma-3 ] ) . in numerical simulations the probability distribution for @xmath72 is still given by eq . ( [ tilde - i - distr ] ) , while for @xmath336 it is p(q_m ) = . [ tilde - q - distr-2 ] for infinitesimal @xmath8 these distributions are equivalent to using eq . ( [ ph - sens - i(t)-2 ] ) for @xmath0 and q(t)=q_0 + _ q ( t ) , s__q= s_q= s_i , for @xmath4 , with uncorrelated white noises in the two channels . averaging of @xmath305 in eqs . ( [ rho - diag-3 ] ) and ( [ rho - offdiag-3 ] ) over random @xmath72 and @xmath336 using the probability distributions ( [ tilde - i - distr ] ) and ( [ tilde - q - distr-2 ] ) produces the ensemble - averaged evolution equations ( [ rho - ea - diag ] ) and ( [ rho - ea - offdiag ] ) . the ensemble - averaged evolution should remain the same as in the phase - sensitive case because of causality . similar to eq . ( [ eta - phase - pres - def ] ) , the quantum efficiency for a phase - preserving measurement can be defined in two ways , = 1- /_d , = /2 , where the first definition is based on the comparison with ideal phase - preserving measurement , while in the second definition we compare the information in @xmath319-channel only with the ideal phase - sensitive case . we emphasize that monitoring of a pure quantum state is still possible with a phase - preserving amplifier if @xmath337 , in spite of the fundamental limitation @xmath338 . we intentionally wrote the evolution equations ( [ rho - diag-2 ] ) , ( [ rho - offdiag-2 ] ) , ( [ rho - diag-3 ] ) and ( [ rho - offdiag-3 ] ) for a finite @xmath8 because this form is more transparent physically , suitable for numerical simulations , and also unambiguous . the differential form for an infinitesimal @xmath8 is significantly more ambiguous because it depends on a chosen definition of the derivative ( as should be for nonlinear stochastic differential equations @xcite ) . if we define the derivative in the symmetric way @xmath339/\delta t$ ] ( the so - called stratonovich form ) , then the standard calculus rules apply , and the differential equations for the evolution can be derived from eqs . ( [ rho - diag-2 ] ) , ( [ rho - offdiag-2 ] ) , ( [ rho - diag-3 ] ) and ( [ rho - offdiag-3 ] ) in a straightforward way ( keeping linear order in @xmath8 ) . thus , for the phase - sensitive measurement we obtain the stratonovich - form evolution as ( see @xcite ) @xmath340 , \qquad \label{dot - rho-11 - 1}\\ & & \dot{\rho}_{10}= -(\rho_{11}-\rho_{00})\ , \frac{\cos{\phi_{\rm d}}\delta i_{\rm max}}{s_i } \left [ i(t)-\frac{i_0+i_1}{2 } \right ] \nonumber \\ & & \hspace{0.9 cm } -i \,\frac{\sin{\phi_{\rm d}}\delta i_{\rm max}}{s_i } \left [ i(t)- \frac{i_0+i_1}{2 } \right ] \rho_{10 } \nonumber \\ & & \hspace{0.9 cm } -\gamma \rho_{10 } -i\,\delta\omega_{\rm q , s}\rho_{10 } , \quad \label{dot - rho-10 - 1}\\ & & \gamma = \gamma_{\rm d } -(\delta i_{\rm max})^2/4s_i , \\ & & i(t ) = \rho_{00}(t ) \ , i_0+\rho_{11}(t ) \ , i_1+\xi_i(t ) , \,\,\ , s_{\xi_i}=s_i , \label{i(t)}\end{aligned}\ ] ] where for convenience we repeated equations for @xmath83 and @xmath0 . we emphasize that @xmath74 , @xmath75 , @xmath79 , and @xmath77 may significantly depend on time during transients . for the phase - preserving measurement we similarly obtain the stratonovich - form equations @xmath341 , \qquad \label{dot - rho-11 - 2}\\ & & \dot{\rho}_{10}= -(\rho_{11}-\rho_{00})\ , \frac{\delta i}{s_i } \left [ i(t)-\frac{i_0+i_1}{2 } \right ] \nonumber \\ & & \hspace{0.9 cm } -i \,\frac{\delta i}{s_i } \ , [ q(t)- q_0 ] \,\rho_{10 } \nonumber \\ & & \hspace{0.9 cm } -\gamma \rho_{10 } -i\,\delta\omega_{\rm q , s}\rho_{10 } , \quad \label{dot - rho-10 - 2 } \\ & & \gamma = \gamma_{\rm d } -2\times ( \delta i)^2/4s_i , \\ & & i(t ) = \rho_{00}(t ) \ , i_0+\rho_{11}(t ) \ , i_1+\xi_i(t ) , \label{i(t)-2}\\ & & q(t ) = q_0 + \xi_q(t ) , \,\,\,\,\ , s_{\xi_q}=s_{\xi_i}=s_i . \label{q(t)}\end{aligned}\ ] ] if we define the derivative in the `` forward '' way , @xmath342/\delta t$ ] ( the so - called it form ) , then the usual calculus rules are no longer correct , and the derivation of the differential equations from eqs . ( [ rho - diag-2 ] ) , ( [ rho - offdiag-2 ] ) , ( [ rho - diag-3 ] ) and ( [ rho - offdiag-3 ] ) should retain the second order in @xmath8 . alternatively , we can use the standard rules of the transformation from the stratonovich form into the it form @xcite , applied to eqs . ( [ dot - rho-11 - 1])([q(t ) ] ) . the resulting it - form equations for the phase - sensitive measurement are @xmath343 , \qquad \ , \label{dot - rho-11 - 3}\\ & & \hspace{-0.4 cm } \dot{\rho}_{10}= -(\rho_{11}-\rho_{00})\ , \frac{\cos{\phi_{\rm d}}\delta i_{\rm max}}{s_i } \ , [ i(t)- ( \rho_{00}i_0+\rho_{11}i_1 ) ] \nonumber \\ & & \hspace{-0.4 cm } \hspace{0.9 cm } -i \,\frac{\sin{\phi_{\rm d}}\delta i_{\rm max}}{s_i } \ , [ i(t)- ( \rho_{00}i_0+\rho_{11}i_1)]\ , \rho_{10 } \nonumber \\ & & \hspace{-0.4 cm } \hspace{0.9 cm } -\gamma_{\rm d } \rho_{10 } -i\,\delta\omega_{\rm q , s}\rho_{10 } , \end{aligned}\ ] ] and the it - form equations for the phase - preserving measurement are @xmath344 , \qquad \\ & & \dot{\rho}_{10}= -(\rho_{11}-\rho_{00})\ , \frac{\delta i}{s_i } [ i(t)- ( \rho_{00}i_0+\rho_{11}i_1 ) ] \nonumber \\ & & \hspace{0.9 cm } -i \,\frac{\delta i}{s_i } \ , [ q(t)- q_0 ] \,\rho_{10 } \nonumber \\ & & \hspace{0.9 cm } -\gamma_{\rm d } \rho_{10 } -i\,\delta\omega_{\rm q , s}\rho_{10 } , \qquad \label{dot - rho-10 - 4 } \end{aligned}\ ] ] while @xmath0 and @xmath4 for numerical simulations are still given by eqs . ( [ i(t ) ] ) , ( [ i(t)-2 ] ) , and ( [ q(t ) ] ) . the it - form equations ( [ dot - rho-11 - 3])([dot - rho-10 - 4 ] ) have two differences compared with the stratonovich equations : ( i ) the combination @xmath345 is replaced with the `` pure noise '' combination @xmath346 and ( ii ) the dephasing rate @xmath83 is replaced with ensemble dephasing @xmath292 . note that it and stratonovich equations have identical solutions when the corresponding definitions of the derivative are used . the drawback of the it form is the loss of intuition based on the standard calculus , because the standard calculus rules are not valid in the it form . however , the advantage is that the ensemble - averaged equations can be obtained by simply replacing the noises @xmath100 and @xmath347 with zero . the quantum trajectory formalism @xcite is based on the it form , while the quantum bayesian formalism @xcite usually uses the stratonovich form ( some formalisms use both forms @xcite ) . we emphasize that while the evolution equations in the differential form are useful in analytical analysis , for numerical calculations a relatively large time step @xmath8 is often preferable . for finite time steps , the formalism discussed in secs . [ sec : imperfect ] and [ sec : ph - pres ] is more useful than the differential equations . the use of non - infinitesimal @xmath8 also avoids possible confusion between stratonovich and it forms . now let us discuss evolution of the qubit - resonator system for an arbitrarily long duration @xmath68 . as in the previous sections , we assume that the qubit does not evolve due to rabi oscillations , energy relaxation , etc . it is not obvious what the solution of the differential equations discussed in sec . [ sec : differential ] is . however , the structure of equations for a small time step @xmath8 derived in secs . [ sec : imperfect ] and [ sec : ph - pres ] permits very simple integration for an arbitrary @xmath68 . this simple solution is also expected from the picture of the `` history tail '' in fig . [ fig : tail ] . the evolution equations ( [ rho - diag-2 ] ) and ( [ rho - offdiag-2 ] ) for the qubit - resonator system ( [ rho - qr - def ] ) can be easily integrated within the time interval @xmath348 $ ] , @xmath349 , \qquad \,\,\ , \label{rho - offdiag - tau}\end{aligned}\ ] ] where @xmath350}{s_i } \,\ , dt ' \\ & & \tilde{i}(t ' ) = i(t ' ) -\frac{i_0(t')+i_1(t')}{2 } , \label{tilde - i(t')}\end{aligned}\ ] ] and the time - dependent dephasing @xmath351 and ac stark shift @xmath352 are given by eqs . ( [ gamma-2])([stark-2 ] ) . note that because parameters are time - dependent , there is no simple relation between the effective measurement results @xmath353 and @xmath354 , which produce `` spooky '' and phase back - actions . the choice of notations @xmath355 and @xmath356 relate to quadratures that are parallel or perpendicular to the informational quadrature . if we need to generate measurement results numerically , then @xmath353 can be picked from the probability distribution @xmath357 , which consists of two gaussians , as usual in the bayesian formalism , @xmath358 , \qquad \label{p(r)-2}\\ & & \bar{r}_{\rm 1}^\parallel = - \bar{r}_{\rm 0}^\parallel = \int_{t}^{t+\tau } \frac{[\delta i(t')]^2}{s_i } \ , dt ' , \,\,\ , d_r^\parallel = 2 \bar{r}_{\rm 1}^\parallel . \label{p(r)-3}\end{aligned}\ ] ] the validity of this formula can be checked by analyzing a composition of two evolutions for @xmath359 and @xmath360 , and by checking consistency with formulas in sec . [ sec : imperfect ] for small @xmath68 . from eqs . ( [ p(r)-1])([p(r)-3 ] ) we see that the qubit will eventually be collapsed onto the state @xmath6 or @xmath7 ( unless @xmath361 ) , as expected for a measured qubit with no additional evolution . note that eqs . ( [ p(r)-1])([p(r)-3 ] ) for @xmath357 can be written in this simple way because eq . ( [ rho - diag - tau ] ) is essentially the classical bayes rule . unfortunately , @xmath354 can not be generated in a similar way . therefore , we need to numerically generate the whole record @xmath362 . the output realization @xmath362 within the interval @xmath348 $ ] can be generated by dividing @xmath68 into small pieces @xmath8 and using eq . ( [ tilde - i - distr ] ) . the probability of a realization @xmath362 will then be @xmath363 ^ 2}{s_i } \ , dt ' \bigg ] \nonumber \\ & & \hspace{0.5 cm } + \rho_{11}(t ) \ , \exp \bigg [ - \int_t^{t+\tau } \frac{[i(t')-i_1(t')]^2}{s_i } \ , dt ' \bigg ] , \qquad \label{i(t')-1}\end{aligned}\ ] ] with an appropriate overall normalization . alternatively , the probability distribution can be obtained by applying eq . ( [ ph - sens - i(t)-2 ] ) , i.e. , taking into account the randomness `` locally '' instead of `` globally '' , which produces @xmath364 ^ 2}{s_i } \ , dt ' \bigg ] , \qquad \label{i(t')-2}\\ & & \hspace{0 cm } i_{\rm av}(t')= \rho_{00}(t ' ) \ , i_0(t ' ) + \rho_{11}(t ' ) \ , i_1(t ' ) , \qquad \label{i(t')-3}\end{aligned}\ ] ] where @xmath365 and @xmath366 should be calculated using eq . ( [ rho - diag - tau ] ) for the previous period @xmath367 $ ] . even though this gives the same probability distribution , it is easier to use the `` global '' method ( [ i(t)-1 ] ) . integrating eqs . ( [ rho - diag-3 ] ) and ( [ rho - offdiag-3 ] ) , we obtain the evolution during the time interval @xmath348 $ ] , @xmath368 , \qquad \,\,\ , \label{rho - offdiag - tau-2}\end{aligned}\ ] ] where @xmath369 @xmath370 is given by eq . ( [ tilde - i(t ) ] ) , @xmath371 is given by eq . ( [ gamma-3 ] ) , and @xmath352 is given by eq . ( [ stark-2 ] ) . we emphasize that the outputs @xmath0 and @xmath4 correspond to the informational and non - informational quadratures , which change in time . in terms of the `` fixed '' experimental quadratures @xmath372 and @xmath373 from the iq mixer they are @xmath374 + q^{\rm fix } ( t ) \sin [ \phi_{\rm opt}(t ) ] , \qquad \\ & & q(t)= q^{\rm fix } ( t ) \cos [ \phi_{\rm opt}(t ) ] - i^{\rm fix } ( t ) \sin [ \phi_{\rm opt}(t ) ] , \end{aligned}\ ] ] where @xmath375 $ ] corresponds to the informational quadrature . if the measurement results are not taken from an experiment , but have to be generated numerically , then it is always possible to generate @xmath376 and @xmath377 without explicitly generating the signals @xmath0 and q(t ) . for @xmath376 we can still use eqs ( [ p(r)-1])([p(r)-3 ] ) , just replacing the superscript @xmath355 with @xmath319 . the probability distribution for @xmath378 is the zero - mean gaussian , @xmath379 , \qquad \label{p(r)-4}\\ & & d_r^q = d_r^i = 2 \int_{t}^{t+\tau } \frac{[\delta i(t')]^2}{s_i } \ , dt ' . \label{p(r)-5}\end{aligned}\ ] ] the probability distribution for a realization of @xmath362 is still given by eqs . ( [ i(t)-1 ] ) or ( [ i(t)-2 ] ) , while the similar probability distribution for @xmath380 is p\{q(t ) ] } . the evolution equations derived in this paper describe the evolution of an entangled qubit - resonator state ( [ rho - qr - def ] ) . however , there is an important special case when we can discuss the state of the qubit alone . if the measurement is of a relatively short duration and the microwave drive is switched off after that , then several decay times @xmath139 later ( or after the rapid driven reset procedure @xcite ) the resonator field is practically vacuum for both qubit states . in this case our formulas give the resulting qubit state , unentangled from the resonator state . in this paper we have developed a simple quantum bayesian formalism for the qubit measurement in the circuit qed setup with a moderate bandwidth of the measurement resonator , so that transients are important . the simplification comes from three assumptions : ( i ) we assume that the qubit evolves only due to measurement ( in particular , there are no rabi oscillations or qubit energy relaxation ) , ( ii ) we assume that the measurement resonator is driven by a classical , i.e. , coherent field ( in particular , no squeezed fields are applied ) , and ( iii ) the resonator is initially in a coherent state ( e.g. , vacuum ) . in this case the entangled qubit - resonator state developing in the process of measurement can be described as the density operator , eq . ( [ rho - qr - def ] ) , in which each of the two qubit states corresponds to its own coherent state of the resonator . therefore , the entangled qubit - resonator state at any moment of time is fully characterized by only 4 numbers : @xmath285 , @xmath286 , @xmath109 , @xmath93 , and two field amplitudes of the resonator : @xmath50 and @xmath51 . the field amplitudes evolve according to the classical equations ( [ alpha0-dot ] ) and ( [ alpha1-dot ] ) . the elements of the @xmath381 matrix @xmath382 evolve according to eqs . ( [ rho - diag-2 ] ) and ( [ rho - offdiag-2 ] ) if a phase - sensitive amplifier is used in the measurement or according to eqs . ( [ rho - diag-3 ] ) and ( [ rho - offdiag-3 ] ) if a phase - preserving amplifier is used . these evolution equations in differential form ( in both stratonovich and it forms ) are presented in sec . [ sec : differential ] . integrated equations for an arbitrary long evolution are discussed in sec . [ sec : arb - duration ] . the equations depend on parameters that are directly measurable in an experiment . the evolution equations for @xmath382 [ eqs . ( [ rho - diag-2 ] ) , ( [ rho - offdiag-2 ] ) , ( [ rho - diag-3 ] ) , ( [ rho - offdiag-3 ] ) ] have exactly the same form as in the `` bad cavity '' limit @xcite and have a simple physical meaning . we see that the diagonal elements @xmath285 and @xmath286 evolve as probabilities , i.e. , they follow the classical bayes rule , which updates the probabilities according to the information on the qubit state acquired from the measurement result . therefore , this `` spooky '' back - action is sensitive to the `` informational '' quadrature of the microwave field . the evolution of @xmath109 ( and @xmath383 ) necessarily depends on the evolution of @xmath285 and @xmath286 ( at least because @xmath384 ) . besides that , there are three more effects producing evolution of @xmath109 : ( i ) phase back - action , which depends on the measurement result sensitive to the `` non - informational '' quadrature of the microwave field , ( ii ) dephasing due to non - ideality of the measurement ( essentially loss of potential information ) , and ( iii ) ac stark shift of the qubit frequency . as discussed in appendix b , the phase back - action can be physically interpreted as being due to fluctuations of the ac stark shift because of a fluctuating number of photons in the resonator . even though the evolution equations ( [ rho - diag-2 ] ) , ( [ rho - offdiag-2 ] ) , ( [ rho - diag-3 ] ) , and ( [ rho - offdiag-3 ] ) are the same as in the `` bad cavity '' regime @xcite , the time step @xmath8 is no longer arbitrary , since the parameters entering the equations ( response @xmath124 , amplified phase difference @xmath77 , ensemble dephasing @xmath292 , etc . ) change during the transients , and therefore @xmath8 should be smaller than the time scale of this change . we emphasize that in the case of non - changing parameters these equations are exact for an arbitrary long @xmath8 . this may be beneficial for numerical simulations in comparison with the quantum trajectory formalism @xcite based on a wiener process , which assumes infinitesimal @xmath8 . in particular , our evolution equations can be easily integrated for an arbitrarily long duration [ eqs . ( [ rho - diag - tau ] ) , ( [ rho - offdiag - tau ] ) , ( [ rho - diag - tau-2 ] ) , and ( [ rho - offdiag - tau-2 ] ) ] . we note that the evolution equations in the phase - sensitive case are also exactly the same as for a qubit measurement by qpc or set @xcite , except now we consider a significantly entangled qubit - resonator state , with classically evolving resonator fields . the case of a phase - preserving amplifier is different because there are two output signals , @xmath0 and @xmath4 , instead of only one signal @xmath0 . nevertheless , the evolution equations are almost the same , and the only significant difference is that the phase back - action is governed by the non - informational quadrature @xmath4 , while the `` spooky '' back - action ( evolution of @xmath285 and @xmath286 ) is governed by the informational quadrature @xmath0 . the derivation in this paper has been based on elementary quantum mechanics and basic facts related to coherent states . in general , the idea is similar to the idea of `` microscopic '' derivation @xcite used to describe a qubit measurement by qpc or set . we solve exactly the quantum evolution due to interaction between the qubit and resonator ( which is very simple because the qubit does not evolve by itself and measurement is of the qnd type ) , and then apply the textbook collapse postulate to the pieces of microwave field , leaking from the resonator . the formalism developed in this paper is equivalent to the `` polaron frame approximation '' used in the quantum trajectory formalism @xcite , even though our language is significantly different . we hope that our derivation is physically transparent and therefore more easily understandable . also , as mentioned above , our formalism may have advantages in numerical calculations . for an evolving qubit ( e.g. , due to rabi oscillations ) it is tempting to simply include additional evolution into the differential equations for evolution derived in sec . [ sec : differential ] . however , this is formally incorrect because in this case the approach based on coherent states is no longer applicable ( though this is still possible in the `` bad cavity '' limit @xcite ) . the reason is the following . when the additional evolution of the qubit is comparable to or faster than @xmath5 , the resonator state @xmath385 or @xmath386 may correspond to the `` wrong '' qubit state produced by this evolution . since for a resonator the evolution of a superposition of coherent states ( a `` cat state '' ) can not be easily described with coherent states , the simple approach based on coherent states fails . therefore , for measurement of an evolving qubit the simple formalism discussed in this paper is not applicable and should be replaced with a more complicated formalism . the derivation of the quantum bayesian formalism for measurement of an evolving qubit is similar ideologically ( using the measurement of the `` history tail '' ) , but much more cumbersome technically . the result is equivalent to `` full '' quantum trajectory formalism @xcite , but uses an explicit fock - space evolution in the schrdinger picture instead of the language of superoperators . we will discuss this formalism in another paper . the formalism developed in this paper can be easily generalized to measurement of a multi - level transmon or measurement of several qubits , which evolve only due to measurement . such a generalization is useful to describe the process of entanglement of superconducting qubits by measurement @xcite . for @xmath387 qubits the state of the system can be described in the way similar to eq . ( [ rho - qr - def ] ) , so that each of @xmath388 qubit basis states corresponds to particular coherent states of the resonators , obtained via the classical field evolution . therefore , we only need to describe the evolution of @xmath389 matrix of coefficients , for which we can easily use the quantum bayesian approach to update the coefficients , depending on the measurement results . this will also be the subject of a future publication . the author thanks justin dressel , eyob sete , mark dykman , farid khalili , and konstantin likharev for useful discussions . the author also thanks justin dressel and juan atalaya for critical reading of the manuscript . the work was supported by aro grant no . w911nf-15 - 1 - 0496 . in this appendix we review basic facts related to coherent states . most of them are very well known in the quantum optics community . however , some of these facts [ e.g. , eq . ( [ phi - dot-2 ] ) ] are usually not discussed in optical textbooks . in contrast to the notation used in the main text , in this appendix we will use hat symbols for operators . as known from undergraduate quantum mechanics , for an oscillator with frequency @xmath12 and mass @xmath148 , the ground state in the @xmath390-representation is |0= _ gr(x ) = ( ) ^1/4 ( - x^2 ) . if we want to describe the classical state of this oscillator with coordinate @xmath391 and momentum @xmath392 ( still taking into account the uncertainty of the ground state ) , we need to shift the ground - state wavefunction by @xmath391 , producing @xmath393 , and also apply the momentum shift by adding the factor @xmath394 . this produces the so - called `` coherent state '' @xmath395 , which is widely used in optics : @xmath396 where @xmath397 and @xmath398 are the ground - state uncertainties . the normalization by doubled uncertanties @xmath399 and @xmath400 in eq . ( [ re - im ] ) as well as the overall phase factor @xmath401 in eq . ( [ alpha - def ] ) are to some extent arbitrary , but this conventional choice simplifies most of the formulas discussed below . note that the phase @xmath402 is exactly in between what we would obtain by first shifting @xmath390 , and then @xmath403 [ in this case we would obtain @xmath404 and , instead , first shifting @xmath403 and then @xmath390 [ in this case we would obtain @xmath405 . equation ( [ alpha - def ] ) can be rewritten in a more standard form @xcite @xmath406 where @xmath407 is the raising ( creation ) operator , @xmath408 ) satisfies the relations @xmath409\ , |\alpha\rangle$ ] and @xmath410\ , confusion between the notations for the stationary states @xmath198 and the coherent state @xmath411 ( for example , @xmath412 with @xmath413 is not the first excited level @xmath7 ) ; to avoid the confusion , we can use greek letters for coherent states and roman letters or integer numbers for the stationary states ( fock states ) . for the ground state the notations coincide , @xmath414 . if the oscillator state rotates with frequency @xmath2 ( for example , due to drive with this frequency ) , @xmath415 , @xmath416 , then from eq . ( [ re - im ] ) we find @xmath417 . in this case it is useful to introduce the rotating frame by defining @xmath418 , so that @xmath419 does not change in time . in the general case @xmath419 changes with time slowly , while @xmath420 rapidly oscillates . the rotating frame frequency @xmath2 can be chosen arbitrarily ; in the case with a drive , the most natural choice is the drive frequency @xmath22 ( because then @xmath419 does not change in the steady state ) ; in the absence of the drive , a natural choice is the oscillator frequency @xmath12 . note that the time dependence for the stationary states is @xmath421 ( counting the energy from the ground state energy ) , so for a `` non - evolving '' oscillator ( i.e. , evolving only naturally ) , from eq . ( [ alpha - def-2 ] ) we find @xmath422 . note that in the main text we always use the rotating frame based on the drive frequency @xmath22 and omit the tilde sign in the notation of the rotating - frame @xmath44 . in contrast , in this appendix we explicitly write @xmath423 for the rotating frame . so far we considered a textbook mechanical oscillator . if we consider a microwave resonator , then the role of @xmath390 and @xmath403 is played by properly normalized voltage and current ( at some point in the resonator ) or by flux and charge ; the effective mass @xmath148 can also be appropriately introduced . the formalism does not change . in quantum optics it is often preferred not to introduce coordinates and effective mass explicitly , and instead to start with the commutation relation @xmath424=1 $ ] , then producing fock states @xmath198 from vacuum @xmath6 with the creation operator . * 1 . * from eq . ( [ alpha - def-2 ] ) it is easy to see that |= | , [ prop-1]since @xmath425 for the lowering ( annihilation ) operator @xmath426 . the property ( [ prop-1 ] ) is sometimes used as a definition of the coherent state @xmath395 . note , however , that it does not specify the overall phase and normalization , while the overall phase if often important in analysis ( when more than one coherent state is involved ) . also note that @xmath427 does not have a simple formula , though @xmath428 from conjugation of @xmath429 . * 2 . * from eq . ( [ alpha - def-2 ] ) , the probability to measure @xmath26 photons in the state @xmath412 is p(n ) = e^-||^2|^2|^n / n ! , [ prop-2 - 1]which is the poissonian distribution with average @xmath430 . this proves that the wavefunction ( [ alpha - def-2 ] ) is normalized and shows that the mean photon number is * 3 . * the inner product of two coherent states @xmath395 and @xmath431 can be easily calculated using eq.([alpha - def-2 ] ) , giving the result @xcite |=e^-(||^2+||^2 ) e^^ * = e^-|-|^2 e^ -iim(^ * ) . [ inner - product]note that a shift of the coherent states by the same value changes the inner product , @xmath432 , since this changes the phase factor . it is useful to introduce the ( unitary ) displacement operator @xmath433 @xcite , ( ) ( ^-^ * ) , ( ) |0= | . a composition of two displacement operators has a phase factor @xcite similar to the phase factor in eq . ( [ inner - product ] ) , ( ) ( ) = ( + ) , [ displacement - comp]as follows from the baker - campbell - hausdorff formula @xmath434 for @xmath435=c$ ] . also note the useful relations @xmath436 * 5 . * let us introduce the ( hermitian ) quadrature operators @xmath437 and @xmath438 as @xcite _ q== , _ q== , [ _ q , _ q]=. [ quadrature - def]note that the quadrature operators are often defined as @xmath439 and @xmath440 ; then their commutator is @xmath441 ; another possible definition @xcite is @xmath442 and @xmath443 ; then the commutator is @xmath444 . the definition ( [ quadrature - def ] ) gives simpler formulas for the average values for the coherent states , |_q|= re ( ) , |_q|= i m ( ) , which follow from the relation @xmath445 . the variance in this case is |_q^2|- |_q|^2 = |_q^2|- |_q|^2 = . the quadrature operator at an angle @xmath446 can be defined as _ _ q + _ q . * 6 . * an important property of a coherent state is that it splits into two _ unentangled _ coherent states after passing through a beam splitter , in full analogy with a classical optical wave or microwave . actually , so far we defined a coherent state only for a resonator , and it is not obvious how to introduce it for a propagating wave . we will not discuss how to do it rigorously @xcite , just implying that a piece of propagating wave can be described in a way , similar to a resonator description . there is a rather simple rigorous way to describe transformation of an arbitrary quantum state passing through a beam splitter ( see , e.g. , @xcite ) . the idea is essentially to write classical field relations , but for the annihilation operators ( conjugated relations are for the creation operators ) , then express the initial state via vacuum and creation operators of the input arms , and then substitute these input - arms operators with their expressions via output - arms operators . this gives the resulting output state . applying this procedure to a beam splitter with transmission and reflection amplitudes @xmath447 and input state @xmath448 , we obtain the output state @xmath449 , exactly as we would expect for a classical field . technically , this follows from the formula @xmath450 [ see eq . ( [ alpha - def-3 ] ) ] and relation @xmath451 , with commuting output - arms operators @xmath452 and @xmath453 , so that @xmath454 . note that if we apply coherent fields to both input arms , @xmath455 , then the resulting output state is also an unentangled product of classically - expected coherent states , @xmath456 , without an overall phase . we can think about field leakage from a microwave resonator to a transmission line through a `` mirror '' ( coupler ) as transmission through a beam splitter . therefore , from the discussed above property , if the initial state in the resonator is a coherent state @xmath395 , then it remains a coherent state @xmath457 , with @xmath35 given by the classical field evolution , ( t)=(0 ) e^-i_r t e^-t/2 , [ kappa - only]where @xmath5 is the energy dissipation rate and @xmath12 is the resonator frequency . we emphasize that this property is highly unusual for a quantum system ( thus indicating that coherent states are classical to a significant extent ) . dissipation usually leads to decoherence , so that an initially pure quantum state becomes a mixed state . in this case we have an exception : a pure state remains pure during the whole evolution . this makes quantum analysis very simple for an evolution involving coherent states . note that eq . ( [ kappa - only ] ) is still applicable when the energy loss rate @xmath5 has a contribution from intrinsic energy relaxation ( at zero temperature ) . now let us for a moment neglect the energy relaxation , and instead consider a classical drive with frequency @xmath22 and ( complex ) amplitude @xmath23 ( in some normalization ) . this is usually described by the hamiltonian = _ r ^ + e^-i_d t ^ + ^ * e^i_d t , which already assumes rotating wave approximation , requiring @xmath458 and sufficiently slowly changing drive @xmath23 . using this hamiltonian , we can find the evolution of an arbitrary quantum state of the resonator @xmath459 via the schrdinger equation @xmath460 . it is easy to see by solving this equation that if the initial state is a coherent state , then it remains a coherent state , though with a _ nontrivial overall phase _ @xmath461 , |(t)= e^-i(t ) |(t ) , so that the evolution is described by two equations , @xmath462 now let us combine the drive @xmath60 and dissipation @xmath5 . since both of them keep the state coherent ( with an overall phase ) , their combination will also keep it coherent ( with an overall phase ) . introducing the rotating frame based on the drive frequency , ( t ) e^i_d t ( t ) , from eqs . ( [ kappa - only ] ) , ( [ alpha - dot ] ) , and ( [ phi - dot-1 ] ) we obtain @xmath463 equation ( [ tilde - alpha - dot ] ) is the standard result for the evolution of a resonator under the drive and dissipation , while eq.([phi - dot-2 ] ) is usually not discussed in quantum optics , even though it is very important for quantum dynamics involving more than one coherent state ( for example , for measurement of a qubit in the circuit qed setup ) . note that eqs . ( [ tilde - alpha - dot])([phi - dot-2 ] ) rely on the fact that for coherent states the dissipation @xmath5 does not introduce decoherence and only brings the term @xmath464 into eq.([tilde - alpha - dot ] ) . we have derived this fact by considering the problem of a coherent state passing through a beam splitter . another ( lengthier ) way to prove it , is to consider the lindblad equation for the density matrix and to show that ( surprisingly ) a pure initial state remains pure if initially it was a coherent state . one of the ways to show it , is to separate the lindblad evolution into `` jump '' and `` no jump '' scenarios ( e.g. , @xcite ) . then the `` jump '' scenario ( application of operator @xmath465 ) brings no evolution because of eq . ( [ prop-1 ] ) , so all the evolution comes from the `` no jump '' scenario ( essentially the bayesian update ) , which keeps a coherent state coherent , with decreasing @xmath35 . this is why _ there is no randomness _ @xcite , normally leading to decoherence . note that the derivation via the lindblad equation can not easily reproduce important equation ( [ phi - dot-2 ] ) , because the overall phase is lost in the density matrix language . in this appendix we derive the results for _ phase back - action _ in the process of qubit measurement using the picture of vacuum noise , which is incident on the resonator from the transmission line ( fig . [ fig : vacuum ] ) . we assume the `` bad cavity '' limit and phase - sensitive amplification . the vacuum noise is treated in a simple classical way . let us start with assuming for simplicity that the resonator damping @xmath5 is only due to coupling with the transmission line carrying the outgoing wave , @xmath142 ; in particular , this requires @xmath143 ( later this assumption will be removed ) . then the vacuum noise enters the resonator only from the output line ( fig . [ fig : vacuum ] ) , and the wave equations for the resonator field @xmath44 and the outgoing field @xmath40 in the rotating frame based on the drive frequency @xmath22 are @xmath466 where @xmath467 is the vacuum noise , which is normalized in the same way as @xmath40 . in this normalization @xmath468 is the average number of photons in the resonator , while @xmath43 is the average number of propagating photons per second . note that the reflection coefficient in eq . ( [ appb - f ] ) is @xmath469 , while the transmission through the `` mirror '' is characterized by the coupling @xmath470 @xcite , as well as in eq . ( [ appb - alpha - dot ] ) . the drive term @xmath471 can also be written via the properly normalized incoming field @xmath472 as @xmath473 . also note that for the two qubit states we have slightly different resonator frequencies , @xmath474 ; however , in this appendix we will mostly use notation @xmath12 for brevity and because the resonator frequency shift is not important for the phase back - action , which is our focus here . in quantum optics the vacuum noise is treated as an operator @xcite with correlator @xmath475 , and eqs . ( [ appb - alpha - dot ] ) and ( [ appb - f ] ) are written for annihilation operators in the heisenberg representation . however , in our simple derivation we will treat the noise @xmath467 classically ( i.e. , as a complex number ) and consider evolution of classical fields ( which corresponds to the schrdinger picture ) . it is simple to see that the photon shot noise is properly reproduced if we assume that _ for any quadrature _ ( so that @xmath476 is real ) v_qu(t ) v(t)_qu= ( t - t ) , [ appb - v - qu - corr]which is equivalent to v(t ) v(t)^*= ( t - t ) , v(t ) v(t)=0 , [ appb - v - corr]if @xmath477 is treated as a complex number , describing both quadrature components ( obviously , @xmath478 ) . for example , this relation can be obtained by considering a propagating wave @xmath479 with constant @xmath480 . then the fluctuating photon number @xmath481 within duration @xmath482 should have the same variance @xmath483\ , dt ' |^2 \rangle$ ] as the mean @xmath484 . therefore , | _ 0^t v_qu(t ) dt |^2 = [ appb - v - qu - int]for the quadrature @xmath476 along @xmath480 , and eq . ( [ appb - v - qu - corr ] ) follows from ( [ appb - v - qu - int ] ) . note that eq . ( [ appb - v - qu - corr ] ) can be interpreted as following from the standard operator correlator , using the correspondence @xmath485 . as another check of this noise formalism , let us derive the correlator for the fluctuating number of photons in the resonator from eq . ( [ appb - v - qu - corr ] ) . using eq . ( [ appb - alpha - dot ] ) , we find the fluctuation ( t)= _ -^t e^-[/2+i(_r-_d)](t - t ) v(t ) dt , [ appb - dalpha]due to the noise @xmath467 . for a fixed stationary value @xmath486 , this leads to photon number fluctuation @xmath487 . then using eq . ( [ appb - dalpha ] ) , performing the double - integration using eq . ( [ appb - v - corr ] ) , and denoting @xmath488 , we find n(t ) n(t+)= |n ( - || ) , [ appb - dn - corr]which is the standard result for the photon number correlator @xcite . note that the photon number fluctuation decays with the rate @xmath25 instead of naively expected @xmath5 . it is also interesting to note that at time @xmath489 only the quadrature @xmath476 along @xmath490 with the fluctuations ( [ appb - v - qu - corr ] ) contributes to the correlator ( [ appb - dn - corr ] ) , while the orthogonal quadrature does not contribute . it is equally possible to say that the contribution comes only from the quadrature @xmath476 along @xmath491 , while the orthogonal quadrature does not contribute . also note that from eq . ( [ appb - dalpha ] ) we obtain ||^2= 1/2 , corresponding to the variance of @xmath492 for any quadrature . now let us consider the qubit measurement , assuming the `` bad cavity '' regime , as in sec . the fluctuation @xmath467 leads to the fluctuating ac stark shift _ q ( t ) = 2n = 4re [ _ st^ * ( t ) ] with @xmath493 given by eq . ( [ appb - dalpha ] ) , and to the fluctuating outgoing field f(t ) = -v(t)+ ( t ) . by integrating these effects over the time period @xmath348 $ ] with @xmath494 , so that the exponential dependence in eq . ( [ appb - dalpha ] ) has sufficient time to fully decay , we find @xmath495 , \label{appb - dwq - int}\\ & & \int_t^{t+\tau } \delta f ( t')\ , dt ' = \frac{\kappa/2 -i(\omega_{\rm r}-\omega_{\rm d})}{\kappa /2 + i(\omega_{\rm r}-\omega_{\rm d } ) } \nonumber \\ & & \hspace{2.3 cm } \times \ , \int_t^{t+\tau } v(t')\ , dt ' . \label{appb - df - int}\end{aligned}\ ] ] we see that these fluctuating integrals are _ proportional _ to each other . obviously , the first integral determines the phase back - action on the qubit state , while the second integral is related to the measurement result . this is how we can relate the phase back - action to the measurement result . using eq . ( [ alpha - st ] ) for the steady - state values @xmath496 and @xmath181 corresponding to the qubit states @xmath6 and @xmath7 , and assuming @xmath497 , we find _ 1,st-_0 , st= _ st , and therefore from eqs . ( [ appb - dwq - int ] ) and ( [ appb - df - int ] ) we obtain @xmath498 . \label{appb - dwq - int2}\end{aligned}\ ] ] this relation shows that the phase back - action is determined by the output quadrature which is _ orthogonal _ to the informational quadrature along @xmath499 . note that the vacuum fluctuations @xmath467 , which produce the output fluctuations along the informational quadrature , do not affect the qubit state , so the corresponding evolution ( [ ph - sens - diag - bc ] ) of the qubit state ( diagonal matrix elements ) is only due to `` spooky '' back - action and can not be explained as an effect of @xmath467 . let us first consider an ideal phase - sensitive amplification of the `` orthogonal '' ( non - informational ) quadrature , so that @xmath126 [ see eq . ( [ phi - d - def ] ) ] . in this case we need to associate the output noise with the effect of @xmath467 fluctuations ( no added noise due to amplifier ) , and therefore _ t^t+ f_qu(t ) dt = , [ appb - dfqu - int]where @xmath110 is the measurement result [ eq . ( [ ph - sens - im - bc ] ) ] , @xmath259 is its variance , and real @xmath500 is the fluctuation along the measured quadrature . note that the left hand side of this relation is for a particular realization of the noise @xmath500 , while the last term in the right hand side assumes averaging over all noise realizations . since @xmath501 should have the usual vacuum noise statistics , we can use eq . ( [ appb - v - qu - int ] ) , which gives @xmath502 ^2 \rangle=\tau /4 $ ] . following eq . ( [ appb - dfqu - int ] ) , we can do the similar conversion for the response , |_1,st -_0,st | = . [ appb - response]finally , multiplying eqs . ( [ appb - dfqu - int ] ) and ( [ appb - response ] ) and noticing that this product corresponds to the right hand side of eq . ( [ appb - dwq - int2 ] ) multiplied by @xmath503 , we obtain _ t^t+ _ q ( t ) dt = , [ appb - dwq - int3]which is exactly the result for phase back - action @xcite presented in sec . [ sec : ph - sens - bc ] , when @xmath504 see eqs . ( [ ph - sens - off - bc ] ) , ( [ d - def - bc ] ) , and ( [ ph - sens - k - bc ] ) . the non - fluctuating part of the ac stark shift can be simply added . if we consider an ideal phase - sensitive amplification of an arbitrary quadrature , @xmath505 , then the derivation for the fluctuating phase shift @xmath506 is similar , except the amplified quadrature @xmath501 is no longer along @xmath507 , and therefore from eq . ( [ appb - dwq - int2 ] ) we obtain an extra factor @xmath508 , which appears in eq . ( [ ph - sens - k - bc ] ) but is absorbed by @xmath124 in eq . ( [ appb - dwq - int3 ] ) . however , it is not obvious if @xmath72 in eq . ( [ appb - dfqu - int ] ) should be counted from @xmath73 or from @xmath509 , and correspondingly if the phase back - action term in eq . ( [ ph - sens - off - bc ] ) should be @xmath510 or @xmath511\tau \}$ ] . we can find the answer by requiring that the phase shift due to the phase back - action term in eq . ( [ ph - sens - off - bc ] ) is zero on average . counterintuitively , the phase shift of the averaged @xmath512 in eq . ( [ ph - sens - off - bc ] ) is zero when the phase back - action term is @xmath510 , even though @xmath513 obviously has a non - zero phase if @xmath514 . this occurs due to a compensating effect from the first term in eq . ( [ ph - sens - off - bc ] ) , which contains @xmath285 and @xmath286 : for example , if @xmath515 , then a positive @xmath110 occurs more often , but produces smaller @xmath516 than for a negative @xmath110 . ( this somewhat counterintuitive compensation is related to the difference between the it and stratonovich approaches . ) thus , using the approach of the vacuum noise we derived the phase back - action term in eq . ( [ ph - sens - off - bc ] ) in the case of ideal phase - sensitive measurement . let us briefly discuss how in this approach we can take into account non - ideality due to additional resonator damping ( e.g. , because of coupling to other transmission lines ) and the loss of the microwave signal before it reaches amplifier ( which is still ideal ) . then eqs . ( [ appb - alpha - dot ] ) and ( [ appb - f ] ) can be replaced with @xmath517 \nonumber \\ & & \hspace{0.7 cm } + \sqrt{1-\kappa_{\rm col}/\kappa_{\rm out}}\ , v_{\rm add,2 } , \label{appb - f-2}\end{aligned}\ ] ] where @xmath518 is the vacuum noise entering the resonator from other transmission lines , the ratio @xmath90 characterizes the energy loss between the resonator and amplifier ( which can be modeled via a beam splitter ) , and because of this loss ( at zero temperature ) an additional vacuum noise @xmath519 contributes to the field @xmath40 , which reaches the amplifier . the noises @xmath520 , @xmath521 , and @xmath519 are uncorrelated and all satisfy eq . ( [ appb - v - corr ] ) ; then the noise of @xmath40 has the same statistics . the calculation becomes more complicated , but it still can be done explicitly . it shows that the correlation ( [ appb - dwq - int3 ] ) between the ac stark shift and the measurement result fluctuations is reduced by the factor @xmath522 , which is the same factor as for the reduction of @xmath124 . therefore , eq . ( [ appb - dwq - int3 ] ) and the corresponding eq . ( [ ph - sens - k - bc ] ) remain valid . analysis of imperfection due to a non - ideal amplifier can be performed as in ref . @xcite ; in this case eqs . ( [ ph - sens - off - bc ] ) and ( [ ph - sens - k - bc ] ) still remain valid . note that even though this approach based on vacuum noise gives a natural description of the physical mechanism responsible for the phase back - action , it still can not explain why in the ideal case with @xmath108 there are no fluctuations of the photon number in the resonator . the fact that in the ideal case only the observed quadrature fluctuates ( and the orthogonal quadrature does not fluctuate ) is a `` spooky '' property of quantum measurement and can not have a realistic interpretation . derivation of the phase back - action coefficient for the phase - preserving measurement can be done in a similar way . alternatively , as discussed in sec . [ sec : ph - pres - bc ] , the results for the phase - preserving case can be obtained from the results for the phase - sensitive measurement . j. y. mutus , t. c. white , e. jeffrey , d. sank , r. barends , j. bochmann , y. chen , z. chen , b. chiaro , a. dunsworth , j. kelly , a. megrant , c. neill , p. j. j. omalley , p. roushan , a. vainsencher , j. wenner , i. siddiqi , r. vijay , a. n. cleland , and j. m. martinis , appl . . lett . * 103 * , 122602 ( 2013 ) . e. jeffrey , d. sank , j. y. mutus , t. c.white , j. kelly , r. barends , y. chen , z. chen , b. chiaro , a. dunsworth , a. megrant , p. j. j. omalley , c. neill , p. roushan , a. vainsencher , j. wenner , a. n. cleland , and j. m. martinis , phys . * 112 * , 190504 ( 2014 ) .

we consider continuous quantum measurement of a superconducting qubit in the circuit qed setup with a moderate bandwidth of the measurement resonator , i.e. , when the `` bad cavity '' limit is not applicable . the goal is a simple description of the quantum evolution due to measurement , i.e. , the measurement back - action . extending the quantum bayesian approach previously developed for the `` bad cavity '' regime , we show that the evolution equations remain the same , but now they should be applied to the entangled qubit - resonator state , instead of the qubit state alone . the derivation uses only elementary quantum mechanics and basic properties of coherent states , thus being accessible to non - experts .

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Proceed to summarize the following text: spin - glasses are simple models for disordered systems . they can be defined very easily in mathematical terms ; on the hand , the properties of such simple models usually are quite rich . statistical physics of spin - glasses has been studied for more than thirty years since the concept of spin - glasses was first presented by edwards and anderson @xcite in 1975 , but there are still many unsolved and heavily debated issues . in the last decades there have been a lot of theoretical investigations concerning models defined on a finite - connectivity random graph . these later models are more realistic than conventional spin - glass models ( e.g. , the sherrington - kirkpatrick model @xcite ) on a complete graph , in the sense that each spin interacts only with a finite number of other spins . as direct analytical studies of spin - glass models on three - dimensional ( regular ) lattices are still beyond reach , people hope that a deep understanding of ( mean - field ) models on finite - connectivity random graphs will shed much light on the properties of 3d systems . for a spin - glass system of very large size , it is generally believed that ergodicity is broken when the environmental temperature @xmath0 becomes lower than a certain value @xmath5 , the spin - glass transition temperature . in this low - temperature spin - glass phase , the time average of a given physical quantity no longer equals to the ensemble average . the whole configurational phase space of the system breaks into many separated domains . ergodicity is still preserved within each such configurational domains . if the system initially is in a spin configuration that belongs to domain @xmath1 of the configurational space , it will eventually visit all other configurations in this domain as time elapses . on the other hand , due to the existence of very high free energy barriers , the system is unable to transit spontaneously from one configurational space domain to another different domain . in this ergodicity - broken situation , each such configurational phase space domain is regarded as a _ macroscopic state _ or _ thermodynamic state_. the free energy @xmath6 of a macroscopic state @xmath1 is related to the microscopic configurations in @xmath1 through the following fundamental formula of statistical mechanics , @xmath7 where @xmath8 is the total number of spins in the system ; @xmath9 is the inverse temperature ( we set boltzmann s constant to unity throughout this paper ) ; and @xmath10 is the partition function for the macroscopic state @xmath1 as defined by @xmath11 \ .\ ] ] in eq . ( [ eq : partitionfunctionalpha ] ) , @xmath12 denotes a microscopic spin configuration , and @xmath13 is the total energy of this configuration . the summation in eq . ( [ eq : partitionfunctionalpha ] ) is over all those microscopic configurations belonging to macroscopic state @xmath1 . for a given spin - glass system , although we can formally write down the expressions of the partition function and free energy for a macroscopic state @xmath1 , the principal difficulty of spin - glass statistical physics is that we do not know _ a priori _ how the configurational space of the system is organized and which are the constituent spin configurations of each macroscopic state . to overcome this difficulty , one possibility is to first assume certain structural organization of the system s configurational space and then try to derive a self - consistent theory . for the sherrington - kirkpatrick model , the full - step replica - symmetry - broken ( frsb ) theory of parisi with an ultrametric organization of macroscopic states @xcite has met with great success . for finite - connectivity mean - field spin - glasses , a cavity approach was also developed by mzard and parisi @xcite . this cavity approach combined the bethe - peierls approximation @xcite for a ferromagnetic ising model with the physical picture that there is a proliferation of macroscopic states in a spin - glass system . this approach , which has been shown @xcite to be equivalent to the first - step replica - symmetry - broken ( 1rsb ) replica theory , can give very good predictions concerning the low - temperature free energy density of a system on a random graph . later on , this cavity approach was extended by mzard and parisi @xcite to the limiting case of zero temperature . the zero - temperature cavity method was applied to some hard combinatorial optimization problems with good performances ( see , e.g. , refs . these interdisciplinary applications in return also call for further understanding on the mean - field cavity approach and its possible extensions ( see , e.g. , refs . @xcite ) . for a spin - glass system at a low temperature @xmath0 , the total number of macroscopic states with a given free energy @xmath14 is denoted as @xmath15 . although it is natural to anticipate that the logarithm of @xmath15 should be an extensive quantity , the exact relationship between @xmath16 and @xmath14 is unknown and is system - dependent . inspired by parisi s frsb solution , in the cavity approach of mzard and parisi @xcite one assumes that the total number @xmath17 of macroscopic states with free energy @xmath14 diverges exponentially with @xmath14 with respect to a reference free energy @xmath18 @xcite , i.e. , @xmath19 where @xmath20 is a dimensionless constant to be determined self - consistently . the parisi parameter @xmath21 stems from the replica theory of infinite - connectivity spin - glasses @xcite , its physical meaning in the cavity framework is not transparent . for some spin - glass systems with many - body interactions , even if the exponential form of eq . ( [ eq : mp01 ] ) is valid in certain range of free energy values , the parameter @xmath21 in this equation may exceed unity . the original cavity iterative equations of ref . @xcite diverge for this situation of @xmath22 , where the statistical physical property of the spin - glass system is not determined by those macroscopic states with the globally minimal free energy density . since the cavity approach of mzard and parisi is a very useful theoretical tool in studying the low - temperature properties of many spin - glass systems , it might be helpful for us to interpret the approach from an another slightly different angle . complementary interpretations of the 1rsb cavity theory will also facilitate its further development . in this paper we demonstrate that , the 1rsb cavity formalism of mzard and parisi can be derived in an alternative way without using eq . ( [ eq : mp01 ] ) . this slightly revised cavity theory is based on the gedanken experiment of repeated temperature heating - annealing . the final macroscopic state reached at the end of an annealing process is anticipated to follow the boltzmann distribution of free energies . this theoretical approach is applied to the @xmath3 spin - glass model on a random regular graph of degree @xmath4 to test its validity . the results of the present work are in close agreement with earlier simulational and numerical results . the mathematical format of the present cavity theory is the same both for non - zero temperatures and for zero temperature . if the macroscopic states of a spin - glass system further organize into clusters of macroscopic states , it can be easily extended to take into account this situation . this paper is organized as follows . the next section introduces the @xmath3 spin - glass model and the ensemble of random regular graphs of degree @xmath23 . section [ sec : mfep ] describes the free energy boltzmann distribution of macroscopic states . in sec . [ sec : cavity - field - distribution ] the concepts of cavity field and cavity magnetization are re - introduced , and the distribution of a vertex s cavity magnetization among all the macroscopic states is calculated . various thermodynamic quantities , including the grand free energy density of the whole system and the free energy density of a macroscopic state , are calculated in sec . [ sec : densities ] and sec . [ sec : single - instance ] for an ensemble of systems and for a single system , respectively . the numerical results for the @xmath3 spin - glass model are reported and analyzed in sec . [ sec : numerics ] . we conclude the present work and discuss its possible extensions in sec . [ sec : discussion ] . the appendix gives an explicit expression for the mean energy density . in this paper we focus on just a single example , the @xmath3 spin - glass model @xcite on a random regular graph of degree @xmath23 . this model was also studied in ref . @xcite , lending us the opportunity to directly compare the results of both treatments . let us consider the graph @xmath24 which is obtained by randomly choosing with equal probability a graph from the set of all regular graphs of size @xmath8 and vertex - degree @xmath23 . @xmath24 is a random regular graph of degree @xmath23 . each of the @xmath8 vertices of @xmath24 is connected to @xmath23 other vertices , but there is no structure in the connection pattern of @xmath24 . for each edge @xmath25 of graph @xmath24 , we assign a coupling constant @xmath26 or @xmath27 with equal probability . once the coupling @xmath28 for an edge is assigned , it no longer changes . therefore , we have a _ network _ with random quenched connection pattern and random quenched coupling constants . ( in the remaining part of this paper , when we use the term @xmath29network , we mean the connection pattern of the graph plus the quenched couplings ; when we use the term @xmath29graph , we mean only the connection pattern . ) on top of such a network we define the following energy function @xmath30 where @xmath31 ( @xmath32 ) is the spin variable of vertex @xmath33 , @xmath31 can take two values , @xmath34 and @xmath35 . equation ( [ eq : energy01 ] ) is the @xmath3 spin - glass model . the total number of microscopic configurations @xmath36 in such a system is simply @xmath37 . the vertex degree of a random regular graph of degree @xmath23 is equal to that of a regular square lattice in @xmath38 dimensions . one hopes that some statistical physical properties of a spin - glass model on a random regular graph will also hold for the same model on a regular lattice . on the other hand , in a regular lattice , there exist many short loops , which make analytical calculations extremely difficult . in a random regular graph of very large size there is no such short loops . the graph is locally tree - like ; and the typical loop length in the graph scales as @xmath39 . one can exploit this absence of short loops to construct a self - consistent mean - field theory . for the benefit of later discussions , we also mention the concept of a random regular cavity graph @xmath40 @xcite . in this graph of size @xmath8 , there are @xmath41 ( @xmath42 ) vertices of degree @xmath43 and @xmath44 vertices of degree @xmath23 . a vertex of degree @xmath43 in graph @xmath40 is referred to as a _ cavity vertex_. like @xmath24 , the connection pattern of @xmath40 is also completely random ; and the quenched coupling constants for the edges of @xmath40 are also independently and equally distributed over @xmath3 . the spin - glass hamiltonian eq . ( [ eq : energy01 ] ) can also be defined for this cavity network . consider the spin - glass system eq . ( [ eq : energy01 ] ) . at high temperatures , the system is in the paramagnetic phase . each vertex of the network does not have any spontaneous magnetization , its spin fluctuates over the positive and negative directions and stays in each orientation with equal probability . the configurational phase space of the system is ergodic ( see left panel of fig . [ fig : ergodicitybreaking01 ] ) . when the temperature @xmath0 is decreased below a spin - glass transition temperature @xmath5 , the system is in the spin - glass phase . in this phase , a vertex of the system might favor one spin orientation over the opposite orientation , but this orientation preference is vertex - dependent . when the number @xmath8 of vertices is sufficiently large , the configurational phase space of this spin - glass system splits into many separated domains ( see right panel of fig . [ fig : ergodicitybreaking01 ] ) . each domain of the phase space , which corresponds to a macroscopic state or thermodynamic state of the system , contains a set of microscopic spin configurations . the system is ergodic within each macroscopic state , while ergodicity is broken at the level of macroscopic states . however , the system is able to transit from one macroscopic state to another different macroscopic state if the following temperature heating - annealing process is performed : the system is first heated to a high temperature beyond @xmath5 and waited for a long time till it reaches equilibrium . the system now stays in a high - temperature ergodic phase . afterwords , it is cooled infinitely slowly till the final low temperature @xmath0 is reached . ( a simulated - annealing idea was previously explored by kirkpatrick and co - workers @xcite to tackle hard optimization problems . ) since in the high - temperature phase the system loses memory about its prior history , at the end of the annealing process it may reach any a macroscopic state . the probability @xmath45 of the system end up being in a particular macroscopic state @xmath1 , however , is in general different for different macroscopic states . we argue in the following that , if an infinite number of the above - mentioned annealing experiment are performed , @xmath45 should depend only on the free energy @xmath6 of the macroscopic state @xmath1 . a schematic picture of ergodicity breaking in a spin - glass system . ( left ) when the temperature is high , all the microscopic configurations ( denoted by dots ) which contribute to the free energy of the system are in a single domain of the configurational phase space . ( right ) when the temperature is low enough , the microscopic configurations which contribute to the free energy of the system are contained in different domains in the configurational phase space . each domain contains a number of microscopic configurations . ergodicity is still preserved within each phase space domain , while the configuration of the system is unable to jump spontaneously from one domain to another different domain of the phase space . ] the total partition function of the system is defined as @xmath46 where @xmath13 is the total energy expression as given by eq . ( [ eq : energy01 ] ) . when ergodicity is broken , the partition function @xmath47 can be re - written as a sum over all the macroscopic states @xmath1 : @xmath48 in eq . ( [ eq : partitiondomain02 ] ) , @xmath10 and @xmath49 are the partition function and free energy of the macroscopic state @xmath1 as defined in eq . ( [ eq : partitionfunctionalpha ] ) and eq . ( [ eq : falpha ] ) , respectively . by way of repeated temperature annealing , the whole configurational space of a spin - glass system is explored ( and ergodicity is recovered ! ) . the total partition function eq . ( [ eq : partitiontotal ] ) can therefore be understood as containing all the information of the system as measured by an infinite number of temperature annealing experiments . from eq . ( [ eq : partitiondomain02 ] ) we see that each macroscopic state @xmath1 contributes a term @xmath50 to the total partition function of the system . therefore we can anticipate that , at temperature @xmath0 , the probability of the system being in macroscopic state @xmath1 at the end of an annealing experiment is given by the following distribution @xmath51 this is a free energy boltzmann distribution over all the macroscopic states @xmath1 . at this point , we introduce an artificial inverse temperature @xmath52 at the level of macroscopic states . ( such an artificial inverse temperature @xmath52 was first introduced in the early work of ref . @xcite for @xmath53 spin - glasses . ) this is better explained by constructing the following artificial single - particle system : the particle has a set of @xmath29energy levels , the @xmath29energy of level @xmath1 is equal to the free energy @xmath54 of the macroscopic state @xmath1 of the actual spin - glass system ; this one - particle system is in a heat bath with inverse temperature @xmath52 . then the partition function of the artificial system is @xmath55 the probability of this artificial system being in energy level @xmath1 is given by @xmath56 when the inverse temperature @xmath52 of the artificial system is set to @xmath57 , then the partition function @xmath58 reduces to the total partition function @xmath59 of the original spin - glass system . the partition function @xmath58 of eq . ( [ eq : newpartitionfunction ] ) can be written in another form as @xmath60 \ . \label{eq : zyb03}\ ] ] in eq . ( [ eq : zyb03 ] ) , @xmath61 is the free energy density of a macroscopic state ; the function @xmath62 is called the complexity @xcite , which is related to the total number @xmath15 of macroscopic states by the following equation @xmath63 the complexity @xmath62 , which should be non - negative , is a measure of the total number of macroscopic states with free energy density @xmath64 . when the spin - glass system is very large ( @xmath65 ) , eq . ( [ eq : zyb03 ] ) indicates that the partition function @xmath58 is contributed exclusively by those macroscopic states whose free energy density @xmath64 satisfying the equation @xmath66 equation ( [ eq : y ] ) gives an implicit relationship between the inverse temperature @xmath52 and the observed mean free energy density @xmath64 of the system . for the benefit of later discussions , we define a grand free energy @xmath67 by the following equation @xmath68\ .\ ] ] from eq . ( [ eq : zyb03 ] ) we see that the grand free energy density @xmath69 is @xmath70 where @xmath71 is the solution of eq . ( [ eq : y ] ) . in the next three sections we will study the statistical physical properties of the @xmath3 spin - glass system using @xmath72 and @xmath52 as a pair of control parameters . the free energy at the microscopic level ( within a macroscopic state ) will be calculated with the inverse temperature @xmath72 , while the grand free energy at the macroscopic state level will be calculated with the inverse temperature @xmath52 . although in the actual spin - glass system , the two inverse temperatures are identical ( @xmath73 ) , they are decoupled in the following analytical theory . this decoupling gives us extra freedom in the theoretical development . finally , to go from the artificial system to the original system , we will choose the largest value of @xmath52 in the interval of @xmath74 which satisfies the requirement that the complexity @xmath75 is non - negative . to analytically study the statistical physical property of a spin - glass model on a random graph , there are basically two different but equivalent approaches . the first one is the replica method @xcite and the second one is the cavity method @xcite . the present paper exploits the cavity approach . it corresponds to the 1rsb approximation of the replica method . basically , one assumes that macroscopic states of the spin - glass system are distributed evenly in the whole configurational phase space , and there is no further organization of the macroscopic states or further structures within each macroscopic state . consider the spin - glass system eq . ( [ eq : energy01 ] ) on a cavity network @xmath76 of @xmath8 vertices and @xmath43 cavity vertices ( see fig . [ fig : cavity01m1]a ) . let us suppose that the configurations of such a cavity system are in a macroscopic state @xmath1 . cavity vertices in a random regular cavity graph of vertex degree @xmath23 ( here @xmath77 ) . ( a ) the neighborhoods for three cavity vertices @xmath78 , @xmath79 and @xmath80 . these three vertices have vertex degree @xmath43 ( @xmath81 ) while all the other @xmath82 vertices in the graph have vertex degree @xmath23 . in a macroscopic state @xmath1 , the three cavity vertices feel the cavity fields @xmath83 , @xmath84 and @xmath85 , respectively . ( b ) a new cavity graph with @xmath86 vertices is generated by adding a new vertex @xmath33 and connecting it to @xmath43 old cavity vertices . the cavity vertex @xmath33 of the new graph feels the cavity field @xmath87 in the corresponding macroscopic state @xmath1 of the new cavity system . ] at given inverse temperature @xmath72 , the spin value @xmath88 of a cavity vertex @xmath78 fluctuates over time around certain mean value @xmath89 which depends on the macroscopic state @xmath1 . since @xmath90 is a binary variable , the marginal distribution @xmath91 of @xmath88 can be expressed in the following form @xmath92 the magnetization of the cavity vertex @xmath78 is @xmath93 from eq . ( [ eq : magnetization01 ] ) we know that @xmath83 is the magnetic field experienced by the cavity vertex @xmath78 due to the spin - spin interactions between vertex @xmath78 and its @xmath43 nearest - neighbors . we call @xmath83 the _ cavity field _ on vertex @xmath78 and @xmath89 the _ cavity magnetization _ of vertex @xmath78 . we emphasize again that @xmath83 and @xmath89 depends on the macroscopic state @xmath1 . since the connection pattern in a cavity graph with @xmath43 cavity vertices is completely random , the typical length @xmath94 of a shortest - distance path between two cavity vertices @xmath78 and @xmath79 in the cavity graph @xmath76 is a large value . actually , it can easily be shown @xcite that this distance scales logarithmically with the cavity graph size @xmath8 : @xmath95 denote @xmath96 as the joint probability distribution of the spin values @xmath97 of a group of cavity vertices in a cavity network . in a large random graph , according to eq . ( [ eq : lengthscale ] ) the shortest path length between any two randomly chosen vertices is long . therefore , as the zeroth - order approximation , one may assume that this joint probability distribution can be written as the following factorized form @xmath98 where @xmath99 is vertex @xmath78 s marginal spin value distribution as given by eq . ( [ eq : cavityfield01 ] ) . equation ( [ eq : factorization ] ) is called the bethe - peierls approximation @xcite in the literature . it assumes statistical independence among the spin states of the cavity vertices within a macroscopic state @xmath1 . ( for recent references on extensions of the bethe - peierls approximation , see refs . @xcite . ) the cavity magnetization @xmath89 of the cavity vertex @xmath78 depends on the identity of the macroscopic state @xmath1 . its value may be different in different macroscopic states . let us denote @xmath100 as the fraction of macroscopic states in which the cavity magnetization of vertex @xmath78 takes the value @xmath89 , and denote @xmath101 as the fraction of macroscopic states in which the cavity magnetization of vertex @xmath102 take the value @xmath103 , respectively . we extend the bethe - peierls approximation eq . ( [ eq : factorization ] ) to the level of macroscopic states and assume that @xmath104 equation ( [ eq : factorization - magnetization ] ) is equivalent to saying that , the fluctuations ( among all the macroscopic states ) of the cavity magnetization of two different cavity vertices are mutually independent of each other . due to the absence of short loops in a random graph , eq . ( [ eq : factorization - magnetization ] ) turns out to be a rather good approximation . to obtain a self - consistent equation for the marginal probabilities @xmath105 , we add a new vertex @xmath33 and connect it to the @xmath43 cavity vertices of @xmath76 . the quenched coupling constant @xmath28 of each newly added edge is set to be @xmath106 with equal probability . this results in a new cavity network @xmath107 of @xmath86 vertices and one single cavity vertex @xmath33 ( see fig . [ fig : cavity01m1]b ) . in the corresponding macroscopic state @xmath1 of the new cavity system @xmath107 , the cavity vertex @xmath33 feels a cavity field @xmath87 . to calculate the cavity field @xmath87 in the new cavity system , we first notice that the energy difference between the cavity network @xmath107 and the old cavity network @xmath76 is @xmath108 where @xmath109 denotes the set of @xmath43 nearest - neighbors of vertex @xmath33 in the cavity graph @xmath107 . in the macroscopic state @xmath1 , the partition function for the cavity system @xmath107 is @xmath110 \sum\limits_{\sigma_i } \exp\bigl ( \beta h_i \sigma_i \bigr ) \label{eq : zacv03m1 } \\ & = & e^{-\beta f_\alpha \bigl ( { \cal g}_k(n , k-1)\bigr ) } \prod\limits_{j\in \partial^\prime i } \biggl[\frac { \cosh \beta j_{i j } } { \cosh \beta u(j_{i j } , h_{j } ) } \biggr ] ( 2 \cosh \beta h_{i } ) \ . \label{eq : zacv03}\end{aligned}\ ] ] in going from eq . ( [ eq : zacv03m3 ] ) to eq . ( [ eq : zacv03m2 ] ) , we have used the bethe - peierls approximation eq . ( [ eq : factorization ] ) . @xmath111 is the free energy of the cavity system @xmath76 in its macroscopic state @xmath1 . in eq . ( [ eq : zacv03m1 ] ) , the quantity @xmath112 is defined as @xmath113 and the quantity @xmath87 is calculated according to @xmath114 from eq . ( [ eq : zacv03m1 ] ) we know that @xmath87 as expressed by eq . ( [ eq : hia ] ) is just the cavity field felt by vertex @xmath33 in the cavity network @xmath107 . if we know all the cavity fields on the nearest - neighbors of vertex @xmath33 , then we obtain the cavity field on vertex @xmath33 through the iterative equation ( [ eq : hia ] ) . consequently , the magnetization of the cavity vertex @xmath33 in the macroscopic state @xmath1 of the cavity system @xmath107 is calculated as @xmath115 - \prod\limits_{j\in \partial^\prime i } \bigl [ 1 - v_{i j } m_j \bigr ] } { \prod\limits_{j\in \partial^\prime i } \bigl [ 1 + v_{i j } m_j \bigr ] + \prod\limits_{j\in \partial^\prime i } \bigl [ 1 - v_{i j } m_j ] } \ , \ ] ] where @xmath116 is a shorthand notation for @xmath117 , i.e. , @xmath118 equation ( [ eq : mi ] ) is an iterative equation for the cavity magnetization @xmath119 _ within one macroscopic state @xmath1_. as we have emphasized in sec . [ subsec : single01 ] , the input cavity magnetization @xmath89 in eq . ( [ eq : mi ] ) may be different in different macroscopic states . as a consequence , the cavity magnetization @xmath119 of vertex @xmath33 does not necessarily take the same value in different macroscopic states . on the contrary , its value may fluctuate a lot among different macroscopic states of the new cavity system @xmath107 . the task is to obtain an expression for the marginal distribution @xmath120 of @xmath119 among different macroscopic states . from eq . ( [ eq : zacv03 ] ) , we know that , after the addition of the vertex @xmath33 , the free energy difference @xmath121 between the macroscopic state @xmath1 of the system @xmath107 and that of the system @xmath76 is @xmath122 + \prod\limits_{j\in \partial^\prime i } \bigl[1- v_{i j } m_j \bigr ] \biggr ) \ . \label{eq : deltaf102}\end{aligned}\ ] ] according to the free energy boltzmann distribution eq . ( [ eq : boltzmanndistribution02 ] ) , each macroscopic state is weighted by the boltzmann factor @xmath123 . after the addition of vertex @xmath33 , the total partition function of the system @xmath107 is @xmath124 \prod\limits_{j\in \partial^\prime i } \biggl [ \int { \rm d } m_j { \cal p}_j(m_j ) \biggr ] \exp(-y \delta f_1 ) \ . \label{eq : rho - m1}\end{aligned}\ ] ] we have used the factorization approximation eq . ( [ eq : factorization - magnetization ] ) in going from eq . ( [ eq : rho - m1a ] ) to eq . ( [ eq : rho - m1 ] ) . similarly , the total weight of those macroscopic states of the system @xmath107 in which the cavity magnetization of vertex @xmath33 being equal to @xmath119 is expressed as @xmath125 \times \nonumber \\ & & \prod\limits_{j\in \partial^\prime i } \biggl [ \int { \rm d } m_j { \cal p}_j(m_j ) \biggr ] \exp(-y \delta f_1 ) \delta\biggl(m_i-\frac { \prod\limits_{j\in \partial^\prime i } \bigl [ 1 + v_{i j } m_j \bigr ] - \prod\limits_{j\in \partial^\prime i } \bigl [ 1 - v_{i j } m_j \bigr ] } { \prod\limits_{j\in \partial^\prime i } \bigl [ 1 + v_{i j } m_j \bigr ] + \prod\limits_{j\in \partial^\prime i } \bigl [ 1 - v_{i j } m_j ] } \biggr ) \ . \label{eq : rho - m2}\end{aligned}\ ] ] from eq . ( [ eq : rho - m1 ] ) and eq . ( [ eq : rho - m2 ] ) we realize that , in the new system , the fraction of macroscopic states in which vertex @xmath33 bearing a cavity magnetization @xmath119 is equal to @xmath126 e^{-y \delta f_1 } \delta\biggl ( m_i - \frac { \prod\limits_{j\in \partial^\prime i } \bigl [ 1 + v_{i j } m_j \bigr ] - \prod\limits_{j\in \partial^\prime i } \bigl [ 1 - v_{i j } m_j \bigr ] } { \prod\limits_{j\in \partial^\prime i } \bigl [ 1 + v_{i j } m_j \bigr ] + \prod\limits_{j\in \partial^\prime i } \bigl [ 1 - v_{i j } m_j ] } \biggr ) \ .\ ] ] equation ( [ eq : rho ] ) is a self - consistent iterative equation for the cavity distributions @xmath127 . a steady state solution of eq . ( [ eq : rho ] ) can be obtained by population dynamics @xcite . an array of @xmath128 probability distributions @xmath129 are stored . at each step of the population dynamics , @xmath43 probability distributions are randomly chosen from this array of @xmath128 stored distributions , and a new probability distribution is generated by using eq . ( [ eq : rho ] ) . this new distribution then replaces a randomly chosen old probability distribution in the array . this iteration process is repeated many times until the population dynamics reaches a steady state . the free energy of a macroscopic state @xmath1 is defined formally by eq . ( [ eq : falpha ] ) . as we noted before , this expression is not directly applicable since we do not know what are the microscopic configurations of state @xmath1 . the cavity approach @xcite circumvents this problem by calculating the grand free energy difference between a system of @xmath8 vertices and an enlarged system of @xmath130 vertices . as demonstrated in fig . [ fig : cavity01 ] , a random network @xmath24 can be constructed from a random cavity network @xmath131 by adding @xmath43 new edges . the grand free energy difference between these two systems can be calculated . similarly , one can add two new vertices and @xmath132 new edges to change the same random cavity network @xmath133 into a random regular network @xmath134 . the grand free energy difference between these two systems can also be calculated . from the two grand free energy differences , one can obtain the grand free energy density @xmath135 for a random regular system @xmath24 . the mean free energy density of a macroscopic state and the complexity of the system can then be obtained from @xmath135 . the cavity approach to the spin - glass model eq . ( [ eq : energy01 ] ) on a random regular graph of connectivity @xmath136 . ( a ) : part of a cavity graph with @xmath137 cavity vertices . ( b ) and ( c ) : construction of a random regular graph of size @xmath8 ( b ) and @xmath130 ( c ) from the cavity graph with @xmath8 vertices and @xmath137 cavity vertices . , title="fig : " ] the cavity approach to the spin - glass model eq . ( [ eq : energy01 ] ) on a random regular graph of connectivity @xmath136 . ( a ) : part of a cavity graph with @xmath137 cavity vertices . ( b ) and ( c ) : construction of a random regular graph of size @xmath8 ( b ) and @xmath130 ( c ) from the cavity graph with @xmath8 vertices and @xmath137 cavity vertices . , title="fig : " ] the cavity approach to the spin - glass model eq . ( [ eq : energy01 ] ) on a random regular graph of connectivity @xmath136 . ( a ) : part of a cavity graph with @xmath137 cavity vertices . ( b ) and ( c ) : construction of a random regular graph of size @xmath8 ( b ) and @xmath130 ( c ) from the cavity graph with @xmath8 vertices and @xmath137 cavity vertices . , title="fig : " ] the difference @xmath139 between the configurational energy of the system on the random graph @xmath24 in fig . [ fig : cavity01]b and that of the system on the random cavity graph @xmath140 in fig . [ fig : cavity01]a is @xmath141 where @xmath142 denotes the set of @xmath43 newly added edges in going from @xmath140 to @xmath24 . each edge in set @xmath142 connects two cavity vertices of @xmath133 . following the analytical procedure of sec . [ subsec : field - distribution ] , we know that difference between the free energy of a macroscopic state @xmath1 of the system @xmath24 and that of the same macroscopic state @xmath1 of the cavity system @xmath131 is @xmath143 where @xmath144 can be understood as the free energy increase caused by the addition of an edge ( with coupling @xmath28 ) between two cavity vertices @xmath33 and @xmath78 . the grand free energy @xmath145 of the new system @xmath24 is related to the grand free energy @xmath146 of the old cavity system @xmath147 by the following equation @xmath148 where @xmath149 \\ ] ] is the increase to the grand free energy caused by adding an edge @xmath25 . the random network @xmath134 in fig . [ fig : cavity01]c is constructed from the random cavity network @xmath140 of fig . [ fig : cavity01]a by adding two vertices ( @xmath151 and @xmath152 ) . vertex @xmath151 is connected to @xmath43 cavity vertices ( @xmath153 ) of @xmath133 , and vertex @xmath152 is connected to the remaining @xmath43 cavity vertices ( @xmath154 ) of @xmath147 . vertex @xmath151 and @xmath152 are directly connected by a new edge @xmath155 , so that every vertex in the graph @xmath134 has degree @xmath23 . the energy difference between the system @xmath134 and the cavity system @xmath147 is @xmath156 the increase @xmath157 in the free energy of macroscopic state @xmath1 due to the addition of two new vertices and @xmath158 new edges can be obtained following the same procedure as given in sec . [ subsec : field - distribution ] . we find that @xmath159 in eq . ( [ eq : freeenergy03 ] ) , @xmath160 is the free energy increase caused by adding vertex @xmath151 and connecting it to the set @xmath161 of @xmath23 vertices . the explicit expression for @xmath160 is @xmath162 + \prod\limits_{j\in \partial i_0 } \bigl[1- v_{i_0 j } m_j \bigr ] \biggr ) \ . \label{eq : deltafi0}\ ] ] the expression for @xmath163 has the same form as eq . ( [ eq : deltafi0 ] ) . let us emphasize that , in eq . ( [ eq : deltafi0 ] ) , @xmath164 is the cavity magnetization of vertex @xmath152 when vertex @xmath151 is not added , i.e. , @xmath165- \prod\limits_{j_s\in \partial i_0 \backslash j_0 } \bigl [ 1 - v_{i_0 j_s } m_{j_s } \bigr ] } { \prod\limits_{j_s\in \partial i_0 \backslash j_0 } \bigl [ 1 + v_{i_0 j_s } m_{j_s } \bigr ] + \prod\limits_{j_s\in \partial i_0 \backslash j_0 } \bigl [ 1 - v_{i_0 j_s } m_{j_s } \bigr ] } \ , \label{eq : m0val } \\ m_{j_0 } & = & \frac { \prod\limits_{i_s \in \partial j_0 \backslash i_0 } \bigl [ 1 + v_{j_0 i_s } m_{i_s } \bigr]- \prod\limits_{i_s \in \partial j_0 \backslash i_0 } \bigl [ 1 - v_{j_0 i_s } m_{i_s } \bigr ] } { \prod\limits_{i_s \in \partial j_0 \backslash i_0 } \bigl [ 1 + v_{j_0 i_s } m_{i_s } \bigr ] + \prod\limits_{i_s\in \partial j_0 \backslash i_0 } \bigl [ 1 - v_{j_0 i_s } m_{i_s } \bigr ] } \ . \label{eq : m0pval}\end{aligned}\ ] ] the term @xmath166 of eq . ( [ eq : freeenergy03 ] ) is the free energy increase caused by setting up an edge between vertex @xmath151 and vertex @xmath152 ; its expression is given by eq . ( [ eq : freeenergy02 ] ) , with @xmath167 and @xmath164 being determined by eq . ( [ eq : m0val ] ) and eq . ( [ eq : m0pval ] ) , respectively . the free energy increase eq . ( [ eq : freeenergy03 ] ) can be intuitively understood as follows : since the contribution of the edge @xmath168 is counted twice in @xmath160 and @xmath163 , the free energy increase should be corrected with an edge term . with these preparations , we can calculate the total grand free energy @xmath169 of the system @xmath134 . similar to eq . ( [ eq : gfd - i ] ) , we find that @xmath170 in eq . ( [ eq : delta - g-2 - 01 ] ) , @xmath171 is the increase to the grand free energy caused by adding vertex @xmath151 and connecting it to @xmath23 vertices , with @xmath172 where @xmath173 is the cavity magnetization distribution of vertex @xmath152 in the absence of vertex @xmath151 . @xmath174 in eq . ( [ eq : delta - g-2 - 01 ] ) is calculated through eq . ( [ eq : dgij ] ) using @xmath175 and @xmath176 . the grand free energy density of the spin - glass system is @xmath177 an explicit expression for @xmath135 can be written down by applying eq . ( [ eq : gfd - i ] ) and eq . ( [ eq : delta - g-2 - 01 ] ) , which is a function of the quenched randomness in the system . the grand free energy density has the nice property of self - averaging @xcite , namely the value of @xmath69 as calculated for a typical system is equal to the averaged value of @xmath69 over many systems with different realizations of the quenched randomness in the graph connection pattern and in the edge coupling constants . when the quenched randomness is averaged out , we obtain that @xmath178 where @xmath179 and @xmath180 are calculated through eq . ( [ eq : dgi0 ] ) and eq . ( [ eq : dgij ] ) , respectively ; and an overline indicates averaging over the quenched randomness of the spin - glass system . the mean free energy density of a macroscopic state of the system is related to @xmath135 by @xmath181 and the complexity of the system at given value of the reweighting parameter @xmath52 is @xmath182 the discussion in sec . [ sec : densities ] was concerned with the typical properties of an ensemble of spin - glass systems governed by a given distribution of quenched randomness . one important advantage of the cavity approach is that , for a single instance of the quenched randomness , the thermodynamic properties can also be calculated . under the bethe - peierls approximation , the total grand free energy of a spin - glass system eq . ( [ eq : energy01 ] ) on a graph @xmath183 with couplings @xmath184 is equal to @xmath185 it is easy to see that the above equation is consistent with eq . ( [ eq : g - final ] ) . in eq . ( [ eq : gfe - single-01 ] ) , @xmath186 denotes the contribution of vertex @xmath33 and its associated edges to the total grand free energy of the system ; its expression has the same form as eq . ( [ eq : dgi0 ] ) : @xmath187 e^{-y \delta f^{(i ) } } \biggr ) \ , \ ] ] where @xmath188 is the cavity magnetization of vertex @xmath78 ( with respect to vertex @xmath33 ) in a macroscopic state @xmath1 , and @xmath189 is the probability distribution of this cavity magnetization among all the macroscopic states ; @xmath190 is the free energy contribution ( in a given macroscopic state ) of vertex @xmath33 and its associated edges , which has the same form as eq . ( [ eq : deltafi0 ] ) but with @xmath89 replaced by @xmath188 . similarly , the term @xmath191 in eq . ( [ eq : gfe - single-01 ] ) is the contribution of an edge @xmath25 to the total grand free energy of the system ; it is expressed as @xmath192 where @xmath193 is given by eq . ( [ eq : freeenergy02 ] ) but with @xmath119 replaced by @xmath194 and @xmath89 replaced by @xmath188 . the set of @xmath195 ( @xmath196 being the total number of edges in the graph @xmath183 ) probability distributions @xmath189 in eq . ( [ eq : gfe - single-01 ] ) should be carefully chosen such that the total grand free energy achieves a minimal value at given @xmath197 values . in other words , for any edge @xmath25 of the graph @xmath183 , the variation of @xmath198 with respect to both @xmath199 and @xmath200 should vanish : @xmath201 equation ( [ eq : gfe - single-04 ] ) results in the following self - consistent bethe - peierls equation for the @xmath202 s : @xmath203 e^{-y \delta f_j^{(i ) } } \delta\biggl ( m_{i\to j}-\frac { \prod\limits_{k\in \partial i \backslash j } [ 1 + v_{i k } m_{k\to i } ] - \prod\limits_{k\in \partial i \backslash j } [ 1 - v_{i k } m_{k\to i } ] } { \prod\limits_{k\in \partial i\backslash j } [ 1 + v_{i k } m_{k\to i}]+ \prod\limits_{k\in \partial i\backslash j } [ 1 - v_{i k } m_{k\to i } ] } \biggr ) \ , \ ] ] where @xmath204 + \prod\limits_{k\in \partial i\backslash j } \bigl[1- v_{i k } m_{k\to i } \bigr ] \biggr ) \ .\ ] ] @xmath205 has the same physical meaning as @xmath121 of sec . [ subsec : field - distribution ] . equation ( [ eq : gfe - single-06 ] ) is consistent with eq . ( [ eq : rho ] ) of sec . [ subsec : field - distribution ] . similar to eq . ( [ eq : f - final ] ) and eq . ( [ eq : sigma - final ] ) , the mean free energy of a macroscopic state of the sample is expressed as @xmath206 and the complexity of the system is calculated through @xmath207 because of eq . ( [ eq : gfe - single-04 ] ) , the first derivative of @xmath208 with respect to @xmath52 can easily be expressed . we find that eq . ( [ eq : fe - single-02 ] ) can be re - written as @xmath209 where @xmath210 e^{-y \delta f^{(i ) } } \delta f^{(i ) } } { \prod\limits_{j\in \partial i } \bigl[\int { \rm d } m_{j\to i } { \cal p}_{j\to i}(m_{j\to i } ) \bigr ] e^{-y \delta f^{(i ) } } } \ , \label{eq : average - dfi } \\ \bigl\langle \delta f^{(i j ) } \bigr\rangle & = & \frac{\int { \rm d } m_{j\to i } \int { \rm d } m_{i\to j } { \cal p}_{j\to i}(m_{j\to i } ) { \cal p}_{i\to j}(m_{i\to j } ) e^{-y \delta f^{(i j ) } } \delta f^{(i j ) } } { \int { \rm d } m_{j\to i } \int { \rm d } m_{i\to j } { \cal p}_{j\to i}(m_{j\to i}){\cal p}_{i\to j}(m_{i\to j } ) e^{-y \delta f^{(i j ) } } } \ . \label{eq : average - dfij}\end{aligned}\ ] ] the mean free energy of a macroscopic state is also decomposed into vertex contributions and edge contributions . according to the discussion in sec . [ sec : mfep ] , in the mean - field theory , the reweighting parameter @xmath52 should be chosen such that its value is closest to the physical inverse temperature @xmath72 . therefore , at a given value of @xmath72 , the mean free energy @xmath211 of the system is obtained by setting @xmath57 in eq . ( [ eq : fe - single-02 ] ) or eq . ( [ eq : free - energy-1rsb ] ) , provided that the complexity @xmath212 is always non - negative in the range of @xmath213 . if , on the other hand , @xmath212 changes from being positive to being negative at a point of @xmath214 , we should set @xmath52 to @xmath215 . in this later situation , the mean free energy of the system is equal to the maximal value of the grand free energy , @xmath216 . after the mean free energy @xmath211 of the spin - glass system is obtained , the mean energy of a macroscopic state is calculated through @xmath217 and the mean entropy of a macroscopic state is calculated through @xmath218 ensemble - averaged densities of mean energy and mean entropy can also be obtained from eqs . ( [ eq : mean - energy ] ) and ( [ eq : mean - entropy ] ) by averaging out the quenched randomness . an alternative derivation for the mean energy density is given in the appendix . we have applied the mean - field treatments of sec . [ sec : cavity - field - distribution ] and sec . [ sec : densities ] to the @xmath3 spin - glass model eq . ( [ eq : energy01 ] ) on a random regular graph of vertex degree @xmath4 . ( in what follows , energy is in units of @xmath219 . ) at temperature @xmath220 ( i.e. , @xmath221 ) , carrus , marinari and zuliani ( as cited in ref . @xcite ) estimated that the energy density of this system is @xmath222 . the mean - field theory of ref . @xcite also predicted the same result . this reference also predicted a mean - free energy density of @xmath223 . in the present population dynamics simulation , each probability profile @xmath120 in eq . ( [ eq : rho ] ) was represented by an vector of @xmath224 elements . a total number of @xmath128 such probability profiles were stored in the computer and they were updated using eq . ( [ eq : rho ] ) . in our work , various @xmath128 values ranging from @xmath225 to @xmath226 were used . in each elementary update of the probability distribution @xmath120 , a metropolis importance sampling @xcite with inverse temperature @xmath52 was exploited to generate a total number of @xmath227 magnetizations @xmath119 . these magnetizations were stored in an ordered set . the averaged value of the first ( second , third , @xmath228 ) @xmath229 magnetizations in this set is assigned to the first ( second , third,@xmath228 ) element of the new @xmath129 . we chose @xmath230 in our simulations . the grand free energy density @xmath69 of the @xmath3 spin - glass model on a random regular graph of vertex degree @xmath4 . the inverse temperature is fixed to @xmath221 . symbols are results obtained by population dynamics simulations using a population size of @xmath231 . the dashed line is a fit to the data using @xmath232 , with the fitting parameters @xmath233 and @xmath234 . ] mean free energy density of the @xmath3 spin - glass model on a random regular graph of vertex degree @xmath4 at @xmath221 as obtained by population dynamics with different population size @xmath128 . ( inset ) the optimal reweighting parameter @xmath235 . ] the energy density , free energy density , entropy density , and complexity as a function of the reweighting parameter @xmath52 for the @xmath3 spin - glass model on a random regular graph of vertex degree @xmath4 at inverse temperature @xmath221 . energy unit is @xmath219 . the symbols represent numerical population dynamics results using @xmath236 . the dashed line in the bottom panel is a fitting to the numerical data with @xmath237 in the range of @xmath238 . to reduce computational time , for the simulations reported in this figure , a faster but less precise numerical recipe was used to generate a new probability profile @xmath120 using eq . ( [ eq : rho ] ) . this might be the reason for the relatively large fluctuations of the mean densities with the reweighting parameter @xmath52 . ] the grand free energy density @xmath69 of the system as calculated according to eq . ( [ eq : g - final ] ) is shown in fig . [ fig : gy ] . the grand free energy density @xmath239 first increases with the reweighting parameter @xmath52 till @xmath52 reaches @xmath234 , at which point @xmath135 attains a maximal value of @xmath240 . this maximal value of @xmath135 corresponds to the best estimate of the mean free energy density of the system by the present mean - field method . it is in close agreement with the prediction of ref . we have checked that the estimated mean free energy density value is not sensitive to the population size @xmath128 ( see fig . [ fig : mfd ] ) . we have also calculated the mean energy density and mean entropy density for the model system at @xmath221 using a population size of @xmath241 , with the mean energy density being @xmath242 and the mean entropy density being @xmath243 ( see fig . [ fig : result01 ] ) . these predictions are also in good agreement with the numerical results of ref . @xcite and with the simulational work of carrus , marinari and zuliani . these good agreements of the present theoretical results with earlier simulational and numerical work suggest that the present cavity approach based on the concept of cyclic heating and annealing is feasible . as a summary , in this paper we have calculated the thermodynamic properties of a spin - glass model by combining the physical idea of repeated heating - annealing and the cavity approach of mzard and parisi @xcite . we have assumed that , during a cyclic annealing experiment , all the thermodynamic ( or macroscopic ) states of the spin - glass system at a given low temperature @xmath0 will be reachable , but with different frequencies which decrease exponentially with the free energy values of the macroscopic states . by using this free energy boltzmann distribution and by using the bethe - peierls approximation , the grand free energy of the spin - glass system can be calculated as a function of a reweighting parameter @xmath52 ; and from the knowledge of the grand free energy , the mean free energy , energy , and entropy of a macroscopic state of the system can also be obtained . for the @xmath3 spin - glass model on a random regular graph of vertex degree @xmath4 , the theoretical predictions of the present work are in good agreement with the results of earlier simulational and numerical calculations . for the @xmath3 model at @xmath221 , we found that the complexity @xmath75 becomes negative before @xmath52 reaches @xmath72 . therefore , the thermodynamic properties of the system are contributed by those macroscopic states which have the global minimal free energy density . however , it may exist other systems for which the complexity is still positive even when the reweighting parameter @xmath52 reaches @xmath72 . for such systems , the thermodynamic properties are not determined by those macroscopic states of the minimal free energy density , but by a set of @xmath29metastable macroscopic states . on the one hand , these metastable macroscopic states have a higher free energy density ; on the other hand , the number of such macroscopic states greatly exceeds the number of macroscopic states of the global minimal free energy density . in the competition between these two factors , the @xmath29metastable macroscopic states may win . then the system will reach one of these metastable macroscopic states almost surely in each round of the temperature annealing experiment . we will work on a model system with many - body interactions to check whether this is really the case . as in the work of ref . @xcite , the present theoretical treatment also assumes that the configurational space of the spin - glass system breaks into exponentially many macroscopic states , but there are no further organizations of these macroscopic states ; and it is assumed that each macroscopic state is ergodic . as pointed out in ref . @xcite , this first - step replica - symmetry - broken ( 1rsb ) cavity solution might be unstable . two types of instability is conceivable . the type i instability concerns with possible clustering of macroscopic states into @xmath29super - macroscopic states ; the type ii instability is caused by splitting of each macroscopic state into many @xmath29sub - macroscopic states . to account for the further clustering of macroscopic states appears to be relatively easy in the present mean - field framework : for each cluster of macroscopic states , a vertex has a probability profile concerning its cavity magnetization ; this probability profile is different in different clusters , and we can introduce a probability distribution of probability profiles to characterize this variation [ h. zhou , submitted to comm . ( beijing ) ] . on the hand , to take into consideration the possibility of splitting of a macroscopic state , maybe one has to introduce other reweighting parameters in the mean - field theory . this is an important issue waiting for further explorations . another important issue is related to the updating of the probability profiles through the iterative equation ( [ eq : rho ] ) or ( [ eq : gfe - single-05 ] ) . in the present paper , we used metropolis importance sampling technique to get an updated probability profile of cavity magnetization . this method appears to be rather precise but time - consuming . if faster algorithms with comparable precision could be constructed , it will be highly desirable . the numerical simulations reported in this paper were carried out at the pc clusters of the state key laboratory for scientific and engineering computing , cas , china . in the main text we did not give an explicit formula for the mean energy density @xmath244 of a spin - glass system . instead , the mean energy density was calculated from first knowing the mean free energy density of the system [ see , e.g. , eq . ( [ eq : mean - energy ] ) ] . in this appendix , we derive an explicit expression for the mean energy density @xmath244 . the assumptions used in this derivation will be clearly pointed out . consider again the two systems , @xmath133 in fig . [ fig : cavity01]a and @xmath24 in fig . [ fig : cavity01]b . ( for notational simplicity , let us hereafter referr to these two systems as system a and b , respectively . ) as was mentioned in sec . [ subsec : nton ] , for a given configuration the energy difference between these two systems is @xmath139 . when averaged over all macroscopic states , the mean total energy of the system a is @xmath245 where @xmath246 is the total free energy of macroscopic state @xmath1 of system a , and @xmath247 is the the configurational energy of system a. similarly , the mean total energy of the system b is @xmath248 \exp\bigl(-\beta h^a(\vec{\sigma } ) - \beta \delta h_2 \bigr ) } { \sum_{\vec{\sigma}\in \alpha } \exp\bigl(-\beta h^a(\vec{\sigma } ) - \beta \delta h_2 \bigr ) } } { \sum\limits_\alpha \exp\bigl(-y f_\alpha^a - y \delta f_2 \bigr ) } \ , \ ] ] where @xmath249 is defined by eq . ( [ eq : freeenergy01 ] ) . the difference of these two mean energy expressions is @xmath250 in the above equation , @xmath251 where @xmath252 in going from eq . ( [ eq : v1m1 ] ) to eq . ( [ eq : v1 ] ) we have used the bethe - peierls approximation both at the level of microscopic configurations [ eq . ( [ eq : factorization ] ) ] and at the level of macroscopic states [ eq . ( [ eq : factorization - magnetization ] ) ] . 1 . the distribution of the configurational energy of the cavity system a and that of the energy increase can be regarded as mutually independent within a macroscopic state . 2 . the distributions of the free energy of the cavity system a and that of the free energy increase can also be regarded as mutually independent . now consider the two systems , @xmath133 in fig . [ fig : cavity01]a ( system a ) and @xmath134 in fig . [ fig : cavity01]c ( system c ) . following the same analytical procedure as given in the preceding subsection , we find that , if the two assumptions ( i ) and ( ii ) listed above are valid when going from system a to system c , then the mean total energy of system c is related to that of system a by the following simple equation @xmath257 ( notice that in calculating @xmath258 through eq . ( [ eq : deij ] ) , @xmath167 should be the cavity magnetization of vertex @xmath151 in the absence of vertex @xmath259 . ) . the same analytical procedure can be applied to calculate the mean free energy density . under the assumption ( ii ) of the preceding subsection , we obtain that the mean free energy density @xmath64 is equal to @xmath261 which is just the same as eq . ( [ eq : free - energy-1rsb ] ) . based on the assumption of exponentially many macroscopic states , it was argued in ref . @xcite that , the assumption ( ii ) of the preceding subsection should be valid under the bethe - peierls approximation . in other words , the joint distribution of the free energy of the old cavity system @xmath14 and the free energy increase @xmath262 is factorized . the consistence between eq . ( [ eq : f - another - final ] ) and eq . ( [ eq : free - energy-1rsb ] ) also confirmed this point . since each macroscopic state ifself contains an exponential number of microscopic configurations , we believe that the assumption ( i ) is also valid in the bethe - peierls mean - field framework . the mean energy density of the @xmath3 spin - glass model as shown in fig . [ fig : result01 ] was calculated using eq . ( [ eq : epsilon - final ] ) , and the output was in good agreement with earlier simulation result . maybe one can further check this point by working on some simple models which are exactly solvable both analytically and computationally . the maximum matching problem as studied in refs . @xcite could be a good candidate model system .

at sufficiently low temperatures , the configurational phase space of a large spin - glass system breaks into many separated domains , each of which is referred to as a macroscopic state . the system is able to visit all spin configurations of the same macroscopic state , while it can not spontaneously jump between two different macroscopic states . ergodicity of the whole configurational phase space of the system , however , can be recovered if a temperature - annealing process is repeated an infinite number of times . in a heating - annealing cycle , the environmental temperature is first elevated to a high level and then decreased extremely slowly until a final low temperature @xmath0 is reached . different macroscopic states may be reached in different rounds of the annealing experiment ; while the probability of finding the system in macroscopic state @xmath1 decreases exponentially with the free energy @xmath2 of this state . for finite - connectivity spin glass systems , we use this free energy boltzmann distribution to formulate the cavity approach of mzard and parisi [ eur . phys . j. b * 20 * , 217 ( 2001 ) ] in a slightly different form . for the @xmath3 spin - glass model on a random regular graph of degree @xmath4 , the predictions of the present work agree with earlier simulational and theoretical results .

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Proceed to summarize the following text: exclusive @xmath19 decays proceed dominantly via a @xmath20 tree process and can be used to determine @xmath18 , one of the smallest and least known elements of the cabibbo - kobayashi - maskawa matrix @xcite . however , the need to translate the observed rate to a @xmath18 value using model - dependent decay form - factors ( ff ) has resulted in large theoretical uncertainties . the recent release of ff results for @xmath17 calculated by unquenched lattice qcd ( lqcd ) @xcite makes possible the first model - independent determination of @xmath18 . since lqcd results are available only in the high @xmath12 region ( @xmath21gev@xmath16 ) , a clean measurement of the partial @xmath17 branching fraction in the same high @xmath12 region is needed . there have been several measurements in the past by cleo , babar and belle for the @xmath17 , @xmath22 , @xmath23 and @xmath24 modes @xcite . the analyses in these measurements utilize the method , originally developed by cleo , where the @xmath6 decays are reconstructed by inferring the undetected neutrino mass from missing energy and momentum ( `` @xmath25-reconstruction method '' ) @xcite . in the @xmath6-factory era , we will improve the statistical precision by simply applying the @xmath25-reconstruction method using a large amount of data . however , the poor signal - to - noise ratio will limit the systematic uncertainty of the measurement . in this paper we present measurements of @xmath26 and @xmath27 decays using @xmath28 decay tagging . we reconstruct the entire decay chain from the @xmath4 , @xmath29 , @xmath30 and @xmath31 with several @xmath32 sub - modes . the back - to - back correlation of the two @xmath6 mesons in the @xmath4 rest frame allows us to constrain the kinematics of the double semileptonic decay . the signal is reconstructed in four modes , @xmath26 and @xmath27 . yields and branching fractions are extracted from a simultaneous fit of the @xmath33 and @xmath34 samples in three intervals of @xmath12 , accounting for cross - feed between modes as well as other backgrounds . we have applied this method to @xmath35 decays for the first time , and have succeeded in reconstructing these decays with significantly improved signal - to - noise ratios compared to the @xmath25-reconstruction method . inclusion of charge conjugate decays is implied throughout this paper . the analysis is based on data recorded with the belle detector at the kekb collider operating at the center - of - mass ( c.m . ) energy for the @xmath4 resonance @xcite . the @xmath4 dataset that is used corresponds to an integrated luminosity of 253 fb@xmath36 and contains @xmath2 @xmath3 events . the belle detector is a large - solid - angle magnetic spectrometer that consists of a silicon vertex detector ( svd ) , a 50-layer central drift chamber ( cdc ) , an array of aerogel threshold erenkov counters ( acc ) , a barrel - like arrangement of time - of - flight scintillation counters ( tof ) , and an electromagnetic calorimeter comprised of csi(tl ) crystals ( ecl ) located inside a super - conducting solenoid coil that provides a 1.5 t magnetic field . an iron flux - return located outside of the coil is instrumented to detect @xmath37 mesons and to identify muons ( klm ) . the detector is described in detail elsewhere @xcite . two inner detector configurations were used . a 2.0 cm beam pipe and a 3-layer silicon vertex detector was used for the first sample of @xmath38 @xmath39 pairs , while a 1.5 cm beam pipe , a 4-layer silicon detector , and a small - cell inner drift chamber were used to record the remaining 123 million @xmath39 pairs @xcite . a detailed monte carlo ( mc ) simulation , which fully describes the detector geometry and response and is based on geant @xcite , is applied to estimate the signal detection efficiency and to study the background . to examine the ff dependence , mc samples for the @xmath35 signal decays are generated with different form - factor models : a quark model ( isgw ii @xcite ) , light cone sum rules ( lcsr for @xmath17 @xcite and @xmath40 @xcite ) and quenched lattice qcd ( ukqcd @xcite ) . we also use unquenched lattice qcd ( fnal @xcite and hpqcd @xcite ) for @xmath17 and a relativistic quark model ( melikhov @xcite ) for @xmath40 . to model the cross - feed from other @xmath19 decays , mc samples are generated with the isgw ii model for the resonant components ( @xmath17 and @xmath40 components are excluded in this sample ) and the defazio - neubert model @xcite for the non - resonant component . to model the @xmath39 and continuum backgrounds , large generic @xmath39 and @xmath41 monte carlo ( based on evtgen @xcite ) samples are used . charged particle tracks are reconstructed from hits in the svd and the cdc . they are required to satisfy track quality cuts based on their impact parameters relative to the measured interaction point ( ip ) of the two beams . charged kaons are identified by combining information on ionization loss ( @xmath42 ) in the cdc , herenkov light yields in the acc and time - of - flight measured by the tof system . for the nominal requirement , the kaon identification efficiency is approximately @xmath43 and the rate for misidentification of pions as kaons is about @xmath44 . hadron tracks that are not identified as kaons are treated as pions . tracks satisfying the lepton identification criteria , as described below , are removed from consideration . neutral pions are reconstructed using @xmath45 pairs with an invariant mass between 117 and 150mev/@xmath46 . each @xmath45 is required to have a minimum energy of @xmath47 mev . @xmath48 mesons are reconstructed using pairs of tracks that are consistent with having a common vertex and that have an invariant mass within @xmath49mev/@xmath46 of the known @xmath48 mass . electron identification is based on a combination of @xmath42 in the cdc , the response of the acc , shower shape in the ecl and the ratio of energy deposit in the ecl to the momentum measured by the tracking system . muons are identified by their signals in the klm resistive plate counters , which are interleaved with the iron of the solenoid return yoke . the lepton identification efficiencies are estimated to be about 90% for both electrons and muons in the momentum region above 1.2gev/@xmath50 , where leptons from prompt @xmath6 decays dominate . the hadron misidentification rate is measured using reconstructed @xmath51 and found to be less than 0.2% for electrons and 1.5% for muons in the same momentum region . for the reconstruction of @xmath31 , the lepton candidate is required to have the correct sign charge with respect to the @xmath52 meson flavor and a laboratory momentum ( @xmath53 ) greater than 1.0gev/@xmath50 . the @xmath52 meson candidates are reconstructed by using seven decay modes of @xmath54 : @xmath55 , @xmath56 , @xmath57 , @xmath58 , @xmath59 , @xmath60 , @xmath61 ; and ten decay modes of @xmath62 : @xmath63 , @xmath64 , @xmath65 , @xmath66 , @xmath67 , @xmath68 , @xmath69 , @xmath70 , @xmath71 , @xmath72 . the candidates are required to have an invariant mass @xmath73 within @xmath74 ( @xmath75 is a standard deviation ) of the nominal @xmath52 mass , where the mass resolution @xmath75 is dependent on the decay mode . @xmath76 mesons are reconstructed in the modes @xmath77 , @xmath78 and @xmath79 by combining a @xmath52 meson candidate and a charged or neutral pion . each @xmath80 candidate is required to have a mass difference @xmath81 within @xmath82 of the nominal values . for the reconstruction of @xmath83 , the lepton candidate is required to have the right sign charge with respect to the @xmath84 system and @xmath53 greater than 0.8 gev/@xmath50 . the @xmath84 system may consist of one pion or two pions ( @xmath85 or @xmath86 for a @xmath87 tag and @xmath88 or @xmath89 for a @xmath90 tag ) . the event is required to have no additional charged tracks or @xmath91 candidates . we also require that the residual energy from neutral clusters be less than 0.15 gev ( @xmath92gev ) . the two leptons on the tag and the signal sides are required to have opposite charge . the loss of signal due to @xmath93 mixing is estimated by mc simulation . we then impose a constraint based on the kinematics of the double semileptonic decay in the @xmath4 rest frame . in the semileptonic decay on each side , @xmath94 ( @xmath95 and @xmath96 ) , the angle between the @xmath97 meson and the detected @xmath98 system @xmath99 is calculated from the relation , @xmath100 ( @xmath101 : 4-momentum vector ) and the known @xmath102 ( the absolute momentum of the mother @xmath6 meson ) . this means that the @xmath97 direction is constrained on the surface of a cone defined with the angle @xmath103 around the direction of the @xmath98 system , as shown graphically in fig . [ fig : double_cone ] . the back - to - back relation of the two @xmath6 meson directions then implies that the real @xmath6 direction is on the intersection of the two cones when one of the @xmath6 systems is spatially inverted . denoting @xmath104 the angle between the @xmath105 and @xmath106 , the @xmath6 directional vector @xmath107 is given by , @xmath108 , @xmath109 , and @xmath110 with the coordinate definition in fig . [ fig : double_cone ] , where the @xmath105 and @xmath111 are aligned along the @xmath112-axis and in the @xmath113 plane , respectively . if the hypothesis of the double semileptonic decay is correct and all the decay products are detected except for the two neutrinos , @xmath114 must range from 0 to 1 . events passing a rather loose cut @xmath115 are used for signal extraction at a later stage of the analysis . since the direction of the @xmath6 meson is not uniquely determined , we calculate , @xmath12 as @xmath116 , using the beam energy ( @xmath117 ) , energy ( @xmath118 ) and momentum ( @xmath119 ) of the @xmath84 system and neglecting the momentum of the @xmath6 meson in the c.m . system . the signal monte carlo simulation finds that the @xmath12 resolution depends on the reconstructed @xmath12 and is in the range 0.32 - 0.95gev@xmath16 . according to monte carlo simulation , the largest backgrounds originate from @xmath120 and non - signal @xmath19 decays , where some particles escape detection . there are sizable contributions from cross talk between the @xmath87 and @xmath34 tags . the contribution from @xmath41 processes is found to be negligible . for events selected as described above , the signal mc simulation indicates that the total detection efficiency ( @xmath121 ) , averaged over electron and muon channels , is @xmath122 for @xmath123 and @xmath124 for @xmath125 , @xmath126 for @xmath127 and @xmath128 for @xmath129 assuming the lcsr ff model . here , @xmath121 is defined with respect to the number of @xmath3 pairs , where one @xmath6 decays into the signal mode , and includes the loss of signal due to @xmath93 mixing . because of the loose lepton momentum cut ( @xmath53 @xmath130gev/@xmath50 ) , the variation of efficiency with different ff models is relatively small . table [ tbl : matrix_sig ] gives the matrix @xmath131 , the efficiency for a signal event generated with true @xmath12 in the bin @xmath132 to be reconstructed in the bin @xmath133 . table [ tbl : summary_vub ] summarizes the results , where the first and second errors are the experimental statistical and systematic errors , respectively . the third error is based on the error on @xmath135 quoted by the lqcd authors . these theoretical errors are asymmetric because we assign them by taking the variation in @xmath18 when @xmath135 is varied by the quoted errors . the values are in agreement with those from inclusive @xmath136 decays @xcite . to summarize , we have measured the branching fractions of the decays @xmath137 and @xmath138 in @xmath139 @xmath140 events using a method which tags one @xmath6 in the mode @xmath141 . our results are consistent with previous measurements , and their precision is comparable to that of results from other experiments . the ratios of results for neutral and charged @xmath6 meson modes are found to be consistent with isospin . the partial rates are measured in three bins of @xmath12 and compared with distributions predicted by several theories . from the rate in the region @xmath142 gev@xmath16 and recent results from lqcd calculations , we extract @xmath18 : @xmath143 the experimental precision on these values is 13% , currently dominated by the statistical error of 11% . by accumulating more integrated luminosity , a measurement with errors below 10% is feasible . with improvements to unquenched lqcd calculations , the present method may provide a precise determination of @xmath18 . we thank the kekb group for the excellent operation of the accelerator , the kek cryogenics group for the efficient operation of the solenoid , and the kek computer group and the national institute of informatics for valuable computing and super - sinet network support . we acknowledge support from the ministry of education , culture , sports , science , and technology of japan and the japan society for the promotion of science ; the australian research council and the australian department of education , science and training ; the national science foundation of china and the knowledge innovation program of chinese academy of sciencies under contract no . 10575109 and ihep - u-503 ; the department of science and technology of india ; the bk21 program of the ministry of education of korea , and the chep src program and basic research program ( grant no . r01 - 2005 - 000 - 10089 - 0 ) of the korea science and engineering foundation ; the polish state committee for scientific research under contract no . 2p03b 01324 ; the ministry of science and technology of the russian federation ; the slovenian research agency ; the swiss national science foundation ; the national science council and the ministry of education of taiwan ; and the u.s.department of energy . h. kakuno _ et al . _ ( belle collaboration ) , phys . lett * 92 * , 071802 ( 2004 ) ; the inclusive branching fraction used in the fitting is based on the partial branching fraction @xmath144gev/@xmath145 gev@xmath16 ) and a calculation of @xmath146 shown in hep - ex/0408115 .

we report measurements of the charmless semileptonic decays @xmath0 and @xmath1 , based on a sample of @xmath2 @xmath3 events collected at the @xmath4 resonance with the belle detector at the kekb @xmath5 asymmetric collider . in this analysis , the accompanying @xmath6 meson is reconstructed in the semileptonic mode @xmath7 , enabling detection of the signal modes with high purity . we measure the branching fractions @xmath8 , @xmath9 , @xmath10 and @xmath11 , where the errors are statistical , experimental systematic , and systematic due to form - factor uncertainties , respectively . for each mode we also present the partial branching fractions in three @xmath12 intervals : @xmath13 , @xmath14 , and @xmath15gev@xmath16 . from our partial branching fractions for @xmath17 and recent results for the form factor from unquenched lattice qcd calculations , we obtain values of the ckm matrix element @xmath18 . 2.cm -1.5 cm belle prerpint 2006 - 10 + kek preprint 2006 - 4 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and , semileptonic , b decay , exclusive 12.15.hh , 12.38.gc , 13.25.hw

You are an expert at summarizing long articles. Proceed to summarize the following text: the optical or conductivity properties of generally anisotropic material are described by ( generally complex ) permittivity @xmath11 or conductivity @xmath10 tensor , being mutually related by ( in si units ) @xmath13 , where @xmath14 being kronecker delta and @xmath15 light frequency ( i.e. @xmath16 corresponds to d.c . current ) . onsager relations require antisymmetry in magnetization direction , @xmath17 . as various magnetization direction causes small perturbation to the crystal optical properties , @xmath11 , they can be expressed as taylor series of permittivity tensor in direction of @xmath18 @xcite @xmath19 where superscripts denote zeroth , first , second etc . order in magnetization . optical permittivity independent on magnetization direction is @xmath20 and in case of cubic crystal , it has unitary form @xmath21 . the term linear in magnetization @xmath22 is described by third - order tensor @xmath23 , where onsager relations assures @xmath24 . in case of cubic crystals , it has simple form @xmath25 , where @xmath26 is levi - civita symbol and @xmath27 is linear magnetooptic element . the term quadratic in magnetization @xmath28 is described by fourth order tensor @xmath29 , with onsager relations assuring @xmath30 . in case of cubic crystal with @xmath31 $ ] , @xmath32 $ ] , @xmath33 $ ] , the @xmath29 tensor can be written in compact form as @xcite @xmath34 = \left [ \begin{array}{cccccc } g_{11 } & g_{12 } & g_{21 } & 0 & 0 & 0 \\ g_{21 } & g_{11 } & g_{12 } & 0 & 0 & 0 \\ g_{12 } & g_{21 } & g_{11 } & 0 & 0 & 0 \\ 0 & 0 & 0 & 2g_{44 } & 0 & 0 \\ 0 & 0 & 0 & 0 & 2g_{44 } & 0 \\ 0 & 0 & 0 & 0 & 0 & 2g_{44 } \end{array } \right ] \ , \left [ \begin{array}{c } m_x^2 \\ m_y^2 \\ m_z^2\\ m_ym_z\\ m_zm_x\\ m_xm_y \end{array } \right]\ ] ] in order to express @xmath35-elements for crystals with cubic symmetry , the elements can be rearranged defining @xmath36 and @xmath37 . note that alternative way used to express anisotropy of the permittivity is @xmath38 @xcite . in case of cubic crystals owning point o@xmath39 symmetry , @xmath40 and hence @xmath41 , @xmath42 @xcite . in case of isotropic medium , then both @xmath43 , @xmath42 . furthermore , notice , that spectra of @xmath44 element can not be determined ( only value of @xmath45 ) as for the ferromagnetic materials , the magnitude of magnetization vector @xmath46 is constant for any magnetization direction and hence @xmath44 becomes inseparable part of @xmath47 . hence , spectra of any permittivity element @xmath11 ( up to second order in magnetization ) can be expressed as weighted sum of principal spectra , where the weights are determined by crystallographic and magnetization directions . table [ t : scans ] demonstrates those weights for @xmath48 $ ] , @xmath32 $ ] and @xmath33 $ ] oriented crystal for two magnetization scans , used later in ab - initio calculations . namely , the angular dependence of @xmath11 permittivity elements is expressed for two magnetization scans , in - plane ( @xmath49 , @xmath50 ) and out - of - plane ( @xmath51 , @xmath52 ) scans as the magnetization passes through all high symmetry directions [ 001 ] , [ 110 ] and [ 111 ] . angles @xmath53 , @xmath54 define magnetization direction , being @xmath55 = [ \sin\theta \cos\varphi , \sin\theta \sin\varphi , \cos\theta ] $ ] . see ref . @xcite providing weighting factors of principal spectra for general magnetization direction and several crystallographic orientations . cc in - plane scan : @xmath56 , @xmath57 : & out - of - plane scan : @xmath58 , @xmath59 : + [ cols="^,^",options="header " , ] to determine all principal spectra we first separate linear and quadratic contributions , by extracting off - diagonal linear in magnetization ( @xmath60 , @xmath61 ) , off - diagonal quadratic in magnetization ( @xmath62 , @xmath61 ) and diagonal quadratic in magnetization ( @xmath63 , @xmath61 ) contributions from which the principal spectra are determined . the examples of angular dependencies of those contributions are demonstrated in fig . [ f : epsdiff ] for fe at photon energies 1.3ev and 56ev . in figs . [ f : specfe][f : specni ] we show principal spectra of @xmath47 , @xmath27 , @xmath64 , @xmath2 for bcc fe , fcc co and fcc ni . as those materials own point o@xmath39 symmetry @xmath65 . the parts of the spectra related with light absorption are shown in first four lines , whereas last four lines are related to light dispersion ( determined by kramers - kronig relations @xcite ) . two approaches to determine principal spectra are used . ( i ) we determine principal spectra from ab - initio calculated @xmath11 for in - plane and out - of - plane magnetization scans . for both scans , the angular dependences as predicted by symmetry arguments for cubic crystals are presented in tab . [ t : scans ] . knowing those dependences we can extract all principal spectra . two pairs of spectra of @xmath45 and @xmath2 are determined , each pair from one scan . for computational details , see appendix [ app : comp ] . ( ii ) we determine principal spectra solely from @xmath11 calculated for magnetization pointing in fundamental symmetry directions such as [ 100 ] , [ 110 ] or [ 111 ] being sufficient to determine all principal spectra . [ t : fund ] summarizes analytical expressions of principal spectra from known permittivity elements with magnetization at fundamental magnetization directions . two pairs of spectra are determined , spectra of @xmath45 from [ 100 ] and [ 110 ] magnetization directions , and spectra of @xmath2 from [ 110 ] and [ 111 ] magnetization directions . obviously , when calculated angular dependences follow relations predicted by symmetry arguments , and when we neglect contributions of higher order effects , both approaches providing four pairs of spectra should be identical . all those principal spectra are shown in figures [ f : specfe]-[f : specni ] for @xmath45 and @xmath2 . one can see that although all elements and spectral ranges were calculated from equal electronic structure , there is very good agreement between all approaches in case of 2@xmath0- and 3@xmath0-edges spectral range . on the other hand , agreement between @xmath35-spectra in visible range is much smaller ( particularly visible for @xmath45 in fe ) . this can be also seen on fig . [ f : epsdiff ] , where demonstrated example ( fe at 1.3ev ) where , in case of both in - plane scan and out - of - plane scans , the values of e.g. @xmath66 or @xmath67 are shifted . the discrepancy stems from the numerical precision of the brillouin zone integration technique used to calculate optical properties for photon energies below about 12ev . for more details see the appendix [ app : comp ] . finally , recall , for isotropic materials @xmath68 . however , in 2@xmath0-edge and 3@xmath0-edge spectra , the peaks of @xmath2 and @xmath64 corresponding to l@xmath4 and m@xmath4 edges have roughly opposite signs , demonstrating very strong ( dominating ) anisotropy of quadratic effects in cubic bcc fe , fcc co and fcc ni in their respective spectral ranges . very strong anisotropy signal in xmld is demonstrated and analysed by both experimental and theoretical calculations at both 2@xmath0-edge and 3@xmath0-edge in case of the bcc fe and fcc co @xcite and compare the corresponding experimental spectral shapes with spectral shape of here predicted g - tensor elements ( @xmath69 ) related to the quadratic contributions of magneto - optical effects . note also the determined im@xmath70 , element related to the linear magneto - optical effects ( middle column of fig . [ f : specfe ] ) has the same spectral shape as recently measured t - moke spectra of bcc fe at the 3@xmath0 edges @xcite . furthermore , fig . [ f : specfe ] demonstrates ( visible particularly for 3@xmath0-edges ) that light absorption @xmath71 has form of sum of lorentzian peaks . then the absorption of linear element @xmath27 is roughly proportional to the first derivation of @xmath47 : @xmath72 , and quadratic elements are approximately proportional to second derivative of @xmath47 : @xmath73 @xcite . notice also different spectral features in both @xmath35-spectra , e.g. , in case of fe at 4.7ev ( or ni at 1.9ev ) , there is a strong peak in @xmath64 spectra , but no such a feature in @xmath2 spectra . similar can be told about feature for ni at 0.4ev , presented only in @xmath2 spectra and absent in @xmath64 spectra . finally notice that for ( 001 ) oriented crystals , ( a ) the real / imaginary part of @xmath64 is proportional to magnetic linear birefringence / dichroism ( mlb / mld ) , respectively , with @xmath74 $ ] . namely , it corresponds to difference of light speed / absorption for @xmath75 $ ] and @xmath76 $ ] . similarly , @xmath2 is proportional to mlb / mld with @xmath77 $ ] , i.e. @xmath78 $ ] and @xmath79 $ ] . once more let us stress that spectral shapes of g - tensors can be directly compared with recorded xmld spectra w.r.t . various m and e vectors of refs . this can be seen also seen on expressions of @xmath80 and @xmath2 using kubo formula ( [ eq : sigs][eq : sig44 ] ) below . ( b ) in case of both @xmath18 and @xmath81 vectors laying in ( 001)-plane ( i.e. in - plane magnetization and normal light incidence ) , the isotropic part of mlb / mld , is proportional to the term @xmath82 , whereas the anisotropic part of mlb / mld ( i.e. varying with crystallographic orientation ) is given by @xmath83 @xcite . we express analytically all principal spectra ( @xmath47 , @xmath27 , @xmath64 , @xmath2 ) by quantum mechanical description ( i.e. by modified kubo formula ) within the electric dipole approximation . in cartesian system defined @xmath31 $ ] , @xmath32 $ ] , @xmath33 $ ] and for magnetization along @xmath84 $ ] direction , the absorption ( dissipative ) part is the well - known non - magnetic ( isotropic ) term @xmath85 is @xcite @xmath86 \left|\langle i|\vec{\pi}\cdot\hat{z}|f\rangle\right|^2 \delta(\omega-\omega_{fi})\ , . \label{eq : sig0}\ ] ] where @xmath87 is matrix elements for light polarization ( electric field ) along @xmath88 with @xmath89 , @xmath90 being initial and final electron states , respectively . kinetic momentum operator @xmath91 is sum of canonical momentum @xmath92 , vector potential of the magnetic field @xmath93 and spin - orbit coupling between electron potential @xmath94 and its spin @xmath95 , @xmath96 , @xmath97 are occupation functions . the dispersive part of the dielectric functions are determined by kramers - kronig relations @xcite . note that according to tab . [ t : scans ] , eq . ( [ eq : sig0 ] ) calculates imaginary part of @xmath98 , not solely @xmath99 . however , @xmath44 can not be determined , as length of magnetization vector is constant for ferromagnetic materials . hence , @xmath44 is inseparable part of @xmath47 and we neglect it compared to @xmath47 . when both magnetization and light propagation is along @xmath88 axis , @xmath84 $ ] , then eigenmodes of light propagation are circularly left and circularly right polarizations . therefore , the linear term @xmath27 is given as a difference of light absorption for circularly - left and circularly - right polarizations as determined by bruno _ et al _ @xcite @xmath100 \left ( |\langle i|\vec{\pi}\cdot(\hat{x}+i\hat{y})|f\rangle|^2 - |\langle i|\vec{\pi}\cdot(\hat{x}-i\hat{y})|f\rangle|^2 \right ) \delta(\omega-\omega_{fi})\ , . \label{eq : sigk}\ ] ] in case magnetization is along @xmath101 $ ] direction ( i.e. along @xmath102 ) , the symmetry arguments predict @xmath103 ( tab . [ t : scans ] ) . by employing eq . ( [ eq : sig0 ] ) , @xmath45 can be expressed as @xmath104 \left ( |\langle i|\vec{\pi}\cdot\hat{x}|f\rangle|^2 - |\langle i|\vec{\pi}\cdot\hat{y}|f\rangle|^2 \right ) \delta(\omega-\omega_{fi})\ , . \label{eq : sigs}\ ] ] this equation corresponds to the case when light propagation is along @xmath88 axis and @xmath105 , then eigenmodes of light propagation are linearly polarized waves parallel and perpendicular to magnetization direction . hence , the quadratic term @xmath45 is given by difference of absorptions for those two light polarizations . similarly , in cartesian system defined @xmath106 $ ] , @xmath107 $ ] , @xmath108 $ ] and for magnetization @xmath109 $ ] , the element @xmath110 is expressed formally in the same way as @xmath45 @xcite . transforming back to cartesian system @xmath31 $ ] , @xmath32 $ ] , @xmath33 $ ] , quadratic term @xmath2 is expressed by difference of absorption of linear light polarization along [ 110 ] and @xmath111 $ ] directions when magnetization is along @xmath112\parallel \hat{x}+\hat{y}$ ] @xmath113 \left ( |\langle i|\vec{\pi}\cdot(\hat{x}+\hat{y})|f\rangle|^2 - |\langle i|\vec{\pi}\cdot(-\hat{x}+\hat{y})|f\rangle|^2 \right ) \delta(\omega-\omega_{fi})\ , . \label{eq : sig44}\ ] ] finally note , that here expressed @xmath45 and @xmath2 corresponds to principal spectra calculated from [ 100 ] and [ 110 ] magnetization directions , as denoted in tab . [ t : fund ] and figs . [ f : specfe][f : specni ] the permittivity tensor phenomenologically describes optical properties of matter . in this section , we provide a brief relation between elements of permittivity tensor and measured effects in common magnetooptical techniques . from experimental point of view , the measured quantities are straightforwardly related with the reflectivity matrix , usually expressed in jones formalism @xcite . then , measured optical quantities , such as kerr effect , magnetic circular dichroism ( mcd ) , mld , reflectivity , magnetoreflectivity , ellipsometry etc . are directly related to this reflection matrix . in case of multilayer structure , the relation between reflection matrix and permittivities of constituent layers are routinely described by yeh @xcite or berreman @xcite formalisms , based on propagation of light eigenmodes in the layer , where eigenmodes intensities are determined by boundary conditions on the interfaces ( continuity of transversal component of electric and magnetic fields through the interfaces ) . as optical anisotropies are usually week , the perturbation optical approach has been developed as well @xcite , handling optical anisotropy as a perturbation from optical isotropic response . analytical solution of optical response of multilayer structure with generally optically anisotropic layers is very complicated . hence , analytical solution is mostly expressed for two simple structures : ( i ) reflection on interface of bulk material . however , even in this simple case the analytical expression of reflection matrix from generally optically anisotropic media is still very complicated @xcite . ( ii ) ultrathin approximation , where thickness of anisotropic layer is assumed to be much thinner than wavelength of light in the material @xcite . in this approximation , optical response can be expressed analytically rather simply . although different multilayer structures provides different relations between constituent permittivity tensor and reflection matrix , some relations can be generalized . in following we assume one of the following simplifications : * many magnetooptical technique ( e.g. moke , transverse moke ( tmoke ) or xmcd ) are selectively sensitive to off - diagonal permittivity elements of the fm layer . in this case , to describe effect measured by the setup , we neglect difference between the diagonal permittivity elements , and assume all diagonal permittivity elements to be equal . * on contrary , mld employs change in reflection / transmission of linearly polarized light . hence , when handling intensities of those reflected or transmitted linearly polarized light , we can neglect influence of the off - diagonal permittivity elements and assume them vanished . in case of ( a ) approximation , the reflection matrix element can be written in general form ( neglecting higher orders in @xmath11 , @xmath61 ) as @xmath114 where @xmath115 , @xmath116 , @xmath117 , @xmath118 and @xmath119 are optical terms given by details of optical and geometrical properties of the multilayer structure , being independent on the off - diagonal permittivity of the fm layer @xmath11 , @xmath61 . note that for zero incidence angle ( @xmath120 ) , @xmath121 as experimentally measured magnetooptical quantities are directly related to reflection matrix , we demonstrate relation between off - diagonal permittivity elements and few magnetooptic techniques : the complex moke angle @xmath122 ( @xmath53 is kerr rotation , @xmath123 is kerr ellipticity ) can be expressed as @xmath124 where @xmath125 , @xmath126 are again optical terms depending on all other optical properties of the constituent layers @xcite . note that for @xmath120 , @xmath127 and @xmath128 , and for small incidence angle @xmath129 . tmoke is change of p - reflectivity due to off - diagonal elements is described by term @xmath130 , being usually induced by transverse magnetization . then , tmoke writes @xmath131 where again @xmath132 is optical term depending on optical details of the multilayer . mcd ( and obviously also xmcd ) is difference of light absorption for circularly left and right polarized light . to be consistent with previous part , we express mcd in reflection , mcd=@xmath133 , where @xmath134 , @xmath135 is reflection for circularly right , left polarized light , respectively : @xmath136 where @xmath137 denotes complex conjugation . in any case , the relations between off - diagonal permittivity elements and mcd in transmission are formally identical . assuming small incidence angle ( @xmath138 ) , and hence @xmath139 , we got @xmath140 where again @xmath141 and @xmath142 are optical terms depending on optical details of the structure . in case of mld ( or xmld ) , the approximation ( b ) is employed , assuming off - diagonal term vanished . then , off - diagonal reflectivity coefficients vanishes as well ( @xmath143 ) and diagonal reflectivities can be written as @xmath144 where @xmath145 , @xmath146 and @xmath147 are again optical terms and @xmath148 , @xmath149 @xmath150 are differences of diagonal permittivity elements from from @xmath47 ( i.e. differences between anisotropic and isotropic permittivities ) . note in case @xmath120 term @xmath147 vanish and @xmath151 . also , in case of metals , term @xmath147 can be neglected as well even for non - zero incidence angle due to large refractive index of metals . mld is defined as a difference of reflectivity ( or transmittivity ) between incident @xmath152 and @xmath0 polarized light . further assuming @xmath120 , mld writes @xmath153 where @xmath94 is again an optical term . in conclusion , we provide principal spectra , _ i_.e . decomposition of isotropic , linear and quadratic contributions in magnetization of magnetooptic effects up to the second order in magnetization . the values of magnetooptic permittivity elements are determined as weighted sum of principal spectra , where weights are determined solely by magnetization direction . although in x - ray range , this spectroscopy is already established , it was not introduced for visible range . in case of ferromagnetic cubic crystals owning point symmetry , all magneto - transport up to second - order effects are described by spectra of four complex principal spectra , denoted @xmath154 , @xmath27 , @xmath64 and @xmath2 , as predicted by symmetry arguments . this approach expresses second - order magnetooptic properties as bulk material property , independent on sample crystallographic orientation , layer thickness , structure details etc . the principal spectra were expressed from ab - initio calculations for bcc fe , fcc co and fcc ni . to extract principal spectra from ab - initio , we have used two methods ( i ) by comparing ab - initio calculated angular dependence of the permittivity tensor , with dependences predicted by symmetry arguments and ( ii ) using permittivity elements calculated for fundamental symmetry magnetization directions . we also express principal spectra analytically using modified kubo formula . although the spectroscopy presented here is for cubic crystals owning @xmath155 point symmetry , it can also be simply modified for materials with different crystal symmetries , higher orders in magnetizations or even different magneto - transport effects . we expect that experimental and theoretical determination of principal spectra will be routinely used in magnetooptic spectroscopy of ferromagnetic materials . financial support by czech grant agency ( 13 - 30397s ) , by it4innovations centre of excellence project ( cz.1.05/1.1.00/02.0070 ) , funded by the european regional development fund and the national budget of the czech republic via the research and development for innovations operational programme , as well as czech ministry of education via the project large research , development and innovations infrastructures ( lm2011033 ) and student grant competitions of vsb - tu ostrava ( sp2015/61 , sp2015/71 and sp2016/182 ) as well as helpful discussions with peter m. oppeneer and karel vborn are well acknowledged . the angular dependence of @xmath11 on magnetization direction with respect to the crystal axes is determined by ab - initio calculations . the electronic structure is determined by full potential linearised augmented plane wave method ( wien2k code @xcite ) , where the exchange energy is based on local spin density approximation @xcite . the spin - orbit coupling was included in a second - variation scheme @xcite . the following parameters were employed : the energy cutoff , given as the product of the muffin - tin radius and the maximum reciprocal space vector was @xmath156 , the largest reciprocal vector in the charge fourier expansion , @xmath157 , was set to 14 ry@xmath158 , and the maximum value of partial waves inside the muffin - tin spheres , @xmath159 . the brillouin zone sampling was done on grid of 27@xmath16027@xmath16027 @xmath161-points , whereas for the optical transition matrix elements @xcite a finer grid of 81@xmath16081@xmath16081 was used . the optical calculations are based on kubo formula @xcite , with omitted drude term , hence we do not present dc transport . at present , also core - hole effects are omitted . the lattice constants of studied materials are fixed being @xmath162(bcc fe ) , @xmath163(fcc co ) and @xmath164(fcc ni ) @xcite . the numerical precision shows to be limited for the beginning of the optical energy range below about 12ev,_i.e . _ the states in vicinity of the fermi level . there may be two reasons for it : ( 1 ) the calculated transition probabilities are calculated for each pair of energy levels and for each @xmath161-point of the mesh of the reciprocal space . the integration of the kubo formula is implemented by blchl - enhanced tetrahedron integration @xcite , which determines mean value of a given property ( e.g. transition probability ) inside each tetrahedron . this mean value is calculated by weighted sum of a given property at each corner of the tetrahedron , where the weights for each stationary state ( i.e. for each energy level ) are calculated from eigenenergy of each k - point and value of the fermi energy . this works well , when either initial or final tetrahedron are fully above or below fermi level . however , in case when both initial and final states are within tetrahedrons cutting fermi level , this approach may not work properly . this may explain , why dependence of permittivity elements on magnetization direction for photon energy below 12ev are rather different from expected symmetry dependence compared to those calculated for higher photon energies . ( 2 ) in blchl - enhanced tetrahedron integration , the irreducible @xmath161-points are selected to fulfil that the symmetry operation copies those irreducible @xmath161-points to all @xmath161 points inside the unit cell of the reciprocal space . obviously , in the case of high symmetry structure , such as cubic crystal , the position of @xmath161-points in the unit cell follows most of symmetry operations . however , the integration employs tetrahedrons , which does not follow so many symmetry operations . for example , in case of cubic crystal , the rotation by 90@xmath165 copy @xmath161-points to themselves , but the tetrahedrons are not copied to themselves . therefore , the numerical error of the integration from different parts of the unit cell does not follow structural symmetry , which may increase error described in the previous paragraph . showcase of imaginary part of the permittivity for bcc fe at 1.3ev ( left columns ) and 56ev ( right columns ) for differences of permittivity tensor elements on magnetization direction for ( top line ) in - plane scans and ( bottom line ) out - of - plane scans . line is fit to dependence as predicted by symmetry arguments ( tab . [ t : scans ] ) . diagonal elements ( blue ) corresponds to difference of diagonal permittivity elements @xmath166 , @xmath61 , being proportional to @xmath41 . lower triangle ( green ) @xmath62 is proportional to linear magnetooptic @xmath27 element . upper triangle ( red ) @xmath60 is proportional to @xmath2 element . ] 65ifxundefined [ 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 * * , ( ) link:\doibase 10.1103/physrev.117.689 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevlett.87.047401 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.104.187401 [ * * , ( ) ] link:\doibase 10.1103/physrevb.79.052401 [ * * , ( ) ] link:\doibase 10.1103/physrevb.65.054415 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.109.196602 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.111.187201 [ * * , ( ) ] @noop ( ) @noop * * , ( ) @noop * * , ( ) , http://stacks.iop.org/0038-5670/18/i=6/a=r02 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1002/pssb.2220490128 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase dx.doi.org/10.1103/physrevlett.83.1862 [ * * , ( ) ] link:\doibase 10.1126/science.287.5455.1014 [ * * , ( ) ] link:\doibase 10.1103/physrevb.67.024431 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevb.82.184415 [ * * , ( ) ] link:\doibase 10.1103/physrevb.67.024431 [ * * , ( ) ] link:\doibase 10.1103/physrevb.74.094407 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.98.197201 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.100.067403 [ * * , ( ) ] link:\doibase 10.1103/physrevb.78.064427 [ * * , ( ) ] link:\doibase 10.1103/physrevb.82.094403 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevb.89.140404 [ * * , ( ) ] link:\doibase 10.1109/tmag.2014.2321632 [ * * , ( ) ] @noop * * , ( ) @noop _ _ ( , , ) @noop _ _ ( , , ) @noop _ _ ( , , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevb.83.064409 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevb.53.9214 [ * * , ( ) ] @noop _ _ ( , ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevb.64.235421 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * ( ) link:\doibase 10.1103/physrevb.43.6423 [ * * , ( ) ] @noop _ _ ( , ) link:\doibase 10.1103/physrevb.67.155411 [ * * , ( ) ] link:\doibase 10.1103/physrevb.52.1090 [ * * , ( ) ] @noop ( ) @noop * * , ( ) link:\doibase 10.1103/physrevb.63.205111 [ * * , ( ) ] @noop * * , ( ) @noop _ _ ( , ) @noop * * , ( )

we provide unified phenomenological description of magnetooptic effects being linear and quadratic in magnetization . the description is based on few principal spectra , describing elements of permittivity tensor up to the second order in magnetization . each permittivity tensor element for any magnetization direction and any sample surface orientation is simply determined by weighted summation of the principal spectra , where weights are given by crystallographic and magnetization orientations . the number of principal spectra depends on the symmetry of the crystal . in cubic crystals owning point symmetry we need only four principal spectra . here , the principal spectra are expressed by _ ab - initio _ calculations for bcc fe , fcc co and fcc ni in optical range as well as in hard and soft x - ray energy range , _ i.e. _ at the 2@xmath0- and 3@xmath0-edges . we also express principal spectra analytically using modified kubo formula . there is a vast number of physical phenomena proportional to quadratic form of magnetization . in case of dc transport phenomena , the most well - known examples are anisotropy magnetoresistance ( amr ) @xcite or longitudinal hall effect @xcite . within the magnetooptic community , the field of magnetooptic effects quadratic in magnetization is complicated by incredible number of nomenclature , being called cotton - mouton effect , voigt effect , quadratic magnetooptic kerr effect ( qmoke ) @xcite , magnetic linear birefringence , x - ray magnetic linear dichroism ( xmld ) @xcite , magnetic double refraction , magnetooptic orientation effect , magnetooptic anisotropy , hubert - schfer effect or magnetorefractive effect @xcite . the nomenclature is not strictly defined , however it refers either to type of samples ( liquid , gas , solid state ) or it refers to experimental configurations of the setup ( namely detecting change of light intensity or detecting change of polarization state upon variation of magnetization direction ) . although those effects are usually not considered as single phenomena , they originate from equal parts of permittivity tensors ( i.e. from equal symmetry breaking ) . notice , that recently new types of quadratic - in - magnetization effects arose , for example anisotropic magneto - thermopower @xcite in spin - caloritronics . within some generalization , one can expect that any magneto - transport linear in magnetization will have its quadratic - in - magnetization counterpart . the first observation of magnetooptic effects quadratic in magnetization dates back more than century ago in works of kerr @xcite , majorana @xcite and cotton and moutton @xcite , where magnetic birefringence was observed in liquids and colloids . later , those quadratic magnetooptic response was observed in gases , solid state materials and obviously also in ferromagnetic materials . see large reviews of smolenskii _ et al _ @xcite and ferr , gehring @xcite from 80 s . the anisotropy of magnetooptic effects ( i.e. dependence of qmoke on crystal and field orientations ) was investigated for various systems . however , the investigation were mostly done for a single photon energy , for example fe @xcite or antiferromagnetic rbmnf@xmath1 @xcite . buchmaier _ et al _ have even determined values of principal spectra ( namely values of @xmath2 and @xmath3 ) , however also only for a single photon energy @xcite . the spectral investigations were rare , for example spectroscopy of qmoke for different orientations of ni surfaces by krinchik @xcite and parker @xcite . the evolution of quadratic magnetooptic effects in x - ray photon energy range was rather independent on those in extended visible range . the x - ray magnetic linear dichroism were theoretically predicted by thole and van der laan @xcite . due to experimental restrictions , the quadratic - in - magnetization effects in x - ray range were measured in absorption , _ i.e. _ dichroic effects @xcite and later in polarization , voigt effect @xcite . the xmld was analytically predicted for [ 100 ] and [ 111 ] magnetization axis by kunes _ et al _ @xcite . they showed that for the correct shape of the xmld spectra at l@xmath4 edges the exchange splitting of the core levels is necessary @xcite . later , with direct comparison with the experiment the angular dependence of the xmld of bcc fe shown that the role of the valence spin - orbit interaction is much weaker @xcite . the simple model of kunes _ et al _ that accounts for the difference of the xmld spectra for the high symmetry crystal directions , _ et . al _ with magnetization along [ 001 ] and [ 111 ] axis showed that the origin is expressed as different weighted spin densities of @xmath5 and @xmath6 orbitals @xcite . shortly after , arenholz _ et . al _ @xcite have shown that in case of cubic crystals having point symmetry , xmld along any quantization axis can be expressed as weighted sum of two fundamental spectra . this approach was extended to tetragonal crystal field by laan _ et al _ , extending number of fundamental spectra to four , demonstrated on cofe@xmath7o@xmath8 @xcite . krug _ et al _ @xcite have generalize this approach from xmld fundamental spectra to fundamental spectra of scattering form - factor @xmath9 , where imaginary part of @xmath9 corresponds to light absorption . later , havekort _ et al _ have expressed symmetry arguments also for other crystal symmetries @xcite . recently , the quadratic effects in magnetization like voigt in reflection or xmld were detected and theoretically explained also in the soft x - ray energy range , _ i. e. _ at the 3@xmath0 edges of transition metals @xcite . from phenomenological point of view , all magnetooptic effects are described by conductivity @xmath10 , permittivity @xmath11 or scattering @xmath12 tensors , being simply convertible between each other . in following , we use tensor of permittivity , being mostly employed in optical community . from quantum - mechanical point of view , the fundamental part of the permittivity tensor is optical absorption , related to optical dispersion by kramers - kronig relations @xcite . the relations of magnetooptic response ( or in general to any magneto - transport ) to crystallographic orientation is given by symmetry arguments , predicting dependence of the permittivity tensor elements on magnetization direction for effects being independent , linear , quadratic , cubic etc in magnetization . in case of magnetooptic effects quadratic - in - magnetization , the form of such tensors can be found in works of birss @xcite , nye @xcite and bhagavantam @xcite and visnovsky @xcite , whose notation is used in this article . symmetry arguments predict , that the permittivity tensor is given by a weighted sum of constituent single dielectric tensor elements , where their weights are given solely by direction of magnetization . in this article , we express principal spectra , _ i.e. _ the non - magnetic , linear and quadratic contributions in magnetization for permittivity tensors , from which the complete spectra of permittivity tensor can be obtained for any magnetization direction . then , we express principal spectra calculated from first principles of cubic material bcc fe , fcc co and fcc ni for three spectral ranges , extended visible range , soft and hard x - rays ( at the 2@xmath0-edges and 3@xmath0-edges , respectively ) . also , we express principal spectra directly using modified kubo formula . finally , we briefly express relations between signal measured by various magnetooptical techniques and off - diagonal permittivity elements .

You are an expert at summarizing long articles. Proceed to summarize the following text: last decade has seen a major shift in stellar spectroscopy : a slow collection of individual spectra has been accelerated by massive surveys , mostly using fiber - fed spectrographs with hundreds of spectra observed simultaneously . the past and ongoing efforts include rave @xcite , gaia - eso @xcite , segue @xcite , apogee @xcite , lamost @xcite , galah @xcite , and of course gaia @xcite . up - to - date overviews of the state and results of these surveys are given elsewhere in this volume . the main goal of stellar spectroscopic surveys is to study galactic structure and evolution . but the collected spectra allow for a significant auxiliary science . the three examples discussed below are an illustration of a vast range of posibilities and are by no means exhaustive . we believe that every observer could add further relevant uses of hundreds of thousands of stellar spectra , which were in most cases selected for observation only following simple positional and magnitude constraints . the first example illustrates research of the multi - dimensional structure of the interstellar medium . the next one helps with identifying young stars in the field . the last one is an example on how even a single spectrum obtained by a stellar survey can improve the solution of an astrometric binary which is being derived by gaia . in 2020 , the gaia mission ( launched in december 2013 ) is expected to release 6-dimensional ( spatial position + velocity ) vectors for a significant fraction of stars on our side of the galactic centre , thus allowing a computation of stellar orbits and of evolution of the galaxy as a whole . traditional studies of the galactic interstellar medium ( ism ) can not yield information equivalent to stars , as absorption studies get only a 2-dimensional ( column density ) information by observing one hot star at a time . but ism allows to open up its 3-rd and 4-th dimension by studying diffuse interstellar bands ( dibs ) , weak but numerous absorption lines seen in spectra of background stars which are likely caused by distinct macromolecular carriers . high dimensionality requires measurement of the strength of these weak interstellar lines also for cool stars which by far outnumber hot stars in the galaxy . recent new approaches divide out the cool star spectrum by use of synthetic models of stellar atmospheres @xcite or in a self - calibrated way by using spectra of similar stars with negligible ism absorption observed at high galactic latitudes by the same survey @xcite . by observing a given dib toward many stars which are nearly in the same direction but at different and known distances one can reconstruct absorption sites along the line of sight . joining observations in many directions on the sky then gives their spatial distribution . finally , measurement of radial velocity shift yields a 4-dimensional picture of the ism for each dib , and can even constrain placement of multiple clouds along each line of sight . interstellar absorption lines of sodium and potassium atoms yield information equivalent to dibs , but emission lines or dust absorptions are limited to up to 3 dimensions . ism is the place of violent collisions of supernova shells , plus winds from asymptotic giant branch stars and hot - star associations . head - on collisions in the galactic plane are difficult to interpret , though an expected galactic rotation pattern has been nicely identified @xcite . but observations of the on - going galah and partly gaia - eso surveys are away from the plane where interactions generally result in a net motion perpendicular to the plane . if any shells of absorbing material are identified we can assume that their motion is perpendicular to shell surfaces and reconstruct a complete velocity vector from its radial velocity component . such information for ism is then equivalent to the one collected for stars by gaia . this information can be used to study past events in the interstellar medium . @xcite published a quasi 3-dimensional map of intensity of diffuse interstellar band at 8620 which shows that distribution of dib extinction is thicker than the one of dust and that it is different on either side of the galactic plane , a witness to asymmetries in placement of recent explosions of supernovae and to incomplete vertical mixing . observations with the gaia - eso and galah surveys could be used to increase the dimensionality of ism studies to 4 dimensions ( for an example of radial velocity measurements see * ? ? ? they could also identify and characterize galactic fountains blown away by supernovae in the last million years . such flows are thought to sustain star formation in the disk by entraining fresh gas from the halo , so they provide a mechanism which explains why star formation in our and other similar galaxies did not stop when gas present in the disk has been used up @xcite . figure [ figdibsgalah ] plots a dozen dibs and the k i interstellar atomic line at 7699 in a stellar spectrum observed by galah . spectrum of tyc 4011 - 102 - 1 , a hot star with strong interstellar absorptions close to the galactic plane , is shown . each 20 wide panel is centred on the dib wavelength as listed in @xcite . plotted wavelengths are heliocentric . right - most panel identifies two interstellar clouds for k i at different velocities . for a majority of galah objects , which lie away from the galactic plane , such complications are rare ( but can be detected ) . properties of a star are entirely determined by its initial composition , mass and current age if one neglects rotation , magnetism or multiplicity . as noted by david @xcite `` age is not a direct agent of change and can not be measured like mass or composition . also , age affects the core of the star , but we observe the surface which is complex . '' large spectroscopic surveys have the possibility to measure some empirical age indicators , i.e. rotation , activity , and lithium depletion boundary . the galah survey will bring studies of these age indicators to industrial scale with its hundreds of thousands of observed spectra . lithium depletion studies have been motivated by lithium observations of main - sequence stars in young clusters and the halo ( e.g. * ? ? ? the galah survey includes the li i 6708 line in its red channel . its resolving power of @xmath2 and a typical s@xmath3n ratio of 100 per resolution element allow for efficient measurement of stellar rotation and for studies of profiles of h@xmath4 and h@xmath5 lines which are sensitive to chromospheric activity . measurement of these youth indicators for field stars is important , as it may point to stars recently ejected from young stellar environments . parallaxes and proper motions measured by gaia , together with spectroscopically derived radial velocities permit to reconstruct their galactic orbits and so to identify recently dispersed stellar clusters . multidimensional chemistry studies , which are within the scope of galah , are then the final check of the emerging picture based on chemical tagging @xcite . stellar activity identification is now entering the era of massive studies . figure [ figactivityrave ] summarizes active star candidates found in rave data using equivalent width of the emission components of the ca ii infrared triplet lines ( ew@xmath6 , * ? ? ? grey histogram is a distribution of ew@xmath6 for stars with active morphology in rave , as identified by a locally linear embedding technique @xcite . solid line marks normal stars which are assumed to be inactive , while dashed line marks rave stars classified by simbad to be pre - main sequence stars . @xmath7 is a logarithmic measure for the probability that a star with a given ew@xmath6 differs from an inactive spectrum . its values from left to right correspond to the probabilities of 5 and 2 @xmath8 below zero , zero , and 2 , 5 and 10 @xmath8 above zero . altogether the work identifies @xmath9 stars with chromospheric flux in ca ii lines detectable at least at a 2 @xmath8 confidence level . [ tablefractions ] crrrrrrr prevalent & + category & @xmath10&@xmath11&@xmath12&@xmath13&@xmath14&@xmath15&@xmath16 + @xmath10 & 80.2 & 0.8 & 3.0 & 0.4 & 0.5&13.5 & 1.5 + @xmath11 & 1.4&80.0 & 0.1 & 0.5 & 4.0 & 3.6&10.4 + @xmath12 & 4.0 & 0.1&73.6 & 1.7 & 1.3&18.6 & 0.6 + @xmath13 & 1.9 & 9.7 & 1.0&64.0 & 3.5 & 3.3&16.7 + @xmath14 & 1.4 & 5.2 & 0.1 & 0.2&77.2 & 5.3&10.5 + @xmath15 & 12.1 & 2.5&13.2 & 1.5 & 1.8&67.1 & 1.9 + @xmath16 & 0.3 & 4.3 & 0.1 & 1.7 & 5.3 & 1.6&86.7 + presence of emission components in the ca ii infrared triplet ( rave , gaia , and gaia - eso ) or in balmer lines ( galah and gaia - eso ) do not prove that the object is young : interacting binaries are an obvious example of old objects with emission type spectra . but such objects are not very common in the field . rave ( fig . [ figactivityrave ] ) found that strong emissions suggest a pre main - sequence evolutionary phase . this is consistent with results of the gaia - eso survey , where @xcite studied 22,035 spectra of stars in young open cluster fields and found that 7698 spectra ( 35% ) belonging to 3765 stars have intrinsic emission in h@xmath4 . again , such a large fraction of emission type spectra in a young stellar environment suggests that emission is related to youth . but emission is a transient property and morphological classification of emission may be changing with time . @xcite shows that most profiles are composed and classifies such profiles by properties of fits using two gaussians : @xmath10 stands for blended emission components , @xmath11 have double sharp peaks , @xmath12 are double emission , @xmath13 are p - cygni profiles , @xmath14 are inverted p - cygni , @xmath15 is self - absorption and @xmath16 is emission within absorption . off - diagonal elements in table [ tablefractions ] report correlations betweeen individual composed profile types . when emission blend is the prevalent category for an object , it is most often in combination with self - absorption , which is best explained by one of the two components being in transition between absorption and emission . similarly , the double sharp peaks can change to emission with absorption if a relatively weak absorption is constantly present and one of the emission peaks diminishes . the largest off - diagonal elements connect double peaks and self - absorption . the distinction between the two categories is largely influenced by the inclination of the slopes in the profile that are liable to change in the presence of additional weaker components or they are harder to retrieve in the case of more noisy spectra . the most frequently identified morphological categories from @xcite are emission blend ( 1729 spectra ) , emission in absorption ( 1652 spectra ) , and self absorption ( 1253 spectra ) . we conclude that many stars have their emission transient in time or in morphological type , so that activity detected through emission is an indication of youth which is not always present and should be used in connection with the absolute position of the star on the h - r diagram , a frequently known property in the gaia era . gaia will observe huge numbers of different types of binaries @xcite and study them with a wide range of techniques ( fig . [ figbinaries ] ) . one of its core strengths will be a derivation of accurate astrometric solutions even for binaries with extreme mass ratios @xcite . on the other hand spectroscopy from ground based surveys will be the source of detailed chemistry for any type of binary or multiple system . astrometry has been frequently used to supplement spectroscopic observations in the past ( e.g. * ? ? ? * ) , but in gaia the opposite will be a common case ( e.g. * ? ? ? * ) . many astrometric binaries will have components of similar mass and luminosity . the reach of astrometry is limited in this case : the two stellar images are usually not spatially resolved , so that gaia will be able to trace only the astrometric motion of the photocenter of the two components . such studies yield accurate orbital period , but since the photocenter is located somewhere between the two stars individual masses can not be derived from astrometry alone . here even a single spectrum obtained during a spectroscopic survey can be extremely valuable . radial velocities of individual components in an sb2 at an orbital phase known from astrometry allow to derive the true sizes of both orbits , and so the complete solution of the system . a proper bayesian analysis of simultaneous astrometric and spectroscopic information will be needed for this task @xcite . bland - hawthorn , j. 2009 , in the galaxy disk in cosmological context , iau symp . 254 , 241 de silva , g. , et al . 2015 , , 449 , 2604 eyer , f. , et al . 2015 , in living together : planets , host stars and binaries , asp conf . , arxiv:1502.03829 fraternali , f. 2014 . in setting the scene for gaia and lamost , iau symp 298 , 228 freeman , k. , bland - hawthorn , j. 2002 , , 40 , 487 gilmore , g. , et al . 2012 , the messenger , 147 , 25 jenniskens , p. , dsert , f .- x . 1994 , a&as , 106 , 39 kordopatis , g. , et al . 2013 , , 146 , 134 kos , j. , et al . 2013 , , 778 , 86 kos , j. , et al . 2014 , science , 345 , 791 kos , j. 2015 , this volume luo , a .- 2015 , arxiv e - prints:1505.01570v1 matijevi , g. , et al . 2012 , , 200 , 14 prusti , t. 2014 , in asteroids , comets , meteors 2014 puspitarini , l. , et al . 2015 , , 573 , 35 sahlmann , j. , fekel , f.c . 2013 , , 556 , a145 steinmetz , m. , et al . 2006 , , 132 , 1645 schulze - hartung , t. , launhardt , r. , henning , t. 2012 , , 545 , a79 siebert , a. , et al . 2011 , , 141 , 187 soderblom , d. 1995 , mem . italiana , 66 , 347 soderblom , d. 2014 , in astrophysical calibration of gaia and other surveys , ringberg castle , 7 - 11 july 2014 , http://www2.mpia-hd.mpg.de/gaia/gaiacal2014/ sderhjelm , s. 2004 , in spectroscopically and spatially resolving the components of the close binary stars , asp conf . , 318 , 413 torres , g. 2006 , , 131 , 1022 traven , g. , et al . 2015 , , submitted yanny , b. , et al . 2009 , , 137 , 4377 zasowski , g. , et al . 2013 , , 146 , 81 zasowski , g. , et al . 2015 , apj , 798 , 35 zwitter , t. , munari , u. 2004 , in the environment and evolution of double and multiple stars , iau coll . 191 , rev . mex . conf . , 21 , 251 zwitter , t. , et al . 2008 , , 136 , 421 erjal , m. , et al . 2013 , , 776 , 127

current ongoing stellar spectroscopic surveys ( rave , galah , gaia - eso , lamost , apogee , gaia ) are mostly devoted to studying galactic archaeology and structure of the galaxy . but they allow for important auxiliary science : ( i ) galactic interstellar medium can be studied in four dimensions ( position in space @xmath0 radial velocity ) through weak but numerous diffuse insterstellar bands and atomic absorptions seen in spectra of background stars , ( ii ) emission spectra which are quite frequent even in field stars can serve as a good indicator of their youth , pointing e.g. to stars recently ejected from young stellar environments , ( iii ) astrometric solution of the photocenter of a binary to be obtained by gaia can yield accurate masses when joined by spectroscopic information obtained serendipitously during a survey . these points are illustrated by first results from the first three surveys mentioned above . these hint at the near future : spectroscopic studies of the dynamics of the interstellar medium can identify and quantify galactic fountains which may sustain star formation in the disk by entraining fresh gas from the halo ; rave already provided a list of @xmath1 14,000 field stars with chromosperic emission in ca ii lines , to be supplemented by many more observations by gaia in the same band , and by galah and gaia - eso observations of balmer lines ; several millions of astrometric binaries with periods up to a few years which are being observed by gaia can yield accurate masses when supplemented with measurements from only a few high - quality ground based spectra .

You are an expert at summarizing long articles. Proceed to summarize the following text: recently , studies in numerical general relativity have shown that merged binary black holes can have a large recoil velocity through anisotropic emission of gravitational waves ( e.g. * ? ? ? * ; * ? ? ? the maximum velocity would reach @xmath3 , although the actual distribution of kick velocities is very uncertain . the discovery of such recoiled black holes is important for studies about the growth of black holes as well as the general relativity . supermassive black holes ( @xmath4 ) at the centers of disk galaxies would be kicked through this mechanism . although they would be bright just after the kick because of the emission from the accretion disk carried by the black holes , they would soon get dim as the disk is consumed by the black holes @xcite . however , if the kick velocity of a black hole is not large enough to escape from the host galaxy , it will eventually fall back onto the galactic disk . if the time - scale of dynamical friction is large enough , it will revolve around the galactic center many times before it finally spirals into the galactic center . when the black hole passes the galactic disk , it will accrete the interstellar medium ( ism ) in the disk @xcite . the accretion of the surrounding gas onto an isolated black hole has been studied by several authors ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * and references therein ) . most of the previous studies focused on stellar mass ( @xmath5 ) or intermediate mass black holes ( imbhs ; @xmath6 ) . the accretion rate and thus the luminosity of a black hole depend on the mass of the black hole and the density of the gas surrounding it ( see equation [ [ eq : dotm ] ] ) . since the mass of a stellar - mass black hole is small , its luminosity becomes large enough to be observed only when it plunges into a high density region such as a molecular cloud . on the other hand , in this letter , we show that a recoiled supermassive black hole can shine even in the ordinary region of a galactic disk because of its huge mass . we calculate the orbit of a recoiled black hole in a fixed galaxy potential . the galaxy potential consists of three components , which are @xcite disk , hernquist spheroid , and a logarithmic halo : @xmath7 @xmath8 @xmath9\:,\ ] ] where @xmath10 ( @xmath11 ) and @xmath12 are cylindrical coordinates aligned with the galactic disk , and @xmath13 . we adopt the parameters for the galaxy . we take @xmath14 , @xmath15 , @xmath16 kpc , @xmath17 kpc , @xmath18 kpc , @xmath19 kpc , and @xmath20 ; @xmath21 is determined so that the circular velocity for the total potential is @xmath22 at @xmath23 kpc ( see * ? ? ? we solve the equation of motion for the supermassive black hole : @xmath24 where @xmath25 is the velocity of the black hole , and @xmath26 . the density of the disk is given by @xmath27^{5/2}(z^2+b^2)^{3/2}}\ ] ] @xcite . we assume that part of the disk consists of the ism ; its density is represented by @xmath28 and @xmath29 . the circulation velocity of the disk is given by @xmath30 the accretion rate of the ism onto the supermassive black hole is given by the bondi - hoyle accretion @xcite : @xmath31 where @xmath32 is the mass of the black hole , @xmath33 ( @xmath34 ) is the sound velocity of the ism , and @xmath35 is the relative velocity between the black hole and the surrounding ism . we assume that the orbit of the black hole is confined on the @xmath36-@xmath12 plane ( @xmath37 ) . thus , the relative velocity is simply given by @xmath38 . the x - ray luminosity of the black hole is given by @xmath39 where @xmath40 is the efficiency . since the accretion rate is relatively small for the mass of the black hole , the accretion flow would be a radiatively inefficient accretion flow ( riaf ; * ? ? ? * ; * ? ? ? * ) . in this case , the efficiency follows @xmath41 for @xmath42 , where @xmath43 is the eddington luminosity ( e.g. * ? ? ? therefore , we assume that @xmath44 for @xmath45 and @xmath46 for @xmath47 , where @xmath48 @xcite . we assume that @xmath49 . we solved equation ( [ eq : motion ] ) by mathematica 6.0 using a command ndsolve . the black hole is ejected on the @xmath36-@xmath12 plane at @xmath50 . the direction of the ejection changes from @xmath51 to @xmath52 , where @xmath51 corresponds to the @xmath12-axis . we calculate the orbit until @xmath53 , which is chosen to be much larger than the period of revolution and to be smaller than the time - scale of dynamical friction . the latter is estimated to be @xmath54 @xcite , where @xmath55 is the total density ( disk@xmath56sphere@xmath56halo ) . the halo component does not much affect the dynamical friction . since @xmath57-body simulations for a spherically symmetric potential showed that the coulomb logarithm is @xmath583 @xcite , we take @xmath59 . the effects of dynamical friction on orbits in a complex potential like the one we adopted would be complicated and ideally should be studied with high - resolution @xmath57-body simulations . thus , equation ( [ eq : dynfrc ] ) should be regarded as a rough estimate of the time - scale of the dynamical - friction . the black hole is placed at the center of the galaxy at @xmath50 . since we do not know the distributions of mass and initial velocity ( @xmath60 ) of the black hole , we consider situations in which the emission from it would be observed easily . that is , the luminosity of the black hole would be large , and the observable time would be long . we consider five combinations of @xmath32 and @xmath60 shown in table [ tab : par ] . if we take larger @xmath32 and/or smaller @xmath60 , the dynamical friction becomes more effective and the black hole quickly falls into the galaxy center . on the other hand , if we take smaller @xmath32 and/or larger @xmath60 , the luminosity of the black hole becomes too small to be observed ( equation [ [ eq : dotm ] ] ) . moreover , the black hole is not bound to the galaxy , if @xmath60 is too large . the dynamical friction is most effective when @xmath61 . in table [ tab : par ] , we show the time - average of the time - scale , @xmath62 , for @xmath63 and @xmath61 . fig . [ fig : orbit ] shows the orbit of the black hole when @xmath64 and @xmath65 . [ fig : lum ] shows the luminosity of the same black hole ( @xmath66 ) . in table [ tab : par ] , we present the distance of the apastrons from the center of the galaxy ( @xmath67 ) when @xmath61 . it is to be noted that @xmath67 is not much dependent on @xmath68 for a given @xmath60 . we also present the maximum x - ray luminosity of the black hole ( @xmath69 ) when @xmath61 in table [ tab : par ] . for a given @xmath32 and @xmath60 , the x - ray luminosity is larger when @xmath68 is closer to @xmath52 , because the orbit is included in the galactic disk , where @xmath70 is large . we found that for @xmath71 , the luminosity reaches its maximum when the black hole passes apastrons and when the apastrons reside in the disk of the galaxy . this is because @xmath72 decreases , @xmath70 increases , and thus @xmath73 increases there ( equation [ [ eq : dotm ] ] ) . on the other hand , for @xmath74 , the distance of the apastrons from the galactic center ( @xmath67 ) is always large ( table [ tab : par ] ) . thus , even if apastrons reside in the disk , @xmath70 is small there . therefore , the luminosity of the black hole reaches its maximum between the apastron and periastron , and @xmath69 is smaller compared with the models of @xmath71 ( table [ tab : par ] ) . assuming that black holes are ejected in random directions at the centers of galaxies , we estimate the probability of observing black holes with luminosities larger than a threshold luminosity @xmath75 . for given @xmath32 and @xmath60 , we calculate 91 evolutions of the luminosity by changing @xmath68 from @xmath76 to @xmath52 by one degree . then , we obtain the period during which the relation @xmath77 is satisfied for each @xmath68 , and divide the period by @xmath78 . this is the fraction of the period during which the black hole luminosity becomes larger than @xmath79 . we refer to this fraction as @xmath80 and show it in fig . [ fig : ftheta ] when @xmath81 , @xmath64 , and @xmath82 . we average @xmath80 by @xmath68 , weighting with @xmath83 , and obtain the probability of observing black holes with @xmath84 . in table [ tab : par ] , we present the probability @xmath85 when @xmath86 ; for the parameters we chose , @xmath870.56 . we have found that a supermassive back hole that had been recoiled at the center of a disk galaxy could be observed in the galactic disk with an x - ray luminosity of @xmath88 . one of the candidates of such objects is ultraluminous x - ray sources ( ulxs ) observed in disk galaxies @xcite . they are found in off - nuclear regions of nearby galaxies and their x - ray luminosities exceed @xmath89 , which are larger than the eddington luminosity of a black hole with a mass of @xmath90 . if ulxs are stellar mass black holes , they might be explained by anisotropic emission @xcite , slim - disks @xcite or thin , super - eddington accretion disks @xcite . on the other hand , there is some evidence that they are imbhs at least for some of them @xcite . considering their x - ray luminosities and off - center positions , some of the ulxs might be the recoiled supermassive black holes . however , the fraction of supermassive black holes in the ulxs would not be large . @xcite estimated that for comparable mass binaries with dimensionless spin values of 0.9 , only @xmath91% of all mergers are expected to result in an ejection speed of @xmath92@xmath93 . since the ejection speed is smaller for mergers with large mass ratios and smaller spin values , the actual fraction would be smaller . moreover , in our model , the time - corrected probability of observing black holes with @xmath94 is @xmath95 , where @xmath96 is obtained by averaging @xmath97 f(\theta)/t_{\rm age}$ ] by @xmath68 , weighting with @xmath83 , and @xmath98 ( @xmath91 gyr ) is the age of a galaxy ( table [ tab : par ] ) . here , we note that @xmath99 should be regarded as the upper - limit of the actual time - scale , because @xmath100 should decrease through the dynamical friction every time the black hole passes the dense region of the galaxy . furthermore , our model indicates that a traveling supermassive black hole needs to have a mass comparable to the one currently observed at the galactic center in order to have large @xmath101 . it is unlikely that a galaxy would have undergone many mergers of black holes with such masses . the number of such mergers that a galaxy has undergone would be @xmath102 . thus , the probability that a galaxy has a traveling supermassive black hole with a luminosity comparable to that of ulxs is @xmath103 . in fact , current radio observations seem to show that ulxs observed so far are not supermassive black holes . our model predicts that the x - ray luminosity of a supermassive black hole traveling through the galaxy is comparable to the typical x - ray luminosity of a liner ( @xmath104@xmath105 ; * ? ? ? liners seem to show core radio emission and many even have detectable jets @xcite . on the other hand , radio observations have shown that no ulx has been detected with a unresolved radio core @xcite . moreover , it has been shown that the optical luminosities of ulxs tend to be smaller than their x - ray luminosities ( e.g. * ? ? ? * ) , which is inconsistent with typical riaf spectra ( e.g. * ? ? ? thus , it is unlikely that most of the ulxs are the supermassive black holes traveling through the galaxies . however , the recoiled supermassive black holes could be found through future extensive surveys . our model predicts that the x - ray luminous black holes should not be observed far from the centers of the host galaxies ( say @xmath106 kpc ) , because @xmath70 should be small there ( [ sec : results ] ) . our model also predicts that the relative velocity between the x - ray source and the surrounding ism and stars is @xmath107 . if atomic line emission associated with the x - ray source is observed , the velocity could be estimated through the doppler shift . instead of x - rays , @xcite argued that radio detections may be best to search for isolated accreting black holes . the detailed analysis of the spectra and the time variability would be useful to determine the masses of the black holes @xcite . in the future , statistical studies could observationally constrain the probability of the mergers of black holes and the recoil . we have shown that a supermassive black hole ejected from the center of the host disk galaxy will return to the galactic disk , if the initial velocity is smaller than the escape velocity of the galaxy . the black hole accretes the surrounding ism and the resultant x - ray luminosity can reach @xmath108 , when it passes the apastrons in the disk . although the luminosity of a recoiled supermassive black hole is comparable to that of ultra - luminous x - ray sources ( ulxs ) , it is unlikely that many of the observed ulxs are the supermassive black holes . i would like to thank the anonymous referee for useful comments . i am grateful to h. tagoshi and t. tsuribe for useful discussion . yf was supported in part by grants - in - aid from the ministry of education , culture , sports , science and technology of japan ( 20540269 ) . cccccccc @xmath109&500 & 0.3 & 4.7 & 1 & @xmath110 & 0 & 0 + @xmath111&500 & 0.3 & 1.4 & 1 & @xmath112 & 0.064 & 0.010 + @xmath111&600 & 1 & 4.5 & 3 & @xmath112 & 0.016 & 0.012 + @xmath111&700 & 2 & 27 & 13 & @xmath113 & 0 & 0 + @xmath114&500 & 0.3 & 0.47 & 1 & @xmath115 & 0.56 & 0.031 + @xmath114&600 & 1 & 1.5 & 3 & @xmath115 & 0.15 & 0.11 + @xmath114&700 & 2 & 9.0 & 13 & @xmath116 & 0 & 0 +

recent studies have indicated that the emission of gravitational waves at the merger of two black holes gives a kick to the final black hole . if the supermassive black hole at the center of a disk galaxy is kicked but the velocity is not large enough to escape from the host galaxy , it will fall back onto the the disk and accrete the interstellar medium in the disk . we study the x - ray emission from the black holes with masses of @xmath0 recoiled from the galactic center with velocities of @xmath1 . we find that their luminosities can reach @xmath2 , when they pass the apastrons in the disk . while the x - ray luminosities are comparable to those of ultra - luminous x - ray sources ( ulxs ) observed in disk galaxies , ulxs observed so far do not seem to be such supermassive black holes . statical studies could constrain the probability of merger and recoil of supermassive black holes .

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Proceed to summarize the following text: the concept of logical depth was introduced by c. bennett @xcite to differentiate useful information ( such as dna ) from the rest , with the key observation that non - useful information pertains in both very simple structures ( for example , a crystal ) and completely unstructured data ( for example , a random sequence , a gas ) . bennett calls data containing useful information logically deep data , whereas both trivial structures and fully random data are called shallow . the notion of useful information ( as defined by logical depth ) strongly contrasts with classical information theory , which views random data as having high information content . i.e. , according to classical information theory , a random noise signal contains maximal information , whereas from the logical depth point of view , such a signal contains very little useful information . bennett s logical depth notion is based on kolmogorov complexity . intuitively a logically deep sequence ( or equivalently a set ) is one for which the more time a compressor is given , the better it can compress the sequence . for example , both on trivial and random sequences , even when given more time , a compressor can not achieve a better compression ratio . hence trivial and random sequences are not logically deep . several variants of logical depth have been studied in the past @xcite . as shown in @xcite , all depth notions proposed so far can be interpreted in the compression framework which says a sequence is deep if given ( arbitrarily ) more than @xmath10 time steps , a compressor can compress the sequence @xmath6 more bits than if given at most @xmath10 time steps only . by considering different time bound families for @xmath10 ( e.g. recursive , polynomial time etc . ) and the magnitude of compression improvement @xmath6 - for short : the _ depth magnitude _ - ( e.g. @xmath11 ) one can capture all existing depth notions @xcite in the compression framework @xcite . e.g. bennett s notion is obtained by considering all recursive time bounds @xmath12 and a constant depth magnitude , i.e. , @xmath13 . several authors studied variants of bennett s notion , by considering different time bounds and/or different depth magnitude from bennett s original notion @xcite . in this paper , we study the consequences these changes of different parameters in bennett s depth notion entail , by investigating the computational power of the deep sets yielded by each of these depth variants . * we found out that the choice of the depth magnitude has consequences on the computational power of the corresponding deep sets . the fact that computational power implies bennett depth was noticed in @xcite , where it was shown that every high degree contains a bennett deep set ( a set is high if , when given as an oracle , its halting problem is at least as powerful as the halting problem relative to the halting problem : @xmath14 is high iff @xmath15 ) . we show that the converse also holds , i.e. , that depth implies computational power , by proving that if the depth magnitude is chosen to be `` large '' ( i.e. , @xmath7 ) , then depth coincides with highness ( on the turing degrees ) , i.e. , a turing degree is high iff it contains a deep set of magnitude @xmath7 . * for smaller choices of @xmath5 , for example , if @xmath5 is any recursive order function , depth still retains some computational power : we show that depth implies either highness or diagonally - non - recursiveness , denoted dnr ( a total function is dnr if its image on input @xmath16 is different from the output of the @xmath16-th turing machine on input @xmath16 ) . this implies that if we restrict ourselves to left - r.e . sets , recursive order depth already implies highness . we also show that highness is not necessary by constructing a low order - deep set ( a set is low if it is not powerful when given as an oracle ) . * as a corollary , our results imply that weakly - useful sets introduced in @xcite are either high or dnr ( set @xmath17 is weakly - useful if the class of sets reducible to it within a fixed time bound @xmath18 does not have measure zero within the class of recursive sets ) . * bennett s depth @xcite is defined using prefix - free kolmogorov complexity . two key properties of bennett s notion are the so - called slow growth law , which stipulates that no shallow set can quickly ( truth - table ) compute a deep set , and the fact that neither martin - lf random nor recursive sets are deep . it is natural to ask whether replacing prefix - free with plain complexity in bennett s formulation yields a meaningful depth notion . we call this notion plain - depth . we show that the random is not deep paradigm also holds in the setting of plain - depth . on the other hand we show that the slow growth law fails for plain - depth : every many - one degree contains a set which is not plain - deep of magnitude @xmath19 . * a key property of depth is that `` easy '' sets should not be deep . bennett @xcite showed that no recursive set is deep . we give an improvement to this result by observing that no @xmath4-trivial set is deep ( a set is @xmath4-trivial if the complexity of its prefixes is as low as possible ) . our result is close to optimal , since there exist deep ultracompressible sets @xcite . * in most depth notions , the depth magnitude has to be achieved almost everywhere on the set . some feasible depth notions also considered an infinitely often version @xcite . bennett noticed in @xcite that infinitely often depth is meaningless because every recursive set is infinitely often deep . we propose an alternative infinitely often depth notion that does nt suffer this limitation ( called i.o . we show that little computational power is needed to compute i.o . depth , i.e. , every hyperimmune degree contains an i.o . deep set of magnitude @xmath8 ( a degree is hyperimmune if it computes a function that is not bounded almost everywhere by any recursive function ) , and construct a @xmath9-class where every member is an i.o . deep set of magnitude @xmath8 . for hyperimmune - free sets we prove that every non - recursive , non - dnr hyperimmune - free set is i.o . deep of constant magnitude , and that every nonrecursive many - one degree contains such a set . in summary , our results show that the choice of the magnitude for logical depth has consequences on the computational power of the corresponding deep sets , and that larger depth magnitude is not necessarily preferable over smaller magnitude . we conclude with a few open questions regarding the constant magnitude case . due to lack of space , some proofs are ommitted and will appear in the journal version of this paper . we use standard computability / algorithmic randomness theory notations see @xcite . we use @xmath20 to denote less or equal up to a constant term . we fix a recursive 1 - 1 pairing function @xmath21 . we use sets and their characteristic sequences interchangeably , we denote the binary strings of length @xmath22 by @xmath23 and @xmath24 denotes the set of all infinite binary sequences . the join of two sets @xmath25 is the set @xmath26 whose characteristic sequence is @xmath27 , that is , @xmath28 and @xmath29 for all @xmath22 . an order function is an unbounded non - decreasing function from @xmath30 to @xmath30 . a time bound function is a recursive order @xmath12 such that there exists a turing machine @xmath31 such that for every @xmath22 , @xmath32\!\downarrow\ , = t(n)$ ] , i.e. , @xmath33 outputs the value @xmath10 within @xmath10 steps of computation . set @xmath14 is left - r.e . iff the set of dyadic rationals strictly below the real number @xmath34 ( a.k.a . the left - cut of @xmath14 denoted @xmath35 ) is recursively enumerable ( r.e . ) , i.e. , there is a recursive sequence of non - decreasing rationals whose limit is @xmath34 . all r.e . sets are left - r.e . , but the converse fails . we consider standard turing reductions @xmath36 , truth - table reductions @xmath37 ( where all queries are made in advance and the reduction is total on all oracles ) and many - one reductions @xmath38 . two sets @xmath25 are turing equivalent ( @xmath39 ) if @xmath40 and @xmath41 . the turing degree of a set @xmath14 is the set of sets turing equivalent to @xmath14 . we fix a standard enumeration of all oracle turing machines @xmath42 ; the jump @xmath43 of a set @xmath14 is the halting problem relative to @xmath14 , i.e. , @xmath44 . the halting problem is denoted @xmath45 . a set @xmath14 is high ( that is , has high turing degree ) if its halting problem is as powerful as the halting problem of the halting problem , i.e. , @xmath46 . high sets are equivalent to sets that compute dominating functions ( i.e. , sets @xmath14 such that there is a function @xmath47 with @xmath48 such that for every computable function @xmath49 and for almost every @xmath22 , @xmath50 ) , i.e. , a set is high iff it computes a dominating function @xcite . a set @xmath14 is low if its halting problem is not more powerful than the halting problem of a recursive set , i.e. , @xmath51 . note that @xmath45 is high relative to every low set . if one weakens the dominating property of high sets to an infinitely often condition , one obtains hyperimmune degrees . a set is of hyperimmune degree if it computes a function that dominates every recursive function on infinitely many inputs . otherwise the set is called of hyperimmune - free degree . another characterization of computational power used in computability theory is the concept of diagonally non - recursive function ( dnr ) . a total function @xmath49 is dnr if for every @xmath16 , @xmath52 , i.e. , @xmath49 can avoid the output of every turing machine on at least one specified input . a set is of dnr degree , if it computes a dnr function . it is known that every r.e . dnr degree is high , actually even turing equivalent to @xmath45 @xcite . if one requires a dnr function to be boolean , one obtains the pa - complete degrees : a degree is pa - complete iff it computes a boolean dnr function . it is known that there exists low pa - complete degrees @xcite . fix a universal prefix free turing machine @xmath53 , i.e. , such that no halting program of @xmath53 is a prefix of another halting program . the prefix - free kolmogorov complexity of string @xmath54 , denoted @xmath55 , is the length of the length - lexicographically first program @xmath56 such that @xmath53 on input @xmath56 outputs @xmath54 . it can be shown that the value of @xmath55 does not depend on the choice of @xmath53 up to an additive constant . @xmath57 is the length of a shortest program that outputs the pair @xmath58 , and @xmath59 is the length of a shortest program such that @xmath53 outputs @xmath54 when given @xmath60 as an advice . we also consider standard time bounded kolmogorov complexity . given time bound @xmath12 ( resp . @xmath61 ) , @xmath62 ( resp . @xmath63 ) denotes the length of the shortest prefix free program @xmath64 such that @xmath65 outputs @xmath54 within @xmath66 ( resp . @xmath18 ) steps . replacing @xmath53 above with a plain ( i.e. , non prefix - free ) universal turing machine yields the notion of plain kolmogorov complexity , and is denoted @xmath67 . we need the following counting theorem . [ t : counting ] there exists @xmath68 such that for every @xmath69 , @xmath70 . a set @xmath14 is martin - lf random ( mlr ) if none of its prefixes are compressible by more than a constant term , i.e. , @xmath71 for some constant @xmath72 , where @xmath73 denotes the first @xmath22 bits of the characteristic function of @xmath14 . a set @xmath14 is @xmath4-trivial if its complexity is as low as possible , i.e. , @xmath74 . see the books of downey and hirschfeldt @xcite and nies @xcite for more on @xmath75 and @xmath4-complexity , mlr and trivial sets . effective closed sets are captured by @xmath9-classes . a @xmath9-class @xmath76 is a class of sequences such that there is a computable relation @xmath77 such that @xmath78 . let @xmath79 be an order . a set @xmath17 is @xmath80 if for every recursive time bound @xmath12 and for almost all @xmath81 , @xmath82 . a set @xmath17 is @xmath0 ( resp . order@xmath2 ) if it is @xmath83 ( resp . @xmath80 ) for every @xmath84 ( resp . for some recursive order @xmath49 ) . a set is said bennett deep if it is @xmath0 . we denote by @xmath85 the above notions with @xmath4 replaced with @xmath75 . it is easy to see that for every two orders @xmath86 such that @xmath87 , every @xmath80 set is also @xmath88 . bennett s slow growth law ( sgl ) states that creating depth requires time beyond a `` recursive amount '' , i.e. , no shallow set quickly computes a deep one . [ l.sgl ] let @xmath89 be a recursive order , and @xmath90 be two sets . if @xmath14 is @xmath91 ( resp . @xmath92 ) then @xmath93 is @xmath94 ( resp . @xmath92 ) for some recursive order @xmath95 . furthermore given indices for the truth - table reduction and for @xmath89 , one can effectively compute an index for @xmath95 . the symmetry of information holds in the resource bounded case . [ l.sym.inform . ] for every time bound @xmath12 , there is a time bound @xmath96 such that for all strings @xmath97 with @xmath98 , we have @xmath99 . furthermore given an index for @xmath12 one can effectively compute an index for @xmath96 . [ c.sym.inform . ] let @xmath12 be a time bound and @xmath100 be strings . then there exists a time bound @xmath96 such that for every prefix @xmath60 of @xmath54 we have latexmath:[$c^t(y \ | \ a ) \geq^+ c^{t'}(x \ index for @xmath12 one can effectively compute an index for @xmath96 . given @xmath64 a @xmath102-minimal program with advice @xmath103 for @xmath60 , @xmath104 remaining bits , and a delimiter after @xmath64 , one can reconstruct @xmath54 in time @xmath105 steps given a. thus @xmath106 . bennett s original formulation @xcite is based on @xmath4-complexity . in this section we investigate the depth notion obtained by replacing @xmath4 with @xmath75 , which we call plain depth . we study the interactions of plain depth with the notions of martin - lf random sets , many - one degrees and the turing degrees of deep sets . the following result is the plain complexity version of bennett s result that no mlr sets are bennett deep . due to lack of space , the proof will appear in the journal version of this paper . a direct proof is given , though this result is also a corollary of theorem [ t : hdeepc - is - hdeepk ] . [ mlrdeepc ] for every mlr @xmath14 and for every recursive order @xmath89 , @xmath14 is not @xmath107 . suppose by contradiction that set @xmath14 is mlr and @xmath107 , for some @xmath89 as above . we claim that @xmath108 . to prove the claim , let @xmath109 . then given @xmath110 with @xmath111 , the program @xmath64 : `` print the smallest @xmath112 such that @xmath113 . '' is a program for @xmath22 of size @xmath114 , i.e. , @xmath115 suppose @xmath116 is a @xmath75-minimal program for @xmath73 of size @xmath117 , then appending @xmath118 bits to @xmath116 yields a prefix free program @xmath119 for @xmath73 of size @xmath120 . since @xmath14 is mlr we have @xmath121 , i.e. , @xmath122 which implies @xmath123 ( for @xmath124 sufficiently large ) . if @xmath111 , then @xmath125 , thus @xmath126 , i.e. , @xmath127 , which proves the claim . since for all @xmath81 we have @xmath128 ( via a `` print '' program ) , it follows that for every @xmath111 , @xmath129 which contradicts that @xmath14 is @xmath107 . sequences that are mlr relative to the halting problem are called @xmath130-random . equivalently a sequence @xmath14 is @xmath130-random iff there is a constant @xmath72 such that @xmath131 for infinitely many @xmath22 @xcite . since there is a constant @xmath132 such that @xmath133 is a trivial upper bound on the plain kolmogorov complexity of any string of length @xmath22 , it is clear that no @xmath130-random sequence can be @xmath1 . thus most mlr sequences are not @xmath1 . the following result shows that the slow growth law fails for plain depth . due to lack of space , the proof will appear in the journal version of this paper . [ t : manyone - deg - deepc ] every many - one degree contains a set which is not @xmath134 . the recursive many - one degrees consist only of sets which are not @xmath134 . so consider any set @xmath14 different from both @xmath135 and @xmath30 and let @xmath136 . given any @xmath137 , choose @xmath138 and let @xmath22 be any number between @xmath139 and @xmath140 which has @xmath75-complexity @xmath124 . now on one hand @xmath141 and on the other hand , one can compute @xmath124 from the @xmath124-digit binary number representing @xmath22 and one can compute @xmath142 from @xmath124 and using @xmath143 one can compute @xmath144 from the binary representation of @xmath22 and its length @xmath124 so that @xmath145 for some time bound @xmath12 and some constant @xmath72 independent of @xmath22 and @xmath124 . this shows that @xmath14 is not @xmath134 . clearly @xmath146 . furthermore , @xmath147 by mapping all values of form @xmath148 to @xmath64 and all other values to a fixed non - element of @xmath14 . note that this result shows that order@xmath2 does not imply order@xmath3 : all the sets in the truth - table degree of any order@xmath2 set are all order@xmath2 ( by the sgl ) , but this degree contains a non order@xmath3 set by the previous result . however , the converse is true , every order@xmath3 set is order@xmath2 . [ t : hdeepc - is - hdeepk ] if @xmath14 is @xmath149 for some recursive order @xmath89 then @xmath14 is also @xmath150 for some recursive order @xmath95 . there is a recursive sequence @xmath151 of numbers such that @xmath152 ( up to some constant ) ; for example one can take @xmath153 . furthermore , assume that some time - bounded version @xmath154 is given . then , for every @xmath137 and every @xmath155 , @xmath156 up to a constant . now for @xmath137 and each @xmath155 , let @xmath157 and note that these @xmath158 sum up to a number below @xmath159 , hence there is a prefix - free machine @xmath53 with recursive domain such that @xmath160 for all @xmath161 whose length is some @xmath162 . as this is a recursive measure which can be evaluated with some time - bound @xmath96 , there is a constant @xmath72 such that for almost all prefixes @xmath161 of @xmath14 of some length @xmath162 , @xmath163 and thus @xmath164 . one can therefore construct a new prefix - free machine @xmath165 where one redistributes , for those @xmath161 of length @xmath162 where @xmath164 , the weight back to the @xmath166 with @xmath167 and obtains that @xmath168 , where @xmath132 is again a constant independent of the @xmath161 and @xmath162 considered . since this step will be taken for almost all prefixes of @xmath14 and as @xmath169 can be translated into @xmath4 with a constant off - set , one has for almost all @xmath22 that @xmath170 where @xmath171 it can easily be seen that @xmath95 is a recursive order and that taking @xmath172 in place of @xmath173 allows to absorb , for almost all @xmath22 , the various constants . the following result shows that being constant deep for @xmath75 implies computational power . [ t.deep.high.or.dnc ] let @xmath14 be an @xmath134 set . then @xmath14 is high or dnr . we prove the contrapositive . suppose that @xmath14 is neither dnr nor high . let @xmath174 be ( a coding of ) @xmath175 . because @xmath176 , there are infinitely many @xmath124 where @xmath177 is defined and equal to @xmath174 . hence there is an @xmath14-recursive increasing function @xmath49 such that , for almost every @xmath124 , @xmath178 is the time to find an @xmath179 with @xmath180 and to evaluate the expression @xmath181 to verify the finding . as @xmath14 is not high , there is a recursive increasing function @xmath89 with @xmath182 for infinitely many @xmath124 . now consider any @xmath124 where @xmath182 . then for the @xmath183 found for this @xmath124 , it holds that @xmath184 and @xmath185 is also larger than the time to evaluate @xmath181 . hence @xmath186 is larger than the time to evaluate @xmath177 for infinitely many @xmath124 where @xmath177 codes @xmath175 . for each such @xmath124 , let @xmath22 be a number with @xmath187 and @xmath188 . starting with a binary description of such an @xmath22 , one can compute @xmath124 from @xmath22 and run @xmath177 for @xmath186 steps and , in the case that this terminates with a string @xmath161 of length @xmath140 , output @xmath189 . it follows from this algorithm that there is a resource - bounded approximation to @xmath75 such that there exist infinitely many @xmath22 such that , on one hand @xmath190 while on the other hand @xmath191 can be described in @xmath192 bits using this resource bounded description . hence @xmath14 is not @xmath134 . since there are incomplete r.e . turing degrees which are high , these are also not dnr and , by theorem [ t.high.iff.deep ] , they contain sets which are @xmath193 . thus the preceeding theorem can not be improved to show that `` @xmath1 sets are dnr '' . [ t : deep - not - dnr ] there exists a set @xmath14 such that @xmath14 is @xmath194 ( for any @xmath195 ) but @xmath14 is not dnr . there is a degree which is high but not dnr @xcite . thus we can , by theorem [ t.high.iff.deep ] , select a set @xmath14 in this degree which is @xmath194 for every @xmath196 . bennett s original depth notion is based on prefix free complexity . he made important connections between depth and truth - table degrees ; in particular he proved that the @xmath0 sets are closed upward under truth - table reducibility , which he called the slow growth law . in the following section we pursue bennett s investigation by studying the turing degrees of deep sets . in the first subsection , we investigate the connections between linear depth and high turing degrees . we then look at the opposite end by studying the interactions of various lowness notions with logical depth . the following result shows that at depth magnitude @xmath8 , depth and highness coincide on the turing degrees . the result holds for both @xmath4 and @xmath75 depth . [ t.high.iff.deep ] for every set @xmath14 the following statements are equivalent : 1 . the degree of @xmath14 is @xmath197 for some @xmath198 . the degree of @xmath14 is @xmath194 for every @xmath195 . @xmath14 is high . we prove @xmath199 using the contrapositive : let @xmath200 and @xmath201 such that @xmath202 with @xmath203 . let @xmath137 be the limit inferior of the set @xmath204 such that there are infinitely many @xmath22 with @xmath205 . now one can define , relative to @xmath14 , an @xmath14-recursive function @xmath49 such that for each @xmath22 there is an @xmath124 with @xmath206 and @xmath207 . as @xmath14 is not high , there is a recursive function @xmath89 with @xmath208 for infinitely many @xmath22 ; furthermore , @xmath209 for all @xmath22 . it follows that there are infinitely many @xmath22 with @xmath210 which is also at most @xmath211 away from the optimal value , hence @xmath14 is not @xmath212 deep , which ends this direction s proof . let us show @xmath213 . let @xmath198 , @xmath14 be high , and let @xmath214 be dominating . we construct @xmath215 such that @xmath93 is @xmath194 . by definition , if @xmath12 is a time bound and @xmath112 an index of @xmath12 then for every @xmath216 @xmath217\!\downarrow\ , = t(m)$ ] . since @xmath49 is dominating , we have for almost every @xmath216 , @xmath218\!\downarrow$ ] . we can thus use @xmath49 to encode all time bounds that are total on all strings of length less than a certain bound into a set @xmath219 , where _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ @xmath220 iff @xmath221\!\downarrow$ ] for all @xmath222 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ thus @xmath12 is a ( total ) time bound iff for almost every @xmath223 , @xmath224 ( where @xmath112 is an index for @xmath12 ) . we have @xmath225 and we choose the pairing function @xmath226 such that @xmath227 encodes the values @xmath228 let @xmath81 and suppose @xmath229 is already constructed . given @xmath230 and @xmath227 , we construct @xmath231 . from @xmath227 , we can compute the set @xmath232 , i.e. , a list eventually containing all time bounds that are total on strings of lengths less or equal to @xmath233 . let @xmath234 find the lex first string @xmath235 of length @xmath236 such that @xmath237 let @xmath238 : = a(n ) x_n.$ ] by construction we have @xmath215 . also , @xmath239 , i.e. , @xmath240 . let us prove @xmath93 is @xmath241 ; we then extend the argument to show @xmath93 is @xmath194 . let @xmath12 be a time bound . let @xmath22 be large enough such that @xmath242 and @xmath243 where the @xmath244 s are derived from @xmath12 as described below . let @xmath223 be such that @xmath245 and @xmath246 , i.e. , @xmath247 ends with the first @xmath248 bits of @xmath235 ( one bit is `` lost '' due to the first bit used to encode @xmath249 ) . we consider two cases , first suppose @xmath250 . let @xmath251 be a time bound ( obtained from @xmath12 ) such that @xmath252 , where neither the constant nor @xmath251 depends on @xmath253 . let @xmath254 be derived from @xmath251 using lemma [ l.sym.inform . ] . we have @xmath255 for the second case , suppose @xmath256 . we have @xmath257 note that each iteration of the argument above yields a @xmath258 term ( @xmath259 ) , therefore for any @xmath195 , there is a number @xmath260 of iterations , such that @xmath93 can be shown @xmath194 , for all @xmath22 large enough such that @xmath261 . [ c : kalso - holds ] theorem [ t.high.iff.deep ] also holds for @xmath4-depth . because every set @xmath14 is @xmath197 ( for some @xmath262 ) iff it is @xmath263 for some @xmath264 , since for every @xmath54 , @xmath265 . an analogue of theorem [ t.deep.high.or.dnc ] holds for @xmath4 . [ t.deep.high.or.dnc.k ] let @xmath14 be a @xmath266 set for some recursive order @xmath89 . then @xmath14 is high or dnr . we prove the contrapositive . suppose that @xmath14 is neither dnr nor high . let @xmath174 be ( a coding of ) @xmath267 , where @xmath268 . the rest of the proof follows the proof of theorem [ t.deep.high.or.dnc ] , with @xmath139 replaced with @xmath269 . as a corollary , we show that in the left - r.e . case , depth always implies highness . [ c : l - re - then - high ] if @xmath14 is left - r.e . and @xmath266 ( for some recursive order @xmath89 ) then @xmath14 is high . let @xmath14 be as above . by definition of @xmath14 being left - r.e . , the left cut @xmath35 of @xmath14 is r.e . and @xmath270 . by lemma [ l.sgl ] , @xmath35 is @xmath94 ( for some recursive order @xmath95 ) . by theorem [ t.deep.high.or.dnc.k ] , @xmath35 is high or dnr . since every r.e . dnr set is high , @xmath14 is high . as a second corollary , we prove that every weakly - useful set is either high or dnr . a set @xmath14 is weakly - useful if there is a time - bound @xmath18 such that the class of all sets truth - table reducible to @xmath14 with this time bound @xmath18 is not small , i.e. , does not have measure zero within the class of recursive sets ; see @xcite for a precise definition . in @xcite , it was shown that every weakly - useful set is @xmath0 ( even order@xmath2 as observed in @xcite ) thus generalising the fact that @xmath45 is @xmath0 , since @xmath45 is weakly - useful . [ t : wudeep ] every weakly - useful set is order@xmath2 . it is shown in @xcite that every high degree contains a weakly - useful set . our results show some type of converse to this fact . [ t : wu - then - hi ] every weakly - useful set is either high or dnr . this follows from theorem [ t.deep.high.or.dnc.k ] , since every weakly - useful set is order@xmath2 by theorem [ t : wudeep ] . we showed in theorem [ t.high.iff.deep ] that every @xmath271 set is high . also theorem [ t.deep.high.or.dnc.k ] shows that every order@xmath2 set is either high or dnr . thus one might wonder whether there exists any non - high order@xmath2 set . we answer this question affirmatively by showing there exist low order@xmath2 sets . [ t : pawu ] if @xmath14 has pa - complete degree , then there exists a weakly - useful set @xmath215 . let @xmath48 be a boolean dnr function and let @xmath272 . it follows that if @xmath273 is boolean and total , then @xmath274 . one can thus encode @xmath49 into a set @xmath41 such that for every @xmath16 such that @xmath273 is boolean and total and for every @xmath54 , @xmath275 . one can also encode @xmath14 into @xmath93 ( for example , @xmath276 ) so that @xmath39 . thus for every recursive set @xmath277 there exists @xmath16 such that for every string @xmath54 , we have @xmath278 , where @xmath279 is computable within @xmath280 steps ( by using a lookup table on small inputs ) . it follows that every recursive set is truth - table reducible to @xmath93 within time @xmath280 . because the class of recursive sets does not have measure zero within the class of recursive sets @xcite , it follows that @xmath93 is weakly - useful . [ c : padeep ] if @xmath14 has pa - complete degree , then there exists an order@xmath2 set @xmath215 . furthermore , there is a @xmath9-class only consisting of order@xmath2 sets . this corollary follows from theorems [ t : wudeep ] and [ t : pawu ] . recall that a set @xmath14 is said low for @xmath281 iff chaitin s @xmath281 is martin - lf random relative to @xmath14 ; a set @xmath14 has superlow degree if its jump @xmath43 is truth - table reducible to the halting problem . well - known basis theorems @xcite given now the following corollary . [ c : low - deep ] for each of the properties low , superlow , low for @xmath281 and hyperimmune - free , there exists an order@xmath2 set which also has this respective property . there exists low sets @xmath14 of pa - complete degree @xcite . by theorem [ c : padeep ] there exists an order@xmath2 set @xmath215 . since @xmath14 is low it follows that @xmath93 is low . here recall that a set @xmath14 is said low for @xmath281 iff chaitin s @xmath281 is martin - lf random relative to @xmath14 ; a set @xmath14 has superlow degree if its jump @xmath43 is truth - table reducible to the halting problem . this corollary follows from theorems [ t : wudeep ] and [ t : pawu ] as well as from the well - known basis theorems for @xmath9-classes @xcite . the reason one uses pa - complete sets instead of merely martin - lf random sets ( which also satisfy all basis theorems ) , is that martin - lf random sets are not weakly - useful ; indeed , it is known that they are not even @xmath0 . this stands in contrast to the following result . there are two martin - lf random sets @xmath14 and @xmath93 such that @xmath282 is order@xmath2 . barmpalias , lewis and ng @xcite showed that every pa - complete degree is the join of two martin - lf random degrees ; hence there are martin - lf random sets @xmath25 such that @xmath282 is a hyperimmune - free pa - complete set . thus , by theorem [ t : pawu ] there is a weakly - useful set turing reducible to @xmath282 which , due to the hyperimmune - freeness , is indeed truth - table reducible to @xmath282 . it follows that @xmath282 is itself weakly - useful and therefore order@xmath2 by theorem [ t : wudeep ] . the next observation shows that this can not be generalised to @xmath4-trivial sets and that they are not @xmath0 . a trivial set is not @xmath0 . let @xmath14 be trivial ( with respect to the prefix - free kolmogorov complexity @xmath4 ) . there is a recursive function @xmath12 and constant @xmath72 such that @xmath283 for infinitely many @xmath22 ; let @xmath93 be the set of these @xmath22 . as @xmath14 is trivial , there is another constant @xmath132 such that @xmath284 for all @xmath22 . for @xmath22 ranging over @xmath93 , let @xmath137 be the limit inferior of the number of strings @xmath161 of length @xmath22 with @xmath285 ; this limit inferior exists . now there is a recursive bound @xmath96 such that , for almost all @xmath22 , @xmath286 and either @xmath287 or @xmath137 strings @xmath288 of length @xmath22 with @xmath289 are found . note that there are infinitely many @xmath290 for which there are exactly @xmath137 strings @xmath288 of length @xmath22 with @xmath291 for all @xmath292 . these @xmath293 also satisfy @xmath294 . the string @xmath295 is among these @xmath137 strings @xmath288 for all of these @xmath22 and therefore @xmath296 for infinitely many @xmath22 . as @xmath297 for some constant @xmath298 and all @xmath22 , it follows that @xmath299 for infinitely many @xmath22 and @xmath14 is not @xmath0 . a key property of depth is that `` easy '' sets should not be deep . bennett @xcite showed that no recursive set is deep . here we improve this result by observing that no @xmath4-trivial set is deep . it follows easily from equivalent characterisations of @xmath4-triviality ( see @xcite ) , but our proof is self - contained . as we will see this result is close to optimal . [ t : ktshallow ] no @xmath4-trivial set is @xmath0 . let @xmath14 be @xmath4-trivial and @xmath68 such that @xmath300 , @xmath301 . let @xmath302 be such that for every string @xmath54 , @xmath303 and let @xmath304 . there exists a constant @xmath305 such that the set @xmath306 is infinite ( see @xcite p. 139 ) . note that @xmath307 is co - r.e . , i.e. , there exists uniformly recursive approximations @xmath308 of @xmath307 . let @xmath309 . by theorem [ t : counting ] ( with @xmath310 ) , @xmath311 . consider the function @xmath312 by modifying @xmath47 on the finitely many values before the liminf is reached , @xmath47 is recursive . wlog @xmath47 is bounded by a time bound which we also denote @xmath47 . we have @xmath313 such that @xmath314 thus for each of these infinitely many @xmath22 s we have @xmath315 , i.e. , @xmath14 is not @xmath0 . call a set @xmath14 ultracompressible if for every recursive order @xmath49 and all @xmath22 , @xmath316 . the following theorem shows that our result is close to optimal . there is an ultracompressible set @xmath14 which is @xmath0 . there is a set @xmath14 which is not @xmath4-trivial but which satisfies that for every @xmath317 order @xmath49 and all @xmath22 , @xmath316 . it would be interesting to know whether such sets as found by herbert can be @xmath0 . the result of herbert is optimal , csima and montalbn @xcite showed that such sets do not exist when using @xmath318 orders and baartse and barmpalias @xcite improved this non - existence to the level @xmath319 . we also point to related work of hirschfeldt and weber @xcite . there is a @xmath319 order @xmath49 such that a set @xmath14 is @xmath4-trivial iff @xmath320 for all @xmath22 . that is , the difference between a @xmath4-trivial and an ultracompressible set is less than the difference of two orders of different complexity . the existence of an ultracompressible @xmath0 set was proved in @xcite . this shows that theorem [ t : ktshallow ] is close to optimal . bennett observed in @xcite that being infinitely often bennett deep is meaningless , because all recursive sets are infinitely often deep . a possibility for a more meaningful notion of infinitely often depth , is to consider a depth notion where the length of the input is given as an advice . we call this notion i.o . depth . a set @xmath14 is i.o . @xmath92 if for every @xmath321 and for every time bound @xmath12 there are infinitely many @xmath22 satisfying @xmath322 if we replace @xmath4 with @xmath75 in the above definition , we call the corresponding notion i.o . @xmath134 . the fact that all recursive sets are infinitely often deep in bennett s approach does no longer hold for i.o . depth as defined above . let @xmath14 be recursive . then @xmath14 is neither i.o . @xmath134 nor i.o . @xmath92 . let @xmath14 be recursive and @xmath12 be a time bound . wlog @xmath14 is recursive in time @xmath12 , i.e. , for every @xmath81 we have @xmath323 for some constant @xmath72 , thus @xmath324 . the @xmath4 case is similar . the following shows that very little computational power is needed to compute an i.o . deep set . [ t : iodeep - hypimmune ] 1 . there is a @xmath9-class such that every member is i.o . @xmath197 for all @xmath196 . in particular there is such a set of hyperimmune - free degree . furthermore , every hyperimmune turing degree contains such a set . every nonrecursive many - one degree contains an i.o . @xmath134 set . if @xmath14 is neither recursive nor dnr , then @xmath14 is i.o . @xmath134 . this result is obtained by splitting the natural numbers recursively into intervals @xmath325 such that @xmath326 . now one defines the @xmath9-class such that for each @xmath327 where @xmath328 is defined up to @xmath329 , a string @xmath330 is selected such that for all @xmath331 , @xmath332 and then it is fixed that all members @xmath14 of the @xmath9-class have to satisfy @xmath333 for all @xmath334 . since there are @xmath335 strings @xmath293 and for each program of size below @xmath336 can witness that only @xmath337 many @xmath293 are violating @xmath338 for some @xmath331 , there will be less than @xmath339 many @xmath293 that get disqualified and so the search finds such a @xmath293 whenever @xmath340 is defined up to @xmath329 . hence , for every total @xmath328 , there are infinitely many intervals @xmath341 with @xmath22 of the form @xmath342 such that on these @xmath341 , @xmath343 and @xmath344 for a constant @xmath72 , as the program only needs to know how @xmath14 behaves below @xmath345 and can fill in the values of @xmath293 on @xmath341 . so the complexity improves after time @xmath346 from @xmath347 to @xmath345 and , to absorb constants , one can conservatively estimate the improvement by @xmath348 . by the choice of @xmath349 , the ratio @xmath350 tends to @xmath159 and therefore every @xmath14 in the @xmath9-class is @xmath351 for every @xmath196 . note that there are hyperimmune - free sets inside this @xmath9-class , as it has only nonrecursive members . furthermore , one can see that the proof also can be adjusted to constructing a single set in a hyperimmune turing degree rather than constructing a full @xmath9-class . in that case one takes some function @xmath47 in this degree which is not dominated by any recursive function and then one permits for each @xmath352 the time @xmath353 in the case that @xmath354 and chooses @xmath293 accordingly and one takes @xmath355 in the case that @xmath340 does not converge on all values below @xmath329 within time @xmath356 otherwise . this construction is recursive in the given degree and a slight modification of this construction would permit to code the degree into the set @xmath14 . for the second item , consider a set @xmath357 . every many - one degree contains such a set . for each binary string @xmath161 , let _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ @xmath358 and @xmath359 for all @xmath360 and @xmath361 for all @xmath362 which are not a power of @xmath363 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ in other word , for every @xmath357 , @xmath364 contains those @xmath293 which are a prefix of @xmath14 and for which @xmath365 is defined but not @xmath366 . for each @xmath367 where @xmath340 is a total function @xmath12 , we now try to find inductively for @xmath368 strings @xmath369 such that whenever @xmath370 is found then it is different from all those @xmath371 which have been found for some @xmath372 and the unique @xmath373 satisfies @xmath374 . note that due to the resource - bound on @xmath102 one can for each @xmath372 check whether @xmath371 exists and take this information into account when trying to find @xmath370 . therefore , for those @xmath124 where @xmath370 exists , the @xmath373 can be computed from @xmath124 , @xmath16 and @xmath137 and hence @xmath375 for some constant @xmath72 independent of @xmath376 . now assume that @xmath14 is not infinitely often @xmath1 . then there is a total function @xmath328 and a @xmath377 such that @xmath378 for all prefixes @xmath293 of @xmath14 . it follows that in particular never a @xmath370 with @xmath379 consisting of prefixes of @xmath14 is selected in the above algorithm using @xmath380 . this then implies that for almost all @xmath22 and the majority of the @xmath124 in the interval from @xmath381 to @xmath382 ( which are those for which @xmath370 does not get defined ) it holds that @xmath383 for the unique @xmath384 . there are at most @xmath385 many strings @xmath386 such that at least half of the members @xmath293 of @xmath387 satisfy that @xmath388 and there is a constant @xmath132 such that for almost all @xmath22 the corresponding @xmath161 satisfy @xmath389 . it follows that @xmath390 for some constant @xmath298 and almost all @xmath22 and all @xmath391 ; in other words , @xmath390 for some constant @xmath298 and almost all prefixes @xmath293 of @xmath14 . hence @xmath14 is recursive ( * ? ? ? * exercise 2.3.4 on page 131 ) . for the third item , let @xmath14 be as above , and let @xmath12 be a time bound . let @xmath392 . since @xmath393 and @xmath14 is hyperimmune - free , there exists a recursive @xmath89 such that @xmath394 . wlog we can choose @xmath89 such that @xmath395 . let @xmath214 be defined by @xmath396 . because @xmath14 is not dnr , we have @xmath397 . thus , @xmath398 @xmath399 and latexmath:[$c^t(a{\upharpoonright}m \ let @xmath16 be an index of a program such that @xmath401 with @xmath22 satisfying @xmath399 . thus @xmath402 @xmath399 and @xmath403 and @xmath404 , i.e. , @xmath14 is i.o . @xmath134 . for the third item , note that merkle , kjos - hanssen and stephan ( * ? ? ? * theorem 2.7 ) showed that a set @xmath14 has dnr turing degree iff there is a function @xmath405 such that @xmath406 for all @xmath22 . it will be shown that sets which are neither recursive nor i.o . @xmath1 will permit to construct such a function @xmath405 and are thus dnr . assume now that there is a time bound @xmath12 and a constant @xmath72 such that , for all @xmath22 , @xmath407 . now , for input @xmath22 , one searches relative to @xmath14 for an @xmath124 such that @xmath408 and lets @xmath409 for the so found @xmath124 . as @xmath14 is not recursive this search terminates for every @xmath22 and obviously @xmath405 . by assumption , @xmath410 and , assuming a suitable compatibility between conditional and normal kolmogorov complexity , @xmath411 , that is , @xmath406 . due to these connections , non - recursive and non - high r.e . sets are a natural example of sets where all members of the turing degree satisfy that they are i.o . @xmath1 but not @xmath1 . results of franklin and stephan @xcite imply that for every schnorr trivial set and every order @xmath89 it holds that @xmath14 is not i.o . @xmath149 , as for every order @xmath89 there is a time bound @xmath12 such that the function @xmath412 grows slower than @xmath89 . thus , the second and third points can not be generalised to i.o . order@xmath3 . it also follows that there are high truth - table degrees and hyperimmune - free turing degrees which do not contain any i.o . order@xmath3 set . these are obtained by considering examples for schnorr trivial sets such as the following ones : all maximal sets and , for every partial recursive @xmath413-valued function @xmath414 whose domain is a maximal set , all sets @xmath14 satisfying @xmath415 . we conclude that the choice of the depth magnitude has consequences on the computational power of the corresponding deep sets , and that larger magnitudes is not necessarily preferable over smaller magnitudes . therefore choosing the appropriate depth magnitude for one s purpose is delicate , as the corresponding depth notions might be very different . when the depth magnitude is large , we proved that depth and highness coincide . we showed that this is not the case for smaller depth magnitude by constructing a low order deep set , but the set is not r.e . we therefore ask whether there is a low @xmath92 r.e . set . by taking the complement of a layer of a universal martin - lf test , one obtains a @xmath9-class containing only mlr reals . are there @xmath9-classes containing only @xmath75-deep reals ? for @xmath4-depth , note that there is a @xmath9-class of weakly useful sets which then , by theorem [ t : wudeep ] , is also a @xmath9-class of order@xmath2 sets . david doty and philippe moser . feasible depth . in s. barry cooper , benedikt lwe , and andrea sorbi , editors , _ computability in europe _ , volume 4497 of _ lecture notes in computer science _ , pages 228237 . springer , 2007 .

we study bennett deep sequences in the context of recursion theory ; in particular we investigate the notions of @xmath0 , @xmath1 , order@xmath2 and order@xmath3 sequences . our main results are that martin - lf random sets are not order@xmath3 , that every many - one degree contains a set which is not @xmath1 , that @xmath1 sets and order@xmath2 sets have high or dnr turing degree and that no @xmath4-trival set is @xmath0 . a sequence is bennett deep @xcite if every recursive approximation of the kolmogorov complexity of its initial segments from above satisfies that the difference between the approximation and the actual value of the kolmogorov complexity of the initial segments dominates every constant function . we study for different lower bounds @xmath5 of this difference between approximation and actual value of the initial segment complexity , which properties the corresponding @xmath6-deep sets have . we prove that for @xmath7 , depth coincides with highness on the turing degrees . for smaller choices of @xmath5 , i.e. , @xmath5 is any recursive order function , we show that depth implies either highness or diagonally - non - recursiveness ( dnr ) . in particular , for left - r.e . sets , order depth already implies highness . as a corollary , we obtain that weakly - useful sets are either high or dnr . we prove that not all deep sets are high by constructing a low order - deep set . bennett s depth is defined using prefix - free kolmogorov complexity . we show that if one replaces prefix - free by plain kolmogorov complexity in bennett s depth definition , one obtains a notion which no longer satisfies the slow growth law ( which stipulates that no shallow set truth - table computes a deep set ) ; however , under this notion , random sets are not deep ( at the unbounded recursive order magnitude ) . we observe that all @xmath4-trivial sets are shallow . for bennett s depth , the magnitude of compression improvement has to be achieved almost everywhere on the set . bennett observed that relaxing to infinitely often is meaningless because every recursive set is infinitely often deep . we propose an alternative infinitely often depth notion that does nt suffer this limitation ( called i.o . depth ) . we show that every hyperimmune degree contains a i.o . deep set of magnitude @xmath8 , and construct a @xmath9-class where every member is an i.o . deep set of magnitude @xmath8 . we prove that every non - recursive , non - dnr hyperimmune - free set is i.o . deep of constant magnitude , and that every nonrecursive many - one degree contains such a set .

You are an expert at summarizing long articles. Proceed to summarize the following text: very recently the scientific community has paid a lot of attention in understanding the supercondictivity of the non - centrosymmetric superconductors , since the superconducting properties of such materials are expected to be unconventional @xcite . in a lattice with inversion symmetry , the orbital wave function of the cooper pair has a certain symmetry and the spin paring will be simply in either the singlet or triplet state . the noncentrosymmetry in the lattice may bring a complexity to the symmetry of orbital wave function . this effect with the antisymmetric spin - orbital coupling gives rise to the broken of the spin degeneracy , thus the existence of the mixture of spin singlet and triplet may become possible@xcite . so there might be something unconventional , such as spin triplet pairing component , existing in the non - centrosymmetric superconductors . recently , a spin - triplet pairing component was demonstrated in li@xmath6pt@xmath7b both by penetration depth measurement@xcite and nuclear magnetic resonance ( nmr)@xcite , as was ascribed to the large atomic number of pt which enhances the spin - orbit coupling . re@xmath0w is one of the rhenium and tungsten alloys family . up to now , two superconducting phases of re@xmath0w were reported with @xmath8k@xcite and @xmath97k@xcite . both phases belong to the @xmath1-mn phase ( a12 , space group i43m)@xcite , which has a non - centrosymmetric structure . moreover , atomic numbers of re and w are 75 and 74 , respectively , being close to that of pt . therefore , similar spin - triplet pairing component as that in li@xmath6pt@xmath7b are expected in re@xmath0w . most recently , it was found that the superconducting phase of re@xmath0w with @xmath10k is a weak - coupling s - wave bcs superconductor by both penetration depth @xcite and andreev reflection measurements @xcite . in this paper , we report the measurements of the ac susceptibility and low - temperature specific heat of re@xmath0w alloys . both the measurements imply that our samples have two superconducting phases with critical temperatures near @xmath11k and @xmath12k , respectively , and the high temperature phase near @xmath11k accounts for nearly 78%-87% in total volume . the specific heat data can be fitted very well by the simple two - component model , which is based on the isotropic s - wave bsc theory . furthermore , a linear relationship is found between the zero - temperature electronic specific heat coefficient and the applied magnetic field . these results suggest that the absence of the inversion symmetry does not result in novel pairing symmetry in re@xmath0w . the re@xmath0w alloys are prepared by arc melting the re and w powders ( purity of 99.9% for both ) with nominal component @xmath13 in a ti - gettered argon atmosphere . normally , the obtained alloy is a hemisphere in shape with a dimension of @xmath14 mm ( radius ) @xmath15 @xmath14 mm ( height ) . some pieces of the alloy had been cut from the original bulk ( e.g. sample @xmath16 and sample @xmath17 ) . the ac susceptibility of these samples has been measured at zero dc magnetic field to identify their superconducting phases , whereas , all of them have two superconducting transitions at about @xmath11k and @xmath12k , as shown in fig . [ fig : fig1 ] . the specific heat was measured by a physical property measurement system ( ppms , quantum design ) . the data at a magnetic field were obtained with increasing temperature after being cooled in field from a temperature well above @xmath18 , namely , field cooling process . under different dc magnetic fields , with ac field @xmath19oe and frequency @xmath20hz.,width=302 ] the temperature dependence of ac susceptibility ( @xmath21 ) at different dc magnetic fields from @xmath22 t to @xmath12 t is shown in fig . [ fig : fig1 ] . one can see that two distinct superconducting transitions occur at @xmath23 and @xmath24 k in @xmath25 curve at @xmath26 [ fig . [ fig : fig1](b ) ] , and double peaks in @xmath27 show up at the corresponding temperatures . these two phases are consistent with the previous reports in which they are proofed to be non - centrosymmetric@xcite . the peaks of @xmath28 shift to lower temperatures as the magnetic field increases , showing the continuous suppression of superconductivity by the magnetic field . the low-@xmath29 peak shifts to lower temperatures more slowly than the high-@xmath29 one , indicating distinct behaviors of the upper critical fields in these two superconducting phases . as @xmath30 increases to @xmath31 7 t , the @xmath32 curves are completely flat , showing no sign of superconducting transition . similar results were obtained on sample @xmath33 and other samples . plotted as @xmath34 versus @xmath35 at various fields.,width=302 ] we thus measured the specific heat of sample @xmath33 and in fig . [ fig : fig3 ] we present the data of @xmath34 versus @xmath35 at various magnetic fields . on each curve , there are two jumps related to the superconducting transitions consistent with the measurements of ac susceptibility . from the zero field data in low temperature region , one can see that the residual specific heat coefficient @xmath36 is close to zero , implying the absence of non - superconducting phase . the superconducting anomaly is suppressed gradually with increasing magnetic field , and from the curve at @xmath12 t there is no sign of superconductivity above @xmath37k , consistent with the observation in @xmath32 curve . the low temperature part of the normal state specific heat at @xmath38 t in fig . [ fig : fig3 ] is not a straight line , implying that the specific heat of phonon does not satisfy the debye s @xmath39 law . we may need a @xmath40 term to fit the normal state specific heat well : @xmath41 the first term is the electronic specific heat in the normal state , and the others are the contributions of the phonons . fitting the data of @xmath12 t to eq . ( [ eq : eq1 ] ) , the coefficients @xmath42mj / mol@xmath43k@xmath44 , @xmath45mj / mol@xmath43k@xmath46 , @xmath47mj / mol@xmath43k@xmath48 , and @xmath49mj / mol@xmath43k@xmath50 are determined . from the relation : @xmath51 where @xmath52 is the avogadro constant , and @xmath53 the number of atoms in one unit cell , we obtained the debye temperature of our alloys @xmath54k . these coefficients and debye temperature are all very close to the results of other works on re - w alloys@xcite . versus t. the solid lines are the calculating results which separate the electronic specific heat into two components with different @xmath18 by using specific heat formula based on the bcs theory.,width=302 ] by subtracting the phonon contribution , the electronic specific heat @xmath55 is obtained , which is shown in fig . [ fig : fig4 ] as @xmath56 versus @xmath29 . before a quantitative analysis , the low temperature specific heat at low fields has presented a strong evidence that re@xmath0w has a nodeless gap function . for a nodal superconductor ( expected by the strong mixing of spin - singlet and spin - triplet pairing components in a heavily non - centrosymmetric superconductor such as li@xmath57pt@xmath0b ) , the low temperature @xmath34 vs. @xmath29 relation should be a power law like . however , as denoted by the dashed lines in fig . [ fig : fig4 ] , if a linear relationship is assumed , the specific heat at zero field would be negative when the temperature approaches to zero . in the following section , by using a quantitative analysis , we will demonstrate that both phases of re@xmath0w have an isotropic gap function , which is in good agreement with the expectation of an @xmath5-wave superconductor . on which the specific heat have been measured . ( b ) shows the zero field specific heat data , and the black line is the calculating result based on the bcs theory.,width=302 ] figure [ fig : fig5 ] shows the ac susceptibility and specific heat data at zero dc field measured on the same sample(@xmath17 ) . the ac susceptibility data have been normalized . the high temperature phase occupies nearly @xmath58% in the whole superconducting volume . in order to fit the zero field electronic specific heat , we attempt to use the formula derived from thermodynamic relations based on the bcs theory@xcite @xmath59 where @xmath60 , and @xmath61 is an isotropic @xmath5-wave gap which depends on temperature in the same way as expected by bcs theory . since there are two coexistent phases in our samples , we use two separate terms of @xmath62 and @xmath63 to take into account the contributions of the high @xmath18 and low @xmath18 phases , respectively . thus the total specific heat can be expressed as follows : @xmath64 in which @xmath65 and @xmath66 indicate the weight of the contributions for the two phases . according to eq . ( [ eq : eq3 ] ) and eq . ( [ eq : eq5 ] ) we can nicely simulate the experimental data very well as presented in fig . [ fig : fig5](b ) by a solid line . the parameters for the best fit are @xmath67mev , @xmath68 for @xmath69k and @xmath70mev , @xmath71 for @xmath72k and @xmath73 is the gap value at zero temperature . interestingly , @xmath68 found here is very close to the relative weight 85% of the high temperature phase which was obtained from the ac susceptibility data in fig . [ fig : fig5](a ) . furthermore , @xmath74mev is in good agreement with that from the penetration depth and andreev reflection experiments@xcite . these results give a strong evidence that there is no novel pairing symmetry in our alloys . to get further evidence for this argument , we did similar calculations for the specific heat in the mixed state using the same weights of the two phases obtained from the zero field calculation . in the mixed state , there are two different regions , namely the core region and the outside core region . therefore we adopted a simple two - component model@xcite which separates the electronic specific heat into two components . the electronic specific heat is thus written as @xmath75 here @xmath1 is an adjustable parameter . the first part on the right hand side is the quasi - particle density of states ( dos ) coming from the normal vortex core regions , and the second part comes from the superconducting regions outside the cores . the results of the quantitative calculations are plotted as solid lines in fig . [ fig : fig4 ] , and they are in good agreement with the experimental data for all magnetic fields . at zero temperature obtained from the calculation based on bcs theory.,width=302 ] in the superconducting state , @xmath76 , where @xmath77 is the electronic specific heat coefficient that is dependent on temperature and field . according to eq . ( [ eq : eq4 ] ) , the zero temperature electronic specific heat coefficient @xmath78 is equal to @xmath79 , which is shown in fig . [ fig : fig7 ] as solid squares , and the solid line is a linear fit to the data . the obvious linear relationship of @xmath77 vs. @xmath30 presents further evidence that re@xmath0w is a conventional superconductor in which @xmath78 is proportional to the number of vortex cores and hence to the applied field . for a nodal superconductor with novel pairing symmetry , on the other hand , a nonlinear @xmath78 relation should be expected , which is obviously not the case in our present samples@xcite . in summary , we have synthesized re@xmath0w alloys by arc melting . from the measurements of ac susceptibility and specific heat on the alloys two distinct superconducting phases were found . both the qualitative and quantitative analysis were done on the specific heat data in zero field and the mixed state . we found that the simple two - component model based on the bcs theory with an isotropic @xmath5-wave gap can fit our experimental data very well , and we obtained a linear @xmath78 relationship . all these results indicate that the absence of the inversion symmetry does not result in any novel pairing symmetry in re@xmath0w for both @xmath10k and @xmath8k phases . this work was supported by the national science foundation of china , the ministry of science and technology of china ( 973 project : no . 2006cb601000 , no . 2006cb921802 , no . 2006cb921300 ) , the knowledge innovation project of chinese academy of sciences ( itsnem ) .

the alloys of non - centrosymmetric superconductor , re@xmath0w , which were reported to have an @xmath1-mn structure [ p. greenfield and p. a. beck , j. metals , n. y. * 8 * , 265 ( 1959 ) ] with @xmath2k were prepared by arc melting . the ac susceptibility and low - temperature specific heat were measured on these alloys . it is found that there are two superconducting phases coexisting in the samples with @xmath3k and @xmath4k , both of which have a non - centrosymmetric structure as reported previously . by analyzing the specific heat data measured in various magnetic fields , we found that the absence of the inversion symmetry does not lead to the deviation from a @xmath5-wave pairing symmetry in re@xmath0w . , lei shan , qiang luo , weihua wang , re@xmath0w , non - centrosymmetric , superconductivity , specific heat

You are an expert at summarizing long articles. Proceed to summarize the following text: it is frequently declared that only lower order formulae can be deduced for the effective conductivity problem which can not be analytically solved in general case because of the complicated random geometrical structures . after such an announce hard numerical computations are applied to solve such a problem . of course , advanced computational approaches can be useful in mechanical engineering . but an exact or approximate analytical formula is always better because it can exactly show asymptotic behavior near singular points when numerics usually fails . in the present paper , we deduce such a formula for a 2d , two - component composite made from a collection of non - overlapping , identical , circular discs , embedded randomly in an otherwise uniform locally isotropic host ( see fig.[figdisksrandom ] ) . the conductivity of the host is normalized to unity . the effective conductivity problem for an insulating or ideally conducting inclusions is called the conductivity and superconductivity problem , respectively @xcite . the problem and its approximate solution go back to maxwell , see e.g. @xcite . there are two important unresolved problems in the theory of random composites : \1 . what quantity should stand for the maximum volume fraction @xmath0 of random composites @xcite , and \2 . theoretical explanation of the values of critical indices for conductivity and superconductivity denoted by @xmath1 and @xmath2 , respectively @xcite . recently , a novel technique for deriving expansions in concentration was suggested @xcite . it combines analytic and numeric methods for solving the conductivity problem directly in the 2d case . it is applicable both for regular @xcite and random cases . thus , we proceed to the case of a 2d random composite , where rather long series in concentration for the effective conductivity by itself , will be presented and analyzed systematically , following generally to @xcite . the series will be used to estimate the index and the threshold in 2d random case . the considered problem can be equivalently formulated as follows . given the polynomial approximation of the function @xmath3 , to estimate the convergence radius @xmath4 of the taylor series of @xmath3 , and to determine parameters of the asymptotically equivalent approximation near @xmath5 . the problem of defining the threshold is highly non - trivial , since the random closest packing of hard spheres turned out to be ill - defined , and can not stand for the maximum volume fraction . it depends on the protocol employed to produce the random packing as well as other system characteristics @xcite . the problem seems less acute in two dimensions , where various protocols seems to agree on what quantity should stand for the maximum volume fraction of random composites @xcite . namely it is the concentration of @xmath6 , attained only for the regular hexagonal array of disks . the sought value for a long time was thought to be close to @xmath7 , and considered as random close packing value @xcite . it was recognized recently , that it does not correspond to the maximally random jammed state @xcite . for volume fractions above @xmath7 some local order is present and irregular packing is polycrystalline , forming rather large triangular coordination domains - grains . in present paper , a protocol with @xmath8 is used , although our method can be applied with another protocol with unknown @xmath4 . all attempts to explain the value of critical indices through geometrical quantities of percolation problem , i.e. universally @xcite , had failed so far and the indices are considered independent . from the phase interchange theorem @xcite it follows that in two - dimensions , the superconductivity index is equal to the conductivity index @xcite , @xcite , @xcite . while it is clear that using expansions in concentration for the conductivity , one should be able to address the two problems , in practice there are no more than two terms available for random systems @xcite , because of the serious technical difficulties . no method even such powerful as renormalization , or resummation approaches can draw reliable conclusions systemically , based on such short series @xcite . `` in fact , the age - old method of series expansions is also blocked by the same difficulties ... ' ' @xcite . this concerns also self consistent methods ( scms ) which include maxwell s approach , effective medium approximations , differential schemes etc . scms are valid only for dilute composites when the interactions between inclusions do not matter @xcite . the idea to correct a self consistent method ( scm ) result @xmath9 in all dimensions remained , therefore , theoretically unattainable ( see , nevertheless , @xcite ) . we should also mention an indirect approach to estimating @xmath1 for resistor networks from resistive susceptibility via scaling relations @xcite . this approach also dwells heavily on resummation techniques . in order to correctly define the effective conductivity tensor @xmath10 of random composites , the probabilistic distribution of disks of radius @xmath11 must be introduced , since already the second order term of @xmath10 in concentration depends on the distribution @xcite . for macroscopically isotropic composites , the third order term begins to depend on the distribution @xcite . in the present paper , we consider the uniform non - overlapping distribution when a set of independent and identically distributed ( i.i.d . ) points @xmath12 are located in the plane in such a way that @xmath13 . for @xmath14 we arrive at the poisson distribution and for the maximally possible concentration @xmath8 , the distribution degenerates to the unique location , the hexagonal array . the tensor @xmath10 is expressed through the scalar effective conductivity @xmath15 as follows @xmath16 , where @xmath17 is the unit tensor . in the present paper , the numerical computations are performed only for the hexagonal representative cell . this assumption does not restrict our investigation since the number of inclusions per cell can be taken arbitrary large , hence , the shape of the cell does not impact on the final result . consider sufficiently large number of non - overlapping circular disks of radius @xmath11 with the centers @xmath18 . the formal definition of the random variable has to be statistically realized to get numerical results . the protocol for the data is based on the monte carlo simulations @xcite and can be shortly described as follows . at the beginning , the centers @xmath18 are located at the nodes of the regular hexagonal lattice and further randomly moved without overlapping . after sufficiently long random walks the centers form a statistical event satisfying the considered distribution . using these locations of disks we compute coefficients of @xmath15 in @xmath19 many times and take the average . detailed description of the computational method and all relevant parameters for simulations can be found in @xcite . the method yields @xmath20 since we are dealing with the limiting case of perfectly conducting inclusions when the conductivity of inclusions tends to infinity , the effective conductivity is also expected to tend to infinity as a power - law , as the concentration @xmath19 tends to the maximal value @xmath0 for the hexagonal array , @xmath21 the critical superconductivity index ( exponent ) @xmath2 believed to be close to @xmath22 @xcite . this value is known from numerical simulations , while rigorously it can be anywhere between one and two @xcite . the critical amplitude @xmath23 is an unknown non - universal parameter . for regular arrays of cylinders the index is much smaller , @xmath24 @xcite and the critical amplitude is also known with good precision . overall effective conductivity of random systems is expected to be higher by order(s ) of magnitude as the threshold is approached @xcite . probably the simplest way to estimate the position of a critical point , is to apply the diagonal pade approximants @xcite , but their direct application leads to poorly convergent , practically random results , with the best estimate for the threshold @xmath25 . we attribute the problem to the trivially flat " starting orders in the series ( [ seriesrand ] ) . in order to compensate for the unchanging values of the coefficients in the starting orders , we consider another sequence of approximants @xmath26 obtained as follows . let us divide the original series ( [ seriesrand ] ) by the function @xmath27 and call the new series @xmath28 . then @xmath29,\ ] ] employing again only the diagonal pade approximants . there is now a reasonably good sequence of approximations for the critical point , @xmath30 , @xmath31 , @xmath32 , @xmath33 . the percentage error given by the @xmath34 equals to @xmath35 . assuming that @xmath4 is unknown , let us estimate from the value of threshold , employing general idea of corrected approximants @xcite . factor approximations of @xmath15 can be always represented as a product of two factors : critical part @xmath36 and of the rest , i.e. regular part @xmath37 . so one can most generally express the threshold @xmath38 the subsequent steps are described below . suppose we found explicitly the solution as a factor approximant @xcite , @xmath39 with approximate threshold value of @xmath40 . such approximant satisfy the three starting terms from ( [ seriesrand ] ) @xcite , and leads to the value of @xmath41 for the index within accepted bounds @xcite . let us look for another solution in the same form , but with an exact , yet unknown threshold @xmath42 , @xmath43 from here one can express formally , @xmath44 since @xmath45 is also unknown . all we can do is to use for @xmath46 the series ( [ seriesrand ] ) , so that instead of a true threshold , we have an effective threshold , @xmath47 which should become a true threshold @xmath42 as @xmath48 ! moreover , let us apply re - summation procedure to the expansion ( [ series1 ] ) using again factor approximants @xmath49 , and define the sought threshold @xmath50 self - consistently , @xmath51 as we approach the threshold , the rhs of ( [ thr ] ) should become the threshold . since factor approximants are defined as @xmath52 for arbitrary number of terms @xmath53 , we will also have a sequence of @xmath54 . e.g. @xmath55 expression matches up to the 5-order terms included . solving ( @xmath56 ) , we obtain @xmath57 . in the next even order there is no real solution for @xmath58 and natural stop - sign is generated . the percentage error of such estimate is just @xmath59 . conventionally , one would first apply the following transformation , @xmath60 to the original series , to make calculations with different approximants more convenient . the most straightforward way to estimate index @xmath2 is to apply factor approximants @xcite ( in terms of the variable @xmath61 ) , so that possible corrections to the mean - field " value unity , appear additively , by definition . following the standard procedure , the simplest factor approximant is written as follows , @xmath62 , where @xmath63 , @xmath64 , @xmath65 , and the critical index @xmath66 . in the next order the value of critical index improves to @xmath67 . using even more terms , we obtain @xmath68 with @xmath64 , @xmath69 , @xmath70 , @xmath71 , @xmath72 , @xmath73 , @xmath74 , and the critical index value is good , @xmath75 . the critical amplitude is equal to @xmath76 . let us again transform the original series in terms of @xmath61 , and to such transformed series @xmath77 let us apply the @xmath78 transformation @xcite and call the transformed series @xmath79 . in terms of @xmath79 one can readily obtain the sequence of approximations @xmath80 for the critical index @xmath2 , @xmath81,n , n+1]).\label{seq1}\ ] ] unfortunately , in the case of ( [ seriesrand ] ) , this method is not accurate . namely , the best result is @xmath82 . let us again apply factor approximants , but this time to @xmath79 . the only positive - valued factor approximant appears to be given as follows , @xmath83 where @xmath84 , @xmath85 , @xmath86 , @xmath87 , @xmath88 , @xmath89 , @xmath90 , @xmath91 . the critical index is simply , @xmath92 effective conductivity can be reconstructed numerically @xcite , @xmath93 also numerically , the critical amplitude evaluates as @xmath94 . ( [ numer ] ) will be compared below with other formula for the effective conductivity valid everywhere . let us look for the solution first in the form of a simple pole , @xmath95 , so that our zero approximation @xmath96 for the critical index , is typical for various scms . let us divide then the original series ( [ seriesrand ] ) by @xmath97 , express the newly found series in term of variable @xmath61 , then apply @xmath78 transformation and call the transformed series @xmath98 . finally one can obtain the following sequence of corrected scm approximations for the critical index , @xmath99,n , n+1 ] ) , \label{cor}\ ] ] the corrected " sequence of approximate values for the critical index can be calculated readily and we have now three good estimates , @xmath100 , @xmath101 and @xmath102 . applying different approximants , such as iterated roots @xcite , one can obtain the following sequence of corrected approximations to the critical index , @xmath103 where @xmath104 stands for the iterated root of @xmath105-th order @xcite , constructed for the series @xmath98 with such a power at infinity that defines constant correction to @xmath106 . calculations with iterated roots are really easy since at each step we need to compute only one new coefficient , while keeping all preceding from previous steps . namely , @xmath107 and so on iteratively . the two starting values for the critical index can be calculated readily , giving @xmath108 and @xmath109 , but in the next orders one obtains complex results . in order to continue we define the new series @xmath110 , and apply the technique of iterated approximants to satisfy the new series asymptotically , order - by - order . we can continue the sequence of ( [ seq1 ] ) ( terms @xmath111 and @xmath112 are trivial and not shown ) , @xmath113 and so on iteratively , so that @xmath114 , @xmath115 , @xmath116 , @xmath117 , @xmath118 , @xmath119 , @xmath120 , @xmath121 , @xmath122 , @xmath123 , @xmath124 , @xmath125 , @xmath126 . conversely , the sequence for amplitude is monotonously increasing and ends up in @xmath127 . similar techniques were applied also to the regular case @xcite . in the random case we proceed by extrapolating from the side of a diluted regime and to the high - concentration regime close to @xmath0 ; while in the regular case we first derived an approximation to the high - concentration regime and then extrapolated to the less concentrated regime . there are indications that physics of a 2d regular and irregular composites is related to the so - called `` necks '' , certain areas between closely spaced disks @xcite . we discuss briefly some formulae for the effective conductivity from @xcite . the first formula ( eq.(22 ) , @xcite ) is nothing else but an improved pade approximant conditioned by appearance of a simple pole at @xmath0 . we also employ eq.(5 ) from @xcite , adjusting it with regard to the threshold and critical index values . it exemplifies a crossover from the diluted regime where scm is valid , to the percolation regime with typical critical behavior . closed - form expression for the effective conductivity of the regular hexagonal array of disks is presented in @xcite . since it is defined in the same domain of concentrations as in the random case , a comparison can explicitly quantify the role of a randomness ( irregularity ) of the composite . but in the most interesting region of large @xmath19 , the relevant formula ( eq 14 from @xcite ) fails . in order to estimate an enhancement factor due to randomness we can still use the numerical results tabulated in @xcite . in particular , the enhancement factor at @xmath128 , is about @xmath129 , compared with ( [ numer ] ) and ( [ 3 ] ) . the two formulae also happen to be very close to each other everywhere . our suggestion for the conductivity valid for all concentrations in the random case is based on @xcite . let us apply to @xmath130 another transformation to get @xmath131 , with @xmath132 , in order to get rid of the power - law behavior at infinity . in terms of @xmath133 one can readily obtain the sequence of approximations @xmath134 for the critical amplitude @xmath23 , @xmath135 , n , n + 1])^{-s}.\ ] ] there are only few reasonable estimates for the amplitude , @xmath136 , @xmath137 , @xmath138 and @xmath139 . following the prescription above , we obtain explicitly , @xmath140 ) ( dotted ) , ( [ 3 ] ) ( solid ) are compared with improved pade approximant from @xcite ( dashed ) and expression from @xcite ( dot - dashed ) . ] various expressions are shown in fig.[figure ] . note , that significant deviations of the pade formula from @xcite ( with typical value of @xmath141 ) compared to our results , start around @xmath142 . in this paper , we developed a direct approach to the effective conductivity of the random 2d arrangements of an ideally conducting cylinders , based on series ( [ seriesrand ] ) . we confirm the position of a threshold for the effective conductivity , calculate the value of a superconductivity critical index , and obtain a crossover expression , valid for arbitrary concentrations . resummation techniques involved to achieve these goals are original extension of @xcite . they are in the same mold as the traditional renormalization group @xcite . our main achievement is a direct ( independent on other indices ) , calculation of the critical index for superconductivity @xmath132 . our methods allow thus to correct effectively the value of the critical index given by the large family of self consistent methods , the most popular among them being ever useful effective medium approximation @xcite . we can not yet completely exclude the possibility that @xmath2 may depend ( weekly ) on the protocol . further studies are needed with different protocols . in a separate paper , we intend to present a generalization of , i.e. the transition formula from the regular hexagonal array to the random array . we expect to obtain a dependence of the critical index on the degree of randomness . we are grateful to wojciech nawalaniec for computer derivation of the formula . s.torquato , f. h. stillinger , jammed hard - particle packings : from kepler to bernal and beyond , reviews of modern physica , * 82 * , 2634 ( 2010 ) , r.czapla , w.nawalaniec and v.mityushev , effective conductivity of random two - dimensional composites with circular non - overlapping inclusions , comput . mat . sci.*63 * , 118 ( 2012 ) r.czapla , w. nawalaniec and v. mityushev , simulation of representative volume elements for random 2d composites with circular non - overlapping inclusions , theoretical and applied informatics , * 24 * , 227 ( 2012 ) j.b . keller , a theorem on the conductivity of a composite medium j. math . phys . * 5 * , 548 ( 1964 ) n. rylko , transport properties of the regular array of highly conducting cylinders . _ j , engrg . math _ * 38 * , 1 ( 2000 ) . s. gluzman , d.a . karpeev , l.v . berlyand , effective viscosity of puller - like microswimmers : a renormalization approach . j. r. soc . interface * 10 * : 20130720 ( 2013 ) i.v . andrianov , v.v . danishevskyy , a. l. kalamkarov , analysis of the effective conductivity of composite materials in the entire range of volume fractions of inclusions up to the percolation threshold , composites : part b * 41 * , 503 ( 2010 )

effective conductivity of a 2d random composite is expressed in the form of long series in the volume fraction of ideally conducting disks . the problem of a _ direct _ reconstruction of the critical index for superconductivity from the series is solved with good accuracy , for the first time . general analytical expressions for conductivity in the whole range of concentrations are derived and compared with the regular composite and existing models .

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Proceed to summarize the following text: over the years , the study of current noise and noise correlations has become a respected and useful diagnosis for transport measurements on mesoscopic conductors . theoretically , noise was first computed mostly for non interacting systems @xcite . however , it soon became clear that low frequency noise could be used to isolate the quasiparticle charge @xcite and to study the statistical correlations @xcite in specific quasi one dimensional correlated electron systems , such as the edge waves in the quantum hall effect . in these chiral luttinger liquids , the charge of the collective excitations along the edges corresponds to the electron charge multiplied by the filling factor . attention is now turning towards conductors individual nano - objects which occur naturally , and which can be connected to current / voltage probes in order to perform a transport experiment . the crucial advantage of such nano objects is that they are essentially free of defects and in some circumstances they have an inherent one dimensional character . carbon nanotubes constitute the archetype of such 1d nano - objects : single wall armchair nanotubes have metallic behavior , with two propagating modes at the fermi level . incidentally , electronic correlations are known to play an important role in such systems . carbon nanotubes seem to constitute good candidates to study luttinger liquid behavior . in particular , their tunneling density of states and thus the tunneling @xmath0 characteristics is known to have a power law behavior @xcite in accordance with luttinger liquid theory . 8 cm luttinger models for nanotubes differ significantly from their quantum hall effect counterpart , because of their non - chiral character . forward and backward fields describing collective excitations effectively mix , because the interactions between electrons are spread along the whole length of the nanotube . for this reason , a straightforward transposition of the results obtained for chiral edge system proves difficult . nevertheless , non chiral luttinger liquids can be described with chiral fields @xcite . such chiral fields correspond to excitations with anomalous ( non - integer ) charge , which has eluded detection so far . in the present work , we propose an experimental geometry which allows to probe directly the underlying charges of the collective excitations . the setup consists of a nanotube whose bulk is contacted by a scanning tunneling microscope ( stm ) tip which injects electrons , while both extremities of the nanotube collect the current ( fig . [ fig1 ] ) . the current , the noise and the noise correlations are computed , and the effective charges are determined by comparison with the schottky formula @xcite for an `` infinite '' nanotube , the striking result is that noise correlations contribute to second order in the electron tunneling , in sharp contrast with a fermionic system which requires fourth order . the noise correlations are then positive , because the tunneling electron wave function is split in two counter propagating modes of the collective excitations in the nanotube . we conjecture that in the presence of 1d fermi liquid leads , modeled as in ref . @xcite , the absence of renormalization / interaction effects of the nanotube is recovered . a recent two terminal experiment studied the current current fluctuations in ropes of nanotubes @xcite . there , it is pointed out that the strong reduction of the low frequency noise can not be understood within the context of scattering theory @xcite . naive comparison with existing non - chiral luttinger liquid models @xcite would imply an interaction parameter much inferior to the free electron case . also , we mention that other multi - terminal geometries where a nanotube or a one dimensional wire is attached to more than two leads , have been considered @xcite . our proposal deals with the same geometry as ref @xcite , where a renormalization analysis identified the exponents of the current voltage characteristics . however , here the emphasis is put on the low frequency current fluctuation spectrum , both for the autocorrelation and the cross correlations between the two ends of the nanotube . the paper is organized as follows : the hamiltonian of our setup is specified in the next section , followed by a general non - equilibrium scheme based on the keldysh formalism to study transport in this device , which is independent of the type of leads chosen ( sect . results for a nanotube connected to leads are then presented in sect . 4 . a connection with the effective charges of refs . @xcite is established in sect . the transport geometry ( fig . [ fig1 ] ) implies tunneling from the tip ( normal or ferromagnetic metal ) to the nanotube , and subsequent propagation of collective excitations along the nanotube . in the absence of tunneling , the hamiltonian is thus simply the sum of the nanotube hamiltonian , described by a two mode luttinger liquid , together with the tip hamiltonian . using the standard conventions @xcite , the operator describing an electron with spin @xmath1 moving along the direction @xmath2 , from mode @xmath3 is specified in terms of a bosonic field : @xmath4 with @xmath5 a short distance cutoff , @xmath6 the fermi momentum , @xmath7 the momentum mismatch associated with the two modes , and the convention @xmath8 , @xmath9 and @xmath10 are chosen for the direction of propagation , for the nanotube branch , and for the spin orientation . it is convenient to express this bosonic phase in terms of the conventional non - chiral luttinger liquid fields @xmath11 and @xmath12 , with @xmath13 identifying the charge / spin and total / relative fields : @xmath14 with @xmath15 , @xmath16 , @xmath17 et @xmath18 . @xmath11 and @xmath12 are dual non - chiral fields . a plausible alternative would have been to express @xmath19 in terms of the chiral luttinger liquid fields . however , the present choice will be simpler later on when dealing with inhomogeneous luttinger liquids ( in order to include the leads ) , as the green s functions for @xmath11 , @xmath12 are known . the hamiltonian which describes the collective excitations in the nanotube has the standard form : @xmath20 with an interaction parameter @xmath21 and velocity @xmath22 . for the stm tip , one assumes for simplicity that only one electronic mode couples to the nanotube . the tip can thus be described by a semi - infinite luttinger liquid , as in kondo type problems . this turns out to be convenient in this problem where both bosonized nanotube fermions operators and tip fermions operators intervene . for the sake of generality , we allow the two spin components of the tip fields to have different fermi velocities @xmath23 , which allows to treat the case of a ferromagnetic metal . the fermion operator at the tip location @xmath24 is then : @xmath25 here , @xmath26 is the chiral luttinger liquid field , whose keldysh green s function at @xmath24 is given by @xcite : @xmath27 u_f^{\sigma}(t_1-t_2)/2a\right\}~ , \end{aligned}\ ] ] where @xmath28 refer to the upper or lower branch of the keldysh contour . the tunneling hamiltonian is a standard hopping term : @xmath29^{(\varepsilon ) } ~.\label{tunnel_hamiltonian}\end{aligned}\ ] ] here the superscript @xmath30 leaves either the operators in bracket unchanged ( @xmath31 ) , or transforms them into their hermitian conjugate ( @xmath32 ) . the voltage bias between the tip and the nanotube is included using the peierls substitution : the hopping amplitude @xmath33 acquires a time dependent phase @xmath34 , with the bias voltage identified as @xmath35 . we will use the convention @xmath36 . similarly , the tunneling current is defined as : @xmath37^{(\varepsilon ) } ~.\end{aligned}\ ] ] in eqs . ( [ fermion_nanotube ] ) and ( [ fermion_stm ] ) , we have omitted the klein factors which guarantee the anti - commutation of the 3 types of fermions operators written in terms of bosonic fields for this problem : the two nanotube branches and the stm single mode . it has been established @xcite that klein factors are in principle necessary to treat multi - luttinger system , as illustrated in the computation of noise correlations between three edge states in the fqhe . in the present work , klein factors can be dropped because we intend to work with lowest order perturbation theory . to order @xmath38 , statistical correlations between the three luttinger systems do not occur . however , they should show up when calculating higher order corrections ( @xmath39 ) . for this problem which implies propagation along the nanotube , it is also necessary to compute the ( total ) charge and ( total ) spin currents using the bosonized fields of eq . ( [ fermion_nanotube ] ) : @xmath40 similarly , we consider the spin current in the @xmath41 direction : @xmath42 note that the contribution from terms containing @xmath43 oscillations has been dropped . this is equivalent to requiring that the current measurement along the nanotube is effectively a spatial average over a length scale larger than @xmath44 . in practice , @xmath43 terms are necessary in order to establish a connection between current fluctuations and density fluctuations . in this section , the general approach used to calculate the tunneling current and noise , as well as the current and noise in the nanotube is described . all quantities are computed at zero temperature for simplicity . the calculation of the tunneling current and noise is quite similar to the perturbative results in ref . @xcite for the fqhe . here it is summarized in order to compare with the nanotube transport quantities . the keldysh technique is used to compute the average tunneling current and noise . we adopt the convention that the coefficients @xmath45 identify the upper / lower branch of the keldysh contour : @xmath46 which applies in typical tunneling situations where the product of the current averages is of order @xmath39 . in order to collect the lowest order contribution in the tunneling amplitude , the exponential is expanded to first order for the current , and to zeroth order for the noise : @xmath47 where the last factor in eqs . ( [ intermediate_tunneling_current ] ) and ( [ intermediate_tunneling_noise ] ) is the tip fermion green s function . next the nanotube and tip fields are specified in terms of the bosonized fields ( nonchiral and chiral ) , and the two keldysh ordered exponential products are computed : @xmath48 as expected , the stationary current and the real time current correlator call for the time differences @xmath49 , @xmath50 only . integrating over time , the zero frequency noise is introduced . further using the symmetry properties of the green s functions @xmath51 and @xmath52 ( similarly for @xmath53 , @xmath54 and @xmath55 ) , only @xmath56 is retained for the current : @xmath57 the tunneling current and noise imply the knowledge of the green s functions at the tunneling location only . the operator averages along the nanotube require a perturbative calculation up to second order in the tunneling hamiltonian for the tunneling current and for the noise . tunneling of an electron from the stm tip is followed by propagation of the collective excitations of the luttinger liquid towards both ends of the nanotube . @xmath58 where the contribution to the noise coming from @xmath59 has been dropped because it contributes to order @xmath39 . expressing the hamiltonian in terms of the fields , the limit @xmath60=\partial_x \phi_{c+}$ ] is used in order to cast the time ordered averages into correlators of exponentials only : @xmath61 where the contribution from the stm tip is the same as before . the two time ordered products ( one for the tip and one for the nanotube ) are expressed in terms of luttinger liquid green s functions . taking the spatial derivative , one obtains an expression with green s functions as prefactors implying propagation as well as exponentiated green s functions at the tunneling location . operating variable changes in the integrals and noticing that only @xmath62 contributes , the current and noise become : @xmath63 note the temporal decoupling ( which occurs after operating variable changes ) in these expressions . the integral over @xmath64 contains information on electron tunneling at @xmath24 , while the remaining integrals involve propagation , thus the spatial dependence in the green s functions arguments . in the previous section , general expressions were derived for the current and noise , which are independent of the form of the green s functions @xmath65 , @xmath66 , @xmath66 and @xmath67 . the green s functions are described in appendix a and are used to compute the tunneling noise and current as well as the nanotube noise and current . after substitution of the green s function of a nanotube , the tunneling current and noise read : @xmath68 with the exponent : @xmath69 @xmath70 is the bulk tunneling exponent of the current voltage characteristics @xmath71 the integrals are computed in appendix b , we obtain : @xmath72 where we used the definition of the gamma function @xmath73 . only electrons can tunnel from the tip to the nanotube , so one can check that the classical schottky formula holds always : @xmath74 some of the time integrals in eq . ( 22 ) has already been encountered when computing the tunneling current and noise . the current and noise thus become : @xmath75\nonumber\\ & & = -\frac{2e^2v^2_f\gamma^2}{\pi a}\left(\sum_{\sigma}\frac{1}{u_f^{\sigma}}\right ) \left(\frac{a}{v_f}\right)^{\nu}\frac{|\omega_0|^{\nu}}{{\bf \gamma}(\nu+1)}(i^{\phi\phi}(x , x')+i^{\phi\theta}(x , x'))~ , \nonumber\\\end{aligned}\ ] ] where the last factors are computed in appendix b. the standard assumptions of the calculation of the tunneling current and noise are recalled , as the same expressions appear in both results . we obtain : @xmath76 current conservation @xmath77 discussion ---------- one accepted diagnosis to detect effective or anomalous charges is to compare the noise with the associated current with the schottky formula in mind . a striking result is that despite the fact that electrons are tunneling from the stm tip to the bulk of the nanotube , the zero frequency current fluctuations are proportional to the current for @xmath78 : @xmath79 with an anomalous effective charge for an infinite nanotube . more can be learned from a measurement of the noise correlations . noise correlations have been proposed to detect statistical correlations in quantum transport @xcite . indeed , our geometry can be considered as a hanbury - brown and twiss correlation device . such experiments have now been completed for photons and more recently for electrons in quantum waveguides . here the novelty is that electronic excitations do not represent the right eigenmodes of the nanotube . for @xmath80 the noise correlations read : @xmath81 this is a priori negative . however , if the current direction is chosen to be positive from the tip to the extremities of the nanotube , the sign of the cross correlations is positive . recall that the fermionic version of the hanbury - brown and twiss experiment yields negative noise correlations @xcite . so far , positive noise correlations have been attributed in priority to bosonic systems @xcite . nevertheless , there are at least two other situations where they are encountered . first , when the source of particle is a superconductor , noise correlations can also be positive depending on the junction configuration@xcite . second , they also occur in systems with floating voltage probes @xcite . in the case of a superconductor , the emission of electron pairs through separate quantum dots guarantee that the noise correlations are always positive : a ( singlet ) entangled electron pair is generated outside the superconductor @xcite . note that the prefactors in eq . ( [ positive_noise_correlations ] ) can readily be interpreted using the language of ref . @xcite . a tunneling event to the bulk of a nanotube is accompanied by the propagation of two counter - propagating charges @xmath82 . recall that the subscript @xmath83 identifies the charge ( as opposed to spin ) excitation given by the total ( rather than relative ) contribution of the two modes propagating in the nanotube . each charge is as likely to go right or left . according to ref . @xcite electron injection in a luttinger liquid is characterized by chiral charges @xmath84 and chiral spin charges @xmath85 which describe the elementary excitations of the nanotube . @xmath86~,\ ] ] with integers @xmath87 ( @xmath88 ) . in particular , the addition of an electron with spin @xmath1 corresponds to the choice @xmath89 and @xmath90 . the current noise and noise correlations can be interpreted as an average over the two types of excitations : @xmath91 8 cm a drawing where the two types of charges `` flow away '' from the tip while propagating along the nanotube is depicted in the lower part of fig . both charges @xmath84 are equally likely to go right or left , and they are emitted as a pair with opposite labels . the noise correlations of eq.([correlations_quasiparticle ] ) are rendered positive if one adopts the standard convention for measuring the current in multi - terminal conductors @xcite . here these `` positive '' noise correlations resulting from charges moving toward both extremities of the nanotube have the added particularity that they occur to second order in a perturbative tunneling calculation . in superconducting - normal systems , the two electrons which emanate from the same cooper pair and which propagate in the two luttinger liquids provide a manifestation of the non local character of quantum mechanics . in the present case , only one electron is injected , but it is split into left and right excitations , unless one imposes one dimensional fermi liquid leads . here , we are dealing with entanglement between collective excitations of the luttinger liquid . written in terms of the chiral quasiparticle fields , the addition of an electron with given spin @xmath1 on a nanotube in the ground state @xmath92 gives : @xmath93 \label{fermion_nanotube_entanglement}\ ] ] with @xmath94 the chiral bosonic fields of the ( nonchiral ) luttinger liquid . this wave function is characterized by right and left movers @xmath8 whose fields appear explicitly in the phase operator of this many - particle wave function . these fields are independent of each other , therefore the exponential can be written as a product of fields : @xmath95 \label{entangled_wave_function}\ ] ] where for each sector ( charge / spin , total / relative mode ) the charges @xmath96 have been introduced , and chiral fractional operators are defined as : @xmath97~. \label{quasiparticle_chiral}\ ] ] the wave function described by eq . ( [ entangled_wave_function ] ) has all the characteristics of an entangled state . because the two types of excitations travel towards opposite ends of the nanotube , the time evolution of this `` injected electron '' state is simply obtained with the substitution @xmath98 . consequently , quantum mechanical non - locality is quite explicit here . the detection of a charge @xmath84 in one arm is necessarily accompanied by the simultaneous detection of a charge @xmath99 in the other extremity of the nanotube . this entanglement is the direct consequence of the correlated state of the luttinger liquid . when additional electrons are injected , these break up into the specific modes which can propagate in either direction in the nanotube . it therefore differs significantly from its analogs which use superconductors as electron injectors , where two electrons from the same cooper pair are dissociated . when considering only one sector , such as @xmath100 , it is interesting to note that the wave function has the same structure of say , a triplet spin state ( a symmetric combination of `` up '' and `` down '' states , or `` plus '' and `` minus '' charges ) for electrons , with the electrons being replaced by chiral quasiparticle operators . indeed , one has to recognize that each chiral field @xmath94 can be written as a superposition of boson operators : @xmath101 where @xmath102 creates a boson with nanotube mode @xmath3 , spin @xmath1 and momentum @xmath103 , and characterizes the collective modes of the one dimensional liquid . according to the state written in eq . ( [ quasiparticle_chiral ] ) , this linear superposition of boson operators appears in an exponential . this expresses that non - local `` many boson '' correlations are created when an electron is injected in a nanotube , and these many - body states are entangled in the present geometry . effects similar to the detection of effective charges show up in the spin sector when time reversal symmetry ( @xmath104 ) does not hold . the spin current and spin noise are obtained in a similar manner : @xmath105 so that at large distances : @xmath106 in practice , when time reversal symmetry holds ( @xmath107 ) , spin noise correlations vanish to order @xmath38 independently from the presence or the nature of the leads . in the case where the tip is non magnetized , the spin current and spin noise correlations also vanish . in the presence of one - dimensional fermi liquid leads , where the leads are considered to be luttinger liquids whose interaction parameters are set to @xmath108 , quasiparticles suffer andreev type reflections @xcite at both exterminites of the nanotubes . multiple reflections of quasiparticles in the fabry - perot geometry fermi liquid / nanotube / fermi liquid are expected to lead to a concellation of the interaction effects in the nanotube , as in the two terminal calculations of conductance and noise @xcite . although the detailed calculation is not presented here , dimensional analysis of the time integrals suggest that , the nanotube current and noise read : @xmath109 for @xmath110 , this whould give the classical schottky formula , in the very same spirit as in ref . for @xmath111 and @xmath112 on opposite ends of the nanotube , this noise correlator should vanish , to this order : the scattering theory result has a lowest non vanishing contribution of order @xmath39 . this low voltage result is modified by a higher power law behavior at higher voltage , with a threshold voltage specified by the size of the system @xmath113 as in ref . in summary , a diagnosis for detecting the chiral excitations of a luttinger liquid nanotube has been presented , which is based on the knowledge of low frequency current fluctuation spectrum in the nanotube . typical transport calculations either address the propagation in a nanotube , or compute tunneling i(v ) characteristics . here , both are addressed because they constitute the key for obtaining the quasiparticle charges . both the noise ( autocorrelation ) and the noise correlations ( cross - correlations ) are needed to identify the charges @xmath114 . independently , note that this measurement could also be confronted to other diagnoses of the nanotube interaction parameter using tunneling current voltage characteristics . this result relies on the assumption that one dimensional fermi liquid leads are avoided . such leads have been treated in different approaches @xcite , and are also labelled radiative contacts . radiative contacts imply equilibration with the electrons . in ref . @xcite , both radiative contacts and equilibration with dressed eigenmodes were studied , with the obvious result that luttinger liquid renormalization shows up in the conductance in the latter case . in special circumstances such as the case of ref . @xcite the nanotube is embedded in the metallic contacts , and it is suspended by its ends . here , the absence of a screening gate is explicit . electron transport between these two entities likely occurs in multiple electron scattering processes as studied in ref . @xcite . in these , or other contacts fabricated by growing techniques @xcite , quantities such as current and noise may not be affected by the presence of the contacts . standard fermion results should be recovered when the system is connected on one dimensional fermi liquid leads . the auto - correlation noise in one end of the nanotube should be related to the charge current with the standard schottky formula . the noise correlation signal should also vanish as expected and the next order correction @xmath115 then needs to be computed . a crucial test of the contacts is in order . it should be possible in practical situations to analyze the type of contacts which one has between the nanotube and its connections . if the ratio of the cross - correlations to the current @xmath116 does not depend on the tunneling distance ( @xmath117 ) , both contributions are of order @xmath38 and this constitutes an indication that the contacts do not affect this quasiparticle entanglement . if we are dealing with a fermi liquid behavior , the noise correlation current ratio should behave like @xmath38 , rather than a constant . finally , we have remarked that the many - body wave function which describes a luttinger liquid with an added electron has necessarily epr @xcite entangled degrees of freedom . both electrons chiralities contribute to the emission of quasiparticle pairs moving in opposite direction . this entanglement involves many particle states , unlike its electron counterpart . a suggestion for detection of such luttinger liquid entanglement without perturbing the system with leads is nevertheless needed . the issue how to detect this many - body entanglement should be addressed while taking into account different models for the leads , possibly involving multiple reflections within one contact @xcite . multiple reflections of the quasiparticles from one contact to the other kill this entanglement in one dimensional fermi liquid leads , which is implicit in the vanishing of the noise correlations to this order . at any rate , this is the first time that collective excitations entanglement is discussed in a condensed matter setting . discussions with a. lebedev , i. safi on the keldysh green s function and with m. bttiker are gratefully acknowledged . in this appendix , the green s functions are computed , assuming a luttinger liquid with an homogeneous interaction parameter @xmath21 and velocity @xmath22 . the product @xmath118 corresponds to the fermi velocity @xmath119 . the finite temperature action associated with this problem has the general form : @xmath120 which implies that the fourier transform of the time ordered green s functions @xmath121 and @xmath122 , define as : @xmath123 where t is the time ordered operator , satisfies the differential equations@xmath124 : @xmath125 the green s function @xmath121 is continuous everywhere , and @xmath126/k_{j\delta}$ ] has a discontinuity at @xmath110 . the similarity between eq . ( [ green_theta ] ) and ( [ green_phi ] ) results from the duality properties of the underlying fields . all information on @xmath122 is obtained by dividing @xmath121 by @xmath127 . according to eqs . ( 14 ) and ( 15 ) , there are additional green s functions in our problem which involve the fields @xmath128 and @xmath129 . for instance : @xmath130 using the action ( [ action ] ) one can show that ( @xmath131 is a real time variable ) : @xmath132 and similarly for @xmath133 . from these real time green s functions , we further specify the keldysh matrix elements which two times @xmath134 are assigned to the upper / lower branch ( @xmath135 ) . given an arbitrary real time green s function @xmath136 a general procedure@xmath137 for obtaining these elements is as follows : @xmath138 where : @xmath139 the same applies to @xmath140 for which we have : @xmath141 the mixed correlators read : @xmath142 where : @xmath143 the same applies to @xmath144 for which we have : @xmath145 we now compute the integrals involved in the tunneling current and noise . the general integrals which will be required to compute the current and noise read : @xmath146 we now write the integral which appears in the nanotube current , which refer to propagation along the nanotube : @xmath147 using the expressions for the green s functions ( appendix [ green_appendix ] ) : @xmath148 where the approximate sign holds at large distances . m. blanter and m. bttiker , phys . rep . * 336 * , 1 ( 2000 ) . c. kane and m.p.a . fisher , phys . lett . * 72 * , 724 ( 1994 ) ; c. de c. chamon , d.e . freed , and x.g . wen , phys . b * 51 * , 2363 ( 1995-ii ) ; p. fendley , a.w.w . ludwig , and h. saleur , phys . lett . * 75 * , 2196 ( 1995 ) . l. saminadayar , d. c. glattli , y. jin , and b. etienne , phys . 79 * , 2526 ( 1997 ) ; r. de - picciotto _ et al . _ , nature * 389 * , 162 ( 1997 ) . h. saleur and u. weiss , phys . rev . b * 63 * , 201302 ( 2001 ) . i. safi , p. devillard , and t. martin , phys . lett . * 86 * , 4628 ( 2001 ) . c. kane , l. balents , and m. p. a. fisher , phys . lett . * 79 * , 5086 ( 1997 ) . m. brockrath , d.h . cobden , j. lu , a.g . rinzler , r.e . smalley , l. balents , and p. mceuen , nature * 397 * , 598 ( 1999 ) . r. egger , a. bachtold , m.s . fuhrer , m. bockrath , d. cobden , and p. mceuen , in _ interacting electrons in nanostructures _ , edited by r. haug , and h. schoeller ( springer , 2001 ) . i. safi , ann . fr . * 22 * , 463 ( 1997 ) . k .- v . pham , m. gabay , and p. lederer , phys . b * 61 * , 16397 ( 2000 ) . w. schottky , ann . 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transport through a metallic carbon nanotube is considered , where electrons are injected in the bulk by a scanning tunneling microscope tip . the charge current and noise are computed both in the absence and in the presence of one dimensional fermi liquid leads . for an infinite homogeneous nanotube , the shot noise exhibits effective charges different from the electron charge . noise correlations between both ends of the nanotube are positive , and occur to second order only in the tunneling amplitude . the positive correlations are symptomatic of an entanglement phenomenon between quasiparticles moving right and left from the tip . this entanglement involves many body states of the boson operators which describe the collective excitations of the luttinger liquid .

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Proceed to summarize the following text: measurement of the surface impedance @xmath8 in high-@xmath9 superconductors ( hts ) in the microwave frequency range is one of the most effective and frequently used methods to study electron properties and mechanisms of superconductivity in these materials . such measurements , performed on high quality hts single crystals or perfect single - crystalline films , allow to obtain in a straightforward way the temperature and frequency dependencies of the complex ac conductivity of the materials @xmath10 in the microwave frequency range , that in turn yield a complimentary information on microscopic electron properties of hts , such as low - energy quasiparticle excitations from the superfluid condensate , their scattering rate and density of states , the symmetry of cooper pairing , etc . numerous experimental and theoretical studies of the microwave response carried out during the last decade have revealed a lot of interesting features of the superconducting state in hts metal - oxide compounds and partly shed a light on the nature of superconductivity in these materials ( e.g. , _ d_wave type of cooper pairing).@xcite however , up to date there is no comprehensive understanding of microwave response in hts . in particular , this concerns the temperature dependence of surface resistance @xmath11 in highly perfect single crystals and epitaxial films , where nonmonotonous character of this dependence with a wide peak below @xmath9 was observed by many investigators . unfortunately , there are still some difficulties in its understanding and explanation in the framework of existing theoretical models.@xcite in the present work we demonstrate for the first time that the nonmonotonous character of @xmath4 in epitaxially - grown single - crystalline ybco films can be even more complicated than it was suggested before : in our experiments @xmath4 curves have two distinct rather narrow peaks at quite different temperatures @xmath12 ( @xmath13 = 2530 k , @xmath14 k ) . this observation clearly indicates that the microscopic scenario of electron properties in hts ( ybco ) is more intriguing and sophisticated than it was assumed before . the temperature dependence of microwave surface resistance , @xmath4 , in ybco perfect single crystals@xcite and epitaxially grown single - crystalline thin films@xcite observed in a number of experiments performed by different groups , turned out to be nonmonotonous and revealed a pronounced broad peak at @xmath15 . the temperature position and height of the peak depend on frequency and crystal quality . it was shown also that the peculiarity of @xmath4 is very sensitive to the crystal defect density . for instance , impurities ( point defects ) suppress the peak of @xmath4.@xcite analysis of these experimental data , based on the phenomenological approach assuming the drude form of microwave conductivity for thermally excited quasiparticles @xmath16 ^{-1}$],@xcite sheds a light on the nature of observed @xmath17 peaks and explains also ( at least qualitatively ) its frequency dependence and suppression of the peak by impurities.@xcite this approach allows also to extract the value of quasiparticle relaxation time @xmath18 directly from microwave measurements of @xmath4 . the @xmath18 value in perfect single crystals appears to be strongly increasing with the temperature lowering and reaching the saturation value of order @xmath19@xmath20 s at low temperatures ( below 20 k).@xcite in the present work the @xmath21 dependencies are studied experimentally and theoretically for the most perfect ybco films in order to establish relation between the microwave response and the defect nanostructure . the first observation of two - peak behavior of the @xmath11 as well as @xmath22 dependencies is presented and a relevant theoretical model is developed . two - peak temperature dependencies of microwave surface resistance , @xmath4 , have been observed for the first time in _ c_oriented perfect ybco thin films of various thickness ( @xmath23 150480 nm ) deposited by off - axis dc magnetron sputtering onto ceo@xmath1-buffered r - cut single - crystalline sapphire substrates of 14 @xmath24 14 mm@xmath25 size . microwave measurements were performed with a use of cylindrical pure copper cavities 2 , 4 and 8 mm in diameter . one of flat bases of the cavity was a film under study . the measurements were performed using @xmath2 mode at the frequencies of 134 , 65 and 34 ghz . several dc off - axis magnetron sputtered ( ms ) , as well as pulse laser deposited ( pld ) , ybco films have been measured in the temperature range from 18 to 100 k. some films revealed a distinct two - peak structure of @xmath4 and @xmath5 dependencies with peaks at 2530 k and 4851 k ( figs . [ fig.1 ] , [ fig.4 ] ) . the peaks are much more pronounced at the lower frequency , while their temperature positions remain almost unchanged at any frequency . for less perfect films ( e.g. , pld ) characterized by a higher density of crystal lattice defects , the @xmath4 and @xmath5 dependencies appear to be monotonous ( power law ) and similar to those obtained in previous works.@xcite the two - peak peculiarity observed for both @xmath4 and @xmath5 is believed to be an intrinsic electronic feature of perfect quasi - single - crystalline ybco films . the two - peak behavior is not detected in much smaller ybco single crystals and in experiments with a use of strip - line resonator measurement technique , which requires film patterning . temperature dependencies of the surface resistance @xmath26 for three films ( # 48 , # 35 and # 10 ) at 34 ghz . @xmath26 for cu is shown for comparison . ] temperature dependencies of the penetration depth @xmath27 for the film # 35 at three different frequencies . ] the observed dependence differs from that for perfect ybco single crystals , for which only one much broader frequency dependent peak of @xmath4 was detected.@xcite some peculiarities of the two - peak character of @xmath4 and @xmath5 dependencies are shown in figs . [ fig.4][fig.5 ] . the two - peak dependencies of @xmath28 for one of the most perfect ybco film are presented in fig . [ fig.2 ] for three different frequencies , while fig . [ fig.3 ] shows these dependencies normalized by @xmath29 . the corresponding dependencies of penetration depth @xmath30 for the same film are shown in fig . [ fig.4 ] for the same three frequencies . [ fig.5 ] demonstrates the effect of aging and the influence of applied dc magnetic field on the two - peak structure of @xmath4 . one can see that the specimen aging as well as application of dc magnetic field lead to smearing of the peaks . moreover , dc magnetic field shifts the peak positions to slightly higher temperatures . @xmath26 dependencies for the most perfect film ( # 35 ) at three different frequencies . ] @xmath26 dependencies for the same film and frequencies as in fig . [ fig.2 ] normalized by @xmath29 . ] effects of aging and applied dc magnetic field on the @xmath26 dependence of the film # 35 at 34 ghz . ] it is well known@xcite that in superconductors at microwave frequencies and not too high temperatures ( @xmath31 ) usually one has @xmath32 , where @xmath33 is the real part of microwave conductivity of superconductor @xmath34 . so , the observed scaling of the @xmath35 curves shown in fig . [ fig.3 ] means that @xmath33 , which is determined by the contribution of thermally excited quasiparticles , almost does not depend on frequency within the studied frequency range : @xmath36 except the vicinity of peaks , where it is falling down rapidly with the increase of frequency . the @xmath37 behavior is rather similar as it is shown in fig . [ fig.4 ] . thus , our experiments on perfect single - crystalline ybco films have revealed the new type of nonmonotonous two - peak temperature dependence of @xmath38 , essentially different from the single - peak nonmonotonous behavior observed before for perfect single crystals and films . it should be noted also , that these observed for the first time two peaks of @xmath7 dependence for the most perfect films are much more narrow in the temperature scale than the single peak of @xmath7 for single crystals mentioned above.@xcite quite similarly to the case of single crystals , these peaks become suppressed , when frequency increases or for less perfect specimens . the substantial difference between @xmath11 values of ybco epitaxially - grown highly biaxially - oriented films and ybco single crystals is shown to exist , increasing with temperature , @xmath39 . this difference is supposed to be due to essentially different crystal defect spectra in ybco epitaxial films and single crystals . perfect quasi - single - crystalline yba@xmath40cu@xmath41o@xmath42 films exhibit several times higher microwave surface resistance than ybco single crystals . a different dimensionality of crystal defects in ybco single crystals and thin films is supposed to be responsible for the difference . in general , two major types of crystal defects ( point and planar , i.e. , oxygen vacancies and twins ) are known to be most essential for electromagnetic behavior of ybco single crystals . the nonmonotonous @xmath43 dependence with a large broad peak is mostly pronounced in untwined crystals , where only point defects are essential for electron scattering at low temperatures.@xcite in a contrast , _ c_oriented extended defects , such as out - of - plane dislocations@xcite and twin boundaries , which in the case of ybco films usually form much more dense network than in single crystals,@xcite are currently shown to be the most important ones for perfect epitaxially grown ybco films . despite the perfect crystallinity , different types of linear defects ( dislocations ) , as well as dislocation arrays , have been identified by tem / hrem in these films.@xcite in order to understand the high - frequency electromagnetic behavior of these films , the existence of edge dislocations arrays should be taken into account , in particular , out - of - plane edge dislocations , associated with low - angle tilt dislocation boundaries . tem / hrem / xrd / afm characterization of the films under study revealed a smooth surface ( peak - to - valley is 2 nm ) , high average in - plane density of out - of - plane edge dislocations ( @xmath44@xmath45 @xmath46 ) and a big size of single - crystalline domains ( @xmath47 = 250 nm ) , which are separated by low - angle dislocation boundaries . in - plane misalignment of the domains is as low as 0.51.0@xmath48.@xcite basing on this difference of defect structures of ybco perfect films and single crystals , in the next section we will suggest a model , which can explain ( at least qualitatively ) the main features of the observed two - peak nonmonotonous behavior of @xmath4 dependence in perfect ybco films , as well as the difference from a single - peak @xmath4 dependence in single crystals . the value of microwave surface resistance @xmath11 in a linear regime of microwave response of superconductor at zero applied dc magnetic field is directly determined by the real part @xmath33 of high - frequency electron conductivity @xmath10 of superconductor : @xmath49 while the surface reactance @xmath22 is determined by the ac penetration depth @xmath37 , which in principle may be different from the london penetration depth @xmath50 : @xmath51 @xmath52 where @xmath53 is the contribution of excited quasiparticles to the screening properties of superconductor . thus , the observed peaks of @xmath4 dependence reveal also the temperature dependence of @xmath33 , because the london penetration depth @xmath50 in eq . ( [ rs ] ) is a monotonous function of temperature . similarly , peaks in @xmath5 and @xmath27 dependencies accordingly to eqs . ( [ xs ] ) , ( [ lambda ] ) are related also to the contribution of the normal component of electron fluid . the @xmath54 value is generally ascribed to the contribution of `` normal '' component of electron fluid to the ac conductivity in the framework of so - called `` two - fluid '' model of superconductor . it should be mentioned that the phenomenological `` two - fluid '' model , which is frequently used for description of microwave properties of superconductors , @xcite also follows from the microscopic bcs theory . a very essential feature of the microscopic approach is that the role of normal component of electron fluid in superconductor is played by a gas of bogolyubov quasiparicles , which are determined as a superposition of electron and hole states in a normal fermi liquid . due to this circumstance , the normal electron fluid in superconductors has quite different properties comparatively to those in a normal metal.@xcite we will take into account the peculiarities of the `` normal '' electron fluid of bogolyubov quasiparticles in superconductor and will show that the principal features of hts microwave response , including the observed two peaks of @xmath4 and @xmath5 dependencies , can be qualitatively explained using the boltzman kinetic equation approach for bogolyubov quasiparticles with a few additional assumptions about the symmetry of superconducting state in hts and its dependence on the concentration of static defects ( impurities , oxygen vacancies , dislocations , etc . ) . namely , we will assume that the case of anisotropic _ s_+_d _ pairing is realized@xcite ( see fig . [ fig.6 ] ) : schematic representation of anisotropic pairing and scattering processes : ( a ) 3d scattering on point - like defects ; ( b ) in - plane scattering on extended defects ; ( c ) scattering of quasiparticles in the momentum space in the case of _ s_+_d _ pairing . ] @xmath55 and the relation between angle - dependent components @xmath56 of the pairing potential @xmath57 described by eq . ( [ delta ] ) is rather sensitive to the defect concentration and to different kind of borders ( e.g. , film surfaces and twin boundaries).@xcite in order to calculate the contribution of bogolyubov quasiparticles to the conductivity value at microwave frequencies we will use the boltzman kinetic equation for nonequilibrium distribution function of quasiparticles @xmath58 . this approach is well known and was widely used for theoretical consideration of electron kinetic properties in normal metals.@xcite due to its relative simplicity it allows in principle to take into account peculiarities of electron spectrum and fermi surface in hts@xcite as well as different mechanisms of electron scattering . this approach was argued on the base of microscopic bcs theory in the quasiclassical limit for quasiparticles in superconducting state.@xcite the boltzman equation in this case may be written in the form:@xcite @xmath59 where @xmath60 is the energy of quasiparticles ; @xmath61 is the energy of a normal electron in a vicinity of the fermi surface , @xmath62 and @xmath63 are the fermi velocity and momentum , respectively , @xmath64 is the equilibrium fermi distribution function of quasiparticles , @xmath65 is the group velocity of quasiparticles , @xmath66 is the quasiparticle charge , which is different from the normal electron charge @xmath67 , because a bogolyubov quasiparticle is a superposition of electron and hole states in the fermi liquid,@xcite @xmath68 is the relaxation time of quasiparticles , which in the case of elastic scattering by static defects can be calculated from the following well known expression:@xcite @xmath69 integration in eq . ( [ tau ] ) is performed in the vicinity of the fermi surface . @xmath70 is the matrix element of electron scattering on the defect . @xmath71 is the so - called `` coherence factor '' : @xmath72 which describes the difference between scattering of quasiparticles comparatively to usual electrons in the framework of bcs theory@xcite and provides a strong dependence of the relaxation time on the quasiparticle energy even in the case of elastic scattering on static defects . in the case of isotropic @xmath73wave pairing and scattering on point - like defects this dependence can be estimated as:@xcite @xmath74 so , @xmath75 diverges at the edge of the gap when @xmath76 and the group velocity @xmath77 goes to zero . for the sake of simplicity we have assumed in eq . ( [ tau ] ) that all defects in the crystal lattice are identical and can be characterized by the same matrix element @xmath70 . if there are several kinds of scattering defects , e.g. , point - like ( oxygen vacancies ) and extended ones ( dislocations , twin boundaries ) oriented along the @xmath78axis , as they usually are in the case of epitaxial films and as it was discussed in the previous section , it follows from eq . ( [ tau ] ) that : @xmath79 the energy dependencies of the relaxation rates in the right hand side of eq . ( [ tau1 ] ) for point - like and extended defects may be quite different because point like defects can scatter electrons in all directions of momentum space , while scattering on extended defects can proceed only with a conservation of momentum along the @xmath78axis . this becomes especially important for layered hts materials , where electrons move mainly within cu - o layers . in the case of _ d_wave or assumed _ s_+_d _ ( or other type ) anisotropic pairing , an additional strong dependence of the relaxation time on the quasiparticles energy @xmath80 has to appear due to a confinement of momentum space , where the quasiparticle can be scattered , with a decrease of @xmath80 . this effect follows directly from eq . ( [ tau ] ) due to the @xmath81function term in the integrand . in a certain sense , this effect can be considered as an analogue of andreev reflection of quasiparticles in the momentum space ( fig . [ fig.6 ] ) . the confinement of momentum space for quasiparticle scattering is mostly essential for the scattering on extended defects , leading to a very rapid increase of @xmath82 , when @xmath83 decreases and approaches the value @xmath84 ( @xmath85 ) as it is shown in fig . [ fig.7 ] : @xmath86 energy dependence of the relaxation rate @xmath87 for the case of _ d_ or _ s_+_d_electron pairing and predominant scattering on extended linear defects . ] the general solution for the quasiparticle ac conductivity can be obtained from eq . ( [ boltz ] ) by a usual manner.@xcite the real part @xmath88 can be written in form : @xmath89 a strong temperature dependence of @xmath33 as well as its frequency dependence arise usually due to the @xmath90 term in the integrand of eq . ( [ sigma ] ) . quite formally this solution can be rewritten in the drude - like form : @xmath91 where @xmath92 is the effective concentration of thermally excited quasiparticles , @xmath93 denotes thermal averaging . as it was originally supposed in refs.@xcite , the nonmonotonous character and appearance of @xmath26 peak dependence as well as its frequency dependence can be explained properly by a strong increase of quasiparticle relaxation time with temperature lowering according to the drude expression for the ac conductivity @xmath33 . the peak position corresponds to the condition @xmath94 . we suppose that this explanation is valid also in our case , when two peaks are observed . the emergence of two peaks instead of one can be explained by existence of two different @xmath84 values for the assumed case of _ s_+_d _ pairing ( see fig . [ fig.6 ] ) , while the sharpness of these peaks comparatively to the peak in single crystals is determined by a very strong energy dependence of the relaxation time ( eq . ( [ tau2 ] ) ) , when quasiparicles are scattered preferably by extended defects ( see fig . [ fig.7 ] ) . this strong energy dependence of @xmath95 transforms into sharp peaks of @xmath96 according to eq . ( [ drude ] ) . the present model can explain also some additional features of the microwave response of ybco films , such as ( i ) smearing of peaks with the frequency increase , ( ii ) lowering of @xmath26 and smearing of peaks with an increase of point - like defect concentration , ( iii ) general quasi - linear behavior of @xmath26 at moderate temperatures . the obtained results shown in figs . [ fig.1][fig.5 ] , that is the nonmonotonous two - peak structure of @xmath38 in perfect single - crystalline ybco epitaxial films , is a strong argument for the scenario of anisotropic electron pairing in hts . the observed for the first time two - peak peculiarity of @xmath7 dependence for the most perfect single - crystalline ybco films , as well as the difference from nonmonotonous @xmath38 dependence for perfect single crystals ( and also some less perfect films ) , can be explained , using just two assumptions : ( i ) the anisotropic _ s_+_d _ character of electron pairing and ( ii ) the dominant role of extended _ c_oriented defects in electron scattering processes . these assumptions look quite natural with regard to thin films , where surfaces and/or twin boundaries can lead to more complicated character of electron pairing than the pure _ d_wave pairing in perfect single crystals.@xcite on the other hand , the extended @xmath78oriented linear or planar defects ( most probably , out - of - plane edge dislocations and twins ) can play a dominant role in electron scattering . in the case of untwined single crystals there are no extended defects . therefore , only point defects are essential for electron scattering at low temperatures . the above two assumptions , which seem to be specific for thin films , distinguish them from single crystals and , thus , provide the difference in microwave response : one broad peak of @xmath4 for single crystals and two sharp peaks for perfect films . for less perfect films with a higher number of point defects and in the case when only pure _ d_wave pairing takes place , their behavior at microwave frequencies is rather similar to that of single crystals . the observed two peaks of @xmath4 are much more narrow comparatively to the single peak for perfect single crystals due to different defect structures in films and single crystals as it was discussed above . namely , a large number of extended defects along with an anisotropic pairing can lead to emergence of a sharp peak at @xmath97 as it follows from eq . ( [ sigma ] ) . the second peak at a lower temperature is caused by the anisotropy of pairing potential ( existence of two different values @xmath84 for different directions in momentum space in the case of _ s_+_d _ electron pairing as it is shown schematically in fig . [ fig.6 ] ) . the present model for quasiparticle conductivity allows also to understand the frequency dependence of the observed peculiarities and their smearing , when the number of point - like defects increases leading to an increase of @xmath98 . it should be noted , that in a contrast to suggestions made in some theoretical works , it does not seem to be necessary to take into account a contribution of inelastic quasiparticle scattering by collective excitations ( magnons , phonons , etc . ) . the two - peak temperature dependence of @xmath33 can occur in accordance to eq . ( [ drude ] ) even in the case of elastic scattering by extended static defects due to additional effect of anisotropy caused by anisotropic _ s_+_d_wave pairing , which leads to the confinement of momentum space available for scattered quasiparticles as it was discussed above . the observed two - peak character of @xmath26 dependence is an feature of the most perfect quasi - single - crystalline ybco films , characterized by a smooth surface , low concentration of defects , large domain size and low - angle boundaries between them . the @xmath26 dependence in less perfect films is monotonous . we suppose that the two - peak character of @xmath26 , observed experimentally for the first time , is an intrinsic fundamental property and reveals the peculiarities of anisotropic electron pairing , which manifests in the microwave electron response and can be properly described using the boltzman kinetic equation approach for bogolyubov quasiparticles . p. j. turner , r. harris , s. kamal , m. e. hayden , d. m. broun , d. c. morgan , a. hosseini , p. dosanjh , g. mullins , j. s. preston , r. liang , d. a. bonn , and w. n. hardy , phys . * 90 * , 237005 ( 2003 ) . v. m. pan , v. s. flis , v. a. komashko , o. p. karasevska , v. l. svetshnikov , m. lorenz , a. n. ivanyuta , g. a. melkov , e. a. pashitskii , and h. w. zandbergen , ieee trans . supercond . * 11 * , 3960 ( 2001 ) . b. dam , j. m. huijbregtse , f. c. claassen , r. c. f. van der geest , g. doornbos , j. h. rector , a. m. testa , s. freisem , j. c. martinez , b. stauble - pumpin , and r. griessen , nature ( london ) * 399 * , 439 ( 1999 ) . a. g. aronov , yu . m. galperin , v. l. gurevich , and v. i. kozub , in _ nonequilibrium superconductivity _ , edited by d. n. langenberg and a. i. larkin ( elsevier science publishers b.v . , amsterdam , 1986 ) , p. 325 phys . * 30 * , 539 ( 1981 ) .

temperature dependencies of microwave surface impedance , @xmath0 , were measured for perfect _ c_-oriented ybco thin films deposited on ceo@xmath1buffered sapphire substrates . the measurements were performed with a use of three copper cylindrical resonators operating at @xmath2 mode ( @xmath3 = 34 , 65 , 134 ghz ) , which incorporated the studied ybco films as end plates . the measurements revealed a distinct two - peak structure of @xmath4 and @xmath5 dependencies with peaks at 2830 k and 50 k. the peaks become smeared at higher frequencies as well as in applied dc magnetic field ( @xmath6 koe ) , while the peak positions remain almost unchanged . for less perfect , e.g. , pld films , @xmath4 and @xmath5 dependencies are monotonous ( power law ) . the two - peak @xmath7 dependencies for ybco films differ from those for high quality ybco single crystals , where only one much broader frequency - dependent peak of @xmath4 was detected earlier . the two - peak @xmath7 behavior is believed to be an intrinsic electron property of extremely perfect quasi - single - crystalline ybco films . a theoretical model is suggested to explain the observed anomalous @xmath7 behavior . the model is based on the boltzman kinetic equation for quasiparticles in layered hts cuprates . it takes into account the supposed _ s_+_d_wave symmetry of electron pairing and strong energy dependent relaxation time of quasiparticles , determined mainly by their elastic scattering on extended defects parallel to the _ c_axis ( e.g. , _ c_oriented dislocations and twin boundaries ) .

You are an expert at summarizing long articles. Proceed to summarize the following text: let @xmath0 be a proper birational map between smooth varieties over a field of characteristic zero . a commutative diagram @xmath1 where @xmath2 and @xmath3 are sequences of blowups of smooth centers , is called a strong factorization of @xmath4 . the existence of a strong factorization is an open problem in dimension @xmath5 and higher . the local version of the strong factorization conjecture replaces the varieties by local rings dominated by a valuation on their common fraction field , and the smooth blowups by monoidal transforms along the valuation . the local strong factorization was proved by c. christensen @xcite in dimension @xmath6 for certain valuations . a complete proof of the @xmath6-dimensional case was given by s. d. cutkosky in @xcite , where he also made considerable progress towards proving the conjecture in general . we prove the local factorization conjecture in any dimension ( see section [ sec - ring ] for notation ) : [ thm1 ] let @xmath7 and @xmath8 be excellent regular local rings containing a field @xmath9 of characteristic zero . assume that @xmath7 and @xmath8 have a common fraction field @xmath10 and @xmath11 is a valuation on @xmath10 . then there exists a local ring @xmath12 , obtained from both @xmath7 and @xmath8 by sequences of monoidal transforms along @xmath11 . the toric version of the strong factorization problem considers two nonsingular fans @xmath13 and @xmath14 with the same support and asks whether there exists a common refinement @xmath15 @xmath16 obtained from both @xmath13 and @xmath14 by sequences of smooth star subdivisions . again , this is not known in dimension @xmath6 or higher . the local toric version replaces a fan by a single cone and a smooth star subdivision of the fan by a smooth star subdivision of the cone together with a choice of one cone in the subdivision . we assume that the choice is given by a vector @xmath17 in the cone : we choose a cone in the subdivision containing @xmath17 . if @xmath17 has rationally independent coordinates , then it determines a unique cone in every subdivision ( all cones are rational ) . we call such a vector @xmath17 a _ valuation _ and the subdivision with a choice of a cone a _ subdivision _ along the valuation . we prove : [ thm2 ] let @xmath18 and @xmath19 be nonsingular cones , and let @xmath20 be a vector with rationally independent coordinates . then there exists a nonsingular cone @xmath21 obtained from both @xmath18 and @xmath19 by sequences of smooth star subdivisions along @xmath17 . the proof of theorem [ thm2 ] is a generalization of the proof given by c. christensen @xcite in dimension @xmath6 . theorem [ thm1 ] follows directly from theorem [ thm2 ] and the monomialization theorem proved by s. d. cutkosky @xcite . theorem [ thm1 ] ia also stated in @xcite , but the proof refers to the strong factorization theorem in @xcite which contains a gap @xcite . we use the same reduction to the toric case , but replace the reference to strong factorization by theorem [ thm2 ] . one can define a more general version of local toric factorization . consider a game between two players @xmath22 and @xmath23 , where the player @xmath22 subdivides the cone @xmath19 or @xmath18 and the player @xmath23 chooses one cone in the subdivision ( and renames it again @xmath19 or @xmath18 ) . then the strong factorization conjecture states that @xmath22 always has a winning strategy : after a finite number of steps either @xmath24 or the interiors of @xmath19 and @xmath18 do not intersect . the proof of theorem [ thm2 ] given in section [ sec - toric ] does not extend to this more general case . a positive answer to the global strong factorization conjecture for toric varieties would imply that @xmath22 always has a winning strategy . conversely , a counterexample to the local factorization problem would give a counterexample to the global strong factorization conjecture . * acknowledgments . * i have benefited a great deal from discussions of the factorization problem with dan abramovich , kenji matsuki and jarosaw wodarczyk . it was jarosaw s suggestion to look for a counterexample in dimension @xmath25 that motivated the current proof . let @xmath26 be a lattice and @xmath18 a rational polyhedral cone in @xmath27 generated by a finite set of vectors @xmath28 @xmath29 we say that @xmath18 is _ nonsingular _ if it can be generated by a part of a basis of @xmath30 . a nonsingular @xmath31-dimensional cone has a unique set of minimal generators @xmath32 , and we write @xmath33 we consider nonsingular cones only . when we draw a picture of a cone , we only show a cross - section . thus , a @xmath6-dimensional cone is drawn as a triangle . let @xmath34 be a nonsingular @xmath35-dimensional cone , and let @xmath36 be a vector @xmath37 such that @xmath38 are linearly independent over @xmath39 . if @xmath40 , then precisely one of the cones @xmath41 contains @xmath17 . the cone containing @xmath17 is called a _ star subdivision of @xmath18 at @xmath42 along @xmath17_. the subdivision is again a nonsingular cone . we often denote a star subdivision of a cone @xmath18 again @xmath18 , and name its generators also @xmath43 . let us consider the situation of theorem [ thm2 ] . it is easy to see that after star subdividing @xmath19 sufficiently many times we may assume that @xmath44 . we say that a configuration @xmath44 is _ factorizable _ if the statement of theorem [ thm2 ] holds . we say that @xmath44 is _ directly factorizable _ if the statement of theorem [ thm2 ] holds with @xmath45 . the vector @xmath17 is not needed for direct factorizability . the following lemma is well - known : [ lem - dim2 ] if the dimension @xmath46 , then @xmath44 is directly factorizable . [ lem - dim3 ] let @xmath5 and consider @xmath47 , where @xmath48 are nonsingular cones such that @xmath49 are linearly dependent . then @xmath50 is directly factorizable . * proof . * let @xmath51 be the quotient map . we claim that @xmath52 are nonsingular cones with respect to the lattice @xmath53 . this is clear for the cone @xmath18 ; for @xmath19 note that the generators @xmath54 of @xmath30 map to generators of @xmath53 . ( more precisely , @xmath55 , @xmath56 , where @xmath57 is primitive , @xmath58 . ) now we apply lemma [ lem - dim2 ] to the configuration @xmath59 . then after a finite sequence of star subdivisions of @xmath18 at vectors lying in @xmath60 , we may assume that @xmath61 if we express @xmath62 , then it follows from the nonsingularity of @xmath19 that @xmath63 . in other words , the cone @xmath19 lies in the subdivision @xmath64 of @xmath18 . performing a sequence of such star subdivisions , we get to the situation where @xmath65 . finally , @xmath66 is strongly factorizable by lemma [ lem - dim2 ] , thus a sequence of star subdivisions of @xmath18 at vectors lying in @xmath67 finishes the proof . by the previous lemma , to show that @xmath50 is factorizable , we have to find a sequence of star subdivisions of @xmath19 such that the condition of the lemma is satisfied . we prove this in any dimension . [ lem - align ] let @xmath68 and consider a configuration @xmath69 . there exists a cone @xmath70 , obtained from @xmath19 by a sequence of smooth star subdivisions along @xmath17 , such that @xmath49 are linearly dependent . moreover , one can find @xmath71 such that @xmath49 satisfy the relation @xmath72 * proof . * the first part of the proof is again due to c. christensen . let us start with the case @xmath5 and prove the first half of the lemma . the algorithm for constructing @xmath71 is as follows . let @xmath51 be the projection and let the generators @xmath54 of @xmath19 be ordered so that @xmath73 . if @xmath74 or @xmath75 , then we are done . otherwise star subdivide @xmath19 at @xmath76 and repeat . to see that this algorithm always terminates , let @xmath77 be defined by : @xmath78 here @xmath79 , @xmath80 , and @xmath81 , @xmath82 . then the algorithm can be described as follows . consider the matrix @xmath83 if some @xmath84 , then we are done . otherwise , choose columns @xmath85 and @xmath86 such that @xmath87 and @xmath88 have the same sign and subtract the @xmath85th column from the @xmath86th if @xmath89 and @xmath86th column from the @xmath85th if @xmath90 . since we always choose columns where @xmath87 and @xmath88 have the same sign , it is clear that @xmath91 does not increase in this process , and it suffices to prove that either the algorithm terminates or @xmath91 drops after a finite number of steps . suppose that @xmath92 , and @xmath93 does not change as we run the algorithm . then @xmath94 also does not change , and every time we choose columns @xmath85 and @xmath6 , we subtract @xmath94 from @xmath95 . it is clear that columns @xmath96 and @xmath97 can be chosen only a finite number of times in a row , hence we choose column @xmath6 infinitely many times . since we can not subtract @xmath94 from @xmath98 or @xmath99 infinitely many times and have a positive result , we get a contradiction . this proves the first half of the lemma for @xmath5 . next let us prove the `` moreover '' part for @xmath5 . we start with a matrix @xmath100 if also @xmath101 , then by nonsingularity of @xmath19 we have @xmath102 . we choose columns @xmath96 and @xmath97 the necessary number of times to get @xmath103 : @xmath104 if both @xmath105 and @xmath106 are nonzero then they must have different signs . hence , if @xmath107 then we are done . otherwise , since @xmath108 , we may assume that @xmath109 . we perform star subdivisions of @xmath19 by choosing columns @xmath97 and @xmath6 the necessary number of times to get to the matrix @xmath110 after this , we run the algorithm as before . for instance , since @xmath105 and @xmath111 have the same sign , at the next step we choose columns @xmath96 and @xmath6 . if we subtract the third column from the first , then latexmath:[$\max_i first column from the third . as before , if @xmath91 does not decrease , then we are subtracting @xmath98 from @xmath99 or @xmath94 infinitely many times , and this gives a contradiction . for @xmath113 we have a matrix @xmath114 we can apply the @xmath5 case to the last three columns and achieve @xmath115 ; then apply the same algorithm to columns @xmath116 to get @xmath117 , and so on , until all but two of the @xmath87 are nonzero . to prove the second half of the lemma , we apply the @xmath5 case to three columns , including the ones with @xmath118 . * proof of theorem [ thm2 ] . * we may assume that @xmath119 , and using lemma [ lem - align ] , we may also assume that @xmath49 satisfy the relation @xmath120 let @xmath51 be the projection . then @xmath121 are both nonsingular with respect to the lattice @xmath53 . the relation @xmath122 implies that @xmath123 is a minimal generator of @xmath124 . in particular , @xmath125 restricts to isomorphisms of cones and lattices : @xmath126 by induction on the dimension @xmath35 , we have a factorization of @xmath127 . unlike the case @xmath5 , we may also have to subdivide @xmath124 . consider a star subdivision of @xmath124 at @xmath128 , @xmath129 , and define @xmath130 and @xmath131 by : @xmath132 now star subdividing @xmath19 first at @xmath130 and then at @xmath131 along @xmath17 , the resulting cone again satisfies the relation @xmath122 ( after possibly reordering the generators ) , and its image under @xmath125 is the star subdivision of @xmath124 at @xmath133 along @xmath134 . in other words , every star subdivision of @xmath124 can be lifted to a subdivision of @xmath19 . thus after a finite sequence of star subdivisions of @xmath19 we may assume that @xmath135 is directly factorizable . the remaining proof is the same as in the @xmath6-dimensional case . star subdividing @xmath18 at vectors lying in the face @xmath136 , we may assume that @xmath137 if @xmath138 , then @xmath139 for @xmath140 , hence after star subdividing @xmath18 at vectors lying in the face @xmath141 , we may assume that @xmath142 for @xmath143 . now @xmath144 are nonsingular cones , hence directly factorizable by lemma [ lem - dim2 ] . a sequence of star subdivisions of @xmath18 at vectors lying in the face @xmath145 finishes the proof . j. wodarczyk has noted that is makes sense to consider the local toric factorization problem also for a vector @xmath17 with rationally dependent coordinates , and this problem can be reduced to the rationally independent case . we bring here an argument for such a reduction . similar reduction appears in s. d. cutkosky s proof of the monomialization theorem @xcite . consider a nonsingular @xmath35-dimensional cone @xmath18 and a vector @xmath36 , with possibly rationally dependent coordinates . a star subdivision of @xmath18 along @xmath17 is a star subdivision of @xmath18 and a choice of an @xmath35-dimensional cone in the subdivision containing @xmath17 ( i.e. , in case there are more than one such cone , we are free to choose any one of them ) . the factorization problem then is : given two @xmath35-dimensional nonsingular cones @xmath146 and a vector @xmath20 , there exists a nonsingular cone @xmath21 obtained from both @xmath19 and @xmath18 by sequences of star subdivisions along @xmath17 . it is clear that the factorization problem has a solution only if the interiors of @xmath18 and @xmath19 intersect nontrivially . we assume that this is the case initially and after every subdivision we choose a cone containing @xmath17 such that this condition again holds . an extreme case of the factorization problem is when @xmath147 . then a factorization along any vector @xmath148 ( for instance , @xmath149 with rationally independent coordinates ) is also a factorization along @xmath17 . if @xmath150 , we reduce the factorization problem to the case of @xmath17 with rationally independent coordinates as follows . the first reduction step is to star subdivide both @xmath19 and @xmath18 along @xmath17 to get to the situation where @xmath151 , @xmath34 and @xmath152 such that the coordinates of @xmath17 with respect to @xmath153 ( hence also with respect to @xmath154 ) are rationally independent . for this write @xmath155 and consider the vector @xmath156 with nonnegative entries . it is a simple exercise to show that after a finite sequence of column operations where one subtracts @xmath95 from @xmath157 for @xmath158 , we get to the vector ( after reordering the components ) @xmath159 , such that @xmath160 are linearly independent over @xmath39 . after a similar sequence of star subdivisions of @xmath18 , we get @xmath161 . note that @xmath162 is the smallest subspace of @xmath163 spanned by rational vectors and containing @xmath17 . the next step is to use the rationally independent case and factor @xmath164 . thus after a finite sequence of star subdivisions of @xmath18 and @xmath19 we may assume that the two cones have a common face @xmath165 containing @xmath17 . after additional subdivisions of @xmath19 we may also assume that @xmath166 . the final step is to consider the projection @xmath167 , and proceed by induction on dimension the same way as in the proof of theorem [ thm2 ] . we recall in this section the monomialization theorem of s. d. cutkosky , and then prove theorem [ thm1 ] . let @xmath168 be a regular local ring of dimension @xmath35 containing a field @xmath9 of characteristic zero , and let @xmath11 be a valuation on the fraction field of @xmath7 , such that the valuation ring @xmath169 dominates @xmath7 . let @xmath170 be a subset of a system of regular parameters @xmath171 of @xmath7 . then the homomorphism @xmath172)_p,\ ] ] for some @xmath173 , and @xmath174 a prime ideal lying over @xmath175 , is called a _ monoidal transform _ of @xmath7 . if @xmath176 is again dominated by the valuation @xmath11 , we say that the monoidal transform is a transform _ along the valuation @xmath11 . _ geometrically , a monoidal transform is obtained by blowing up a smooth center and localizing at a point @xmath174 above @xmath175 determined by the valuation . in the following , we will be interested in monoidal transforms with @xmath177 . let @xmath7 and @xmath8 be two excellent regular local rings of dimension @xmath35 containing a field @xmath9 of characteristic zero , both dominated by a valuation @xmath11 on their common fraction field . s. d. cutkosky proved in @xcite that after a sequence of monoidal transforms of @xmath7 and @xmath8 , one can express a system of regular parameters of @xmath8 as monomials in regular parameters of @xmath7 . more precisely , if @xmath11 has rank @xmath96 and rational rank @xmath35 ( i.e. , the value group can be embedded in @xmath178 and it contains @xmath35 rationally independent elements ) , then after a finite sequence of monoidal transforms , we may assume that a system @xmath179 of regular parameters of @xmath8 can be expressed in terms of regular parameters @xmath180 of @xmath7 as : @xmath181 where @xmath182 are nonnegative integers and @xmath183 . note that @xmath184 are rationally independent positive real numbers . if @xmath11 is an arbitrary valuation , then the matrix @xmath185 is block diagonal , with @xmath186 corresponding to the same block having rationally independent values @xmath187 ( @xcite , theorem 4.4 ) . in the following proof we will perform monoidal transforms with centers @xmath188 or @xmath189 , with @xmath85 and @xmath86 lying in the same block , hence we may assume that @xmath11 has rank @xmath96 and rational rank @xmath35 . now let us consider the situation of theorem [ thm1 ] . we assume that an embedding @xmath190 is given by monomials as above , and we have to show that after a sequence of monoidal transforms along @xmath11 , we get ( renaming parameters ) @xmath191 for @xmath192 . this follows directly from theorem [ thm2 ] , once we express the problem in terms of cones and subdivisions . let @xmath193 be a nonsingular cone , and let @xmath194 be the cone defined by @xmath195 where @xmath185 is the matrix of exponents above . since @xmath183 , the cone @xmath19 is nonsingular . we also let @xmath196 be a vector @xmath197 . now one can easily check that the monoidal transform of @xmath7 with center @xmath198 along @xmath11 corresponds to the star subdivision of @xmath19 at @xmath199 along @xmath17 ( which in terms of the matrix @xmath185 corresponds to adding one column to another ) , and similarly for @xmath8 and @xmath18 ( in terms of @xmath185 , subtract one row from another ) . applying theorem [ thm2 ] , after a finite sequence of monoidal transforms of @xmath7 and @xmath8 , the matrix @xmath182 is the identity matrix , hence @xmath200 . k. matsuki , _ correction : `` a note on the factorization theorem of toric birational maps after morelli and its toroidal extension '' [ tohoku math . j. ( 2 ) 51 ( 1999 ) , no . 4 , 489537 ] by d. abramovich , matsuki and s. rashid _ , tohoku math . j. ( 2 ) 52 ( 2000 ) , no . 4 , 629631 .

the strong factorization conjecture states that a proper birational map between smooth algebraic varieties over a field of characteristic zero can be factored as a sequence of smooth blowups followed by a sequence of smooth blowdowns . we prove a local version of the strong factorization conjecture for toric varieties . combining this result with the monomialization theorem of s. d. cutkosky , we obtain a strong factorization theorem for local rings dominated by a valuation .

You are an expert at summarizing long articles. Proceed to summarize the following text: investigating deuterium chemistry is useful to put constraints on the ionization fraction , temperature , density and thermal history of dense molecular clouds @xcite . the observations of multiply deuterated molecules in space , e.g. @xcite and references therein , have shown the necessity to reexamine some reaction rates in chemical networks @xcite , elemental d / h ratio in cold dense gas @xcite and the density structure in sources such as l1544 and @xmath11 oph d @xcite , as well as the effects of accretion on grains @xcite , possible effects of internal dynamical motion @xcite , and the evolution of ice mantles in dense clouds and cores @xcite . the first multiply deuterated interstellar molecule detected has been d@xmath1co almost twenty years ago @xcite . since then , the study of deuterated molecules in the ism has rapidly increased as they have been proven to be a unique observational probe of the early stages in low - mass star formation . multiply deuterated species such as triply deuterated ammonia have been detected with a surprisingly high abundance ratio of 10@xmath12 with respect to their fully hydrogenated forms @xcite . by comparing this ratio with the elemental d / h ratio ( 1.65@xmath13 , linsky et al . 1993 ) it is easily seen that there is a remarkable enrichment in deuterium in this and other molecules . in iras 16293 - 2422 ceccarelli et al . ( 1998 ) measured d@xmath1co / h@xmath1co = 5% . in the same source parise ( 2004 ) measured cd@xmath0oh / ch@xmath0oh = 1.4% . this enrichment in deuterium has mainly been explained by the exothermicity of the h - d exchange reactions , and by the depletion of co and o onto dust grains @xcite , which enhances the abundance of the multiply deuterated forms of h@xmath14 and , upon dissociative recombination , the d / h ratio . roberts et al . ( 2003 ) calculated d / h abundance ratios close to 0.3 ( 20,000 times the cosmic value ) , in regions with large amount of co freeze - out . despite extensive studies of nearly 30 deuterated molecules in the interstellar gas , the processes regulating their formation are not completely understood , in particular the relative importance of gas - phase versus grain - surface chemistry . for example , simple gas phase chemical models do not reproduce the deuterium enrichment observed in some of these species . in particular , the formation of methanol on grain mantles is presented as an explanation for its enhanced deuteration @xcite ; for h@xmath1co instead , the gas - phase reaction path can not be excluded . in fact , in the orion bar photodissociation region hdco seems to be formed solely in the gas phase @xcite . deuteration of ammonia can be reproduced with gas - phase models ( e.g. roueff et al . 2005 ) , although nh@xmath0 and its deuterated forms are also expected to form on the surface @xcite . the first detection of c - c@xmath0h@xmath1 in the laboratory @xcite allowed the identification of several u - lines previously detected in space @xcite , namely the strong lines at 85338 and 18343 mhz . since then , cyclopropenylidene has been proven to be one of the most abundant and widespread molecules in our galaxy . it has been observed in the diffuse gas , cold dark clouds , giant molecular clouds , photodissociation regions , circumstellar envelopes , and planetary nebulae @xcite . given the high abundance of the normal species , both c - c@xmath0hd and the singly substituted @xmath9c species ( off axis ) have been observed with a good signal to noise in cold dark clouds . furthermore cyclopropenylidene shows an enhancement in deuterium fractionation in cold dark clouds , for example gerin et al . ( 1987 ) measured a 1:5 ratio of the 2@xmath6 - 1@xmath15 lines of c - c@xmath0hd and c - c@xmath0h@xmath1 in tmc1 . the reactions which lead to such a high deuteration are still poorly understood . some rates of reactions which may be involved in the formation of deuterated c - c@xmath0h@xmath1 have been measured by savi et al . ( 2005 ) but to our knowledge they have not been included in models so far . last year the centimeter and millimeter wavelength spectra of doubly deuterated c - c@xmath0h@xmath1 have been measured in the laboratory @xcite , allowing for the first time a search for c - c@xmath0d@xmath1 in space . like its fully hydrogenated counterpart , c - c@xmath0d@xmath1 presents the spectrum of an oblate asymmetric top with b - type transitions . furthermore c - c@xmath0d@xmath1 shows a deuterium quadrupole splitting resolvable at very low @xmath16 . given the presence of two equivalent off axis bosons , it has _ ortho _ and _ para _ symmetry species with relative statistical weight of 2:1 . unlike several of the six known multiply deuterated species observed in the radio band ( d@xmath1h@xmath17 , chd@xmath1oh , nhd@xmath1 , d@xmath1co , d@xmath1s , and d@xmath1cs ) , c - c@xmath0h@xmath1 is believed to form solely by gas phase reactions @xcite . the interplay between the gas phase and grain surface reactions in the deuteration of interstellar molecules is not clear so far , partially because there are not many probes available for testing the models : c - c@xmath0h@xmath1 is an ideal molecule for this purpose because of its easily observable transitions and because it has the possibility of double deuteration . assuming c - c@xmath0d@xmath1 is formed in the gas - phase like its fully hydrogenated counterpart , cyclopropenylidene will be a unique probe for the deuteration processes happening in the gas - phase . furthermore , since c - c@xmath0h@xmath1 is , in terms of cloud evolution , an early - type molecule @xcite , it is a particular useful tool to investigate early stages of a molecular cloud . this makes observations of its deuterated forms particularly important to test time - dependent chemical codes which include deuteration processes . here we report on the positive detection of three emission lines of c - c@xmath0d@xmath1 in the 3 mm band , namely the @xmath2 , @xmath18 , and @xmath4 transitions , towards tmc-1c and l1544 : to our knowledge this is the first search for doubly deuterated cyclopropenylidene undertaken.d@xmath1 towards l1527 in the framework of the nobeyama 45 m telescope survey at 3 mm . ] the observations have been carried out from 2012 september 28 until october 2 at the iram 30 m telescope , located in pico veleta ( spain ) , towards the starless cores tmc-1c and l1544 . the choice of the sources has been made on the basis of two simple criteria : the abundance of the normal species ( c - c@xmath0h@xmath1 ) and a high deuterium fractionation . both sources are in the taurus molecular cloud , one of the closest dark cloud systems and low - mass star forming regions in our galaxy . l1544 is a perfect test bed to investigate the initial conditions of protostellar collapse : its structure is consistent with a contracting bonnor - ebert sphere , with central densities of about 10@xmath19 @xmath20 and a peak infall velocity of @xmath210.1 km s@xmath22 at about 1000 au from the center ( e.g. keto & caselli 2010 ) . its centrally concentrated structure and measured kinematics suggests that this is a pre - stellar core at a late stage of evolution , toward star formation . tmc-1c is a relatively young core , with evidence of accreting material towards a core and immersed in a cloud with densities higher than those surrounding the l1544 core @xcite . the coordinates that were used are @xmath23 = 04@xmath2441@xmath2516@xmath26.1 @xmath27 = + 25@xmath2849@xmath2943@xmath30.8 for tmc-1c , and @xmath23 = 05@xmath2404@xmath2517@xmath26.21 @xmath27 = 25@xmath2810@xmath2942@xmath30.8 for l1544 . in the case of tmc-1c these are the same coordinates as reported by bell et al . ( 1988 ) and gerin et al . ( 1987 ) , while in the case of l1544 they correspond to the coordinates of the peak of the 1.3 mm continuum dust emission from ward - thompson et al . ( 1999 ) . in both cores we observed two lines of c - c@xmath0h@xmath1 , one line of c - h@xmath9cc@xmath1h ( off axis ) , two lines of c - c@xmath0hd and three lines of c - c@xmath0d@xmath1 , using three different tuning settings . a summary of the observed lines is reported in table 1 . the emir receivers in the e090 configuration were employed , and observations were performed in a frequency switching mode with a throw of @xmath31 mhz . all four emir sub - bands were connected to the fts spectrometer set to high resolution mode ; this delivered a final spectrum with 50 khz channel spacing ( corresponding to 0.15 km s@xmath22 at 3 mm ) and a total of 7.2 ghz of spectral coverage ( nominal bandpass of 1.8 ghz per sub - band ) . telescope pointing was checked every two hours on jupiter and was found accurate to 3 - 4 arcsec . lines of the isotopologues of c - c@xmath0h@xmath1 listed in table 1 have been detected in both sources with very high signal to noise ratio . a selection of spectra of c - c@xmath0h@xmath1 and isotopologues in tmc-1c and l1544 is shown in figure 1 . table 1 lists the observed line parameters . even the weakest line of the doubly deuterated species , 2@xmath32 - 1@xmath33 at 108 ghz , is detected at a 7.5@xmath10 level in tmc-1c ( t@xmath34 = 2.5 mk ) , and at a 9@xmath10 level in l1544 ( t@xmath34 = 4.6 mk ) . the gildas software @xcite was employed for the data processing : high order polynomials had to be used for baseline subtraction given the strong baseline produced by the frequency switching observing mode . the column densities and optical depths given in table 1 were calculated using the expressions given in the appendix . as was already pointed out by bell et al . ( 1988 ) , c - c@xmath0h@xmath1 shows two velocity components towards tmc-1c , one more intense at 6 km s@xmath22 and one less intense at 5.4 km s@xmath22 , see figure 1 . there is a hint of detection of the component at 5.4 km s@xmath22 also in c - h@xmath9cc@xmath1h , but no clear presence in the deuterated species . assuming that all lines have the same excitation temperature in both components , we expect for the component at 5.4 km s@xmath22 a line intensity of 0.06 k for c - c@xmath0hd ( 3@xmath35 - 2@xmath36 ) and 0.013 k for c - c@xmath0d@xmath1 ( 3@xmath37 - 2@xmath38 ) . comparing these estimates with the noise level in our spectra ( 0.007 k for c - c@xmath0hd and 0.002 k for c - c@xmath0d@xmath1 ) we can say that the lower velocity component is absent in the deuterated species of cyclopropenylidene : this behavior may suggest that the lower velocity component traces a hotter region , where the deuterated molecules are not present in detectable amounts . the observed line intensity ratios of c - c@xmath0d@xmath1 pose the question whether local thermal equilibrium is a valid assumption for its excitation . only for _ para_-c@xmath0d@xmath1 more than one optically thin line was detected . the ratios of the integrated 3@xmath5 - 2@xmath6/2@xmath39 - 1@xmath40 lines should be 4.3 and 3.3 for tmc-1c ( t@xmath41 = 7 k ) and l1544 ( t@xmath41 = 5 k ) , respectively , assuming thermalization of both lines to their assumed excitation temperature , while they are 1.5 and 1.35 . to gain insight into the excitation of the lines we performed radiative transfer calculations with radex @xcite . collision rates of c@xmath0h@xmath1 with h@xmath1 calculated by chandra et al . ( 2000 ) and supplied by the lamda database @xcite were used together with molecular constants of _ para_-c@xmath0d@xmath1 . calculations were done on a grid in density of molecular hydrogen , n(h@xmath1 ) , and column density of _ para_-c@xmath0d@xmath1 , n(_para_-c@xmath0d@xmath1 ) , and the results are shown in figure 2 . it is in principle possible to read the column density of c - c@xmath0d@xmath1 from figure 2 , by knowing the density of molecular hydrogen . the nominal observed line ratios would , particulary for l1544 , result in very high column densities . the lines would be very subthermally excited and very optically thick . however , given the fact that the observed line strengths run parallel to each other , a small change in the observed value would result in a substantial change in the ratio . the line intensities and @xmath42 contours of both lines show that the line strengths are very close to each other along a diagonal line spanning a large range of densities and column densities ( filling factors smaller than unity would move this line to the right in the plot ) , so that much smaller values of column densities and opacities are also in agreement with the data . the predicted excitation temperatures tend to be rather low , in the range 33.5 k , which introduces a considerable uncertainty in the column density determinations . since the transitions used for the excitation calculation for c - c@xmath0hd and c - c@xmath0d@xmath1 have similar upper state energy , the excitation behavior will naturally be degenerated . therefore , in the absence of data for more transitions for c - c@xmath0d@xmath1 , c - c@xmath0hd or c - h@xmath9cc@xmath1h , it is difficult to derive conclusive results . despite these uncertainties , we have assumed local thermodynamic equilibrium with @xmath43 values as described in the appendix . in addition to the observation of two lines of c - c@xmath0h@xmath1 , one line of c - h@xmath9cc@xmath1h ( with the @xmath9c off of the molecular axis ) , and two lines of c - c@xmath0hd , we also claim the detection of three lines of c - c@xmath0d@xmath1 in both tmc-1c and l1544 . this first interstellar detection of c - c@xmath0d@xmath1 is validated by the following reasons : * we detected all favorable transitions of c - c@xmath0d@xmath1 available in the covered frequency range . * the rest frequencies employed have laboratory accuracy @xcite , and in both sources the line shapes and velocities are in agreement with each other and with those observed for more abundant isotopologues ( see figure 1 ) . * the intensities of the c - c@xmath0d@xmath1 lines are consistent with what is expected from the deuteration of the ring in these sources , i.e. c - c@xmath0h@xmath1/c - c@xmath0hd is consistent with c - c@xmath0hd / c - c@xmath0d@xmath1 , as will be discussed below . in table 2 we present the relative abundances of the deuterated species with respect to the hydrogenated ones for cyclopropenylidene over the 27@xmath30 beam for tmc-1c and l1544 obtained from this work , and also for h@xmath1co , hco@xmath17 , n@xmath1h@xmath17 and nh@xmath0 obtained from previous work . the abundance of doubly deuterated cyclopropenylidene with respect to the normal species is ( 0.4 - 0.8)% in tmc-1c and ( 1.2 - 2.1)% in l1544 . this interval has been determined considering the differences in @xmath44 obtained from different lines . the deuteration of c - c@xmath0h@xmath1 follows the same trend observed for other molecules in both sources . it is interesting to note that the ratios [ c - c@xmath0d@xmath1]/[c - c@xmath0hd ] and [ c - c@xmath0hd]/[c - c@xmath0h@xmath1 ] are quite similar in both sources . we calculated the d / h ratio of c - c@xmath0h@xmath1 in prestellar cores using the network model of aikawa et al . ( 2012 ) . for the physical structure of the core , we adopt the collapsing core model of aikawa et al . ( 2005 ; the @xmath46=1.1 model ) and also a static model of l1544 from keto & caselli ( 2010 ) . for co depletion factors consistent with those of the two objects , i.e. @xmath47 = 3.8 for tmc-1c @xcite and @xmath47 = 14 for l1544 @xcite , the calculated column density ratio of c - c@xmath0d@xmath1/c - c@xmath0h@xmath1 is @xmath48 , consistent with the observed value of 0.6% for tmc-1c and 1.5% for l1544 . there is no need for any deuterium fractionation reactions of c - c@xmath0h@xmath1 on grain surfaces to account for the observed d / h ratio : the deuteration of cyclopropenylidene can be explained solely by gas - phase reactions . the main route of formation of deuterated cyclopropenylidene is the successive deuteration of the main species via reaction with h@xmath1d@xmath17 , d@xmath1h@xmath17 , and d@xmath14 . an example of the reaction scheme is sketched in figure 3 , considering only h@xmath1d@xmath17 as reaction partner . the depicted cycle of reactions starts with c - c@xmath0h@xmath1 and h@xmath1d@xmath17 , producing in the first step c - c@xmath0hd and subsequently c - c@xmath0d@xmath1 . the same reactions happen with d@xmath1h@xmath17 and d@xmath14 . the overall process is a series of two reactions : the proton - deuteron transfer ( slow step , red arrows ) , and the subsequent dissociative recombination with electrons ( fast step , blue arrows ) . the presence of this deuteration cycle results in a time dependent deuterium fractionation . assuming low levels of deuteration at the start , it is expected that this level increases as a function of time , reaching a stationary level after some time . other deuteration processes , e.g. the formation of c - c@xmath0hd from the reaction of c@xmath0h@xmath17 with hd , were found to be negligible . the d / h ratio of cyclopropenylidene is , therefore , directly related to that of h@xmath14 , the main deuterium donor in dark interstellar clouds . recently huang & lee ( 2011 ) have calculated highly accurate spectroscopic constants for @xmath9c and d isotopologues of c - c@xmath0h@xmath14 in order to guide the laboratory and astronomical search . since these species are intermediates in the formation of isotopic species of c - c@xmath0h@xmath1 , their detection would be useful to put more constraints on the models . doubly deuterated cyclopropenylidene appears to be a very interesting probe for the earliest stages of star formation . its formation mechanism puts important constraints on gas - phase deuteration models , and suggests the possibility of using c - c@xmath0d@xmath1 as a chemical clock . furthermore the brightness of the @xmath49 line of c - c@xmath0d@xmath1 at 97 ghz in l1544 would allow to make an on the fly map ( otf ) of the core in a reasonable amount of time , for the first time for a doubly deuterated molecule . during our observations , the line was observed with a s / n of more than 6 after just 30 minutes . emission from c@xmath50s is spread over 10 arcmin@xmath51 in l1544 @xcite . we estimated that an otf map of 6 arcmin@xmath51 with a noise level of 20 mk at the iram 30 m telescope with the emir receivers would be possible in 14 hours , allowing to map at the same time c - c@xmath0h@xmath1 and c - c@xmath0d@xmath1 . by mapping the core it will be possible to locate the deuteration peak , and put more constraints on current gas - grains models . _ acknowledgement _ + the authors thank the referee for the useful comments . this work has been supported by sfb956 . s. spezzano has been supported in her research with a stipend from the international max - planck research school ( imprs ) for astronomy and astrophysics at the universities of bonn and cologne . s. schlemmer and s.b . acknowledge support by the deutsche forschungsgemeinschaft ( dfg ) through project br 4287/1 - 1 . l.b . acknowledges support from the science and technology foundation ( fct , portugal ) through the fellowship sfrh / bpd/62966/2009 , and he is grateful to the sfb956 for the travel allowance . the column densities and optical depths given in table 1 were calculated using the following expressions . the line center opacity @xmath52 is where k is the boltzmann constant , @xmath56 is the frequency of the line , @xmath57 is the planck constant , @xmath58 is the speed of light , @xmath59 is the einstein coefficient of the transition , @xmath60 is the full width at half maximum , @xmath61 is the degeneracy of the upper state , @xmath62 is the energy of the upper state , @xmath63 is the partition function of the molecule at the given temperature @xmath43 . @xmath43 , @xmath64 , @xmath65 are the excitation , the background ( 2.7 k ) and the main beam temperatures respectively , in k. to calculate @xmath44 and @xmath66 we assumed a @xmath43 of 7 k for tmc-1c and 5 k for l1544 for all deuterated isotopologues , following gerin et al ( 1987 ) , and 8 k for tmc-1c and 6 k for l1544 for the main species and the @xmath9c isotopologues as they trace also warmer regions of the cloud . the effect of the excitation temperature on the derived column densities ratios in table 2 was found to be small , with a change of few percent upon a variation of @xmath671 k. by using these expressions we assumed that the source fills the beam , and optically thin emission obeying lte . since lines of c - c@xmath0h@xmath1 are optically thick , we derived its total column density from the total column density of c - h@xmath9cc@xmath1h assuming a @xmath68c/@xmath9c ratio of 77 , determined by wilson & rood ( 1994 ) from h@xmath1co and co as a function of distance from the galactic center , @xmath69 and @xmath66 were calculated backwards . ccccccccccccccc molecule & transition & frequency & ref.@xmath70 & e@xmath71 & t@xmath72&rms & w&b@xmath73&@xmath74 & v@xmath75&@xmath76 & n@xmath77 & n@xmath78@xmath79 & @xmath80 + & ( ortho / para)&(ghz ) & & ( cm@xmath22)&(k)&mk & ( k km s@xmath22)&%&(arcsec)&(km s@xmath22 ) & ( km s@xmath22 ) & ( @xmath82 @xmath83 ) & ( @xmath84 @xmath83 ) & + * tmc- 1c + c - c@xmath0h@xmath1 & 2@xmath36 - 1@xmath85 ( o ) & 85.338 & 1 & 4.48 & 2.91&7&1.05(1)&81&29 & 5.996(2 ) & 0.338(4 ) & 25(1 ) & 22(1 ) & 1.887 + & 2@xmath36 - 1@xmath85 ( o ) & 85.338 & 1 & 4.48 & 1.27 & 7&0.414(9)&81&29&5.361(4 ) & 0.307(8 ) & & & + & 3@xmath86 - 3@xmath37 ( p ) & 84.727 & 1 & 11.21 & 0.16&7&0.047(2)&81&29 & 5.984(6 ) & 0.27(2 ) & 3.5(2 ) & 22(1 ) & 0.148 + c- h@xmath9cc@xmath1h & 2@xmath36 - 1@xmath85 & 84.185 & 2 & 4.40 & 0.22 & 7&0.060(3)&81&29 & 5.977(7 ) & 0.26(1 ) & 0.52(2 ) & 0.62(3 ) & 0.056 + c - c@xmath0hd & 2@xmath87 - 1@xmath33 & 95.994 & 2&5.25 & 0.1&7 & 0.033(2)&80&27 & 6.03(1 ) & 0.33(3 ) & 1.8(1 ) & 2.8(2 ) & 0.026 + & 3@xmath35 - 2@xmath36 & 104.187 & 2 & 7.54 & 0.34&7&0.091(2 ) & 79&25 & 6.034(3 ) & 0.250(6 ) & 0.63(1 ) & 1.1(3 ) & 0.092 + c - c@xmath0d@xmath1 & 3@xmath35 - 2@xmath36 ( p ) & 94.371 & 3 & 6.84 & 0.04 & 2&0.009(1)&80&27 & 6.07(1 ) & 0.23(2)&0.063(5 ) & 0.17(1)&0.010 + & 3@xmath37 - 2@xmath38 ( o ) & 97.761 & 3 & 6.87 & 0.07 & 2&0.017(1)&80&26 & 6.062(5 ) & 0.233(9 ) & 0.11(4 ) & 0.15(6)&0.018 + & 2@xmath88 - 1@xmath33 ( p ) & 108.654 & 3 & 5.49 & 0.02&2&0.006(1)&78&24&6.06(1 ) & 0.25(3 ) & 0.032(4 ) & 0.09(1 ) & 0.005 + * l1544 + c - c@xmath0h@xmath1 & 2@xmath36 - 1@xmath85 ( o ) & 85.338 & 1 & 4.48 & 2.44&10&1.35(1 ) & 81&29&7.180(2 ) & 0.520(4 ) & 50(2 ) & 37(1 ) & 3.579 + & 3@xmath86 - 3@xmath37 ( p ) & 84.727 & 1 & 11.21 & 0.21&10&0.10(1)&81&29 & 7.210(8 ) & 0.46(1 ) & 4.7(2 ) & 37(1 ) & 0.172 + c- h@xmath9cc@xmath1h & 2@xmath36 - 1@xmath85 & 84.185 & 2 & 4.40 & 0.19 & 10&0.093(3)&81&29 & 7.154(8)&0.44(2 ) & 0.92(4 ) & 0.96(4 ) & 0.096 + c - c@xmath0hd&2@xmath87 - 1@xmath33 & 95.994&2 & 5.25 & 0.13 & 10&0.065(3)&80&27 & 7.17(1 ) & 0.48(3 ) & 4.1(2 ) & 6.2(3 ) & 0.066 + & 3@xmath35 - 2@xmath36 & 104.187 & 2 & 7.54 & 0.48&10&0.238(4 ) & 79&25&7.181(4 ) & 0.468(9 ) & 2.1(4 ) & 4.5(9 ) & 0.278 + c - c@xmath0d@xmath1 & 3@xmath35 - 2@xmath36 ( p ) & 94.371 & 3 & 6.84 & 0.07&5&0.032(2)&80&27&7.20(1 ) & 0.45(3 ) & 0.26(2 ) & 0.77(5 ) & 0.035 + & 3@xmath37 - 2@xmath38 ( o ) & 97.761 & 3 & 6.87 & 0.13&5&0.059(2)&80&26&7.181(7 ) & 0.43(2 ) & 0.44(2)&0.66(2 ) & 0.067 + & 2@xmath88 - 1@xmath33 ( p ) & 108.654 & 3 & 5.49 & 0.04 & 5&0.023(2)&78&24&7.17(2 ) & 0.54(5 ) & 0.16(1 ) & 0.45(4 ) & 0.020 + * * c|cc & * tmc-1c&*l1544 + & ( 0.4 - 0.8)%&(1.2 - 2.1)% + /&(3 - 15)%&(7- 17)% + @xmath89&(5 - 13)%&(12 - 17)% + @xmath89 & - & 4%@xmath90 + @xmath89 & 2%@xmath91&4%@xmath92 + @xmath89 & 8%@xmath91 & 20%@xmath92 + @xmath89 & 1%@xmath91 & 13%@xmath70 + * * d@xmath1 . the color scale gives the 3@xmath5 - 2@xmath6/2@xmath39 - 1@xmath40 line ratio , while the observed value is drawn in as a red contour . the observed integrated line intensities of 3@xmath5 - 2@xmath6 and 2@xmath39 - 1@xmath40 are shown as blue and green contours , respectively , and the calculated @xmath93 values as grey scale . ]

we report the first interstellar detection of c - c@xmath0d@xmath1 . the doubly deuterated cyclopropenylidene , a carbene , has been detected toward the starless cores tmc-1c and l1544 using the iram 30 m telescope . the @xmath2 , @xmath3 , and @xmath4 transitions of this species have been observed at 3 mm in both sources . the expected 1:2 intensity ratio has been found in the 3@xmath5 - 2@xmath6 and 3@xmath7 - 2@xmath8 lines , belonging to the para and ortho species respectively . we also observed lines of the main species , c - c@xmath0h@xmath1 , the singly deuterated c - c@xmath0hd , and the species with one @xmath9c off of the principal axis of the molecule , c - h@xmath9cc@xmath1h . the lines of c - c@xmath0d@xmath1 have been observed with high signal to noise ratio , better than 7.5@xmath10 in tmc-1c and 9@xmath10 in l1544 . the abundance of doubly deuterated cyclopropenylidene with respect to the normal species is found to be ( 0.4 - 0.8)% in tmc-1c and ( 1.2 - 2.1)% in l1544 . the deuteration of this small hydrocarbon ring is analysed with a comprehensive gas - grain model , the first including doubly deuterated species . the observed abundances of c - c@xmath0d@xmath1 can be explained solely by gas - phase processes , supporting the idea that c - c@xmath0h@xmath1 is a good indicator of gas - phase deuteration .

You are an expert at summarizing long articles. Proceed to summarize the following text: let @xmath0 be a two - sided standard brownian motion in @xmath6 having @xmath8 . if @xmath7 is a stopping time with respect to the filtration @xmath9 , then the shifted process @xmath10 is a one - sided brownian motion independent of @xmath4 . however , the two - sided shifted process @xmath11 need not be a two - sided brownian motion . moreover , the example of a fixed time shows that even if it is , it need not be independent of @xmath4 . we call a random time @xmath2 an _ unbiased shift _ ( of a two - sided brownian motion ) if @xmath2 is a measurable function of @xmath1 and @xmath11 is a two - sided brownian motion , independent of @xmath4 . we say that a random time @xmath2 _ embeds _ a given probability measure @xmath5 on @xmath6 , often called the _ target distribution _ , if @xmath4 has distribution @xmath5 . in this paper we discuss several examples of nonnegative unbiased shifts that are stopping times . however , we wish to stress that nonnegative unbiased shifts are not assumed to have the stopping time property , see for instance example [ exnonstop ] . the paper has three main aims . the first aim is to characterise all unbiased shifts that embed a given distribution @xmath5 . the second aim is to construct such unbiased shifts . in particular , we solve the skorokhod embedding problem for unbiased shifts : given any target distribution we find an unbiased shift which embeds this target distribution ( and is also a stopping time ) . the third and final aim is to discuss some properties of unbiased shifts . in particular , we discuss optimality of our solution of the skorokhod embedding problem for unbiased shifts . the case when the embedded distribution is concentrated at zero is of special interest . let @xmath12 be the local time at zero . its right - continuous ( generalised ) inverse is defined by @xmath13 = r\ } , \quad & r \geq 0,\\ \sup\{t < 0 : \ell^0 [ t,0 ] = -r\ } , \quad & r < 0.\end{cases } \end{aligned}\ ] ] note that @xmath14 and @xmath15 if @xmath16 . we prove the following theorem . [ littletwin ] let @xmath17 . then @xmath18 is an unbiased shift embedding @xmath19 . this result formalises the intuitive idea that two - sided brownian motion looks globally the same from all its ( appropriately chosen ) zeros , thus resolving an issue raised by mandelbrot in ( * ? ? ? * , p. 385 ) and reinforced in @xcite . another way of thinking about this result is that if we travel in time according to the clock of local time we always see a two - sided brownian motion . this is analogous to a well - known property of the two - sided stationary poisson process with an extra point at the origin : the lengths of the intervals between points are i.i.d . ( exponential ) and therefore shifting the origin to the @xmath20th point on the right ( or on the left ) gives us back a two - sided poisson process with an extra point at the origin . in fact , much of the work behind the present paper was inspired by recent developments for spatial point processes and random measures @xcite . in the terminology of @xcite , brownian motion is _ mass - stationary _ with respect to local time , see section [ secmassstat ] . theorem [ littletwin ] is relatively elementary . to state the further main results of this paper , we need to introduce some notation and terminology . it is convenient to define @xmath1 as the identity on the canonical probability space @xmath21 , where @xmath22 is the set of all continuous functions @xmath23 , @xmath24 is the kolmogorov product @xmath25-algebra , and @xmath26 is the distribution of @xmath1 . define @xmath27 , @xmath28 , and the @xmath25-finite and stationary measure ( see section [ secpalm ] ) @xmath29 expectations ( resp . integrals ) with respect to @xmath30 and @xmath31 are denoted by @xmath32 and @xmath33 , respectively . for any @xmath34 the shift @xmath35 is defined by @xmath36 an _ allocation rule _ @xcite is a measurable mapping @xmath37 that is _ equivariant _ in the sense that @xmath38 a _ random measure _ @xmath39 on @xmath6 is a kernel from @xmath22 to @xmath6 such that @xmath40 for @xmath31-a.e . @xmath41 and all compact @xmath42 . if @xmath39 and @xmath43 are random measures , and @xmath44 is an allocation rule such that the image measure of @xmath39 under @xmath44 is @xmath43 , that is , @xmath45 then we say that @xmath44 _ balances _ @xmath39 and @xmath43 . if @xmath44 balances @xmath39 and @xmath43 and @xmath25 is an allocation rule that balances @xmath43 and another random measure @xmath46 , then the allocation rule @xmath47 balances @xmath43 and @xmath46 . let @xmath48 be the random measure associated with the _ local time _ of @xmath1 at @xmath28 ( under @xmath26 ) , see also section [ secpalm ] . if @xmath5 is a probability measure on @xmath6 , then below implies that @xmath49 defines a random measure @xmath50 . this is an _ additive functional _ of brownian motion , that is , it has the invariance property , see section [ secpalm ] for more details . in fact , any diffuse additive functional must be of this form with a @xmath25-finite _ revuz measure _ @xmath5 , see e.g. ( * ? ? ? * chapter 22 ) . for any random time @xmath2 we define an allocation rule @xmath51 by @xmath52 since @xmath53 , there is a one - to - one correspondence between @xmath2 and @xmath51 . our key characterisation theorem is based on a result in @xcite , which will be recalled as theorem [ th2 ] below . [ main1 ] let @xmath2 be a random time and @xmath5 be a probability measure on @xmath6 . then @xmath2 is an unbiased shift embedding @xmath5 if and only if @xmath51 balances @xmath12 and @xmath50 . for any probability measure @xmath5 on @xmath6 we denote by @xmath54 the distribution of a two - sided brownian motion with a random starting value @xmath55 with law @xmath5 . we show in section [ secmassstat ] that all these distributions coincide on the invariant @xmath25-algebra . a general result in @xcite ( see also ( * ? ? ? * theorem 10.28 ) ) then implies that there is a random time @xmath2 ( possibly defined on an extension of @xmath56 ) such that @xmath57 has distribution @xmath58 ( under @xmath26 ) . the next two theorems yield a much stronger result . they show that @xmath2 can be chosen as a _ factor _ of @xmath1 , that is , as a measurable function of @xmath1 , see @xcite for a similar result for poisson processes . moreover , this factor is explicitly known . the proof is based on theorem [ main1 ] and on a general result on the existence of allocation rules balancing jointly stationary orthogonal diffuse random measures on @xmath6 with equal conditional intensities , see theorem [ main3 ] . [ main2 ] let @xmath5 be a probability measure on @xmath6 with @xmath59 . then the stopping time @xmath60=\ell^\nu[0,t]\big\}\end{aligned}\ ] ] embeds @xmath5 and is an unbiased shift . the stopping time was introduced in @xcite as a solution of the _ skorokhod embedding problem_. this problem requires finding a stopping time @xmath7 embedding a given distribution @xmath5 , see @xcite for a survey . it has apparently not been noticed before that is an unbiased shift . if @xmath5 is of the form @xmath61 where @xmath62 and @xmath63 , then theorem [ main2 ] does not apply . in fact , if @xmath64 then is an unbiased shift embedding @xmath65 . still we can use theorem [ main2 ] to construct unbiased shifts without any assumptions on @xmath5 : [ main2a ] let @xmath5 be a probability measure on @xmath6 . then there exists a nonnegative stopping time that is an unbiased shift embedding @xmath5 . in theorem [ littletwin ] we have @xmath66 and @xmath67 if @xmath16 . it is interesting to note that unbiased shifts @xmath2 ( even if they are not stopping times ) are almost surely non - zero as long as the condition @xmath68 is fulfilled : [ prop3 ] let @xmath5 be a probability measure on @xmath6 such that @xmath64 . then any unbiased shift @xmath2 embedding @xmath5 satisfies @xmath69 in contrast to the previous theorem , if @xmath2 is an unbiased shift with @xmath70 , then the probability @xmath71 may take any value : [ p11 ] for any @xmath72 $ ] there is an unbiased shift @xmath7 embedding @xmath19 and such that @xmath73 . a solution @xmath2 of the skorokhod embedding problem is usually required to have good moment properties , but some restrictions apply . for instance , if the target distribution @xmath5 is not centered , by ( * ? ? ? * theorem 2.50 ) , we must have @xmath74 . if the embedding stopping time is also an unbiased shift the situation is worse , even when @xmath5 is centred . [ t14complete ] suppose @xmath5 is a target distribution with @xmath59 , and @xmath75 , and the stopping time @xmath76 is an unbiased shift embedding @xmath5 . then @xmath77 under the same assumptions on @xmath5 the unbiased shift constructed in satisfies @xmath78 dropping the stopping time assumption we show in theorem [ pmoment ] that @xmath79 for any unbiased shift @xmath2 embedding a target distribution @xmath5 with @xmath64 . if the target distribution is concentrated at zero and @xmath2 is nonnegative but not identically zero , we show in theorem [ posmoment ] that @xmath80 . nonnegativity is important in this result , example [ exexpon ] provides an unbiased shift with @xmath81 that has exponential moments . theorem [ pminimal ] further shows that , in addition to the nearly optimal moment properties stated above , the stopping times @xmath2 defined in are also _ minimal _ in a sense analogous to the definition in @xcite ( see also @xcite , or @xcite for a survey ) . this means that if @xmath82 is another unbiased shift embedding @xmath5 such that @xmath83 , then @xmath84 . our discussion of minimality is based on a notion of stability of allocation rules , which is similar to the one studied in @xcite . the structure of the paper is as follows . section [ secpalm ] provides some background on palm measures and local time . section [ secmassstat ] presents a general result on mass - stationarity for diffuse random measures on the line , implying theorem [ littletwin ] . section [ sexinv ] contains the proof of a more general version of theorem [ main1 ] . section [ secexistence ] presents the key general result on balancing diffuse jointly stationary random measures , theorem [ main3 ] , implying theorem [ main2 ] . section 6 contains the proofs of theorems [ main2a ] , [ prop3 ] and [ p11 ] . in sections [ secstability ] and [ secproperties ] we discuss minimality and moment properties of unbiased shifts , including the proof of theorem [ t14complete ] section [ secremarks ] concludes with some remarks . recall the definition of the measure @xmath31 on @xmath85 . since @xmath86 for any compact @xmath42 , @xmath31 is @xmath25-finite . we also note the invariance property @xmath87 the proof of is only based on the stationary increments of @xmath1 , see @xcite . corollary [ timep ] provides an alternative definition of @xmath31 . a random measure @xmath39 is called _ invariant _ if @xmath88 where @xmath89 is the borel @xmath25-algebra on @xmath6 . in this case the _ palm measure _ @xmath90 of @xmath39 ( with respect to @xmath31 ) is defined by @xmath91}(s){{\bf 1}}_a(\theta_sb)\ , \xi(ds ) , \quad a\in\mathcal{a}.\end{aligned}\ ] ] this is a @xmath25-finite measure on @xmath85 . if the _ intensity _ @xmath92 of @xmath39 is positive and finite , @xmath90 can be normalised to yield the palm probability measure of @xmath39 . even though @xmath90 is generally not a probability measure , we denote integration with respect to @xmath90 by @xmath93 . the invariance property implies the _ refined campbell theorem _ @xmath94 for any measurable @xmath95 . we recall the following result from @xcite . [ th2 ] consider two invariant random measures @xmath39 and @xmath43 on @xmath6 and an allocation rule @xmath44 . then @xmath44 balances @xmath39 and @xmath43 if and only if @xmath96 where @xmath97 is defined by @xmath98 , @xmath99 . recall that @xmath48 is the random measure associated with the _ local time _ of @xmath1 at @xmath28 ( under @xmath26 ) . this means that @xmath100 for all measurable @xmath101 . we assume that @xmath12 is @xmath30-a.e . diffuse for any @xmath28 and that @xmath102 the global construction in @xcite guarantees the existence of a version of @xmath12 with these properties , see also ( * ? ? ? * proposition 22.12 ) or ( * ? ? ? * theorem 6.43 ) . this construction implies , moreover , that it is no restriction of generality to assume that @xmath103 and @xmath104 equation implies that @xmath105 is @xmath30-a.e . diffuse for any @xmath28 and has the invariance property . the following result is essentially from @xcite , see also @xcite . [ palmlocal ] let @xmath106 . then @xmath107 is the palm measure of @xmath105 . _ let @xmath95 be measurable . by definition of @xmath31 we have @xmath108 by this equals @xmath109 where the equality comes from a change of variables and fubini s theorem . by this equals @xmath110 since @xmath111 , we obtain @xmath112 and hence the assertion . equation is the refined campbell theorem in the case @xmath113 . in particular , it implies that @xmath105 has intensity @xmath114 : @xmath115)=1.\end{aligned}\ ] ] in this section we prove theorem [ littletwin ] . moreover , we show that the invariance property in theorem [ littletwin ] characterises mass - stationarity ( defined below ) of general diffuse random measures on the line . let @xmath39 be a diffuse random measure on @xmath6 and @xmath116 be a random element in a space on which the additive group @xmath6 acts measurably . let @xmath117 denote the shift of @xmath118 by @xmath119 . let the pair @xmath120 be defined as the identity on a canonical measurable space equipped with a @xmath25-finite measure @xmath121 . let @xmath122 denote lebesgue measure on @xmath6 . the pair @xmath118 is called _ mass - stationary _ if , for all bounded borel subsets @xmath123 of @xmath6 with @xmath124 and @xmath125 and all nonnegative measurable functions @xmath126 , @xmath127 here we set @xmath128 whenever @xmath129 . mass - stationarity is a formalisation of the intuitive idea that the origin is a typical location in the mass of a random measure . the invariance property can be interpreted probabilistically as saying that if the set @xmath123 is placed uniformly at random around the origin and the origin shifted to a location chosen according to the mass distribution of @xmath39 in that randomly placed set then the distribution of the pair @xmath118 does not change . the invariance property in the following theorem is a new characterisation of mass - stationarity . it is similar to the well - known characterisation in the simple point process case ( see e.g. ( * ? ? ? * theorem 11.4 ) ) and is certainly more transparent than . it is however restricted to the diffuse case on the line while works for general random measures in a group setting . the result below is also new , but the equivalence of mass - stationarity and palm measures was established in @xcite for abelian groups and in @xcite for general locally compact groups . [ mass - stat ] assume that @xmath130 and let @xmath131 , @xmath132 , be the generalised inverse of the diffuse random measure @xmath39 defined as in . then @xmath133 if and only if @xmath118 is mass - stationary and if and only if the distribution @xmath121 of @xmath120 is the palm measure of @xmath39 with respect to a @xmath25-finite stationary measure @xmath134 . the measure @xmath134 is uniquely determined by @xmath121 as follows : for each @xmath135 and each bounded nonnegative measurable function @xmath126 , @xmath136 first assume . then , @xmath137=0\}=\bq\{\xi[s_1,s_1+\varepsilon]=0\}=0 $ ] , for any @xmath138 , where the second identity comes from @xmath139 @xmath121-a.e.and the definition of @xmath140 . this easily implies that @xmath141 let @xmath142 be a bounded borel with @xmath124 and @xmath125 . changing variables and noting that , for any @xmath143 in the support of @xmath39 , we have @xmath144 for @xmath122-a.e . @xmath145 , we obtain that the left - hand side of equals @xmath146 where we have changed variables to get the equality . the key observation and assumption yield that the above equals @xmath147 thus holds , that is , @xmath118 is mass - stationary . by ( * ? * theorem 6.3 ) equation is equivalent to the existence of a stationary measure @xmath134 such that @xmath121 is the palm measure of @xmath39 with respect to @xmath121 . @xcite inversion formula ( see also ( * ? ? ? * section 2 ) ) implies that @xmath134 is uniquely determined by @xmath121 and that , moreover , @xmath148 . fix @xmath135 . for the claim that @xmath134 defined by is stationary when holds , see lemma [ secondq * ] below . to show that @xmath121 is then the palm measure of @xmath39 with respect to this @xmath134 let @xmath126 be nonnegative measurable and use for the first step in the following calculation , @xmath149}(s ) f(\theta_s(x,\xi))\,\xi(ds ) = \be_{\bq}\iint { { \bf 1}}_{[0,1]}(s ) { { \bf 1}}_{[0,s_w]}(t ) f(\theta_s\theta_t(x , \xi ) ) \,\theta_t\xi(ds)\,dt\\ & = \be_{\bq}\iint { { \bf 1}}_{[0,1]}(v - t ) { { \bf 1}}_{[0,s_w]}(t ) f(\theta_v(x , \xi))\,\xi(dv)\,dt\\ & = \be_{\bq}\iint { { \bf 1}}_{[0,1]}(s_r - t ) { { \bf 1}}_{[0,s_w]}(t ) f(\theta_{s_r}(x,\xi))\,dr\,dt\\ & = \be_{\bq}\iint { { \bf 1}}_{[0,1]}(-s_{-r}-t ) { { \bf 1}}_{[0 , s_{w}(\theta_{s_{-r}}(x , \xi))]}(t ) f(x,\xi)\,dr\,dt\\ & = \be_{\bq}f(x , \xi)\iint { { \bf 1}}_{[-1,0]}(u ) { { \bf 1}}_{[s_{-r},\ , s_{-r+w } ] } ( u)\ , dr\,du\\ & = w\,\be_{\bq}f(x , \xi),\end{aligned}\ ] ] where we have used and for the fourth identity and the final identity holds since the double integral equals @xmath150 . finally , if @xmath121 is the palm measure of @xmath39 with respect to a @xmath25-finite stationary measure @xmath134 , then theorem [ th2 ] implies once we have shown for any @xmath17 that the allocation rule @xmath151 defined by @xmath152 balances @xmath39 with itself , that is , @xmath153 asssume @xmath154 . then , outside the @xmath134-null set @xmath155 we obtain for any @xmath156 ( interpreting @xmath157 $ ] as @xmath158 $ ] for @xmath159 ) that @xmath160\le r , \xi[s , b ] > r\ } \ , \xi(ds)\\ & = \int { { \bf 1}}\{s\le b , r < \xi[s , b]\le r+\xi[a , b ] \ } \ , \xi(ds)= \xi[a , b],\end{aligned}\ ] ] which implies the desired balancing property . the case @xmath161 can be treated similarly . [ secondq * ] let @xmath82 be a random time and @xmath134 be the measure defined by setting , for each bounded nonnegative measurable function @xmath126 , @xmath162 if @xmath163 has the same distribution as @xmath118 under @xmath121 then @xmath118 is stationary under @xmath134 . for each @xmath126 as above and @xmath34 , @xmath164 where the third identity follows from the assumption that @xmath163 has the same distribution as @xmath118 under @xmath121 . as a corollary we obtain an alternative construction of the stationary measure by integrating over time rather than space . [ timep ] let @xmath135 and consider @xmath165 defined by . then @xmath166 , where @xmath167 _ proof of theorem [ littletwin ] . _ the result follows from theorem [ mass - stat ] and lemma [ palmlocal ] . the _ invariant @xmath25-algebra _ is defined by @xmath168 we now apply theorem [ littletwin ] to prove the following result which we need in the proof of theorem [ main2 ] in section [ secexistence ] . [ h1 ] let @xmath169 . then either @xmath170 for any @xmath28 ( in which case @xmath171 ) or @xmath172 for any @xmath28 ( in which case @xmath173 ) . we first show that @xmath174 we use here the random times @xmath175 ( see ) for integers @xmath20 . by theorem [ littletwin ] , for any integer @xmath20 , the processes @xmath176 and @xmath177 are independent one - sided brownian motions . this implies that the processes @xmath178 are independent under @xmath26 . since , by , @xmath179)=1\ } = \inf\{t\ge 0 \colon \ell^0(b,[t_n , t_n+t])=1\ } = t_{n+1}-t_n\end{aligned}\ ] ] holds @xmath26-a.s . for any @xmath180 , the @xmath181 have the distribution of a one - sided brownian motion stopped at the time its local time at @xmath182 reaches the value @xmath114 . clearly we have that @xmath183 for a suitably defined measurable function @xmath184 . by invariance of @xmath185 and definition of the family @xmath186 , @xmath187 where the final equation holds @xmath26-a.s . since iid - sequences are ergodic ( by the law of large numbers ) , we obtain . the refined campbell theorem implies ( with @xmath122 denoting lebesgue measure ) @xmath188 provided that @xmath189 . assume now that @xmath190 . then implies that @xmath191 for all compact @xmath42 . letting @xmath192 , we obtain @xmath193 , that is , @xmath194 on the other hand , by , @xmath195 for @xmath122-a.e . @xmath196 so that @xmath170 for @xmath122-a.e . @xmath196 . therefore @xmath171 . by this implies @xmath170 for all @xmath28 . in this section we prove a result that is more general than theorem [ main1 ] . recall that @xmath197 for a probability measure @xmath65 on @xmath6 . [ main1a ] let @xmath2 be a random time and @xmath198 be probability measures on @xmath6 . then the following two assertions are equivalent . 1 . it is true that @xmath199 @xmath200 , and that @xmath201 and @xmath4 are independent under @xmath202 . the allocation rule @xmath51 defined by balances @xmath203 and @xmath50 . we start by noting that the random measures @xmath203 and @xmath50 are invariant in the sense of . this follows from the invariance of local time , and fubini s theorem . let us first assume that ( i ) holds . then we have for any @xmath204 that @xmath205 lemma [ palmlocal ] and fubini s theorem imply that @xmath58 is the palm measure of @xmath50 . therefore we obtain from theorem [ th2 ] that @xmath51 balances @xmath203 and @xmath50 . assume now that ( ii ) holds . by theorem [ th2 ] we obtain for any @xmath204 that @xmath206 this implies @xmath207 for any @xmath208 and any @xmath209 . this yields ( i ) . [ remsub]an _ extended _ allocation rule is a mapping @xmath210 $ ] that has the equivariance property . the balancing property can then be defined as before . using these concepts , theorem [ main1a ] can be proved for a subprobability measure @xmath211 . the conditions in ( i ) have to be replaced with @xmath212 , @xmath213 and the independence of @xmath201 and @xmath4 under @xmath214 . in this section we prove theorem [ main2 ] . the proof is based on the following new balancing result for general random measures on the line , which is inspired by @xcite . [ main3 ] let @xmath39 and @xmath43 be jointly stationary orthogonal diffuse random measures on @xmath6 with finite intensities . assume further that @xmath215 \big| \mathcal i \big ] = \be\big [ \eta[0,1 ] \big| \mathcal i \big ] \quad \bp\text{-a.e.}\ ] ] then the mapping @xmath37 , defined by @xmath216=\eta[s , t]\},\quad s\in\r,\end{aligned}\ ] ] is an allocation rule balancing @xmath39 and @xmath43 . for convenience , theorem [ main3 ] has been formulated for random measures defined on our canonical @xmath217 . however , the special structure of that space is of no importance here . we start the proof of theorem [ main3 ] with an analytic lemma . here and later it is convenient to work with the continuous function @xmath218 , defined by @xmath219-\eta[0,t],&\text{if $ t\ge 0$,}\\ \eta[t,0]-\xi[t,0],&\text{if $ t < 0$}. \end{cases}\end{aligned}\ ] ] [ analytic ] suppose @xmath39 and @xmath43 are orthogonal diffuse measures . then @xmath220$},\ ] ] provided that @xmath221 for all @xmath222 . the proof of lemma [ analytic ] rests on three further lemmas . [ des ] + * for @xmath39-almost every @xmath143 there exists @xmath223 with @xmath224 . * for @xmath43-almost every @xmath143 there exists @xmath223 with @xmath225 . it suffices to prove @xmath226 , as @xmath227 follows by reversing the roles of @xmath39 and @xmath43 . recall that @xmath39 and @xmath43 are orthogonal and hence there exists a borel set @xmath185 with @xmath228 and @xmath229 . we need to show that , for each @xmath230 , @xmath231 given any @xmath232 we may choose an open set @xmath233 with @xmath234 . we can cover @xmath235 by a countable collection @xmath236 of nonoverlapping intervals @xmath237 $ ] , @xmath238 , @xmath239 , such that @xmath240 . indeed , suppose that @xmath241 is a connected component of @xmath242 , which intersects @xmath235 . if there is a minimal element @xmath143 in @xmath243 let @xmath244 be the minimum of @xmath245 and the distance of @xmath143 to the right endpoint of @xmath241 . add the interval @xmath237 $ ] to the collection @xmath236 and remove it from @xmath235 and @xmath242 . if no such minimum exists we can pick a strictly decreasing sequence @xmath246 , @xmath247 , converging to the infimum . let @xmath248 be the minimum of @xmath245 and the distance of @xmath249 to the right endpoint of @xmath241 , and , for @xmath250 , let @xmath251 be the minimum of @xmath245 and @xmath252 . add all intervals @xmath253 $ ] to the collection @xmath236 and remove their union from @xmath235 and @xmath242 . note that after one such step ( performed in every connected component ) all of @xmath235 in connected components of length at most @xmath245 will be removed , and the lower bound of the intersection of all other connected components with @xmath235 , if finite , is increased by at least @xmath245 . also , after one step , the intersection of any connected component with @xmath235 is either empty or bounded from below . therefore , every set of the form @xmath254\cap a_\epsilon$ ] will be completely covered after finitely many steps by nonoverlapping intervals , as required . observe that @xmath255 for every interval in the collection , and hence @xmath256 the result follows as @xmath232 was arbitrary . we now fix @xmath257 and decompose @xmath126 on @xmath258 $ ] according to its backwards running minimum @xmath259 given by @xmath260 see figure 1 for illustration . the nonegative function @xmath261 can be decomposed on @xmath258 $ ] into a family @xmath262 of excursions @xmath263 with starting times @xmath264 $ ] . note that an excursion @xmath263 is a function such that there exists a number @xmath265 , called the lifetime of the excursion , such that @xmath266 , @xmath267 for all @xmath268 , and @xmath269 for all @xmath270 . formally putting @xmath269 for all @xmath271 the decomposition can be written as @xmath272 note that the intervals @xmath273 , @xmath274 , are disjoint . we denote by @xmath123 the complement of their union in @xmath258 $ ] , i.e.@xmath275 \colon f(t)=m(t)\}.$ ] for every @xmath274 we have @xmath276 we only have to show that @xmath277 for @xmath39-almost every @xmath278 . by lemma [ des ] ( a ) , for @xmath39-almost every @xmath278 , there exists @xmath223 such that @xmath224 . as @xmath279 , by continuity of @xmath126 , we infer that there exists @xmath280 such that @xmath281 . therefore @xmath282 as required . [ rest ] we have @xmath283 first observe that if @xmath284 , then @xmath285 for all @xmath286 $ ] . if @xmath287 for all @xmath286 $ ] then @xmath288 . otherwise there exists a maximal @xmath286 $ ] with @xmath289 . then @xmath290 is a true local minimum of @xmath126 in the sense that there exists @xmath291 with @xmath292 for all @xmath293 and @xmath294 for all @xmath295 . in particular there are at most countably many levels @xmath296 where this can happen . fixing such a level @xmath297 we note that @xmath298 . summing over all these levels we see that @xmath299 . we conclude the proof by showing that @xmath300 . lemma [ des ] ( b ) ensures that , for @xmath43-almost every @xmath301 $ ] there exists @xmath223 such that @xmath225 , which implies that @xmath302 . hence the stated equality follows . _ proof of lemma [ analytic ] . _ taking the sum over the equations in the previous two lemmas we obtain @xmath303.$ ] this implies @xmath304 , \quad a\geq 0,\ ] ] as any @xmath271 with @xmath305 satisfies @xmath306 $ ] , and lemma [ des ] ( a ) implies that @xmath39-almost every @xmath271 with @xmath307 satisfies @xmath308 , and so @xmath309 . _ proof of theorem [ main3 ] . _ define @xmath310 recall that by lemma [ analytic ] we have @xmath311 on @xmath312 provided @xmath313\ge \eta[s , t]$ ] for all @xmath314 . by lemma [ des ] this holds for @xmath315-a.e . moreover , by stationarity of @xmath315 we have that @xmath316=0,\xi^\infty\ne 0\}=0 $ ] for all @xmath317 . we infer that @xmath318 using the refined campbell theorem twice , we obtain @xmath319\ } \ , ds\\ & = \be \int { { \bf 1}}\{\xi^\infty\not=0 , \tau(s)\in[0,1]\ } \ , \xi(ds)\\ & = \be { { \bf 1}}\{\xi^\infty\not=0\ } \ , \eta^*[0,1 ] . \end{aligned}\ ] ] using first and then our assumption gives @xmath320 = \be { { \bf 1}}\{\xi^\infty\not=0\ } \eta[0,1 ] = \be { { \bf 1}}\{\xi^\infty\not=0\ } \xi[0,1],\ ] ] and together with we infer that @xmath321,\ ] ] and therefore @xmath322 for @xmath39-a.e . @xmath143 , @xmath31-a.e . in particular , this implies that @xmath44 is a well - defined allocation rule . an analogous argument implies that @xmath323 where @xmath324=\eta[t , s]\}\ ] ] is the inverse of @xmath44 . we now use this to show that @xmath44 balances @xmath39 and @xmath43 . fixing @xmath156 we aim to show that @xmath325=\eta[a , b]$ ] . if @xmath326 for all @xmath327 $ ] this holds by lemma [ analytic ] . otherwise we apply this lemma to suitably chosen alternative intervals . to this end let @xmath328 \colon \text{$f(s)\leq f(t)$ for all $ a\le t \le b$ } \}\ ] ] be the leftmost minimiser of @xmath126 on @xmath329 $ ] . as @xmath330\geq f(a^*-\frac1n)-f(a^ * ) > 0 $ ] for all sufficiently large @xmath247 , we find a decreasing sequence @xmath331 with @xmath332 and hence @xmath333 . then @xmath334 $ ] and @xmath335 if @xmath336 . assuming first that @xmath336 we obtain from lemma [ analytic ] that @xmath337$},\ ] ] which implies the statement . now assume that @xmath338 . in this case we get @xmath311 on @xmath339 $ ] and on @xmath340 $ ] for every @xmath20 , and the result follows as @xmath341 . _ proof of theorem [ main2 ] . _ by theorem [ h1 ] , @xmath342| \mathcal i ] = \be [ \ell^\nu[0,1]| \mathcal i]$ ] almost surely . since @xmath12 and @xmath50 are orthogonal , we can combine theorems [ main3 ] and [ main1 ] to obtain the result . assume in theorem [ main2 ] that @xmath5 is a subprobability measure . then @xmath2 takes the value @xmath343 with positive @xmath26-probability . indeed , by remark [ remsub ] , defining the extended allocation rule @xmath44 by @xmath344 we get that @xmath44 balances the restriction of @xmath12 to @xmath345 and @xmath50 . assertion ( i ) of theorem [ main1a ] remains valid in the sense explained in remark [ remsub ] and gives @xmath346 . now assume in theorem [ main2 ] that @xmath5 is a locally finite measure with @xmath347 . then @xmath348 and @xmath44 balances @xmath12 and @xmath349 . the proof of theorem [ main3 ] still yields the inequality @xmath350 . in particular @xmath43 is a diffuse ( and invariant ) random measure and must therefore be of the form @xmath351 for some measure @xmath352 , see e.g. ( * ? ? * theorem 22.25 ) . in fact , @xmath352 is a probability measure smaller than @xmath5 . hence @xmath2 is an unbiased shift embedding @xmath352 . some properties of @xmath352 can be found in @xcite . the nonnegative unbiased shifts in theorem [ main2 ] , theorem [ main2a ] and in theorem [ littletwin ] are all stopping times . in the next example we construct a nonnegative unbiased shift embedding a distribution not concentrated at zero , which is not a stopping time . [ exnonstop]let @xmath353 . we define an allocation rule @xmath44 that balances @xmath12 and @xmath48 and such that @xmath354 is nonnegative but not a stopping time . the mapping @xmath44 is the composition of the following five allocation rules . let @xmath355 balance @xmath12 and @xmath48 according to theorem [ main2 ] . let @xmath356 balance @xmath48 and @xmath48 by shifting forward one mass - unit , that is , let @xmath357 be defined by with @xmath358 and with @xmath12 replaced with @xmath48 . let @xmath359 balance @xmath48 and @xmath12 according to theorem [ main2 ] . finally define @xmath360 by shifting _ backward _ one mass - unit in the local time at @xmath196 , that is , let @xmath360 be defined by with @xmath361 and @xmath12 replaced with @xmath48 . the composition @xmath44 of these allocation rules balances @xmath12 and @xmath48 moreover , @xmath362 . however , @xmath2 is not a stopping time . this example can be extended to a general target distribution @xmath5 . in this section we prove theorems [ main2a ] , [ prop3 ] , and [ p11 ] . in contrast to the previous section we allow here for an atom at @xmath182 . _ proof of theorem [ main2a ] . _ let @xmath363 such that @xmath364 and define @xmath365 theorems [ main1 ] and [ main2 ] imply that the allocation rule @xmath366=\ell^\mu[s , t]\big\},\quad s\in\r,\end{aligned}\ ] ] balances @xmath12 and @xmath203 . the same theorems imply that there is an allocation rule @xmath367 that balances @xmath105 and @xmath12 . define @xmath368 then we have for any borel set @xmath42 outside a fixed @xmath31-null set that @xmath369 where we have used ( and @xmath364 ) in the penultimate equation . hence @xmath44 balances @xmath12 and @xmath50 . theorem [ main1 ] now implies that @xmath354 is an unbiased shift embedding @xmath5 . _ proof of theorem [ prop3 ] . _ let @xmath2 be any unbiased shift embedding @xmath5 and define @xmath370 . outside a fixed @xmath31-null set we obtain for any borel set @xmath142 that @xmath371 where we have used to obtain the final identity . this implies that @xmath372 assuming now that @xmath64 we obtain @xmath373 for @xmath12-a.e . @xmath143 , @xmath31-almost everywhere . lemma [ palmlocal ] now implies . _ proof of theorem [ p11 ] . _ let @xmath374 , where @xmath375 is given by with @xmath376 . define a stationary random measure @xmath39 by @xmath377 . the allocation rule @xmath378=1\big\}\ ] ] balances @xmath39 with itself . define @xmath379 it is easy to see that @xmath44 balances @xmath12 with itself . lemma [ palmlocal ] and theorem [ main1 ] ( or a direct calculation ) implies that @xmath354 satisfies @xmath380 since @xmath2 is an unbiased shift , the proof is complete . the following definition is a one - sided version of the notion of stability introduced in @xcite for point processes . we call an allocation rule @xmath37 balancing @xmath39 and @xmath43 _ right - stable _ if @xmath381 for all @xmath317 and @xmath382 roughly speaking this means that the mass of pairs @xmath383 such that @xmath143 would prefer the partner of @xmath384 over its own partner , while @xmath385 would prefer @xmath143 over @xmath384 as a partner , vanishes . [ rstable ] let @xmath39 and @xmath43 be invariant random measures satisfying the conditions of theorem [ main3 ] , and suppose @xmath37 is the allocation rule constructed in the theorem . then @xmath44 is right - stable . by lemma [ des ] ( a ) and continuity of @xmath126 , we have for @xmath39-a - e . @xmath143 that @xmath386 for all @xmath387 . hence @xmath388-almost every pair @xmath383 with @xmath389 satisfies @xmath390 contradicting the definition of @xmath44 . right - stable allocation rules have a useful minimality property . [ mini ] any right - stable allocation rule @xmath44 balancing two measures @xmath39 and @xmath43 is minimal in the sense that if @xmath25 is another allocation rule balancing @xmath39 and @xmath43 such that @xmath391 for @xmath39-almost every @xmath317 , then @xmath392 by right - stability of @xmath44 we have , _ for @xmath39-almost every @xmath393 , _ @xmath394 \longleftrightarrow \tau(s)\in[a,\tau(a ) ] \quad \xi\text{-a.e.\ $ s$}.\end{aligned}\ ] ] from the assumption @xmath391 and we obtain for any @xmath395 $ ] that @xmath396 $ ] implies @xmath397 $ ] for @xmath39-almost every @xmath143 . therefore @xmath398=\int { { \bf 1}}\{\tau(s)\in[a , t]\}\ , \xi(ds ) \le \int { { \bf 1}}\{\sigma(s)\in[a , t]\}\ , \xi(ds ) = \eta[a , t].\end{aligned}\ ] ] this implies @xmath399\}= { { \bf 1}}\{\sigma(s)\in[a , t]\}\quad \xi\text{-a.e.\ $ s\in\r$}.\ ] ] therefore @xmath44 and @xmath25 coincide @xmath39-almost everywhere on @xmath400)$ ] . now fix some @xmath401 and recall the definition of the backwards running minimum @xmath402 and the set @xmath403 . we have seen that the complement of @xmath123 consists of countably many intervals @xmath404 as above and therefore @xmath44 and @xmath25 coincide @xmath39-almost everywhere on @xmath405\setminus c)$ ] . on the other hand , by lemma [ rest ] we have @xmath406 , as required to finish the argument . one could define an allocation rule @xmath44 to be _ stable _ if latexmath:[\[\xi\otimes\xi\big\ { ( s , t ) \colon @xmath44 of theorem [ main3 ] does not satisfy this . we do not know if stable allocation rules in the above sense exist , or if they are unique . an unbiased shift @xmath2 is called _ minimal unbiased shift _ if @xmath408 and if any other unbiased shift @xmath409 such that @xmath410 and @xmath411 satisfies @xmath84 . the following theorem provides more insight into the set of all minimal unbiased shifts . the result and its proof are motivated by proposition 2 in @xcite . [ pminexist ] let @xmath2 be an unbiased shift embedding the probability measure @xmath5 and such that @xmath408 . then there exists a minimal unbiased shift @xmath412 embedding @xmath5 and such that @xmath413 . let @xmath414 denote the set of all unbiased shifts @xmath409 embedding @xmath5 and such that @xmath410 . this is a partially ordered set , where we do not distinguish between elements that coincide @xmath26-a.s . by the hausdorff maximal principle ( see , e.g. ( * ? ? ? * section 1.5 ) ) there is a _ maximal chain _ @xmath415 . this is a totally ordered set that is not contained in a strictly bigger totally ordered set . let @xmath416 then there is a sequence @xmath417 , @xmath247 , such that @xmath418 as @xmath341 . since @xmath419 is totally ordered it is no restriction of generality to assume that the @xmath417 are decreasing @xmath26-a.s . define @xmath420 . by construction and monotone convergence @xmath421 we also note that @xmath422 . we claim that @xmath412 is a minimal unbiased shift embedding @xmath5 and first show that @xmath412 is an unbiased shift . let @xmath423 , and consider continuous and bounded functions @xmath424 and @xmath425 . let @xmath426 . since @xmath427 for any @xmath247 we have that @xmath428 by bounded convergence the above left - hand side converges towards @xmath429 as @xmath341 . the monotone class theorem implies that @xmath412 is an unbiased shift embedding @xmath5 . it remains to show the minimality property of @xmath412 . assume on the contrary that there is some unbiased shift @xmath409 embedding @xmath5 such that @xmath430 and @xmath431 . the last two relations imply that @xmath432 by this means that @xmath433 . on the other hand , since @xmath434 , we have that @xmath435 , contradicting the maximality property of @xmath419 . as announced in the introduction the stopping time is a minimal unbiased shift : [ pminimal ] let @xmath5 be a probability measure on @xmath6 with @xmath59 . then @xmath2 defined by is a minimal unbiased shift . let @xmath409 be an unbiased shift embedding @xmath5 and such that @xmath410 . theorem [ main1 ] implies that the allocation rules @xmath436 and @xmath51 balance @xmath12 and @xmath50 . by theorem [ rstable ] , @xmath51 is right - stable @xmath31-a.e . the assumptions yield @xmath437 @xmath31-a.e . by theorem [ mini ] we therefore have @xmath438 @xmath31-a.e . this readily implies that @xmath84 . in this section we discuss moment properties of unbiased shifts . the following two theorems together were stated as theorem [ t14complete ] in the introduction . [ t14lower ] suppose @xmath5 is a target distribution with @xmath59 , and @xmath75 , and the stopping time @xmath76 is an unbiased shift embedding @xmath5 . then @xmath77 we start the proof with a reminder of the barlow - yor inequality @xcite , which states that , for any @xmath439 there exist constants @xmath440 such that , for all stopping times @xmath2 , @xmath441^{p } \le c\ , \be_0 t^{p/2}.\ ] ] hence it suffices to verify that @xmath442^{1/2}=\infty$ ] . the proof of this fact uses an argument similar to that in the proof of theorem 2 in @xcite . let @xmath443 be the allocation rule associated with @xmath2 and set @xmath444=t\}$ ] . then , on the one hand , @xmath445\}\,\ell^0(ds ) = \be_0\int^{t_t}_0{{\bf 1}}\{\tau(s)-s > t_t - s\}\ , \ell^0(ds)\\ & = \int^t_0\bp_0\{\tau(t_s)-t_s > t_t - t_s\}\,ds = \int^t_0\bp_0\{t\circ\theta_{t_s}>t_{t - s}\circ\theta_{t_s}\}\,ds\\ & = \int^t_0\bp_0\{t > t_s\}\,ds=\be_0[\ell^0[0,t]\wedge t],\end{aligned}\ ] ] where we have used the strong markov property at @xmath446 ( or theorem [ littletwin ] ) for the fourth step and change of variable for the second and fifth steps . on the other hand , the fact that @xmath44 balances @xmath12 and @xmath50 easily implies that @xmath447\}\,\ell^0(ds)\ge ( \ell^0[0,t_t]-\ell^\nu[0,t_t])_+.\ ] ] hence , combining these two facts , @xmath448\wedge t]\ge \be_0 ( \ell^0[0,t_t]-\ell^\nu[0,t_t])_+.\end{aligned}\ ] ] observe that @xmath449=t=\be\ell^\nu[0,t_t]$ ] , where the second identity is a trivial consequence of the second ray - knight theorem ( see theorem 2.3 in chapter xi of @xcite ) and in fact a consequence of general palm theory . jensen s inequality and again the second ray - knight theorem imply that @xmath450)^2\le \be_0 \int ( \ell^x[0,t_1])^2\,\nu(dx ) = \int ( 1+|x|)\,\nu(dx)\end{aligned}\ ] ] which is finite by assumption . hence , by the central limit theorem , @xmath451-\ell^\nu[0,t_t])_+=:c_1>0.\ ] ] assume for contradiction that @xmath452^{1/2}]<\infty$ ] . since @xmath453\wedge t)\le \ell^0[0,t]^{1/2}$ ] , dominated convergence implies that @xmath454\wedge t]\to 0 $ ] as @xmath455 , contradicting the positivity of @xmath456 . note that the unbiased shifts defined in satisfy the conditions of theorem [ t14lower ] if @xmath5 has finite mean . the next result shows that they have nearly optimal moment properties . [ t14 ] let @xmath5 satisfy @xmath75 , and let @xmath2 be the stopping time constructed in . then , for all @xmath457 , @xmath458 the proof of theorem [ t14 ] uses a result similar to theorem4 ( ii ) in @xcite and theorem 2 in @xcite , which is of independent interest and may also serve as another example for theorem [ main3 ] . we consider the ` clock ' @xmath459+\ell^\nu[0,t]=r\big\}\ ] ] and random measures @xmath39 and @xmath43 on the positive reals given by @xmath460:=\ell^0[0,u_r],\qquad \eta[0,r]:=\ell^\nu[0,u_r ] , \quad r\geq 0.\ ] ] [ t41 ] let @xmath39 and @xmath43 be defined as above and let @xmath461=\eta[0,t]\}$ ] . then @xmath462 , but for some @xmath463 we have @xmath464 for all @xmath34 . the proof of @xmath462 is very similar to theorem 2 in @xcite and is therefore omitted . we prove here the upper bound for the tail asymptotics ( only this part is needed ) . this result is similar to theorem 6 ( ii ) in @xcite , but due to the specific form of @xmath409 we can use a more direct argument . for any @xmath465 let @xmath466\leq i+1\}$ ] . using the second ray - knight theorem and the assumption on the first moment of @xmath5 as in , we see that the sequence @xmath467 is an i.i.d . sequence of random variables with mean one and finite variance . define , for @xmath247 , @xmath468 let @xmath469 and fix @xmath470 . then , for any @xmath471 , @xmath472 by a classical result of spitzer @xcite , see also ( * ? ? ? * theorem 1a in section xii.7 ) , the first term on the above right - hand side is bounded by a constant multiple of @xmath473 . by chebyshev s inequality we have @xmath474 = \frac{\lfloor at\rfloor}{(t-2\lfloor at\rfloor)^2 } \,\be_0[(1-y_1)^2],\ ] ] which is bounded by a constant multiple of @xmath475 . this completes the proof . _ proof of theorem [ t14 ] . _ the variable @xmath409 , defined in proposition [ t41 ] , satisfies @xmath476+\ell^\nu[0,t]=2\ell^0[0,t].\ ] ] it remains to relate the tail behaviour of @xmath477 $ ] ( which we know ) to that of @xmath2 ( which we require ) . to this end we observe that for @xmath478 and @xmath479 , using ( * ? ? ? * theorem 6.10 ) , @xmath480 < 1/\theta\big\ } & = \bp_0\big\{\inf_{s > t } \frac1{\sqrt{s/\log s } } \max_{0\leq r\leq s } |b_r| < 1/\theta\big\ } \\ & \leq \sum_{k=0}^\infty \bp_0\big\ { \frac1{\sqrt{t+k } } \max_{0\leq r\leq t+k } |b_r| < \frac{2}{\theta\sqrt{\log(t+k)}}\big\}.\end{aligned}\ ] ] by a step in the proof of chung s law of the iterated logarithm , see e.g. ( * ? ? ? * ( 2.1 ) ) , @xmath481 and hence we have @xmath482 < 1/\theta\big\ } \le t^{-1/4},\ ] ] for a sufficiently large constant @xmath483 . for sufficiently large @xmath384 we have @xmath484>\sqrt{t}\ } + \bp_0\big\{\inf_{s > t } \frac1{\sqrt{s/\log s}}\ell^0[0,s ] < 1/\theta \big\},\ ] ] and the right hand side in this inequality is bounded by a constant multiple of @xmath485 . the result follows directly by integration . next we turn to unbiased shifts @xmath2 embedding a measure @xmath486 , which need neither be stopping times , nor nonnegative . we conjecture that any such shift satisfies @xmath487 . at the moment we can only prove the following weaker result . [ pmoment ] if @xmath2 is an unbiased shift embedding a probability measure @xmath486 , then @xmath488 the idea of this proof is due to alex cox . we work under the probability measure @xmath26 . by definition of an unbiased shift @xmath489 and @xmath490 are independent brownian motions . moreover , the pair @xmath491 is independent of @xmath4 . assume that @xmath492 , where @xmath493 is chosen such that @xmath494 . ( if there is no such @xmath493 we find an @xmath495 such that @xmath496>0 $ ] and assume @xmath497 . ) if @xmath498 , then @xmath499 , so that @xmath500 if @xmath501 , then @xmath502 , so that @xmath503 hence @xmath504 . it is well - known that @xmath505 and @xmath506 . since @xmath507 and @xmath508 are independent , this property transfers to @xmath409 . it follows that @xmath509 unbiased shifts embedding @xmath19 also have bad moment properties if they are nonnegative ( or , by time - reversal , nonpositive ) but not identically zero . the result can be compared with theorem 3(i ) in @xcite . however , the proofs are very different . [ posmoment ] if @xmath7 is an unbiased shift such that @xmath70 and @xmath510 , then @xmath511 we assume for contradiction that @xmath512 . define a probability measure @xmath513 on @xmath22 by setting @xmath514 for each bounded nonnegative measurable function @xmath126 . by lemma [ secondq * ] , @xmath513 is stationary . to show that , on the invariant @xmath25-algebra @xmath236 , the process @xmath1 has the same distribution under @xmath513 as under @xmath26 , take @xmath169 and recall from theorem [ h1 ] that @xmath515 . but @xmath516 or @xmath114 according as @xmath517 or @xmath114 , as required . by ( * ? ? ? * theorem 2 ) we infer from this that @xmath518 with respect to the total variation norm . on the other hand , for every @xmath291 , @xmath519 implying @xmath520 for all @xmath291 , which is a contradiction . in contrast to the two theorems above , we shall see below that unbiased shifts can have _ good _ moment properties if they can assume both signs . [ exexpon ] we construct a nonzero unbiased shift @xmath2 embedding @xmath19 , which has @xmath521 for some @xmath522 . let @xmath523 be the countable collection of maximal nonempty intervals @xmath524 with the property that @xmath525 for all @xmath526 and @xmath527 for some @xmath528 . we assume that the collection is ordered such that @xmath529 for all @xmath530 . we define an allocation rule @xmath44 by the requirement that , for @xmath531 , @xmath532 it is easy to see that @xmath44 balances @xmath12 with itself , and hence by theorem [ main1 ] , we have that @xmath533 is an unbiased shift embedding @xmath19 . moreover , we have @xmath534 where @xmath535 and @xmath536 . @xmath140 and @xmath537 are obviously independent and identically distributed , and it is easy to see that they , and hence @xmath538 , have the required moment property . if @xmath76 is an unbiased shift such that @xmath70 and @xmath510 , then we conjecture that @xmath74 ( strengthening theorem [ posmoment ] ) , but we can not prove this without additional assumptions . one such assumption ( covering @xmath18 defined in for @xmath291 ) is that @xmath539 for some @xmath540 such that @xmath541 is @xmath26-almost surely in the @xmath25-algebra generated by @xmath542 . indeed , in this case we have @xmath543 where @xmath544 . by the markov property @xmath545 since @xmath546 for all @xmath547 and @xmath548 . note that this argument does not use that @xmath2 is unbiased . in this section we make some comments on our results and their possible extensions . we begin with the following counterpart of theorem [ main3 ] for simple point processes . [ tpp ] let @xmath39 and @xmath43 be invariant simple point processes on @xmath6 defined on some probability space equipped with a flow and an invariant @xmath25-finite measure . assume that @xmath39 and @xmath43 have finite intensities and that @xmath549\mid \mathcal{i}]=\be[\eta[0,1]\mid \mathcal{i}].\end{aligned}\ ] ] then @xmath550=\eta[s , t]\},\quad s\in\r,\end{aligned}\ ] ] is an allocation rule balancing @xmath39 and @xmath43 . theorem [ tpp ] is a one - sided ( and one - dimensional ) version of the stable matching procedure described in @xcite . it can be proved by adapting the ideas of theorem [ main3 ] to a discrete and therefore much simpler setup . in the point process case the allocation rule is right - stable in the sense of section [ secstability ] and it is not difficult to show that it is the unique right - stable allocation balancing @xmath39 and @xmath43 . we conjecture that this uniqueness property also holds in the general case and therefore can be added to theorem [ rstable ] . [ statincr ] let @xmath551 be a right - continuous real - valued stochastic process with left - hand limits and @xmath552 , defined on its canonical probability space @xmath21 . assume that @xmath116 has stationary increments . defining @xmath30 , @xmath28 , as before , the definition still yields a stationary measure . assume that there is a ( measurable ) family @xmath48 , @xmath28 , of local times satisfying . under technical assumptions on this family ( weaker than and ) theorem [ main1 ] still holds . one does not need to assume that the local times are diffuse . however , one can not expect the existence results ( e.g. theorem [ main2a ] ) to hold in such a general setting . consider the setting of example [ statincr ] and assume in addition that @xmath116 is a lvy process . there are assumptions on @xmath116 that guarantee the existence of diffuse local time measures that are perfect in the sense of and , see @xcite . under these assumptions theorems [ main2 ] , [ main2a ] , [ littletwin ] , [ h1 ] , and [ timep ] all hold . also the minimality properties stated in theorems [ pminexist ] and [ pminimal ] as well as the stability assertion of theorem [ rstable ] remain true in this more general setting . * acknowledgements : * this research started during the oberwolfach workshop ` new perspectives in stochastic geometry ' and was supported by a grant from the _ royal society_. all this support is gratefully acknowledged . we wish to thank sergey foss and takis konstantopoulos for helpful discussions of theorem [ tpp ] and alex cox for providing the idea of the proof of theorem [ pmoment ] . kagan , y.a . and vere - jones , d. ( 1996 ) . problems in the modelling and statistical analysis of earthquakes . in athens conference on applied probability and time series , vol 1 : applied probability ( springer lecture notes in statistics 114 ) , eds c.c . heyde , y .. v . prohorov , r. pyke , s.t . rachev , springer , berlin , 398425 .

let @xmath0 be a two - sided standard brownian motion . an _ unbiased shift _ of @xmath1 is a random time @xmath2 , which is a measurable function of @xmath1 , such that @xmath3 is a brownian motion independent of @xmath4 . we characterise unbiased shifts in terms of allocation rules balancing additive functionals of @xmath1 . for any probability distribution @xmath5 on @xmath6 we construct a stopping time @xmath7 with the above properties such that @xmath4 has distribution @xmath5 . in particular , we show that if we travel in time according to the clock of local time we always see a two - sided brownian motion . a crucial ingredient of our approach is a new theorem on the existence of allocation rules balancing jointly stationary diffuse random measures on @xmath6 . we also study moment and minimality properties of unbiased shifts . _ 2000 mathematics subject classification . _ 60j65 ; 60g57 ; 60g55 . _ key words and phrases . _ brownian motion , local time , unbiased shift , allocation rule , palm measure , random measure , skorokhod embedding .

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Proceed to summarize the following text: the electron dephasing time @xmath0 is a very important quantity that governs the mesoscopic phenomena at low temperatures . particularly , the behavior of the dephasing time near zero temperature , @xmath2 , has recently attracted vigorous experimental @xcite and theoretical @xcite attention . one of the central themes of this renewed interest is concerned with whether @xmath3 should reach a finite or an infinite value as @xmath4 0 . the connection of the zero - temperature dephasing behavior with the very fundamental condensed matter physics problems such as the validity of the fermi - liquid picture , the possibility of the occurrence of a quantum phase transition , and the persistent currents in metals , etc . , has been addressed . conventionally , it is accepted that @xmath3 should reach an infinite value if there exist only the inelastic electron - electron and electron - phonon scattering . however , several recent measurements performed on different mesoscopic conductors have revealed that @xmath3 depends only very weakly on @xmath5 , if at all , when @xmath5 is sufficiently low . there is no generally accepted process of electron low - energy - excitation interactions that can satisfactorily explain the saturation " of @xmath3 found in the experiments . it should be noted that those experiments @xcite have ruled out electron heating , external microwave noises , and very dilute magnetic impurities as the origins for the observed finite dephasing as @xmath6 . to unravel the issue of electron dephasing , systematic information of @xmath3 over a wide range of sample properties is very desirable . bearing this in mind , we have in this work performed systematic measurements of @xmath3 on several series of _ as - sputtered _ and subsequently _ annealed _ aupd and sb _ thick _ films . the low - field magnetoresistances of the as - sputtered samples are first measured . the samples are then annealed , and their magnetoresistances measured . the annealing and magnetoresistance measurement procedures are repeated a few times . @xmath0 is extracted by comparing the measured magnetoresistances with the three - dimensional ( 3d ) weak - localization ( wl ) theoretical predictions @xcite . generally , thermal annealing causes a decrease in the sample resistivity , signifying a reduction in the amount of defects in the microstructures . controlled annealing measurements are thus crucial for testing the theoretical models of electron dephasing invoking magnetic impurities and dynamical defects @xcite . thick film samples were prepared by dc sputtering deposition onto glass substrates held at room temperature . the deposition rate was varied to tune the amount of disorder , i.e. , the residual resistivity @xmath7 (= @xmath8(10 k ) ) of the films . the aupd films were typically 6000 @xmath9 @xmath10 0.3 mm @xmath10 17 mm , while the sb films were typically 3000 @xmath9 @xmath10 0.3 mm @xmath10 17 mm . thermal annealing of the aupd ( sb ) films was performed in a 99.999% pure ar atmosphere at moderate temperatures of @xmath11 100@xmath12300@xmath13c ( @xmath11 150@xmath13c ) for about one half to several hours until @xmath7 changed by a desirable amount . the use of an extremely high purity ar atmosphere greatly minimized the presence of any oxygen residual gas in the annealing . the values of the relevant parameters for our as - sputtered films are listed in table [ t.1 ] . .resistivities , diffusion constants , and dephasing times of the as - sputtered aupd and sb thick films . the value of @xmath3 is extracted by least - squares fitting the measured @xmath14 to eq . @xmath15 . [ cols="<,^,^,^,^",options="header " , ] [ t.1 ] we notice that the four aupd films listed in table [ t.1 ] were _ newly _ made from a _ new _ au@xmath16pd@xmath16 target different from that used in our previous study @xcite . moreover , a _ different _ sputtering gun and a different vacuum chamber were utilized . previously , we had studied @xmath0 in a series of dc sputtered aupd thick films prepared and measured in a different laboratory @xcite . by applying these new samples , we are able to perform a close comparison study of @xmath0 in the same material prepared under different conditions . such a comparison is indispensable for clarifying the possible role , if at all , of magnetic scattering on @xmath3 . if there were any noticeable magnetic contamination during this experiment , it is natural to expect an _ unintentional _ magnetic concentration , @xmath17 , that differs from that in our previously samples @xcite . consequently , a distinct value of @xmath3 should be observed . on the other hand , if a similar value of @xmath3 is measured , this result must then bear important information about an intrinsic material property . in addition to the newly prepared aupd samples , we have studied two aged " sb films . the two sb films listed in table [ t.1 ] were first deposited and studied two years ago in ref . @xcite . during the past two years , they were exposed to air all the time . one might have naively speculated that these two samples must be heavily contaminated by ( magnetic ) impurities and , thus , have a shorter @xmath0 with a much weaker @xmath5 dependence , compared with that measured two years ago . to the contrary , this experiment indicates that 3d aupd and sb are _ not _ as vulnerable to contamination as speculated . our results point to an experimental fact that suggests that the observed saturation of @xmath3 can not be readily explained by magnetic scattering . we have measured the magnetoresistances and compared with 3d wl predictions @xcite to extract the values of @xmath0 . our experimental method and data analysis procedure had been discussed previously @xcite . here we emphasize that , in the limit of strong spin - orbit scattering ( which applies for both aupd and sb ) , @xmath0 is the _ only _ adjusting parameter in the least - squares fits of the measured magnetoresistances with wl predictions . this great advantage makes the extraction of @xmath0 highly reliable . empirically , @xmath0 can be written in the form @xmath18 where @xmath3 dominates at the lowest measurement temperatures , and @xmath19 is the relevant inelastic scattering time which is usually important at a ( few ) degree(s ) kelvin and higher . in three dimensions , electron - phonon scattering is the predominant inelastic process while the nyquist electron - electron scattering is negligibly small @xcite , i.e. @xmath20 in eq . the electron - phonon scattering rate @xmath21 varies as @xmath22 , with 2 @xmath23 4 . figure [ f.1](a ) shows our measured @xmath0 as a function of temperature for four as - sputtered aupd films . this figure demonstrates that @xmath0 first increases with decreasing @xmath5 at a few degrees kelvin , where the electron - phonon scattering dominates the total dephasing and @xmath24 in aupd @xcite . below about 2 k , the inelastic process is much less effective and a new mechanism progressively takes over , resulting in a very weak temperature dependence of @xmath0 as @xmath25 . to our knowledge , there is no generally accepted process of electron low - energy - excitation interactions that can account for such a weak @xmath5 behavior . a weak temperature dependence of @xmath3 is also observed in the two sb thick films listed in table [ t.1 ] . in fact , an ( almost ) absence of temperature dependence of @xmath0 has previously been found in numerous three - dimensional polycrystalline metals @xcite . we notice that the values of @xmath3 in fig . [ f.1](a ) follow _ closely _ the scaling relation of @xmath26 s established in fig . 3 of @xcite , where the diffusion constant @xmath27 is in @xmath28/s . this result suggests that the behavior of @xmath0 found in fig . [ f.1](a ) is material intrinsic . it should be emphasized that the weak @xmath5 dependence or the so - called saturation " of @xmath3 in fig . [ f.1](a ) is observed in a temperature regime where the sample resistance varies as @xmath29 all the way down to our lowest measurement temperatures ( fig . [ f.1](b ) ) . this @xmath29 dependence of resistance is well described by the 3d electron - electron interaction effects @xcite . this result indicates that the saturation of @xmath3 is _ not _ caused by electron heating . a similar assertion of non - hot - electron effects has also been reached in previous experiments @xcite . information about the effect of annealing is crucial for clarifying the nature of magnetic scattering and dynamical defects . figure [ f.2](a ) shows the variation of @xmath0 with temperature for the as - prepared and subsequently annealed aupd1e thick film . this figure clearly indicates that @xmath0 increases with annealing . similar behavior of increasing @xmath0 with annealing has also been found in the as - prepared and annealed aupd6e thick film . at first glance , this observation is easily explained . suppose that annealing results in the rearrangement of lattice atoms and relaxation of grain boundaries and , hence , makes the film less disordered . because two - level systems ( tls ) are closely associated with the presence of defects in the microstructures , their number concentration would be reduced by annealing . by assuming that dynamical defects are effective scatterers , one can then understand fig . [ f.2](a ) in terms of a reducing tls picture . however , our further measurements indicate that the nature of low - temperature dephasing in real metals is not so straightforward . we find that the effect of annealing on @xmath0 is distinctly different in strongly disordered samples ( fig . [ f.3 ] ) . for the convenience of discussion , the aupd1e and aupd6e ( aupd4a and aupd5e ) thick films are referred to as moderately ( strongly ) disordered , because they have @xmath30 120 ( 470 ) @xmath31 cm before annealing . figure [ f.2](b ) shows the variation of @xmath0 with @xmath5 for the as - prepared and annealed sb01b thick film . this figure clearly indicates that @xmath0 increases with annealing . similar effect of annealing has also been found in the sb12 thick film . the results of figs . [ f.2](a ) and [ f.2](b ) suggest that an enhancement of @xmath0 by thermal annealing is common to different _ moderately _ disordered metals . ( we notice that the high resistivities in sb films arise from a low carrier concentration instead of a short electron mean free path @xcite . our sb thick films are thus moderately disordered . ) one of the widely accepted explanations for the saturation " behavior of @xmath3 invokes magnetic spin - spin scattering due to a low level contamination of the sample . this explanation has been challenged in several recent careful experiments @xcite . however , despite this experimental situation , there is still an insisting opinion that argues for non - zero magnetic scattering in the sample . to completely reject such an opinion is non - trivial , because it is argued that the level of unintentional contamination is so low that it can not be detected by the state - of - the - art material analysis techniques . the situation becomes more serious when reduced - dimensional systems are involved . in the case of low - dimensional structures , surface effects due to interfaces , substrates , and paramagnetic oxidation are non - negligible . therefore , it is not straightforward to ascribe the observed saturation of @xmath3 to either intrinsic material properties or surface / interface effects . on the other hand , this kind of ambiguity does _ not _ occur in our _ three - dimensional _ measurements . in fact , we believe that magnetic scattering can at most play a subdominant role in our experiment . our reasons are given as follows . ( i ) suppose that there is a low level of magnetic contamination in our as - sputtered films . upon annealing , the magnetic impurity concentration @xmath17 should be left unchanged . if the original saturation " in our as - sputtered samples is caused by spin - spin scattering , one should then expect the same value of @xmath3 ( @xmath32 ) after annealing . however , we find increasing @xmath3 with annealing . our result is thus in disagreement with this assumption . ( ii ) our argument for a non - magnetic origin is supported by the observation of an _ increased _ @xmath3 in the aged and annealed sb films . since our sb01b and sb12 thick films were aged in air for two years , one might have naively expected a large decrease in @xmath3 due to magnetic contamination . nevertheless , this is not the case found in fig . [ f.2](b ) . ( iii ) moreover , if our samples do contain an appreciable level of unintentional magnetic impurities , the contaminated concentration @xmath17 should be basically the same in all films , because similar fabrication and measurement procedures were involved . one should then expect a similar @xmath3 in all _ as - prepared _ samples , regardless of disorder . this is certainly inconsistent with the observed scaling relation @xmath33 discussed above . therefore , magnetic scattering in its current form can not easily explain our overall results in a consistent manner . in order to explain the widely observed saturation behavior of @xmath3 , it has recently been proposed that dynamical defects can be important @xcite . the low - energy excitations of the dynamical defects are usually modelled by two - level systems . we already discussed that tls might be partly responsible for the saturated dephasing found in our moderately disordered films . however , it is impossible to perform a quantitative comparison of our experiment with the tls theories . the difficulties lie on the facts that ( i ) the number concentration of tls in a particular sample is not known , ( ii ) the strength of coupling between conduction electrons and a tls is poorly understood , and ( iii ) the dynamical properties of real defects ( impurities , grain boundaries , etc . ) are even less clear . experimentally , we also find other features of thermal annealing ( fig . [ f.3 ] ) that seem incompatible with a tls picture of dephasing . in addition to the moderately disordered samples , we have performed measurements on thick films containing much higher levels of disorder . surprisingly , we discover that the effect of annealing is completely different . in the _ strongly _ disordered aupd4a and aupd5e thick films , we find that annealing causes _ negligible _ effect on @xmath0 . figure [ f.3 ] shows the variation of @xmath0 with @xmath5 for the as - prepared and annealed aupd4a thick film . this figure clearly demonstrates that the values of @xmath0 for the as - prepared and annealed samples are essentially the same , even though the resistance , and hence diffusion constant @xmath27 changed by a factor of more than 6 . the absence of an appreciable annealing effect in this case implies that , in addition to the usual tls addressed above , these two films contain other defects that can not be readily cured by thermal annealing . such a null effect of annealing seems to suggest that , despite a large effort in this direction , no real defects of any nature can be found to have dynamical properties that may explain the saturation behavior of @xmath3 . a comparison of figs . [ f.2 ] and [ f.3 ] strongly indicates that low - temperature dephasing is very sensitive to the microstructures . the observation of fig . [ f.3 ] deserves further discussion . first , we recall that our measured @xmath0 in the as - sputtered , strongly ( and moderately ) disordered films follows the scaling relation of @xmath33 mentioned above @xcite , implying that the result of fig . [ f.3 ] is material intrinsic . secondly , in the context of magnetic scattering , blachly and giordano recently found that kondo effect is very _ sensitive _ to disorder , namely , an increase in disorder suppresses kondo effect @xcite . along this line , if the original saturated @xmath3 found in fig . [ f.3 ] were due to magnetic scattering , one should argue that thermal annealing that suppresses disorder should enhance kondo effect . then , a decreased @xmath3 should be expected with annealing . since our @xmath3 does not change even when the sample resistivity is reduced by a factor of more than 6 by annealing , fig . [ f.3 ] thus can not be easily reconciled with a magnetic scattering scenario . besides , this picture of a weakened kondo effect by disorder is also incompatible with our result for the moderately disordered films ( fig . [ f.2 ] ) where an increased , instead of a decreased , @xmath3 with annealing is found . thirdly , one might argue that annealing in the presence of oxygen can lead to the oxidation of magnetic impurities , and hence to the disappearance of kondo resistivity @xcite . we argue that this is unlikely in our case , since we had taken precautions by annealing our films in a 99.999% pure ar atmosphere , where the amount of oxygen residual gas was greatly minimized . moreover , since we do not find a rapid increase in the @xmath5 dependence of @xmath3 after annealing , there is thus no clear evidence of the presence of oxygen or magnetic impurities in our films . in any case , it would be interesting to study if annealing in oxygen would have a drastic effect on @xmath3 . lastly , we discuss the advantage of using three- , instead of lower - dimensional , mesoscopic structures for @xmath0 measurements . in 3d , the dominating inelastic process is the electron - phonon scattering for which @xmath34 obeys a strong @xmath35 ( 2 @xmath23 4 ) dependence @xcite . such a temperature variation is much stronger than the dominating @xmath36 in one dimension and @xmath37 in two dimensions ( which are both due to electron - electron scattering @xcite ) . inspection of the solid lines , which are drawn proportional to @xmath38 , in figs . [ f.1 ] and [ f.3 ] reveals that our measured @xmath3 at 0.3 k is already @xmath11 _ two orders of magnitude _ lower than that as would be extrapolated from the measured @xmath34 at a few degrees kelvin . such a huge discrepancy is well outside any experimental uncertainties . on the contrary , in the case of narrow wires , the dominating inelastic dephasing time obeys a much weaker @xmath39 law just mentioned . in this case , any discrepancy between the measured and extrapolated values of @xmath3 would be less dramatic in the attainable experimental temperature range , rendering a discrimination of the presence or absence of a saturated @xmath3 less clear - cut . we have studied the influence of thermal annealing on low - temperature electron dephasing in polycrystalline aupd and sb thick films . we find that @xmath3 reveals an extremely weak temperature dependence in both as - sputtered and annealed samples . the effect of annealing is non - universal , depending strongly on the amount of disorder quenched in the microstructures during deposition . the observed saturation behavior of @xmath3 can not be easily explained by magnetic - scattering in its current form . we also find that the disorder behavior of @xmath3 in as - prepared and annealed samples is very different . a complete theoretical explanation would need to take the microstructures into account . the authors are grateful to m. bttiker , c. s. chu , p. mohanty , r. rosenbaum , j. von delft , g. y. wu , and a. zawadowski for valuable discussion . this work was supported by the taiwan nsc through grant no . nsc 89 - 2112-m-009 - 054 . mohanty p. _ lett . * 78 * 3366 , 1997 ; webb r. a. _ et al . _ , fortschr . phys . * 46 * 779 , 1998 ; webb r. a. _ et al . _ , in _ quantum coherence and decoherence _ , ed . by k. fujikawa and y. a. ono ( elsevier , amsterdam , 1999 )

we have studied the effect of thermal annealing on electron dephasing times @xmath0 in three - dimensional polycrystalline metals . measurements are performed on as - sputtered and annealed aupd and sb thick films , using weak - localization method . in all samples , we find that @xmath0 possesses an extremely weak temperature dependence as @xmath1 . our results show that the effect of annealing is non - universal , and it depends strongly on the amount of disorder quenched in the microstructures during deposition . the observed saturation " behavior of @xmath0 can not be easily explained by magnetic scattering . we suggest that the issue of saturation can be better addressed in three - dimensional , rather than lower - dimensional , structures .